Home Equity Insurance

Transcription

1 Maser Thesis M.Sc. in Finance Auhor Peer Kasen [sud.nr ] Academic Supervisor Sefan Hirh Home Equiy Insurance Aarhus School of Business Universiy of Aarhus Deparmen of Business Sudies 008

2 Conen Inroducion. Problem Saemen. Moivaion.3 Srucure 3.4 Mehodology 4.5 Limiaions 5 Home Equiy Insurance 6. The Hisory of Home Equiy Insurance 7. Academic Lieraure on Home Equiy Insurance 8 3 Descripion of he Danish Real Esae Marke 9 3. Hisorical Developmen in he Danish Real Esae Prices 0 3. Valuaion of he Danish Real Esae Marke 3.. Deerminaion of he Demand for Real Esae 3.. Deerminaion of he Supply of Real Esae The Dynamics of he Real Esae Marke 3.3 Economeric Techniques Applied o Analyze he Real Esae Marke Empirical Resuls Conclusion: Valuaion of he Danish Real Esae Marke 7 4 Measuring he Underlying Asse 8 4. Indices on Real Esae Prices Simple Average and Median Mix Adjusmen Hedonic Regression Repea Sales Hedonic Repea Sales Indices Used in Pracice oday Discussion: Implemening Indices Conclusion: Indices on Real Esae Prices 38 5 Analysis of The Underlying Asse Saionariy or Uni Roo Specifying he Appropriae Model The Auoregressive Process Forecasing wih he Auoregressive Process Discussion: Analysis of he Underlying Asse Conclusion: Analysis of he Underlying Asse 48 6 Valuaion of Home Equiy Insurance Producs Fuures Opions Mone Carlo Simulaion Valuaion of Opions Discussion: Valuaion of Home Equiy Insurance Producs 54 II

3 6.3. Basic Risk Conrac Lengh Cusomers Providers Conclusion: Valuaion of Home Equiy Insurance Producs 58 7 Alernaive Ways o Hedge he Real Esae Risk Invesing in Real Esae Socks Appropriae Morgage Choice Ineres Rae Derivaives Invesing in a Real Esae Invesmen Trus Conclusion: Alernaive Ways o Hedge he Real Esae Risk 6 8 Behavioral Issues 6 8. Moral Hazard Adverse Selecion 63 9 Perspecive 64 0 Conclusion 64 References 68 Appendices 7 III

4 Inroducion One of he mos imporan opics in finance oday is risk and risk managemen. Risk is presen in almos every decision we make, so i is no hard o ge an idea of he imporance of he abiliy o esimae and conrol he risk associaed wih decision making. When invesing in he sock marke, he academics and marke paricipans have developed a grea knowledge on how o inves in he sock marke and especially on how o minimize he risk associaed wih invesing in he sock marke, bu oher areas have been somewha negleced by he academics and marke paricipan. One of hese areas is he real esae marke especially he privae marke for real esae invesmens. When invesing in real esae, he mos common advice is o acquire a condiion repor on he propery and o make sure ha here are no deb or oher forms of securiy in he propery. This should make sure ha he invesmen in he propery is relaively safe. - Bu is his he only risk luring when making a real esae invesmen? - Is here any oher risk ha should be aken ino accoun when making real esae invesmen decisions? This hesis will concenrae on he imporance of being able o manage and conrol he risk associaed wih making invesmens in he privae real esae marke. The concep of home equiy insurance will be he cenre of aenion in his hesis as his concep deals wih he abiliy o hedge he risk incurred when invesing in real esae (beside he already menioned risk associaed wih real esae invesmens). Like every oher asse he value of real esae is volaile, which enails ha deerioraion in he value of he real esae is an acual risk. Should his risk no be aken ino consideraion when making a large undiversified invesmen in a single propery? Unforunaely here are no public raded hedging derivaes available o he privae invesor, who is concerned wih deerioraing real esae prices his is where home equiy insurance eners he picure.

5 . Problem Saemen In his hesis he emphasis will be on home equiy, more precisely on he insurance of he home equiy value. Today here is a very limied access for he privae household in Denmark o insure he home equiy. Why is his so, and wha needs o be done in order o make insurance of home equiy available o he privae household? These quesions are he cener of aenion in his hesis, and will be answered hrough he following cenral quesions: - Why insure home equiy value? - Why is i no possible for he privae invesor o insure he home equiy value? - How should a financial insrumen be consruced o make i possible o insure home equiy value? The cenral quesions are very broad and a number of sub quesions will emerge as he cenral quesions are answered. These sub quesions will be highlighed and answered in he appropriae conex. This hesis seeks o develop a framework in which i will be possible o consruc a hedge agains declining real esae prices; his would make risk managemen possible for he privae invesor in he real esae marke. Today he opporuniy o manage he risk is very scarce and he possibiliies ha do exis are no feasible o he general public.. Moivaion Why insure home equiy value? The problem saemen raises he quesion as one of he core quesions o be answered and i is herefore obvious o begin by conemplaing some of he moivaional houghs and ideas behind he use of home equiy insurance. The fuure is uncerain. The uncerainy linked o he fuure financial oulook is of general concern o he public, and can have a large impac on he privae household economy and welfare, bu why should he privae household be concerned abou he fuure financial oulook, especially when he privae household only has a limied Home equiy is in his hesis defined as he curren marke value of a home minus he ousanding morgage balance.

6 abiliy o influence and hedge he risk facing he household due o changes in he economy? The decision wheher or no o purchase home equiy insurance could be compared o he decision o purchase fire insurance, he risk of a fire bursing ou is o some degree beyond he conrol of he individual, and he cos associaed wih fire damage can be very cosly, as a resul mos households purchase a fire insurance. Bu he risk of a decline in he marke value of a propery is far greaer han he risk of a fire, why no insure agains his paramoun risk and why no make home equiy insurance mandaory when obaining a real esae loan? The proposal of mandaory home equiy insurance could become an advanage o boh he privae real esae owner and he financial insiuions 3. The financial insiuions would have increased securiy in he value of he home if he credi risk were reduced. As a resul of he minimized credi risk he financial insiuions will need less risk compensaion resuling in more favorable morgage condiions o he privae real esae invesor..3 Srucure The hesis begins wih an inroducion o he concep of home equiy insurance, wih special aenion o he hisory and he academic lieraure on home equiy insurance. Afer he inroducion o home equiy insurance he aenion is drawn owards a descripion of he real esae marke, where an equilibrium model is developed, in an aemp o acquire knowledge and undersanding of he governing dynamics in he real esae marke. The equilibrium model is updaed wih he laes Danish developmen and an indicaion of he fuure price developmen is given. Afer he inroducion o home equiy insurance and he (Danish) real esae marke, chaper 4 describes and analyses he muliple ways in which o measure changes in he value of real esaes. The chaper begins wih a descripion of he mos basic measuring echniques before moving on o more complex and sophisicaed mehodologies. In he end of he chaper acual real esae indices are inroduced including he available Danish real esae indices. The acual indices are compared and he mos suiable mehodologies o measure he real esae marke are recommended. Fire insurance is ofen mandaory when obaining a real esae loan. 3 Shiller, R.J and Weiss, A.N., (999), p. 3

7 In chaper 5, he acual Danish indices are analyzed using economerics echniques. The knowledge acquired from his analysis of he indices is used in chaper 6, where home equiy insurance producs (fuures and opions) are priced based on he ime series analysis of he Danish real esae indices. The hesis is ended wih a chaper concerning alernaive ways o hedge some of he risk facing invesmens in real esaes and a chaper abou behavioural issues in connecion wih a poenial inroducion of home equiy insurance producs o he public. Home equiy insurance is a he ime of he hesis wriing no ye available in Denmark, and alernaive ways of minimising he risk exposure is worh considering while waiing for home equiy insurance. Inroducing insurance produc should always be accompanied by an evaluaion of he behavioural issues seaming from moral hazard and adverse selecion problems..4 Mehodology This secion is divided in subsecions, which corresponds o he acual chapers in he hesis. I should be noed ha addiional mehodologies are applied in he hesis, besides he mehodologies menioned in he secion. Descripion of he Danish Real Esae Marke: Vecor auoregressive (VAR) and vecor error correcion models (VECM) are he mos widely applied economeric echniques o esimae equilibrium models in he real esae marke, hese echniques are inroduced and empirical resuls are depiced. Measuring he Underlying Asse: In measuring he changes in he value of he underlying asse, here are several compeing mehodologies, ranging from basic measures of cenral endency (average and median), o more sophisicaed mehodologies (mix adjusmen, hedonic, repea-sales and hedonic repea-sales mehodologies), all hese echniques are described and compared in he chaper. Analysis of he Underlying Asse: The daa used in he analysis are based on observaions from where he Danish real esae prices are given in grea deail, wih quarerly observaions daed back o 995. Eviews are applied 4

8 o analysis and ess of he ime series daa. Time series analysis of he available Danish real esae indices are underaken, where several economeric echniques are applied. Among he more imporan is he augmened Dickey-Fuller es; idenificaion of ARand MA- processes and various oher model selecion crierions are included in he analysis. Valuaion of Home Equiy Insurance Producs: The programming language Visual Basic for Applicaions (VBA) is applied o simulae he developmen in he real esae indices. Mone Carlo simulaion is used o price home equiy insurance producs; o improve he efficiency of he simulaion aniheic variables are incorporaed in he code. In addiion o he Mone Carlo simulaion a modified Black & Scholes pu opion formula is wrien in VBA, o be able o compare he resuls..5 Limiaions The hesis focuses on describing and analyzing condiions and facors surrounding he Danish real esae marke, bu in some of he chapers i has been necessary o broaden he focus o include condiions and knowledge from oher counries, mainly due o lack of knowledge abou Home Equiy Insurance in Denmark. Descripion of he Danish Real Esae Marke: An acual equilibrium model will no be consruced and analyzed, he purpose of he chaper is o ge an undersanding of he mos imporan dynamics of he real esae marke, o esimae an equilibrium model will be o exensive. The dynamics of he real esae marke is likely o have a differen impac on differen propery ypes, locaions and regions hese differences will no be emphasized, only he general dynamics will be included in he descripion. Analysis of he Underlying Asse: The indices analysed in he chaper are based on observaions covering all of Denmark (geographically), his is no he bes soluion when an acual home equiy insurance produc is offered o he public, bu in his conex he large coverage of he indices will imply ha he daase conains enough observaions o be unbiased and reliable. Valuaion of Home Equiy Insurance Producs: The fuures and opions are chosen on he basis of heir simpliciy and heir appealing characerisics. The derivaives are kep 5

9 simple (simple conrac erms) o avoid unnecessary complicaions. Tha is o give a good indicaion of he cos of insurance he home equiy value. Several oher derivaives would properly also be suiable in a home equiy insurance seing (i.e. barrier opions), bu hey are no priced in his hesis. Home Equiy Insurance The concep of home equiy insurance will be inroduced in his secion, and as he opic is discussed, several issues in connecion wih home equiy insurance will unfold. These issues will be deal wih in he following secions of he hesis. Home equiy insurance is as he name saes an insurance agains decline in home equiy value caused by exernal facors. A financial derivaive founds he basis for he insurance and is consruced specifically o hedge he risk facing he home equiy value. The derivaive is based upon an index ha mimics he movemens of he real esae prices. The bigges challenge lies in he index consrucion phase, where i is imporan o consruc an index, which is highly, correlaed wih he movemens in he acual real esae prices. I is imporan o make a close o perfec hedging insrumen. The home equiy insurance is basically creaed in an aemp o make hedging of he risk associaed wih privae invesmens in real esae possible. If he goal is reached i will give he privae invesor he opporuniy o minimize he risk associaed wih he highly undiversified and highly leveraged invesmen ha real esae invesmens ofen are o he privae invesor 4. The descripion of home equiy insurance opens up o a lo of unanswered quesions: - Which exernal facors have an impac on he value of home equiy? - Which underlying asse should be used o resemble he movemens in he real esae prices? - Which derivaives should be used o insure he risk facing he home equiy value? - Is home equiy insurance he bes and only way o hedge he risk facing home equiy value? 4 Inernaional Moneary Fund (008), p

10 These quesions are oo broad o be answered horoughly in his secions and are lef o he following chapers, insead his chaper coninues wih a descripion of he more hisorical aspec of home equiy insurance.. The Hisory of Home Equiy Insurance The hisory of home equiy insurance has undergone a developmen from being moivaed by governmen incenives o reduce he consequences of shudown of governmenal aciviies, which can have grea impac on he value of he homes. Laer on home equiy insurance ook on a more social aspec, insuring agains deerioraing neighbourhood condiions. The laes endency moves owards a more broad adopion of he concep o make home equiy insurance more commercial and applicable in a more general seing. The firs sep owards he developmen of home equiy insurance was aken in California in year 95, where a civil code was added o regulae he insurance of he value of land, his code was evenually desered. There were o go some years before he firs acual home equiy insurance program saw he day of ligh in 966, where he U.S. Deparmen of Defence sared a program which should proec he privae household from loss in home value, caused by he closing of miliary faciliies 5. Oher small-scale programs i.e. he Oak Park program, Illinois [978] 6, and he Chicago program [987] are examples of projecs where focus was on insuring he home equiy value. The Oak Park program was creaed as a par of a deermined effor o preven he area from declining due o racial changes. The paricipan in his program had o be enrolled in five years o ge reimbursemen, when selling heir home for less han he appraised value. One cenral condiion o ge he compensaion was ha he loss of value should be caused by he decline in he area and no a propery specific condiion. The propery owners and a small ax paymen financed he program. The Chicago program was very similar o he Oak Park program in ha i was an insurance agains decline in he home value due o changes in neighbourhood condiions. In conras o he Oak Park program his program was based on individual enrolmen and paymen of 5 Caplin, A. e al., (003), p.5 6 Caplin, A. e al., (003), p.5 7

11 an insurance fee. I is imporan o emphasize ha he Oak Park and Chicago programs did no insure agains municipal decline in home value he insurance only covered neighbourhood relaed deerioraion. I is imporan o reflec on he experience from hese early aemps o inroduce home equiy insurance o make sure ha home equiy insurance is more likely o succeed in he fuure. There are some general problems wih hese firs aemps. The risk definiion was oo narrow, he neighbourhood condiion is a subjecive condiion and quanificaion of he acual condiion is difficul. The risk should be defined broad enough so ha i is possible o acually measure he changes accuraely. Furhermore i should be noed ha he Chicago program excluded losses due o changes in marke condiions. Oher issues wih he programs have been idenified; he programs were no managed professionally, lack of financial incenives o sell he insurance and lack of markeing. These issues migh be some of he reasons for he lack of success for he home equiy insurance concep 7. The firs seps oward he developmen of a general home equiy insurance produc have also been aemped by several financial insiuions. One of he firs insiuions o inroduce he producs was he London Fuures and Opion exchange in he early 990ies. Unforunaely heir early aemps failed. In recen ime, new effors have been made and he marke shows signs of being ready o embrace he propery derivaive producs (see also inro o chaper 6) 8.. Academic Lieraure on Home Equiy Insurance The firs academic lieraure on home equiy insurance originaes back o Marcus and Taussig [970] and Yarmolinsky [97]. They proposed general home equiy insurance programs, where he difference beween he insured value and he acual sales price would be reimbursed if he acual sales price were below he insured value. The main difference beween heir proposals is found in he financing of he programs. Marcus and Taussig proposed public financed insurance whereas Yarmolinsky had a more commercial oulook, and believed ha an insurance company would be more suiable o 7 Shiller, R.J and Weiss, A.N., (999), p Deusche Bank Research (007) 8

12 manage he insurance, bu Yarmolinsky also believed ha he insurance company should be reinsured by he sae 9. Laer Shiller and Weiss [994] conribued o he academic lieraure wih heir aricle he Home Equiy Insurance. Many of heir houghs are in line wih he preceding academics, bu wih a more commercial view ha includes houghs on how o price home equiy insurance producs and how o avoid behavioural issues in connecions wih he insurance producs. In 003 Caplin e al. published Home Equiy Insurance: A Pilo Projec, which is very similar o he work by Shiller and Weiss [994]. In Caplin el al. addiional documenaion on he Syracuse (New York) home equiy insurance program is included wih updaed remarks on he developmen of he program. The imporance of he home equiy insurance concep is backed up by academics emphasizing he benefis obained from risk sharing in he real esae marke. Iacoviello and Oralo-Magne [00] along wih Englund, Hwang and Quigley [00] are some of he academics ha have emphasized he imporance of risk sharing and he imporance of he abiliy o hedge he risk facing he real esae marke 0. 3 Descripion of he Danish Real Esae Marke Home equiy insurance is based on knowledge abou he real esae prices, in ha conex i is imporan o ge an idea abou he dynamics and behaviour of he price movemens in he real esae markes. The developmen in he real esae prices has radiionally been explained by he developmen in he underlying fundamenal economic facors, which in urn conrols he demand and supply of real esae o he marke. Wih a long run equilibrium where he cos of consrucing new real esae is equal o he price of he exising properies in he marke. Building a srong knowledgebase is essenial o ge greaer insigh of he governing dynamics of he real esae marke. The real esae prices are influenced by a large number of facors and he complex dynamics and ineracions beween hese facors make an economeric explanaion of he real esae price movemens a very difficul and absrac ask. 9 Caplin, A. e al., (003), p.5 0 Furher discussion in chaper 7 Wagner, R., (005), p. European Cenral Bank (003), p.- 9

13 This chaper begins wih a hisorical oulook of he price developmen in he Danish real esae marke. The chaper hen moves on o idenifying he facors and dynamics ha conrol he movemen and changes in he real esae prices. Pleny of academic research is devoed o he opic and appropriae references will be made hroughou his secion. Following his preliminary descripion of he real esae marke, he mos applied valuaion echniques are described. These valuaion echniques fuel he developmen of accurae and reliable measures (indices) of he value of real esaes and especially he measure of he changes in value of he real esae. These differen measuring echniques are described and he acual uses of he indices are discussed in chaper Hisorical Developmen in he Danish Real Esae Prices Several insiuions supply daa on he residenial real esae marke in Denmark. Among he mos recognized are Danmarks Saisik and Realkrediråde 3, who publish he price developmen in he Danish real esae marke. In describing he hisorical developmen in he real esae prices in Denmark, daa from hese insiues are used. In he period spanning from 966 o 973 real esae prices were increasing. Around year here were economic uncerainy in Denmark, which were mainly fuelled by an oil crisis, resuling in an increase in he general price level, he real esae prices were no o follow his developmen alhough a sagnaion in he real esae prices where unavoidable, he sagnaion in he real esae prices were shor-lived and in he following years he real esae prices coninued o increase, reaching is maximum level around year In he years o come large deerioraion in he value of residenial real esae were realised. The real esae markes were is his period synonymous wih dropping prices. In 98 he real esae prices had hi rock boom, a his ime he real esae price had dropped wih abou /3 of he value compared o he price level in 978 measured in real erms. The deerioraion in he marke was largely due o hisorical high ineres raes reaching an all ime high in 98. In he nex 5 years o come he real esae prices increased due o declining ineres raes and posiive economic oulook. In 986 he real esae prices once again exhibied declining prices, his ime he facors a faul were a ax reform where he value of he ax deducion on ineres raes were dropped, and more resricive rules concerning real esae morgages were founded, on 3 and 0

14 op of he ax changes his period where characerised by an increasing unemploymen rae. In 993 he rend were changed and he norm of declining real esae prices were ransformed ino increasing real esae prices 4. This new rend was o las for a long period, which can be seen from he char 3.. In he growh rae in he real esae prices has slowed down and he real esae prices have exhibied decreasing prices in some areas of he counry, especially properies, which previously have exhibied large price increases 5. Char 3.: Danish Real Esae Price (006 = 00) Single-family house Holiday coage Owner-occupied fla 99K 99K4 993K3 994K 995K 995K4 996K3 997K 998K 998K4 999K3 000K 00K 00K4 00K3 003K 004K 004K4 005K3 006K 007K 007K4 Source: - see appendix A Valuaion of he Danish Real Esae Marke The pricing and valuaion of properies and real esae markes have been sudied exensively for decades, using a variey of differen approaches and mehods. Sudies on he valuaion of individual properies have radiionally been carried ou using he sales comparison approach he income approach and he cos approach 6, while sudies of he enire real esae marke have been done by using equilibrium models. The sales comparison approach uilizes, he informaion conained in he prices of previously sold properies o esimaes he marke value of he subjec propery being valued. The income approach is basically a discouned cash flow model (also known from he analysis of sock prices) where he fuure expeced income generaed from a specific propery is discouned in an aemp o verify and explain he acual price of he propery. The raionale behind he income approach is ha an invesor is only willing o 4 Andersen T.M. e al., (00), p Danmarks Naionalbank, (007), p Brueggeman, W.B. and Fisher, J.D., (993), p

15 pay an amoun equivalen o he income generaed from holding he propery. The cos approach looks a he cos of buying land and building a propery similar o he propery being valued, he raionale behind his approach is ha an invesor will pay no more han i will cos o consruc a similar propery 7. Models of he enire real esae marke are occupied wih he sudy of he driving forces behind he price seing in he real esae marke. In describing he dynamics and driving forces of he real esae marke a classical demand-supply framework is ofen consruced in which he facors ha affec demand and supply in he real esae marke are deermined and heir dynamics are described by a se of equaions. The observed marke prices are given by he inersecion of he supply and demand, where he prices serve as a marke clearing mechanism. The fundamenal hypohesis is ha he markeclearing prices in he long run will be dependen upon he cos of consrucing new real esae. In he shor run he clearing prices can deviae from his equilibrium, due o he fac ha he supply of real esae is fixed in he shor run. This enails ha he shor-erm demand deermines he shor-erm prices, while he long-erm prices are equilibrium prices. While i is eviden ha he various valuaion approaches of individual properies are feasible, he knowledge and undersanding of he real esae marke gained from hese models are limied. Therefore his chaper coninues wih he developmen of an economic framework where he facors deermining he demand and supply of real esae are idenified and he governing dynamics of he real esae marke are emphasized. The developmen of an economic framework is inducive and should herefore be in accordance wih general acceped economic heory. 3.. Deerminaion of he Demand for Real Esae The demand for real esae is deermined by a variey of facors. Among he mos influenial facors are he household income, he inflaion, he user cos, and he demographic composiion. Increasing household income will have a posiive effec on he demand, which will resul in increasing real esae prices. Increasing cos associaed wih owning real esae (user cos), will all else being equal diminish he demand and 7 Schulz, R. (003), p.-7

16 herefore force he real esae prices down. Poenial housing demands are deermined by he acual demographic composiion and changes in he demographic composiion will have an impac on he demand for real esae. These facors will in he following secion be described in greaer deails o give beer undersanding of heir relaive imporance and he effec ha hey have on he demand for real esae. Household income: Household income is in he lieraure described as he single mos significan explanaory variable of real esae prices. The income forms a naural limi for he amoun of money ha he household can afford o use on consumpions of goods and services including real esae. The acual consumpion of housing is limied by he income of he household, and an increase in he households income can be used o increase he consumpion and hereby increase he households demand for housing. Empirical research has shown ha he elasiciy of real esae prices wih respec o household income is close o uniy 8. The household income is largely deermined by he economic developmen in he economy. In prospers markes here is a clear endency of an increasing household income, which in urn will boos he real esae prices. In periods of recession he household income will drop and so will he real esae prices. This is why empirical research ofen uses he growh in gross domesic produc (GDP) as a proxy for he growh in he household income. Table 3.: GDP-growh ( growh in Household Income) (expeced) 009 (expeced) 00 (expeced) GDP-growh 3.9%.8%.3% 0.8% 0.7% Source: De Økonomiske Råd, (008), p. From he economic councils (De Økonomiske Råd) forecas of he GDP-growh in Denmark i is clear ha he income will coninue o increase in he years o come, bu wih a more moderae pace. The economic council has furhermore forecased he annual increase in wages o be around 5% in 009 and 4,4% in 00, which is above he counries ha Denmark normally compares o 9. I should be noed ha he nominal income is no ineresing by iself only when compared wih he acual inflaion, he nominal income makes sense, in ha i is he real income which deermines he acual 8 European Cenral Bank (003), p De Økonomiske Råd, (008), p.- 3

17 demand in he economy. The household income is deermined by various facors. Among he more imporan deerminans are he employmen (and he unemploymen rae) and he inflaion. The acual housing income migh be a oo simple measure of he demand for real esae. The overall consumpion is no only deermined by he presen income, bu also based on he expecaion o he fuure income, which are ofen described in erms of he Life- Cycle model, which sipulaes ha he human capial and he permanen income associaed wih he human capial are imporan in deermining he acual consumpion. The basic idea is ha he consumpion is equally divided hroughou he life and herefore no only deermined by he acual household income, bu also he expeced fuure earnings, and he wealh 0. Employmen: The behavior of he unemploymen rae is radiionally described by Okun s law, which describes he relaion beween changes in oupu growh and changes in he unemploymen rae. When he unemploymen rae is low hen he bargaining power of he workforce increases, which in urns will have a posiive effec on he household income. In prospers economies he unemploymen rae ends o decrease due o increasing economic aciviy and vise versa. The overall effec of a decreasing unemploymen rae will be an increase in he oal income making he populaion wealhier and more capable of invesing in real esae, resuling in an increase in he real esae prices. The acual Danish unemploymen rae is a a hisorical low level, wih abou,9% unemploymen in March 008, wih only persons unemployed in Denmark, which is a very low unemploymen rae. The unions have uilized he hisorically low unemploymen rae and negoiaed hisorically collecive bargaining agreemens o increase he wages of he public workers in he nex 3 years by up o around 3% in he period. Table 3.: Unemploymen in Denmark (,000 persons) (expeced) 009 (expeced) Unemploymen Source: De Økonomiske Råd, (008), p. 00 (expeced) 0 Bodie, Z. and Meron, R.C., (000), p Blanchard, O., (003), p.0-4, and p.8-84 Srejken er slu lønnedgang på vej, Børsen 4/

18 The unemploymen rae is pressured by he increased wages, which will lead o increased inflaion, resuling in deerioraing compeiive advanages. This could poenially lead o job losses in he end, which is also indicaed by he forecased increase in he unemploymen rae in Denmark in he years 009 and Inflaion: The connecion beween he unemploymen rae and inflaion is well known; he relaion is bes described by he Phillips curve where a decreasing unemploymen rae resuls in increased inflaion and vise versa. The inflaion is ofen described by he following equaion: e π = π α( u un ) (E.3.) e Where π and π is he inflaion and he expeced inflaion. The expeced inflaion (he expeced inflaion is ofen subsiued by las years inflaion) and he deviaion of he unemploymen from he naural rae of unemploymen is used as a proxy for he acual inflaion 4. From he equaion i is seen ha an unemploymen rae below he naural unemploymen rae will cause he acual inflaion o increase, which is he siuaion in Denmark a presen ime, wih an acual unemploymen rae below he naural unemploymen rae, indicaing an upward pressure on he inflaion in he ime o come. Anoher facor affecing he inflaion is he increasing cos of living i.e. he increasing fuel (crude oil) and food prices, which also increase he inflaion levels 5. Table 3.3: Inflaion in Denmark (expeced) Inflaion (%) Source: De Økonomiske Råd, (008), p. In Denmark he general financial oulook poins in he direcion of increasing inflaion. Wih a low unemploymen rae and increasing cos of living he inflaion can only move in one direcion 6. The curren economic siuaion poins in he direcion of a sagflaion, which is characerized by a combinaion of low growh and high inflaion, his usually leads o a higher unemploymen rae 7. 3 De Økonomiske Råd, (008), p Blanchard, O., (003), p Oliens opur sender rener i årsrekord, Børsen / De Økonomiske Råd, (008), p. and p.3 7 Sagflaion får økonomien i knæ, Børsen 3/

19 User cos: Up unil now he facors deermining he demand for real esae have been occupied by he income of he household and how much he household have o spend on housing. This is of course no he only imporan aspec of he demand funcion, anoher imporan aspec is he cos associaed wih owning real esae, which is usually described by he erm user cos. I is eviden ha he demand for real esae will decrease wih increasing user cos. The erm user cos describes he opporuniy cos of invesing in real esae. In deermining he user cos here are four main facors, he firs is he cos associaed wih he purchase and financing of he propery, he second facor is he axaion and he ax relief associaion wih he purchase and ownership of he propery. The hird facor in deermining user cos is he depreciaion and oher cos associaed wih mainenance of he propery, he las facor represen he expeced capial gain, seaming from changes in he real esae prices 8. Financing cos: When purchasing a propery he household ypically divided he purchase price in hree pars, a down paymen par corresponding o 5% of he purchase price (from own savings, or obained via a bank loan), a morgage conribuion wih around 80% of he purchase price, while he remaining 5% are ypically financed hrough a bank loan 9. The loan-o-value (LTV) raio is abou 80 percen in Denmark, which enails ha around 80% of he value of all properies is financed via morgages 30. From he appendix A.., i is clear ha he financing is mainly obained from fixedrae morgages and adjusable-rae morgages. The fixed-rae morgages is ypically based on bonds wih 30 years o mauriy, bu bonds wih mauriies spanning from 0 o 30 years are available, he adjusable-rae morgages varies from a shor erm bond wih a mauriy of year up ill around 0 years mauriy 3. Wih he liberalizaion of he morgage marke he sensiiviy of house prices o changes in he ineres rae has increased, primarily due o he reduced credi consrains ha followed he liberalizaion of he financial markes. An increasing number of differen morgage producs have become available as a resul of he liberalizaion and he 8 De Økonomiske Råd, (00) Inernaional Moneary Fund (008), p

20 morgage marke is oday characerized by an increased variaion in conrac ypes, which implies ha he effec of ineres rae changes is more difficul o predic 3 The acual ineres rae is deermined in he equilibrium beween he supply of money and he demand for money. I is clear ha he lower he ineres rae, he more money people wan o hold. Increasing ineres raes have on he oher hand a negaive effec on he demand for money caeeris paribus 33. Dropping ineres raes have he impac o increase he real esae prices due o he fac he morgages become less expensive and he price of buying a propery is cheaper all oher hings being equal (favorable financing condiions), ha is he iniial deb servicing cos decreases. Increasing ineres raes will increase he deb servicing cos resuling in deerioraing real esae prices. The laes financial developmen in Denmark (crisis in he financial marke) has made he shor-erm ineres rae as well as he long-erm ineres rae increase, and increasing credi demand are making i harder o obain a loan o finance real esae purchases 34. Furhermore he increasing income, he low unemploymen rae and he expeced increase in inflaion all indicae ha he ineres rae should increase, which is also seen from char 3., depicing he developmen in he morgage ineres raes in Denmark 35. Char 3.: Developmen in he morgage ineres rae 9,00 % 8,00 7,00 6,00 5,00 4,00 3,00 Shor erm ineres rae ( year) Long erm ineres rae (30 years),00,00 0, Year Source: see also appendix A..3 3 European Cenral Bank (003), p.5 and p.4 33 Blanchard, O., (003), p De Økonomiske Råd, (008), p. 35 Oliens opur sender rener i årsrekord, Børsen /

21 The increasing shor- and long-erm ineres rae is of course bad news o he real esae marke where he prices will deeriorae as a resul of he increased deb servicing cos. Real Esae Taxes: Taxes, subsidies and oher public policies are facors ha influence he individual decision making behavior oward invesing in real esae, especially in periods of major reforms, he behavior can be changed significanly due o changes in policies. Changes in policies can have boh a posiive and a negaive effec on he real esae prices 36. Wih he inroducion of he ax sop in and he ax relief in 008 and 009, he consumpion is simulaed and he household income is expeced o increase in he years o come mainly due o ax relief and he coninuaion of he ax sop. The ax relief amouns o around 9 billion Danish kroners in 009, and is expeced o increase he privae consumpion by around 0.5% 38. The resul of he general ax policies is an increase in he real esae prices. The ownership of real esae is subjec o axaion. In Denmark here are wo ypes of propery axes, a ax on real esae grundskyld and a propery value ax. The ax on real esae are calculaed as per mille of he official appraisal value of he propery, in 008 he ax on real esae lies beween and of he appraisal value dependen upon he geographical locaion 39. The propery value axes are progressive, which means ha properies wih a value below 3 million Danish kroners are axed wih % of he value whereas properies appraised above 3 millions are axed wih % of he value below 3 million and 3% axaion of he value exceeding 3 millions 40. Depreciaion: Almos every asse is subjec o depreciaion, bu he acual depreciaion rae of real esae is very hard o deermine. According o IAS 6 and 40, he depreciaed of a propery should be allocaed sysemaic over he useful live of he propery. Typically he depreciaion of a propery is spread ou over a ime period of years, which corresponds o depreciaion rae beween % and 3,3% per year. The acual depreciaion rae is properly lower han %, empirically % is ypically applied 36 European Cenral Bank (003), p Finansminiserie (00), p.3 38 De Økonomiske Råd, (008), p

22 as he rae of depreciaion. The depreciaion rae can also be inerpreed as mainains cos per year o uphold he sandard of he propery. Expeced capial gains: The long run endency is ha he real esae prices follow he inflaion in consrucion cos, which can be inerpreed as a capial gain from owning real esae, which of course should be included in he overall cos associaed wih real esae possessions. Demographic changes: The demographic composiion has an imporan impac on he real esae prices. In he demographic field hey deal wih a erm called he poenial housing demand. The poenial housing demand is deermined by various demographic rends, such as he size of he populaion, he populaion s age, living arrangemens and so forh. The changes in demographics will have an impac on he demand for housing and herefore have an impac on he real esae prices 4. DREAM 4 a Danish insiuion concerned wih demographic rends and changes have divided he Danish populaion according o heir age group and ried o forecas he demographic changes. The populaion is divided ino 3 separae age groups, he firs group consiss of persons under he age of 5, he second group consiss of persons beween 5 and 64 years and a hird group conains persons of age 65 and above. Table 3.4: Forecased Demographic Composiion in DK (in millions) Age Toal Source: Hansen, M.F., Egger, M. and Sephensen, P., (007), p De Økonomiske Råd, (008), p.73 and own calculaions (number are in millions of people). From DREAMs forecas i is apparen ha he group consising of children below he age of 5 years are a somewha consan populaion size hroughou he forecasing period. The mos ineresing age group in connecion wih he developmen in he real esae marke is he age group beween 5 and 64 year, who are he mos acive in he real esae marke. Their size decreases from a populaion around 3,6 million in 007 o 4 De Økonomiske Råd, (00), p Danish Raional Economic Agens Model - DREAM 9

23 3.3 million in 040, which would affec he poenial housing demand in a negaive direcion. On he oher hand he age group above he age of 65 increases in size in he period saring a 0,8 million and reaching around,4 million in 040. Wih increasing age he demand for housing decreases, bu he housing demand is sill presen for he age group above 65. Overall he forecased demographic changes seem o be in favor of a sligh increase in housing demand in Denmark in he fuure. I should be emphasized ha i is no only he absolue size of he populaion ha deermines he demand, bu also he living arrangemens and so for. The populaion increases from 5.4 million inhabians in 006 o approximaely 5.7 million in 040, which all oher hings kep equal will increase he housing demand. 3.. Deerminaion of he Supply of Real Esae An increasing demand for real esae will creae an upward pressure on he real esae prices. The increased real esae prices will evenually exceed he cos of consrucing new real esae, which will give he consrucion indusry an inciemen o sar consrucing new properies. The price will in long run converge owards he consrucion cos as new properies are supplied o he marke. The increased invesmen aciviies in he real esae marke will evenually force he real esae prices down. Bu in he shor-erm, he ime facor becomes imporan. Supply of new properies: The ime i akes o consruc new properies is an imporan facor in deermining he volailiy and rends of real esae prices. Compeiion in he consrucion indusry and he availabiliy and cos of specialized labour, building regulaions, vacancies, he land planning sysem, axes and subsidies, paricular for new housing all influence he supply of real esae. The irreversibiliy of housing invesmen plays a significan role in he relaively slow responsiveness of he supply of new houses 43. The invesmens in real esae have been very high during he las years due o he high real esae prices, wih many new properies supplied o he marke. The demand for new real esae is relaively small due o he high prices and he economic oulook. The decreasing real esae prices are expeced o have a negaive effec on he 43 Quigley, J.M., (999) 0

24 fuure invesmens in he real esae marke, bu a posiive effec on he demand for new real esae The Dynamics of he Real Esae Marke Equilibrium in he real esae prices is ofen described using he framework proposed by James Tobin [969]. Tobin s Q-heory implies ha in he case where he acual real esae prices exceed he coss of consrucing new properies (Tobin s Q > ), he consrucion indusry will sar consrucing new properies, which will force he real esae prices down owards he cos of consrucing new real esae 45. To give an idea of he dynamics of he equilibrium in he real esae prices he cycle of he real esae prices is described. An iniial increase in he real esae prices boos he aciviy in he consrucion indusry resuling in more new properies being consruced, he increased aciviy is primarily fueled by increased selling prices of properies and he increased profi opporuniy, inheren in he increased real esae prices. The increased aciviy in he consrucion indusry resuls in a larger supply of real esae causing he real esae prices o drop. When he real esae prices reach he consrucion prices, he real esae marke is said o be in equilibrium. Char 3.3: Consrucion of new Real Esae and Tobin s Q (Denmark, ),9 4000,4 9000,9, ,9 Tobin's Q ,4 Consrucion K4 993K4 994K4 995K4 996K4 997K4 998K4 999K4 000K4 00K4 00K4 003K4 004K4 005K4 006K4 007K4 Source: and own calculaions, see appendix A..4 for furher deails. Descripion: he y-axis on he lef measures Tobin s Q, while he righ axis measures he number of consrucion sar-ups. From char 3.3 i is apparen ha here is a posiive correlaion beween Tobin s Q and he consrucion on new properies in Denmark, ha is as he real esae prices increase 44 De Økonomiske Råd, (008), p.9 45 Danmarks Naionalbank, (003), p.46-49

25 (Tobin s Q increases) more properies are consruced, which are in good compliance wih he heory. I should be noed ha here seems o be a slow down in he aciviy in he consrucion indusry, his could poenially be an imporan indicaion of a cyclical downurn in he Danish economy 46. I is imporan o emphasize he fac ha in describing equilibrium in real esae prices i is imporan o remember he ime facor. The consrucion of new real esae akes ime due o consrains in he consrucion indusry. This means ha in he shor run i is possible ha he real esae prices will exceed he consrucion cos, bu in he long run he real esae prices will converge owards he cos of consrucing a new propery. Consrains on he availabiliy of consrucion sies, especially aracive sies, can make he real esae prices deviae from equilibrium even in he long run. The long run equilibrium is one of he main reasons why here is a connecion beween real esae prices and consrucion cos on one-side and consumpion prices on he oher side 47. The Imporance of he financial markes: I has become more and more clear ha he financial markes play an imporan role in he dynamics of he real esae marke and he overall economy. In his connecion here are several imporan dynamics ha are worh menioning: The Credi Channel The credi channel is he erm used o describe he siuaion where an iniial drop in ineres raes increases he real esae prices and hereby he wealh of he homeowners. The increased prices increase he collaeral value, giving he household access o increase heir deb. This rend pus even more upward pressure on he real esae prices, boosing real esae even furher. During his process he LTV raio is kep approximaely consan, bu in a siuaion wih subsequen increase in he ineres rae he servicing of he deb will become more expensive. The increased deb servicing cos will have a negaive impac on he real esae prices. The effec is ha he LTV raio increases due o dropping real esae prices. One of he major reasons for he unforunae developmen hrough he credi channel is ha newly obained deb is used 46 Inernaional Moneary Fund (008), p.4 47 De Økonomiske Råd, (00), p.8

26 on consumpion and no on improving he propery. I is also believed ha he credi channel can be an imporan facor in riggering boom-burs cycles in he real esae marke 48. Equiy Wihdrawal House equiy wihdrawal is acually closely relaed o he credi channel. I refers o he siuaion where he real esae prices have increased and hus making he homeowners wealhier. In his siuaion here are wo ways o realize a capial gain; one way is o borrow more agains he curren real esae value. Anoher way o realize he gains are in he second-hand marke beween ransacions of houses, where he homeowners can reduced his/her own conribuion and borrow more in he financial markes; in boh cases house equiy is being wihdrawn, bu he mechanism in he wo siuaions is quie differen 49. The credi channel and house equiy wihdrawal are more a resul of he increased real esae prices han a source o he increased real esae prices, he poin is ha hese wo phenomenon are imporan in ha hey channel capial back ino he sysem boosing he economy and affecing he macroeconomic facors in he enire economy. 3.3 Economeric Techniques Applied o Analyze he Real Esae Marke The economeric echniques applied in he empirical lieraure are basically idenically, and hey also have he same goal, which is o capure he facors and he dynamics affecing he pricing of he enire real esae marke as well as he individual propery. In describing he dynamics of he enire real esae marke praciioners ofen use a vecor auoregressive (VAR) framework or a srucural VAR framework. This framework builds on he basic auoregressive process (AR-process), where he acual observaions in he marke is explained in erms of pas informaion on prices and various oher inpus ha can help o explain he acual prices. The main difference beween he simple AR process and he VAR is, ha in he VAR framework several series of daa are modelled a he same ime. In he lieraure srucural models are ofen uilized, where he difference beween a srucural and a non-srucural model simply is ha in he 48 European Cenral Bank (003), p European Cenral Bank (003), p

27 srucural model he equaions are derived from and in agreemen wih general economic heory 50. Where he simple AR(p)-process is occupied wih describing he behaviour of a single ime series process, he VAR framework can be used o analyse he more complex siuaion where muliple ime series are analysed in an aemp o describe he srucure in he behaviour of he ime series o be able o undersand he governing dynamics. The basic formula of he VAR(p) model is he same as he AR(p)-process wih he imporan excepion ha he variables included in he model are vecors and marices o encompasses he mulivariae naure of he echnique. Furhermore he use of lag operaors makes he noaion simpler. The VAR(p)-model is ofen depiced as: Where y + Θ(L) y µ + ε (E.3.) = µ + Θy Θpy p ε or in shor as: = Θ (L) is a marix of polynomials in he lag operaor. I is assumed ha he error erm behave like whie noise, having E(ε ) = 0 and E(ε ε -j) = Σ for j = 0 oherwise zero. Σ being a posiive definie marix 5. The VAR analysis is appropriae when he impulse responses from he included variables are going o be analyzed. If he variables included in he VAR analysis are difference saionary he es is invalid. This means ha uni roo esing is necessary o rule ou he presence of a non-saionary daa series. In he case of a uni roo, a vecor error correcion model (VECM) has o be implemened o ake he characerisics of he uni roo and a possible coinegraion relaionship ino accoun. Aemp o analyse he long-erm relaionship beween he variables can be done via a coinegraion analysis, i.e. he VECM. This ype of analysis is used o indicae long-erm connecion beween variables. Moving from he VAR model o he VECM some modificaion has o be done. The VAR model has o be reparameerized ino differences and are lagged (single level erm) o obain he VECM model, which can be formulaed as: y = µ + Πy + Φ y Φp y p+ + ε (E.3.3) 50 Greene, W.H., (008), p.355 and p Greene, W.H., (008), p

28 Where Π a marix describing he long run connecions beween he variables 5. I should be emphasized ha he model specificaion and consrucion is a he cener of aenion when empirical research on he real esae marke is underaken. I is imporan o undersand which facor heoreically should have an impac on he developmen of he real esae prices. The resul obained from he model is only as good as he model specificaion, so misspecificaion of he model could yield wrong resuls and resul in misinerpreaion. 3.4 Empirical Resuls In Denmark here are a variey of heoreical macroeconomic models, describing he dynamics and developmen in he real esae marke. Among he mos respeced official real esae models are MONA (Danmarks Naionalbank ADAM (Danmarks Saisik - and SMEC (De Økonomiske Råd which are based on somewha similar assumpions as sipulaed in he above demand-supply framework. Several academics have used he VAR approach in an aemp o describe he developmen of prices in he real esae marke. Tsasaronis and Zhu [004] made a cross-counry analysis of he driving forces behind he price dynamics. In heir srucural VAR model hey included he growh rae of GDP, he rae of inflaion, he real shor-erm ineres rae, he erm spread and he growh rae in inflaion adjused bank credi of 7 indusrialised economies including Denmark. They decomposed he observed variabiliy of he endogenous variables over he sample o he six innovaions provides a measure of heir relaive imporance in he deerminaion of he overall dynamics of he sysem. When all he counries in he analysis are pooled he resuls indicae ha he mos imporan facor in deermining he innovaion in he oal house price variabiliy is inflaion, explaining approximaely 53% of he variabiliy. Bu also he GDP, Bank Credi and he shor rae and erm spread are imporan deerminan of he real esae price developmen. I should also be emphasized ha heir resul differ from counry o counry, mainly due o differences in he morgage sysems Verbeek, M., (004), p Tsasaronis, K and Znu, H., (004) 5

29 Tsasaronis and Zhu s findings are in good compliance wih an analysis performed by Rober Wagner [005] who conduced a similar empirical analysis of he Danish real esae marke. Wagner s resuls show ha he fundamenal economic facors explain abou 90% of he change in he real esae prices since 993, Wagner has among oher facors included inflaion, income, ineres raes, number of firs ime home buyers, new consrucion of houses and changes in axes. The resul from he analysis underlined ha he mos imporan facors are inflaion (36%), income (39%) and ineres rae (.6%) conribuing o he increase in he real esae prices in he period 54. Wagner s analysis is no as clear cu as Tsasaronis and Zhu s analysis due o he fac ha Wagner s model includes more explanaory facors wih adverse effecs. The empirical resuls emphasized in his secion are represenaive for empirical research made on he basis of he real esae marke, for addiional resuls of empirical research see appendix A..5. The analyical resuls included in his secion and he appendix underscore he imporan poin ha here are numerous ways in which o build an empirical model on he price developmen in he real esae marke. I should furhermore be noed ha he resuls varies quie a bi according o he chosen model and especially he acual model specificaion. Even hough he acual resuls vary, he imporan economic facors in deermining he real esae prices are similar in he various models. The variaion in he empirical resuls are undoubedly due o he differen ways here is o formulae he relaionships beween he variables and he numerous ways in which o measure hese variables. Furhermore he choices of ime period have an imporan impac on he resuls. In April 008 he Inernaional Moneary Fund (IMF) have in heir laes World Economic Oulook, analyzed he Danish real esae marke using a VAR approach. They have prediced ha he marke could be overvalued by abou 7,5% (indicaing a bubble in he real esae prices). Their resuls seam from unexplained observed price increases in he Danish real esae marke. Their model could no explain he price increase by changes in he fundamenal facors included in heir valuaion model of he Danish real esae marke. IMF are weak in heir predicions on he fuure developmen and conclude ha he Danish real esae prices could reac in wo ways, eiher decline or 54 Wagner, R., (005) 6

30 adjus hrough he fundamenal facors, driving he real esae prices, i.e. hough inflaion, bu hey also emphasize ha a furher deerioraion in he fundamenal facors would increase he probabiliy of declining real esae prices in he near fuure Conclusion: Valuaion of he Danish Real Esae Marke The economic facors and heir dynamics described in his chaper can seem a bi overwhelming and he resuls obained from he analysis of he condiions in he Danish real esae marke are far from clear-cu. In able 3.5, he resuls from he analysis of he demand and supply facors are dividend ino macroeconomic and microeconomic facors, where he macroeconomic facors are characerized as he sudy of broad aggregaions of markes and in conras he microeconomic facors are focused on describing he individual decision-making and behavior o give an alernaive synopsis of he facors and heir effec on he real esae prices 56. Table 3.5: Synopsis of Macro- and Microeconomic Facors Macroeconomic facors Microe conomi c Conclusion Household income The low unemploymen rae and he expeced increase in he price level poin in he direcion of increased household income in he years o come, which also are confirmed by he economic councils forecas of GDPgrowh. Employmen The hisorically low unemploymen rae resuls in an increasing income and wealh accumulaion. Inflaion The low unemploymen rae and he increasing living coss boos he inflaion, so he expecaions are ha he inflaion will increase in he years o come. Ineres rae changes The increase in he ineres rae increases he deb servicing cos resuling in he downward pressure on he real esae prices. Demographic changes The demographic developmen sends mixed signals, he age group beween 5-64 year decreases in he coming years, bu he age group above 65 increases. Supply of real esae The supply of real esae is a a hisorically high level pressuring he real esae prices down. Impac on real esae prices Increasing real esae price. Increasing real esae prices. Increasing real esae prices. Decreasing real esae prices. Unchanged real esae prices. Decreasing real esae prices. 55 Inernaional Moneary Fund (008), p.3-6 and p Frank, R.H., (003), p.5-6 7

31 Financial markes Taxes, subsidies and oher public policies The presen financial crisis creaes uncerainy in he marke. The uncerainy abou he fuure ineres level has a negaive effec on he real esae prices, mainly due o he increased ineres raes. The inroducion of he ax sop in 00 and he presen ax relief, he household income is boosed having a posiive effec on he real esae prices. Decreasing real esae prices. Increasing real esae prices. From he synopsis i is eviden ha he facors have very differen implicaions on he real esae prices. There are no uniform predicions on he fuure price movemens in he Danish real esae marke. However, here are some indicaions of a slowdown or decrease in he real esae prices in he ime o come. This slowdown is mainly brough on by he uncerain economic oulook, he increasing ineres raes and he large supply of new properies o he Danish real esae marke. The fac ha he real esae prices have been developing very opimisic for a long period resuling in a likely bubble in he Danish real esae prices, is also an imporan facor, which mos likely will resul in a slowdown or in he wors case deerioraions in he Danish real esae prices in he near fuure. Two likely oucome for he fuure Danish real esae prices seem predicable eiher a decrease in he real esae prices or an adjusmen via he fundamenal facors. 4 Measuring he Underlying Asse I is of crucial imporance o be able o measure he developmen in he prices in he real esae marke accuraely o build a srong foundaion for he consrucion of home equiy insurance produc, which are dependen upon a reliable measure of he developmen in he real esae marke. Before coninuing he chaper, i is imporan o emphasize ha he objecive of he measure of real esae prices is o make he measure (he index) resemble he value associaed wih a sandard claim on fuure income associaed wih owning real esae. This objecive is complicaed by he naure of real esaes. There are hree imporan issues which enails a serious obsacle in reaching he goal of consrucing a reliable measure of he developmen in he real esae prices; he heerogeneous naure of real esaes, he infrequen ransacions and he mere fac ha he adverised prices ofen is a poor guide o he acual selling price 57. The goal of he 57 The acual selling price is ofen reached hrough negoiaion or a an aucion. 8

32 index consrucion herefore becomes an aemp o sandardize he claim on he residenial real esae asse. If he sandardizaion of he index is no obained he price changes observed as a change in he index may no be associaed wih real changes in he price of real esae, he changes could hen be brough on by he simple fac ha he characerisics of he real esaes sold in a given period is no represenaive for he marke as a hole, making he index biased Indices on Real Esae Prices There is a wide range of differen measures of real esae prices, varying from he mos basic measuremen of cenral endency in he real esae prices as average or median sales prices, o more sophisicaed economeric echniques as mix adjusmen, hedonic, repea-sales and hedonic repea-sales measures. These echniques and he heory behind will be he cenre of aenion in his chaper, saring wih he mos basic measures. 4.. Simple Average and Median The simpliciy of he average or median price indices is heir sronges argumen. The index is easily calculaed and he daa requiremens are limied o informaion on he sales prices a which all properies ransaced during a given ime period 59. Furhermore he saisics are easily undersood and available in mos geographic areas also in Denmark 60. Generally he median price is preferred o he average, due o he sharply skewed disribuion of real esae prices, which mean ha he average price will be grealy influenced by changes in he mix of properies sold in a given period, whereas changes in he mix of properies will only have a mued impac on he median price. The mix of properies sold in any given period is no consan over ime. This means ha he simple measures end o oversae he price increase. The major problem wih hese simple measures is quie subsanial, hey do no make adjusmen for he changes in he qualiy of properies sold. Improvemen in qualiy of a given propery should of course increase he value of he propery, bu he simple average and median prices are no able o separae ordinary price inflaion from price increases due o qualiy improvemens, which resuls in posiive biased resul from hese simple measures Shiller, R.J., (998), p.6 59 Hansen, J., (006), p.0 60 Li, W., Prud homme, M. and Yu, K., (006), p.3 6 Case, B. and Wacher, S., (000), p

33 Due o he shorfalls of he simple average and median indices, several mehodologies have been developed o improve he reliabiliy of he indices. The evoluion of real esae indices have moved on o improve on he fac ha he basic indices do no accoun for changes in qualiy of he properies. Among he mos imporan mehodologies are he mix-adjusmen, he hedonic and he repea-sales indices. 4.. Mix Adjusmen Mix-adjusmen measures refer o a simple modificaion of he average or median indices, where aemps are made o sraify he sample ino subgroups (cells) wih similar locaion and physical aribues making he subgroups more homogenous and hereby making he index more reliable and less biased, wihou complicaing he mehodology unnecessarily 6. The average prices of each subgroup are calculaed and every subgroup is weighed ogeher resuling in a mix adjusmen price. The force of his echnique is ha he mix adjusmen prices are no sysemaic affeced by changes in he mix of properies sold in a given period as were he case wih he simple indices Hedonic Regression The hedonic regression is basically an aemp o overcome he issue of he composiional bias arising from he simple cenral endencies measures. Cour [939] was among he firs o apply he hedonic echnique. Cour applied a mulivariae regression echnique in he auomoive indusry o evaluae car prices. The concep was fundamenally very similar o he one laer used in he real esae marke. Griliches [97] and Rosen [974] were he firs o acually apply he hedonic approach o he real esae marke. The hedonic index is based on ordinary leas square regression, where regressions are underaken in each ime period. The hedonic mehodology suggess ha he price of a propery is given as a funcion of he ime period in which he propery is ransaced plus he hedonic characerisics of he propery. A hedonic price funcion on a given propery could be given by: β β Y + γ T, + γ T γ T i. (E.4.) i i i, n i, n P = αx e i, 6 Hansen, J., (006), p.0-63 Wood, Rober (005), p.4 30

34 The hedonic price funcion is ofen illusraed using logs which resuls in he following expression: ln P = lnα + β ln X + β Y + γ T + γ T, γ T i. i i i, i (E.4.) n in Where P i, is he ransacion price of propery i, in period. X i and Y i are hedonic aribues of he propery (X measured coninuously and Y measured discreely), T i,n are dummy variables indicaing he ime period in which he propery was ransaced. α, β j and γ τ are he parameers o be esimaed. There are several advanages o he hedonic approach compared o average, median and he represenaive mehod. By decomposing he propery ino srucural and locaional characerisics, he qualiy variables can be conrolled and priced by he hedonic echnique. Tha is by decomposing he individual propery ino characerisics where he prices are relaively sable he hedonic echnique are able of incorporaing he heerogeneous naure and he changes in qualiy of he real esaes ino he index 64. One of he major disadvanages wih he hedonic approach is he daa requiremen, where daa on he characerisics, he sale prices and he ime of sales are needed o consruc a hedonic real esae index 65. I should also be noed ha even hough he hedonic mehodology incorporaes informaion of changes in qualiy of he propery ino he index, here is no guaranee ha imporan qualiy characerisics are no lef ou. This will resul in an increasing index, where he acual increase was caused by qualiy improvemens insead of acual price increases in he properies 66. Anoher unforunae feaure of he hedonic framework is ha a large number of qualiy variables may be needed in order o obain a saisfacory explanaion of he real esae prices; his enails some problems (i.e. mulicollineariy see appendix A..) and can make esimaion of he model almos impossible 67. The acual finished hedonic model is based on empirical research concenraing on which characerisics o include and he form of he relaionship beween he real esae 64 Case, B. and Wacher, S., (000), p Rappapor, J., (007), p Case, B. and Wacher, S., (000), p.0 67 Clapp, J.M. and Giaccoo, C., (998), p.7-8 3

35 price and he characerisics 68, his enails poenial model specificaion issues, which could resul in biased resul if he funcional form is incorrecly specified, emiing relevan variables in he hedonic equaion will heoreically biased he resuls 69. See he appendix A.. for examples of which hedonic variables praciioners include in hedonic indices based on real esae prices. The lesson o be learned from he disadvanage characerisics of he hedonic mehodology is ha one should ry o find models ha are more robus o omission of variables and model specificaion. The repea-sales mehod is in ha conex a more sable mehod compared o he hedonic mehod Repea Sales The change in he composiion of properies is a major obsacle in consrucing a reliable index, where he hedonic mehod ry o overcome his hurdle by decomposing he value of he propery, he repea-sales measure aims o conrol he composiional change hrough ime by maching pairs. This implicaes ha a propery needs o be sold more han once in order o be include in he index calculaions. The reasoning behind he mehodology is ha he qualiy of one specific propery is on average consan, making he change in price of he specific propery an unbiased indicaor of a price change. In conras o he hedonic approach he repeaed measure does no need o measure he qualiy variables, he repeaed measure mehod simply assumes ha he propery sold does no change physically during he observaions 7. The repea-sales measure was firs inroduced by Bailey, Muh and Nourse [963]. Case and Shiller [987] developed he mehodology furher o incorporae assumpion on he reurn generaed from he real esae. In he original model by Bailey, Muh and Nourse i was assumed ha he dispersion of reurn generaed by real esae was consan across ime. Case and Shiller developed his model furher o incorporae a relaionship beween he dispersion of reurn and he ime beween sales, see appendix A..3 for furher deails on he incorporaion of dispersion in reurn. The repea-sales index can be 68 Griliches, Z., (97), p.4 69 Case, B. and Wacher, S., (000), p.0 and Hansen, J., (006), p.7 70 Shiller, R.J., (998), p.9 7 Clapp, J.M. and Giaccoo, C., (998), p.8 3

36 consruced in numerous ways. The repea-sales model can be derived from he hedonic model, by expressing he raio of he prices for wo ransacions of he same propery in he hedonic model ( indicaing observaions from he previous sale): P i, P ' i. αx = αx β i ' β i e e β Y γ T i + i, + β ' ' Yi + γti, + γ T i, γ ' Ti, γ T n i, n γ ' nti, n (E.4.3) The raio are ofen exemplified by he use of he naural logarihm, which yields: Pi ln P, ' i. Xi ' ' ' = β ln + β ( ) (,, ) (,, )... ( ' Yi Yi + γ Ti Ti + γ Ti Ti + + γ n T X i i, n T ' i, n ) (E.4.4) The repea-sales mehodology assumes as menioned ha he characerisics of he propery are kep consan hroughou ime (ha is simplifies he equaion: X = X and i ' i ' Y i = Y i ), which Pi ln P, ' i. ' i, ' i, = γ ( T T ) + γ ( T T ) γ ( T T ) (E.4.5) i, i, n i, n ' i, n ' I should be noed ha he expression T i, n T i, n akes on he value in he period of he iniial sale, while + in he period of he second ransacion, oherwise he expression is 0. Every propery ha is sold more han wice is included in he model, wih he excepion of he properies ransaced muliple imes wihin he same ime period, in which case all he erms on he righ-hand side are 0, and he propery is dropped from he sample. The key advanage of he repea-sales mehod compared o he hedonic mehod is he grealy reduced daa requiremen. The only informaion needed is he ransacion prices; he ime of sales and an indicaion of wheher or no he qualiy of he propery has been changed during he ime beween he sales. The issue wih his advanage is ha he analys needs o gain insigh ino he qualiy characerisic of he properies, which in pracice means ha he daa requiremens are roughly he same as in he hedonic framework 7, bu heoreically he requiremens are lesser han in he hedonic case. Anoher advanage of he repea-sales echnique is ha i uses all informaion conained 7 In pracice he analys ofen assumes ha he qualiy is unchanged 33

37 in he ransacion prices, where he hedonic only accouns for he characerisics ha are idenified, he repea-sales include he observed as well as he unobserved characerisics, as long as he characerisics are unchanged hroughou ime. The main disadvanage of he repea-sales echnique is ha is wases informaion; all he informaion conained in he ransacion prices of properies ha are sold only once are los in he repea-sales framework. This implies ha a lo of properies are lef ou of he sample; his is a grea cos for he advanages ha he model enails 73, furhermore i should also be kep in mind ha he mehodology requires a large daabase so ha here are sufficien observed pairs of ransacions o make he esimaion reliable, if here are oo few paired ransacions his will resul in high sandard errors Hedonic Repea Sales The disadvanages of he repea-sales mehod sill leave wishes o be graned, especially he wase of informaion in he repea-sales mehod was a moivaional facor of he hedonic repea-sales echnique, which is a hybrid model ha combines he hedonic and he repea-sales echnique. The hybrid uses he informaion conained in single sales and muliple sales of he individual propery. Case and Quigley [99] were he firs o sugges he hybrid model; Quigley [995] renewed he hybrid - his ime based on an explici error srucure 75. The noaion of he hybrid differeniaes in ha here are no ime dummies included in he noaion. The basic hybrid model can be derived using hree fundamenal scenarios, he firs scenario is based on one observed ransacion price, where i is assumed ha he propery value a ime 0, is given by: a a a3x 3 i, 0 = X e (hybrid ) (E.4.6) P αx The same propery is sold a ime, where he ransacion price is given by: b b b3x3 P i, = Pi, 0X X e (hybrid ) (E.4.7) These equaions pu ogeher yield ha if we observe a ransacion a ime, hen he price of he propery is given by: 73 Case, B. and Wacher, S., (000), p.0-03 and Case, K.E. and Shiller, R.J., (987), p Shiller, R.J., (998), p Quigley, J.M., (995), p. 34

38 ln, = lnα ln ln ln ln (E.4.8) P i + a X + a X + a3x 3 + b X + b X + b3x 3 In he second scenario, wo ransacions of he same propery wih unchanged qualiy are observed, a ime and τ, where > τ implying ha observed ransacion price a ime is given by: ln P, i, ln Pi τ + b( τ)ln X + b( τ)ln X + b3( τ) X 3 = (E.4.9) The las scenario assumes, like he previous scenario, ha wo sales are observed a ime and τ, where > τ, wih he modificaion ha he qualiy of he propery is changed inbeween he wo observaions a ime * from (X,X,X 3 ) o (X *,X *,X * 3 ), where > * > τ. This resuls in he following equaion: ln P i, = ln P + b [ ln X * i, τ τ ln X * + a ln( X / X ) + a ] + b [ ln X * ln( X * τ ln X / X ) + a ( X ] + b [ X 3 3 * 3 * 3 τx X 3 ] 3 ) (E.4.0) Equaion are sacked and he coefficiens are esimaed via ordinary leas squares. The hybrid is in effec a weighed average of he hedonic and he repea-sales mehod. Imposing resricing on he equaions will make esimaion more efficien wih he use of generalized leas squares see appendix A for a more deailed descripion of he esimaion echnique. The reasoning behind he hree scenario equaions is ha in he firs scenario only a single ransacion is need o esimae he model (similar o he hedonic regression), whereas scenario equaion wo is able o incorporae muliple ransacions where he hedonic characerisics are unchanged (similar o he repea sales mehod) and finally in he hird scenario equaion where he characerisics are changed beween he muliple ransacions, his model is able o incorporae he feaures of he hedonic and repea-sales mehodology ino one model he hedonic repea sales mehodology 76. The main advanage of he hedonic repea-sales approach is ha i uses all he informaion available in he sample period, which is accomplished by combining he hedonic and he repea-sales echniques. Where he repea-sales mehod would exclude properies where he qualiy had changed beween observaions, he hybrid is able o make use of hese observaions as well as informaion 76 Case, B. and Quigley, J.M., (99), p

39 on properies only sold once. This is clearly an advanage when informaion is a limied resource Indices Used in Pracice oday In an aemp o ge an idea of which mehodology is mos widely used in pracice oday an overview of he mos popular indices are depiced in he following. There are basically wo sources o real esae indices; governmenal agencies and privae agencies. The governmenal agencies ofen provide he informaion free of charge whereas he privae agencies sell he informaion. In he following able he mos widely used residenial real esae indices are colleced. I should be menioned ha only Denmark, Unied Kingdom and he Unied Saes are included. The decision o choose hese paricular counries is based on he argumen ha he mos developed residenial real esae indices are found in hese counries (UK and US). Table 4.: Main Residenial Real Esae Indices (Denmark, UK and US) Provider Mehodology Denmark Realkrediråde Average Danmarks Saisik Average The UK Land Regisry Average ODPM (old) Mix adjusmen ODPM (new) Mix adjusmen Homerack Mix adjusmen Righmove Mix adjusmen Halifax Hedonic Naionwide Hedonic The US The Naional Associaion of Realors Median The Census Bureau (Governmenal) Median and Hedonic The Bureau of Labor Saisics (Governmenal) Represenaive propery S&P/Case-Shiller Home Price Index Repea-sales OFHEO House Price Index Repea-sales Source: See he bibliography for references o he various suppliers of residenial indices From he able i is clear ha a variey of differen mehodologies are applied in pracice in he consrucion of residenial indices, from he knowledge abou he indices included in he able, he mos successful indices seem o be he indices, which uilize he mos sophisicaed mehodologies including he Halifax House Price Index in UK and S&P/Case-Shiller Home Price Index in he US. Derivaives are currenly rading on he Halifax and he Case-Shiller (mainly fuures and opions) indices, which clearly indicae 77 Case, B. and Quigley, J.M., (99), p

40 ha he mehodologies used o consruc hese indices are superior when i comes o derivaive rading and reliance upon he indices. In he commercial segmen here are many more suppliers of indices 78, which indicaes ha he marke paricipans in he commercial segmen are more occupied wih hedging of he risk associaed wih invesing in real esae. I should be noed ha he risk he insiuional invesor akes on could be far greaer han ha he privae invesor akes on invesing in a single propery. The mos widely used indices in he commercial segmen is he Invesmen Propery Daabank Index (IPD) in UK 79 and he Naional Council of Real Esae Fiduciaries Propery Index (NPI) in US 80. In he commercial segmen an addiional mehodology is used o consruc indices. This mehodology is called he porfolio approach, for furher deails on he porfolio approach see appendix A..7. When comparing he public available indices in Denmark o he indices available overseas, i is clear ha he mehodologies, which are applied on he Danish residenial marke are inferior o heir counerpars in UK and US. In Denmark here are mainly wo providers of public available real esae indices, Realkrediråde and Danmarks Saisik. Realkrediråde publishes updaes on heir real esaes indices on a quarerly basis. Their saisics are based on observaions made by morgage insiuions. They publish prices on he average price per square meer divided ino differen regions of he counry and differen ypes of residence (deached and errace houses, owner-occupied fla, weekend coage). Their daabase covers a period of more han 0 years saring he s of January 995. The indices are sraified ino cells according o he size and locaion of he residence. Indices covering larger geographical areas are weighed, where he areas are weighed according o heir respecive share of he oal populaion of properies 8. Danmarks Saisik also publishes quarerly indices on real esae prices, heir saisics are very similar o he daa supplied by Realkrediråde, in ha he indices are divided ino subgroups according o heir geographical locaion, and he ype and size of propery. Danmarks Saisik s daabase on real esae prices daes back o See appendix A..6 for a represenaive lis of commercial indices. 79 Deusche Bank Research (007) 80 Merrill Lynch (007) 8 Meodenoa ejendomsprissaisikkens opbygning og grundlag 37

41 49 bu he basis of he calculaions has changed and he curren mehodology has been used since Discussion: Implemening Indices The discussion on how o acually consruc he indices wih respec o geographical coverage and real esae ype and which approaches are preferable is a quesion ha is no easily answered. The variaion in real esae prices beween various real esae ypes and various geographical locaions can be subsanial, which could speak for a consrucion of finer deailed indices o incorporae he variaion ino he indices. In he exreme case he individual real esae could become he index. Anoher issue wih consrucing finer deailed indices could pose furher problems wih illiquidiy 83. In addiion o he level of deail (ype and geographic), he quesions of which index mehodologies are mos suied o use in a home equiy insurance seing arise. From he analysis of he indices used in pracice, here seems o be a movemen owards he more sophisicaed mehodologies when derivaive rading has o be applicable, which is dependen on a very reliable index mehodology. I is also clear ha o make home equiy insurance available in Denmark; he sophisicaion of he applied mehodologies has o improve in order o make home equiy insurance a realiy and applicable in Denmark. 4.4 Conclusion: Indices on Real Esae Prices The choice of he mos reliable and suiable real esae index includes many consideraions. One of he mos imporan consideraion o ake ino accoun is he daa requiremens and he complexiy of he index including consideraion of which feaures he index mehodology should incorporae, in able 4., he advanages and disadvanages are lised. Table 4.: Comparison of Real esae Indices Index ype Advanage Disadvanage Mean (average) or median + Daa requiremens + Simpliciy + Easy inerpreaion - No correcion for qualiy changes - Misleading Mix Adjusmen + Conrol for qualiy changes - Daa requiremens + No model specificaion Hedonic Regression + Conrol for qualiy changes - Daa requiremens Adminisraive oplysninger om saisikproduke: Ejendomssalg 83 Shiller, R.J and Weiss, A.N., (999), p

42 Repea Sales + No jus one represenaive propery + Daa requiremens + No model specificaion necessary - Poenial bias from incorrec model specificaion - Assumes no qualiy changes - Wases daa (a leas wo sales required) From he able i is clear ha here is no single index mehodology ha sands ou as he ulimae mehodology. Bu he indices mehodologies, which encompass qualiy adjusmen echniques, are favored compared o he more basic indices. The hedonic mehod includes qualiy adjusmen echniques bu demands high qualiy daa whereas he repea-sales mehod offers an easier way o conrol he qualiy changes wih favorable daa requiremens. The hybrid offers he beer of he wo echniques 84 bu he gain from he hedonic repea-sales hybrid is negligible 85. All in all he hedonic and he repeaed measures (and combinaions of he wo) are more sophisicaed and accurae in describing he price and price change of real esae prices. These sophisicaed mehodologies will also be preferable when rading in derivaes is anicipaed. 5 Analysis of The Underlying Asse The indices on Danish real esae prices (Realkrediråde) will be horoughly analysed via economeric mehods in an effor o be able o price home equiy insurance producs wih he Danish real esae indices as he underlying asse. The economeric analysis is necessary due o he fac ha he pricing of derivaives is buil on ses of assumpions abou he characerisics of he underlying asse. The previous chaper indicaes ha i would be preferable o use indices based on a hedonic, a repea-sales or a hedonic repea-sales mehodology, unforunaely he only daa available in Denmark are indices based on simple average sales prices of properies sold in a given period in a given geographical locaion. The pracical discussion on he implemenaion of indices in he real esae marke indicaed a payoff beween he abiliy o rack he acual changes in prices of he individual propery and he liquidiy of he indices. For one hing he index needs o rack he local price developmen, o make he insurance produc an efficien hedge agains he risk facing he homeowners, and for anoher here have o be enough observaions o make he index unbiased and reliable. In his hesis he indices cover larger geographical areas o make sure ha he daase conains enough observaions o be unbiased and reliable. In pracice his would no be he opimal soluion for he issuer 84 Case, B. and Quigley, J.M., (99), p Case, B. and Wacher, S., (000), p.0 39

43 of home equiy insurance producs. For furher discussion on his issue see secion Three indices will be analysed, hey all cover he same geographical area (all of Denmark), bu varies in he ype of residence. Index P, are based on observed prices of deached- and errace-house, whereas Index E are based on ransacions prices of owner-occupied flas and Index F are based on he prices of weekend coages, see appendix A Saionariy or Uni Roo The concep of saionariy is of crucial imporance when i comes o undersanding and inerpreing ime series daa. If a daa series is saionary hen he expecaions and variance of he daa can be consan. When he covariance of he daa series is only dependen on he ime difference and no dependen on he acual poins in ime, hen he process is said o be covariance saionary. The concep of saionariy is imporan when he relaionship beween ime series daa is o be analyzed. There have o be sabiliy in he sysem o be able o analyze he relaionship beween variables. If he daa series change arbirarily due o insabiliy in he sysem, hen here is no much informaion in he daa series analysis and he knowledge obained from he analysis is minimal. If he sysem is unsable i is said o be non-saionary. In he case of nonsaionariy he daa have o be ransformed in order o obain saionariy and reliable resuls from he analysis. This ransformaion process is called inegraion. Inegraion is he degree of differencing needed in order o obain saionary daa. The indices are depiced in appendix A.3.. From he graphs i is clear ha he indices a level are no saionary bu evolving around a deerminisic rend. Aemps are made o ransform he indices; he simple log 86 (naural logarihm) ransformaion does no seem o remove he deerminisic rend. The firs differences (inegraion of order one) are aken from he daa a level, and from level daa wih log-ransformed daa. The graphs depicing he firs difference (dp, de and df) seem o behave more saionary, bu he variance seems o be unsable. The firs difference of he log-ransformed daa 86 Log: is he naural logarihm in he res of he hesis 40

44 (dlogp, dloge and dlogf), seems o be more saionary and he variance seem more homogenous. A non-saionary process is said o conain a uni roo, in order o deermine if he indices conains a uni roo he augmened Dickey-Fuller (ADF) es is carried ou. The basic Dickey-Fuller (DF) es is robus agains some degrees of heeroskedasiciy while auocorrelaion causes problems. The use of he ADF es solves he issue wih auocorrelaion in he error erm, by included lagged variables of he dependen variable in he regression, when enough lagged variables are included in he model, he auocorrelaion in he error erm disappears. In he ADF es he equaion o be esimaed can ake he following form: Y = α βy + γ Y + γ Y + + γ n Y n e (E.5.) Wih mean and n-lagged differences included. A decision has o be made on he number of lagged differences included in he ADF es; here are several suggesed procedures, one approach is o chose he appropriae model on he basis of he Akaike and he Schwarz informaion crierion, where he esimaed model wih he lowes es score is he model ha fis he daa bes, when he number of esimaed parameers and he sample size are aken ino accoun (see appendix A.3.3 for furher deails on he DF and ADF es and appendix A.3.4 for informaion on he selecion crierion). Anoher approach is o use a sequenial esing procedure saring from a high-order model and eliminae insignifican coefficiens. The criical values used in he DF and ADF are smaller han he criical values obained from he normal approximaion acquired from he -disribuion. The deviaion from he normal -saisic is due o he fac ha he es saisic in he ADF has a non-sandard limiing disribuion, and he criical values are obained from simulaions 87. From he ADF ess on he level daa i is clear ha he null hypohesis of a uni roo canno be rejeced, i should be noed ha when a rend is included in he ADF es, he null hypohesis is rejeced in he case of ADF(3) 88 for he owner occupied flas and deached- and errace houses indices. In he case of he weekend coages he ADF(5) wih a rend shows significan resuls, his indicaes ha he daa in level are likely o 87 Lükepohl, H. and Kräzig, M. (004), p Wih 3 lagged differences included 4

45 exhibi a linear rend componen 89. Tesing for a uni roo in he differenced daa (de, df, dp, dloge, dlogf and dlogp), he ADF ess resuls are generally more significan, indicaing ha he ime series have o be differenced in order o obain saionary ime series, he choice beween he differenced series and he log-differenced series is no clear cu. The evidence is poining in he direcion of he log-differenced series being more efficien in ransforming he ime series. The ADF ess show more significan es resuls in he case of he log-differenced series compared o he ess conduced on he ordinary differenced series. I should be noed ha he ADF es ha looks he mos promising in boh cases (d and dlog) is wih a consan and no rend included. Indicaing ha he differenced series have a nonzero mean and are no likely o have a linear rend componen. When he resuls from he ADF es is compared o he graphical presenaion of he ime series, he log-difference daa seem o have superior characerisics. The logdifferenced ime series will be used in he res of he analysis of he ime series. 5. Specifying he Appropriae Model The mos widely used process o describe ime series daa is he auoregressive (AR) and he moving average (MA) processes. The AR-process ries o explain he curren observaion (y ) wih he use of pervious values (y -p ). The simple AR(p)-process are ypically represened via he following equaion: p y = δ + θiy-i + ε (E.5.) i= The MA-process is bes illusraed by he following MA(q) process, which is given by he following equaion: y q = µ + ε + α ε j= j j (E.5.3) I is obvious ha MA-processes use a weighed average of he error erms in describing he acual value of y. The error erm ε in he moving average is, like in he AR(p)- processes, also resriced o exhibi whie noise. The wo ypes of processes have 89 See appendix A.3.3 for ADF resuls and derivaion 4

46 differen feaures and i is imporan o know hese aribues in order o choose he mos appropriae model. In he appendix 3.5, he mos basic characerisics of he wo ypes of processes are depiced 90. One way o differeniae beween he AR- and he MAprocesses is by inspecion of he auocorrelaion and he parial auocorrelaion funcions of he ime series Auocorrelaion: The auocorrelaion funcion describes he correlaion beween y and is lagged values. The AR()-process has an exponenial decaying funcion, whereas he MA(q) auocorrelaion funcion cus off afer he q h auocorrelaion 9. I should be emphasized ha i is easier o deermine he MA-process by he use of he auocorrelaion funcion, han i is o deermine AR-processes. The auocorrelaion funcion becomes complex in he case of a higher order (more han wo lags) ARprocess and Yule-Walker equaions 9 needs o be solved in order o deermine he auocorrelaion for higher order auoregressive processes 93. Preliminary resuls from inspecions of he auocorrelaion funcions of he acual indices yield he following resuls: Table 5.: Preliminary resuls from he auocorrelaion funcions dloge dlogf dlogp MA() MA() MA(4) Parial auocorrelaion: To deermine he correc AR-process he auocorrelaion funcions are no as convenien o use as he parial auocorrelaion funcion. The parial auocorrelaion for he AR() model is simply he coefficien θ, he parial auocorrelaion simply measures he addiional correlaion beween Y and Y -k, afer adjusmens are made o he inermediae values. This enails ha he parial auocorrelaion funcion can be used o deermine he appropriae order of he ARprocess. The moving average process canno be deermined from he parial auocorrelaion funcion, he parial auocorrelaion funcion for he moving average process shows a paern corresponding o he paern exhibied by he auoregressive 90 Verbeek, M., (004), p Verbeek, M., (004), p Yule-Walker equaions are a se of linear equaions connecing he parameers of AR-processes wih he auocorrelaion coefficiens. 93 Greene, W.H., (008), p

47 process auocorrelaion funcion 94. Preliminary resuls from inspecions of he parial auocorrelaion funcions of he indices resul in he following model specificaions: Table 5.: Preliminary resuls from he parial auocorrelaion funcions dloge dlogf dlogp AR() AR() AR() The resuls from he auocorrelaion and parial auocorrelaion are no conclusive, his resul could be due o he fac ha an ARMA-model are more suied o describe he ime series. To become more cerain of which model fis he ime series he bes, he suggesed models are esimaed and he residuals are analysed. In he appendix A.3.6, he suggesed models are esimaed and some basic ess are conduced o see how good he esimaed models fi he acual daa (Akaike and Schwarz informaion crierion) and how he residuals behave. The problem wih serial correlaion is addresses wih he use of he Durbin Wason (DW) saisic, bu due o he limiaions of he DW-saisic he auocorrelaion, parial auocorrelaion funcions and he Ljung-Box Q-saisic of he residuals are esimaed alongside he Breusch-Godfrey serial correlaion LM es. The problem arising from serial correlaion in he error erms are more severe han he issue of heeroskedasiciy, bu ess on heeroskedasiciy is also included in he esing of he esimaions (see appendix A.3.7 for furher deails on he problems concerning serial correlaion and heeroskedasiciy). The owner-occupied fla index (dloge): The coefficiens in he esimaed MA() and AR() are significan a a 5% significance level, whereas he AR()-erm in he ARMA(,) are insignifican. This means ha he MA() and AR() processes are more appropriae in describing he developmen in he log-differenced index on owneroccupied flas. The addiional es saisics included in he appendix A.3.6 (model esimaions) indicae ha he AR() process is superior o he MA() process, in ha he Durbin Wason es saisic indicaes ha he error erm in he AR() are closer o whie noise han he error erm in he MA() process. The informaion crierions indicae ha he AR() process have a beer fi o he daa when he number of esimaes parameers are aken ino accoun. The choice of model herefore falls upon he AR() process. The Q-saisics and he Breusch-Godfrey serial correlaion LM es are no able o rejec he null hypohesis of no auocorrelaion in he error erm. The esimaed AR()-process: 94 Verbeek, M., (004), p

48 d log E log + ε (E.5.4) =. +. d E The weekend coages index (dlogf): The model esimaed on he basis of he log differenced weekend coages index, yields various resuls. In he esimaed moving average and auoregressive equaions all he coefficiens are significan a a 5% level, whereas he AR() and MA() erms in he ARMA(,) are insignifican. The oher es saisics included in he able are very similar for he AR() and MA() process, so he choice of he mos appropriae model are no clear. To follow in he pah of he resuls obained from he index on owner-occupied flas, he AR() model is also chosen as he preferable process. The Q-saisics are insignifican a all lags of he residuals (4 lags included). In addiion o he Q-saisic he Breusch-Godfrey serial correlaion LM es is esimaed including 4 lagged residuals, he LM es is no able o rejec he null hypohesis of no serial correlaion up o 4 lags. The wo ess indicae ha he error erm does no exhibi any serial correlaion. The esimaed AR()-process: d log F log + ε (E.5.5) =. +. d F The deached- and errace-house index (dlogp): The coefficiens are significan in he MA(4) and AR() process wih he excepion of he MA()-erm in he MA(4) process wih a significance level a 6.9%. When he MA(4) are re-esimaed wihou he MA()-erm all he coefficiens lef in he equaion are significan. Again in he ARMA(,4) he MA()-erm are he only insignifican coefficien, when aken ou of he equaion he res of he coefficiens are significan. The informaion crierions shows a slighly beer model fi in he MA(4), han in he AR() process, while he Durbin Wason saisic favours he AR() process. In line wih he wo previous conclusions he AR() process is chosen as he preferred descripion of he log difference index on he deached- and errace houses. An inspecion of he AC- and PAC funcions shows no serious signs of correlaion. The Breusch-Godfrey ess are no able o rejec he null hypohesis of no serial correlaion wih four lagged residual included. The esimaed AR()-process: d log P log + ε (E.5.6) =. +. d P Almos no auoregressive condiional heeroskedasiciy was deeced in he residuals of he esimaed AR()-processes, wih he excepion of ARCH(5) - ARCH(8) being significan in he AR()-process based on he deached- and errace-house index (index 45

49 P). This is ignored in ha he impac on he OLS esimaes are minor and limied o he efficiency of he OLS esimaors. The indices seem o exhibi homoskedasiciy in he residuals, which also can be seen from he residual plos from he fied AR()- processes. I should be noed ha ou of he hree indices analysed in his secion he index based on deached and errace houses, seems o be he index, which have he poores fi o he AR()-process, while he oher indices seems more convincing. 5.3 The Auoregressive Process The analysis of he real esae indices seems o sugges ha he proposed AR() models have a saisfacory fi o he daa. The proposed AR()-models indicae ha he indices (dlog-indices) on he real esae prices do no follow a random walk. This in urns means ha he AR()-process can be used o forecas he price changes in he real esae marke. Anoher appealing characerisic abou he AR()-process 95 is ha i has a mean revering endency. When he process exhibis shocks, i will only have a ransiory effec on he AR()-process 96, which is in good compliance wih he equilibrium model, where divergence is possible in he shor run while he long run endency is o converge owards he inflaion. The uncondiional mean can be calculaed on he basis of he esimaed auoregressive models, which yields an 8.8% (dloge), 3.53% (dlogf) and 5.0% (dlogp) uncondiional increase per period of he respecive indices. When he acual price changes differ from he uncondiional mean in he respecive indices, he coefficien of he lagged erm (0.799, and 0.60) implies ha beween /3 and /3 of he deviaion will have an impac on he price changes in he nex period. A well known feaure of he AR()-process is ha i can be rewrien as a infinie moving average MA( ), see he appendix A.3.8 for he derivaion of he infinie MA-process, rewriing he equaions yields he following resuls: dloge dlogf = ε ε +... ε (E.5.7) = ε ε +... ε (E.5.8) 95 θ < 96 Franses, P.H., (998), p

50 dlogp = ε ε +... ε (E.5.9) 5.4 Forecasing wih he Auoregressive Process The fac ha he log-reurn generaed from real esae can be described as an auoregressive process implies ha here is some forecasabiliy in he reurn. By subsiuing successively ino he AR() process and using he fac ha he error erms are independen and uncorrelaed 97. The expeced forecas one period ahead yields 98 : y δ θ ε (E.5.0) + = + y + + Repeaing he subsiuion ino he AR()-process resuls in a general expression for forecasing n periods using he AR()-process yields: n n E y+ n) = δ ( + θ + θ θ ) + θ ( y (E.5.) The abiliy o forecas is an imporan feaure of he AR()-process, bu he abiliy o forecas canno sand alone, i needs o be accompanied by an esimae on he accuracy of he forecas. The forecasing error one period ahead is given as: f ) = [ δ + θy + ε ] [ δ + θy ] = ε + ( (E.5.) In he general case he forecasing error n-periods ahead is given by: f n n = ε + n + θ ε+ n + θε+ n θ ε + ( ) (E.5.3) This implies ha he squares predicion error (SPE) of he AR()-process corresponding o he variance of he -period ahead predicion error is given by σ ε. The SPE corresponding o he forecasing error n-periods ahead is given by: 4 n σ ε ( + θ + θ θ ) (E.5.4) When saionariy is assumed he SPE of he AR()-process approaches a limi, as n ): σ ε θ (E.5.5) 97 Davidson, R. and MacKinnon, J.G., (999), p Verbeek, M., (004), p

51 The SPE implies ha he variance is no linearly dependen upon ime, as is he case in he GBM, his needs o be incorporaed ino he expression for he variance of he process. The variance of he forecas one period ahead is given asσ, he expression for he variance addiional periods ahead is more complex, which can be seen from he esimaed wo periods ahead variance of he process as: ε σ ( θ θ ) ε + + (E.5.6) I is assumed ha he variance of he forecas wo periods is he accumulaed variance including he auocorrelaion beween he wo periods. In appendix A.3.9 he expecaions, he forecas error, he squares predicion error and he variance of he AR()-process are derived and described in greaer deail. The indicaion ha he real esae indices analyzed in his hesis do no follow a random walk process are in good agreemen wih he academic lieraure on he opic. Poerba [99], Kuo [996] and Shiller and Weiss [999] have found similar evidence ha he house price movemens are predicable on he basis of lagged informaion. They basically explain he predicabiliy wih reference o he equilibrium model, where he long-erm dynamics dominaes (consrucion cos, user cos, household income). 5.5 Discussion: Analysis of he Underlying Asse The uncondiional increase seems o be high compared o he descripive saisics on he log difference indices, his needs o be invesigaed furher before inroducing an acual home equiy insurance produc o he marke. The uncondiional increase in he respecive indices has an impac on he acual price seing of he derivaives and is herefore imporan. Anoher opic ha could be ineresing o invesigae furher is he forecasabiliy of he real esae indices, ha fac ha he developmen in he real esae prices o some degree is known, will have imporan implicaions, i.e. if he real esae prices are expeced o decline, hen i would be expensive o insure he home equiy value. 5.6 Conclusion: Analysis of he Underlying Asse The economeric analysis of he Danish real esae indices showed ha an auoregressive inerpreaion of he differenced log-reurn generaed from he real esae indices yielded a reasonable approximaion o he observaions. The AR()-process is 48

52 characerised by a degree of forecasabiliy in he fuure log-reurn, in addiion he AR()-process has a mean revering endency, and if exposed o shocks hey will only have a ransiory effec on he process, he characerisics of he AR()-process are in good compliance wih he descripion of he dynamics in he real esae marke, where he real esae prices can flucuae in he shor run, while he long-run endency is clearly correlaed wih he inflaion in he consrucion coss. From his chaper i is clear ha he characerisics of he AR()-process needs (he forecas and he variaion of he forecas) o be incorporaed ino he pricing of he derivaes on which o build home equiy insurance produc, o incorporae all he informaion available in he real esae indices. 6 Valuaion of Home Equiy Insurance Producs The real esae marke is an obvious marke in which o use derivaives. In a marke where direc invesmen is characerized by high ransacion coss, illiquidiy and uncerain fuure oulook, he use of derivaives is a useful and inexpensive ool o manage he risk exposure. Trading real esae is associaed wih high ransacion coss, which makes is prohibiively expensive o manage he risk exposure and he porion of real esae by rading direcly in he real esae marke. The illiquidiy of real esae is primarily due o is heerogeneous naure, which makes i difficul o manage he iming of buying and selling real esae. The purchase or selling decision is furher prolonged by he necessiy o collec informaion abou he acual propery before being able o make invesmen decisions. The fac ha he fuure is unknown makes invesmens in real esae risky; which makes he opporuniy o hedge he risk aracive o he real esae invesor. Home equiy insurance belongs o he group of financial insrumens classified as propery derivaives. This marke has exised for some years and is divided in a commercial and a privae segmen. The commercial segmen is he mos developed whereas he privae segmen is sill in is infan years and here are very few examples of privae households using propery derivaives. Propery derivaives are of growing imporance and he marke value of he raded produc is increasing rapidly as a resul 99. The mos developed markes for propery derivaives are found in he Unied Kingdom (London) and in he Unied Saes (Chicago), bu also in Neherlands, Sweden, France and Germany here are indicaions of increasing ineres in he propery derivaive 99 See appendix A.4. for a shor descripion of he mos basic use of propery derivaes 49

53 marke. The main propery derivaes raded in he commercial marke oday are swaps, bonds and opions 00. The marke is developing quickly and more exoic producs are being offered o he marke 0. From he shor inroducion of he propery derivaive marke i is clear ha here exis a variey of derivaives, which are applicable in connecion wih he inroducion of home equiy insurance. The emphasis will be on opions and fuures as he derivaes used o insure home equiy value. In pracice he provider of home equiy insurance is free o choose among he various producs and pool he risk of all he insurances. I would properly be an advanage for he provider of home equiy insurance o use more complex derivaes bu, o ge an inuiive feeling abou home equiy insurance and he cos of insuring he home equiy value, fuures and opions seem appropriae. 6. Fuures The value of a fuure conrac is zero when enered ino, during he life of he conrac a margin accoun will be debied and credied according o he movemen in he real esae price index and he posiion hold (shor vs. long). A expiraion of he fuures conrac he conrac can be closed ou by buying an offseing conrac or delivery can ake place. The fuures price is normally given by he following equaion: rt F0 = S0e (E.6.) Where F 0 is he fuure price, S 0 is he curren value of he real esae index, r is he riskfree rae and T is he ime o mauriy. A shor posiion in a fuures conrac would be he mos suiable posiion. A shor posiion is appropriae when he holder of he posiion own he propery and wans o sell he propery a a cerain price a a cerain ime in he fuure. The shor fuure posiion would insure he acual value of he real esae oday, by making sure ha he real esae could be sold a a given value (F 0 ) a he expiraion of he conrac. The conrac would lead o a gain in he case of a decrease in he real esae index and a loss would occur in he case he index increases. There are some problemaic issues conneced wih he use of fuures conracs (shor posiion) in a home equiy insurance seing. For one hing he owner of he propery has o give up 00 Deusche Bank Research (007), p.- and p Merrill Lynch (007) 50

54 he upside change. In a siuaion where he real esae prices increase, he homeowner will have o give up he price increase. For anoher he margin accoun makes he fuures more demanding, where deposi has o be made in case of an increase in he real esae index 0. The issues wih he fuure conrac avoided wih a posiion in a pu opion, which seems o be a more appealing and aracive derivaive o use in his seing. 6. Opions Basically he opion can be used o hedge he downside risk (pu opion) or expose he holder o he upside change (call opion), opions can be used o a variey of differen sraegies, in his conex he pu opion is he mos obvious o uilize in he home equiy insurance produc. The pu opion gives he holder he righ o sell he underlying asse, which in his insurance seing means ha he holder will be able o sell he real esae index a a cerain ime (a mauriy for he European opion) for a cerain exercise price. This mean ha using he real esae index, he insurance holder will be able o insure he value of his propery 03. In he classic Black & Scholes (BS) seing he price of a pu opion is given by he following equaion 04 : p = Ke rt ln( S / K) + ( r σ / ) T qt ln( S0 / K) + ( r + σ / ) T N S e N 0 σ T σ T 0 (E6.) The BS equaion is derived upon a se of assumpions 05. The indices and he characerisic of real esaes violaes several of hese assumpions, where he mos severe violaions are he fac ha he reurn generaed from real esae o some degree are predicable, anoher violaion is he large ransacions cos associaed wih real esae rading. In he BS opion formula i is assumed ha he underlying asse follows a geomeric Brownian moion (GBM). In The GBM process only he presen value of he underlying asse is relevan (he Markov propery), his is obviously violaed by he AR()-processes of he real esae indices, in which previous value of he underlying 0 Hull, J.C., (006), p.48, and p.0-03 Assumes ha he index are perfecly correlaed wih he value of he acual propery, see chaper 6.3. concerning basic risk 04 Hull, J.C., (006), p.95 and p See he BS assumpions in appendix A.4.. 5

55 impac he fuure behavior of he underlying asse. The ransacions cos associaed wih real esae rades are no neglecable, which clearly violaes he BS assumpion 06. In an aemp o price he opions based on he real esae indices a VBA-code is derived, where Mone Carlo simulaions are capable of incorporaing he forecasabiliy of he AR-process ino he pricing of he opions. 6.. Mone Carlo Simulaion In secion 5.4 he naure of he AR()-process were described, in he Mone Carlo simulaions he feaures of he AR()-process (he forecasabiliy and he variance) are incorporaed so ha he opion prices include all he available informaion. The forecasabiliy and he variance of he reurn has major implicaions on he price seing of he opions, which is incorporaed ino he VBA-code of he Mone Carlo simulaion 07. In he acual Mone Carlo simulaion i is assumed ha he error erm is normally disribued, he Marsaglia-Bray algorihm is uilized, his mehod is a variaion of he Box-Muller algorihm, bu avoids rigonomeric calculaions used in he original Box-Muller algorihm 08, when generaing normally disribued random variables. To improve he convergence speed of he Mone Carlo simulaion, aniheic variables are included in he algorihm. The aniheic variable echnique uses he symmerical disribuion of he normal disribuion o obained increased convergences speed wih he same number of simulaed pahs 09. The value of he (European) real esae opions using he Mone Carlo simulaion echnique is calculaed as he presen value of he expeced payoff of he pu opion. The expeced payoff is calculaed as he srike price minus he expeced value of he underlying asse, he presen value of he expeced payoff equals he value of he pu opion. 6.. Valuaion of Opions The prices of he various real esae opions using he Mone Carlo echnique are depiced in appendix A.4.4, and in able 6. he prices of he 3 monhs opions based on he owner-occupied fla index is given, from he able i is clear ha he real esae opions in some aspecs behave as regular sock opions, i is clearly very expensive o 06 Hull, J.C., (006), p and p See appendix A.4.3 for deails on he VBA-code for he Mone Carlo simulaion 08 Paul Wilmo Inroduces Quaniaive Finance, p Paul Wilmo Inroduces Quaniaive Finance, p.597 5

56 insure he propery a a higher exercise value han he acual value of he propery (he pu opion is in he money), i.e. if you wan o insure an average fla a a value of. million in he nex hree monhs, when he fla is only worh million you will have o pay a huge premium. I is imporan o emphasize ha he acual premium paid are dependen upon he acual change observed in he real esae marke, ha is if he marke for flas have jus exhibied a decrease of 0% in he log-reurn, hen he premium of he above pu opion will cos 44,37 kroners, bu on he oher hand if he marke had jus observed increasing reurn of +5%, hen he same pu opion will only cos 35,808 kroners. This shor example highlighs a very imporan aribue of he real esae opions, ha is if he marke experience deerioraing condiions hen he pu opions will be very expensive due o he forecasabiliy of he reurn, he same pu opions will be inexpensive in he case of improving marke condiions. In a more realisic example where he fla is insured a he acual value (a he money), he real esae pu opion for 3 monh will cos,437 kroners when he observed change is 5% in he log-reurn, while if posiive changes is observed he pu opion will be very cheap. Table 6.: Pu Opion Prices - 3 monhs conrac Acual Real Esae value: d.kr A regular sock opion become more expressive wih increase ime o expiraion, bu his in no always he case for he real esae pu opion, from he appendix A.4.4, i is seen ha wih increasing ime o expiraion he opion prices acually decreases in mos cases, which is cause by he mean revering endencies of he indices, which resuls in ha, he reurn generaed from he real esae will be posiive and he pu opion will end ou of he money due o he expecaions inheren in he AR()-process in he long run. Shiller and Weiss [999] have proposed a closed-form soluion o he real esae pu opion and argued ha wih some simple modificaion o he original BS formula, hey 53

57 are able o incorporae he naure (AR()-process) of he indices 0. The modified BS equaion (pu opion) is : p = Ke σ µ + rt ln( K / S 0) µ ln( K / S0) N S0e N σ σ µ σ rt (E.6.3) In he appendix A.4.6 several European pu opions based on he real esae indices are prices using his formula. From he formula i is clear ha he expression of he sandard deviaion ( σ T ) from he original BS equaion is subsiued wih an expression for he sandard deviaion (σ ) of he AR()-process, which as menioned is no direcly proporional o ime as were he case in he GBM. Furhermore he expression for he rae of reurn is subsiued wih an expression for he expeced oal ( r σ / ) T growh in he real esae index (µ). The opions values obained from he modified BS formula show a similar behavior o he behavior seen from he opion prices based on he Mone Carlo simulaions. 6.3 Discussion: Valuaion of Home Equiy Insurance Producs I should be noed ha he acual pu opion prices given in he appendices and he able above is clearly dominaed by he forecasabiliy of he real esae reurn, he variance of he AR()-processes used in he pu opion calculaions is relaively small and is herefore no as imporan in he pricing of he pu opions as he forecasabiliy, i would be recommendable o invesigae he variance of he AR()-processes furher before launching an acual home equiy insurance o he public. There are several addiional issues ha need some houghs in connecing wih he launching of a home equiy insurance produc. Among some of he more imporan issues are: - The divergence beween he value of he propery and he index value; basic risk - The conrac lengh; how long should he insurance conrac be? - Who are he cusomers? - Who are he providers of he Produc? 0 Shiller, R.J and Weiss, A.N., (999), p See appendix A.4.5 for VBA-code of he modified BS equaion 54

58 6.3. Basic Risk Basic risk is a erminology ofen used in connecion wih he use of fuures as hedging insrumens, bu he concep is feasible o highligh some poenial problems in connecion o home equiy insurance producs. The basis is usual defined as he difference beween he spo price and he fuures price. The issue wih basic risk is ha he price of he asse o be hedged may no be exacly he same as he price of he underlying asse. The uncerainy abou which dae he asse will be raded furhermore complicae he issue. Home equiy insurance producs would ypically by based on real esae indices, here are no guaranee ha he indices will behave exacly like he price of specific propery i.e. if he index increase in value while he price of he specific propery deeriorae, he insurance (pu opion) will no yield any compensaion o he homeowner. The divergence beween he price of he acual propery being insured and he value of he index can be minimized by using finer indices eiher via a more narrow geographical coverage of he indices and/or o define he indices wih respec o ype, value and oher aribues of he properies 3. I should be noed ha here are some of he divergences ha could no be covered by he indices, especially discrepancy arising from facors ha are under he influence of he homeowners (see moral hazard issues). Derivaes are usually raded wih fixed expiraion daes, bu he homeowners do ofen no known when hey will need o ge reimbursemen from he insurance and here are some risk ha he derivaive will no able o mee heir needs. In he case of he pu opion i would be possible o supply American opions, which have he feaure ha hey can be exercised a any ime prior o expiraion. Fuures on he oher hands need o be rolled over or closed ou in order o be able o adjus he expiraion dae, a soluion could be o purchase shor-erm fuures and roll over or close hem o ge he needed cover Conrac Lengh The choice of conrac lengh is subjecive and here is no righ or wrong conrac lengh, bu some ime horizons migh be more appealing han ohers. The risk associaed wih invesmens in real esae is by large shor-erm risk. In he long run he real esae prices will follow he inflaion, which implies ha over a longer ime period, Hull, J.C., (006), p Shiller, R.J and Weiss, A.N., (999), p.7 55

59 he inflaion will elevae he nominal real esae prices, decreasing he probabiliy of a nominal loss. The speculaion abou bubbles in he real esae prices also have shor erm characerisics, ha is ha a possible bubble in he real esae prices will manifes iself wihin a few years. In able 6., calculaions illusrae how long i will ake before he inflaion has caugh up wih a plausible bubble in he real esae prices. From he able i is clear ha if we anicipae ha he real esae prices are overvalued by approximaely 0% i will ake 3.9 years before an inflaion of.5% per year will have caugh up wih he overvalued real esae prices. On he oher hand if we expec ha he real esae prices o be overvalued by 0% i will ake 7.4 years before he inflaion has caugh up wih he prices. Remember ha IMF suggesed ha he Danish real esae prices can be overvalued by up o 7,5% 4. The inflaion is as previously menioned expeced o increase in he coming years (3.% in 008 and.6% in 009), which will shoren he cach up ime. This simple example indicaes ha conrac duraion of less hen 4-5 years migh be appropriae. Table 6. 5 : Inflaion Cach Up Expeced Bubble Expeced +0% +5% +0% Inflaion.50% 6.4 years (approx.) 9.4 years. years.75% 5.5 years 8. years 0.5 years.00% 4.8 years 7. years 9. years.5% 4.3 years 6.3 years 8. years.50% 3.9 years 5.7 years 7.4 years.75% 3.5 years 5. years 6.7 years 3.00% 3. years 4.7 years 6. years 3.5% 3.0 years 4.4 years 5.7 years Shiller and Weiss [999] argue for an even shorer conrac lengh beween one and wo years. Their argumenaion is based on a planning horizon beween one and wo years, and ha decision o sell or buy a propery is made wihin ha ime frame 6. The discussion poins in he direcion of an opimal conrac lengh of less han five years. I could also be seen from he pu opion calculaions ha an opion wih duraion of more han a couple of years would be inexpensive, his also indicaes ha he need for insurance of home equiy over longer ime periods is no presen. 4 Inernaional Moneary Fund (008), p See appendix A Shiller, R.J and Weiss, A.N., (999), p.34 56

60 6.3.3 Cusomers Homeowners are a heerogeneous group and heir risk exposure in he real esae marke differs. I is eviden ha many homeowners would benefi from he opporuniy o hedge he real esae risk by purchasing home equiy insurance, bu here are groups of homeowners who would benefi more from home equiy insurance han ohers. Homeowners ha would benefi he mos from home equiy insurance are homeowners wih no or lile wealh, young firs-ime homeowners, and homeowners who hold a heavily leveraged posiion in real esae. These homeowners are ofen paricular sensiive o he risk of declining real esae prices. Englund, Hwang and Quigley [00] idenify he same groups of homeowners as he homeowners who will benefi he mos from increased hedging opporuniies. Anoher group of poenial cusomers would be homeowners who expec o sell heir propery wihin a few years, and wan o lock in he selling price. These poenial cusomers could benefi from home equiy insurance, by minimizing heir risk exposure. Anoher imporan facor o ake ino accoun when considering he poenial cusomers are he individual cusomers risk aversion, which vary from homeowner o homeowner, ha is some people will wan o speculae in he real esae marke while ohers will wan o play i more safe Providers Several academics have emphasized, ha an inermediae insiuion is needed in order o make home equiy insurance available o he households. Their argumen is moivaed by he complexiy of he home equiy insurance producs. There are wo obvious insiuions wih he knowledge and compeences o provide home equiy insurance producs; hese are financial insiuions (i.e. banks) and insurance companies. Financial insiuions have grea knowledge abou derivaives and hedging. The banks provides loans o finance real esae, and he average Danish bank have abou 0% of heir lending porfolio placed in real esae 8, so hey should know he real esae marke. By he same oken heir large exposure in he real esae marke could also be used as an argumen agains banks as providers of home equiy insurance, in ha when he real esae marke show signs of deerioraions, he lending porfolio will deeriorae and a he same ime he probabiliy of insurance paymens will increase, which in he 7 Frank, R.H., (003), p.-4 8 Ejendomsudlån lægger pres på bank-ledere, Børsen 8/

61 wors case will resul in he banks no being able o fulfil heir obligaions. Insurance companies could also be suppliers of home equiy insurance. Their compeences in consrucing insurances and insurance policies would be an advanage. Furhermore he home equiy insurance produc could be launched as insurance and no as a derivaive or speculaive produc. The launch of home equiy insurance via banks could have he negaive effec of he public seeing he insurance as a speculaive produc, which would no benefi he promoion of he insurance. The mos suiable supplier of he home equiy insurance produc is a subjecive quesion, bu here mus be some securiy ha he provider will be able o pay he insurance paymens, a soluion would be o make he governmen reinsure he provider. If he home equiy derivaes where raded in he marke he supplier would be able o hedge he risk in he marke and could pass on he risk o he marke. Eiher way here mus be some securiy, so ha he public has rus in he insurance producs. 6.4 Conclusion: Valuaion of Home Equiy Insurance Producs The use of derivaives in conrolling he risk exposure in he real esae marke is obvious (high ransacion coss, illiquidiy and uncerainy). The characerisics of he indices showed in chaper 5 Analysis of he Underlying Asse ha he Danish real esae indices did no behave as regular socks, his implied ha he regular framework for pricing derivaives had o be modified. To accomplish his a Mone Carlo simulaion incorporaing he characerisics of he AR()-process was applied alongside a modified Black & Scholes equaion o price pu opion based on he real esae indices. The acual prices of he pu opions showed some imporan feaures, he forecasabiliy of he indices implied ha i would be cosly o insure home equiy value when he marke expec deerioraion real esae prices, which can be seen as a limiaion o he use of opion in connecion wih home equiy insurance, in ha when he need for insurance is bigges, hen he insurance is cosly, by he same oken he prices also showed ha he mean revering endency implied ha insurances covering longer ime periods would be relaively inexpensive. The chaper highlighed several imporan consideraions in connecion wih a public launching of he home equiy insurance producs, among oher issues were consideraions on how o consruc an appealing produc wih consideraions of he basic risk, he conrac lengh, he poenial cusomers and suppliers of he producs. 58

62 7 Alernaive Ways o Hedge he Real Esae Risk The possibiliy o hedge he risk exposure on real esae is one of he main reasons for he developmen of Home Equiy Insurance. Bu hedging real esae canno only be accomplished wih he use of derivaives. In his chaper some of he oher possibiliies o hedge he risk facing he homeowners will be described. The possibiliies include in his secion includes real esae socks, he choice of morgage, he use of ineres rae derivaives and REITs. 7. Invesing in Real Esae Socks Iacoviello and Oralo-Magne [00] along wih Englund, Hwang and Quigley [00] agree on he poenial benefis gained from he opporuniy o exploi real esae socks in order o hedge he risk facing he homeowners. Boh aricles use he mean-variance approach o invesigae he hedging opporuniies when invesing in real esae socks. Empirical research has shown ha he real esae socks are posiive correlaed wih he value of he real esae value. In Denmark here are several real esae socks lised on he OMX 9 i.e. Sjælsø Gruppen and TK Developmen. The correlaion beween he Danish real esae socks and he real esae indices are lised in able From he able i is seen ha he correlaion beween he socks and he indices is posiive, which is in good compliance wih exising research. Table 7. : Correlaion beween Danish Peal Esae Indices and Danish Real Esae Socks Sjælsø TK Develop. P E F Sjælsø TK Develop P E F In he mean-variance framework one sraegy ha could be used o uilize he posiive correlaion beween he real esae sock and he real esae prices would be o sell he real esae sock shor, which would minimize he risk exposure, by urning he posiive See appendix A.5. for he acual calculaion I is assumed ha he reurn generaed from he invesmen in real esae can be expressed as he reurn generaed from he indices. and 59

63 correlaion ino negaive correlaion and hereby increasing he diversificaion effec 3. If he homeowners are no able o go shor in real esae socks an alernaive sraegy would be o buy pu opions on he sock. The posiive correlaion indicaes ha he sock and real esae reurn move in he same direcion, which in pracise means ha a negaive developmen in he real esae prices also would resul in a negaive developmen in he real esae sock prices, so o insure agains dropping real esae prices one could ge compensaion by buying a suiable number of pu opions on he real esae socks 4. A hird alernaive would be o ener ino a shor posiion in a fuures conrac wih he real esae sock as he underlying asse 5. I should be noed ha he real esae price indices would have a higher posiive correlaion wih he acual real esae prices and herefore be a beer hedging alernaive han he real esae socks Appropriae Morgage Choice There are broadly wo ypes of loans available o he privae household considering invesing in real esae 7, he fixed-rae morgage (FRM) and he adjusable-rae morgage (ARM). The choice of morgage has a large effec on he oal risk acually endured by he homeowners. Campbell and Cocco [003] analysed he risk associaed wih he choice of morgage in financing real esae purchases 8. They divided he risk associaed wih obaining a morgage in wo ypes of risks; he wealh risk and he income risk. The Wealh risk: The wealh risk maerializes from he poenial change in he real capial value of a morgage. A nominal FRM is especially exposed o he wealh risk, in ha i is sensiive o changes in inflaion whereas an ARM is largely immune o changes in he inflaion due o he fac ha expecaion on he fuure inflaion is incorporaed ino he fuure ineres paymen on he ARM 9. The value of a FRM will change when inflaion and ineres raes change, in he case of high inflaion he ineres raes are normally also high. This siuaion will resul in a decreasing value of he deb. In he 3 Elon, E.J. e al., (003), p Hull, J.C., (006), p.8 5 Hull, J.C., (006), p.48 6 Englund, P., Hwang, M. and Quigley, J.M., (00) 7 Many differen combinaions of fixed-rae and adjusable-rae morgages. 8 Campbell, J.Y. and Cocco, J.F., (003) 9 Campbell, J.Y. and Cocco, J.F., (003), p.37 60

64 opposie siuaion wih low inflaion and low ineres raes, he value of he deb will increase, compared o he siuaion wih high ineres. I is clear ha he value of he deb behaves similar o he value of real esae. Tha is when he real esae prices decrease so does he value of he deb, which indicaes ha he risk of a FRM is seen in connecion wih he risk of declining value of he propery, in a sense seems o sabilize and minimize he oal risk exposure. The ARM on he oher hand is more dangerous, he value of he deb of an ARM is approximaely sable (depend on he underlying srucure, year ARM or longer periods). In he case of increasing ineres raes he ARM, in he shor erm he value of he deb will decline, bu in he long run he value of he deb will be more consan (no including morgage paymens). This enails ha when he value of he propery declines due o high user cos, he morgage value is consan, which in he wors case could resul in a siuaion where he deb is more worh han he propery, increasing he risk as a resul. The income risk: The major downside of an ARM is he shor-erm income-risk associaed wih a sudden increase in he monhly ineres paymen, which will ulimaely force he consumpion of he household down emporarily 30. If high ineres raes coincide wih low income and low real esae prices, he risk is even more severe, especially when combined wih borrowing consrains 3. From his shor descripion of he wealh and income risk, he FRM seems o be less risky compared o he ARM. The risk associaed wih he ype of morgage is of course no as simple as depiced above and in a horough analysis of he households oal risk exposure more facors needs o be included i.e. he household income, borrowing consrains and so for. 7.3 Ineres Rae Derivaives The ineres rae has as previously menioned in he equilibrium model a huge impac on he real esae prices. Increasing ineres raes have hisorically been followed by decreasing real esae prices and vice versa (have a mean revering endency). An ineres rae derivaive which value increases wih increasing ineres rae would reduce he risk associaed wih he falling real esae prices when ineres raes are decreasing. 30 Campbell, J.Y. and Cocco, J.F., (003), p.3 3 Campbell, J.Y. and Cocco, J.F., (003), p.37 6

65 Typically an ineres rae derivaive could be consruced as a converible ineres rae swap, where he invesor receives he ineres rae spread beween a long fixed rae and a shor variable rae, if he ineres increases he value of he swap would increase in value and vice versa. In Denmark Nykredi has markeed a produc very similar o he above menioned, alhough heir ineres rae produc is argeed commercial purposes Invesing in a Real Esae Invesmen Trus A real esae invesmen rus (REIT) is a corporae eniy ha buys, sells and manages properies. A REIT is somewha similar o an invesmen in real esae socks in ha he asse base in he corporaions and russ is similar, and herefore he hedging opporuniies are comparable o ha of he real esae socks. The feaure ha differeniaes REITs from ordinary sock in real esae corporaions is he ax reamen of he REITs. The advanages of he REITs is ha if he rus allocaes all of he axable income o he uniholders, hen he REIT will no be subjec o axaion. The REIT concep originaes from Unied Saes and have since been adoped in several oher counries. Since he concep is no used in Denmark he hedging opporuniies for privae homeowners in Denmark are no obvious Conclusion: Alernaive Ways o Hedge he Real Esae Risk The hedging vehicles give significan diversificaion and risk managemen opporuniies, bu here are several shorfalls wih he hedging alernaives menioned in his secion. The complexiy of he hedging sraegies oulined above makes he opporuniies somewha difficul o manage for he privae household and would need professional risk managemen, o be a pracical soluion. Anoher imporan issue is ha he hedge obained from he use of he alernaive hedging possibiliies is no a perfec hedge, which means ha he risk is no enirely eliminaed by he use of he hedging sraegies. 8 Behavioral Issues In consrucing and promoing home equiy insurance producs he supplier of he producs can conrol he vas majoriy of decisions and facors affecing he producs, 3 Nykredi (004) 33 Deloie (004) 6

66 bu here are a number of facors, which are beyond he conrol of he supplier. Behavioral issues like moral hazard and adverse selecion issues confron he issuer of home equiy insurance produc, and he issues needed o be deal wih before inroducing he insurance. 8. Moral Hazard Moral hazard is a general phenomenon, and is a well-known problem in he insurance indusry. In he case of home equiy insurance he moral hazard can arise in several siuaion for insance when he homeowner neglecs o mainain his/her propery as a resul of he proecion he homeowner has acquired from he home equiy insurance. The home equiy insurance will cover any losses he homeowner exhibis when selling he propery wih a loss. This could have he unforunae effec o moivae he neglecing of mainenance of he propery. To avoid hese siuaions he insurance policy should include erms, which sae ha he reimbursemen would be reduced in case of neglec o mainain he individual propery. Anoher problem is o prove neglec, evidence of neglec is no objecively quanifiable, and a judgmen is needed o deermine if he homeowner purposefully negleced o mainain he propery. Oher examples of moral hazard could be he case where a homeowner for insance remodels he propery, his aciviy can creae value, bu if he remodeling is very disincive (read special) i could quie as easily desroy value. A hird siuaion could arise in he case where he homeowner is in a hurry o sell he propery and selling he propery a a discoun o ge i of her/his hands. There are several oher siuaions ha poenially could cause similar problems, bu he underlying problem is he same; he owner canno lose money on he propery and herefore acs o maximize his/her own uiliy wihou considering he cos o he insurance company, he insurance supplier needs o be aware of he moral hazard risk and ake counermeasures o avoid hem Adverse Selecion The homeowners have more informaion abou he individual propery han he insurance company; his means ha if he homeowner knows ha he propery is in danger of depreciaing for whaever reason, he owner would be moivaed o ake ou a policy o avoid losing money. The asymmery of informaion can lead o severe losses 34 Shiller, R.J and Weiss, A.N., (999), p.5-6 and Caplin, A. e al., (003), p. and p.5 63

67 for he insurance company, and hey need o ake acions o avoid hese selecion bias issues. If for insance he propery is overvalued, conaminaed or of a bad build qualiy he owner can insure he oal value of he propery, hereby puing loses ono he insurance company 35. I should be noed ha mos of he behavioral issues are avoided by basing he home equiy insurance reimbursemens on he acual deerioraion of he underlying index and no on he depreciaion in he value of he individual propery. The behavioral issues could be seen as he opponen o he basic risk issue previously menioned. Too much basic risk is unforunae, ha is he insurance will fail o insure he value of he home, bu he oher siuaion where here is no basic risk he behavioral issues become a poenially big problem for he insurance company here is a radeoff in-beween. 9 Perspecive The idea o inroduce home equiy insurance producs o he public seems o be a good and reasonable idea, bu here are issues ha need o be deal wih before he insurance can be launched. The lack of sophisicaed real esae indices in Denmark, will pose a serious obsacle in he launching of he insurance. Acions need o be aken o develop more accurae real esae indices o incorporae he aribues of he real esae marke. The inroducion of more sophisicaed mehodologies, could also have posiive effec on he discussion of he economic condiion in Denmark by making i more deailed and accurae in describing he developmen in he real esae marke. 0 Conclusion The conclusion will concenrae on answering he cenral quesions asked in he problem saemen alongside he addiional quesions arising from he descripion of he home equiy insurance concep. Why insure home equiy value? Privae invesmens in he residenial real esae are ofen undiversified and highly leveraged, which pus he privae invesor in grea risk of incurring a relaively large loss in a shor-erm perspecive. The privae invesor has limied abiliy o influence he 35 Shiller, R.J and Weiss, A.N., (999), p

68 risk associaed wih he real esae invesmens due o he fac ha he facors affecing he value of home equiy are unconrollable o he invesor (macro- and microeconomic facors). I would seem o be a good idea o consruc an insrumen ha could improve he privae invesors abiliy o manage he risk associaed wih invesmens in he privae real esae marke. A risk managemen device of his ype would have he power o improve he welfare of many privae households. Why is i no possible for he privae invesor o insure he home equiy value? I is possible o insure (some of) he value of he home equiy by using he proposed alernaive hedging opporuniies. The skills needed o insure he home equiy value using he alernaive hedging devises prohibi he general public in insuring he home equiy value. In addiion he real esae marke is characerized by high ransacion cos and illiquidiy, which complicaes he insurance of home equiy value even furher. There need o be a simple produc, which insures he home equiy value, o make insurance feasible o he general public. In Denmark here are no insurance producs available of ha kind, and here seem o be several obsacles in he way of making insurance of home equiy value available. One of he main obsacles is he indices used in Denmark. The available indices, which measures he developmen in Danish real esae prices, are obsolee and more refined mehodologies are needed in order o be able o consruc a reliable foundaion for he insurance. How should a financial insrumen be consruced o make i possible o insure home equiy value? To give a horough answer o his quesion he following sub quesions will be answered: ) Which exernal facors have an impac on he value of home equiy? ) Which underlying asse should be used o resemble he movemens in he real esae prices? 3) Which derivaives should be used o insure he risk facing he home equiy value? Ad ) From he definiion of home equiy value i is clear ha he developmen and changes in he value of he individual properies will have an effec on he value of he home equiy. This enails ha he facors deermining he demand for real esae and he 65

69 supply of real esae will have an impac on he value of home equiy. Among he mos influenial demand driving facors are he household income, he inflaion, he user cos and he demographic composiion. From he descripion and he updaes on he Danish real esae marke i is clear ha he mos imporan developmen in he demand driving facors are he increased uncerainy surrounding he economic oulook, which among oher hings already has had an impac on he inflaion and ineres raes, resuling in deerioraing condiions on he real esae marke. Among he supply driving facors he aciviies in he consrucion indusry are of crucial imporance. The consrucion indusries have been very acive in he previous years, which have resuled in a large supply of vacan properies. The siuaion where deerioraing marke condiions collide wih a large supply of properies will resul in a downward pressure on he real esae prices, which again sresses he imporance of he home equiy insurance concep. Ad ) I is eviden ha he asse underlying he home equiy insurance should follow he movemen in he real esae prices. This hesis emphasizes he use of real esae indices o measure he movemen in he real esae prices. The index should incorporae he feaures characerizing properies, which is he heerogeneiy of he properies, he infrequen ransacions and he poor guidance of adverised selling prices. The indices need o rack he developmen in he acual real esae prices, o be able o serve as an underlying asse in he home equiy insurance produc. The mos advanced index mehodologies (hedonic, repea-sales and hedonic repea-sales) are also he mos likely mehodologies o be found used in connecing wih home equiy insurance, due o he fac ha hey can incorporae mos of he feaures characerizing properies. I is clear ha here are many consideraions ha need o be made before being able o choose he mos appropriae index mehodology. I addiion o deciding on he index mehodology houghs need o go ino he acual level of deail in he index. If he index covers a large area wih respec o locaion (geography) and ype of propery, here is an imminen danger ha he index will no be able o rac he movemen in he properies ha he home equiy insurance ries o cover. Ad 3) Obviously here are many differen derivaives, which could be used in connecion wih he insurance. This hesis highlighs he use of fuures and pu opions as he derivaive o use when insuring he value of home equiy. The pu opion being he superior derivaive, by using he fuure o insure he home equiy value, he insured 66

70 needs o give up he poenial capial gain when real esae prices develop posiively. In conras he pu opion seems o be more suied o fi he needs, in ha he insurance premium only need o be paid once. Furhermore he opion insures agains deerioraing condiions while he insurance holder will receive he poenial capial gains. The simpliciy of he derivaive makes he pu opion superior o he alernaive derivaives. In addiion o he choice of index mehodology and derivaive o use in he insurance produc, i is vial ha he behavioral issues surrounding he insurance is aken ino accoun. The moral hazard and adverse selecion issues need o be considered and he appropriae counermeasures need o be incorporaed ino he insurance erms o avoid misuse. I is obvious ha here are many consideraions ha need o be aken ino accoun when deciding on how o consruc a financial insrumen o make home equiy insurance produc available. Bu when all his is said and done home equiy insurance sill seems o be a good idea, which could improve he welfare of many households. Is home equiy insurance he bes and only way o hedge he risk facing home equiy value? While he concep of home equiy insurance is he mos simple and mos suied o hedge he risk facing home equiy value, i is no he only hedge opporuniy available. The alernaive hedging opporuniies include among ohers invesmens in real esae sock, choosing he appropriae morgages and ineres rae derivaives. The complexiy of hese alernaive-hedging opporuniies favor he home equiy insurance concep as he mos appropriae. 67

71 References Andersen T.M. e al., (00), Beskrivende dansk økonomi, bogforlage HandelsVidenskab,.udgave,.oplag. Bailey, M.J., Muh, R.F. and Nourse, H.O., (963), A Regression Mehod for Real Esae Price Index Consrucion, Journal of he American Saisical Associaion, Vol. 58, No. 304, pp Blanchard, O., (003), Macroeconomics, Third Ediion, Prenice Hall Bodie, Z. and Meron, R.C., (000), Finance, Inernaional Ediion, Prenice Hall Inernaional, Inc. Borio, C. and McGuire, P., (004), Twin Peaks in Equiy and Housing Prices? BIS Quarerly Review, March 004 Brueggeman, W.B. and Fisher, J.D., (993), Real Esae Finance and Invesmen, 9 h Ediion Calhoun, C.A., (996), OFHEO House Price Indexes HPI Technical Descripion, OFHEO Campbell, J.Y. and Cocco, J.F., (003), Household Risk Managemen and Opimal Morgage Choice Caplin, A. e al., (003), Home Equiy Insurance: A Pilo Projec, Yale Inernaional Cener for Finance, Working Paper No.03- Case, K.E. and Shiller, R.J., (987), Prices of Single Family Homes Since 970: ew Indexes for Four Ciies, Crowles Foundaion Discussion Paper No. 85, Yale Universiy Case, B. and Quigley, J.M., (99), The Dynamics of Real Esae Prices, The Review of Economics and Saisics, Vol. 73, No., pp Case, B. and Wacher, S., (000), Residenial Real Esae Price Indices as Financial Soundness indicaors: Mehodological issues, BIS Papers No Clapp, J.M. and Giaccoo, C., (998), Price Indices Based on he he Hedonic Repea- Sales Mehod: Applicaion o he Housing Marke, Journal of Real Esae Finance and Economics, 6:, 5-6. Cour, A.T., (939), Hedonic Price Indexes wih Auomoive Examples, The Dynamics of Auomobile Demand, General Moors Corporaion, New York Danmarks Naionalbank, (003), Mona En Kvaralsmodel af Dansk Økonomi Danmarks Naionalbank, (007), Finansiel Sabilie 007 Davidson, R. and MacKinnon, J.G., (999), Economeric Theory and Mehods Deloie (004), REIT Guide, 8 h Ediion, Second Prin Deusche Bank Research, (007), Propery derivaives marching across Europe De Økonomiske Råd, (00), Boligmarkede Skæv og Ineffekiv De Økonomiske Råd, (008), Konjunkurvurdering Opsparing Elon, E.J. e al., (003), Modern Porfolio Theory and Invesmen Analysis, 6 h Ediion, John Wiley and Sons, Inc. Englund, P., Hwang, M. and Quigley, J.M., (00), Hedging Housing Risk, Journal of Real Esae Finance and Economics, 4:/, European Cenral Bank (003), Srucural Facors in he EU Housing Markes. Finansminiserie (00), Væks, velfærd og fornyelse Finanslovforslage 003 Fleming, M.C. and Nellis, J.G., (008), The Halifax House Price Index Technical Deails Frank, R.H., (003), Microeconomics and Behavior, 5 h Ediion, McGraw-Hill Franses, P.H., (998), Time Series models for business and economic forecasing, Cambridge Universiy Press 68

72 Greene, W.H., (008), Economeric Analysis, 6 h Ediion, Pearson Inernaional Ediion, Prenice Hall Griliches, Z., (97), Price Indexes and Qualiy Change Sudies in ew Mehods of Measuremen, Harvard Universiy Press Hansen, M.F., Egger, M. and Sephensen, P., (007), Danmarks fremidige befolkning Befolkningsfremskrivning 007, Dream Hansen, J., (006), Ausralian House Prices: A Comparison of Hedonic and Repea- Sales Measures, Research Discussion Paper, Economic Research Deparmen, Reserve Bank of Ausralia Hull, J.C., (006), Opion, Fuures and Oher Derivaives, 6 h Ediion, Prenice Hall Iacoviello, M. and Oralo-Magne, F., (00), Hedging Housing Risk in London Inernaional Moneary Fund (008), World Economic Oulook Housing and he Business Cycle, World Economic and Financial Surveys Jus, T. and Feil, J., (007), Propery Derivaives Marching Across Europe, Deusche Bank Research, Inernaional Topics Kuo, C., (996), Serial Correlaion and Seasonaliy in he Real Esae Marke, Journal of Real Esae Finance and Economics, : 39-6 Li, W., Prud homme, M. and Yu, K., (006), Sudies in Hedonic Resale Housing Price Indexes, OECD-IMF Workshop Real Esae Price Indexes Lükepohl, H. and Kräzig, M. (004), Applied Time Series Economerics, Cambridge Universiy Press Marcus, M. and Taussig, M.K., (970), A Proposal for Governmen Insurance of Home Values agains Locaional Risks, Land Economics, Vol. 46, No.4 (Nov., 970), pp Merrill Lynch (007) Propery Derivaives Nykredi (004), ykredi VærdiSikring Effekiv sikring af egenkapialen i ejendommen Poerba, J.M., Weil, D.N. and Shiller R., (99), House Price Dynamics: The Role of Tax Policy and Demography, Brookings Papers on Economic Aciviy, Vol. 99, No., pp Quigley, J.M., (999), Real Esae Prices and Economic Cycles, Inernaional Real Esae Review, 999 Vol., No.: pp.-0 Quigley, J.M., (995), A Simple Hybrid Model for Esimaing Real Esae Price Indexes, Journal of Housing Economics 4, -. Rappapor, J., (007), A Guide o Aggregae House Price Measures, Economic Review, Second Quarer 007, Federal Reserve Bank of Kansas Ciy Rosen, S., (974), Hedonic Prices and Implici Markes: Produc Differeniaion in Pure Compeiion, The Journal of Poliical Economy, Vol. 8, No. (Jan-Feb., 974), pp Schulz, R. (003), Valuaion of Properies and Economic Models of Real Esae Markes Wirschafswissenschaflichen Fakulä, Humbold-Universiä zu Berlin Shiller, R.J. and Weiss, A.N. (994), Home Equiy Insurance, Cowles Foundaion Discussion Paper 074, Yale Universiy Shiller, R.J., (998), Macro Markes Creaing Insiuions for Managing Sociey s Larges Economic Risks, Oxford Universiy Press Shiller, R.J and Weiss, A.N., (999), Home Equiy Insurance, Journal of Real Esae Finance and Economics, 9:, -47. Tobin, J., (969), A General Equilibrium Approach To Moneary Theory, Journal of Money, Credi and Banking, Vol., No., pp

73 Tsasaronis, K and Znu, H., (004), Wha Drives Housing Price Dynamics: Cross- Counry Evidence, BIS Quarerly Review, March 004 Verbeek, M., (004), A Guide o Modern Economerics, nd Ediion, John Wiley & Sons, Ld Wagner, R., (005), En Model for de danske Ejerboligpriser, Økonomi- og Erhvervsminiserie Washam, T.J. and Parramore, K., (997), Quaniaive Mehods in Finance, Thomson Wilmo, P., (007), Paul Wilmo Inroduces Quaniaive Finance, nd Ediion, John Wiley & Sons, Ld Wood, Rober (005), A Comparison of UK Residenial House Price Indices, BIS Paper No Wooldridge, J.M., (006), Inroducory Economics A Modern Approach, 3 rd Ediion, Thomson Yarmolinsky, A., (97), Reassuring he Small Homeowner, The Public Ineres, Number, p.06 ewspaper aricles Ejendomsudlån lægger pres på bank-ledere, Børsen 8.july 008 Srejken er slu lønnedgang på vej, Børsen 4.june 008 Oliens opur sender rener i årsrekord Børsen.may 008 Sagflaion får økonomien i knæ Børsen 3.june 008 Web pages Danmarks Naionalbank Danmarks Saisik De Økonomiske Råd Halifax House Price Index Homerack Land Regisry Naionwide ODPM OFHEO House Price Index OMX - Realkrediråde Realkredi Danmark Righmove S&P/Case-Shiller Home Price Index Sjælsø Gruppen - TAX.DK ska & afgif The Bureau of Labor Saisics The Census Bureau The Naional Associaion of Realors TK Developmen and

74 Appendices A. Descripion of he Danish Real Esae Marke 73 A.. Danish Real Esae Prices 73 A.. Lending Porfolio 73 A..3 Developmen in he Morgage Ineres Rae 74 A..4 Tobin s Q and Consrucion of new Real Esae 75 A..5 Empirical Resuls on he Dynamics of he Real Esae Marke 75 A. Measuring he Underlying Asse 79 A.. Mulicollineariy an Issue wih he Hedonic Regression 79 A.. Examples of Variables included in Acual Hedonic Real Esae Indices 80 A..3 Dispersion of Reurn Repea Sales Mehodology 80 A..4 Derivaion of he Hedonic Repea Sales Mehodology. 8 A..5 Generalized Leas Squares 8 A..6 Represenaive Lis of Commercial Indices 84 A..7 The Porfolio Mehodology 84 A.3 Analysis of he Underlying Asse 87 A.3. The Time Series Daa Used in he Analysis 87 A.3. Presenaion of he Indices (Including Differen Levels of he Indices) 87 A.3.3 Uni Roo and Augmened Dickey-Fuller es 9 A.3.4 The Akaike and he Schwarz Informaion Crierion 97 A.3.5 Basic Characerisics of he AR() and he MA() processes 98 A.3.6 Model Esimaions 98 A.3.7 Problems Concerning Serial Correlaion and Heeroskedasiciy 0 A.3.8 Derivaion of he MA( ) Process from he AR() Process 05 A.3.9 Forecasing wih he AR()-Process 06 A.4 Valuaion of Home Equiy Insurance Producs 08 A.4. Basic Propery Derivaives 08 A.4. Black & Scholes Assumpions 09 A.4.3 Mone Carlo Simulaion 09 A.4.4 Valuaion of Opions using Mone Carlo Simulaion 4 A.4.5 VBA-code Modified Black & Scholes Equaion 8 A.4.6 Valuaion of Opions using Modified Black & Scholes Equaion 9 A.4.7 Inflaion Cach Up A.5 Alernaive Ways o Hedge he Real Esae Risk A.5. Mean-Variance Calculaions 7

75 A. Descripion of he Danish Real Esae Marke A.. Danish Real Esae Prices See also he excel file 3 Danish Real Esae Prices for he complee daase. A.. Lending Porfolio See also he excel file 0 Lending Porfolio 73

76 A..3 Developmen in he Morgage Ineres Rae See also he excel file Developmen in he Morgage Ineres Raes for a complee daase. 74

77 A..4 Tobin s Q and Consrucion of new Real Esae See also he excel file Tobins Q and Consrucion Aciviy for deails on calculaions and complee daase. A..5 Empirical Resuls on he Dynamics of he Real Esae Marke In suppor of he conclusions on he dynamics of he real esae prices, several empirical resuls are reprined here. I should be menioned ha he resuls depiced in he following are boiled down o he mos imporan conclusions. For a comprehensive descripion of he resuls he reader is referred o he acual lieraure. Wha drives housing price dynamics: cross-counry evidence By Tsasaronis and Zhu [004] Mehodology: Vecor auoregression model (VAR) Facors included in he empirical model: - House price growh - The growh rae of GDP (indicaion of he sae of he business cycle and household income) 75

78 - The inflaion rae (measured by consumer prices) - The real shor erm ineres rae - The erm spread (difference beween yield of long-mauriy governmen bond and he shor rae) - The growh rae in inflaion adjused bank credi Resuls obained from he empirical analysis: Tsasaronis and Zhu decomposes he observed variabiliy in accordance wih he conribuion from each of he six variables, heir relaive imporance in deermining he overall dynamic of he model is given by heir conribuion o he variaion. Table: Tsasaronis and Zhu Variance decomposiion Impac on housing prices from a shock o All counries Inflaion 53.0% Bank Credi.4% Shor rae 0.8% Term Spread 9.8% GDP 7,6% Housing Prices 7,4% Denmark 4.3% 9.% 8.7% 4.% 6.9% 8.9% The differen resuls are according o Tsasaronis and Zhu brough on by differences in he morgage finance srucures in he respecive counries. En model for de danske ejerboligpriser By Rober Wagner [005] Mehodology: Co-inegraed VAR model In he model several funcions are esimaed: A long-erm demand funcion: p r =. 9( y h ) 7. 7uc +. 9demo + cons an Where: - p he real esae prices in real erms - y household income - h he supply of real esae - uc is he user cos, which in he model is given by: o uc = (ineres rae afer ax expeced inflaion) + Real esae ax rae + depreciaion rae expeced real capial gains - demo is he number of new house owners. 76

79 From he esimaed long-erm demand relaion i is seen an increase of % in he user cos, will resul in a deerioraion of he real esae prices by abou 7.7%, furhermore a decreased number of new house owners will decrease he real esae prices. A shor-erm relaion in difference level given by: d r p = 0. 05ecm p 0. 0 ( y h ). 6 uc dum85: 3, dum86:, + ε Where ecm is he long-erm demand, esimaed from he previous funcion. The longerm demand, which is included in he funcion, is used as an error correcion erm. The coefficien of he error correcion erm indicaes ha i will ake 5 years (/0,05 = 0 quarers) before he shor-erm price level converges owards he long-erm equilibrium. Furhermore i is clear ha real esae prices will deeriorae by.6% in he shor run, if he user cos increases by %. Two dummies are included in he model, he firs is included due o emporary large deviaions in he observed real esae prices, and he second is included due o large flucuaions in he inflaion. Wagner s model is capable of explaining 9/0 h of he increase in he Danish real esae prices. The resul from he co-inegraed VAR-model is depiced by he decomposing of conribuions from he various facors o he oal growh seen in he real esae prices since 993. Table: Conribuion o he oal growh in he real esae prices since 993 ominal growh in he real esae prices 53% Divided beween: General price increase 55% Increase from fundamenal facors: (divided beween) 79% - Increase in disposal income 59% - Decreasing real ineres afer ax 33% - Decreasing value of ax shield -% - Increasing numbers of new home owners 6% - Increasing supply of real esae -6% - Changes in real esae axes 8% Unexplained increase in real esae prices 9% 77

80 House Prices and Business Cycles in Europe: a VAR Analysis By Maeo Iacoviello [00] Mehodology: Vecor auoregression model (VAR) Facors included in he empirical model: - Real GDP (y ) - A measure of real money (mp ) - A real house price index i.e. a nominal house price index deflaed by he consumer price level (q ) - A shor erm nominal ineres rae (i ) - An annualized quarerly (consumer price) inflaion (π ) Real variables are specified in naural logarihms, ineres rae and consumer prices in percenage erms. The acual resuls from his repor are hard o inerpre due o he lack of saisical resuls, and ha he resuls from he VAR analysis is only presened graphically. The resuls are presened as impulse responses and variance decomposiion, in he following he resuls are summarized. The impulse response analysis: - Moneary shocks. A moneary ighening implies an upward pressure on he ineres rae, and he resuls show ha his have a srong negaive effec on house prices, which responds wih decreasing prices. - Demand shocks. The response o a shock in he demand is o increase he real house prices, wih peaking prices afer years, and dies ou afer 5 o 6 years. - Inflaion shock. The responses o inflaion shock are wo fold; he demand for real esae increases due o favourable ax reducion due o morgage ineres paymens, which are ax deducible, while capial gains are essenially unaxed. Bu increased inflaion also has a negaive effec on oupu and ineres raes wih counerac he posiive impac from he favourable ax affec. The variance decomposiion: - Moneary shocks only plays a minor role in explaining he variance of he real esae prices where beween 5% and 40% of he variance in he real esae prices can be explained by he moneary facor in he shor run. 78

81 - Demand shocks explain up ill 60% of he variaion in he shor run (UKresuls), which indicaes ha demands play an imporan role, especially in he shor run. The resuls are somewha weak in House Prices and Business Cycles in Europe: a VAR Analysis, and here are no surprising findings, bu he empirical resuls are in good accordance wih oher research resuls. A. Measuring he Underlying Asse Time series daa are likely o exhibi some fundamenal problems when esimaion and inerpreaion of he economeric mehodologies are underaken such as mulicollineariy, heeroskedasiciy and auocorrelaion. The fundamenal problems are of general concern when consrucing indices on real esae prices. A.. Mulicollineariy an Issue wih he Hedonic Regression Mulicollineariy refers o he issue where some or all of he explanaory variables are highly correlaed, which enails ha he variables are no independenly disribued. The inerrelaionships beween explanaory variables can cause problems wih esimaion of regression coefficiens ha may no be uniquely deermined and he coefficiens may flucuae markedly, and he coefficiens become less reliable. The basic problem arise due o he calculaion of he OLS parameers in which he normal equaions are calculaed assuming ha he X X marix is inverible: y = Xβ + To esimae he unknown bea s one need o use he normal equaions: u ˆβ = ( X X) X y If mulicollineariy is presen (exac mulicollineariy) hen he marix is no inverible making he esimaes of he parameers highly inaccurae. A soluion o he problem is o omi one of he variables 36. Where mulicollineariy direcly impacs he esimaed parameers β, heeroskedasiciy and auocorrelaion indirecly impacs he parameers by influencing he error erm, which in urn may no acually bias he parameers bu violae he underlying Gauss-Markov assumpions of he OLS Verbeek, M., (004), p.9 and p Verbeek, M., (004), p

82 A.. Examples of Variables included in Acual Hedonic Real Esae Indices Table: Examples of Variables included in Acual Hedonic Real Esae Indices Characerisic Halifax aionwide Deached house Terrace house Deached bungalow only one bungalow Semi-deached bungalow dummy Purpose-buil fla/maisonee or new convered only one fla dummy Convered fla/maisonee Tenure Number of bedrooms Number of habiable rooms X Double garage X Number of garages X Number of garage spaces X Parking space or no garage X Cenral heaing ype Floorsize (sq.f.) X Number of acres X More han one bahroom X Number of bahrooms X Number of oiles X Garden X Subjec o a road charge X Propery age X New X Region ACORN classificaion 38 X Preliminary consiuency X A..3 Dispersion of Reurn Repea Sales Mehodology As menioned in he hesis he original repea-sales index assumes consan dispersion of reurn generaed from real esae, bu Case and Shiller did no wan o accep his assumpion and proposed ha he dispersion where dependen of he ime inerval beween sales. They assumed ha he sampling variance of changes in housing value could be expressed in erm of a marke price index β, a Gaussian random walk H i and whie noise N i, They expressed he log price of he individual house as: ln( P ) = β + H + N i i i Their assumpions implies ha he percenages change in he real esae price is given by: 38 A classificaion of residenial neighborhoods 80

83 ln( P i, ) ln( Pi, + s) = β β+ s + Hi, Hi, + s + Ni, Ni, + s Where s is he ime of he second sale. β describes he average behavior of he marke prices, while H describes how he variaions in individual propery prices behave around he rae of change in he average real esae marke. The whie noise erm represen he difference in how he individual properies are valuaed 39. Today i is sandard pracice o use hese assumpions abou he behavior of individual propery prices 40. A..4 Derivaion of he Hedonic Repea Sales Mehodology 4. Scenario : One Observed Transacion Price (Similar o he Hedonic Regression) A ime = 0, he propery value is given by: P i, 0 = α X a X a e a 3 x 3 And a ime =, he same propery is observed a a ransacion price given by: P i, = Pi, 0 X b X b e b x 3 3 From he second expression i is seen ha he observed ransacion price is dependen upon he propery value a ime = 0, which by inserion ino he second equaion yields he following expression: P = αx i, a X a e a x 3 3 X b X b e b x 3 3 Taking naural logarihm o he expression yields he following expression: ln, α P i = ln + a ln X + a ln X + a3x 3 + b ln X + b ln X + b3x 3 This expression for he observed ransacion price is he one found in he hesis. Scenario : Two Transacions of he same propery, wih unchanged qualiy (similar o he repea sales) The fac ha he qualiy of he propery is unchanged means ha he expression of he propery value can be expressed as he observed ransacion price a ime, wih he simple modificaion ha he ime beween he ransacions has o incorporaed ino he expression: 39 Case, K.E. and Shiller, R.J., (987), p.4-5 and p Calhoun, C.A., (996), p Case, B. and Quigley, J.M., (99), p

84 8 3 3 x b b b i i e X X P P ) ( ) ( ) (,, τ τ τ τ = Expressed wih aking he naural logarihm o he expression yields he equaion depiced in he hesis: 3 3 X b X b X b P P i i ) ( )ln ( )ln ( ln ln,, τ τ τ τ = Two Transacions of he same propery, wih changes qualiy (hedonic and repea sales) The qualiy of he propery is changed in-beween he wo observaions a ime * from (X,X,X 3 ) o (X *,X *,X 3 * ), where > * > τ. Expressing his using he above noaion yields he following equaion for he propery value a ime = * jus afer he firs sale bu before changes in he qualiy of he propery is given by: 3 3 x b b b i i e X X P P ) * ( ) * ( ) * (, *, τ τ τ τ = (h) Using he approach in he case of observed ransacion prices in he firs scenario his yields ha he new value of he propery is given as: ] *][ [ * * * * *, *, * ) / ( ) / ( x x b a b a b a i i e x x X x P P = (h) And a ime he price of he propery is given by: * *) ( *) ( *) ( *,, * * 3 3 x b b b i i e X X P P = (h3) Subsiuion of (h) and (h) ino (h3) yields he following expression: * * *) ( *) ( *) ( ] *][ [ * * * * ) * ( ) * ( ) * (,, * * ) / ( ) / ( x b b b x x b a b a b a x b b b i i e X X e X X X X e X X P p = τ τ τ τ Taking he naural logarihm and rearranging he equaion yields he equaion depiced in he hesis: ] [ ] ln ln [ ] ln ln [ ) ( ) / ln( ) / ln( ln ln * * * * * *,, X X b X X b X X b X X a X X a X X a P P i i τ τ τ τ = A..5 Generalized Leas Squares The generalized Leas squares (GLS) mehod is a generalizaion of he ordinary leas squares esimaor (OLS). The OLS echnique is widely used o sudy he relaionship beween variables, i is especially he simpliciy of he echnique ha makes i so popular o use in pracice. To obain unbiased resuls from he use of he OLS mehods and several oher economeric mehods he underlying daa have o mee cerain crieria.

85 Assumpion underlying he OLS. Linear in parameers The sochasic process {(x,x,x k, y ): =,, n} follows he linear model: a. y = β 0 + βx βk xk + u, where {u : =,,,n} is he sequence of errors or disurbances. Here, n is he number of observaions (ime periods).. o perfec collineariy in he sample, no independen variable is consan or a perfec linear combinaion of he ohers. 3. Zero condiional mean For each, he expeced value of he error u, given he explanaory variables for all ime periods, is zero. Mahemaically, a. E(u X) = 0, =,,,n 4. Homoskedasiciy Condiional on X, he variance of u, is he same for all : Var(u X) = Var(u ) = σ, =,,,n. The variance of u is independen of X and he variance of u is consan. 5. o serial correlaion Condiional on X, he errors in wo differen ime periods are uncorrelaed: Corr(u,u s X)= 0, for all s. When a series suffer from serial correlaion i is also said o exhibi auocorrelaion, he series are correlaed across ime. 6. ormaliy The errors u are independen of X and are independenly and idenically disribued as normal(0,σ ) Under he above-menioned assumpions -6 for ime series, he OLS esimaors are normally disribued, condiional on X. Under he null hypohesis, each -saisic has a -disribuion, and each F-saisic has an F-disribuion. The usual consrucion of confidence inervals is also valid 4. To obain he resul from he regression: y = xβ + The coefficiens of β needs o be esimaed, which are done via he following calculaion: u ˆβ = ( x x) x y This leas squares esimaor minimizes he sum of he squared residuals 43. When he elemens in y have unequal variance and/or are correlaed here are no guaranee ha he OLS esimaor is he bes BLUE esimaor. The GLS mehod improves he efficiency of he esimaor. The GLS esimaor is given by: 4 Wooldridge, J.M., (006), p Wooldridge, J.M., (006), p

86 ˆβ = ( x Ω x) x Ω y Which is he efficien GLS esimaor of β, where Ω is a posiive definie symmerical marix. There are no requiremens ha he variance has o be consan or ha he requiremen of uncorrelaion need has o be fulfilled under he GLS mehod 44. A..6 Represenaive Lis of Commercial Indices Compared o he residenial real esae indices he commercial marke is more developed and herefore many insiuions supply indices on he commercial real esae marke. Among some of he larges supplier is he Invesmen Propery Daabank (IPD), he Naional Council of Real Esae Invesmen Fiduciaries (NCREIF) and a lo of oher insiuions menion in he able below. Table: Represenaive Lis of Commercial Indices Supplier Mehodology Counry - IPD-indices NCREIT Propery Index (NPI) Naional Associaion of Realors (NAR) index RP Daa-Rismark real esae indices The Office of Federal Housing Enerprise Oversigh (OFHEO) index S&P/GRA Commercial Real Esae Indices (SPCREX ) Appraisal-Based commercial propery indices region UK Germany Ausralia France Japan Denmark Websie Porfolio Measure U.S. Median value U.S. Median value Hedonic Repea-sales Ausralia Repea-sales U.S. U.S. A..7 The Porfolio Mehodology The mos widely used indices used for conrac selemen and herefore he mos successful indices in he marke oday are sock indices. The use of sock indices for conrac selemen is widespread and numerous derivaes are oday raded on hese indices. The sock marke and he individual socks qualiies are very hard o define, bu 44 Greene, W.H., (008), p

87 forunaely he sock marke is se up in a way so ha he liquidiy is conserved allowing he marke o form public valuaions on he socks. The Sandard & Poor s Composie Sock Price Index is one of he major sock indices and is consruced as a value weighed arihmeic sock price index. The idea behind he Sandard & Poor s Composie Sock Price index consrucion is ha he index value should correspond o he value of a porfolio consising of all he socks in he marke in proporion o heir acual marke value. This consrucion involves rebalancing of he porfolio each period o incorporae he changes in acual marke value. The change in he index value is hen largely given by acual changes in he value of he socks. The value of he index is hen based on repeaed sales daa. The index is aracive o conrac selemen due o he fac ha he value of he index changes when he value of he underlying socks changes and herefore he index is reliable a good vehicle for hedging he risk associaed wih he socks in he index. The value for he value weighed arihmeic price index is given by he following formula: I = Q Q i, i, P P i, i, I Where: - P i, is he price of a share of a sock in company i, a ime. - Q i,- is he number of shares ousanding of he ih sock a ime -. - The index is usually se o 00 a he base period ( = 0). () This paricular index is called a chain index due o he fac ha he value index a ime is chained back o he value of he index a ime -. Furhermore i is apparen form he formula ha he raio I /I - is a value weighed arihmeic average of he price raios. The chain index is ofen conras by he fixed base indices characerized wih he fac ha he number of shares is fixed ha is Q is held consan, in his siuaion he index have he following formula: I = i q i q P P i, / I i, () Where: 85

88 - q is he se of all shares ousanding a ime - - i represen he share raher han he corporaion company. From equaion () i is seen ha he value weighed arihmeic index is based on he sum of he sock prices a ime, divided by he sum of sock prices a ime - deflaed wih he index a ime -. The porfolio value represened by he wo above-menioned indices is physically replicable, which means ha index arbirages can consruc a porfolio ha rack he value of he index. Which is in conras o some oher indices, i.e. indices based on geomeric averages 45. A a firs look if would be obvious o ry o use he chain indices jus inroduced in an aemp o creae an index on real esae prices, bu here are issues ha has o be aken ino accoun before he simple chain index can be used as he basis for a real esae index. One issue is ha real esaes are illiquid asses, which in urns means ha he prices on he real esaes are no observable in every period due o infrequen rades. This means ha we lack informaion on which o calculae he value of he chain index 46. I would be possible o adop he chain index framework () building on a repeaed sales framework, bu here are several unforunae consequences of doing so; The porfolio of he real esae included in he index will no be physically replicable in real ime (canno buy shares in illiquid individual real esae). One canno know when he real esae are bough and sold (he value of he real esae is only included in he index in he period hey are raded). Anoher unforunae consequence of he chain index consrucion is ha he newes real esae is less likely o be included in he index (small probabiliy of he house having been sold wice) 47. Oher problems ha are likely o arise from he use of a basic chain index is ha he average sales price of he properies in he index will vary more han he acual value of any given propery due o variaions in he qualiy of he properies sold from period o period. Furhermore is here is chance of a progressive change in qualiy of he properies sold a differen imes, which will cause bias in he index over ime Shiller, R.J., (998), p Shiller, R.J., (998), p.3 47 Shiller, R.J., (998), p Bailey, M.J., Muh, R.F. and Nourse, H.O., (963), p

89 A.3 Analysis of he Underlying Asse A.3. The Time Series Daa Used in he Analysis See also he excel file 4 Danish Real Esae Indices and 5 Danish Real Esae Indices Yearly Observaions A.3. Presenaion of he Indices (Including Differen Levels of he Indices) Table: The Basic Descripive Saisics (F) Series F df LogF dlogf Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Observaions Jarque-Bera

90 Probabiliy Table: Graphical presenaion (F) F LOGF DF DLOGF Table: The Basic Descripive Saisics (E) Series E de LogE dloge Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Observaions Jarque-Bera Probabiliy Table: Graphical presenaion (E) 88

91 E LOGE DE DLOGE Table: The Basic Descripive Saisics (P) Series P dp LogP dlogp Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Observaions Jarque-Bera Probabiliy Table: Graphical presenaion (P) P LOGP 89

92 DP DLOGP One of he mos basic es applied o ime series daa, is he es for normaliy in he reurn generaed form he specific ime series. The Jarque-Bera es is a classical es in his conex; he es uses esimaions of he skewness and kurosis of a given disribuion o es wheher he reurn from he ime series is normal disribuion or no. Table: The Jarque-Bera (JB) es JB = gˆ + hˆ Where g and h is calculaed: T gˆ 6 S ˆ T = ˆ T (ˆ K 3) h T = 6 4 The skewness is given by: T 3 X = i X Sˆ T i= σˆ The kurosis is given by: T 4 X = i X Kˆ T i= σˆ A normal disribuion is characerized by: K=3 and S=0 T T The JB es follows a χ () disribuion. A large es saisic indicaes ha he reurn generaed from he ime series is normally disribued, from he descripive saisics, from he ables above i is seen ha he indices a 5% significan level are insignifican, wih he excepion of he index on owner-occupied flas, which is significan a a 3,9% level. When applying he dlog ransformaion he es saisic improves for he indices on owner-occupied flas dloge (6.36% significance) and he weekend coages dlogf (.94% significance), whereas he es saisic deerioraes on he index based on he deached- and errace-houses dlogp. The dlogp reurn needs o be explored furher before analysis on he ime series, his is done via a uni roo es. 90

93 A.3.3 Uni Roo and Augmened Dickey-Fuller es The mos applied mehod for esing he degree of inegraion and of saionariy is he Dickey-Fuller es or he augmened Dickey-Fuller es. In he Dickey-Fuller es he following equaion is esimaed: Y α + e = Y If α is equal o one, he ime series is said o exhibi a uni roo, ha is he series needs o be inegraed in order o ransform he ime series ino a saionary ime series. The acual Dickey-Fuller es is a one sided es. If α is less han one he series is inegraed of order zero, I(0). In he case ha α =, he ime series is inegraed of order one, I(), and he series needs o be differenced. The alernaive of α being greaer han, is very unlikely in finance, due o he explosive naure of he ime series. The null hypohesis (α =) violaes he underlying OLS assumpions of consan variance in he error erm 49, and he equaion needs o be re-specified o ake his violaion ino accoun, his is accomplished by expressing he original equaion in erms of he change in Y: Y c Y c Y Y = ( α ) Y + e (one) = βy = αy + e Y + e Seing β = ( α ), he hypohesis es hen become a es on: H H 0 : β = 0 : β < 0 If bea is zero hen from he definiion i is seen ha α is one, corresponding o a uni roo process, I(). If he null hypohesis is rejeced, hen he process is saionary in he mean, I(0). This basic es assumes a zero mean and no ime rend. Ofen is i appropriae o include a posiive mean (expec a posiive payoff) and in some financial imes series may furhermore include a ime rend. In he case of a posiive mean, he equaion becomes: 49 The violaion can be seen from recursive subsiuion ino he equaion. 9

94 e Y Y e Y Y e Y Y Y Y e Y Y + + = + + = + + = + + = β α α α α α α α c c c ) ( (wo) And finally when a ime rend is incorporaed he corresponding equaion becomes: e T Y Y e T Y Y e T Y Y Y Y e T Y Y = = = = γ β α γ α α γ α α γ α α c c c ) ( (hird) The radiional -es is no appropriae o use in esing he significance of bea in ha his would sugges ha we assume ha bea is less han zero due o he fac ha regression is used o esimae bea. This assumpion would resul in rejecing bea oo ofen when bea is acually zero. Furhermore he uni roo es using he Dickey-Fuller es is robus agains some degrees of heeroskedasiciy while auocorrelaion causes problems. The use of he augmened Dickey-Fuller es solve he issue wih auocorrelaion in he error erm, by included lagged variables of he dependen variable in he regression, when enough lagged variables are included in he model, he auocorrelaion in he error erm disappear. An example of he augmened Dickey-Fuller could be: n n e Y Y Y Y Y = γ γ γ β α... 0 The es of uni roo can be performed via ree differen models, i.e. a model wih zero mean and zero ime rend (one), a second model including a posiive mean (wo) and

95 finally a hird model including a posiive mean and a ime rend (hird), i should be noed ha he significance es depends upon he chosen model 50. Dickey-Fuller and Augmened Dickey-Fuller Tess The -saisic in he augmened Dickey-Fuller es is no asympoic sandard normal disribued his means ha he criical values are obained using simulaions (5% criical value = -.93 (wihou rend) and 3.50 (wih rend)) DF and ADF es on owner-occupied fla index (E) Table: Dickey-Fuller and augmened Dickey-Fuller Tess (E) Saisic Consan wihou rend ADF(0) (0.9500) ADF() (0.6046) ADF() (0.704) ADF(3) (0.6780) ADF(4) (0.840) ADF(5) (0.9933) ADF(6) (.0000) ADF(7).535 (0.9974) ADF(8).558 (0.999) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend (0.8707) (0.0989) (0.0585) * (0.044) (0.087) (0.649) (0.9833) (0.8867) (0.9404) Akaike info crierion Table: Dickey-Fuller and augmened Dickey-Fuller Tess (de) Saisic Consan wihou rend ADF(0) (0.49) ADF() (0.653) ADF() (0.853) ADF(3) (0.469) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend Schwarz creerion Durbin- Wason sa Akaike info crierion Schwarz creerion Durbin- Wason sa (0.574) (0.6300) (0.6875) (0.4650) ADF(4) Washam, T.J. and Parramore, K., (997), p

96 * (0.0038) ADF(5) * (0.0009) ADF(6) (0.60) ADF(7) (0.56) ADF(8) (0.0984) * (0.03) * (0.0003) (0.8) ,9474 (0.679) * (0.034) Table: Dickey-Fuller and augmened Dickey-Fuller Tess (dloge) Saisic Consan wihou rend ADF(0) (0.0987) ADF() (0.43j) ADF() (0.570) ADF(3) (0.4966) ADF(4) (0.087) ADF(5) * (0.0003) ADF(6) * (0.050) ADF(7) * (0.0353) ADF(8) * (0.064) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend (0.843) (0.6496) (0.7536) (0.7594) (0.598) * (0.008) (0.084) (0.63) (0.0833) Akaike info crierion Schwarz creerion Durbin- Wason sa DF and ADF es on weekend coages index (F) Table: Dickey-Fuller and augmened Dickey-Fuller Tess (F) Saisic Consan wihou rend ADF(0) (0.9999) ADF() (0.9836) ADF() (0.953) ADF(3) (0.756) ADF(4) (0.6849) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend (0.8886) (0.787) (0.679) (0.4470) (0.396) Akaike info crierion Schwarz creerion Durbin- Wason sa

97 ADF(5) (0.904) ADF(6) (0.608) ADF(7) (0.843) ADF(8) (0.8875) * (0.0484) (0.9) (0.374) (0.663) Table: Dickey Fuller and augmened Dickey-Fuller Tess (df) Saisic Consan wihou rend ADF(0) * (0.007) ADF() (0.043) ADF() (0.4044) ADF(3) (0.4958) ADF(4) (0.6648) ADF(5) (0.496) ADF(6) (0.354) ADF(7) (0.443) ADF(8) (0.669) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend * (0.033) (0.30) (0.8836) (0.93) (0.9855) (0.945) (0.8385) (0.7945) (0.3948) Akaike info crierion Schwarz creerion Durbin- Wason sa Table: Dickey Fuller and augmened Dickey-Fuller Tess (dlogf) Saisic Consan wihou rend ADF(0) * (0.000) ADF() * (0.0046) ADF() (0.094) ADF(3) (0.3799) ADF(4) (0.56) ADF(5) (0.570) ADF(6) (0.3455) ADF(7) (0.5683) ADF(8) (0.485) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend * (0.000) * (0.059) (0.3689) (0.86) (0.950) (0.9348) (0.8736) (0.9095) (0.754) Akaike info crierion Schwarz creerion Durbin- Wason sa

98 DF and ADF es on deached- and errace-house index (P) Table: Dickey-Fuller and augmened Dickey-Fuller Tess (P) Saisic Consan wihou rend ADF(0) (0.9996) ADF() (0.853) ADF() (0.8357) ADF(3) (0.5543) ADF(4) (0.787) ADF(5) (0.946) ADF(6) (0.9984) ADF(7) (0.9990) ADF(8) (0.9953) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend (0.9740) (0.798) (0.834) * (0.076) * (0.000) (0.338) (0.989) (0.9587) (0.0967) Akaike info crierion Table: Dickey-Fuller and augmened Dickey-Fuller Tess (dp) Saisic Consan wihou rend ADF(0) (0.303) ADF() (0.43) ADF() (0.583) ADF(3) (0.405) ADF(4) (0.340) ADF(5) * (0.0080) ADF(6) (0.0987) ADF(7) (0.5676) ADF(8) (0.804) Akaike info crierion Schwarz creerion Durbin- Wason sa Consan wih rend (0.588) (0.658) (0.90) (0.8340) (0.408) * (0.0300) (0.69) (0.655) (0.97) Schwarz creerion Durbin- Wason sa Akaike info crierion Schwarz creerion Durbin- Wason sa Table: Dickey Fuller and augmened Dickey-Fuller Tess (dlogp) Saisic Consan wihou Akaike info Schwarz creerion Durbin- Wason Consan wih Akaike info Schwarz creerion Durbin- Wason 96

99 rend crierion sa rend crierion sa ADF(0) * (0.00) (0.0890) ADF() (0.099) (0.3086) ADF() (0.4304) (0.7865) ADF(3) (0.5600) (0.8769) ADF(4) (0.958) ADF(5) * (0.03) ADF(6) * (0.039) ADF(7) (0.34) ADF(8) * (0.039) (0.6669) (0.047) (0.049) (0.639) (0.0766) A.3.4 The Akaike and he Schwarz Informaion Crierion The Akaike (AIC) and he Schwarz (SIC) informaion crierion is used o balance beween he goodness-of-fi and parsimonious specificaion of a specific model: Table: Informaion crierion: Formulas The Akaike informaion crierion (AIC): l + n k n The Schwarz informaion crierion (SIC): l log( n + k ) n n Where k is he number of esimaed parameers, n is he number of observaions and l is he value of he log likelihood funcion using he k esimaed parameers. I is seen ha boh measures use he average log likelihood funcion which is associaed wih a negaive value of minus, he raional behind he average log likelihood funcion is ha he beer fi is associaed wih he lowes value. The wo crierions furhermore uilise a penaly funcion, which akes he number of esimaed parameers and observaions ino accoun. I is obvious ha he value of he es will increase when he number of esimaed parameers in he models are increased and he es value will also increase wih decreasing numbers of observaions. The selecion rule is ha he model wih he lowes informaion crierion is he superior model. 97

100 A.3.5 Basic Characerisics of he AR() and he MA() processes Table: Basic Characerisics of he Auoregressive and Moving Average Processes 5 AR() Variance V ( y ) = σ y = σ θ Covariance k σ cov { Y, Y k} = θ θ Auocorrelaion k ρ k = θ for k = 0,, MA() V ( ) = ( + α ) σ y cov { Y, Y k} = ασ { Y, } = 0 cov Y k for k =,3 α ρ = + α ρ = 0 for k =,3 k A.3.6 Model Esimaions In his secion several models are esimaed in an aemp o explain he movemen in he underlying indices. The processes are modelled on he basis of he log-difference daa from he various indices. In addiion o he model esimaions several saisical ess are conduced o be able o make informed decisions on he mos appropriae model specificaion in describing he indices. Table: Auocorrelaion and parial auocorrelaion funcions of he ime series (dlog) dloge dlogf dlogp The numbers in brackes are he sandard error of he respecive coefficien. Along side he sandard errors are he Durbin Wason, Akaike and Schwarz saisics included o give an idea of he fi of he esimaed models. Table: Models esimaed on he basis of he owner-occupied fla index (dloge) 5 Verbeek, M., (004), p

101 MA() d log E ε ε + ε = ( ) (0.8486) (0.885) DW =.939 * Akaike = * Schwarz = * R = * σ ε = 0.07 AR() d log E = d log E ( ) (0.0370) DW = * Akaike = * Schwarz = * R = * σ ε = ε ARMA(,) d log E = d log E ε ε + ε (0.0055) ( ) ( ) (0.7893) DW = * Akaike = * Schwarz = * R = * σ ε = 0.06 Table: Residuals from he AR() process of he dloge Index AC and PAC Breusch-Godfrey serial correlaion LM es Table: Residual plo from he AR() process of he dloge Index Residual Acual Fied Table: Models esimaed on he basis of he weekend coages index (dlogf) 99

102 MA() d log F ε + ε = ( ) ( ) DW = * Akaike = * Schwarz = * R = * σ ε = AR() d log F = d log F ( ) (0.3877) DW = * Akaike = * Schwarz = * R = 0.09 * σ ε = ε ARMA(,) d log F = ε d log F + ε (0.0043) ( ) ( ) DW = * Akaike = * Schwarz = * R = * σ ε = Table: Residuals from he AR() process of he dlogf Index AC and PAC Breusch-Godfrey serial correlaion LM es Table: Residual plo from he AR() process of he dlogf Index 00

103 Residual Acual Fied Table: Models esimaed on he basis of he deached- and errace-house index (dlogp) MA(4) d log P = ε ε ε ε 4 + ε ( ) ( ) (0.0993) ( ) (0.374) DW =.8404 * Akaike = * Schwarz = * R = * σ ε = 0.03 MA(4) wihou MA() erm d log P = ε ε ε 4 + ε ( ) (0.0054) ( ) ( ) DW = * Akaike = * Schwarz = * R = * σ ε = 0.06 AR() d log P = d log P ( ) (0.88) + ε DW = * Akaike = * Schwarz = * R = * σ ε = ARMA(,4) d log P d log ε ε ε ε + ε = P ( ) ( ) ( ) (0.789) (0.7937) DW = * Akaike = * Schwarz = * R = * σ ε = 0.0 ARMA(,4) wihou MA() erm d log P = d log P ε ε ε 4 + ε ( ) ( ) (0.350) (0.069) (0.5085) DW = * Akaike = * Schwarz = * R = * σ ε = Table: Residuals from he AR() process of he dlogp Index AC and PAC Breusch-Godfrey serial correlaion LM es ( ) 0

104 Table: Residual plo from he AR() process of he dlogp Index Residual Acual Fied A.3.7 Problems Concerning Serial Correlaion and Heeroskedasiciy Serial correlaion (or auocorrelaion) Serial correlaion refers o he violaion of V{ ε } = σ I. When wo or more consecuive error erms are correlaed hen he error erm exhibi auocorrelaion or serial correlaion. Tha is he covariances beween differen error erms are no all equal o zero 5. There are several es on he presence of serial correlaion in he error erm, among hem are he Durbin Wason (DW) es, Q-saisics and he Breusch-Godfrey LM-es. 5 Verbeek, M., (004), p.97,

105 The Durbin Wason (DW) es saisic is a es for firs-order auocorrelaion. The DW saisic measures he linear associaion beween adjacen residuals from he acual regression model. The DW es saisic is given by: DW = T = ( e T = e e ) The DW es saisic can also be approximaed as DW ρˆ ; from his approximaion i is clear ha a DW es saisic close o is an indicaion of zero firs order auocorrelaion in he error erm. When he es saisic deviaes from, i is a sign of problems concerning firs order auocorrelaion. Lower values indicae posiive auocorrelaion (ρ > 0) while higher values indicaes negaive auocorrelaion (ρ < 0) 53. The limiaion o he DW es (only used o es for firs order auocorrelaion) can be deal wih by using Q-saisics and he Breusch-Godfrey LM-es, which can be used o es for he presence of higher orders of auocorrelaion. The AC- and PAC-funcions of he esimaed residuals are ofen supplemened wih he Ljung-Box Q-saisics. In he case of serial correlaion in he residuals he Q-saisics will be significan, wih low p- values. In he Breusch-Godfrey LM-es he null hypohesis saes ha here is no serial correlaion in he residuals included in he es 54. The presence of auocorrelaion in he error erm can be inerpre as a misspecificaion of he model, ha is he model do no capure he behavior of he ime series, which could be caused by a wrong funcional form or explanaory variables missing in he model 55. Heeroskedasiciy Heeroskedasiciy refers o he problem ha arises when he error erms are muually uncorrelaed, while he variance of ε i may vary over he observaions, which is a violaion of he homoskedasiciy assumpion made in OLS 56. The violaion of he homoskedasiciy assumpion does no imply ha OLS parameers are biased or inconsisen, his only have an impacs on he variance of he parameers making he OLS esimaors inefficien. The inefficiency implies ha hypohesis esing on he 53 Verbeek, M., (004), p Verbeek, M., (004), p Verbeek, M., (004), p Verbeek, M., (004), p

106 parameers is no possible due o wrong esimaes of he sandard error. The -es is no longer -disribued and he F-es is also invalid. This resuls in a oo narrow confidence inervals and oo large -saisics 57. The mos widely used es o es for heeroskedasiciy is he Breusch-Pagan and he Whie es 58. Heeroskedasiciy can also be deeced by an inspecion of he disribuion of he residuals ploed agains he esimaed values of he dependen variables or agains he independen variables 59. Condiional Heeroskedasiciy Auoregressive condiional heeroskedasiciy (ARCH) is ofen observed in financial imes series and is referred o as volailiy clusering, where large residuals are followed by large residuals and vice versa. In oher words he variance of he error erm seems o be dependen upon ime. The presence of ARCH(p) in he residuals can be esed via he following regression: ε + α ε + + σ = ϖ + α... α ε p p Where α and ϖ are resriced o posiive values (he variance canno become negaive). The presence of ARCH processes do no invalidae he OLS esimaions, i jus implies ha more efficien esimaors are available. The null hypohesis in he ARCH es is: H α α... α 0 = = = = p = The es saisic asympoically follows a χ disribuion wih p degrees of freedom, under he H he residuals follows an ARCH(p) process Table: ARCH-es compued on he esimaed AR()-processes N R (Probabiliy) dloge dlogf dlogp ARCH() (0.6) 0.09 (0.88) (0.766) ARCH().9975 (0.3) (0.96).34 (0.5) ARCH(3).6575 (0.448) (0.993) (0.0) ARCH(4) (0.79) (0.97) (0.) ARCH(5) (0.44) (0.989).087 (0.035)* ARCH(6) (0.49) (0.997) (0.04)* ARCH (7) 8.39 (0.304) 0.70 (0.998) (0.04)* ARCH (8) (0.98).35 (0.995) (0.049)* ARCH (9) (0.346).398 (0.983) (0.085) ARCH (0).078 (0.35).7694 (0.986) (0.06) ARCH ().049 (0.36) (0.98) (0.30) 57 Wooldridge, J.M., (006), p Wooldridge, J.M., (006), p Fleming, M.C. and Nellis, J.G., (008) 60 Verbeek, M., (004), p

107 ARCH ().586 (0.400) (0.989) (0.97) A.3.8 Derivaion of he MA( ) Process from he AR() Process AR() process is described by he following equaion: Y δ θ + ε = + Y Where ε - is a whie noise process. The expeced value of Y is given by: E ( Y ) = δ + θe( Y ) Assuming he E(Y ) does no depend upon and θ <, he expeced value can be wrien as: E( Y ) = δ + θe( Y ) c E( Y c E( Y c E( Y µ ) θe( Y )( θ) = δ δ ) = θ E ( Y ) = δ δ ) = θ To express he AR()-process as an infinie moving average, repeaed subsiuion ino he AR()-process has o be underaken, using he fac ha Y = δ + θy + ε and ha δ can be expressed as δ = µ( θ) his yields 6 : Y = µ + θ ( Y µ ) + ε + θε Repeaing his procedure yields he following resul: Y µ + θ n ( Y µ ) + j= 0 = θ ε When n, he second expression on he righ hand side approaches zero, which reduces he equaion o he MA( ) process: Y = µ + θ ε j= 0 j j j j 6 Modern Economerics, p

108 A.3.9 Forecasing wih he AR()-Process The abiliy o forecas wih he AR()-process has been indicaed in he hesis, here more deails on he derivaion ec. is given. Forecasing one ime period ahead is simple and given by 6 : y + = δ + θy + ε+ To derive forecasing predicions addiional ime periods ahead expecaions and updaing of he equaion are compued, ha is he expecaions o he forecas one period ahead is: E ( y+ ) = δ + θ Updaing he equaion o forecas wo periods ahead yields: y y + = δ + θy + + ε+ The condiional expecaion given he informaion se (he informaion se conains he value of y and all is lags) yields: E( y+ ) = δ + θe( y+ ) Insering E(y + ) ino he expression yields: E ( y + ) = δ + θ( δ + θy ) = δ( + θ) + θ y This process is coninued n periods ahead where he resuling expression yields: n n E( y+ n) = δ ( + θ + θ θ ) + θ y In he limi (n ), his expression converges owards: E ( y + n ) δ = θ I is assumed ha he AR()-process is saionary ( θ < ), oherwise he process has no mean value. The abiliy o forecas he fuure oucome of he AR() is off course conneced wih some kind of uncerainy. This uncerainy is ofen described using he forecasing error. We need o know how precise he predicions are, o be able o make inferences abou he developmen in he behavior of he process. The one period ahead 6 Modern Economerics, p

109 forecas error (f () is given by: f ) y E ( y ) inserion of y + and E (y + ) yields he following expression: f ( = + + ( ) = [ δ + θy + ε ] [ δ + θy ] = ε + The forecasing error wo periods ahead is given by: f ) y E ( y ) which yields: f ( = + + ( ) = [ δ + θy+ + ε + ] [ δ + θe( y+ )] = θε + + ε+ In he general case he forecasing error for n-periods ahead becomes: f ( n ) n = ε + n + θ ε+ n + θε+ n θ ε + The predicion error gives an indicaion of he uncerainy linked o he poin predicion obain from he AR()-process. The expeced quadraic predicion error corresponds o he variance (σ ) of he n-period ahead predicion error. The squared predicion error (SPE) is given in he able 63. Table: The Squared Predicion Error (SPE) of he AR()-process One period ahead Two periods ahead σ ( ε + ) Three periods ahead 4 σ ε ( + θ + θ ) n-periods ahead ( + θ 4 + θ n θ ) If we le n, his expression becomes σ ε σ ε σ ε θ The squared predicion error gives an indicaion of how well ha esimaed model fis he acual observaions, he squared predicion error can be used o inferences abou he variance of he forecased values. The variance of he forecas one period ahead is given by: σ ε The variance of he forecas covering wo periods, ha is period and, are no jus he accumulaed squared predicion error, he second periods variance is of course influenced by he variance of he firs period. Tha is he parial auocorrelaion 64 of he 63 Franses, P.H., (998), p The parial auocorrelaion equals he coefficien of he AR() erm. 07

110 auoregressive process needs o be included in he calculaions o measure he addiional correlaion beween he successive observaions. The variance of forecasing wo periods can be calculaed using he law of summing variance wih dependence beween observaions: Var(X,Y) = Var(X) + Var(Y) + COV(X,Y) Where Var(X) = σ ε, Var(Y) = σ ( θ ε + ), he COV is found by rewriing he expression for he coefficien of correlaion 65 : ρ cov( X, Y) = cov( X, Y) ρσ xσ y σ σ = From his rewriing i is seen ha ρ = θ and forecasing wo periods ahead as: x y σ σ = x y σ ( θ θ ) ε + + σ ε, which yields he variance of Coninuing in he manner he variance of n-periods ahead forecasing variance can be calculaed. A.4 Valuaion of Home Equiy Insurance Producs A. 4. Basic Propery Derivaives The Swap: The mos popular derivaive in he propery derivaive marke is he swap. A swap is an agreemen beween wo paries o exchange cash flows in he fuure 66. The swap agreemen is bes illusraed by an example; an invesor wih a large porfolio of real esae migh wan o reduce he risk exposure on he real esae by enering ino a swap agreemen. The invesor pays he reurn generaed from he real esae, which is ypically done using a real esae index as a proxy for he real esae reurn. In reurn he/she ges a fixed or floaing ineres, he ineres rae is ypically linked o an inerbank rae as he LIBOR or EURIBOR. This agreemen will reduce he risk he real esae 65 Keller, G. and Warrack, B., (003), p.8 and p Hull, J.C., (006), p49 08

111 invesor is exposed o and hereby sabilize he cash flow generaed from his/her porfolio. The oher pary in he agreemen receives he reurn from he real esae, hereby gaining exposure o he real esae marke 67. The Bonds: The bonds are basically very similar o he swap, in ha he paymen from he cerificaes are calculaed on he basic of he reurn on an real esae index, he invesor buys a cerificae as a way o inves in he real esae marke wihou having o own any properies, ha is o ge exposure o he real esae marke wihou having he rouble of owning real esae 68. The Opions: Opions used in relaion o propery invesmens are ypically linked o a real esae index where he buyer of he opion eiher ges he upside by invesing in a call opion, or he downside by buying a pu opion. There are various alernaive consrucions and sraegies regarding he use of opions 69. A.4. Black & Scholes Assumpions 70 In deriving he BS equaion several assumpions needs o be fulfilled. - The underlying follows he geomeric Brownian moion wih µ and σ consan. - Shor selling of securiies and he use of proceeds permied. - No ransacions coss and axes, and securiies are perfecly divisible. - No dividend during he life of he derivaive. - No riskless arbirage opporuniies - Securiy rading is coninuous - The risk-free rae is consan and he same for all mauriies. A.4.3 Mone Carlo Simulaion One Period ahead The VBA-Code, he following is aken from he acual VBA-code and can also be found in he excel file Real Esae Opions Mone Carlo Simulaions in module. Funcion PuSimARperiod(Spo As Double, Srike As Double, Acualchange As Double, Expiry As Double, Rae As Double, Volailiy As Double, Con As Double, Co As Double, Pahs As Double) As Double 67 Deusche Bank Research (007) 68 Deusche Bank Research (007) 69 Deusche Bank Research (007) 70 Hull, J.C., (006), p

112 (Con is he consan in he acual AR()-process, while Co is he coefficien in he AR()-process) Dim Op As Double sores he sum of he simulaed opion prices Dim Opm As Double sores he sum of he simulaed opion prices aniheic variable Dim Expecedchange As Double Dim Expecedchangem As Double aniheic variable Dim Spoexpiry As Double value of he underlying a expiry Dim Spoexpirym As Double value of he underlying a expiry aniheic variable Dim Pupayoff As Double compound-payoff a he simulaed value of he underlying Dim Pupayoffm As Double compound-payoff a he simulaed value of he underlying aniheic variable Dim n As Long Dim Errorerm As Double Sar he generaion of random numbers Call Rnd(-) Randomize () Seing he iniial value of he opion soring faciliy Op = 0 Opm = 0 Simulaion of he firs pah For n = To Pahs Simulaing he price of he underlying using he auoregressive naure of he index, he second expression uses aniheic variables. Expecedchange = Exp(Con + (Co * Acualchange)) + Exp((MarsagliaBrayNormSRnd()) * Volailiy) Expecedchangem = Exp(Con + (Co * Acualchange)) *Exp((- MarsagliaBrayNormSRnd()) * Volailiy) Calculaed he expeced value of he index using he value of he acual propery i.e. spo mio, second expression uses aniheic variables. Spoexpiry = Expecedchange * Spo Spoexpirym = Expecedchangem * Spo Calculaion of he payoff pu opion on he index, second expression calculaed he payoff from he aniheic variable, see also he definiion of he payoff funcion below. Pupayoff = Payoff(Spoexpiry, Srike, 0) Pupayoffm = Payoff(Spoexpirym, Srike, 0) 0

113 Summaion of he calculaed pu opion prices Op = Op + Pupayoff 'Updaed sum of call values Opm = Opm + Pupayoffm 'Updaed sum of call values Taking he average of he calculaed pu opions he aniheic echnique. Opavg = (Op + Opm) / Generaion of he nex pah, in he Mone Carlo simulaions 0,000 pahs is generaed o price he individual opion prices (ha is n = 0,000) Nex n Pricing he pu opion, by discouning he average pu opion value. Dim Disc As Double Disc = Exp(-Rae * Expiry) Op = Opavg * Disc / Pahs PuSimARperiod = Op End Funcion Mone Carlo Simulaion Addiional Periods Ahead In simulaion addiional periods ahead he VBA code is changed o incorporae he forecasabiliy of he AR()-process ino he simulaion. The only change made in he code is a sligh modificaion of he expeced change formula, in he one period ahead expression he formula used where: Expecedchange = Exp(Con + (Co * Acualchange)) * Exp((MarsagliaBrayNormSRnd()) * Volailiy) In forecasing he expeced change wo periods ahead he following formula is used: Expecedchange = Exp(Con * ( + Co)) + ((Co ^ ) * Acualchange) * Exp((MarsagliaBrayNormSRnd()) * Volailiy) Three periods ahead Expecedchange = Exp(Con * ( + Co + (Co ^ )) + ((Co ^ 3) * Acualchange) * Exp((MarsagliaBrayNormSRnd()) * Volailiy)) 'simulaed price of he underlying as expiry And so forh, in addiion o he changes in he expeced change, he variance is calculaed according o he above derivaion of he variance and he correc variance is used o price he respecive pu opions.

114 Payoff funcion The payoff funcion coded in VBA can be used o find he payoff of he pu opion, by seing Callop as 0, if he Callop is se a, he payoff correspond ha of a call opion on he indices, his means ha i is very simple o calculaed call opions on he real esae indices if his is needed. Funcion Payoff(Spo As Double, Srike As Double, Callop As Ineger) Dim Value As Double Value = Spo - Srike Calculaion of he call opion payoff If callop = Then If Value > 0 Then Payoff = Value Else Payoff = 0 End If Calculaion of he pu opion payoff Else If Value < 0 Then Payoff = -Value Else Payoff = 0 End If End If End Funcion Generaion of Random Normal Variables (Marsaglia-Bray) The Marsaglia-Bray algorihm is coded in VBA o generae efficien random normal variables. Funcion MarsagliaBrayNormSRnd() As Double Dim U As Double Dim U As Double Dim x As Double Dim y As Double If Saved = falsh Then x =

115 While x > U = ( * Rnd() - ) U = ( * Rnd() - ) x = (U * U) + (U * U) Wend y = Sqr(- * Log(x) / x) MarsagliaBrayNormSRnd = U * y SavedNormSRnd = U * y Saved = True Else MarsagliaBrayNormSRnd = SavedNormSRnd Saved = False End If End Funcion 3

116 A.4.4 Valuaion of Opions using Mone Carlo Simulaion See also he excel file 7 - Real Esae Opions Mone Carlo Simulaion where he acual VBA-codes can be found including he valuaion of he various pu opions Conrac lengh: 3 Monhs 4

117 Conrac lengh: 6 Monhs 5

118 Conrac lengh: 9 Monhs 6

119 Conrac lengh: Year 7

120 A.4.5 VBA-code Modified Black & Scholes Equaion Funcion Realesaeopion(Real_esae_value As Double, Exercise_value As Double, Expiraion As Double, Acual_change As Double, Variance As Double, Ineres_rae As Double) As Double Dim Value As Double Dim R As Double Dim R As Double Dim S As Double Dim S As Double R = Exp(-Ineres_rae * Expiraion) * Exercise_value R = Exp(Acual_change + (Variance / ) - Ineres_rae * Expiraion) * Real_esae_value S = ((Log(Exercise_value / Real_esae_value) - Acual_change) / Sqr(Variance)) S = ((Log(Exercise_value / Real_esae_value) - Acual_change) / Sqr(Variance)) - Sqr(Variance) Realesaeopion = R * Applicaion.NormSDis(S) - R * Applicaion.NormSDis(S) End Funcion 8

121 A.4.6 Valuaion of Opions using Modified Black & Scholes Equaion See also he excel file 8 - Real Esae Opions Modified BS Equaion where he acual VBA-codes can be found including he valuaion of he various pu opions 9

122 0

123 A.4.7 Inflaion Cach Up See also excel file 6 Inflaion Cach Up for deails on he acual calculaions.

124 A.5 Alernaive Ways o Hedge he Real Esae Risk A.5. Mean-Variance Calculaions See also he excel file 9 Alernaive Ways o Hedge he Real Esae Risk Real Esae Socks where he acual calculaion (including marix calculaions) can be found.