5.1. General Remarks to Value at Risk

Transcription

1 5. Risk Measuremen on he Basis of Value a Risk 5.1. General Remarks o Value a Risk Slides: Tanja Dresel, Luz Johanning,. Hans-Peer Burghof 61

2 5.1 General Remarks o Value a Risk The Value a Risk of a single or aggregae posiion can be calculaed by he probabiliy bili disribuion ib i of marke value changes derived d from he risk analysis. Value a Risk of normally disribued daily marke value changes V Probabiliy densiy p=5 % VaR(95%) V 6

3 Basics of he VaR Calculaion: Calculaion of Expeced Value, Sandard Deviaion and Correlaion Value a Risk of normally disribued daily marke value changes V Probabiliy densiy Sandard deviaion of V i, Expeced value of V i, p=5 % X S(V i, ) i, VR(95%) VaR(95%) E(V i, ) V i, 63

4 Definiion of Profis and Losses (P&L): generally: price/reurn aleraion To be formal: V i, =V i, V i,1 V i, = Aleraion in marke value of sock i a ime V i, V i,1 = Difference of sock i's marke values in resp. -1 In he common marke risk models, he VaR calculaion relies on daily reurns (wih one day holding period) of financial insrumens. 64

5 Discreely vs. Coninuously Compounded Reurns: Reurn R i, as discree price aleraion: R i, V i, V V i,, 1 i, 1 Reurn r i, as coninuous price aleraion: ln( Vi, / Vi, 1) i, p i,, r For very small reurns, i is: R i, ri, Subsequenly, we predominanly assume logarihmic (resp. coninuous) reurns => Implicaion: sock price can never be negaive 65

6 Example 1: The sock prices of hree socks are given from 06/10/008 unil 10/10/008 Sock A Sock B Sock C 47,90 33,70 164,99 47,83 34,00 164,80 48,80 33,73 165,40 49,1 33,65 17,00 a) Calculae he coninuous and discree reurns of he hree socks for he ime period menioned above. coninuous Sock A Sock B Sock C discree Sock A Sock B Sock C

7 Calculaing he Expeced Value i, of Sock i s s Daily Reurns for Time Assumpion: he (esimaed) expeced value equals he average reurns over a hisoric observaion period K i, 1 K K K k 1 r i, k 1 => Assumpion of saionariy 67

8 Example: DAX 007 Expeced Value of Daily Reurns 0,00 0,0016 0, ,0008 0, => The expeced value of daily reurns in 007 was always greaer han zero 68

9 b) Now calculae cu a he expeced ed values of he socks soc sconsidered sdeedabovebyus by using coninuously compounded reurns. Sock A Sock B Sock C 69

10 Calculaing he Sandard Deviaion i, of Reurns Esimaion for sandard deviaion: 1 K i, ( ri, k 1 i, ) K 1 k 1 Esimaion of he sandard deviaion is also based on hisorical daa (=> assumpion of saionariy) Sandard deviaion describes he mean variaion abou he expeced value (mean) 70

11 Example: DAX 007 Sandard Deviaion of Daily Reurns 0, ,01 0,0095 0,009 0, ,008 0,0075 0,007 0, , Sandard deviaion changes in ime, i.e. no saionariy 71

12 Esimaing he sandard deviaion, i is ofen assumed ha i, =0. => Assumpion is no jusified as even he expeced value of one-day reurns mosly varies from zero (see previous graph) => Bu: misakes esimaing he expeced value can be ruled ou => and he calculaions are simplified, => herefore, VaR calculaion in pracice ofen draws his assumpion K 1 => Esimaed value for sandard deviaion: i, k1 i r i, K k 1 Divided by K, as no degree of freedom is los by esimaing he expeced value 7

13 Alernaively, he sandard deviaion can be esimaed exponenially. Exponenially means ha more weigh is pu on he more recen observaion days compared o days furher back (Keep in mind: so far, every day is weighed equally wih 1/K!). Esimaed value for he sandard deviaion wih i, =0 : i, i, 1 ( 1 ) ri, = Smoohing facor 0 1 Assumpion = 0.94 a RiskMerics 73

14 Example: DAX 007 Exp. Sandard Deviaion of Daily Reurns 0, ,014 0,01 0,01 0,008 0,006 0,004 0, Exp. sandard deviaion is srongly volaile => Considers insaionary naïve calculaion, as he sandard deviaion can be calculaed recursively from he previous day value and squared daily reurns 74

15 Equally Weighed vs. Exponenial Sandard Deviaion 0,014 exponenial sandard deviaion 001 0,01 0,01 0,008 0,006 0, ,00 equally weighed sandard deviaion The exp. sandard deviaion alernaes significanly sronger and is herefore closer a he marke han he equally weighed value Parially, here are huge differences! 75

16 c) Please calculae by neglecing he expeced value he esimaor for he hree sock s sandard deviaion using coninuously compounded reurns for 10/10/008. Apply he mehod of equal weighing and exponenial smoohing (for = 0.94). equal Sock A Sock B Sock C 10-Oc-008 exponenial Sock A Sock B Sock C 10-Oc

17 The correlaion coefficien ij, indicaes how much he reurn of sock i rises (falls), if he reurn of sock j rises (falls). Correlaion coefficien -1 ij, 1: K r r ) ( ) ( j i j i K K k j k j i k i ij r r r r, 1, 1,, 1,, ) ( ) ( ) ( ) ( k k j k j i k i r r 1 1, 1,, 1, ) ( ) ( 77

18 Correlaion Commerzbank - Deusche Bank 0,8 0,75 0, ,65 0,6 0,55 0, Correlaion on he basis of a 50-days observaion period 78

19 d) Now calculae he correlaion marix of he hree socks on dae 10/10/008, based on coninuously compounded reurns. Sock A Sock B Sock C Sock A Sock B Sock C 79

20 The VaR calculaion according o he Variance-Covariance-Mehod and he Mone-Carlo-Simulaion is based on he correlaion beween risk facors. J.P. Morgan provides he correlaion marix of he following facors daily via inerne [hp:// Foreign exchange raes from US-$ o 13 currencies Sock indices of counries Fuure prices of 11 commodiies wih up o en duraions ec... => In oal, over 400 volailiies and abou 75,000 correlaions are provided => For VaR calculaion, he correlaions and volailiies of he risk facor's reurn raes do no have o be calculaed by marke paricipans 80

21 Large rading porfolios usually consis of several housand posiions. The correlaions of all posiions could no be esimaed sufficienly by saisics. Moreover, he mahemaical effor would be inefficienly high. For insance, even for 5,000 rading posiions, (5,000 5,000 5,000)/ = 1,497,500 correlaions need o be calculaed! Resuls: Firs, rading posiions are divided ino several posiions, so called risk facors. Then, he correlaion marix for he risk facors is calculaed (wih much less effor). This procedure is called cash flow mapping. 81

22 5. Value a Risk as Insrumen for Risk Measuremen 5. Cash Flow Mapping Folien: Tanja Dresel, Luz Johanning,. Hans-Peer Burghof 8

23 Cash-Flow-Mapping wih socks All socks of a counry are described as a bea equivalen of a marke index (risk facor). Therefore, he volailiy of he dividend yield in resuls in:, i, i I, Bea-Facor: Ii, Covariance Index I - Sock i Ii, I, i, I, Variance of he index yield => The neglec of idiosyncraic risk is only jusified wih good diversificaion. => Bea facors mus be esimaed or aken from exernal agens as hey are no provided by J.P. Morgan. 83

24 Bonds,,Plain vanilla" bonds/loans/obligaions are repaid a mauriy and characerized by fixed ineres raes. Ineres paymen of fixed income bonds usually occurs annually in Germany, in oher counries predominanly half a year in arrears (e.g. in he USA) Generally, a bond has a fixed mauriy and redempion by he issuer is impossible. Cash Flow Mapping of a Bond Payback of face amoun + ineres a final mauriy Before mauriy, ineres is paid in fixed poins of ime 84

25 Cash-Flow-Mapping wih Ineres Raes RiskMerics provides a yield curve wih maximum 14 mauriy poins (zero bond raes). in % Zero-Rae 1 Mona 3 Monae 6 Monae 1 Jahr Jahre 3 Jahre 4 Jahre 5 Jahre 7 Jahre 9 Jahre 10 Jahre 15 Jahre 0 Jahre 30 Jahre A bond resp. cash flows from fixed income producs mus be spli ino zero bonds (risk facors) ha face a mauriy offered by RiskMerics. 85

26 If a bond generaes a cash flow (i.e. ineres paymen) afer 6 years, he ne presen value of he paricular cash flow (CF) has o be allocaed o zero bonds wih a mauriy of 5 and 7 years, because RiskMerics do no provide ineres raes for a mauriy of 6 years. Ne Presen Value of Cash Flows years 6 years 7 years... ime Quesion: By which proporion ( and1 ) has he ne presen value of he 6 year cash flow o be allocaed o he 5-year and 7-year zero bond? 86

27 Example: An invesor holds a bond whose ime o mauriy is exacly 10 years. The invesor wans o calculae he VaR of his posiion. In he following example, he ineres paymen in 6 years of 100 is supposed o be allocaed o 5-year and 7-year zero bonds. In, he following marke daa are observed: Zero bond rae for 5 years r 5 = % Zero bond rae for 7 years r 7 = % Sandard deviaion of a 5-year zero bond 5 = % Sandard deviaion of a 7-year zero bond 7 = % Correlaion of reurns of 5-year and 7-year zero bonds 57=

28 In case of cash flow mapping, hree condiions have o be considered: 1. The ne presen value of he ineres paymen in 6 years mus equal he sum of he 5 and 7-year zero bond's ne presen values. V 6 V5 V7. A long posiion (ineres paymen) is only allowed o be divided ino long posiions of zero bonds (=> equaliy of algebraic signs) 3. The risk (VaR) of he ineres paymen mus equal he risk (VaR) of he zero bonds. 88

29 Cash flow mapping happens in 5 seps: 1. Sep: Calculaion of he zero bond rae for 6 years by linear inerpolaion 6 5 (1 ) 7 => = 0.5 r6 r5 ( 1 ) r %. Sep: Calculaion of he ne presen value of he cash flow (CF) V CF (1 r ) 100 ( ) => have o be allocaed o 5 and 7-year zero bonds. 89

30 3. Sep: Calculaing he sandard deviaion of price reurns for 6-year posiions by linear inerpolaion of sandard deviaions for 5 and 7 years: 5 ( 1 ) % 4. Sep: Calculaing he proporions for allocaing he 6-year cash flow o 5 and 7-year zero bonds. => Implemening he proporions ( und 1 ) 90

31 The hird condiion requires ha he risk of he composed posiion maches he risk of he iniial posiions. Using he variance as risk measure, i follows ha: ha: ) (1 ) ( wih 0 a c b a c b a, b and c can be calculaed using he observed marke parameers. For his example he resuls are: c b a 91

32 As soluion we ge: b b 4 a c a From boh possible soluions, he one is chosen, which mees all of he hree condiions. Example: = 0,498 and 1 = 0, Sep: Allocaing he real cash flow o 5-year and 7-year zero bonds. => 0,498 67,86 = 33,81 o 5-year zero bond => 0, ,86 = 34,05 o 7-year zero bond 9