Valuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate


 Ami Griffin
 2 years ago
 Views:
Transcription
1 Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology Sockholm
2 Acknowledgmens During he progress of he work on his hesis we have received a grea deal of help, suppor and guidance from a large number of people. Special hanks o Professor Boualem Djehiche as our supervisor a he deparmen of mahemaics, KH for valuable suggesions. Furhermore, we would like o express our graiude o our families. Finally, we would like o hank our girlfriends, Xin Chen and Liu Jiaxin for heir suppors.
3 Absrac We use Black and Scholes opions heory o obain marke valuaion of ypical life insurance conracs. he conracs are specified as in rosen & Loche Jorgensen (001), bu wih a sligh exension where he guaraneed rae is simulaed from marke models of shor ineres rae such as Vasicek, CoxIngesollRoss and HoLee models. Anoher exension is ha we assume he guaraneed rae is equal o he risk free ineres rae. Excep he above exensions, we keep oher facors of model similar as rosen & Loche Jorgensen (001). Firs, he liabiliy holders have prior claim for company asse han equiyholders. Second, a regulaory mechanism is added ino he model in order o reduce insolvency risk. Finally, we derive valuaion formulas and give numerical examples for iniial fair conracs and marke values of conracs a differen ime poins.
4 Conens 1 Inroducion 1 Basic Model 1 3 Conrac Specificaions 3.1 he liabiliyholders Claim 3. he Equiyholder s Claim 7 4 Valuaions 9 5 Numerical Examples and Comparison Fair Conracs Componens of Fair conracs Conrac componen prices 0 Appendix 31
5 1 Inroducion he fair valuaion of life insurance liabiliies has been discussed in he pas decades. One reason is how o give a suiable srucure of liabiliies conracs, which is also he main purpose of his paper. For example, mos insurance companies will promise o give policy holders an explici ineres rae added o heir accoun, named as guaraneed ineres rae. An unsuiable guaraneed rae will cerainly bring insolvency risk o insurance company. he purpose of his paper is o discuss he effecs of he variabiliy of he guaraneed rae on he insolvency risk, and we will make some exension in such poin. In our model, we assume ha he guaraneed rae is a deerminisic funcion, which means we can use Black and Scholes opion heory o derive price formulas. hen, we use shor rae models, such as Vasicek, CoxIngesollRoss, and HoLee models, o simulaed he above guaraneed rae and obain discree ime poin valuaions. Finally, we insered hem ino he formulas and obain corresponding valuaion resuls for his simulaed guaraneed rae. Furhermore, we will presen a reasonable connecion beween i and risk free rae. Anoher mechanism we will inroduce, which was also inroduced in he lieraures, is regulaory resricion. However, he difference here is ha he regulaory line is a random process, raher han deerminisic exponenial funcion as in rosen and Jorgensen (001). Finally, we provide a conclusion abou he valuaions of policyholders claims under he following ineres rae models: Vasicek model, CoxIngesollRoss model and HoLee model. Basic Model In his secion, we will presen a model o deal wih he problem discussed in he inroducion. he basic framework is based on Briys and de Varenne(1997), bu we es he shor rae and guaranee rae by using sochasic process. A =0, policyholders and equiyholders form a muual company, he life insurance company. he oal Asse is A 0, a fracion L 0 = αa 0 (0 α 1) is invesed by policyholders and E 0 = (1 α)a 0 is invesed by equiyholders. he parameer α is called he wealh disribuion coefficien. 1
6 able 1 Asses Liabiliies A 0 L 0 αa 0 E 0 (1 α)a 0 hen he agens acquire a claim for a payoff on or before he mauriy dae. herefore, hese claims are very similar o financial derivaives wih he company s asses as he welldefined underlying asse. Hence, firs, we will give a precise descripion of claims and hen use he powerful apparaus of coningen claims valuaion o price hem. Second, we will choose suiable parameers in he valuaion formula in order o form a fair conrac a =0, which means ha iniial invesmens equal o he ime zero marke value of he claims. 3 Conrac Specificaions In his secion we describe he deails of he sakeholders claims on he company s asses, as suggesed in Briys & Varenne(1997) and Loche & Jorgensen(001). We begin wih a specificaion of he liabiliyholders claim. 3.1 he liabiliyholders Claim As menioned in he inroducion, he life insurance company promises he liabiliyholders a coninuously compounded reurn on he iniial marke value of he liabiliies of r during he life of he conrac: L = L 0 e 0 r (s)ds a ime. Furhermore, anoher imporan assumpion in our model is ha he risk free rae equals o ineres rae guaranee: r r. In Figure1, r is simulaed from he Vasicek shor rae models: dr = a (br ) d+σ r dw, (1) where we fix he parameers a he following values a=0.5, b=0.06, σ r = 0.0, r0=0.04. hese values imply a posiively sloped curve rising from 4% owards
7 approximaely 6% as ime o mauriy ends o infiniy. Figure 1 his ranslaes ino a guaraneed final paymen of L = L 0 e 0 r (s)ds. When here are no any exernal guaranors, he company s promise can be realized only if i urns ou ha A > L a ime. In he opposie case, A L, and in he even ha he company has no been declared insolvey by regulaors, he policyholders receive A and he equiyholders receive nohing. Besides, in addiion o he promised mauriy paymen L, liabiliyholders are also generally given bonus if he marke value of he asses evolves sufficienly favorable. δ[αa L ] () From () i is clear ha in final saes where he policyholders share of oal value A exceeds he promised paymen of L hey will receive a fracion, δ, of his surplus. he parameer δ is called as he paricipaion coefficien and should be in he inerval [0 1]. 3
8 o sum up, he oal mauriy payoff o policyholders, Ψ L (A ), can be described as or A, A < L, Ψ L (A ) = L, L <A < L L +δ[αa L ], A > L α, α, (3) Ψ L (A )= δ[αa L ] + +L +[L A ] +. (4) he firs par of he righhand side has already been named as he bonus opion. he wo remaining erms are fixed mauriy paymen and shored pu opion respecively. Figure is a graphical illusraion of (4), where r is given by he Vasicek model (1). Figure 4
9 In he nex sep, we will impose a regulaory resricion. echnically, suppose ha in he framework above, he asse mus evolve above a cerain level: A > λl 0 e 0 r (s)ds B (5) he curve{b()} 0, will be called he regulaory boundary. he purpose of adding he regulaory boundary is o deermine an absolue lower bound of he asses, under which he company is declared insolvency: L 0 e 0 r (s)ds he policyholders iniial deposi compounded wih he guaraneed rae of reurn up o ime. herefore, only if ha he oal asses A a all imes have been sufficien o cover L 0 e 0 r (s)ds opions keep evolving. muliplied by some prerequisie consan, λ, will he sakeholders In he opposie even, asses will a some poin in ime,τ, equal o B τ, A τ = B τ. In his siuaion regulaory auhoriies close down he company immediaely and disribue he wealh o sakeholders. A his poin we noe ha here are wo ineresing cases. For λ 1 and in he even τ of a boundary hi, liabiliyholders receive L 0 e 0 r (s)ds. A he same ime τ equiyholders receive (λ 1)L 0 e 0 r (s)ds. Conversely, for λ < 1 and in he even of τ a boundary hi, liabiliyholders receive λl 0 e 0 r (s)ds. A he same ime equiyholders receive nohing. he various siuaions are illusraed in he figures below. is 5
10 Figure 3a For λ < 1 Figure 3b For λ 1 6
11 As described above here will be a rebae o liabiliyholders in he even of premaure closure a ime τ. his rebae, Θ L (τ), is given as τ L 0 e 0 r (s)ds, λ 1, Θ L (τ)= (6) τ λl 0 e 0 r (s)ds, λ < 1, or τ Θ L (τ)= (λ 1) L 0 e 0 r (s)ds. (7) Before considering he valuaion of claim, (Ψ L (A ),Θ L (τ)), we describe he deails regarding he equiyholders claim. 3. he Equiyholder s claim As residual claimans, equiyholders will receive a payoff a he mauriy dae condiional on no premaure closure as follows: 0, A < L, Ψ E A = A L, L <A < L A L δ[αa L ], A > L α. α, (8) or Ψ E A = [A L ] + δ[αa L ] +. (9) he above payoff funcion is depiced in Figure 4 below. 7
12 Figure 4 As described above here will be a rebae o equiyholders in he even of echnical insolvency a ime τ. his rebae, Θ E (τ), is given as τ (λ 1)L 0 e 0 r (s)ds, λ 1, Θ E (τ) = (10) 0, λ < 1. or τ Θ E (τ)= [(λ 1) 0] L 0 e 0 r (s)ds. (11) 8
13 4 Valuaions o hese conracs we apply he basic framework of Black and Scholes (1973) where all aciviy occurs on a probabiliy space (Ω, F, F, P). Under such assumpion, he dynamic evoluion of he asses is described by he sochasic differenial equaion: da =μa d + σa dw p (1) where μ and σ are consans and {W p } is a sandard Brownian moion under P. iven he riskless ineres rae, r, we have da = r A d + σa dw Q where {W Q } is a sandard Brownian moion under he riskneural probabiliy measure Q. Here,V i A,, i = L, E, denoe he ime value of he liabiliy and equiyholders claims respecively. We can wrie (see e.g. Ingersoll (1987), p ) (13) V i A, = E Q [e r (s)ds Ψ i A 1 τ + e τ r s ds Θ i (τ)1 τ< F ]= e r (s)ds Ψ i (A ) 0 f(a, ; A, )da + e τ r (s)ds Θ i (τ) g(τ; A, )dτ (14) where we assume r in he above formula a deerminisic funcion and he following heorems are also based on he assumpion. Moreover, f(a, ; A, ) denoes he riskneural densiy for he value of he asses a ime wih an absorbing barrier B 0. Similarly, g(τ; A, ) denoes he riskneural firs hiing ime densiy of asse hrough he absorbing barrier condiional on he posiion A a ime. he explanaion of hese wo densiies is deferred o he Appendix. 9
14 heorem 1. We have he following value of he Liabiliyholders Claim Ψ L (A ): A bonus (call) opion elemen, a fixed paymen, and a shored pu opion elemen. If a premaure barrier is hi here will be a rebae paymen specified by Θ L (τ) in (6). V L A, is he sum of he ime value of hese four elemens as given below in pars IIV where we used he noaion: d γ (x, )= ln x + (r(s) γ 1 σ )ds, σ ( ) I: he ime value of he bonus (call) opion elemen is δα{a N d o + ( A ) [ B + N(d B A o A X, L α e B A X, ) L r(s)ds α e r s ds + N(d o N(d 0 ( A X, )) B A X, )]}. where, X L (λ 1 α ). II: he ime value of he condiional fixed paymen elemen is L e r(s)ds N d r A B, A B N d r B A,. III: he ime value of he shored pu opion elemen is 1 λ<1 L e r(s)ds A N( d 0 L, N d r A B, A N( d 0 + A L, N d + A r, ] B L e A B r(s)ds B N( d 0 A L, N d r B A, + B N( d A 0 A L, B N d + B r, ]}}. A IV: he ime value of he rebae is (λ 1) + A A λ N d r, A 1 + B N d B B r,. A he proof relies on Black & Scholes formula and is presened in he Appendix. 10
15 heorem. We have he following value of he Equiyholders Claim Ψ E (A ): One residual claim call opion and anoher shored bonus opion. if a premaure barrier is hi here will be a rebae paymen specified by Θ L (τ) in (10). V L A, is he sum of he ime value of hese hree elemens as given below. I. he ime value of he long residual claim (call opion) elemen is A B A N d o + B A N d o + A Y, L α e B A Y, L e r(s)ds N(d 0 ( A Y, )) r s ds N d o B A Y,, where, Y = (B L ). II: he ime value of he shored bonus opion is given in par I of heorem 1. III he ime value of he equiy holders rebae is ((λ 1) 0) + A A λ N d r, B A B 1 + B N d r,. A Proof of heorem. Par I is esablished by seing δ = α = 1 in Par I of heorem 1. Par II is precisely Par I of heorem 1 wih he sign reversed. Par III ress on calculaions similar o hose ha esablished Par IV of heorem 1. 5 Numerical Examples and Comparisons 5.1 Fair Conracs he formulas for he values of he liabiliy and equiy claims are closed formulas ha can be calculaed once he relevan parameers are given. Jus as discussed in he inroducion par, i is clear ha no every choice of parameers will represen fair conracs. So he firs quesion o ask is which 11
16 combinaions of parameers will represen fair conracs. A fair conrac should saisfy he iniial equaion: L 0 = αa 0 = V L (A 0, 0; α, δ, λ, σ, r, r ), he equaion is obvious because he liabiliy holders iniial conribuion o he oal asses L 0, should equal o he iniial marke value of coningen claim. Anoher equaion is E 0 = 1 α A 0 = V E A 0, 0; α, δ, λ, σ,, r, r. Nex, we provide some seleced represenaive plos o illusrae some ypical relaions beween parameers of iniially fair conracs, where r=r are simulaed from differen shor ineres rae models such as Vasicek, CIR and HoLee. his exends he work by rosen & Jorgensen(001), who considered he case where r and r are held consan Using Vasicek model Figures 5 illusraes he relaion beween fair values of he paricipaion coefficien, δ, and he wealh disribuion coefficien, α, for some fixed and represenaive values of he remaining parameers. I is noed ha all hese graphs are negaively sloped as a higher wealh disribuion coefficien will be associaed wih a lower paricipaion coefficien in order for he conrac o be fair o boh sides (noe r and r are simulaed from random process and valuaions are averages in hese examples). Figure 5a 1
17 Figure 5b 5.1. Using CIR model Figure 6 and Figure 7 illusrae similar comparison under CIR ineres rae model and HoLee model. We find he resuls are lile differen when we use hree differen ineres rae models: Cox_ingersoll_Ross shor rae model: dr = a(br)d+σ r rdw Here we se parameers as a=0.5, b=0.05, σ r = 0.0 and r 0 =0.1. Under such a seing, he shor rae will sar from r 0 =10% oward o 5% when ime is infinie. he purpose of such an iniial parameer seing is o make shor rae under his model is much more differen o Vasicek model. However, afer careful comparison, we find ha he difference of comparison among hree models is negligible. Anoher hing we wan o menioned here is ha we will keep he above parameer seing in he CIR model when we compare he difference among models in secion 5. and secion
18 Figure 6a Figure 6b 14
19 5.1.3 Using HoLee model HoLee ineres rae model: dr=θ d+σ r dw, Here we se parameers as: Θ = 0.05e, σ r =0.0 and r 0 = 0.1 Under such a seing, he shor rae will sar from r 0 =10% oward o 5% when ime is infinie. he purpose of such an iniial parameer seing is o make shor rae under his model is much more differen o Vasicek model bu similar as CoxIngersollRoss model. We wan o menion is ha we will keep he above parameer seing in he HoLee model when we compare he difference among models in secion 5. and secion 5.3 Figure 7a 15
20 Figure 7b 5. Componens of Fair conracs In Secion 4 we derived value formulas for each of he componens. he following able shows some examples of how he oal conrac value decomposes ino he separae elemens. We will find ha he resuls are lile differen when we use hree differen ineres rae models Using Vasicek model Le r=r be he values simulaed from he following Vasicek shor rae model: dr = a (br)d+σ r dw, r=r. able : Decomposiions of Conrac Values a=0.5, b=0.06, σ r = 0.0, r0=0.04 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) 16
21 λ λ = λ = λ = λ = λ = λ =1.5 All All From he able several ineresing observaions can be made. For example: If volailiy is increased, δ mus be increased o mainain a fair value disribuion A larger volailiy ends o increase he value of he rebae elemen and o decrease he value of he condiional fixed paymen. his is, of course, explained by he fac ha generally a larger volailiy is associaed wih a larger probabiliy of an early barrier hi. As expeced, he value of he equiy rebae elemen is nil when λ 1 and posiive when λ > 1. 17
22 5.. Using CIR model Here, Le r=r be he values simulaed from he following Cox_ingersoll_Ross shor rae model: dr = a(br)d+σ r rdw able 3: a=0.5, b=0.05, σ r = 0.0 and r 0 =0.1 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) λ λ = λ = λ = λ = λ = λ =1.5 All All
23 When Lambda is small he volailiy is large, when Sigma is small he volailiy is large Using HoLee model Here, Le r=r be he values simulaed from he following HoLee shor rae model: dr=θ d+σ r dw able 4: Θ = 0.05e, σ r =0.0 and r 0 = 0.1 A 0 = 100, r = r, α = 0.8, = 0 σ δ BO SP CFP RL V L (A 0, 0) RC SBO RE V E (A 0, 0) λ λ = λ = λ = λ =
24 λ = λ =1.5 All All When Lambda is small he volailiy is large, when Sigma is small he volailiy is large HoLee model has a higher volailiy 5.3 Conrac componen prices I is an imporan propery of our model ha i can idenify fair conracs for a given se of iniial condiions. However, i is equally imporan ha he model can price conracs and heir consiuing elemens a any given poin in ime given he iniially specified erms. Now, we presen he following numerical examples in which he values of he componens of he liabiliy holders conracs as a funcion of he sae variable, A, a differen imes during he life of he conrac. he conrac parameers have been se so ha he conrac was fair a =0 and he conrac elemens values are ploed for =0.1 (righ afer incepion), =10, =19 and =0(see rosen & Jorgensen (001)). We will find ha he resuls are lile differen when we use hree differen ineres rae models Using he Vasicek model he corresponding r : 0
25 Figure 8a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 1
26 Figure 8b Figure 8c
27 Figure 8d Figure 8e 3
28 For each ime =0.1,10,19,0, we assume A sar from corresponding B o 450. For differen simulaed values of r, he corresponding prices are differen a ime =0.1,10,19.0. o obain fair prices a differen ime, we should simulae r sufficien many imes and choose averages Using he Cox_Ingersoll_Ross model he corresponding r : Figure 9a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 4
29 Figure 9b Figure 9c 5
30 Figure 9d Figure 9e 6
31 5.3.3 Using he HoLee model he corresponding r : Figure 10a oal Liabiliy Value Bonus call opion Condiional Fixed Paymen Shored pu opion Rebae 7
32 Figure 10b Figure 10c 8
33 Figure 10d Figure 10e 9
34 Reference [1] Briys, E. and F. de Varenne (1994): Life Insurance in a Coningen Claim Framework: Pricing and Regulaory Implicaions, he eneva Papers on Risk and Insurance heory, 19(1):53 7 [] Briys, E. and F. de Varenne (1997): On he Risk of Life Insurance Liabiliies: Debunking Some Common Pifalls, Journal of Risk and Insurance, 64(4): [3] rosen, A. and P. L. Jørgensen (1997): Valuaion of Early Exercisable Ineres Rae uaranees, Journal of Risk and Insurance, 64(3): [4] Han, L.M.,. C. Lai, and R. C. Wi (1997): A FinancialEconomic Evaluaion of Insurance uarany Fund Sysem: An Agency Cos Perspecive, Journal of Banking and Finance,1(8): [5] Bauer, D., Kiesel, R., Kling, A., Ruß, J., (006). Riskneural valuaion of paricipaing life insurance conracs. Insurance: Mahemaics and Economics 39, [6] Cox, J.C., Ingersoll, J.E., Ross, S.A., (1985). A heory of he erm srucure of ineres raes. Economerica 53, [7] Hull, J.C., (006). Opions, Fuures, and oher Derivaives, 6h ed. PreniceHall, Englewood Cliffs, NJ. [8] Vasicek, O., (1977). An equilibrium characerizaion of he erm srucure. Journal of Financial Economics 5, [9] Bjork,., (004): Arbirage heory in Coninuous ime, nd (Oxford Universiy Press). 30
35 Appendix he process of he liabiliy holders claim valuaion In his appendix we presen a series of corollaries each of which can lead o he heorem which gives he closedformula for fair value of every componen of he liabiliy holders claim in main conex. We begin o research on he ime valuaion of he mauriy paymen elemens. Here mos mehods are based on rosen and Jørgensen (00). Firs refer o rosen and Jørgensen (00) who show he derivaion of he defecive ransiion densiy for Brownian moion wih drif and an absorbing barrier a he origin in appendix A. Here we use heir resuls direcly. Namely: f(z, ; z, ) = 1 σ ( ) {N(Z z μ( ) )e z μ σ σ ( ) N( Z+z μ( ) )} (C.1) σ ( ) Noe ha he above densiy funcion is for Brownian moion of he form like zu ( u ) ( Wu W ) z Recall ha we have A u and he exponenial barrier: u 1 rs ds u Wu W A e, u u u u r ds r ds r ds r ds 0 0 u 0 0 B B e B e L e L e, u, (A.1) hrough he whole aricle we assume ha a he valuaion dae he barrier has no ye been reached i.e A B.Furher we le ha r r are deerminisic funcion. Corollary 1 he bonus (call) opion elemen: E e A L 1 r ds rs ds Q e E Q A L s = 1 rs ds =, ;, e A L f A A da B = e rs ds L ( A L ) f ( A, ; A, ) da (A.) = { AN ( B ) X A X 1 ( ) ln rs ds ( ) 31
36 A 1 ln rs ds ( ) L rs ds e N X ( ) B / 1 / A B A 1 ln ( ) A s ln s ( ) r sds B r ds r ds X L N e N X B A ( ) ( ) where f A, ; A, is he densiy of A wih an absorbing barrier Bu as given in (A.1). } Proof of Corollary 1: We firs perform a change of variables in order o be able o use he resul (C.1). Noe firs ha A B u u u 1 u rs ds uwu W r ds B e A e 1 B r ds u W W r ds u u s u ln A, since we choose r r 1 u W W A ln 0 u B Hence, passage of A() hrough B() is equivalen o passage of he Brownian moion 1 A A Zu u Wu W ln hrough zero. Define now z ln and noe ha: B B L A ( B ) X z rs dsz A e X X z ln z r sds A X z ln r sds q B herefore we can wrie (A.) as 3
37 αe r s d s q B e z + r s d s L α f(z, ; z, ) d z where f ( z, ; z, ) is he defecive densiy of Brownian moion wih drif and absorbing barrier a zero. Subsiuing (C.1) for his, he res is jus edious calculaions. Noe ha 1 he drif is in our formula. So we ge: αe r s d s q B e z + r s d s L α f(z, ; z, ) rs ds z rs ds L = e ( B e ) q 1 1 Z z ( ) ( ) 1 Z z 1 z n e n ( ) ( ) ( ) ( ) d z 1 ( ) 1z z ( ) rs ds rs ds z 1 { q = e B e e dz ( ) 1 ( ) 1z z ( ) 1 L e q ( ) dz 1 ( ) 1z z ( ) rsds z 1 q z e B e e dz ( ) 1 ( ) 1z z ( ) 1 L z e e dz q ( ) 1 q z ( ) rsds rsds = e { Ae N ( ) 1 q z ( ) L N ( ) 1 q z ( ) z B rs ds e e N A ( ) } 33
38 1 q z ( ) L e z N } ( ) A 1 ln rs ds ( ) = { AN X ( ) A 1 ln rs ds ( ) L rs ds e N X ( ) B / 1 / A B A 1 ln ( ) A s ln s ( ) r sds B r ds r ds X L N e N X B A ( ) ( ) } Corollary he condiional fixed paymen elemen As a par of he mauriy payoff, he liabiliy holders receive L on he condiion ha he knockou barrier has no been hi. A ime wih A B and r r are deerminisic his payoff elemen is valued as follows: rs ds EQ e L 1 rs ds = e E L 1 Q B rs ds =, ;, L e f A A da A 1 A 1 ln ( ) ln ( ) rs ds B A B = L e N N ( ) B ( ) Proof of Corollary : Changing variables as in Corollary 1 and using (C.1) yields f A, ; A, da B =, ;, 0 f z z dz 34
39 1 1 z ( ) ( ) 1 z z 1 z = z n e n dz 0 ( ) ( ) ( ) ( ) A 1 A 1 ln ( ) ln ( ) B A B = N N ( ) B ( ) which esablishes he resul. _ Corollary 3 he pu opion elemen Noe firs ha his payoff elemen can only be sricly posiive when 1, i.e. when he u r ds 0 barrier lies below he curve Le 0, u [0, ]. he ime value of he pu opion elemen is given as follows: rs ds 1 1 EQ e L A 1 r ds 1 e E L A 1 s = 1 Q rs ds = 1 1 e L A f ( A, ; A, ) da B s L = r ds 1 1 e L A f ( A, ; A, ) da (A.3) = 1 1 { B L 1 B 1 ln ( ) ln ( ) r sds rs ds A A L e N N ( ) ( ) L 1 B 1 ln rs ds ( ) ln ( ) A A A A N N ( ) ( ) B L A 1 A 1 ln ( ) ln ( ) r sds rs ds B B [ L e N N / ( ) ( ) 35
40 L A 1 A 1 ln ( ) ln ( ) r sds B B B N N ]} A ( ) ( ) Proof of Corollary 3: Change variables as before. Firs A B implies ha z 0.Also we should have A L z rs dsz Ae L L z ln z r sds A z L ln B r ds L seing q ln r sds B we can rewrie (A.3) as = s rs ds q s z r dsz 1 1e L (, ;, ) 0 A e f z z dz rs ds q s z r ds 1 e L 0 Be 1 ( ) 1 1 Z z ( ) ( ) 1 Z z 1 z n e n ( ) ( ) ( ) ( ) =. = 1 1 ( ) 1z z ( ) rs ds 1 q 1 e { L e dz ( ) 1 ( ) 1z z q ( ) z rsds 1 0 B e e dz ( ) 36
41 1 ( ) 1z z ( ) q z 1 e ( L 0 e dz ( ) 0 q 1 ( ) 1z z ( ) rsds z 1 Be e dz ( ) 1 = 1 e rs ds L 1 B 1 ln ( ) ln ( ) r sds rs ds A A { L e N N ( ) ( ) 1 1 ( ) ( ) r sds q z z A e N N ( ) ( ) 1 1 q z ( ) z ( ) z e ( L N N ( ) ( ) 1 1 ( ) ( ) rs ds B q z z e N N )} A ( ) ( ) L 1 B 1 ln ( ) ln ( ) r sds rs ds A A 1 { L e N N ( ) ( ) L 1 B 1 ln rs ds ( ) ln ( ) A A A A N N ( ) ( ) B L A 1 A 1 ln ( ) ln ( ) r sds rs ds B B [ L e N N ( ) ( ) = 1 )} 37
42 L A 1 A 1 ln ( ) ln ( ) r sds B B B N N ]} A ( ) ( ) Corollary 4 (he rebae) he ime value of he rebae paymen o he liabiliy holders assuming A B is rs ds EQ e 1 L 1 e EQ 1 L 1 rs ds = 0 = s r ds rds 1 0 ( ;, ) (A.4) e L e g z d A 1 A 1 ln ( ) ln ( ) 1 B B = A N B N ( ) ( ) where g( ; z, ) is he firs hiing ime densiy which will be derived below. Proof of Corollary 4 Firs we would like o presen he firs passage ime densiy of geomeric Brownian moion hrough an exponenial barrier: Since A u and he exponenial barrier: u 1 rs ds u Wu W A e, u u u u r ds r ds r ds r ds 0 0 u 0 0 B B e B e L e L e, u, A 1 Le denoe he smalles u such ha Au Bu.Se ln z and.do B exacly he same calculaion as rosen and Jørgensen (00) did in appendix B. We can esablish ha z z ( ) g( ; z, ) n (C.) ( ) 3 Now subsiuing for g( ; z, ) in (A.4) we ge 38
43 1 z ( )( ) rsds rds z 1 ( ) 3 1 L e e e d =1 = 1 z ( )( ) rsds rds z 1 ( ) 3 L e e e d A 1 L g ( ; z, ) d B 1 B B = A 1 A 1 ln ( ) ln ( ) A N B N ( ) ( ) 39
PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees.
The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling 1 Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationThe Impact of Surplus Distribution on the Risk Exposure of With Profit Life Insurance Policies Including Interest Rate Guarantees
1 The Impac of Surplus Disribuion on he Risk Exposure of Wih Profi Life Insurance Policies Including Ineres Rae Guaranees Alexander Kling Insiu für Finanz und Akuarwissenschafen, Helmholzsraße 22, 89081
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationAnalyzing Surplus Appropriation Schemes in Participating Life Insurance from the Insurer s and the Policyholder s Perspective
Analyzing Surplus Appropriaion Schemes in Paricipaing Life Insurance from he Insurer s and he Policyholder s Perspecive Alexander Bohner, Nadine Gazer Working Paper Chair for Insurance Economics FriedrichAlexanderUniversiy
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationABSTRACT KEYWORDS. Term structure, duration, uncertain cash flow, variable rates of return JEL codes: C33, E43 1. INTRODUCTION
THE VALUATION AND HEDGING OF VARIABLE RATE SAVINGS ACCOUNTS BY FRANK DE JONG 1 AND JACCO WIELHOUWER ABSTRACT Variable rae savings accouns have wo main feaures. The ineres rae paid on he accoun is variable
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationA TwoAccount Life Insurance Model for ScenarioBased Valuation Including Event Risk Jensen, Ninna Reitzel; Schomacker, Kristian Juul
universiy of copenhagen Universiy of Copenhagen A TwoAccoun Life Insurance Model for ScenarioBased Valuaion Including Even Risk Jensen, Ninna Reizel; Schomacker, Krisian Juul Published in: Risks DOI:
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationDifferential Equations in Finance and Life Insurance
Differenial Equaions in Finance and Life Insurance Mogens Seffensen 1 Inroducion The mahemaics of finance and he mahemaics of life insurance were always inersecing. Life insurance conracs specify an exchange
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationPricing BlackScholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension
Pricing Blackcholes Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zulang Liao a, and HsingHua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationOptimal Consumption and Insurance: A ContinuousTime Markov Chain Approach
Opimal Consumpion and Insurance: A ConinuousTime Markov Chain Approach Holger Kraf and Mogens Seffensen Absrac Personal financial decision making plays an imporan role in modern finance. Decision problems
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationOn the Management of Life Insurance Company Risk by Strategic Choice of Product Mix, Investment Strategy and Surplus Appropriation Schemes
On he Managemen of Life Insurance Company Risk by raegic Choice of Produc Mix, Invesmen raegy and urplus Appropriaion chemes Alexander Bohner, Nadine Gazer, Peer Løche Jørgensen Working Paper Deparmen
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationIMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION. Tobias Dillmann * and Jochen Ruß **
IMPLICIT OPTIONS IN LIFE INSURANCE CONTRACTS FROM OPTION PRICING TO THE PRICE OF THE OPTION Tobias Dillmann * and Jochen Ruß ** ABSTRACT Insurance conracs ofen include socalled implici or embedded opions.
More informationFair Valuation and Risk Assessment of Dynamic Hybrid Products in Life Insurance: A Portfolio Consideration
Fair Valuaion and Risk ssessmen of Dynamic Hybrid Producs in ife Insurance: Porfolio Consideraion lexander Bohner, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichlexanderUniversiy
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationResearch Article Optimal Geometric Mean Returns of Stocks and Their Options
Inernaional Journal of Sochasic Analysis Volume 2012, Aricle ID 498050, 8 pages doi:10.1155/2012/498050 Research Aricle Opimal Geomeric Mean Reurns of Socks and Their Opions Guoyi Zhang Deparmen of Mahemaics
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationEquities: Positions and Portfolio Returns
Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi
More informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: ConinuousTime Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationDoes Option Trading Have a Pervasive Impact on Underlying Stock Prices? *
Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a UrbanaChampaign Allen M. Poeshman Universiy of Illinois a UrbanaChampaign Joshua Whie Universiy
More informationRISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION. 1. Introduction
RISKSHIFTING AND OPTIMAL ASSET ALLOCATION IN LIFE INSURANCE: THE IMPACT OF REGULATION AN CHEN AND PETER HIEBER Absrac. In a ypical paricipaing life insurance conrac, he insurance company is eniled o a
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationBlack Scholes Option Pricing with Stochastic Returns on Hedge Portfolio
EJTP 3, No. 3 006 9 8 Elecronic Journal of Theoreical Physics Black Scholes Opion Pricing wih Sochasic Reurns on Hedge Porfolio J. P. Singh and S. Prabakaran Deparmen of Managemen Sudies Indian Insiue
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationPRICING AND PERFORMANCE OF MUTUAL FUNDS: LOOKBACK VERSUS INTEREST RATE GUARANTEES
PRICING AND PERFORMANCE OF MUUAL FUNDS: LOOKBACK VERSUS INERES RAE GUARANEES NADINE GAZER HAO SCHMEISER WORKING PAPERS ON RISK MANAGEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAGEMEN
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationCarbon Trading. Diederik Dian Schalk Nel. Christ Church University of Oxford
Carbon Trading Diederik Dian Schalk Nel Chris Church Universiy of Oxford A hesis submied in parial fulfillmen for he MSc in Mahemaical inance April 13, 29 This hesis is dedicaed o my parens Nana and Schalk
More informationA Universal Pricing Framework for Guaranteed Minimum Benefits in Variable Annuities *
A Universal Pricing Framework for Guaraneed Minimum Benefis in Variable Annuiies * Daniel Bauer Deparmen of Risk Managemen and Insurance, Georgia Sae Universiy 35 Broad Sree, Alana, GA 333, USA Phone:
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationDynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint
Dynamic Hybrid Producs in Life Insurance: Assessing he Policyholders Viewpoin Alexander Bohner, Paricia Born, Nadine Gazer Working Paper Deparmen of Insurance Economics and Risk Managemen FriedrichAlexanderUniversiy
More informationStochastic Volatility Models: Considerations for the Lay Actuary 1. Abstract
Sochasic Volailiy Models: Consideraions for he Lay Acuary 1 Phil Jouber Coomaren Vencaasawmy (Presened o he Finance & Invesmen Conference, 191 June 005) Absrac Sochasic models for asse prices processes
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationAgnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/2010085
Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/2010085 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen
More information