DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS


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1 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 winess work, and resoluion in wrongful deah cases, requires esimaion of he amoun of loss. Tha loss resuls from loss of income ha he deceased would have earned had he/she remained alive. The crucial modeling problem is he uncerainy of he fuure income, and he change of i resuling from deah or injury. In his work, we apply he heory of sochasic processes o his problem. We consider he case of a single paymen, and a series of level paymens, corresponding o common mehods of paymens of awards in loss liigaion. We esablish he discouning procedure for he value of fuure salary, and apply sochasic simulaion o deermine loss in case of wrongful deah. We consider discouning wih a single ineres rae, and discouning when ineres raes are modeled wih a sochasic process. We also show ha he models can be exended o injury as well as oher liabiliy cases. INTRODUCTION Liigaion of wrongful deah and injury resuls in an imporan problem of deermining he loss amoun o be compensaed. The loss considered usually refers o income loss of affeced wrongful deah or injury persons minus his (or her) consumpion. Because he income loss of affeced persons is dynamic, ha is o say, he income changes over ime, we mus consider he dynamic change of ineres rae and salary income comprehensively. In his aricle, we deermine he loss compensaed by quaniy analysis and esablishing dynamic model of income loss of wrongful deah. I should be noiced ha i is an imporan work o analyze he changing paern of salary rae wih ime and forecas he income loss ha should be compensaed. There are several aricles ha have sudied his aspec. [2] proposed parameer esimaion of salary funcion, [5] proposed log wage equaion and [3] invesigaed income model of auocorrelaion Y() a + by( 1) + e where Y() is he income, a and b are consans and e is random error a ime. He poined ou ha here exised some limiaion for auo correlaion model, because many economic ime series are nonsaionary. The resuls he analysis relaed change of he series daa of wage raes wih years suppored he assumpion of random walk model. Similar research was discussed by Horviz in [4]. He invesigaed he geomeric Brownian described by he following model Y() Y()e a 1 2 b2 +bz( ), where z() N(,). Anoher imporan issue is how o deermine he discoun value of income excep for forecasing he change paern of wage income wih he ime. In his aricle, we esablish dynamic compensaion models of wrongful deah based on he assuming ha discoun rae is a consan and a sochasic process respecively. We also assume ha he increase rae of wage is a sochasic process and here exiss correlaion beween he discoun rae and he increase rae of wage. The models esablished including he forms of a lumpsum paymen and level paymen. The calculaing mehods can be easily exended o he case of wrongful injury.
2 1. Models of a lump sum paymen Assuming ha he ineres rae is a consan Assume ha a person died in life hur a age. The compensaion paid o his legiimae successor should be equal o he presen value of his cumulaive income from he dae of his deah o he dae of his reiremen ha includes premium for reiremen annuiy minus his consumpion. Here we neglec he addiional paymen for pain and suffering o survivors. We also consider he probabiliy ha he dies in oher acciden or illness. Assume ha he increase rae of wage can be wrien as he following form [6]: dw W µ W d + σ W dz W (1) where µ W is he average wage rae, σ W is he sandard deviaion if wage rae and z W is a sandard Wiener process. Then we have W + µ Ws W s ds + σ Ws W s dz Ws (2) where W is he level of he wage of affeced person when wrongful deah even occurs. When µ W µ W and σ W σ W are consan he sochasic differenial equaion (1) has he unique soluion based on Io s Lemma W e µ W 1 2 σ 2 W +σ W z w (3) Assume ha he random variable T x expresses he survival ime if he affeced person aged x does no die in life hur. Le p x+ Pr ( T x < + 1) express he probabiliy ha a person aged x survives years. The values of p x+ can be found from he life ables. Here we also assume ha T x is a random variable ha is independen of he wage and he ineres rae. Assume ha he age of reiremen is rae is and he risk free ineres, hen he discoun facor calculaed by coninuous ineres rae can be expressed as. To simplify analysis, assume ha he consumpion for a person is h percen of his wage. Le P 1 be he presen value of compensaion of wrongful dearh wih a single paymen, hen P 1 E W (1 h) p x+ E where E denoes he expeced value. Le W (1 h)e(µ W 1 2 σ W 2 )+σ W z w p x+ e r W (1 h)eµ W p x+ e r, (4) e µ W p x+ e r k 1, (5) where k 1 is called as he facor of he compensaion of he wrongful deah wih a single paymen. By
3 combining equaion (5) wih (4), we obain he following equaion: P 1 (1 h)k 1 (6) Assuming ha he ineres rae is a sochasic process As he cumulaive calculaing erm of he compensaion for wrongful dearh is very long, i is very imporan o consider he undeermined characerisic of he ineres rae. I will affec he discoun value of he compensaion. Now we assume ha he real ineres rae can be expressed as he following sochasic differenial equaion: dr β(µ r r)d + σ r dz r (7) (see [1]) where dz r is a sandard Wiener process σ r is he sandard deviaion of ineres rae, µ r is he equilibrium ineres rae of long erm and β is he speed ha he real ineres rae recoveries o he equilibrium ineres rae of long erm. The facor of sochasic discoun calculaing wih coninuous ineres rae can be expressed as e r u du. Assume ha he increase rae of he wage saisfies wih equaion (3), he insan correlaion coefficien beween he wage and he ineres rae is ρ rw and oher assumpions are same as in secion 1. Then he presen value of he compensaion for wrongful deah is E W (1 h) p x+ e r u du}. (8) The paper [7] indicaes ha if he sochasic ineres rae r() follows he sochasic process given by equaion (7), he facor of sochasic discoun is e r u du A()e B( )r, where B() 1 e β β, A() exp (B() ) µ r 1 2 σ r 2 β 2 σ 2 B() r 4β, (9) Therefore, E W (1 h) p x+ A()e By combining equaion (3) wih equaion (1), we obained Le E E B( )r (1) W (1 h)e (µ W 1 2 σ 2 W )+σ W z w B( )r p A()e x+ (11) e (µ W 1 2 σ 2 W )+σ W z w p x B( )r A()e k (12) 2 where k 2 is referred as he facor of compensaion for wrongful deah when he ineres rae is a sochasic process. By combining equaion (12) wih equaion (13), we obain
4 Soluion of (13) is obained wih he help of Mone Carlo simulaion. (1 h)k 2 (13) 2. Models of compensaion for wrongful dearh wih level paymen in n years Because he way of a lump sum paymen needs obligaors o pay a lo of money one ime, someimes, i is difficul o redeem. One effecive mehod o solve his problem is o ake he form of periodical paymen. Assume ha he age he obligaor begins o pay he claim is y, p y+ expresses he probabiliy ha he obligaor aged survives years, and he value of periodical paymen paid a he beginning of each year is AP. Assuming he ineres rae is a consan The sum of presen value ha one dollar is paid a he beginning of each year wihin n years equals 1+ p y+ (14) Based on he principles of acuarial science, he value paid a he beginning of each year by he obligaor should be equal o he value paid a he form of a lump sum divided by he sum of presen value ha one dollar is paid a he beginning of each year wihin n years. Tha is AP 1 E W (1 h)e (µ W 1 2 σ W 1+ p y+ 2 )+σ W z w p x+ e r W (1 h)e µ W p x+ 1+ p y+ (15) Le e µ W p x+ 1+ p y+ k 3 (16) where k 3 is called as he facor of compensaion for he wrongful deah when he ineres rae is a consan. By combining equaion (15) and equaion (16), we obain AP 1 (1 h)k 3 (17) Assuming he ineres rae is a sochasic process The sum of presen value of one dollar paid a he beginning of each year wihin n years equals r u du} 1+ E p y+ e B( )r 1+ E p ( y+ A()e ) (18) Divided by equaion (18), he equaion (11) becomes
5 E A W (1 h)e (µ W 1 2 σ 2 W )+σ W z w B( )r p A()e x+ B( )r ( ) 1+ E p y+ A()e (19) Le E e (µ W 1 2 σ 2 W )+σ W z w B( )r p A()e x+ B( )r ( ) 1+ E p y+ A()e (2) where is called as he facor of compensaion for he wrongful deah when he ineres rae is a sochasic process. By combining equaion (19) wih equaion (2), we obained A (1 h). (21) 3. Calculaion of he compensaion for wrongful deah Assume β.1,σ r.15,σ W.1, µ r.35, µ W.1, ρ rw.2, 5, m 6. When he ineres rae is a consan, r.35. When he ineres rae is a sochasic process, he iniial level of he ineres rae r.3. The probabiliy ha he persons aged and survive years can be found from life ables. Table 1 and Figure 1 liss he values of he facors of he compensaion for he wrongful deah k 1,and calculaed wih he help of Mone Carlo simulaion and equaions of (5), (12), (16) and (2), where he deah age x Leing he iniial wage level when he affeced person dies W muliplied by (1 h) and by he facors of he compensaion for he wrongful deah k 1, and respecively, we can ge he values of he compensaion for he wrongful deah P 1,, AP 1 and A, where he meanings of k 1, and are illusraed in Table 2. Figure 2 is he descripion of he simulaion of sochasic processes when calculaing he facors of he compensaion for wrongful deah. Similarly, we can obain he values of he compensaion for he wrongful deah when he parameers ake oher values.
6 TABLE 1: THE VALUES OF k 1,k 2,k 3,and FOR DIFFERENT x VALUES Source: Auhors calculaion x k k k x k k k TABLE 2: FACTORS OF COMPENSATION FOR WRONGFUL DEATH UNDER DIFFERENT ASSUMPTIONS The ineres rae is a consan The ineres rae is a sochasic process A lump sum paymen Way of paymen Level paymen wihin n years k 1 k 3 k 2 FIGURE 1: THE VALUES OF FOR DIFFERENT VALUES Source: Auhors calculaions
7 FIGURE 2: STOCHASTIC SIMULATION OF THE FACTOR Source: Auhors calculaions Conclusions In his aricle, we apply he sochasic process heory and esablish he sochasic models of he compensaion for he wrongful deah wih a lump sum paymen and level paymen. We also calculae he facors of he compensaion for he wrongful deah k 1, and under various assumpions. Muliplying he facors of he compensaion for he wrongful deah by he difference beween he iniial level of he wage and consumpion of he dead person, we can ge he values of he compensaion for he wrongful deah. The modeling and calculaing process discussed above can be applied o he calculaion of compensaion for he injury case. REFERENCES [1] Babbel, D. F., and C. B. Merrill, IneresSensiive Coningen Claim Valuaion, Sociey of Acuaries, [2] Carriere, J. F. and Shand, K. J., 1998, New Salary Funcions for Pension Valuaions, Norh American Acuarial Journal, Vol. 2, No. 2, pp [3] Hosek, W. R, 1982, Problems in he Use of Hisorical Daa in Esimaing Economic Loss in Wrongful Deah and Injury Cases, Journal of Risk and Insurance, Vol.49, No.2, pp [4] Horviz, S. A, 1986, Implicaions of Projecing Fuure Losses of Earning Capaciy wih Deerminisic Models, Journal of Risk and Insurance, Vol. 53, No. 3, pp [5] Mincer, J. S., 1974, Experience and Earnings, New York: Naional Bureau of Economic Research. [6] Pennacchi, G. G, 1999,The Valuaion of Guaranees on Pension Fund Reurns, Journal of Risk and Insurance, Vol. 66, No. 2, pp [7] Vasicek, O. A., 1977, An Equilibrium Characerizaion of he Term Srucure, Journal of Financial Economics, Vol. 5, pp
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