In this section, we first provide brief descriptions of the mortality model and related assumptions. Then we describe the proposed hedging strategy using life settlements. A life insurer that sells whole life insurance to senior people should consider an investment plan to hedge the mortality risk since hedging the risk through the liability side itself may be infeasible.
Suppose that insurer’s product portfolio contains m whole life contracts and the life expectancy of the life contract j is τj
, j = 1, 2, …, m. Further assume that the insurer may
purchase n senior life settlements that have life expectancy ti (i = 1, 2, …, n) with the current valueV
i( t
i)
. According to Stone and Zissu (2006; 2008), the values of a life settlement when considering the life extension or contraction from expectancy Ti can be expressed asfollows:
𝑉𝑉𝑖𝑖(𝑡𝑡𝑖𝑖+ 𝑇𝑇𝑖𝑖) ≈ 𝑉𝑉𝑖𝑖+ 𝑑𝑑𝑖𝑖𝑇𝑇𝑖𝑖+ 12 𝑐𝑐𝑖𝑖𝑇𝑇𝑖𝑖2, 𝑖𝑖 = 1, 2, … , 𝑛𝑛 (1)
The constants di and ci represent modified life-extension7 duration (le-duration) and life-extension convexity (le-convexity), respectively.
Stone and Zissu (2006; 2008) consider only a “static” life extension and thus assume that all life settlements have the same life extension. However, the above assumption may
7 For the convenience of the reading, we shorten the term “life extension or contraction from expectancy” to
“life extension”.
8
not be appropriate when comparing the mortality risk between life settlements and insurance policies. When measuring the hedging effect between life settlements and the insurer’s liabilities associated with life insurance contracts, we should further consider the variation in life extensions in various life settlements and insurance contracts. Therefore, we model the future lifetimes (t1+T1, t2+T2, …,
t
n+T
n) and (τ
1,τ
2, …,τ
m) using random vectors with known marginal distributions8. This setting enables us to have the mortality tables corresponding to each life settlement and insurance contract. More precisely, the morality tablecorresponding to life settlement i describes the marginal distribution of ti +Ti, and the morality table for insurance contract j describes the marginal distribution of
τ
j.The marginal distribution function of ti+Ti is denoted by Fi(.) and their joint behaviors are described by a normal factor copula9 (see Burtschell, Gregory, and Laurent, (2009); Hull, (2011). Burtschell, Gregory, and Laurent, (2009) select normal factor copula to model the dependence structure of times to the defaults of bonds in a credit portfolio. Following their idea, we select normal factor copula to model the dependence structure of times to the deaths of policy holders in a life settlement pool in this paper. In particular, we assume that
𝑇𝑇𝑖𝑖 = 𝐹𝐹𝑖𝑖−1�𝑁𝑁(𝑋𝑋𝑖𝑖)� − 𝑡𝑡𝑖𝑖, i = 1, … , n, (2)
8 We do not make any special assumptions of the marginal distribution of ti+Ti and τj. For valuation purpose, the marginal distribution of ti+Ti is based on the mortality table provided by medical underwriters and the marginal distribution of τj is based on insurers’ internal mortality table.
9 As suggested by Asmussen and Glynn (2007), copulas provide a possible approach for modeling multivariate distributions in which one has a rather well-defined idea of the marginal distributions but a rather vague one of the dependence structure.
9
where N(.) is the cumulative distribution function of the standard normal random variable and
X
i are the latent variables used to model the joint distributions of Ti. The correlations amongX
i are induced through common factors M and Mls as follows:𝑋𝑋𝑖𝑖 = 𝑎𝑎𝑎𝑎 + 𝑏𝑏𝑎𝑎𝑙𝑙𝑙𝑙+ √1 − 𝑎𝑎2− 𝑏𝑏2𝑍𝑍𝑖𝑖, 𝑖𝑖 = 1, 2, … , 𝑛𝑛, (3)
where M, Mls, Z1,…., Zn are independent standard normal random variables and a, b denote constant factor loadings. The common factor M represents the global trend in age
improvement while Mls reflects improvement trend of the life settlement pool10. On the other hand, Z1,…., Zn are specific factors pertaining to each life settlement.
The marginal distribution function of
τ
j is denoted by Gj(.) and their joint behaviors arealso described by a normal factor copula. In particular, we assume that
𝜏𝜏𝑗𝑗 = 𝐺𝐺𝑗𝑗−1�𝑁𝑁�𝑌𝑌𝑗𝑗�� , 𝑗𝑗 = 1, … , 𝑚𝑚, (4)
where Yj are the latent variables used to model the joint distributions of
τ
j. The correlationsamong Yj are induced through common factors M and Mwl as follows:
𝑌𝑌𝑗𝑗 = 𝑐𝑐𝑎𝑎 + 𝑑𝑑𝑎𝑎𝑤𝑤𝑙𝑙+ √1 − 𝑐𝑐2− 𝑑𝑑2𝑊𝑊𝑗𝑗, 𝑗𝑗 = 1, 2, … , 𝑚𝑚, (5)
where Mwl, W1,…., Wm are independent standard normal random variables and constants c and d denote factor loadings. Mwl represents the factor of age improvements for the insurance contract portfolio as a whole and W1,…., Wm are specific factors pertaining to each
10 We use Maximum Likelihood approach to estimate the factor loadings from the related samples of life settlements pool.
10
insurance contract. They are also independent of M, Mls, Z1,…., Zn.
According to above settings, the correlation coefficient
ρ
ik between life settlements i andk is a
2+b
2. The correlation coefficientκ
jl between life contracts j and l is c2+d
2, and the correlation coefficient between life settlement i and insurance contract j is ac.Let the rate of return for the life settlement and required investment return of insurance contracts be rls and rwl, respectively. Assume that the premium payment of life settlement i at time t is Pi(t), its death benefit is Bi, the premium payment of insurance contract j at time t is Qj(t), and its death benefit is Aj. Then the value of the life settlement pool can be
expressed as Equation (6):
𝑉𝑉𝑙𝑙𝑙𝑙= ∑𝑛𝑛𝑖𝑖=1𝑉𝑉𝑙𝑙𝑙𝑙(𝑖𝑖), (6)
where 𝑉𝑉𝑙𝑙𝑙𝑙(𝑖𝑖) =(1+𝑟𝑟𝐵𝐵𝑖𝑖
𝑙𝑙𝑙𝑙)𝑡𝑡𝑖𝑖+𝑇𝑇𝑖𝑖 − � (1+𝑟𝑟𝑃𝑃𝑖𝑖(𝑡𝑡)
𝑙𝑙𝑙𝑙)𝑡𝑡 𝑡𝑡𝑖𝑖+𝑇𝑇𝑖𝑖
𝑡𝑡=1 .
and the value of the insurance liability portfolio is Equation (7):
𝑉𝑉𝑤𝑤𝑙𝑙 = ∑𝑚𝑚𝑗𝑗=1𝑉𝑉𝑤𝑤𝑙𝑙(𝑗𝑗), (7)
where 𝑉𝑉𝑤𝑤𝑙𝑙(𝑗𝑗) =(1+𝑟𝑟−𝐴𝐴𝑗𝑗
𝑤𝑤𝑙𝑙)𝜏𝜏𝑗𝑗 + � (1+𝑟𝑟𝑄𝑄𝑗𝑗(𝑡𝑡)
𝑤𝑤𝑙𝑙)𝑡𝑡 𝜏𝜏𝑗𝑗
𝑡𝑡=1 .
Suppose that the insurer would like to use life settlements as a hedging tool. By
implementing a hedging program, the total value of the hedged liabilities Vh can be expressed as:
V
h=V
wl + h Vls, (8).11
in which h denotes the hedge ratio. Here we assume the life settlement portfolio is a closed-end fund and the insurer can decide to purchase a portion of it. Therefore, h is a number between 0 and 1. Note that, according to Equations (6) and (7), Vwl is negative (liability) and
hV
ls is positive which represents the investment in life settlements (asset). Equation (8) implies that the hedging effect between the liabilities and life settlements depends on the correlation parameters (a, b, c, and d) as well as the hedge ratio h. We can calculate the optimal hedge ratio h under a specified risk measure and correlation structure. Theoptimality is defined by minimizing a certain risk measure on Vh. The risk measures that we consider in this paper include standard deviation, Value at Risk (VaR) and expected shortfall (ES). We use Monte Carlo simulation to generate the distributions of Vwl and Vls and
determine the optimal hedge ratio h based on the simulated distributions of Vh. The insurers can then establish their hedging programs according to different liability portfolios or needs.