國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
6
πP(rt, T) = 1
∏ (1 + rTi=1 i)− P(rt, T)
where P(rt, T) is the bond price calculated by using the closed form bond price formula of CIR model.
2.3.Adjusting mortality table
Without the mortality rate for the insured selling life settlement, we will not be able to analyze the distribution of profit function for life settlement. The available
information about the insured sold life settlement is the age and life expectancy.
Maximum entropy principle provide a reasonable and feasible methodology to adjust the "standard" mortality rates into a adjusted mortality rates by incorporating newly obtained information such as life expectancy, variance, median...etc . For example, Kogure., A. et al.(2010), Johnny Siu-Hang Li et al.(2010) and Johnny Siu-Hang Li et al.(2011) applied maximum entropy principle to change the physical probability measure to the objective probability measure for pricing mortality linked derivatives.
We will applied the method in Brockett, P. L. (1991) to construct the life time distribution of life settlement seller.
Let K(x) be the curtate life time of (x)
According to standard life table the probability mass function of K(x) is (g0, g1, … , gω−x)
where gi= Pr (K(x) = i)
We want to find adjusted mortality table with curtate life time of (x) as (f0, f1, … , fω−x)
s. t � fk
k
= 1 and � kfk
k
= ET
where fi= Pr (K(x) = i) under adjusted mortality table and ET is the expectation of lifetime based on newly obtained information.
‧
To find the adjusted distribution of life time we have to solve the following
optimization problem that minimizes the Kullback–Leibler information(Kullback, S et al.(1951))
The solution can be obtained by Lagrange multiplier method. First we consider the Lagrangian function
we need to solve ∇L(f, β) = 0, which is equivalently to solving the following system of equations.
‧
As defining the profit function of assets and liabilities, we then define the profit function of the surplus to be the profit function of assets minus profit function of liabilities. cima are female annuity and male annuity annual payment amount for the i-th
annuitant.
The insurance company need to manage the profit function of surplus. The mean variance optimization problem will be
N1,…,NnBmax,M1,…,MnS𝐸[𝜋(𝑡)] − 𝜃𝑉𝑎𝑟[𝜋(𝑡)]
The first constraint is to avoid short position of assets and the second constraint
‧
indicates the budget constraint.
Let
Then the budget constraint can be rewritten as
� NiB(rt, Ti)
This optimization problem includes equality constraints and inequality constraints, the Karush-Kuhn-Tucker (KKT) optimality conditions(Kuhn et al.(1951)) in appendix 1 provide a method to solve this problem analytically.
Let u = �M1, … , MnS, NnS+1, … , NnB+nS�′= (u1, … , un)′ be the units column vectors, the first nS components are the units we need to buy life settlements with different ages, gender and life expectancies and the last nB components are the units we need to buy bonds with different maturities.
Our target is to solve the problem:
maxu �[m′, m�] � u−1� − θ[u′, −1] �Σ11 Σ12
Σ21 Σ22� � u−1��
s. t. u′a = 1 and u′≥ 0
where m is the n by 1 mean column vectors of profit function of all the assets, m� is the sum of expected value of profit function of all liabilities. Σ is the (n+1)*(n+1) covariance matrix of all assets and liabilities, we can decompose Σ into 4
sub-matrices
�Σ11 Σ12 Σ21 Σ22�
Σ11 is n*n matrix represent covariance matrix of assets. Σ22 is 1*1 matrix equaling
‧
to the variance of sum of all liabilities. And
a = Our optimization problem becomes
minu f(x)
s. t. u′a − 1 = 0 and −u′ ≤ 0 The optimal solution is
u = 1
The detailed derivation of the solution is in appendix 2.
Mean variance approach is easy to implement and has good properties such as closed-form optimal allocation formula, however using first two moments to
determine hedging strategies may be too simple to capture the characteristics of profit function. We consider further objective functions such as value at risk(VaR) and
‧
conditional tail expectation(CTE) to offer a comparative hedging performance to the mean variance approach.
Set loss function as negative of profit function, that is L = −π The definition of VaR could be written as
VaRβ(L) = inf{ξ|P(L ≤ ξ) = β}
and we apply the result of Trindade et al.(2007) and Pflug, G. (2000) to obtain the value of CTE by solving the following optimization problem
CTEβ(L) = 1
We first consider the mortality is stochastic and the interest rate is non-stochastic.
Therefore the interest rate is assumed to be a constant rate 0.03 in this example, there will be 100,000 generated mortality sample paths for calculating profit function of the liabilities. On the asset side, we choose life settlement of insured aged 65 and with suggested life expectancy 10 for both male and female. On the liability side, we include life contracts of female aged 50 and male aged 65 with benefit payment 100, there are also annuities of female aged 55 and male aged 65 with annual payment 1 in the liability. Table 1 summarizes the assets and liabilities:
‧
Asset Liability
Life settlement:
Male 65
(suggested life expectancy=10) Female 65
(suggested life expectancy=10)
Life: (benefit=100)
Table 1: Assets and liabilitiesFigure 1: The profit functions on the liability side
(top left): Life Female 50, (top right): Life Male 65, (bottom left): Annuity Female 55, (bottom right): Annuity Male 65.
The distribution of profit functions are displayed on Figure 1 and Figure 2. We can discover due to mortality improvement, the expected value of profit function of life
-5 0 5 10 15
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
13
contracts are negative whereas they are positive for annuities. The averaged value of profit function of life settlements are also positive.
Figure 2: The profit functions on the asset side (top):life settlement male aged 65 with life expectancy 10.
(bottom): life settlement female aged 65 with life expectancy 10.
The expected value and covariance matrix of profit functions are shown in Table 2 and Table 3. The life settlements have similar properties to the life insurance contracts, therefore it provide excellent hedging effectiveness against mortality risk from life insurance contracts.
-0.30 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
1000 2000 3000 4000
-0.40 -0.2 0 0.2 0.4 0.6 0.8 1
1000 2000 3000 4000
‧
0.016953 0.008119 -0.29314 -0.75086 0.598889 0.852789 Table 2: Mean of profit functions
Life
0.006401 0.004383 0.055503 0.117803 -0.00458 -0.01429 0.004383 0.01301 0.136444 0.075508 0.001919 0.006579 0.055503 0.136444 3.080249 1.234243 0.035585 0.11696 0.117803 0.075508 1.234243 2.376275 -0.099 -0.3025 -0.00458 0.001919 0.035585 -0.099 0.054546 0.098543 -0.01429 0.006579 0.11696 -0.3025 0.098543 0.308113 0.006401 0.004383 0.055503 0.117803 -0.00458 -0.01429
Table 3: Covariance matrix of profit functions
The optimal hedging strategies according to different objective functions are in Table 4. As the parameter θ increases, the optimal weight for life settlement male will decrease but the optimal weight for life settlement female will increase. The result for VaR objective functions and CTE objective functions are similar, it put more weights on both life settlement of male and female comparing to the result with mean variance objective function.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
15
Life settlement male 65 ET=10
Life settlement female 65 ET=10 MV 𝛉 = 𝟏
Units
17.7247 11.2857Weight
0.1010 0.0746MV 𝛉 = 𝟐
Units
17.0029 11.3729Weight
0.0969 0.0752VaR(0.05)
Units
22.2499 11.8484Weight
0.1268 0.0783CTE(0.05)
Units
21.9820 13.8237Weight
0.1253 0.0914Table 4 optimal hedging strategies
Figure3~6 display the hedging effectiveness of different objective function. In Figure 3 and Figure 4, we can see under the mean variance hedging strategies, the
distributions are less volatile after hedging, because the goal is to reduce the variance of portfolio and maximize the mean of profit function simultaneously. Figure 5 and Figure 6 have different hedging outcomes, the hedged distribution will retain the weight on the right tail and reduce the weight on the left tail. This is the most
desirable result, it means our hedging strategies may reduce the down side risk of our portfolio but at the same time it will not harm the opportunity of making profit.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
16
Figure 3: hedging effectiveness with objective function mv θ = 1
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
Figure 4: hedging effectiveness with objective function mv θ = 2
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
-25 -20 -15 -10 -5 0 5 10
0 1000 2000 3000 4000 5000 6000 7000 8000
-25 -20 -15 -10 -5 0 5 10
0 1000 2000 3000 4000 5000 6000 7000 8000
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
17
Figure 5: hedging effectiveness with objective function VaR
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
Figure 6: hedging effectiveness with objective function CTE
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio Next step, we will discuss the hedging strategies by incorporating both interest risk and mortality risk. The constant interest rate is replaced by100,000 sample paths of interest rate generated according to CIR model with parameters a = 0.2, b = 0.03,
-25 -20 -15 -10 -5 0 5 10
0 1000 2000 3000 4000 5000 6000 7000
-25 -20 -15 -10 -5 0 5 10
0 1000 2000 3000 4000 5000 6000 7000
‧
maturity 20 years to hedge the interest rate risk.Figure 7: The profit functions on the liability side
(top left): Life Female 50, (top right): Life Male 65, (bottom left): Annuity Female 55, (bottom right): Annuity Male 65.
Fig 7 and Fig 8 have similar distribution shape for each asset and liability comparing to the case without interest rate risk but the dispersion is larger due to stochastic interest rate contribute more randomness to the distributions of profit functions.
-20 0 20 40
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
19
Figure 8: The profit functions on the asset side (top): zero coupon bond with maturity 20 years.
(bottom left):life settlement male aged 65 with life expectancy 10.
(bottom right): life settlement female aged 65 with life expectancy 10.
Table 5 is the optimal hedging allocation incorporating additional interest rate risk, we can observe the large portion of weight is put on the zero coupon bond, hence under our assumption, the interest rate risk dominates the mortality risk. Similarly, as the parameter θ increases, the weight on zero coupon bond and life settlement male decrease but the weight on life settlement male increases. This result indicates that life settlement female seems has better effect on reducing portfolio variance. While
considering the VaR and CTE criterion, we find it put more weight on zero coupon bond and life settlement male. This is quite different form the result of mean variance hedging strategies.
-0.40 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
1000 2000 3000 4000
-0.50 0 0.5 1
1000 2000 3000 4000 5000
-0.50 0 0.5 1
1000 2000 3000 4000
‧ female 65 ET=10 MV 𝛉 = 𝟏
Units
183.5250 20.1810 12.7361Weight
0.8008 0.1151 0.0842MV 𝛉 = 𝟐
Units
183.4617 19.7923 13.1131Weight
0.8005 0.1128 0.0867Table 5 optimal hedging strategies
Fig 9~12 show the hedging effectiveness of our example. As mentioned earlier, the interest rate risk dominates the mortality risk. We cannot easily recognize the differences between theses 4 figures.
Figure 9: hedging effectiveness with objective function mv θ = 1
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
-60 -40 -20 0 20 40 60
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
21
Figure 10: hedging effectiveness with objective function mv θ = 2
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
Figure 11: hedging effectiveness with objective function VaR
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
-60 -40 -20 0 20 40 60
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-60 -40 -20 0 20 40 60
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
22
Figure 12: hedging effectiveness with objective function CTE
(Red):hedged profit function of portfolio. (blue): unhedged profit function of portfolio
5.Conclusions
This paper proposes the methodology to hedge mortality risk by life settlement. Using zero coupon bonds and life settlement to hedging the interest rate and mortality risk, we find the risk on the liability side is effectively reduced. Furthermore we have derived the closed-form optimal solution under mean variance assumption. Hedging strategies with mean variance objective function can adjust the parameter θ to reflect their risk aversion. We also investigate alternative objective function such as VaR and CTE, the result is more attractive for insurance companies, it reduces the downside risk without sacrificing upside profit.
Our hedging approaches is flexible. Even we change the interest rate or mortality rate model, the methodology in this paper is still adoptable. This hedging strategy is also applicable in practice for insurance companies which have complicated liabilities structures. Under the mean variance objective function assumption, the larger value θ
-60 -40 -20 0 20 40 60
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
23
is, the more emphasis on reducing variance of portfolio. The mean variance hedging strategy is similar to the strategy of VaR and CTE objective functions, the main target is to control the downside risk. In order to control the downside risk, we not only need to care about the variance but also need to take the mean of portfolio into account.
Therefore life settlement can be regard as effective hedging instrument to controlling the mortality risk for insurance companies.
‧
Blake, D., and Burrows, W., 2001. “Survivor Bonds: Helping to Hedge Mortality Risk”, Journal of Risk and Insurance 68: 339-348.
Brockett, P. L., 1991. Information Theoretic Approach to Actuarial Science: A Unification and Extention of Relevant Theory and Applications, Transactions of the Society of Actuaries, 42: 73-115
Cox, J. C., Ingersoll, Jr., J. E., and Ross, S. A., 1985. “A Theory of the Term Structure of Interest Rates”, Econometrica 53: 385-408.
Cox, S. H. and Y. Lin, 2007. Natural Hedging of Life and Annuity Mortality Risks,North American Actuarial Journal, 11(3): 1-15.
Dowd, K., Blake, D., Cairns, A. J. G., and Dawson, P., 2006. “Survivor Swaps”, Journal of Risk & Insurance 73: 1-17.
Hua Chen, Samuel H. Cox and Zhiqiang Yan, 2010. Hedging Longevity Risk in Life Settlements. Working paper.
Johnny Siu-Hang Li, 2010. Pricing longevity risk with the parametric bootstrap: A maximum entropy approach, Insurance: Mathematics and Economics, 47:176-186.
Johnny Siu-Hang Li and Andrew Cheuk-Yin NG.,2011. Canonical valuation of mortality-linked securities, The Journal of Risk and Insurance, Vol. 78, No. 4, 853-884
Kogure., A., and Kurachi, Y., 2010. A Bayesian Approach to Pricing Longevity Risk Based on Risk-Neutral Predictive Distributions, Insurance: Mathematics and
Economics, 46:162-172.
Kuhn, H. W.; Tucker, A. W., 1951. "Nonlinear programming". Proceedings of 2nd Berkeley Symposium. Berkeley: University of California press. pp. 481-492.
Kullback, S., and R. A. Leibler, 1951. On Information and Sufficiency, Annals of Mathematical Statistics, 22: 79-86.
Lee, R.D., Carter, L.R., 1992. Modeling and forecasting US mortality. Journal of the American Statistical Association 87, 659_675.
Pflug, G., 2000. Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk. S. Uryasev, ed. Probabilistic Constrained Optimization
Methodology and Applications. Kluwer, Dordrecht, The Netherlands, 272–281.
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
25
Trindade, A. A., S. Uryasev, A. Shapiro, and G. Zrazhevsky, 2007. Financial Prediction with Constrained Tail Risk. Journal of Banking and Finance 31 3524–
3538.
Tsai, J.T., J.L. Wang, and L.Y. Tzeng, 2010. On the optimal product mix in life insurance companies using conditional Value at Risk, Insurance: Mathematics and Economics, 46, 235-241.
Wang, J.L., H.C. Huang, S.S. Yang, J.T. Tsai, 2010. An optimal product mix for hedging longevity risk in life insurance companies: The immunization theory approach, The Journal of Risk and Insurance, Vol. 77, No. 2, 473-497.
‧
1.Karush-Kuhn-Tucker (KKT) optimality conditions:
Consider the constrained optimization problem:
minx f(x)
s. t. �gj(x) ≤ 0, 𝑗 = 1, … , 𝑚 hl(x) = 0, l = 1, … , r The Lagrangian Function is given by
L(x, µ, λ) = f(x) + � µjgj(x) + � λlhl(x)
r l=1 m
j=1
IF x* is an optimal solution of the problem, then there exist Lagrange multipliers µ∗ and λ∗ such that This is the KKT condition
2.Solution of the optimal hedging problem
The Lagrangian function can be written asL(u, μ, λ) = f(u) − μ′u + λ(u′a − 1) The KKT conditions imply the following system of equations:
∇f(u) − μ + λa = 0 (1) where
∇f(u) =∂f(u)
∂u = −m +θ[2Σ11u − 2Σ12]
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
27
u′a − 1 = 0 (2)
−u ≤ 0 (3) μ ≥ 0 (4) μiui = 0 i = 1, … , N (5) Case 1: ui > 0 ∀𝑖
(1)=>
−m + θ[2Σ11u − 2Σ12] − μ + λa = 0
⇒ Σ11u = 1
2θ(μ + m − λa) + Σ12
u = 1
2θΣ11−1(μ + m − λa) + Σ11−1Σ12 From (5), IF ui ≠ 0 ∀ i then μi = 0 ∀ i so we have
u = 1
2θΣ11−1(m − λa) + Σ11−1Σ12 (∗) substitute u into (2) we can solve λ easily
a′u = 1
⇒ a′�1
2θΣ11−1(m − λa) + Σ11−1Σ12� = 1
⇒ 1
2θ a′Σ11−1(m − λa) + a′Σ11−1Σ12 = 1
⇒ a′Σ11−1m − λa′Σ11−1a = 2θ�1 − a′Σ11−1Σ12�
λ =a′Σ11−1m − 2θ�1 − a′Σ11−1Σ12�
a′Σ11−1a (∗∗) substitute (**) into (*), we get the desired optimal asset allocation.
u =1
θΣ11−1�m −a′Σ11−1m − 2θ�1 − a′Σ11−1Σ12�
a′Σ11−1a a� + 2Σ11−1Σ12
Case 2:ui = 0 for some i's
suppose there are k u′s being zero say
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
28
u(1)= u(2)= ⋯ = u(k) = 0 {(1), … , (k)} ∈ {1, … , N}
and (1) ≤ (2) ≤ ⋯ ≤ (k) The other u′s are nonzero called
u(1)′, u(2)′, … , u(N−k)′
By (5) of KKT condition, we can say μ(1), … , μ(k) are nonzero, The others are all zero called μ(1)′, … , μ(N−k)′
From the expression of u u = 1
2θΣ11−1(μ + m − λa) + Σ11−1Σ12 Define uA = (u(1), u(2), … , u(k))′,we have
uA= 0 = 1
2θΣ11−1((1): (k), : )(μ + m − λa) + Σ11−1((1): (k), : )Σ12
here Σ11−1((1): (k), : ) denotes the matrix obtained by picking rows (1), (2),...,(k) from Σ11−1.
we also define μA= (μ(1), … , μ(k))′, then 1
2θΣ11−1�(1): (k), (1): (k)�μA+ 1
2θΣ11−1�(1): (k), : �m − λ
2θΣ11−1�(1): (k), : �a + Σ11−1((1): (k), : )Σ12 = 0
⇒ Σ11−1�(1): (k), (1): (k)�μA
= λΣ11−1�(1): (k), : �a − Σ11−1�(1): (k), : �m − 2θΣ11−1((1): (k), : )Σ12
⇒ μA
= Σ11�(1): (k), (1): (k)��λΣ11−1�(1): (k), : �a − Σ11−1�(1): (k), : �m
− 2θΣ11−1((1): (k), : )Σ12�
where Σ11−1�(1): (k), (1): (k)� means picking rows (1), (2),...,(k) and columns (1), (2),...,(k) from Σ11−1 to form the new submatrix.
The remaining part is to solve λ
By the (2) of KKT condition, and define aB= (a(1)′, a(2)′, … , a(N−k)′)′ and
‧ 國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
29
uB= (u(1)′, u(2)′, … , u(N−k)′)′
aB′ uB = 1 we have
aB′ �1
2θΣ11−1((1)′: (N − k)′, : )(m − λa) + Σ11−1((1)′: (N − k)′, : )Σ12� = 1
⇒ aB′ Σ11−1((1)′: (N − k)′, : )(m − λa) = 2θ�1 − aB′ Σ11−1((1)′: (N − k)′, : )Σ12� λ = a′BΣ11−1((1)′: (N − k)′, : )m − 2θ�1 − aB′ Σ11−1((1)′: (N − k)′, : )Σ12�
a′BΣ11−1((1)′: (N − k)′, : )a