Application in Mortality Swap

4.1 Natural hedging and mortality swap

Mortality (longevity) risk is the risk that the number of deaths (survivors) is higher than expected. When there is an unexpected change in mortality rates, either life insurers or annuity providers will experience a loss. If the mortality rates in-crease unexpectedly in a year, the number of deaths during the year is higher than expected so that life insurance companies need to pay more death benefits.

However, in this case, annuity providers gain from the mortality increase. If the mortality rates decline unexpectedly, the impacts on financial situation of life in-surers and annuity providers reverse. Natural hedge is a strategy of hedging two risks responding oppositely to a change in a common factor. Since life insurers and annuity providers face mortality and longevity risks, respectively, both of them can adopt natural hedge by swapping a portion of their business each other. When a life insurer (an annuity provider) owns both life and annuity business at the same time, the mortality and longevity risks of the portfolio can be offset to a lower level no matter how the mortality rates change. Now, a question rises: what are the opti-mal portions of business swapped between the life insurer and the annuity provider such that the risk of the portfolio is minimized?

Let L denote the loss function at time zero, which is the present value of future liabilities less the present value of future premium incomes. The values of both future liabilities and premium incomes depend on the future mortality rates. Before swapping the business, a life insurer (an annuity provider) has a portfolio of life

28

CHAPTER 4. APPLICATION IN MORTALITY SWAP 29

(annuity) business, and the loss function for the portfolio is denoted by Ll(La). To hedge mortality (longevity) risk, the life insurer (annuity provider) would like to swap wl (wa) of life (annuity) business to the annuity provider (life insurer); the resulting loss functions of the life insurer and annuity provider become

LL= (1 − wl) × Ll+ wa× La

and

LA= (1 − wa) × La+ wl× Ll, respectively, with variances

V ar(LL) =(1 − wl)²× V ar(Ll) + w²_a× V ar(La)

+ 2 × (1 − wl) × wa× Cov(Ll, La) (4.1) and

V ar(LA) =(1 − wa)²× V ar(La) + w²_l × V ar(Ll)

+ 2 × (1 − wa) × wl× Cov(Ll, La). (4.2)

There are three aspects to approach the optimal pair of weights which min-imizes the variance of a loss function or the sum of the variances of two loss functions. The first pair of weights,(w^L+A_l , w^L+A_a ), is used to minimize V ar(LL) + V ar(LA) (see Figure 4.1); the second pair of weights, (w^L_l, w_a^L), is used to mini-mized V ar(LL) (see Figure 4.2), and the third pair of weights, (w_l^A, w^A_a), is used to minimize V ar(LA) (see Figure 4.3), where we place superscripts L+A, L and A on wl and wato denote the weights that minimize V ar(LL) + V ar(LA), V ar(LL) and V ar(LA), respectively.

CHAPTER 4. APPLICATION IN MORTALITY SWAP 30

Figure 4.1: Mortality swap: minimizing V ar(LL) + V ar(LA)

Figure 4.2: Mortality swap: minimizing V ar(LL)

Figure 4.3: Mortality swap: minimizing V ar(LA)

CHAPTER 4. APPLICATION IN MORTALITY SWAP 31

In mathematical optimization, the method of Lagrange multipliers is a strat-egy for finding the local maximum (minimum) of a function subject to some con-straint(s). Let Pl (Pa) stand for the present value of the future premiums of all life (annuity) policies in the portfolio before swap. When the life insurer (annuity provider) swaps wl(wa) of life (annuity) policies to the annuity provider (life insurer), the life insurer (annuity provider) loses premium wl×Pl(wa×Pa) and gets premium wa× Pa(wl× Pl). We set a swap condition wl× Pl= wa× Pawhich will be applied as the constraint in the three optimization problems mentioned above using the method of Lagrange multipliers. Specifically, we would like to find (wˆl, ˆwa) which minimizes f(wl, wa) subject to wl× Pl = wa× Pa where f is V ar(LL) + V ar(LA),

According to the method of Lagrange multipliers with a constraint w^L+A_l × Pl = w_a^L+A× Pa, the Lagrange function is defined by

ϕ(w^L+A_l , w_a^L+A, λ) = f (w^L+A_l , w_a^L+A) + λ(w_l^L+A× Pl− w^L+A_a × Pa),

where λ is called a Lagrange multiplier. To obtain the optimal solution, we differen-tiate ϕ with respect to w^L+A_l , w_a^L+A, and λ, respectively, and set all results to zero.

CHAPTER 4. APPLICATION IN MORTALITY SWAP 32 the theoretical expressions of Vl, Vaand σ². Instead, we would like to compute the corresponding sample variances and covariance by simulating thousands of Vl, Va

and σ².

CHAPTER 4. APPLICATION IN MORTALITY SWAP 33

4.2 Numerical illustrations

4.2.1 Assumptions and portfolios

In practice, a life (an annuity) portfolio consists of a variety of life (annuity) products.

For simplicity, we assume that the portfolio for the life insurer consists of(65 − x)-payment whole life insurance issued to the insureds aged x = 25 ∼ 64 and the death benefits are paid at the end of the year of death, and that the portfolio for the annuity provider is composed of(65−x)-payment and (65−x) years deferred whole life annuity due issued to the insureds aged x = 25 ∼ 64. Since life insurance (annuities) are more often purchased by those who have poorer (better) health conditions, and we have no life (annuity) tables for consecutive years for forecasting mortality rates with the models proposed in the preceding chapter, we use the U.S.

male (female) mortality table for the life (annuity) insureds. Because the year span [t0, t0+ n − 1] is used for estimating the parameters of the mortality models, and forecasting the mortality rates for years t0 + n − 1 + τ, τ = 1, 2, ...., we set the beginning of year t1 = t0+ n as time 0. Denote lx,t1,ithe initial number of insureds aged x at time 0 for population i (i = 1 for life and i = 2 for annuity) and we set l25,t1,1 = l25,t1,2 = 10⁷. By lx+1,t1,i = lx,t1,i× px,t1−1,i, x = 25, ..., 63, the initial numbers of insureds for the entire portfolio can be obtained. The death benefit for life insurance is Bl= 100, 000 and the annual survival benefit is Ba= 10, 000. Based on the assumptions, the loss functions of the life insurer and annuity provider at time zero are

CHAPTER 4. APPLICATION IN MORTALITY SWAP 34 interest rate. Note that we set the limiting age equal to100 for both populations, that is, q100,t,i = 1, i = 1, 2. Moreover, Pl and Pa, the total present values of the future premiums for the life insurer and annuity provider, respectively, in(4.3) ∼ (4.8) are (see(4.9) and (4.10))

are pre-determined and do not respond to a change in mortality rates, whereas Ax,1, ¨a_x:65−x|,i and 65−x|¨ax,2 in (4.11), (4.12) and (4.13) vary in response to the realized mortality rates. Therefore, the deterministic mortality rates, ˆqx,t0+n−1+τ, τ = 1, 2, ..., are used to calculate the premiums Px,l and Pa,x, and the stochastic ones,˜qx,t0+n−1+τ, τ = 1, 2, ..., are used for simulations to compute Ax,1,¨a_x:65−x|,iand

65−x|¨ax,2. When the realized mortality rates are different from the expected ones, each of the loss functions Ll and Lais either positive or negative. To forecast de-terministic and stochastic mortality rates for ages25 ∼ 64 and years 2011 ∼ 2086 using the four models given in Chapter3, we set the year span [1981, 2010] and age span[25, 100] (see Figure 4.4) to estimate the parameters with the male and female mortality data from the Human Mortality Database for the life and annuity policies, respectively. Specifically, for the independent model, we give the following steps to forecast deterministic mortality rates for computing the premiums Px,l and Px,a:

CHAPTER 4. APPLICATION IN MORTALITY SWAP 35

Figure 4.4: age span[25, 100] and year span [1981, 2010]

1. compute ln( ˆmx,2010+τ,i) = ˆax,i+ ˆbx,i× (ˆk2010,i+ τ × ˆθi), i = 1, 2, τ = 1, 2, ..., 76, x = 25, ..., 64;

2. transfer ln( ˆmx,2010+τ,i) to ˆqx,2010+τ,i; and

3. take the diagonal entriesˆqx+τ −1,2010+τ,i, i = 1, 2, x = 25, ..., 64, τ = 1, 2, ..., (101−

x).

To generate stochastic mortality rates for simulating Ax,1, ¨a_x:65−x|,i and65−x|¨ax,2, a similar procedure is given as follows:

1. generate s,τ,ifrom N(0, 1) and sε,τ,ifrom N(0, 1), i = 1, 2, τ = 1, 2, ..., 76;

2. multiply s,τ,i by ˆσ,i, and sε,τ,i by ˆσεx,i such that s,τ,i× ˆσ,i ∼ N(0, ˆσ²_,i) and sε,τ,i× ˆσεx,i∼ N(0, ˆσ_ε²_x,i);

3. get simulated ln( ˜mx,2010+τ,i) = ln( ˆmx,2010+τ,i)+√

τ ׈bx,i×s,τ,i׈σ,i+sε,τ,i׈σεx,i, i = 1, 2, τ = 1, 2, ..., 76, x = 25, ..., 64;

4. transfer ln( ˜mx,2010+τ,i) to ˜qx,2010+τ,i;

5. take the diagonal entries˜qx+τ −1,2010+τ,i, i = 1, 2, x = 25, ..., 64, τ = 1, 2, ..., (101−

x); and

6. repeat steps(1) ∼ (5) for N times (N = 1, 000 in this project).

CHAPTER 4. APPLICATION IN MORTALITY SWAP 36

The procedures of forecasting the deterministic and stochastic mortality rates for the joint-k, the co-integrated and the augmented common factor models are similar to those above. With{ˆqx+τ −1,2010+τ,i: i = 1, 2, x = 25, ..., 64, τ = 1, ..., (101 − x)} and N {˜qx+τ −1,2010+τ,i : i = 1, 2, x = 25, ..., 64, τ = 1, ..., (101 − x)}’s, we can calculate N realized values of Ll and La, get the sample variances Vl and Vaand the sample covariance σ², and obtain the optimal weights with(4.3) ∼ (4.8).

4.2.2 Robustness testing

In the preceding subsection, for each of three optimization problems, a pair of opti-mal weights for life and annuity portfolios is produced by some stochastic mortality model which generates N cohort mortality rates from age x to age 100 at time zero for each of x = 25, ..., 64. If we re-run the procedure and generate another set of N cohort mortality rates, can we still produce a pair of optimal weights of close values? In this subsection, we will perform robustness testing. Robustness testing is originally used in computer science whether a computer system can continue to work well in case of invalid inputs. In our case, robustness testing is a way to investigate whether the optimal weights produced by a model is insensitive to the simulated mortality rates.

To complete the robustness testing, we repeat the simulation procedure M (M is set to 50) times, and yield M pairs of optimal weights for each model. Figures 4.5 and 4.6 show scatter plots for the optimal weights ( ˆwl, ˆwa) generated from each model, from which we can see that the 50 pairs of weights obtained through 50 simulation procedures for each model are quite close to each other. That means the four models are robust to simulations. Within each model, the first two pairs of optimal weights ( ˆw^L+A_l , ˆw^L+A_a ) and ( ˆw_l^L, ˆw_a^L) seem to be more consistent than ( ˆw^A_l , ˆw^A_a).

Table 4.1 summarizes the median values of 50 optimal weights for each type obtained from the four models. The weightwˆlis more than twice as big aswˆafor all types because wl/wa= Pa/Pl≈ 2.3. All the optimal weights are within (0, 1) except forwˆ^A_l for the independent model. The pairs of optimal weights( ˆwl, ˆwa) from the joint-k and co-integrated models are quite close to each other. The independent model produces the largest( ˆw^L+A_l , ˆw^L+A_a ) and ( ˆw^A_l , ˆw^A_a) but the smallest ( ˆw_l^L, ˆw^L_a),

CHAPTER 4. APPLICATION IN MORTALITY SWAP 37

L+A 0.8505 0.3623 0.7385 0.3146 0.7431 0.3169 0.7110 0.3029 L 0.4797 0.2044 0.6459 0.2752 0.6384 0.2723 0.6868 0.2926 A 1.2213 0.5203 0.8311 0.3541 0.8479 0.3616 0.7352 0.3132

Table 4.1: the median of 50 optimal weights

Figure4.7 displays V ar(LL)’s and V ar(LA)’s for four models based on 50 cor-responding( ˆw_l^L+A, ˆw^L+A_a )’s, ( ˆw^L_l, ˆw^L_a)’s and ( ˆw^A_l , ˆw_a^A)’s, respectively. These figures further confirm the comments on Table4.1 above, and show the variability of 50 variances. Under the joint-k and co-integrated models, the variances from 50 runs of simulations are quite close and smaller, whereas for the independent and aug-mented common factor models, the sample variances are higher and not as stable as those based on the joint-k and co-integrated models.

Figures4.8, 4.9 and 4.10 exhibit the simulated loss distributions before and af-ter swap using the median optimal weights for all four models. It is obvious that the loss distributions after swap for the life insurer and annuity provider are al-most narrowed, which implies that the variance of the loss distribution is reduced significantly. No matter using ( ˆw^L+A_l , ˆw_a^L+A), ( ˆw_l^L, ˆw_a^L) or ( ˆw_l^A, ˆw^A_a), the loss distri-butions for the joint-k and co-integrated models after swap for the annuity provider are much narrower than those for the independent and augmented common factor models, and for the life insurer, there is not much difference in the loss distribu-tions after swap among the joint-k, co-integrated and augmented common factor models.

To future quantify and compare the performances of hedging mortality and longevity risks after swap for the life insurer and annuity provider, respectively, we give a measure called hedge effectiveness (HE; see Li and Hardy (2011)) as follows:

HE(L + A) = 1 −V ar(LL) + V ar(LA) V ar(Ll) + V ar(La),

CHAPTER 4. APPLICATION IN MORTALITY SWAP 38

The HE measure is a variance reduction (variance of a loss function before hedge less the variance of the loss function after hedge) ratio. Clearly, the larger the HE is, the more effective the hedge is. Table4.2 shows the comparisons of HEs, which are consistent with the results from Figures 4.8, 4.9 and 4.10. The independent model overall performs the worst among all models. The HE(L), HE(A) and HE(L + A) for the joint-k model are the largest among the four models, which implies that the joint-k model is the most effective in hedging mortality and longevity risks. However, the co-integrated model produces the smallest variances of the losses before and after swap for both the life insurer and annuity provider.

Independent V ar(Ll) V ar(LL) V ar(La) V ar(LA) HE(L + A) HE(L) HE(A)

Table 4.2: comparisons of sample variances (×10²³) and HE’s

CHAPTER 4. APPLICATION IN MORTALITY SWAP 39

ˆ

w_l^L+A, ˆw^L+A_a wˆ_l^L+A, ˆw_a^L+A

ˆ

w^L_l, ˆw^L_a wˆ^L_l, ˆw_a^L

ˆ

w_l^A, ˆw^A_a wˆ^A_l , ˆw_a^A

Figure 4.5: optimal weights for the independent and joint-k models

CHAPTER 4. APPLICATION IN MORTALITY SWAP 40

ˆ

w_l^L+A, ˆw^L+A_a wˆ_l^L+A, ˆw_a^L+A

ˆ

w^L_l, ˆw^L_a wˆ^L_l, ˆw_a^L

ˆ

w_l^A, ˆw^A_a wˆ^A_l , ˆw_a^A

Figure 4.6: optimal weights for the co-integrated and augmented common factor models

CHAPTER 4. APPLICATION IN MORTALITY SWAP 41

V ar(LL)

variances using( ˆw^L+A_l , ˆw^L+A_a )

V ar(LA)

variances using( ˆw^L+A_l , ˆw^L+A_a )

variances using( ˆw_l^L, ˆw^L_a) variances using( ˆw_l^L, ˆw^L_a)

variances using( ˆw_l^A, ˆw^A_a) variances using( ˆw^A_l , ˆw^A_a) Figure 4.7: variances after swap

CHAPTER 4. APPLICATION IN MORTALITY SWAP 42

Figure 4.8: simulated loss distributions using( ˆw_l^L+A, ˆw_a^L+A)

CHAPTER 4. APPLICATION IN MORTALITY SWAP 43

Figure 4.9: simulated loss distributions using( ˆw^L_l, ˆw_a^L)

CHAPTER 4. APPLICATION IN MORTALITY SWAP 44

Figure 4.10: simulated loss distributions using( ˆw_l^A, ˆw^A_a)

Chapter 5