In the preceding chapter, we forecast deterministic and stochastic mortality rates with some two-population mortality models to determine the premiums of life and annuity products and simulate the loss functions of portfolios of life and annuity business. The weights of business for swap are calculated by the sample variances and the sample covariance of the loss functions of the life insurer and annuity provider before swap, which can minimize the risk of the portfolio and produce high hedge effectiveness. All the great results are based on that the assumed mortality model is the actual one, which, however, might not be true. Therefore, both the life insurer and annuity provider face model risk and parameter risk that will potentially affect the results. In this chapter, a bootstrap method which is model and parameter free will be applied to generating samples for the future mortality rates to calculate the weighted loss functions and their variances with the weights obtained by each of the four models in Chapter3.
Bootstrap is usually used to resample data with replacement to estimate some statistic of a population from the sampled data. The procedure of the bootstrap (naive bootstrap) is as follows:
1. Draw values from the original data set with replacement to form a new data set of size n∗.
2. Repeat the first step N∗times to obtain N∗ new data sets.
3. Compute the test statistic using the new data sets.
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CHAPTER 5. BLOCK BOOTSTRAP METHOD 46
However, the naive bootstrap fails when applied to the mortality rates. First of all, mortality rates over years can be treated as a time series displaying a decreas-ing trend because of the improvement of medical and environmental conditions, which shows mortality rates are not stationary. Secondly, the naive bootstrap is likely to destroy the dependency of mortality rates on both age and time dimen-sions. Alternatively, the block bootstrap can solve the problems above. First, re-garding the non-stationary problem, differencing is a popular and effective method of removing trend from a time series and making the time series weakly stationary.
The procedure is given below.
• Convert the empirical mortality rates qx,t,ito ln(mx,t,i), i = 1, 2, x = 25, ..., 100, t = 1981, ..., 2009.
• Devide ln(mx,t+1,i) by ln(mx,t,i) to get the ratio, denoted by rx,t,i, that is, rx,t,i=
ln(mx,t+1,i)
ln(mx,t,i) , t = 1981, ..., 2009.
• Subtract rx,t,ifrom rx,t+1,ito get the difference, denoted by dx,t,i, that is, dx,t,i= rx,t+1,i− rx,t,i, t = 1981, ..., 2008.
Figure5.1 shows the time series {dx,t,i}, i = 1, 2, for the U.S. males and females aged 35, 45, 55 and 65, which look stationary.
CHAPTER 5. BLOCK BOOTSTRAP METHOD 47
Figure 5.1:{dx,t,i} for x = 35, 45, 55, 65
CHAPTER 5. BLOCK BOOTSTRAP METHOD 48
To further ensure the time series{dx,t,i} is stationary, the Phillips-Perron (PP) and the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) hypothesis tests are conducted.
For the PP test where an AR(1) model is assumed, the null hypothesis is that the time series has a unit root so that the time series is non-stationary, which implies that a small p-value suggests a stationary time series. For the KPSS test, the null hypothesis is that the time series is stationary; thus, a big p-value indicates a sta-tionary time series. Table5.1 exhibits the results of the hypothesis tests. All the p-values for the PP test are below0.05, and the ones for the KPSS test are larger than0.1, which all suggest that the time series {dx,t,i} is stationary.
35 45 55 65
Male PP 0.01 0.01 0.01 0.01
KPSS >0.1 >0.1 >0.1 >0.1 Female PP 0.01 0.01 0.01 0.01 KPSS >0.1 >0.1 >0.1 >0.1 Table 5.1: p-values for testing stationarity of{dx,t,i}
For the second problem regarding dependency, according to Li and Ng (2011), the two-dimensional mortality rates matrix Mi for population i is converted to a series of column vectors,
mt,i= (mx0,t,i, mx0+1,t,i, ..., mx0+m−1,t,i), t = t0, t0+ 1, ..., t0+ n − 1.
Thus, the matrix can be expressed as Mi = {mt0,i, mt0+1,i, ..., mt0+n−1,i}, which contains n elements. In this way, the age dependency can be retained. Before proceeding to the next step, as discussed in the first problem above, the vector mt,i needs to be converted to ln(mt,i), get the ratio vector rt,iand then form the difference vector
dt,i= (dx0,t0+t−1,i, dx0+1,t0+t−1,i, ..., dx0+m−1,t0+t−1,i), t = 1, 2, ..., n − 2, i = 1, 2.
For the time dependency problem, it’s assumed that k consecutive time series, dt,i, ..., dt+k−1,i, are dependent. While the dependency between dt,i and dt+k−1,i
gradually becomes weaker as k increases, and it’s going to vanish thoroughly for a large k. Thus, the block bootstrapping suggests splitting d into n overlapped blocks
CHAPTER 5. BLOCK BOOTSTRAP METHOD 49
Figure 5.2: a circle diagram of dt,i’s
of size k as follows:
D ={(d1,i, d2,i, ..., dk,i), (d2,i, d3,i, ..., dk+1,i), (d3,i, d4,i, ..., dk+2,i),
...,
(dn−2,i, d1,i..., dk−1,i)}
Figure5.2 gives a circle diagram of dt,i’s. We make all dt,i’s a circle and every con-secutive k dt,i’s form a block. After the(n−2) blocks of size k, which are numbered 1, 2, ..., (n − 2), have been created, the block number will be drawn with replace-ment to form a sample. The values within each block don’t change at all so that the dependent structure among k dt,i’s is maintained. In Chapter4, we simulate 76 time series of{˜q2010+τ,i}’s, τ = 1, 2, ..., 76, for computing one value of loss functions LL and LA. To simulate the same number of time series of {˜q2010+τ,i}’s with the block bootstrap method, we need to draw Z = Wk block numbers with replacement as a bootstrap sample, where W = 76 and k is the block size. IfWk is not an integer, then(z + 1) blocks need to be drawn and take the first s elements of the (z + 1)th block to make the sample size equal to W where z and s are the integer and the remainder ofWk, respectively.
CHAPTER 5. BLOCK BOOTSTRAP METHOD 50
Finally, the block size is determined by the size of observations V . As sug-gested by Hall et al. (1995), the optimal block size, k, can be V13, V14 or V15. Since there are 28{dt,i}’s (V = n − 2 = 28), a block size of 2 is chosen.
The following is the procedure of the block bootstrap method.
1. Take the logarithm on the real central death rates, ln(mx,t,i), for i = 1, 2, x = x0, ..., x0+ m − 1, and t = t0, ..., t0+ n − 1. 8. Obtain simulated mx,t0+n−1+τ,iby
mx,t0+n−1+τ,i= exp{ln(mx,t0+n−1+τ−1,i) × [ln(mx,t0+n−1,i)
9. Convert simulated mx,t0+n−1+τ,i matrix to qx,t0+n−1+τ,i matrix and take the di-agonal entries to form cohort mortality sequences.
10. Repeat (3)-(9) for N times (N = 1000).
CHAPTER 5. BLOCK BOOTSTRAP METHOD 51
The median weights and premiums calculated in the preceding chapter are used in simulating the loss functions before and after swap using the block boot-strap method. By applying the pre-determined premiums and1000 simulated mor-tality paths to (4.9) and (4.10), 1000 La’s and Ll’s can be obtained, from which sample V ar(Ll), V ar(La) and Cov(Ll, La) can be calculated. Plugging V ar(Ll), V ar(La) and Cov(Ll, La) into (4.1) and (4.2) with the pre-determined weights, the variances of the loss functions LL and LA after swap, V ar(LL) and V ar(LA), are obtained.
Figures5.3, 5.4 and 5.5 display the simulated loss distributions before and after swap. Compared with the loss distributions in Figures4.8, 4.9 and 4.10, the overlap-ping between the simulated loss distributions before and after swap becomes far smaller. The loss distributions before and after swap for the life insurer mainly fall in the negative (gain) and positive (loss) territories, respectively, whereas those for the annuity provider largely spread in the negative (gain) territory. Under the block bootstrap method, the life insurer will benefit more gains from swap and the annuity provider will suffer more losses (or less gains) from swap than under the parametric mortality models. Moreover, even though the simulated loss functions generated from the block bootstrap method exhibit big differences from those based on the four mortality models, the pre-determined weights still lower down the risks for both insurance and annuity portfolios, which can be seen from the narrowed loss distribution curves after swap.
CHAPTER 5. BLOCK BOOTSTRAP METHOD 52
Figure 5.3: simulated loss distributions using( ˆwlL+A, ˆwaL+A)
CHAPTER 5. BLOCK BOOTSTRAP METHOD 53
Figure 5.4: simulated loss distributions using( ˆwLl, ˆwaL)
CHAPTER 5. BLOCK BOOTSTRAP METHOD 54
Figure 5.5: simulated loss distributions using( ˆwAl , ˆwaA)
CHAPTER 5. BLOCK BOOTSTRAP METHOD 55
Similar to Table 4.2, Table 5.2 shows the sample variances and hedge effec-tiveness (HE) under the block bootstrap method. In this case, the differences in HE and V ar(L) among four models are not as obvious as those using multi-population mortality models because the same simulated mortality rates from the block bootstrap method are applied to the loss functions before and after swap with different( ˆwl, ˆwa)’s from four mortality models. Compared with the values in Table 4.2, the variances of the loss functions before and after swap for both life insurer and annuity provider are far enlarged except for V ar(LL) based on ( ˆwAl , ˆwaA) for the independent model. The HEs generally increase for the independent model, and decrease for the joint-k and co-integrated models. Among the four models, the co-integrated and independent models achieve the highest and the lowest hedge effectiveness, respectively, for HE(L + A), HE(L) and HE(A) based on ( ˆwL+Al , ˆwaL+A), ( ˆwLl, ˆwLa) and ( ˆwAl , ˆwaA), respectively. The joint-k model can also pro-duce high hedge effectiveness. Thus, under model-free block bootstrap method, the joint-k and co-integrated models still outperform the other two models.
Independent V ar(Ll) V ar(LL) V ar(La) V ar(LA) HE(L + A) HE(L) HE(A) Table 5.2: comparisons of sample variances (×1023) and HE’s (block bootstrap)
Chapter 6 Conclusion
The strategy of natural hedging in this project shows a significant effect on reducing risks of insurance and annuity portfolios by swapping business. The robustness testing exhibits that the optimal weights and the variances of the loss functions after swap for both life insurer and annuity provider obtained from four multi-population models are robust to simulations.
The performances of hedging mortality and longevity risks for each model are compared in two ways; one is based on parametric multi-population mortality mod-els, and the other is based on non-parametric block bootstrap method. Both ways suggest that the joint-k and co-integrated models outperform the independent and augmented common factor models, which implies that the improvement on mortal-ity rates between two populations tend to become more related, and assuming a stronger bond between two time-varying factors for two populations seems to be more reasonable when forecasting future mortality rates.
Although natural hedging can achieve the goal of reducing the variance of the loss, future research can still be carried out so that it can be further applied to more practical situations. For example, both a life insurer and an annuity provider won’t swap their business directly in practice. Generally, companies prefer to transfer their business to a financial intermediary called SPV (special purpose vehicle) who is in charge of business swapping. Another problem is that the portfolio of life (annuity) business in this project consists of only one type of life insurance (annuity) product, the(65 − x)-payment whole life insurance ((65 − x)-payment and (65 − x) deferred whole life annuity due), which is too simple to be practical.
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