Mortality Risk Management Under the Factor Copula Framework—With Applications to Insurance Policy Pools

Abstract

Mortality risk is one of the core risks that life insurers undertake. The uncertain future lifetime of each insured represents one risk factor, and the dependence structure among these risk factors determines the aggregate risk of an insurance policy pool. We propose using factor copulas to describe the dependence structure among the future lifetimes of numerous insureds. This differs from Chen, MacMinn, and Sun (2015) in that their focus is on pricing the securities linked to several mortality indexes. To mitigate the systematic mortality risk associated with an insurance pool, the insurer may purchase an asset exposed to similar systematic risk. We thus set up a two-factor copula framework and solve for the optimal investment amount in the asset. In numerical illustrations, we employ real-case data from a life insurer and a life settlement market maker involving hundreds of policies.

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Notes

1 For brevity, we use the term “mortality risks” in a broad sense to include both mortality and longevity risks hereafter. Note that the risks concerned are systematic risks.

2 The case had legal disputes later over conduct / ethical issues.

3 The literature applies copula to two types of risk factors associated with rates (such as stock returns and mortality rates) and time (e.g., time of default and future lifetime) respectively. Oh and Patton (2015) apply a factor copula to the returns of S&P 100 stocks, while Chen, MacMinn, and Sun (2015) apply a factor copula to the mortality rates of six countries. The formal Oh and Patton paper filters 100 time series with AR(1)–GJR-GARCH models and then derives a proposition to determine the number of common factors in the copula; the latter fits the mortality improvement rates of six countries with an ARMA-GARCH model and imposes the assumption of one common factor upon the copula. The SMM (simulated method of moments) method used by Chen, MacMinn, and Sun (2015) is from Oh and Patton (2013). The pricing method used by Chen, MacMinn, and Sun (2015) is from Stutzer (1996) and had also been used in a few previous studies on pricing mortality-linked securities (see, e.g., Li 2010; Kogure and Kurachi 2010; Li and Ng 2011). In short, Chen, MacMinn, and Sun (2015) build upon the previous literature by providing new pricing and risk mitigation techniques for mortality bonds.

This article and other papers in the literature dealing with default risk apply a copula to the risk factor associated with time. In this research the risk factor is future lifetime, while the others are time to default. Hull and White (2004) may be regarded as a benchmark to be followed by many later papers, including this research, Laurent and Gregory (2005), and even Oh and Patton (2015). Laurent and Gregory (2005) extend Hull and White (2004) in showing that most complex multiname credit derivatives can be priced in a semi-analytical way under a one-factor copula model; we apply the methodology of Hull and White (2004) to the future lifetimes of insureds under the two-factor copula framework for the purpose of analyzing the risk mitigation effects of life settlements. The Hull–White model is for financial instruments, while our model is for new life market instruments. The implications of our results are thus different from those of other results in the literature. In short, we build upon the previous literature by showing the risk mitigation effects of life settlements.

4 Note that Laurent and Gregory (2005) do not conduct parameter estimation either, since they do not have data on the defaults of individual names. On the other hand, Chen, MacMinn, and Sun (2015) applied the factor copula to the residuals of the ARMA-GARCH processes fitted to the mortality rate time series of several countries. Wang, Yang, and Huang (2015) also applied copulas to filtered mortality rate time series, while Zhu, Tan, and Wang (2017) applied copulas to the residuals of fitting Lee–Carter models. Difficulties in model/parameter estimation arise when one tried to apply copulas to individuals’ mortality (as Laurent and Gregory [2005] encountered with individual names’ defaults).

5 Note that V_wl (j) is equivalent to negative policy reserves. Policy reserves are equal to the sum of the present values of expected cash outflows minus that of expected cash inflows, while V_wl (j) is the sum of the present values of expected cash inflows minus that of expected cash outflows.

6 Note that τ_j, T_i, and t need not be integers. The premium payment schedules of life settlements are usually in months.

7 An alternative assumption could be conditional independence in a Cox-process setup as in survival analysis (Hougaard 2000). These types of models correspond to the Clayton copula (a special case of an Archimedean copula) when the marginal distributions are some well-known parametric distributions. Since the marginal distributions are deduced from mortality tables and are thus nonparametric, employing such types of models would be cumbersome. We thus choose the t-copula as the alternative model to the normal copula for the dependence structure.

8 We have no way to speculate which copula model is more appropriate for our data since all insureds were alive at the time when we secured the data. Furthermore, we can only conjecture that tail dependence among the future lifetimes of these insureds may be caused by pandemics, catastrophes, breakthroughs in gene therapies, etc., like the tail dependence among the future lifetimes of the general public.

9 A default on a bond is similar to the death of a policyholder. Both are usually rare events with significant consequences. Applying default risk models to insurance policies can also be seen in Chaplin, Aspinwall, and Venn (2011). They did not, however, employ factor copula to model dependence.

10 We prefer the term “risk mitigation” to “hedging” in this article since we are dealing with liabilities, while standard hedging is usually for owned assets. This preference is also because the setup involves significant basis risk as reflected by group factors M_ls and M_wl.

11 The reasoning for the heterogeneity between two groups of insureds being smaller than that between two groups of company borrowers is as follows. Personal mortality is determined by population biological factors, personal genes, social–economic factors, and personal lifestyles; personal default risk depends on macroeconomics and personal financial conditions. The lack of the biological similarity factor thus may results in the higher heterogeneity across (groups of) corporate borrowers.

12 We may refer to Li and Hardy (2011) for educated guesses on whether the global trend, the trend in a specific insurance pool, or the idiosyncratic risk dominates the mortality dynamics. They applied the augmented common factor model to the female populations of Canada and the United States. From their Figure 8, we may roughly compare B(x) × K(t) with b(x,i) × k(t,i) and say that the global trend seems to be more significant than the population-specific factors. The idiosyncratic risk is probably immaterial since the model has a very high explanation ratio of 0.9931. The forces driving the dynamics of mortality rates may also make the dynamics of future lifetimes subject to significant global trends, moderate population-specific factors, and insignificant idiosyncratic risk.

13 Note that we do not have to establish time-series models for the global factor and group factors since we do not aim to engage in dynamic hedging. Our proposed strategy is to minimize the risk of the insurance pool from the random shocks of risk factors at a given point in time. This is similar to the duration matching in interest rate risk management, and such strategies do not require dynamic models of interest rates.

14 For the sake of brevity, we report the results of one risk measure under one factor copula framework only. All results are consistent across risk measures and copulas.

15 The value at risk of the insurance pool is reduced as well: from $72,183,048 to $64,172,539. Such a reduction is understandable since a smaller d represents a smaller sensitivity to the shocks on the group factor M_wl.

16 On the other hand, VaR(V_wl) should be independent of a since the risk of the life insurance pool should have nothing to do with the risk of life settlements pertaining to the global factor M. Table 4 does display such a result: $72,183,048 vs. $71,127,485. The small difference is due to random variations in simulations.

17 The value 8.6% is the average of the expected IRRs on individual life settlements. The data from Coventry contain the acquisition cost, life expectancy, scheduled premiums, and death benefit for each policy. We are thus able to solve for the expected IRR.

18 Life settlements are probably at the same level of (in)effectiveness as other mortality-linked securities because the underlying populations of these hedging vehicles differ from life insurance policies carried by life insurers and the markets of these vehicles are still at their early development stage. In terms of product form/format, life settlements are like longevity bonds to life insurers and can be used as a mortality risk management tool in the asset-liability management (ALM) framework. Mortality swaps (including q-forwards), futures, and options are used as derivatives that have small or no values at contract inception, on the other hand.

Additional information

Funding

The authors are grateful to the Risk and Insurance Research Center of National Chengchi University for its financial support and for financial support from the Ministry of Science and Technology (project number: 102-2410-H-004-027-MY3).