Applications of Mortality Durations and Convexities in Natural Hedges

Abstract

Defining and deriving the mortality durations and convexities of the prices of life insurance and annuity products with respect to an instantaneously proportional change and an instantaneously parallel shift, respectively, in μs (the forces of mortality), qs (the one-year death probabilities), ps (the one-year survival probabilities), ln (μ)s, (q/p)s, and ln (q/p)s, this article applies 24 proposed duration/convexity matching strategies classified into seven groups to determine the weights of two products in an insurance portfolio. The hedging performances of some qualified matching strategies selected as representatives are evaluated by comparing their Value at Risk (VaR) values and variance reduction ratios for a base scenario. We also test some specific scenarios for the population basis risk, model risk, volatility and jump risks, and interest rate risk to see the impacts on the matching strategies. Numerical examples show that some convexity matching strategies overall outperform the others in the VaR value and in the effectiveness of hedging both longevity and mortality risks for two kinds of insurance portfolios.

APPENDIX A. ADJUSTMENT FUNCTIONS , DURATION FUNCTIONS , AND CONVEXITY FUNCTIONS

TABLE 2 Adjustment Functions for k ⩾ 1 with for _kp_x

TABLE 3 Duration Functions for k ⩾ 1 with for

The sign of some duration functions is very obvious by intuition. Since is the slope of the tangent line to at γ = 0, which can be approximated by the slope of the secant line connecting (0, ) and (), _kp*_x = _kp_x · f^λ_Ux(k, γ) > ( < ) _kp_x or 1 (see Table A.1) will result in a positive (negative) slope for a positive constant or proportional change γ in U_x. Therefore, both and are positive (negative) for U_x = P_x (U_x = μ_x, Q_x and Q_x/P_x), and is negative for U_x = ln (μ_x) and ln (Q_x/P_x).

For the duration functions in Table A.2, decreases from zero at k = 0 for λ = p, c and U_x = μ_x, Q_x, Q_x/P_x and for λ = c and U_x = ln (μ_x), ln (Q_x/P_x), whereas increases from zero at k = 0 for λ = p, c. In general, for U_x = ln (μ_x) and ln (Q_x/P_x). To see this, since ln (p_{x + i − 1}) < 0, the inequality ln [ − ln (p_{x + i − 1})] < 0 or q_{x + i − 1} < 1 − e^{− 1} = 0.632121 for i = 1, …, k ensures . Similarly, ln (q_{x + i − 1}/p_{x + i − 1}) < 0 or q_{x + i − 1} < 0.5 for i = 1, …, k implies . Theoretically, q_{x + i − 1} < 0.5 does not always hold; however, it holds in practice for most mortality rates for the ages near the limiting age. Even if q_{x + i − 1} < 0.5 does not hold for some i, the resulting and are almost certainly positive for U_x = ln (μ_x) and ln (Q_x/P_x). For the convexity functions in Table A.3, increases from zero at k = 0 for λ = p, c and U_x = μ_x, Q_x, P_x, Q_x/P_x, whereas decreases from zero at k = 0 to negative and then increases (to positive for most cases) for λ = p, c and U_x = ln (μ_x), ln (Q_x/P_x). As a result, for most cases for λ = p, c and U_x = ln (μ_x), ln (Q_x/P_x).

TABLE 4 Convexity Functions for k ⩾ 1 with for

APPENDIX B. RELATIONSHIPS AND ORDERING AMONG 12 s

There are some relationships among these 12 . From Table A.2 in Appendix A, we have , , and for each integer k, which can be seen by intuitive reasons as follows:

• ln (U*_x) = ln (U_x) + γ implies U*_x = U_x × e^γ = (1 + γ′) × U_x where γ = ln (1 + γ′) has the same sign as γ′, and γ → 0 if and only if γ′ → 0; thus, for U_x = μ_x and Q_x/P_x.

• Since Q*_x = Q_x + γ if and only if P*_x = 1 − Q_x* = P_x − γ, we have .

• μ*_x = μ_x + γ or − ln (P*_x) = −ln (P_x) + γ gives P*_x = P_x × e^{− γ} = (1 − γ′) × P_x where γ = −ln (1 − γ′) has the same sign as γ′, and γ → 0 if and only if γ′ → 0; therefore, .

Moreover, each of the four equalities above gives and , where ± stands for + or − depending on or for (λ₁, U¹_x, λ₂, U²_x) = (p, μ_x, c, ln (μ_x)), (c, μ_x, p, P_x), (c, Q_x, c, P_x), or (p, Q_x/P_x, c, ln (Q_x/P_x)). Hence, by (3.4(3.4) ).

Next, we can expand ln (p_{x + i − 1}) = ln (1 − q_{x + i − 1}) in and to ≈ −q_{x + i − 1}, and expand − (1/p_{x + i − 1} − 1) = −[1/(1 − q_{x + i − 1}) − 1] in to − ∑^∞_{j = 1}q_{x + i − 1}^j ≈ −q_{x + i − 1}. Similarly, we expand − 1/p_{x + i − 1} in to − ∑^∞_{j = 0}q_{x + i − 1}^j ≈ −(1 + q_{x + i − 1}). Thus, where A⪆B denotes A > B and A ≈ B, that is, A is larger than B by a tiny amount. Moreover, since ln (p_{x + i − 1}) · ln [ − ln (p_{x + i − 1})] > −q_{x + i − 1} · ln (q_{x + i − 1}/p_{x + i − 1}) for q_{x + i − 1} > 0.90855 and both functions can be approximated by ln (p_{x + i − 1}) · ln (q_{x + i − 1}) < [ln (0.5)]² < 1, we have for almost all cases. Combining this with the order above, we get The corresponding mortality durations of follow the same order accordingly.