Natural Hedges With Immunization Strategies of Mortality and Interest Rates

Abstract

In this paper, we first derive closed-form formulas for mortality-interest durations and convexities of the prices of life insurance and annuity products with respect to an instantaneously proportional change and an instantaneously parallel movement, respectively, in μ* (the force of mortality-interest), the addition of μ (the force of mortality) and δ (the force of interest). We then build several mortality-interest duration and convexity matching strategies to determine the weights of whole life insurance and deferred whole life annuity products in a portfolio and evaluate the value at risk and the hedge effectiveness of the weighted portfolio surplus at time zero. Numerical illustrations show that using the mortality-interest duration and convexity matching strategies with respect to an instantaneously proportional change in μ* can more effectively hedge the longevity risk and interest rate risk embedded in the deferred whole life annuity products than using the mortality-only duration and convexity matching strategies with respect to an instantaneously proportional shift or an instantaneously constant movement in μ only.

A new option for mortality–interest rates

“…A linear relationship between two sequences of mortality rates has been used in mortality modeling (see Lin & Tsai, 2015; Lin et al, 2015; Tsai & Yang, 2015) and in natural hedges with mortality immunization (see Tsai & Chung, 2013; Lin & Tsai, 2013, 2014). The force of mortality–interest moves approximately linearly, which also has been used in Lin and Tsai (2020) for natural hedges with mortality–interest immunization. Motivated by the strand of literature, we use a simple linear regression to model and forecast the force of mortality or the force of mortality–interest for each age $$x$$ as follows: $$${{\rm{\Pi }}}_{x,t}={a}_{x}^{n}+{b}_{x}^{n}\times {{\rm{\Pi }}}_{x,t-n}+{\epsilon }_{x,t}^{n},$$$ where $${{\rm{\Pi }}}_{x,t}$$ and $${{\rm{\Pi }}}_{x,t-n}$$ are two sequences of $${\mu }_{x,t}$$ or $${\mu }_{x,t}^{* }$$ data, $$t={t}_{L}+n,{t}_{L}+n+1,\ldots ,{t}_{U},n$$ is a predetermined lag year, $${\epsilon }_{x,t}^{n}$$ is the normal distributed error term with mean zero and variance $${\sigma }_{x,n}^{2}$$, and any two …”

“…A linear relationship between two sequences of mortality rates has been used in mortality modeling (see Tsai & Yang, 2015) and in natural hedges with mortality immunization (see Tsai & Chung, 2013;Lin & Tsai, 2013. The force of mortality-interest moves approximately linearly, which also has been used in Lin and Tsai (2020) for natural hedges with mortality-interest immunization. Motivated by the strand of literature, we use a simple linear regression to model and forecast the force of mortality or the force of mortality-interest for each age x as follows:…”

“…We measure hedge effectiveness in terms of the variance reduction ratio as follows (see Li & Luo, 2012; Lin & Tsai, 2020): $$$HE({\sigma }^{2})=\frac{{\sigma }_{u}^{2}-{\sigma }_{h}^{2}}{{\sigma }_{u}^{2}}=1-\frac{{\sigma }_{h}^{2}}{{\sigma }_{u}^{2}},$$$ where $${\sigma }_{h}^{2}$$ represents the variance of the present values of all net cash flows with a hedge portfolio, and $${\sigma }_{u}^{2}={\sigma }^{2}[{L}_{u}^{A}]$$ ($${\sigma }_{u}^{2}={\sigma }^{2}[{L}_{u}^{A* }]$$) denotes the variance of the present values of all payments made by the annuity provider without a hedge portfolio. The largest hedge effectiveness can be achieved by minimizing $${\sigma }_{h}^{2}={\sigma }^{2}[{L}_{h,i}^{A}]$$ ($${\sigma }_{h}^{2}={\sigma }^{2}[{L}_{h,i}^{A* }]$$) to obtain $${u}_{k}^{A,M}$$ ($${u}_{k}^{A,MI}$$), the units of mortality (mortality–interest) call options purchased by the annuity prov...…”

“…We measure hedge effectiveness in terms of the variance reduction ratio as follows (see Li & Luo, 2012;Lin & Tsai, 2020):…”