On the optimal hedge ratio in index-based longevity risk hedging

Abstract

In an index-based longevity hedge, the so-called longevity basis risk arises from the potential mismatch between the hedging instrument and the annuity portfolio being hedged, due to the differences in the underlying populations and payoff structures. To reduce the impact of this longevity basis risk and increase the hedge effectiveness, an optimal position in the hedging instrument should be determined appropriately with regard to the nature of the two populations and also the timing of the payments. In this paper, we examine some analytical results on the optimal hedge ratio in hedging the longevity exposure of an annuity portfolio with index-based longevity- or mortality-linked securities. This optimal hedge ratio may serve as a convenient starting point for constructing an index-based hedge. We also conduct a cost–benefit analysis using different financial objectives. Our results are based on data of Australian public sector pensioners.

Appendix

The entire modelling and hedging process can be summarised as follows:

• (a) The M7-M5, CAE + Cohorts, and possibly other models are fitted to the book and reference data.

• (b) The most optimal model is chosen based on the BIC and/or other quantitative and qualitative criteria.

• (c) Future book and reference mortality rates are simulated from the selected model. Depending on the portfolio and index-based derivative structures, future book and reference cash flows are then generated from the simulated values.

• (d) The optimal position(s) of the index-based derivative(s) are computed based on the simulated cash flows and the investment or hedging objective(s) being adopted.

To incorporate both process error (variability in the time series) and parameter error (uncertainty in parameter estimation) into the simulation of future mortality rates, the residuals bootstrapping method [14, 16] is applied as below:

• (a) The residuals from fitting the M7-M5 model or CAE + Cohorts model to the actual data are resampled with replacement. The residuals are resampled for each age-time cell within all $x$ and $t$. Note that the standardised deviance residuals are computed by the two formulae $r_{x,t}^{R} = \tfrac{1}{{\sqrt {\hat{\phi }^{R} } }}{\text{sign}}\left( {d_{x,t}^{R} - e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)\sqrt {2\left( {d_{x,t}^{R} \ln \left( {{{d_{x,t}^{R} } \mathord{\left/ {\vphantom {{d_{x,t}^{R} } {\left( {e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}}} \right. \kern-0pt} {\left( {e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}}} \right) - d_{x,t}^{R} + e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}$ and $r_{x,t}^{B} = \tfrac{1}{{\sqrt {\hat{\phi }^{B} } }}{\text{sign}}\left( {d_{x,t}^{B} - e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)\sqrt {2\left( {d_{x,t}^{B} \ln \left( {{{d_{x,t}^{B} } \mathord{\left/ {\vphantom {{d_{x,t}^{B} } {\left( {e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}}} \right. \kern-0pt} {\left( {e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}}} \right) - d_{x,t}^{B} + e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}$ respectively for the reference and book populations. The dispersion parameters are estimated by the two formulae $\hat{\phi }^{R} = \tfrac{1}{{n_{d}^{R} - n_{p}^{R} }}\sum\nolimits_{x,t} {2\left( {d_{x,t}^{R} \ln \left( {{{d_{x,t}^{R} } \mathord{\left/ {\vphantom {{d_{x,t}^{R} } {\left( {e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}}} \right. \kern-0pt} {\left( {e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}}} \right) - d_{x,t}^{R} + e_{x,t}^{R} \, \hat{m}_{x,t}^{R} } \right)}$ and $\hat{\phi }^{B} = \tfrac{1}{{n_{d}^{B} - n_{p}^{B} }}\sum\nolimits_{x,t} {2\left( {d_{x,t}^{B} \ln \left( {{{d_{x,t}^{B} } \mathord{\left/ {\vphantom {{d_{x,t}^{B} } {\left( {e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}}} \right. \kern-0pt} {\left( {e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}}} \right) - d_{x,t}^{B} + e_{x,t}^{B} \, \hat{m}_{x,t}^{B} } \right)}$, in which $n_{d}^{R}$ ($n_{d}^{B}$) and $n_{p}^{R}$ ($n_{p}^{B}$) are the number of observations and the number of effective parameters of the reference (book) population, $e_{x,t}^{R}$ and $e_{x,t}^{B}$ are the central exposed to risk measures, $d_{x,t}^{R}$ and $d_{x,t}^{B}$ are the observed numbers of deaths, and $\hat{m}_{x,t}^{R}$ and $\hat{m}_{x,t}^{B}$ are the fitted central death rates.

• (b) The inverse functions of the residuals formulae above are used to convert the resampled residuals into a pseudo sample of the number of deaths, $d_{x,t}^{R \, \left( i \right)}$ and $d_{x,t}^{B \, \left( i \right)}$, for all $x$ and $t$, in which the superscript $\left( i \right)$ refers to the ith scenario.

• (c) The M7-M5 model or CAE + Cohorts model is fitted to the pseudo data sample from step (b) and the corresponding model parameters ($\kappa_{t,1}^{R \, \left( i \right)}$, $\kappa_{t,2}^{R \, \left( i \right)}$, $\kappa_{t,3}^{R \, \left( i \right)}$, $\gamma_{t - x}^{R \, \left( i \right)}$, $\kappa_{t,1}^{B \, \left( i \right)}$, $\kappa_{t,2}^{B \, \left( i \right)}$, or $\alpha_{x}^{R \, \left( i \right)}$, $\beta_{x}^{R \, \left( i \right)}$, $\kappa_{t}^{R \, \left( i \right)}$, $\gamma_{t - x}^{R \, \left( i \right)}$, $\alpha_{x}^{B \, \left( i \right)}$, $\kappa_{t}^{B \, \left( i \right)}$) are calculated.

• (d) The selected time series processes are fitted to the time-varying model parameters based on the pseudo data sample ($\kappa_{t,1}^{R \, \left( i \right)}$, $\kappa_{t,2}^{R \, \left( i \right)}$, $\kappa_{t,3}^{R \, \left( i \right)}$, $\gamma_{t - x}^{R \, \left( i \right)}$, $\kappa_{t,1}^{B \, \left( i \right)}$, $\kappa_{t,2}^{B \, \left( i \right)}$, or $\kappa_{t}^{R \, \left( i \right)}$, $\gamma_{t - x}^{R \, \left( i \right)}$, $\kappa_{t}^{B \, \left( i \right)}$) from step (c) to simulate their future values across time.

• (e) Samples of future mortality rates, $q_{x,t}^{R \, \left( i \right)}$ and $q_{x,t}^{B \, \left( i \right)}$, for all $x$ and future $t$, are obtained from inserting the calculated parameters and simulated values based on the pseudo data sample from steps (c) and (d) into the M7-M5 or CAE + Cohorts model structure. This set of future mortality rates characterises one random future scenario.

• (f) Steps (a) to (e) are repeated to generate 5000 random future scenarios.

• (g) In each random scenario, the future number of survivors in the annuity portfolio at time t + 1 is simulated by the assumption $l_{x + 1,t + 1}^{B \, \left( i \right)} \sim {\text{Binomial}}\left( { \, l_{x,t}^{B \, \left( i \right)} \, , \, 1 - q_{x,t}^{B \, \left( i \right)} } \right)$. This step can be omitted if the annuity portfolio is assumed to be infinitely large.

The computation time of the above residuals bootstrapping process using Excel VBA can be up to 1 day. If the software R is used instead, the time it takes is like threefold. One way to shorten the computation time is to apply some approximate parameter estimation methods. For instance, the M7-M5 model can be fitted first via a simple linear regression for each year $t$, like ${\text{logit }}q_{x,t}^{R} = \kappa_{t,1}^{R} + \left( {x - \bar{x}} \right) \, \kappa_{t,2}^{R} + \left( {\left( {x - \bar{x}} \right)^{2} - \sigma_{x}^{2} } \right) \, \kappa_{t,3}^{R}$ without the cohort parameter. Then taking the estimated $\hat{\kappa }_{t,1}^{R}$, $\hat{\kappa }_{t,2}^{R}$, and $\hat{\kappa }_{t,3}^{R}$ as given, the residuals for all $x$ and $t$ are fitted with another linear regression as ${\text{logit }}q_{x,t}^{R} - \hat{\kappa }_{t,1}^{R} - \left( {x - \bar{x}} \right) \, \hat{\kappa }_{t,2}^{R}$$- \left( {\left( {x - \bar{x}} \right)^{2} - \sigma_{x}^{2} } \right) \, \hat{\kappa }_{t,3}^{R}$$= \sum\nolimits_{c} {I_{c} \, \gamma_{c}^{R} }$, in which $I_{c}$ is an indicator variable which is one if $c = t - x$ and zero otherwise. Finally, the book component’s parameters for each $t$ are computed by ${\text{logit }}q_{x,t}^{B} - \hat{\eta }_{x,t}^{R} = \kappa_{t,1}^{B} + \left( {x - \bar{x}} \right) \, \kappa_{t,2}^{B}$, which is again a simple linear regression. The estimation may be improved by using the weighted least squares and the central exposed to risk measures as the weights. In the same vein, for the CAE + Cohorts model, the singular value decomposition (SVD) or the principal component analysis (PCA) can be used to calculate the model parameters.

Regarding numerical optimisation, the algorithms used by different software may yield slightly different hedging results. The choice depends on the user’s own preference and experience. In the longevity swap example, there is only one unknown quantity to determine in order to achieve a single objective. If the annuity portfolio has multiple cohorts instead and if longevity swaps for more cohorts and with different maturities are also available and included in the hedge, the numerical optimisation procedure can readily be extended to find the optimal weights of multiple swaps simultaneously. When the number of unknown quantities increases, however, there may be a higher chance of getting trapped in a local optimum rather than reaching the true global optimum. Some simple methods in practice to alleviate this problem include trying other appropriate starting values, applying different numerical algorithms, simplifying the model structures where feasible, and repeating the optimisation procedure with the preceding solutions as the new starting values.

For instance, suppose in an Excel spreadsheet the cells A2 and A3 contain the notional amounts of two swaps and the cell A1 calculates the level of longevity risk reduction based on the simulations and the weights of the swaps. The Excel VBA codes below can be used to find numerically the optimal weights of the swaps that maximise the level of longevity risk reduction: