Robust Mean–Variance Hedging of Longevity Risk

• Published date: 01 April 2017
• Author: Hong Li,Anja Waegenaere,Bertrand Melenberg
• DOI: http://doi.org/10.1111/jori.12201
• Date: 01 April 2017

©2017 The Journal of Risk and Insurance. Vol.84, No. S1, 459–475 (2017).

DOI: 10.1111/jori.12201

Robust Mean–Variance Hedging of Longevity Risk

Hong Li

Anja De Waegenaere

Bertrand Melenberg

Abstract

Parameter uncertainty and model misspecification can have a significant im-

pact on the performance of hedging strategies for longevity risk. Tomitigate

this lack of robustness, we proposean approach in which the optimal hedge is

determined by optimizing the worst-case value of the objective function with

respectto a set of plausible probability distributions. In the empirical analysis,

we consider an insurer who hedges longevity risk using a longevity bond,

and we compare the worst-case (robust) optimal hedges with the classical

optimal hedges in which parameter uncertainty and model misspecification

are ignored. We find that unless the risk premium on the bond is close to

zero, the robust optimal hedge is significantly less sensitive to variations in

the underlying probability distribution. Moreover,the robust optimal hedge

on average outperforms the nominal optimal hedge unless the probability

distribution used by the nominal hedger is close to the true distribution.

Introduction

Realized survival rates in an insured population can deviate substantially from the

ex ante estimates. This source of risk, referred to as longevity risk, can have a signifi-

cant impact on the balance sheets of pension funds and annuity providers (Biffis and

Blake, 2010). We focus on capital market solutions of longevity risk. As compared to

traditional approaches such as reinsurance, mortality-linked derivatives may provide

cheaper and more flexible options of mitigating longevity risk exposure (Cairns et al.,

2014).1

Hong Li is an Assistant Professor in the School of Finance at Nankai University,Tianjin, China.

Anja De Waegenaereand Bertrand Melenberg are both Professors in the Department of Econo-

metrics and Operations Research, TilburgUniversity, Tilburg, Netherlands. Also, they are both

affiliated with Netspar, Netherlands. Li can be contacted via e-mail: hong.li@nankai.edu.cn.

The authors are grateful to Torsten Kleinow, Johnny Li, Andrew Hunt, the editor, two anony-

mous reviewers, and the seminar participants at the 18th International Congress on Insurance:

Mathematics and Economics, Longevity 10, and Nanjing University for their useful comments.

All errors are our responsibility. This work is part of the research programManaging Longevity

Risk, which is financed by the Netherlands Organisation for Scientific Research (NWO).

1A list of existing transactions of mortality-linked derivative can be found on www.artemis.bm/

library/longevity swaps risk transfers.html.

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460 The Journal of Risk and Insurance

Our article contributes to the rapidly growing literatureon longevity risk management

(see, e.g., Cox et al., 2013; Biffis et al., 2014; Wong,Chiu, and Wong, 2015) by explicitly

taking estimation inaccuracy and model risk into account in the optimization prob-

lem. Empirical studies show that the solutions to portfolio optimization problems can

be rather sensitive to small errors in the assumed underlying probability distribution

(Garlappi, Uppal, and Wang, 2007). This lack of robustness is a concern in particular

when longevity risk is involved. As discussed in Cairns et al. (2011), different mortal-

ity models could generate significantly different mortality forecasts, and the choice of

mortality model is often unclear ex ante. Moreover, even for a given model, mortality

forecasts could vary substantially when a different calibration window is chosen (see,

e.g., Booth, Maindonald, and Smith, 2002; Cairns, Blake, and Dowd, 2006; Li, 2015).

In the presence of these potential estimation inaccuracies, it is important that the

insurer’s hedged portfolio would perform well even when the true probability distri-

bution differs from the estimated one. Cairns (2013) evaluates the robustness of static

hedging strategies with respect to several sources of risk, including parameter risk

that results from recalibrating the model as new observations arrive. Cox et al. (2013)

use the method of moments to derive bounds for quantities like the value-at-risk of

portfolios. The bounds apply to a wide class of distributions and are used to inves-

tigate the robustness of their optimal portfolio choices to changes in the underlying

distribution. We propose an alternative approach in which both model misspecifica-

tion and parameter uncertainty are taken into account in the optimization process.

Instead of optimizing the objective function with respect to a single estimated proba-

bility distribution, the insurer considers a set of plausible probability distributions and

optimizes with respect to the worst-case distribution in this set. Like Cairns (2013), we

focus on robustness of static hedging strategies. In a well-functioning market with ac-

tive trade, dynamic hedging strategies (as in, e.g., Biffis, Denuit, and Devolder, 2010)

may be preferable since they could lead to smaller tracking errors of the insurer’s

liabilities. However, when markets are still immature and contracts are often traded

over the counter, dynamic hedging may be infeasible or too costly (Pelsser, 2003).

In our empirical application, we consider an insurer who holds a portfolio of single life

annuities for Dutch males and wishes to hedge the longevity risk in this portfolio using

longevity bonds. We compare the performance of the robust optimal hedge and its

nominal counterpart in which parameter uncertainty and model misspecification are

ignored. In the nominal approach, the insurer estimates the probability distribution

of the (hedged) liabilities using the Lee and Carter (1992) model for the mortality

rates of the total population of Dutch males, and the Plat (2009) model for differences

between mortality rates of annuitants and those of the total population. Then, the

portfolio of longevity bonds is determined as if these estimated distributions are the

true distributions. In the robust approach,the insurer takes into account the possibility

that: (1) the “true” model for mortality rates of Dutch males is the CBD (Cairns, Blake,

and Dowd, 2006) model (i.e., model misspecification), and (2) for both the Lee–Carter

model and the CBD model, the true parameters may deviate from the estimated ones

(i.e., parameter risk). Wefind that the robust insurer in general holds more bonds than

the nominal insurer,indicating that concerns about parameter uncertainty and model

misspecification lead to more conservative hedging strategies. When the true model

is indeed the CBD model, the robust optimization on average yields substantially