Risk Management of Policyholder Behavior in Equity‐Linked Life Insurance

• Author: Maciej Augustyniak,Mary R. Hardy,Anne MacKay,Carole Bernard
• Published date: 01 June 2017
• DOI: http://doi.org/10.1111/jori.12094
• Date: 01 June 2017

©2015 The Journal of Risk and Insurance. Vol.84, No. 2, 661–690 (2017).

DOI: 10.1111/jori.12094

Risk Management of Policyholder Behavior in

Equity-Linked Life Insurance

Anne MacKay

Maciej Augustyniak

Carole Bernard

Mary R. Hardy

Abstract

The financial guarantees embedded in variable annuity contracts expose in-

surers to a wide range of risks, lapse risk being one of them. When policy-

holders’ lapse behavior differs from the assumptions used to hedge variable

annuity contracts, the effectiveness of dynamic hedging strategies can be sig-

nificantly impaired. By studying how the fee structureand surrender charges

affect surrender incentives, we obtain new theoretical results on the optimal

surrender region and use them to design a marketable contract that is never

optimal to lapse.

Introduction

Variable annuities (VAs) and other types of equity-linked insurance products have

grown in popularity over the last 20 years. These products combine a participation

in equity performance with insurance features and are therefore normally sold by

insurance companies. For example, a policyholder entering into a VA contract pays

an initial premium and chooses to invest it in one or more mutual funds.1However,

unlike mutual funds, VAs offer various financial guarantees upon death of the poli-

cyholder or at maturity of the contract that protect the policyholder’s capital against

Anne MacKay is with the Department of Mathematics at ETH Z ¨

urich; e-mail: anne.mackay@

math.ethz.ch. Maciej Augustyniak is with the Department of Mathematics and Statistics at

the University of Montreal; e-mail: augusty@dms.umontreal.ca. Carole Bernard is with the

Department of Accounting, Law and Finance at Grenoble Ecole de Management; e-mail:

carole.bernard@grenoble-em.com. Mary R. Hardy is with the Department of Statistics and

Actuarial Science at the University of Waterloo; e-mail: mrhardy@uwaterloo.ca. All authors

acknowledge support from NSERC. C. Bernard and M. R. Hardy acknowledge support from

research grants awarded by the Global Risk Institute and the Society of Actuaries CAE (Centers

of Actuarial Excellence). A. MacKay also acknowledges the support of the Hickman scholarship

of the Society of Actuaries. The authors would like to thank P. Forsyth, B. Li, A. Kolkiewicz,

and D. Saunders for helpful discussions.

1In some cases, additional premium amounts can be deposited in the VA account at regular

intervals.

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662 The Journal of Risk and Insurance

market downturns. Some VAs also guarantee minimum periodic withdrawals or in-

come amounts for a fixed or varying period of time (see Hardy, 2003, for more details).

Financial options embedded in VAs have payoff structures that are similar to those

of options traded on stock exchanges, but include three distinctive features that com-

plicate their pricing and risk management. First, they include guarantees that are

triggered upon death of the policyholder or after a long-term maturity, which gener-

ally exceeds 5 years. Second, they are financed by a fee, which is typically paid as a

fixed proportion of the account value, as opposed to being paid up front. Third, the

policyholder has the flexibility to withdraw the account value and cancel the contract

before maturity by paying a penalty (which may be zero after an initial period) to

the insurer. These nonstandardfeatures give rise to risks that are difficult to mitigate,

such as mortality risk, equity risk, interest rate risk, and policyholder behavior risk.

As many insurance companies suffered large losses on their VA business during the

recent financial crisis, it is vital to examine original ways to manage these risks.

This article focuses on surrender or lapse risk2and studies how an insurer can miti-

gate this risk with product design. Surrender risk refers to the risk that policyholders

terminate their contract prior to maturity and it is a specific type of policyholder be-

havior risk. Surrenders may impair the insurer’s business for several reasons. First, the

insurance company may not be able to fully recover the initial expenses and up-front

costs of acquiring new business (Pinquet, Guill´

en, and Ayuso, 2011), as well as setting

up a hedge for the guarantees in the contract. Second, surrenders may cause liquidity

issues (and loss of future profits) (Kuo, Tsai, and Chen, 2003) as there is potential for

large cash demands in very short time frames. Finally, surrenders may give rise to an

adverse selection problem because policyholders with insurability issues tend not to

lapse their policies (on this topic, see Benedetti and Biffis, 2013). Due to these nega-

tive consequences, insurers typically include surrender charges in their VA products

to discourage lapsation (Milevsky and Salisbury, 2001). These charges are generally

high in the first few years of the contract to provide a way for the insurer to recover

acquisition expenses. Although surrender charges act as a disincentive for policyhold-

ers to lapse, there are many situations where surrendering can be advantageous for

the policyholder, even after accounting for surrender penalties.

Various methods have been proposed to model policyholder lapse behavior. They

range from simple, deterministic lapse rates to sophisticated models such as De Gio-

vanni’s (2010) rational expectation and Li and Szimayer’s (2014) limited rationality.

Knoller, Kraut, and Schoenmaekers (forthcoming) show that the moneyness of the

2Some authors distinguish between “lapse” and “surrender.” A lapse may refer to the pol-

icyholder not taking advantage of the possibility to renew an expiring policy (by ceasing

premium payments and without receiving any payout from the insurer),or any voluntary ces-

sation without a surrender payment to the policyholder (Dickson, Hardy, and Waters, 2013).

The “surrender” (or cancellation) refers to the specific action by a policyholder during the

policy term to terminate the contract and recover the surrender value. See, for instance, Eling

and Kochanski (2013) for details. Throughout this article, as in Eling and Kochanski, we will

not make this distinction given that we do not consider the renewing option but only the

surrender option.