The Effect of Risk Aversion and Loss Aversion on Equity‐Linked Life Insurance With Surrender Guarantees

• DOI: http://doi.org/10.1111/jori.12297
• Date: 01 September 2020
• Author: Christian Hilpert
• Published date: 01 September 2020

© 2019 American Risk and Insurance Association. Vol. 87, No. 3, 665–687 (2020).

DOI: 10.1111/jori.12297

THE EFFECT OF RISK AVERSION AND LOSS AVERSION ON

EQUITY‐LINKED LIFE INSURANCE WITH SURRENDER

GUARANTEES

Christian Hilpert*

ABSTRACT

We price equity‐linked life insurance with surrender guarantees and ac-

count for risk preferences in the form of risk‐averse and loss‐averse pol-

icyholders in continuous time. Risk‐averse policyholders surrender their

policy for higher equity index values. Compared to optimally surrendered

policies, this behavior creates substantial policy value losses. In contrast,

loss‐averse policyholders surrender once the surrender benefit realizes a

gain but keep under‐performing policies. This disposition effect reduces

the policy value relative to both optimally surrendered policies and policies

surrendered by risk‐averse policyholders. Insurers in competitive markets

need to estimate their policyholders’risk preferences accurately.

INTRODUCTION

Equity‐linked life insurance products combine a classical term life insurance and a

savings component, which invests the policyholder’s premiums in equity. Usually,

they include a capital guarantee plus a minimum interest rate guarantee and an

option to surrender the policy.

1

Policy surrender is a significant risk for insurers as

*Christian Hilpert is at the Lingnan College, Sun Yat‐sen University, Guangzhou 510275, PR

China. Hilpert can be contacted via e‐mail: martin@mail.sysu.edu.cn. We thank Knut Aase

(discussant), Stefan Ankirchner, An Chen, Chunli Cheng, Keith Crocker (the editor), Kon-

stantin Neinstell, Philipp Schaper (discussant), Judith Schneider, Petra Steinorth (discussant),

Alexander Szimayer, and two anonymous referees for helpful comments and discussions.

Additionally, we thank seminar participants of the 41st Simposio de la Asociación Española

de Economía, the 43rd Annual Seminar of the European Group of Risk and Insurance

Economists, the American Risk and Insurance Association Annual Meeting 2016, the 2016

China International Congress on Insurance and Risk Management, the Annual Congress of

the German Insurance Science Association 2016, the 8th Conference in Actuarial Science &

Finance, the 18th Congress on Insurance: Mathematics and Economics, and seminar partici-

pants in Bonn, Hamburg, and Odense for helpful discussions. Financial Support of the

German Research Foundation through the Bonn Graduate School of Economics and the

research training group “Heterogeneity, Risk, and Dynamics in Economic Systems”is

gratefully acknowledged.

665

identified by European Union regulators (Eling and Kochanski, 2013). Empirically,

the surrender rate depends on the policyholder’s economic situation and the pol-

icy’s performance that depends on an underlying equity index or fund. Kuo, Tsai,

and Chen (2003)link the surrender rate to both interest and unemployment rates.

Eling and Kiesenbauer (2013)show the relevance of both policy and policyholder

characteristics for policyholders’surrender decisions.

This paper analyzes the impact of policyholders’risk preferences as on their sur-

render behavior and policy values. We model the surrender decision of a re-

presentative policyholder in the spirit of Albizzati and Geman (1994). The policy-

holder employs mental accounting, as introduced by Thaler (1980, 1985), that is, he

isolates the surrender decision from other portfolio components. We model the

surrender decision as a continuous‐time stopping problem for a policyholder with

both risk‐averse and loss‐averse expected‐utility preferences. We derive the implied

surrender behavior and show that the policyholders’risk preferences strongly in-

fluence surrenders and the policy’s value. For loss‐averse policyholders, a bench-

mark value called the reference point separates payments into gains and losses as in

Tversky and Kahneman (1992). Together with the diminishing utility of gains and

losses implied by risk aversion, the reference point creates an S‐shaped utility

function. If the reference point is zero, we obtain classical constant relative risk‐

aversion preferences, that is, power utility, as a special case.

This paper shows that risk‐averse and loss‐averse policyholders’surrender behavior

deviates from the value‐maximizing strategy that is characterized as an American

option exercise decision. Risk‐averse policyholders surrender for higher equity

index levels compared to the American solution in order to limit the policy risk.

Loss‐averse policyholders surrender systematically different compared to purely

risk‐averse policyholders: because of loss aversion, they do not realize surrender

benefits lower than their reference point. In contrast, they surrender mildly profit-

able policies relative to their reference point to secure gains. Shefrin and Statman

(1985)label this effect as disposition effect. Barberis and Xiong (2009)show that

individual investors commonly display this effect. It influences the policy’s value:

While the early surrender of purely risk‐averse policyholders reduces their con-

tract’s value by up to 15 percent, loss‐averse policyholders also fall short of this

policy value. Surrender behavior substantially affects insurers. Our numerical

analysis shows that misspecification of policyholders’risk preferences leads to

significant mispricing of the policy of up to 20 percent.

Our results are robust with respect to surrender fees and the specification of mor-

tality. Realistic surrender fees strongly discourage surrender because of high costs,

in particular, at early times. The disposition effect, however, is robust with respect to

surrender fees. We specify the mortality as either stochastic mortality in Lee and

1

There are different technical ways of terminating a life insurance contract. One option is to

hand the contract back to the insurance company and collect a surrender benefit, another is

to keep the insurance that the already invested capital provides, but set all future premiums

to zero. In this paper, we consider the first version. The second version is sometimes referred

to as “lapse of the contract.”

666 THE JOURNAL OF RISK AND INSURANCE