Life insurance surrender and liquidity risks

Abstract

Surrender options in endowment life insurance contracts can result in a surrender risk for the insurer. This risk is closely related to investment and liquidity risks. Consequently, the surrender risk is underestimated if it is assessed without consideration of all major risk sources. Using different risk measures, this paper shows that the surrender risk increases if the liquidity constraint is considered. Additionally, in an extreme event, mass surrender and a liquidity crisis can trigger each other. Therefore, the surrender risk could grow even faster. In this case, the solvency capital calculated within the Solvency II regime may not be enough to provide the intended protection. Furthermore, if policyholders surrender their contracts rationally, insurers face an even higher default threat.

Keywords:

• Endowment life insurance
• Risk management
• Liquidity risk
• Surrender risk

JEL Classifications:

• G13
• G1
• G
• G33
• G3
• G22
• G2

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Slight differences between the two terms, lapse and surrender, are defined in Kuo et al. (2003) and Gatzert et al. (2009). The former indicates the early termination of insurance contracts, usually without a payment to the policyholder. The latter refers to the surrender value paid to the policyholders upon termination before the end of maturity. According to Kuo et al. (2003) and Eling and Kiesenbauer (2014), lapse risk generally relates to the uncertainty caused by both surrender and lapse.

2 For the contract setup used in our paper, see Schmeiser and Wagner (2011) and the primary sources given in this contribution.

3 The guaranteed rate is considered as the lower bound of the contract's interest rate. With a participating scheme, policyholders receive a surplus return whenever the insurer's asset return exceeds the guaranteed rate. This surplus is retained in the policy account (see Bacinello 2003a, 2003b, Gatzert and Schmeiser 2008, Schmeiser and Wagner 2011).

4 The Vasicek model allows for a negative interest rate, which has currently occurred in many countries.

5 Note that we assume that the contract can only be started if the policyholder is alive and the first premium is paid at year $t=0.$

6 Assuming that there are no liquidity constraints, rational policyholders will exercise their surrender option at its optimal value (if the option is in the money throughout the contract period).

7 A contract without the surrender option is the basic contract δ ($\delta =0$) defined in Equation (9(9) $\begin{array}{rl}\delta & =E^{\mathbb{Q}}\left(\gamma \sum _{t=0}^{T-1}{}_t{p}_x{q}_{x+t}\prod _{i=1}^{t+1}\right(1+{\hat{r}}_i^f{)}^{-1}+A_T\prod _{i=1}^T(1+{\hat{r}}_i^f{)}^{-1}\\ & -B-B\sum _{t=1}^{T-1}{}_t{p}_x\prod _{i=1}^t(1+{\hat{r}}_i^f{)}^{-1}).\end{array}$(9) ).

8 More details are given in Chang and Schmeiser (2020).

9 In a frictionless market, no transaction cost exists. In a competitive market, buying or selling any amount of a security can be conducted without any restrictions and without influencing its market price (see e.g. Jarrow and Protter 2005).

10 In our model, only the long position is considered.

11 A detailed proof can be found in Acerbi and Scandolo (2008).

12 This paper focuses on the relative change in the default/insolvency probabilities if liquidity is considered. Therefore, an initial capital value can be considered irrelevant in this relative measure.

13 This simplified investment strategy is implemented so that the contract's NPV for the insurers and the policyholders is zero with a predetermined α. In reality, an investment strategy would include duration matching and other hedging positions. The management of the interest rate risk is, however, not the main focus of this paper.

14 As the mortality risk is assumed to be fully diversified, the death and survival benefits are based on expected cash outflows only. These cash flows can be replicated by continuous trading. Given continuous trading, a large-volume trade can be divided into infinitely small transactions, each having a negligible impact on its market price (see Cetin et al. 2004). As a consequence, in our model setting, liquidity risk only plays a role in the context of surrender risk and not in the case of pay-outs for death or survival.

15 Biagini et al. (2021) applied their model to German insurance data. Their result concludes that the $99.5\mathrm{\%}$ quantile of the lapse rate lies between $20\mathrm{\%}$ and $25\mathrm{\%}$, while the mass lapse rate suggested in Solvency II can be overly conservative.

16 The mortality probabilities are for a 30-year-old US woman based on the data from the HMD (Human Mortality Database).

17 This discount is close to the recent empirical studies in Ellul et al. (2011) and Newman and Rierson (2011). The discount has been applied in Feldhütter (2012), Förstemann (2018), and Kubitza et al. (2020). However, the discount estimated in Ellul et al. (2011) and Newman and Rierson (2011) is triggered by the bond market itself. In our model, the fire sale is caused by individual insurers due to their policyholder surrender behaviors. Therefore, we acknowledge that these discount parameters could be overestimated in our model.

18 With $\pi =100\mathrm{\%}$, we focus here for illustrative purpose on the extreme strategy, where the insurer invests in risky assets only. For various reasons – in particular because of regulatory requirements – this is not an asset allocation which we find in insurance practice.

Additional information

Funding

The authors gratefully acknowledge the financial support of the Swiss National Science Foundation [grant number 100018_176437].