A COVID-19 stress test for life insurance: insights into the effectiveness of different risk mitigation strategies

Hanika, Moritz Article — Published Version A COVID-19 stress test for life insurance: insights into the effectiveness of different risk mitigation strategies European Actuarial Journal Provided in Cooperation with: Springer Nature Suggested Citation: Hanika, Moritz (2023) : A COVID-19 stress test for life insurance: insights into the effectiveness of different risk mitigation strategies, European Actuarial Journal, ISSN 2190-9741, Springer, Berlin, Heidelberg, Vol. 14, Iss. 2, pp. 525-550, https://doi.org/10.1007/s13385-023-00371-3 This Version is available at: https://hdl.handle.net/10419/312465 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) European Actuarial Journal (2024) 14:525–550 https://doi.org/10.1007/s13385-023-00371-3 1 3 CASE STUDY A COVID‑19 stress test forlife insurance: insights intotheeffectiveness ofdifferent risk mitigation strategies MoritzHanika1 Received: 28 March 2023 / Revised: 18 September 2023 / Accepted: 30 October 2023 / Published online: 7 December 2023 © The Author(s) 2023 Abstract COVID-19 has affected mortality rates and financial markets worldwide. Against this background, we perform a COVID-19 stress test for life insurance, considering a joint financial and mortality shock, to evaluate the effectiveness of different risk mitigation strategies. Specifically, we conduct a model-based simulation analysis of a life insurer selling annuities and term life insurances. The analysis includes stress scenarios that are calibrated to observations during the first year of the COVID-19 pandemic. We also consider new business and study the risk situation under three different risk mitigation strategies observed in practice as an immediate response to the pandemic: stopping sales, increasing premiums, or adjusting investment strategies. Results show that a life insurer’s risk situation is mainly affected in the short term, selling annuities (in addition to term life insurance) immunizes against the mortality shock, and the immediate use of risk mitigation strategies can help reduce the negative impact. Keywords Life insurance· Stress test· COVID-19· Risk mitigation strategies· Asset–liability model JEL Classification G22· G23· G32· J11 1 Introduction COVID-19 had and will continue to have a major impact on society worldwide. Under the backdrop of globalization, the virus spread quickly across countries, thereby increasing mortality rates and causing many economic problems, such as business interruptions and supply chain issues. The COVID-19 pandemic is thus * Moritz Hanika [email protected] 1 School ofBusiness, Economics andSociety, Friedrich-Alexander-Universität ErlangenNürnberg (FAU), Lange Gasse 20, 90403Nuremberg, Germany 526 M.Hanika 1 3 notable not only for its impact on human health and mortality [29, 36] but also for its economic effects on financial markets [1, 43]. This makes it particularly relevant for the life insurance industry, whose core business is to adequately price mortality risks and ensure stable returns on assets under management. Research by a U.S. company that uses data science to predict disease outbreaks indicates an increasing frequency of pandemics, revealing a 47–57% chance of a COVID-19-like event happening within the next 25years [30]. Hence, managing pandemic shocks is highly relevant for life insurers. Although the previous literature has examined the general impact of COVID-19 on the insurance industry [34], particularly with respect to mortality risk [10, 33, 37] or insurance investing [27], academic research that analyzes the impact of the joint financial and mortality shock caused by COVID-19 on a life insurer’s risk situation in an asset–liability-based simulation analysis is still missing. Further, Harris etal. [22] showed that U.S. life insurers only sparingly responded to COVID-19 by removing specific policies or increasing premiums for newly sold policies for older people. However, the effectiveness of these immediate measures has not been analyzed in academic research so far. Against this background, we examine the impact of a joint financial and mortality shock on a life insurer’s risk situation in a COVID19 stress test. Furthermore, we evaluate the effectiveness of different risk mitigation strategies observed in practice that can be applied on short notice to draw conclusions for potential future pandemics. We find that a temporary pandemic event such as COVID-19 affects a life insurer’s risk situation, especially in the short run, and that the immediate use of risk mitigation strategies can reduce the negative impact. COVID-19 has increased mortality rates differently across population groups, with older individuals or those with comorbidities being the most affected [11, 29]. Spiegelhalter [38] and Sasson [36] proposed that COVID-19 caused an upward loglinear shift in mortality rates, affecting older individuals more. Schnürch etal. [37] described how to embed this shift in the popular Lee–Carter (LC) mortality model. Furthermore, they studied the impact of a COVID-19-like mortality shock on the valuation of life insurance products, similar to Carannante etal. [10] who conversely used an accelerated mortality model. Carannante etal. [9] examined the profitability of variable annuity contracts under a COVID-19-induced mortality shock, similar to Cheng [13] who focused on participating life insurance and considered changes in policyholders’ surrender behavior. While these existing studies observed only a limited impact of mortality shocks on life insurance, they did not consider joint financial and mortality shocks. Even before COVID-19, empirical research indicated that mortality and financial risks could be correlated [16] and some studies have explicitly investigated the consequences for the risk-neutral valuation of life insurance products [2, 17, 23, 28]. In the context of COVID-19, Li etal. [26] used excess mortality and interest rate data from the COVID-19 pandemic in the U.S. to calibrate a bivariate jump diffusion model for the risk-neutral pricing of mortality bonds and observed a high correlation between mortality and interest rates during the pandemic. Furthermore, Arık etal. [3] found only a small relevance of simultaneous financial and mortality shocks for the risk-neutral pricing of annuities and buyout 527 1 3 A COVID‑19 stress test forlife insurance: insights intothe… premiums. In contrast to previous studies, we focus on a life insurer’s risk situation and the effectiveness of risk mitigation strategies rather than risk-neutral valuation. Regarding the observed risk mitigation strategies of life insurers as an immediate response to COVID-19, Harris etal. [22] observed only a few adjustments to product pricing and offerings in the U.S.; these adjustments include slightly increased premiums by low-price leaders, increased prices for high-risk groups (e.g., smokers), and removal of specific policies offered to older people from the market. Similarly, U.S. life insurers during the Spanish flu of 1918 charged higher prices for new policies and gave up business in states with greater exposure to the disease [14]. Regarding investment strategies, Berry-Stölzle et al. [5] observed that U.S. life insurers created cash buffers by raising external capital and cutting dividends in response to the 2008–2009 financial crisis; the same approach was also observed during the COVID-19 pandemic [19]. Another common strategy is to shift assets into low-risk investments (e.g., government bonds) to avoid higher volatility caused by financial crises [24]. However, investments in bonds are exposed to credit risk, which is highly relevant for insurance investments, as downgrades by rating agencies increased during the COVID-19 pandemic [27]. In line with the academic literature, regulatory stress tests by the European and Occupational Pension Authority (EIOPA) and the Australian Prudential Authority (APRA) showed that reactive management actions, such as de-risking of assets or raising capital, can help reduce the negative impact of a COVID-19-like stress scenario [18]; the common management actions used by insurers are capital raising, reduction of new business volumes, and repricing [4].1 Building on these observations, we consider three risk mitigation strategies in our stress test: premium increases for new business, age-specific adjustments of product offerings during the pandemic, and changes in the life insurer’s investment strategy at the beginning of the pandemic. For the model framework of our stress test, we build on previous research that used asset–liability models to study the effectiveness of natural hedging2 [20, 42] or mortality bonds [12, 21] as risk mitigation strategies. In contrast to these studies, we consider a joint financial and mortality shock calibrated to COVID-19 and analyze the effectiveness of immediate measures rather than risk mitigation strategies that had to be initialized in the past. For the mortality model of our simulation analysis, we follow Gatzert and Wesker [20] and use an extension of the LC model proposed by Brouhns etal. [8]. To embed the mortality shock, we employ the approach of Schnürch etal. [37] using mortality data from the first year of the COVID-19 pandemic. For the asset model, we differentiate between highand low-risk investments. Both evolve according to a geometric Brownian motion, and the life insurer specifies the portfolio composition (as in Bohnert etal. [7]) making adjustments in response to the pandemic outbreak possible. To include financial shocks depending on the risk levels of investment types, we use financial data from the first year 1 Note that regulatory stress tests consider joint financial and mortality shocks but apply many simultaneous shocks to complex balance sheets relying on standard formulas, rarely consider multi-timestep internal models, and most often present results in an accumulated way. 2 The literature on natural hedging shows that selling annuities next to term life insurance is an effective tool to hedge against mortality risks and to smooth cash flow structures in general [20, 42]. 528 M.Hanika 1 3 of the COVID-19 pandemic and consider market and credit risks. The life insurer’s asset–liability model accounts for regular dividend payments to shareholders and new business (term life and annuity). Policies are actuarially calculated with additional loadings sold to different age groups, allowing the assessment of the effectiveness of premium increases or age-specific adjustments in product offerings during the pandemic. We run Monte Carlo simulations to analyze the longand short-term effects of individual and joint financial and mortality shocks on a life insurer’s risk situation under different product portfolios (term life vs. annuity). Furthermore, we evaluate the effectiveness of the three risk mitigation strategies to reduce the negative impact of the joint financial and mortality shock for a life insurer particularly focusing on term life insurance. Our results show that financial and mortality shocks reinforce each other. The stress scenario increases the life insurer’s one-year default probabilities during the pandemic but exerts only a small long-term impact on the life insurer’s risk situation. A sufficiently large portion of annuities in the product portfolio immunizes against the mortality shock but only partially immunizes against the financial shock. Furthermore, the risk mitigation strategies reduce the negative short-term impacts, with the effectiveness depending on the respective strategies. The remainder of this paper is organized as follows: Sect.2 introduces the mortality and asset models with embedded shock mechanisms along with the life insurer’s asset–liability model. Section3 explains the calibration of the financial and mortality shock based on COVID-19 and the parameters used in the simulation analyses. The numerical results are presented by first examining the longand short-term effects of the (joint) shocks and then analyzing the effectiveness of different risk mitigation strategies. Finally, Sect.4 summarizes the main findings. 2 Model framework In this section, we describe the model framework used in our simulation analysis to study the impact of a joint financial and mortality shock, as exemplified by COVID19, on life insurance. We first describe the mortality model with a mortality shock, followed by the asset model with a financial shock, and finally, the life insurer’s asset–liability model. 2.1 Mortality model withshock The mortality model describes the development of mortality rates for a given population. The mortality rate m(x,t) of an x-year-old person in year t is defined as where D(x,t) is the death count and E(x,t) is the exposure at risk. Given the mortality rates in Eq. (1), the corresponding one-year survival probability of an x-year-old person in year t can be computed using px(t)=exp (−m(x,t)) , and the (1) m (x,t)= D(x,t) E(x,t), 529 1 3 A COVID‑19 stress test forlife insurance: insights intothe… one-year death probability is qx(t)=1−px(t) [8]. The probability npx(t) that an x-year-old person in year t will survive for the next n years can be computed using n p x (t)= ∏n−1 i=0 p x+i (t+i ) . Building on Gatzert and Wesker [20], we assume that mortality rates follow the LC model [25], i.e., with age-specific constants ax and bx , time-varying trend kt , and homoscedastic error terms 𝜀m x,t with a mean of zero. Although the model in Eq.(2) is underdetermined, adding the two constraints ensures identifiability [25]. Furthermore, we use an extension of the LC model proposed by Brouhns etal. [8], where the death counts D(x,t) are given by a Poisson distribution, i.e., The unknown parameters ax , bx , and kt are estimated by maximizing the corresponding log-likelihood function using Newton’s method, where the constraints in Eq. (3) are applied after each update step [8]. To forecast mortality rates beyond the time of observation, the estimated parameters kt are used to fit a time series model. Lee and Carter [25] employed a simple random walk with drift, i.e., where 𝜙 is a constant drift and 𝜀t denotes independent normally distributed random variables with a mean of zero and constant volatility 𝜎 .3 In year t=  t , a pandemic event occurs and increases mortality in this specific year. To model the mortality shock in year  t , we build on the empirical observations during COVID-19, showing that the increase is age-specific and can be approximated by an upward log-linear shift for the 25–84 age group [36, 37]. Therefore, following Schnürch etal. [37] we consider the adjusted mortality rate (2) m (x,t)=exp ( ax+bx⋅kt+𝜀m x,t ), (3) ∑ x bx=1 and ∑ t kt= 0 D (x,t)∼Poisson(E(x,t)⋅m(x,t)) with m(x,t)=exp ( a x +b x ⋅k t). L( ax,bx,kt ) = ∑ x , t ( D(x,t)⋅ ( ax+bx⋅kt ) −E(x,t)⋅exp ( ax+bx⋅kt )) (4) kt+1=𝜙+kt+𝜀t, (5) m shock(x,t)= { (1+c)⋅m(x,t)if t=  t m(x,t)else 3 Note that more sophisticated time series models (e.g., ARIMA models) can be considered too [20, 25] and that the random walk with drift could be replaced by a Brownian motion with drift in a continuous time model [6]. 530 M.Hanika 1 3 for a strictly positive constant c. Assuming that m(x,t) and mshock(x,t) follow an LC model, Schnürch etal. [37] mathematically proved that the only difference between the two LC models is approximately given by an increase in k t without affecting kt in other years t≠ t or the parameters ax and bx . Therefore, we employ a single LC model in our stress test, where parameter kt is increased by a strictly positive constant kshock in the year of the pandemic event  t , i.e., 2.2 Asset model withfinancial shock The asset model accounts for two types of investments. A high-risk investment S represents stocks while a low-risk investment B represents bonds. As in Bohnert etal. [7], the asset prices Ii t of both investments i∈{S,B} evolve over time t according to the geometric Brownian motion, i.e., with drifts 𝜇i , volatilities 𝜎i , and Brownian motions Wi t . The solutions for the stochastic differential equations are given by with independent standard normally distributed random variables 𝜀i t . Therefore, investments i∈{S,B} yield a continuous one-year return of To account for credit risk, the investments in bonds are reduced by the portfolio loss with exposure at default EADt , loss given default LGDt , and normalized portfolio loss  Lt .4 The normalized portfolio loss is Vasicek distributed, i.e., k shock t= { kt+kshock if t=  t k t else ,ashock x=axand bshock x=bx . dIi t =𝜇 i ⋅I i t ⋅dt +𝜎 i ⋅I i t ⋅dW i t (6) I i t=Ii t−1 ⋅exp ( 𝜇i− 𝜎2 i 2+𝜎i⋅𝜀i t ) (7) ri t =𝜇 i −𝜎 2 i ∕2+𝜎 i ⋅𝜀 i t. (8) Lt=EADt ⋅ LGDt ⋅  Lt  L t=Φ �√ 1−𝜌⋅𝜀L t−Φ −1 � PDt � √ 𝜌 �, 4 The normalized portfolio loss  Lt represents the loss of a large homogeneous portfolio consisting of N non-recoverable loans with equal values 1∕N and equal default probabilities PDt . Assuming that the assets of the borrowing companies are modeled by geometric Brownian motions with equal correlation parameters 𝜌 , the normalized portfolio loss  Lt converges for N → ∞ against the Vasicek distribution [40]. 531 1 3 A COVID‑19 stress test forlife insurance: insights intothe… where Φ denotes the standard normal distribution, PDt denotes the default parameters, 𝜌 denotes the correlation parameter, and 𝜀L t denote independent standard normally distributed random variables.5 The correlations between 𝜀S t , 𝜀B t , and 𝜀L t are denoted as 𝜌SB , 𝜌SL , and 𝜌BL , respectively. To model a financial shock in year  t , we consider a price drop in combination with increased volatility for both investments i∈{S,B} , i.e., for strictly positive constants 𝜎shock i . Consequently, the asset prices in Eq.(6) become and the returns in Eq.(7) become To account for an increased credit risk in year  t , we use two different default parameters, i.e., 2.3 Asset–liability model The asset–liability model of our stress test describes the situation of a life insurer who frequently writes new business over a time horizon of T years. Cash flows only arise at the beginning of year t denoted by t+ or at the end of the year denoted by t− . The life insurer is founded in year 1+ and sells term life insurance contracts with death benefit D as well as temporary annuities with annual payments R. Both types of contracts have a term of T years. The age x of the purchasing policyholders varies, and the number of contracts sold in year t to policyholders with age x is denoted by nD(x,t) for term life insurances and nR(x,t) for annuities. Term life insurance contracts are sold against annual premiums while annuities are sold against single premiums. The premiums are actuarially fair with an additional loading 𝜆 and depend on the year of sale and the age of the purchaser, i.e., (9) 𝜀 i t= { − || 𝜀 i t || if t=  t 𝜀i t else and 𝜎i= { 𝜎i+𝜎 shock iif t=  t 𝜎 i else I i t= ⎧ ⎪ ⎨ ⎪ ⎩ Ii t−1 ⋅exp � 𝜇i−(𝜎i+𝜎shock i)2 2− � 𝜎i+𝜎shock i � ⋅� � 𝜀i t� �� if t=  t Ii t−1 ⋅exp � 𝜇i−𝜎2 i 2 � +𝜎i⋅𝜀i telse r i t= { 𝜇i− ( 𝜎i+𝜎shock i ) 2 / 2− ( 𝜎i+𝜎shock i ) ⋅ || 𝜀i t || if t=  t 𝜇 i −𝜎2 i ∕2+𝜎 i ⋅𝜀i t else. (10) PD t= { PD shock if t=  t PD else. 5 Note that this approach is motivated in the Capital Requirement Regulation of Basel III (see Regulation (EU) No 575/2013, article 153). 532 M.Hanika 1 3 The corresponding actuarial present values are given by where v =1∕ ( 1+r G) denotes the discount factor for some actuarial interest rate rG . When the life insurer is founded in year 1+ , shareholders make an initial contribution E0 , and the first contracts are sold, resulting in the insurer’s initial asset volume of For the transition from t+ to t− , the fraction 𝛼 of assets At+ is invested in stocks, and the fraction (1−𝛼) is invested in bonds. Accounting for the continuous interest rates for the stock investments rS t and bond investments rB t (see Eq.(7)), as well as the portfolio loss Lt (see Eq.(8)), the adjusted assets are given by To define contractual payments at the end of the year, let dD(x,𝜏,t) and dR(x,𝜏,t) respectively denote the number of deaths in year t of the policyholders who bought a term life insurance contract or an annuity at the beginning of year 𝜏 at age x. Furthermore, let denote the number of contracts bought in year 𝜏 by policyholders with age x who are still active in t− .6 Given this notation, the adjusted assets in Eq.(13) are reduced by contractual payments, and some dividend div7 is paid to shareholders, i.e., The assets at the beginning of all years t>1 are given by (11) P D(x,t)=(1+𝜆)⋅D⋅ T|A (t) x a(t) x∶T | | | and PR(x,t)=(1+𝜆)⋅R⋅a(t) x∶T | | | . (12) T |A(t) x= T−1 ∑ k=0 vk+1⋅kpx(t)⋅qx+k(t+k),a(t) x∶T| | | = T−1 ∑ k=0 vk⋅kpx(t ) and a(t) x∶T | | | = T ∑ k=1 vk⋅kpx(t), A 1+=E0+ ∑ x nD(x,1 )⋅PD(x,1 )+ ∑ x nR(x,1 )⋅PR(x,1 ) . (13) Aadj t −=𝛼⋅A t +⋅exp ( rS t) + ( (1−𝛼)⋅A t +−L t) ⋅exp ( rB t). n i(x,𝜏,t)=ni(x,𝜏)− t ∑ k=𝜏 di(x,𝜏,k)∶i∈{D,R } A t−=Aadj t−−D⋅ t ∑ 𝜏=1 ∑ x dD(x,𝜏,t)−R⋅ t ∑ 𝜏=1 ∑ x nR(x,𝜏,t)−div . 6 For 𝜏<1 and t<𝜏 , we set ni (x, 𝜏 ,t)= 0 . 7 The dividend is calculated by rM ⋅E 0 =(1−0.005)⋅div +0.005 ⋅ ( −E 0) based on the assumption that the shareholders assume a constant one-year default probability of 0.5%, with rM denoting the market rate of return [20]. 539 1 3 A COVID‑19 stress test forlife insurance: insights intothe… mortality shock in Spain leads to a higher increase in DP. Meanwhile, the financial shock increases DP approximately equally for Germany and Spain. For both countries, the increase in DP caused by the pure financial shock is greater than that caused by the pure mortality shock. Higher numbers of annually sold annuities reduce the negative impact of the financial shock until a saturation point of approximately 3.5 percentage points. However, the mortality shock can be completely absorbed by natural hedging [20, 42]. If both shocks are applied simultaneously, the increase in DP exceeds the sum of the individual shocks through an additional reinforcement effect (see Fig.4b). The extent of the reinforcement effect decreases as the number of annually sold annuities increases, further emphasizing the benefits of natural hedging. These results show that a diversified life insurer is relatively well immunized against the pure mortality shock caused by COVID-19 but that natural hedging cannot mitigate the risks caused by the simultaneous financial shock. As a life insurer’s Fig. 4 Default probability DP when no financial or mortality shock is applied (a) and the absolute increase in DP caused by the financial and/or mortality shock in year t=10 (b) depending on the number of sold annuities nR for Germany and Spain 540 M.Hanika 1 3 product mix cannot be adjusted at short notice and doing so might not even be possible because of a specific market orientation, the following analyses focus on a life Table 2 Default probabilities DP in case of nR = 50 for Germany and Spain caused by the financial and/ or mortality shock in t=10 Notes: The absolute increase is expressed in percentage points (p.p.) Germany Spain DP Absolute increase DP Absolute increase No shock 9.29% 10.12% Financial shock 15.13% + 5.84 p.p 15.59% + 5.47 p.p Mortality shock 9.47% + 0.18 p.p 10.88% + 0.76 p.p Both shocks 16.11% + 6.82 p.p 19.42% + 9.30 p.p Fig. 5 Absolute increase of one-year default probabilities DPt when the joint financial and mortality shock is applied in t=10 or in t=10 and t=11 for Germany (a) and Spain (b) 541 1 3 A COVID‑19 stress test forlife insurance: insights intothe… insurer with a high share of term life insurance, i.e., we set nR=50 .13 The default probabilities for this scenario are listed in Table2. To investigate the shortand long-term effects of a COVID-19-like event, we present in Fig.5 the absolute increase of the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 (filled bars). The years before the pandemic outbreak (t<10) are not displayed as the one-year default probabilities are not affected until that point of time. For Germany (Fig.5a) and Spain (Fig.5b), the highest increase in the one-year default probabilities can be seen right at the end of that year. For later years, the increase exponentially decreases and converges against zero. The long-term effects result from the life insurer’s reduced equity leading to higher default probabilities in the subsequent years as reserves must first be rebuilt. Although the increase for Spain starts at a higher level (7.90 percentage points) than that for Germany (5.44 percentage points), the convergence toward zero is similarly fast for both countries, falling below 0.2 percentage points four years after the end of the pandemic. As mortality rates are only increased in t=10 , higher payments for term life insurance are limited to this period too, as is the reduction of the life insurer’s assets caused by the temporary financial shock. Therefore, the longterm impact is comparatively small. This result is in line with the findings of Carannante etal. [10], who showed that a temporary mortality shock would have a minor impact on life insurance, whereby the required premium increase to compensate an accelerated mortality decreases for longer contract terms. Motivated by previous events such as the 1918 Spanish flu or the 2004 earthquake, mortality shocks are usually assumed to be short-term catastrophes that do not extend over several years [12, 15]. However, the case of COVID-19 indicates that a large-scale pandemic can increase mortality rates in the span of more than one year. Therefore, Fig.5 also shows the impact of a two-year pandemic event in our stress test (unfilled bars), where the financial and mortality shocks are applied in years t=10 and t=11 . In this case the one-year default probabilities first increase until the end of t=11 and then again converge against zero. The higher increase in t=11 may be explained by the life insurer’s already reduced reserves caused by the shocks in t=10 , resulting in a worse financial starting point for the life insurer to enter the second year of the pandemic. As the decrease after the end of the two-year shock is exponential, as in the case of the one-year shock, the long-term impact for life insurance seems negligible even if the pandemic extends over several years. 3.4 Risk mitigation intheevent ofshocks In this section, we investigate the effectiveness of different risk mitigation strategies to reduce the impact of the joint financial and mortality shock. We restrict our analysis to observed practices that can be applied on short notice, namely, stopping new business, increasing premiums, and adjusting investment strategies. Furthermore, we build on the results described in the previous section, showing that the impact of the shocks is short term regardless of the duration of the pandemic. Thus, we 13 Note that in this case, the premium volume of annuities is approximately 8.7% (11.2%) of the total premium volume in the case of Germany (Spain). 542 M.Hanika 1 3 focus on the joint shock only applied in year t=10 , with the different risk mitigation strategies applied directly at the beginning of that year in t=10+ . Although this approach would mean that the life insurer anticipates the pandemic before it starts and responds immediately, this point in time serves as a threshold, i.e., a lower bound, owing to the used time steps of one year.14 3.4.1 Stopping new business For the first risk mitigation strategy, we analyze the effectiveness of stopping new business for specific age groups during the pandemic. This strategy is motivated by the empirical findings of Harris etal. [22], who showed that some U.S. life insurers stopped selling specific policies for the oldest age groups in response to COVID-19. Figure6 shows the absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 , when in t=10+ , the term life insurance contracts are sold to all age groups, or only to the age groups x=30 and x=40 years, or only to the age group x=30 years, or no term life insurance contracts are sold. The number of sold contracts is 3000 for all age groups and remains unchanged for all years before and after t=10 . The more age groups are excluded, the higher the risk reduction as the height of the four consecutive bars Fig. 6 Absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 for Germany (a) and Spain (b) if term life insurance policies are sold only to specific age groups in t=10+ 14 Note that similar results regarding the effectiveness of the different risk mitigation strategies are obtained when the shocks extend over two consecutive years and the strategies are applied at the beginning of the second year. 543 1 3 A COVID‑19 stress test forlife insurance: insights intothe… decreases. The largest risk reduction can be observed at the end of the pandemic year t=10 , and it then exponentially decreases over time, analogous to the general decline in risk reduction potential. Therefore, the analyzed strategy indeed mitigates the increase in the one-year default probabilities caused by the pandemic shock. Stopping new business during the pandemic reduces the number of sold contracts and, therefore, reduces exposure toward the mortality shock. However, as most contracts are sold before the pandemic and the strategy does not address the financial shock, the effectiveness of this strategy is relatively small and offsets only approximately 14% of the increased one-year default probability in t=10 . Furthermore, solely excluding the oldest age group, in anticipation of the greater increase in mortality for this group, is not optimal. Stopping new business for all the three age groups (unfilled bars with dashed lines) is more than three times as effective as solely excluding the oldest age group x=50 (unfilled bars with solid lines). Table3 in the Appendix shows the overall default probability under all possible combinations of age-specific exclusions for the new term life business in year t=10 . Some benefits are observed with the exclusion of older over younger individuals. However, a robustness test revealed that these benefits stem from the higher premium volumes for the contracts sold to older age groups due to their higher death probabilities rather than the greater increase in their mortality rates given the mortality shock in Eq.(5). When using age-specific death benefits in our simulation analysis to ensure that the premium volumes depend only on the number of sold contracts, we observe a higher risk reduction for excluding the younger age groups than for excluding the older age groups. Specifically, the risk for term life insurance decreases with age as the liability value becomes less volatile owing to the shorter duration until death at older ages [42]. Consequently, the general riskiness of a life insurance product can play a more important role and should not be neglected even if the mortality shock is more pronounced for older people (as in the case of COVID-19). 3.4.2 Increasing premiums For the second risk mitigation strategy, we analyze the effectiveness of increasing actuarial fair premiums for contracts sold during the pandemic event by an additional loading. Therefore, different loadings 𝜆 for the contracts sold in t=10+ are used in Eq.(11) in the simulation analysis. The premiums before and after t=10 are unchanged, i.e., actuarial fair premiums with 𝜆=0% . Figure7 shows the absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 for different loadings 𝜆 . The loading 𝜆=0% (filled bars) serves as a reference point, where actuarial fair premiums are used during the pandemic. For the loadings 𝜆>0% , the increase in the one-year default probabilities is reduced with the largest risk reduction at the end of the pandemic year in t=10 . As higher loadings increase the life insurer’s equity and thus help to avoid liquidity shortage, this strategy addresses the negative impact of the financial and mortality shocks; by contrast, stopping new business during the pandemic only addresses the negative impact of the mortality shock on newly sold contracts. Hence, using a comparatively small loading of 𝜆=2% for the contracts 544 M.Hanika 1 3 sold in year t=10+ reduces the one-year default probabilities more than stopping the sale of all term life insurance contracts during the pandemic (see Figs.6 and 7). Moreover, a sufficiently large loading can fully offset the increase in the one-year default probabilities DPt . For Germany, the loading of 𝜆=23% completely negates the increase in DPt caused by the joint financial and mortality shock in t=10 for all t≥10 (see Fig.7a).15 This mechanism makes premium increases a hypothetically attractive short-term strategy. However, in practice, this strategy carries the risk of losing a leading market position to competitors, which can result in a decline in demand in subsequent years. This assumption is also supported by Harris etal. [22], who observed that the premium response of U.S. life insurers to COVID-19 was minimal and limited to life insurers with a certain price gap toward competitors. Meanwhile, policyholders might even accept premium increases during a pandemic event because of the stronger sensitization toward death and, therefore, a higher willingness to pay for term life insurance [35]. Assuming that the premium increase only leads to a decline in demand during the period when premiums increase, this condition would technically result in a combined strategy of stopping new business and increasing premiums, which would (in our simulation) be superior to the sole strategy of stopping new business. Fig. 7 Absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 for Germany (a) and Spain (b) if additional loadings 𝜆 for the contracts sold in t=10+ are used 15 For further information, Table4 in the Appendix contains values for the overall default probability DP under the different loadings used in t+=10 . 545 1 3 A COVID‑19 stress test forlife insurance: insights intothe… 3.4.3 Adjusting investment strategies For the third risk mitigation strategy, we analyze the effectiveness of changing the investment strategy at the beginning of the pandemic year. We assume that the stock ratio 𝛼 in Eq.(13) is reduced at t=10+ and remains constant for the remaining time horizon to account for the long-term nature of bonds. Higher shares in bonds generally reduce asset volatility and the exposure toward the volatility shock; this effect is less pronounced for bonds than for stocks in our stress test (see Fig.2b, c). However, in our stress test, bond investments are exposed to credit risk, which is increased by the financial shock (see Sect.3.1). Figure8 shows the absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in year t=10 for different adjustments of the stock ratio 𝛼 in t=10+ . The stock ratio 𝛼=15% (filled bars) serves as a reference point, where the investment strategy remains unchanged. Reducing the stock ratio 𝛼 at the beginning of the pandemic decreases the absolute increase in DPt in t=10 to a minimum of 3.67 (5.63) percentage points for Germany (Spain) if all assets are reallocated to bonds, i.e., 𝛼=0% (see Fig.8). Thereby, the largest step-wise reduction effect of 0.80 (0.96) percentage points for Germany (Spain) is obtained with the reduction of 𝛼 from 15 to 10% while the smallest step-wise risk reduction of 0.39 (0.55) percentage points is obtained with the reduction of 𝛼 from 5 to 0%. In later years, the one-year default probabilities DPt start to increase again at some point t≥11 , which happens earlier and is more pronounced for lower stock ratios 𝛼 . This observation is consistent for both countries, showing that a reduction in the stock ratio can have a more pronounced long-term effect than the other two strategies. Fig. 8 Absolute increase in the one-year default probabilities DPt caused by the joint financial and mortality shock in t=10 for Germany (a) and Spain (b) if the stock ratio 𝛼 is adjusted in t=10+ 546 M.Hanika 1 3 Therefore, adjusting the investment strategy might be less suitable for mitigating the negative impact of a short-term pandemic event such as COVID-19. Overall, the strategy seems more effective than stopping new business (see Fig.6) but is more limited than the strategy of increasing premiums (see Fig.7).16 Furthermore, reallocating all assets to bonds may not be worthwhile for shareholder value maximization because of the comparably small further risk reduction at the cost of a lower profit potential. 4 Summary In this study, we examine the effectiveness of different risk mitigation strategies that can be applied on short notice during a pandemic on the basis of a COVID-19 stress test for life insurance. To the best of our knowledge, this study is the first to use an asset–liability model to examine the impact of a joint financial and mortality shock caused by COVID-19 on a life insurer offering annuities and term life insurances. Mortality rates are modeled by an extension of the LC model, where we use mortality data from the first year of the COVID-19 pandemic in Germany (smaller shock) and Spain (larger shock) to embed realistic mortality shocks. Financial shock differs between highand low-risk investments, increases market and credit risks, and is calibrated to financial data from the first year of the COVID-19 pandemic. The life insurer’s asset–liability model accounts for new business (annuity and term life) sold to policyholders of different age groups, actuarial fair premiums, and dividend payments to shareholders. First, we use Monte Carlo simulation to analyze the longand short-term effects of the individual and joint shocks on the life insurer’s risk situation under different product portfolios (annuity vs. term life). Second, we investigate the effectiveness of three risk mitigation strategies observed in practice, focusing on a life insurer with a high share of term life insurance. We consider stopping new business for certain age groups, increasing premiums during the pandemic, and adjusting the investment portfolio at the beginning of the pandemic. The numerical results of our simulation analysis show that financial and mortality shocks can reinforce each other and, therefore, should not be analyzed separately. Product diversification, i.e., selling annuities next to term life insurances, can completely negate the negative impact of the mortality shock but not that of the financial shock. Furthermore, our results indicate that a temporary pandemic event such as COVID-19 might have only a limited impact on the long-term risk situation of life insurers as the increase in one-year default probabilities exponentially decreases immediately after the end of the pandemic in our simulation analysis. Stopping new business during the pandemic reduces the negative impact of the stress scenario; however, its effectiveness is low. Furthermore, we observe that solely excluding policies for older age groups (in anticipation of a larger mortality shock at older ages) is not necessarily beneficial as older policyholders generally carry lower risks due to their shorter duration until death. Premium increases during a pandemic turn out to be a very effective risk mitigation strategy in our stress test as it increases the life insurer’s equity, helping to deal with 16 This is also supported by the values of the overall default probabilities given in Table5 in the Appendix. 547 1 3 A COVID‑19 stress test forlife insurance: insights intothe… the financial and mortality shocks. However, in practice, this strategy bears the risk of losing market position. Adjusting the investment portfolio at the beginning of the pandemic is more effective than stopping sales but leads to more pronounced negative long-term effects owing to the long-term nature of bond investments and might be less suitable for mitigating the risks of a short-term pandemic event such as COVID-19. In conclusion, our simulation analysis shows that either the applicability in practice or the effectiveness of such immediate measures is limited. Therefore, more sophisticated risk mitigation strategies, such as reinsurance or alternative risk transfer seem unavoidable for life insurers next to product portfolio diversification. While investigating these strategies is beyond the scope of the current study, further research could analyze their effectiveness under a COVID-19 stress test. Additionally, our findings are limited to the impact of COVID-19 on mortality and insurer investment; hence, other aspects such as demand or contract cancellation could also be addressed in future research. Our results also rely on the specific impact of COVID-19 on mortality and financial markets, and such effect may differ in future pandemics. Furthermore, the study does not examine the effectiveness of combining different strategies, which might also be interesting to analyze. Although the greatest impact of COVID-19 appears to be over, the next pandemic will likely occur sooner than expected, and life insurers should be prepared for this situation. Even when specific characteristics differ between pandemics, this study provides some general insights into what to expect and how to respond as a life insurer. Appendix See Fig.9, Tables3, 4 and 5. Fig. 9 Estimated parameters ax and bx of the LC model for Germany and Spain 548 M.Hanika 1 3 Acknowledgements The author would like to thank Cassandra Cole, Nadine Gatzert, the participants at the AnnualMeeting of the American Risk and Insurance Association 2023 in Washington D.C., and two anonymous reviewersfor valuable comments on an earlier version of this paper. Funding Open Access funding enabled and organized by Projekt DEAL. Data availability The mortality data used in this study are publicly available in the Human Mortality Database athttps://www.mortality.org/. The other data used in this study are available from the author upon reasonable request. Table 3 Overall default probability DP when in year t=10 , the joint financial and mortality shock is applied for Germany and Spain under different numbers of sold contracts in t=10+ Notes: The number of sold contracts is 3000 for all age groups, and the absolute decrease is expressed in percentage points (p.p.) Number of sold term life contracts in t=10+ Age groups Germany Absolute decrease Spain Absolute decrease 9000 30/40/50 16.11% 19.42% 6000 40/50 16.07% − 0.04 p.p. 19.40% − 0.02 p.p. 6000 30/50 16.04% − 0.07 p.p. 19.25% − 0.17 p.p. 6000 30/40 15.86% − 0.25 p.p. 19.10% − 0.32 p.p. 3000 50 15.97% − 0.14 p.p. 19.15% − 0.27 p.p. 3000 40 15,81% − 0.30 p.p. 18.98% − 0.44 p.p. 3000 30 15.77% − 0.34 p.p. 18.89% − 0.53 p.p. 0 15.16% − 0.95 p.p. 18.12% − 1.30 p.p. Table 4 Overall default probability DP when in year t=10 , the joint financial and mortality shock is applied for Germany and Spain under different loadings 𝜆 in t=10+ Notes: The absolute decrease is expressed in percentage points (p.p.) Germany Absolute decrease Spain Absolute decrease 𝜆=0% 16.11% 19.42% 𝜆=1% 15.45% − 0.66 p.p. 18.69% − 0.73 p.p. 𝜆=2% 14.83% − 1.28 p.p. 17.99% − 1.43 p.p. 𝜆=23% 9.04% − 7.07 p.p. 10.38% − 9.04 p.p. Table 5 Overall default probability DP when in year t=10 , the joint financial and mortality shock is applied for Germany and Spain under different adjusted stock ratios 𝛼 in t=10+ Notes: The absolute decrease is expressed in percentage points (p.p.) Germany Absolute decrease Spain Absolute decrease 𝛼=15% 16.11% 19.42% 𝛼=10% 15.29% − 0.82 p.p. 18.49% − 0.93 p.p. 𝛼=5% 15.03% − 1.08 p.p. 18.11% − 1.31 p.p. 𝛼=0% 15.48% − 0.63 p.p. 18.48% − 0.94 p.p.