Is Fair Pricing Possible? An Analysis of Participating Life Insurance Portfolios

• Author: Hato Schmeiser,Carolina Orozco‐Garcia
• Date: 01 June 2019
• DOI: http://doi.org/10.1111/jori.12223
• Published date: 01 June 2019

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©2017 The Journal of Risk and Insurance. Vol.XX, No. XX, 1–40 (2017).

DOI: 10.1111/jori.12223

Is Fair Pricing Possible? An Analysis of

Participating Life Insurance Portfolios

Carolina Orozco-Garcia

Hato Schmeiser

Abstract

Pooling individual customers with differentinception dates into a single legal

entity may generate intergenerational subsidies that are accentuated when

the insurer has limited liability. This article aims to investigate whether an

insurer can charge fair premiums while simultaneously ensuring identical

levels of default risk—measured by the value of the default put option ratio—

for all generations. The decision variables for achieving these goals are asset

allocation and the amount of the insurer’s equity capital. We propose an

accounting framework where the insurer controls for insolvency positions

annually after the first contract is issued. Additionally, a run-offframework

is developed where the insurer does not declare bankruptcy in case of an

insolvency, but instead stops issuing new policies and runs the company

until the assets are exhausted or the last policyholder is paid. We find that

intergenerational subsidies and differentlevels of default risk per generation

cannot be avoided whenever we face a positive default risk.

Introduction

Much research has been published recently on the subject of participating life insur-

ance contracts. This has largely focused on pricing at the market value of life insur-

ance contracts with embedded options. Doherty and Garven (1986) and Briys and

de Varenne (1997) introduce the default risk of the insurer as a European put option

arising from the insurer’s limited liability that can be exercised on the maturity date if

the company cannot meet the policyholders’ claims. Hence, shareholders do not incur

all the downside costs of default, while still capturing some of the upside earnings,

and policyholders pay part of the insolvency cost. Because of this asymmetric payoff,

both the value of the insolvency put option and the value of the shareholders’ stake

increase as the insurer takes on more risk. Grosen and Jørgensen (2002) develop a

pricing model in which life insurance liabilities and equity are considered composite

contingent claims. These authors examine the fair valuation of life insurance liabili-

ties, including interest rate guarantees and additional participation in the investment

surplus. The authors conclude that the model can be used to determine the set of

CarolinaOrozco-Garcia and Hato Schmeiser are at the Institute of Insurance Economics, Univer-

sity of St. Gallen. Orozco-Garcia can be contacted via e-mail: carolina.orozcogarcia@unisg.ch.

Schmeiser can be contacted via e-mail: hato.schmeiser@unisg.ch

Vol. 86, No. 2, 521–560 (2019).

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parameters that characterizes initially fair contracts (i.e., in which the present value

of the premium equals the present value of future payouts).

Previous research has primarily addressed the fair pricing of individual insurance

contracts (e.g., Bacinello, 2001; Tanskanenand Lukkarinen, 2003; Ballotta, Haberman,

and Wang, 2006; Ballotta, 2009). In general, researchers have focused on finding a

suitable set of design parameters that could lead to a fair valuation of such contracts.

Many studies have also investigated the interactions between the contract parameters

that define the insurer’s and customers’ risk situations (e.g., Gatzert and Kling, 2007;

Kling, Richter, and Ruß, 2007b; Le Courtois and Quittard-Pinon, 2008; Schmeiser and

Wagner, 2015). Other authors, such as Kling, Richter, and Ruß (2007a,2007b), Graf,

Richter, and Ruß (2011), Zemp (2011), and Bohnert and Gatzert (2012), have investi-

gated different schemes for distributing surplus and the effects of such schemes on

both the fair valuation of contracts and the insurer’s risk exposure.

In practice, insurance companies issue portfolios of similar insurance contracts that

are typically not signed simultaneously and do not have coinciding maturity dates.

Instead of using separate legal entities for each policyholder, the insurer collects the

individual premiums and invests them as a whole using a single investment strat-

egy. Few studies have focused on portfolios of participating life insurance policies

(cf. Hansen and Miltersen, 2002; Gerstner et al., 2008; Gollier, 2008). In particular,

Døskeland and Nordahl (2008) comparethe return for different generations, assuming

correct pricing, and show that there are intergenerational cross-subsidization effects

in life insurance contracts with interest rate guarantees when different generations

share the same reserves. Ibragimov, Jaee, and Walden (2010) investigate premium-

to-liability ratios across insurance lines when different sharing rules are applied to

allocate the shortfall in claims when the insurer defaults.

To our knowledge, the fair pricing of individual policies as part of a pool of isurance

policies has not been particularly investigated so far. Hieber, Natolski, and Werner

(2016) theoretically demonstrate the existence of parameter combinations that allow

individual insurance contracts with an annual (cliquet-style) return guarantee to be

incorporated into an existing portfolio of insurance policies (heterogeneous portfolio)

with no wealth transfer between the two groups. Even though it can seemingly be

proven that a fair price for the new contract exists, the specific parameter combination

can only be accessed numerically and the interest rate guarantee must be adjusted for

different contracts, which is not always possible if the model is applied to compul-

sory schemes, for example, the second pillar in Switzerland. Hieber, Natolski, and

Werner’s analysis does not extend to possible cross-subsidization between sequences

of different generations. In addition, the authors concentrate on the default risk of the

whole portfolio rather than the risk profiles for individual generations.

This article investigates whether the following two ideal conditions for premiums

charged to policyholders joining an insurance portfolio can be achieved simultane-

ously: (1) that all generations pay fair premiums, and (2) that all generations face the

same default risk (measured by the value of the default put option ratio). We try to

determine the likelihood of avoiding intergenerational subsidies. If cross-subsidies

Fair Pricing of Insurance Portfolios 3

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are ex ante unavoidable, we try to identify which policyholders (or policyholder gen-

erations) are subsidizing and which policyholders (or policyholder generations) are

being subsidized (i.e., which generation(s) pays higher premiums than the others).

Additionally, solvency standards applicable to private life insurance companies re-

quire continuous supervision of their risk situation. Thus, sophisticated pricing mod-

els must be developed to account for evolving insolvency risk. We investigate the

existence of risk management strategies where all policyholders who join the insur-

ance portfolio face equitable default risk and consequently pay the same cost for risk

of default.

We introduce two portfolio pricing frameworks for participating life insurance poli-

cies. These aim to capture the complex financial interactions between different pol-

icyholder generations within a portfolio. The first framework, hereafter called the

accounting framework, is an extension of the individual pricing model in the litera-

ture (cf. Grosen and Jørgensen, 2002). The model assumes that all policies have the

same contracts’ maturity,even though the policyholders sign their insurance contracts

at different times. The insolvency position can be assessed yearly after the first insur-

ance contract is issued. If the market value of the insurer’s assets is lower than the

market value of its accrued liabilities at the end of the period (year), then the insurer

declares bankruptcy and all policyholders are paid out. The policyholders would then

absorb part of the shortfall and their payoffs would be reduced by the default cost,

which equals the value of a put option on the insurer’s asset value at the moment of

declaring bankruptcy.The insurance company can continue operating, including the

issuance of new insurance policies and payments of maturity benefits, as long as it

has not actually defaulted. The company controls for insolvency at the end of each

period until all generations participating in the portfolio are paid out. The accounting

framework seems to be an acceptable approximation of a portfolio of policies in insur-

ance companies. However,state-provided pension schemes typically do not interrupt

their operation if the value of the liability exceeds the assets but rather continue as

long as payments can be provided. Our second framework, hereafter referredto as the

run-off framework, allows the insurance company to continue operating as long as

the assets’ value is enough to pay the benefits to the policyholders. Under this model,

we assume that the insolvency positions are assessed only at the maturity dates. If the

assets’ value is lower than the accrued liabilities, then the company enters a “run-off”

situation in which it is not allowed to issue any new policies but continues paying

the benefits of the in-force portfolio until the assets account is expired or the last pol-

icyholder receives its maturity payment. Bankruptcy is declared only if the assets are

not enough to pay actual cash flow until the end of maturities.

Even though we believe our results are of general interest for the participating life

insurance sector, our model setup is based on the “second pillar” old-age provi-

sion system in Switzerland, introduced in 1985. As a compulsory system, the em-

ployer/policyholders can choose between providing savings within a pension fund

or an insurance company.For instance, pension funds are subject to the run-off frame-

work while insurance companies have to use the accounting framework. The under-

lying concept is a funding system with a minimum interest rate guarantee and a

participation rate (cliquet-style type). Both parameters are regulated for long periods