Consumption‐Based Asset Pricing in Insurance Markets: Yet Another Puzzle?

• Author: Hato Schmeiser,Daliana Luca,Alexander Braun
• DOI: http://doi.org/10.1111/jori.12230
• Published date: 01 September 2019
• Date: 01 September 2019

©2017 The Journal of Risk and Insurance (2017).

DOI: 10.1111/jori.12230

Consumption-Based Asset Pricing in Insurance

Markets: Yet Another Puzzle?

Alexander Braun

Daliana Luca

Hato Schmeiser

Abstract

Although insurance is the typical textbook example for an asset that

negatively correlates with consumption, the suitability of the classical

consumption-based asset pricing model with power utility to explain his-

torical premiums and claims has not yet been tested. We fill this gap by

fitting it to property–casualty market data for Australia, Italy, the Nether-

lands, the United States, and Germany. In doing so, we reveal yet another

asset pricing anomaly. More specifically, the consumption-based model im-

plies even larger relative risk aversion coefficients in the insurance sectors

than in the equity markets of the aforementioned countries. To solve this

puzzle, we draw on the loss aversion and narrow framing approach by

Barberis, Huang, and Santos (2001) as well as the second-degree expectation

dependence framework by Dionne, Li, and Okou (2015), with encouraging

results.

If you buy an asset whose payoff covaries negatively with consumption, it

helps to smooth consumption and so is more valuable than its expected pay-

off might indicate. Insurance is an extreme example. Insurance pays off ex-

actly when wealth and consumption would otherwise be low—you get a check

when your house burns down. For this reason, you are happy to hold insur-

ance, even though you expect to lose money—even though the price of in-

surance is greater than its expected payoff discounted at the risk-free rate.

–John H. Cochrane

Introduction

The classical consumption-based model, which was established through the work

of Rubinstein (1976), Lucas (1978), Breeden (1979), Grossman and Shiller (1981),

Alexander Braun, Daliana Luca, and Hato Schmeiser are at the University of St. Gallen,

Tannenstrasse 19, 9000 St. Gallen, Switzerland. Braun can be contacted via e-mail:

alexander.braun@unisg.ch. The authors are grateful to two anonymous refereesfor their valu-

able comments and suggestions and to Swiss Re for supporting this research with aggregate

insurance market data.

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2The Journal of Risk and Insurance

as well as Hansen and Singleton (1983), is one of the most influential accom-

plishments in modern asset pricing theory. Utility-maximizing individuals in a

representative-agent economy face the intertemporal choice of either consuming

their wealth or investing it in a risky asset. The first-order condition for this trade-

off leads to the central pricing equation of the model: the demand for the asset

is adjusted until the loss in utility suffered due to a slightly lower consumption

level today equals the gain in expected utility achieved by being able to consume

a little more of the future payoff of the asset. Therefore, equilibrium asset prices

are expectations of payoffs discounted at the representative agents’ marginal rate

of substitution. Based on this idea, it is possible to derive risk adjustments that

rely on the covariance of payoffs or returns with marginal utility and thus, ulti-

mately, consumption. Assets that perform well when the investor is able to con-

sume abundantly, but pay little in times when his consumption is restrained, are

perceived to be risky and sell for prices below their expected payoff discounted

at the risk-free interest rate. Insurance policies, in contrast, indemnify their holder

after the occurrence of a loss in wealth, thus reducing the volatility of consump-

tion. Hence, individuals are prepared to accept a negative expected return on such

contracts.

Despite its theoretical appeal, the consumption-based model repeatedly failed in em-

pirical applications. Its most famous shortcoming is the inability to explain risk pre-

miums observed in postwar stock market data with a reasonable degree of risk aver-

sion. This is the famous equity premium puzzle, which was described by Mehra

and Prescott (1985) for the United States and, since then, has received a lot of schol-

arly attention. Wheatley (1988) as well as Campbell (2003) found evidence for the

puzzle in many developed economies, while Donadelli and Prosperi (2012) revealed

its existence in a number of emerging markets. Early on, Kandel and Stambaugh

(1991) suggested that it might be necessary to contemplate higher values for the

risk aversion coefficient. This, however, leads to the emergence of another well-

known asset pricing anomaly: the risk-free rate puzzle as constituted by Weil (1989).

Given extreme risk aversion, power utility agents are extraordinarily reluctant to en-

gage in intertemporal substitution. This implies that the empirically observed low

and stable risk-free interest rates can only be explained by the consumption-based

model if investors exhibit a subjective time discount factor greater than one. Al-

though such a negative time preference is theoretically possible, it is not very plau-

sible as individuals are typically impatient, favoring earlier over later consumption

(see, e.g., Kocherlakota, 1996).

Thus, ever since the discovery of the equity premium puzzle, economics and fi-

nance researchers have targeted more meaningful risk aversion levels. The domi-

nant strand of literature in this regard centers on model refinements by means of

separated time and risk preferences (see, e.g., Epstein and Zin, 1989, 1991), habit

formation (see, e.g., Abel, 1990; Constantinides, 1990; Ferson and Constantinides,

1991; Campbell and Cochrane, 1999), idiosyncratic consumption shocks (see, e.g.,

Mankiw, 1986; Weil, 1992; Constantinides and Duffie, 1996; Heaton and Lucas,

1996; Gomes and Michaelides, 2008), and rare economic disasters (see, e.g., Rietz,

1988; Barro, 2006, 2009; Gabaix, 2008, 2012; Wachter, 2013). However, none of these

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Consumption-Based Asset Pricing in Insurance Markets 3

solutions is considered to be fully satisfactory (see, e.g., Mehra and Prescott, 2003).

Other research efforts have focused on long-run persistence in consumption and div-

idend growth (see, e.g., Bansal and Yaron, 2004; Bansal, Kiku, and Yaron,2010; Koijen

et al., 2010), loss aversion (see, e.g., Benartzi and Thaler, 1995; Barberis, Huang, and

Santos, 2001; Barberis and Huang, 2001, 2009), disappointment aversion (see, e.g.,

Routledge and Zin, 2010), ambiguity aversion (see, e.g., Chen and Epstein, 2002;

Gollier, 2011; Rieger and Wang, 2012), and most recently, higher-order risk pref-

erences (see Dionne, Li, and Okou, 2015). In addition, there have been attempts

to improve the model’s estimation basis by relying on stockholder samples (see,

e.g., Vissing-Jorgensen and Attanasio, 2003), long-run consumption changes (see,

e.g., Parker and Julliard, 2005), as well as forward-looking survey and option data

(see S¨

oderlind, 2009b). Finally, some authors have explored whether factors such

as transaction costs (He and Modest, 1995; Bansal and Coleman, 1996; Luttmer,

1996), borrowing constraints (see, e.g., Constantinides, Donaldson, and Mehra,

2002), and taxation (see, e.g., McGrattan and Prescott, 2003) drive the equity risk

premium.

Apart from equities, the consumption-based model has also been applied to fixed

income (see, e.g., Backus, Gregory, and Zin, 1989; Wachter, 2006), stock options (see,

e.g., Liu, Pan, and Wang, 2005; Backus, Chernov, and Martin, 2011), currencies (see

Verdelhan, 2010) and even catastrophe bonds (see, e.g., Dieckmann, 2011). Some-

what surprisingly, however, its suitability for insurance contracts has not yet been

tested in the financial economics literature, although they are the typical textbook

example for an asset that is negatively correlated with consumption and therefore

positively correlated with marginal utility. We fill this gap by fitting the classical

consumption-based model with power utility to historical property–casualty insur-

ance market data. In doing so, we consider two alternatives for the estimation of

the relative risk aversion (RRA) coefficient. First, we apply an extended version of

Stein’s Lemma introduced by S¨

oderlind (2009a), which builds on a bivariate mix-

ture normal distribution and thus allows for skewed and leptokurtic asset returns,

given the log stochastic discount factor (SDF) is Gaussian. Second, we follow Hansen

and Singleton (1983) in assuming that consumption growth and asset returns are

jointly lognormally distributed as well as homoskedastic. Both approaches are com-

plemented by Hansen and Jagannathan (1991) volatility bounds. Based on aggregate

annual premiums and claims for Australia, Italy, the Netherlands, the United States,

and Germany, we are able to provide evidence of yet another asset pricing anomaly.

More specifically, the consumption-based model implies even larger RRA coefficients

in the insurance sectors than in the equity markets of the aforementioned countries.

To solve this insurance premium puzzle, we draw on the loss aversion and narrow

framing approach by Barberis, Huang, and Santos (2001) as well as the second-degree

expectation dependence framework by Dionne, Li, and Okou (2015), with encouraging

results.

The rest of the article is organized as follows. In the next section, we briefly revisit

the classical consumption-based model and derive the two procedures that will be

employed for its empirical application. The third section contains a discussion on

the applicability of asset pricing theory in the insurance context. Furthermore, in the

Consumption-Based Asset Pricing in Insurance Markets 3

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