A Note on Risk Aversion, Prudence and Portfolio Insurance

The Geneva Risk and Insurance Review, 2010, 35, (81–92) r 2010 The International Association for the Study of Insurance Economics 1554-964X/10 www.palgrave-journals.com/grir/ A Note on Risk Aversion, Prudence and Portfolio Insurance Philippe Bertranda and Jean-Luc Prigentb a GREQAM, University Aix-Marseille II & Euromed Management, Marseille, France. E-mail: b THEMA, University Cergy-Pontoise, 33 Bd du Port, Cergy-Pontoise 95011, France. E-mail: This paper examines some properties of portfolio insurance that are linked to the risk aversion and the prudence of the investor. We provide explicit conditions to measure portfolio sensitivity to downside risk. We also characterize the degree of portfolio insurance by means of the ratio of absolute prudence to absolute risk aversion. The Geneva Risk and Insurance Review (2010) 35, 81–92. doi:10.1057/grir.2009.8; published online 20 April 2010 Keywords: portfolio insurance; prudence; risk aversion Jel classification: D81; G11 Introduction Since the seminal results of Pratt (1964) and Arrow (1965), it has been recognized that the ratio A(x)¼u00 (x)/u0 (x) (the so-called Absolute Risk Aversion) is an appropriate measure of degree of risk aversion. Indeed, an individual with utility function u1 is more risk-averse than an individual with utility function u2 (i.e. u1 is a concave transformation of u2) if A1XA2. In that case, risk premium p1 is higher than risk premium p2. The literature on precautionary saving (see Leland, 1968) shows that precautionary saving is linked to the convexity of marginal utility u0 , or equivalently to the concavity of – u0 . Assuming the existence of the third derivative u000 , this is equivalent to the positivity of u000 . By analogy with risk aversion, Kimball (1990) introduces Absolute Risk Prudence P(x)¼u000 (x)/u00 (x) to measure the intensity of the precautionary saving motive. The idea is to substitute – u0 for u. An agent with utility function u1 is more prudent than an agent with utility function u2 if P1XP2. Precautionary premium c1 is higher than precautionary premium c2. The notion of prudence has also been introduced to examine the demand for insurance (see Eeckhoudt and Kimball, 1992). Additionally, Eeckhoudt and Gollier (2005) have analysed the link between prudence and optimal prevention. The Geneva Risk and Insurance Review 82 In this note, we examine the joint influence of risk aversion and prudence on portfolio insurance design. The portfolio optimization theory usually considers that investors maximize the expected utility of portfolio values VT at maturity T. These values may be payoff functions h of a given reference portfolio ST (also called the benchmark), usually a financial index such as the S&P 500. For example, one of the standard portfolio insurance techniques is Option Based Portfolio Insurance introduced by Leland and Rubinstein (1976), which consists of a portfolio invested in a risky asset S covered by a listed put written on it. More generally, portfolio insurance has been examined in the partial equilibrium framework by Leland (1980) and by Brennan and Solanki (1981). We provide new insights on the properties of optimal payoff h. Let us recall, as mentioned by Leland (1980), that convexity of payoff is linked to portfolio insurance. First, we give explicit conditions based on the investor’s prudence to analyse the sensitivity of portfolio payoff h to downside risk, as defined by Menezes et al. (1980). Second, we show how the investor must structure her portfolio, according to both her risk aversion and prudence. More precisely, we prove that an individual with utility function u1 has an optimal payoff h1 more convex (resp. more concave) than the optimal payoff h2 of an individual with utility function u2 if P1/A1XP2/A2 (resp. P1/A1pP2/A2). The next section introduces the model and presents the results. The section after that provides concluding remarks. All proofs are in the appendix. The model and our results In the standard optimal insurance literature, the optimal design of insurance contracts is mainly based on the determination of assumptions, under which a straight deductible or coinsurance are optimal. For example, Raviv (1979) proves that, for a risk-averse insurer, increasing and convex administrative costs imply that the optimal insurance policy displays coinsurance above a deductible. The insurance contract is usually modelled by a couple [I(.), P], where I(.) corresponds to the indemnity paid by the insurance company with additional administrative costs c(I) and P is the premium paid by the insured client. When the client faces a loss x, she receives I(x) from the company (usually, we assume: 0pI(x)px).1 The contract [I, P] is optimal if it maximizes the client’s expected utility of final wealth, while allowing the company to keep at least the same utility level. Denote respectively by W0 and W̃0, the initial wealths of the client and the insurance company, and by U and Ũ their utility 1 See Gollier (1987) and Breuer (2006) when this condition is not assumed. Philippe Bertrand and Jean-Luc Prigent A Note on Risk Aversion, Prudence and Portfolio Insurance 83 functions. Let F be the cumulative distribution function of the potential loss. Then, the usual optimal contract [I, P] is the solution of the following problem: xZmax max ½I;P U½W0  x þ IðxÞ  PdFðxÞ; 0 subject to: xZmax     ~ W ~ 0 þ P  IðxÞ  cðIðxÞÞ dFðxÞXU ~ W ~0 : U 0 The determination of the optimal solution allows to analyse the effect of risk aversion on the optimal deductible and on the coinsurance rate. In the finance literature, the optimal design of financial portfolio is studied in a different framework. We search for an optimal payoff schedule h(.) defined as a function of a given financial asset return (usually, one of the main financial indices). Therefore, we do not focus only on losses but also on potential gains.2 Additionally, the standard model of insurance assumes a constant loading, while the financial model supposes that derivative instruments can be used to hedge the portfolio. Finally, the constraint corresponds to a standard budget condition since only one agent is concerned: the investor. The calculation of the optimal solution allows to examine the effect of risk aversion on the convexity/concavity of the optimal portfolio payoff. Optimal payoff as a function of a reference portfolio In this section, we recall the results of Leland (1980) and of Brennan and Solanki (1981). Assume that the investor maximizes the expected utility of her terminal wealth VT under historical probability P. Portfolio value VT is assumed to be a function h of risky asset value ST. As usual, the utility U of the investor is taken to be increasing, concave and twice differentiable. Under the standard condition of no-arbitrage, asset prices are calculated under a riskneutral probability Q. Denote by (dQ)/(dP) the Radon–Nikodym derivative of 2 For the optimal insura (...truncated)