Portfolio Insurance Strategies: A Comparison of Standard Methods When the Volatility of the Stock is Stochastic

Abstract

ABSTRACT We compare the performances of the two standard portfolio insurance methods: the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI), when the volatility of the stock index is stochastic. In this framework, we provide a quite general formula for the CPPI portfolio value. We use criteria such as comparison of payoffs functions at maturity and various quantiles. We emphasize in particular the role of the insured percentage of the initial investment. JEL: G0, G15 Keywords: Portfolio insurance; OBPI; CPPI; Stochastic volatility I. INTRODUCTION One of the more popular strategies of portfolio insurance is the Option Based Portfolio Insurance (OBPI), introduced in Leland and Rubinstein (1976). It consists basically in buying simultaneously the stock (generally a financial index) and a put written on it. The value of this portfolio at maturity is always greater than the strike of the put, whatever the market fluctuations. Thus, this strike is the insured amount, which is often equal to a given percentage of the initial investment. The CPPI method has been analysed in Black and Rouhani (1989) and Black and Perold (1992). This method is based on a particular strategy to allocate assets dynamically over time. The investor starts by choosing a floor equal to the lowest acceptable value of the portfolio. Then, he computes the cushion that is equal to the excess of the portfolio value over the floor. Finally, the amount allocated to the risky asset (usually called the exposure) is determined by multiplying the cushion by a predetermined multiple. The remaining funds are invested in the reserve asset (for example, Treasury bills or other liquid money market instruments.) Initial cushion, multiple, floor and tolerance can be chosen according to the investor's own objective. The higher the multiple, the more the portfolio value increases in a bullish market. Nevertheless, the higher the multiple, the nearest the portfolio will be to the floor in a bearish market. As the cushion approaches zero, the amount invested on the risky asset approaches zero too. This feature implies that the portfolio value is below the floor only when there is a very sharp drop in the market before the investor can modify his investment weights. Therefore, the multiple must be bounded as shown in Bertrand and Prigent (2002a). Thus, this strategy is rather simple with respect to other approaches. The purpose of this paper is to compare these two strategies of portfolio insurance, when the volatility of the risky asset is stochastic. In section 2, we recall the basic properties of these two strategies. In section 3, we examine the impact of stochastic volatility for each portfolio insurance method. Finally in section 4, we analyse their properties and compare them. For this purpose, first we examine their payoffs and compute their expectations, variances, skewness and kurtosis of their returns. Second, we evaluate some of the quantiles of their returns. II. BASIC PROPERTIES OF THE OBPI AND THE CPPI We consider the following financial market: the period of time considered is [0,T]. Denote [B.sub.t] the riskless asset which has the following dynamics: [dB.sub.t] = [B.sub.t] r dt. Assume that the risky asset [S.sub.t] is a diffusion process: [dS.sub.t] = [S.sub.t] [a dt + [[sigma].sub.t] d[W.sup.1.sub.t]], where ([W.sup.1.sub.t])t is a standard Brownian motion. The volatility [[sigma].sub.t] is assumed to be stochastic and is defined as solution of the following stochastic differential equation: d[[sigma].sub.t] = [alpha] (t, [[sigma].sub.t])dt + [beta] (t, [[sigma].sub.t]) d[W.sup.2.sub.t], where [([W.sup.2.sub.t]).sub.t] is another standard Brownian motion independent from [([W.sup.1.sub.t]).sub.t]. We particularize the case of a stochastic volatility that evolves according to an Ornstein-Uhlenbeck process, as introduced in Scott (1987) and Stein and Stein (1991): d[[sigma]. …