# Constant Proportion Portfolio Insurance Effectiveness under Transaction Costs

## Article excerpt

I. INTRODUCTION

Portfolio insurance allows investors to recover, at maturity, a given percentage of their initial investment, in particular when markets are bearish. One of the main standard portfolio insurance methods is the Constant Proportion Portfolio Insurance (CPPI). It has been introduced by Perold (1986), and further developed by Black and Jones (1987) for equity instruments and Black and Perold (1992). This dynamic strategy consists in setting a floor equal to the lowest acceptable value of the portfolio then allocating an amount to the risky asset which is determined as follows: this amount (called the exposure) is equal to the product of the cushion (defined as the excess of the portfolio value over the floor) and of a predetermined multiple. Both the floor and the multiple depend on the investor's risk tolerance. Usually, results about CPPI method are established under the assumption of continuous-time rebalancing. In this framework, the investor can modify his portfolio at any time. For example, if the cushion approaches zero, he reduces his exposure drastically, which keeps portfolio value from falling below the floor.

In this paper, we take account of the impossibility of trading truly in continuous-time. We focus on stochastic-time rebalancing. We prove that the impact of investor's tolerance is important, in particular when transaction costs occur. In Section 2, basic properties about CPPI method are recalled. In Section 3, we consider the case of stochastic time rebalancing with a deterministic target multiple. The investor rebalances his portfolio as soon as the ratio "exposure/cushion" reaches a lower or an upper bound. These bounds can be chosen equal to percentages of a fixed multiple (the target multiple). We provide explicit (or quasi-explicit) formulas for the portfolio values and probability distributions of rebalancing times, when asset price dynamics are driven by a Geometric Brownian motion. In Section 4, we illustrate main properties of such portfolio strategy. Simulations also allow the comparison between these different methods by means of the first four moments and some quantiles. (1)

I. CPPI WITH CONTINUOUS-TIME REBALANCING

A. The Standard Financial Model

The portfolio manager is assumed to invest in two basic assets: a money market account, denoted by B, and a portfolio of traded assets such as a composite index, denoted by S. The period of time considered is [0,T]. The strategies are self-financing. The value of the riskless asset B evolves according to:

d[B.sub.t] = [B.sub.t]rdt,

where r is the deterministic interest rate. The dynamics of the risky asset price S are given by a diffusion process: (2)

d[S.sub.t] = [S.sub.t-][[mu]t, [S.sub.t])dt + [sigma](t, [S.sub.t])d[W.sub.t]],

where W is a standard Brownian motion.

B. The Standard CPPI Method

This strategy consists in managing a dynamic portfolio so that its value is above a floor P at any time t of the management period. The value of the floor indicates the dynamic insured amount. It is assumed to evolve according to:

d[P.sub.t] = [P.sub.t]rdt

Obviously, the initial floor P0 is smaller than the initial portfolio value [V.sub.0]. The difference ([V.sub.0]- [P.sub.0]) is called the cushion. It is denoted by [C.sub.0]. Its value [C.sub.t] at any time t in [0,T] is given by:

[C.sub.t] = [V.sub.t] - [P.sub.t]

Denote [e.sub.t] as the exposure. It is the total amount invested in the risky asset. The standard CPPI method consists in letting [e.sub.t]=m[C.sub.t] where m is a constant called the multiple. The interesting case is when m>1, that is, when the portfolio profile is convex. Thus, the CPPI method is parametrized by [P.sub.0] and m. Note that the multiple must not be too high as shown for example in Prigent (2001a) or in Bertrand and Prigent (2002). The cushion value at any time is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, the guarantee is satisfied since the cushion is always non negative. …

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