EXPECTED PERFORMANCE AND RISKS OF VARIOUS PORTFOLIO INSURANCE STRATEGIES

Guaranteed funds have had great success in France recently. These products are complex, and their behavior is not always well understood. What are their expected performance and risk profiles? While the technique of portfolio insurance (PI) has existed for some time, only the simplest case of the most popular method, the socalled “cushion method” of Black and Fisher, also known as Constant Proportion Portfolio Insurance (CPPI) has received any theoretical analysis. In this paper, we consider this method in a realistic case, where there are constraints on the maximum exposure to the market. We also study PI techniques more sophisticated than this basic method, principally through stochastic simulations. We arrive at an understanding of the performances and risks of PI strategies little developed till now, including sensitivity to parameters such as the multiplier. Our simulation results show that the imposition of a constraint on the maximum exposure significantly diminishes the performance of the method. Under the constraint, the leverage effect of the multiplier appears to be lost; but the introduction of a “ratchet” effect recuperates part of the loss. EXPECTED PERFORMANCE AND RISKS OF . , 1095 Introduction’ This article concerns guaranteed funds that permit investment in the stock market while being assured of recuperating at least the amount invested. The idea behind these funds is the dynamic portfolio strategy called portfolio insurance. Until recently portfolio insurance was generally done by replicating options an idea introduced originally by Rubinstein and Leland [6]. In this technique, the portfolio is insured through a put with the appropriate maturity. In practice, since puts of sufficiently long maturity are not available, the strategy consists of rebalancing positions in the underlying portfolio (let us assume it is an index such as the CAC 250), and cash so as to replicate exactly a put option on the index. In this method, the exposure to the index at any moment is therefore given by the theory of options. An alternative approach, first popularised by Black and Jones [l] but having its origins in earlier work of Met-ton [53, also rebalances positions on the index and cash, but it is based on the following two ideas: first, the portfolio is always maintained above a certain minimum level called the floor, the difference or the “surplus” being called the “cushion”the floor is assumed to grow at a fixed rate (for example, at the risk-less rate of interest) such that at the maturity of the fund, it is at least equal to the guaranteed amount; second, the exposure to the market at any moment is determined as a (non-decreasing) function of the cushion, usually a constant multiple of the cushion (Constant Proportion Portfolio Insurance, or CPPI). The first takes care of the insurance; the second determines the performance of the fund. When it comes to the mutual funds market, the reluctance of many private investors, especially in continental Europe, to invest in stocks led CCF, like many other mutual fund managers, to design appropriate guaranteed funds. The products marketed in the early 90’s at a time when interest rates were high enough to ensure an attractive pay-off, fell into two broad categories. In the first one entry and exit dates were fixed so that the guarantee is actually me same for every subscriber; for the sake of practicality, early withdrawal was generally possible, but with a big penalty on the performance. The second type of fund is open-entry open-ended, and tries to combine the well’ Acknowledgement : We thank Emmanuel Campagne-Simon for extensive programming help 1096 5TH AFIR INTERNATIONAL COLLOQUIUM appreciated advantage of liquidity (without any penalty) and the enticing guarantee. Whereas options technology is suitable to manage the first category, it is no longer appropriate to the second. The CPPI technique on the other hand can accommodate for various entry dates and it is therefore useful to investigate the capacities and the limitations of this technique. CPPI: Guarantee and performance There are two aspects to the study of this technique: its ability to meet the guarantee, and its performance. Under such a strategy, the exposure approaches zero as the cushion approaches zero; when the cushion is zero, the portfolio is completely invested in cash. Thus, in theory, the guarantee is perfect: the strategie of exposure ensures that the portfolio never descends below the floor; in the event that it touch the floor, the fund is “dead” it can deliver no performance beyond the guarantee. In practice, the rebalancement of the portfolio being done at discrete intervals rather than continuously, there is a small risk of the portfolio crashing through the floor in between two rcbalancements, as happened with some assured portfolios during the 1987 crash. In such a case, it is impossible even to meet the guarantee. Thercfore, one objective of management might be to minimise this possibility. With respect to performance, the objective is typically to attain some percentage of the performance that would have been attained if all the funds had been invested in the index. The performance of the method depends clearly on its two defining factors : the evolution of the floor, and the multiplier. In theory, assuming that the portfolio is rebalanced continuously, that the floor grows at the riskless rate and that the price of the risky asset follows a log-normal model, it can be shown that the strategy has an option-like payoff, which can be explicitly given as a function of the terminal value of the index in particular, the value of the portfolio at maturity is pathindependent. It is well known that portfolio insurance modifies the risk profile, in displacing it to the right but “paying” for the guarantee in terms of a lower likelihood of high gains. The attraction of the method lies in the leverage effect of the multiplier. The higher the multiplier, the higher the EXPECTED PERFORMANCE AND RISKS OF . 1097 exposure to the market, and the more the gains in strong market conditions; conversely, the more rapid the fall, accompanying a fall of the market. Ihe portfolio therefore has a return which increases with the multiplier, but so does the risk. Modifications of CPPI This idealised portfolio has some obvious limitations, apart from the assumption of continuous evolution to begin with, the existence of leverage constraints in practice. If we impose the realistic constraint, say, that the exposure to the risky asset cannot exceed a certain proportion of the value of the portfolio, the latter loses its attractive properties; in particular, it is no longer analytically tractable. From the point of view of performance, another potential weakness of the method is the constant growth rate of the floor, since it determines the level of insurance: if the market climbs sufficiently, the floor threatens to become insignificant in relation to the value of the portfolio. A logical modification of the CPPI is therefore to have a variable floor: to incorporate a “ratchet” effect, or a jump in the floor, in order to take advantage of significant market rises. The obverse of this case is when the market falls sufficiently for the cushion to approach zero, in which case the exposure approaches zero. Under the basic CPPI, when the exposure touches zero, the portfolio is completely invested in the non-risky asset a situation to be avoided. A possible approach to tackle this eventuality is to adjust the floor downward to artilicially augment the cushion however, to avoid tampering with the guarantee, the floor could be, for example, initially augmented by some amount beyond what is prescribed by the CPPI method. The floor could in fact be dynamically adjusted in response to market changes, while keeping the basic CPPI rule for the exposure. We may refer to such strategies as CPPI with variable (growth) floor. A variable floor strategy, based on a somewhat different idea, is described in [3]. Such strategies may be too complex to be treated theoretically. For example, it can be shown that unless the floor grows exactly at the risk-less rate,the value of the portfolio at maturity is path-dependent. Thus, while the 5TH AFIR INTERNATIONAL COLLOQUIUM performance promised is in terms of that of the index at maturity, the final value is not a simple function of the index in constrast to the case of put replication, but a function of its entire trajectory. The performance is thus by no means transparent. In addition, if we allow that rc-balancing is not continuous, and the possibility of discontinuitics (jumps) in the price of the risky asset, we have the problem of the guarantee not being perfect. On the other hand, under the idealiscd CPPI discussed above, the portfolio can be completely described analytically. Thus, this strategy, while unrealistic, can serve as a benchmark for the more complex modilicd stratcgics. Theoretical framework The CPPI was in fact shown by Merton [S] to be the optimal strategy corresponding to a certain utility maximization problem: Grossman and Vila [41 even give the optimal “multiplier” as a function of the risk aversion parameter. Further, under leverage constraints, they show that a modiiicd strategy, under which the CPPI is followed until the leverage constraint becomes binding, is still optimal; the optimal multiplier is the same as in the unconstrained case. However, it is not clear that this theory can bc extended to the case where both the multiplier and the floor strategy are parameters. It is also not clear to what extent this expected utility formulation is useful in practice (the choice of the multiplier is not evident, as the risk-aversion parameter of the investor is not easily determinable; nor is even the choice of the appropriate pcrfonnance criterion for an insured portfolio), however, the CPPI is a technique easy to understand and implement, and independent of tim