بهترین بیمه پرتفوی برای استراتژی های بلند مدت سرمایه گذاری در شرایط واقع بینانه

Constant proportion portfolio insurance (CPPI) strategies implemented in continuous time on asset prices following geometric Brownian processes are expected utility maximising for investors with HARA utilities. But, in reality, these strategies are implemented in discrete time and asset prices might jump. We show that under these more realistic circumstances, optimal CPPI strategies are still superior to optimal option based portfolio insurance (OBPI) strategies. The effects of discrete replication and jumps on optimal strategy parameters and certainty equivalent returns (CER) are examined by simulation and turn out to be minor in typical circumstances. Hence the much discussed gap risks are unimportant for investors in both portfolio insurance strategies and comparable for insurers of the gap risks.

Portfolio insurance strategies are designed to limit downside risk by ensuring a predefined floor whilst allowing participation in the upside potential of a risky asset. Their popularity is increasing amongst investors, e.g. pension funds, that seek insurance not only against abrupt falls in the markets, such as the crash in equities after the default of Lehman Brothers, but also against general downturns such as following the collapse of the dot.com bubble in the early 2000s and the 2007–08 subprime crisis. They also attract investors which otherwise would not consider investing in the riskier asset classes such as equities and commodities. This paper examines which of two approaches to portfolio insurance is preferable. The two most popular strategies are option based portfolio insurance (OBPI) and constant proportion portfolio insurance (CPPI). OBPI strategies, that protect an investment with a put option on the risky asset, were first discussed by Leland and Rubinstein (1976). Alternatively, one can secure a floor with an investment in a risk-free asset (a bond or a savings account) and the purchase of a call option on the risky asset. OBPI is a static method if the option can be bought, but in practice the option often needs to be replicated using a dynamic, discretely monitored investment strategy. Merton (1971) and Brennan and Solanki (1981) derive the optimal investment payoff for a HARA utility investor in a Black–Scholes economy with a risky and a risk-free asset. They show that when the risky asset follows a geometric Brownian process the optimal payoff consists of an investment floor plus a power on the underlying risky asset price. Perold (1986) and Black and Jones (1987) introduce the concept of CPPI strategies; Perold and Sharpe (1988) analyse its properties further and show that, in continuous time, CPPI strategies replicate the optimal payoff for HARA investors. A CPPI strategy ensures a predefined floor by dynamically rebalancing allocations between the risky asset and a risk-free asset. A constant proportion or multiplier, m, of the excess value of the investment above the floor (the buffer) is allocated to the risky asset, the rest is invested risk-free. The floor and the multiplier are exogenous variables to the model and are determined by the investor’s risk attitude and the investor’s views on the risky asset price dynamics. The lower the floor and the higher the multiplier, the greater the allocation to the risky asset. The investor then has a higher upside potential but the floor is approached more quickly if the risky asset price falls. As OBPI and CPPI strategies offer alternative downside protection, it is natural to examine under what circumstances an investor should prefer one type of protection to the other. Zhu and Kavee (1988) use Monte Carlo simulation to compare various sample statistics of replicated OBPI and CPPI payoffs. El Karoui et al. (2005) prove a very general result: for any concave utility function and in a complete market, the investment strategy that maximises expected utility (EU) subject to providing a downside protection is the unconstrained optimal strategy with a put option written on it, struck at the desired downside protection level. This result holds for both European and American style downside protection. Of course, the optimal unconstrained strategy is usually a dynamic rather than a buy and hold strategy, for example, a constant mix strategy in the case of CRRA utilities and geometric Brownian price dynamics. Bertrand and Prigent (2005), Annaert et al. (2009), and Zagst and Kraus (2011) compare OBPI and CPPI using stochastic dominance criteria. Bertrand and Prigent (2005) assume the geometric Brownian case and conclude that there is no evidence of strong or weak stochastic dominance between the two strategies, but one strategy may dominate the other in a mean-variance sense, depending on the value of the CPPI multiplier. Zagst and Kraus (2011) extend this analysis to considering second and third order stochastic dominance, deriving conditions for the strategy parameters and market parametrisations such that CPPI stochastically dominates OBPI to the second order at maturity. Annaert et al. (2009) simulate from an empirical distribution and also find no stochastic dominance between OBPI and CPPI. Furthermore, they consider a broader range of performance measures without finding that one strategy type systematically outperforms the other. Bertrand and Prigent (2011) show the dominance of CPPI strategies under Kappa performance measures by comparing theoretical payoffs in the geometric Brownian case and when the risky asset follows a jump process with Poisson distributed jump term.1 When the risky asset price follows a geometric Brownian diffusion process the portfolio value of a continuous time CPPI strategy can, theoretically, never reach the floor. But there is ample evidence for the existence of asset price jumps, if only because most markets trade during limited time periods every day and there are gaps between closing and opening prices. Hence in reality portfolio insurance strategy cannot be implemented in continuous time. Cont and Tankov (2009) examine the gap risk–the risk of falling below the floor–and derive the gap loss distribution and various associated risk measures in the context of a jump-diffusion price process. De Franco and Tankov (2011) maximise the investor’s utility when gap risk is covered by a third party. Zhu and Kavee (1988) compare various sample statistics of simulated CPPI and OBPI portfolios and Bertrand and Prigent (2011) show the outperformance of CPPI strategies over OBPI under the Omega measure, both, for a risky asset following a compound Poisson process. Table 1 gives an overview of the most relevant papers in the portfolio insurance literature and briefly summarises their contributions. There are different approaches to the modelling of discontinuous returns. Widely used is the model by Merton (1976) who adds a Poisson-driven jump term to a standard geometric Brownian process to account for rare sudden moves. As an alternative, Madan and Seneta (1987) introduce time-changed Lévy processes to model long-tailed stock return distributions. The authors consider pure jump processes that allow small moves to occur with a higher probability than large moves. This is a generalisation of the results of Clark (1973) who introduces subordinated processes that make use of a random time-change in a geometric Brownian process. The time-jump and the Brownian process are taken to be independent. The price can jump upwards or downwards, but the geometric process prohibits negative values for the risky asset after the occurrence of a jump. Empirical research supports the time-change modelling of asset returns. For instance Geman and Ané (1996) show that calendar-time returns are not normally distributed, as often assumed, but that returns on a unit trade basis follow a normal distribution. Madan et al. (1998) confirm that a Brownian motion time-changed process that models time with gamma distributed jumps fits historical returns significantly better than standard diffusion models. Our analysis extends previous research as follows. First, we use a certainty equivalent return (CER) to compare the performance of CPPI and OBPI strategies in realistic circumstances. We base this CER on a two-parameter HARA utility function, which encompasses most common types of utility functions, thus representing investors with very diverse risk preferences. Second, we argue that comparing a static OBPI strategy with a fixed payoff function to a dynamic CPPI strategy with replication errors would be unfair. The CPPI payoff profile under continuous replication and a geometric Brownian price process has a fair, path-independent price that can be calculated as for standard options. We can therefore compare the two defined payoffs based on their fair prices in a complete market. Third, we recognise that standard options for portfolio insurance are not always available. Thus we compare the performances of the two strategies when implemented with delta replication in discrete time. Fourth, we search for the optimal payoff under HARA utilities in a market with discontinuous returns modelled via a time-changed geometric Brownian process. In Section 2 we specify the continuous and discontinuous price dynamics for a risky asset price. We introduce portfolio insurance strategies in Section 3, show the optimal payoff under continuous returns and derive an approximation for the optimal payoff profile under discontinuous returns. Section 4 introduces a calibration method for a simple time-changed Brownian process and illustrates its application on an equity index and a single stock price. Despite the simplicity of this process we show that it fits empirical returns better than a geometric Brownian process. Section 5 discusses the results for different time horizons and investors with diverse sensitivities of risk tolerance to wealth. We conclude and comment on the merits of CPPI strategies in Section 6.

This paper compares two popular types of portfolio insurance strategies, CPPI and OBPI, in realistic market settings and shows the optimality of CPPI in all of them. We know from previous research that CPPI strategies are CER maximising and thus optimal for assets with geometric Brownian price processes and HARA utility investors. The investor’s utility function and the market parameters determine the optimal CPPI strategy parameters and hence the curvature of the payoff profile. When returns are discontinuous, we show that the optimal payoff is no longer a CPPI strategy but one that still provides portfolio insurance and that can be approximated by a polynomial function of return. Thus we set up a fair contest to examine if CPPI strategies still outperform OBPI in a discontinuous market setting. The strategies are either both purchased at fair prices or both manufactured via discrete replication strategies. We assume that risk averse investors have two-parameter HARA utilities that span a wide range of risk tolerance sensitivities to wealth. The comparison between CPPI and OBPI strategies is based on their maximum CERs. The payoff parameters are the multipliers and floors for the CPPI strategies and the strikes and floors for the OBPI strategies. We examine continuous and discontinuous return processes, two different time horizons for the investment and different utility parameters. Continuous returns are modelled with a geometric Brownian process whilst discontinuous returns are modelled with a time-changed Brownian process fitting the first three even moments of the return distributions for the equity index S&P 500 and the Apple stock. These two assets are chosen to illustrate relatively low and high risk assets and with few large or many small price jumps. Combining continuous or discontinuous return processes with payoffs that can be purchased at fair price or must be manufactured using discrete delta hedging creates four settings for the comparison of optimal OBPI and CPPI strategies. We confirm the optimality of CPPI over OBPI in a continuous Black–Scholes economy by simulations and assess the accuracy of the parameters obtained through simulations with their theoretical values. In the three other settings, theoretical results are not available but simulations show that CPPI strategies are never inferior to OBPI strategies. Discrete replication has minor effects as CERs are only very slightly reduced. Discontinuous returns reduce the CERs further, especially when the payoffs are discretely replicated. Our findings are applicable to the many pension funds’ and insurance companies’ investment portfolios that provide capital guarantees. In situations involving multiple risky assets, investors optimally invest in a CPPI portfolio where the risky component is a constant mix portfolio of the risky assets. Gap risk which is often named as one of the main disadvantages of portfolio insurance strategies proves to be relatively unimportant for investors in both CPPI and OBPI strategies. A historical advantage of OBPI is that is has been offered in the form of contractual guarantees, whereas CPPI strategies have generally been presented as a target but without contractual guarantee. However, power payoffs, as generated by CPPI strategies, should be favoured over OBPI strategies for their open-endedness. They are also superior from an EU point of view for geometric Brownian price processes and HARA utilities. We have shown that they maintain this superiority in more realistic circumstances where the payoffs have to be discretely replicated and the underlying process is discontinuous. The CPPI payoff is very flexible as it can be convex or concave depending on the market and utility parameters. One would expect a balance between strategies leading to convex and concave payoffs in a market equilibrium.