Piecewise nonlinear goal-directed CPPI strategy

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Abstract

Traditional portfolio insurance (PI) strategy, such as constant proportion portfolio insurance (CPPI), only considers the floor constraint but not the goal aspect. This paper proposes a goal-directed (GD) strategy to express an investor’s goal-directed trading behavior and combines this floor-less GD strategy with the goal-less CPPI strategy to form a piecewise linear goal-directed CPPI (GDCPPI) strategy. The piecewise linear GDCPPI strategy shows that there is a wealth position M at the intersection of the GD and CPPI strategies. This M position guides investors to apply the CPPI strategy or the GD strategy depending on whether current wealth is less than or greater than M, respectively. In addition, we extend the piecewise linear GDCPPI strategy to a piecewise nonlinear GDCPPI strategy. This paper applies genetic algorithm (GA) technique to find better piecewise linear GDCPPI strategy parameters than those under the Brownian motion assumption. This paper also applies forest genetic programming (GP) technique to generate the piecewise nonlinear GDCPPI strategy. The statistical tests show that the GP strategy outperforms the GA strategy which in turn outperforms the Brownian strategy.

Introduction

Preventing from losing too much money in the investment process is also important while pursuing high return. Keeping the current wealth above a certain floor level is one way of portfolio insurance to achieve this objective. The optimal strategy for a constant floor turns out to be the well-known constant proportion portfolio insurance (CPPI) strategy (Black and Perold, 1992, Perold and Sharpe, 1988) and can be expressed as:xt=m1(Wt-F),Wt>F,where xt is the dollar amount invested in the risky asset at time t, Wt is the wealth at time t, m1 is a constant risk multiplier, and F is the floor. This optimal strategy states that one should invest more in the risky asset when the wealth increases. The CPPI strategy is a special case of portfolio insurance (PI) strategies. Most PI strategies consider only the floor and are therefore goal-less. In practice, a mutual fund manager usually has a performance objective or goal in mind at the beginning of an investment period. Then the fund manager has to do his best to achieve this objective or goal before losing too much money. Now if a fund manager follows the CPPI strategy and the current wealth is very close to the goal, he will invest a large portion of capital in the risky asset and will have a greater chance of failing his almost reached goal. This possibility is not desirable to mutual fund managers. The major reason is that the CPPI strategy is goal-less while fund managers do have a goal in mind during the investment process.

In addition, evidences show that an investor will change his risk-attitude under different wealth levels. In particular, these studies showed that fund managers change their risk-attitudes based on their performance compared to the benchmark. However, there are contradictory observations among these studies. Some studies observed that fund managers take risk-averse behavior when their performance is worse than the benchmark (low wealth risk aversion), while some other studies observed that fund managers take risk-seeking behavior when their performance is worse than the benchmark (high wealth risk aversion).

The above contradiction in fact can be explained by portfolio insurance perspective and goal-directed perspective, respectively. The goal-less CPPI strategy demonstrates the low wealth risk aversion phenomenon. Goal-directed perspective proposes that an investor in financial markets will consider certain investment goals. A goal-directed investor will take risk-seeking behavior when the distance from current wealth to the goal (goal distance) is long and will take risk-averse behavior when the goal distance is short.

To exhibit the high wealth risk-averse behavior, we therefore construct a goal-directed (GD) strategy:xt=m2(G-Wt),WtG,where G is the goal and m2 is a constant. The concept of GD strategy can also be supported by Browne’s study (Browne, 1997). The GD strategy is not goal-less but is floor-less.

To consider both floor and goal, we further combine the CPPI strategy and the GD strategy to form a piecewise linear goal-directed CPPI (GDCPPI) strategy as:xt=0,WtF,m1(Wt-F),F<Wt<M,m2(G-Wt),MWtG,where M=m1F+m2Gm1+m2 is a wealth position at which point the CPPI and GD strategies intersect. This M position guides investors to apply the CPPI strategy or the GD strategy depending on whether current wealth is less than or greater than M, respectively. Note that the piecewise linear GDCPPI strategy reduces to the GD strategy when m1  ∞, and becomes the CPPI strategy when m2  ∞. That is, the piecewise linear GDCPPI strategy is a generalization of both the CPPI and GD strategies.

Piecewise linear GDCPPI strategy is linear and makes the Brownian motion assumption. It is reasonable to infer that some better nonlinear strategies can be found in the solution space when the Brownian motion assumption no longer holds. Therefore, we extend the piecewise linear GDCPPI strategy to a piecewise nonlinear GDCPPI strategy. We then apply the forest genetic programming (GP) technique to generate a good piecewise nonlinear GDCPPI strategy.

To compare the performance of the GP generated piecewise nonlinear GDCPPI strategy with the linear one, we need to determine the constants m1 and m2 in the piecewise linear GDCPPI strategy. One way to determine these constants is using their theoretical optimal formulas derived on the basis of the Brownian motion assumption for stock prices as in traditional CPPI strategy (Black and Perold, 1992, Perold and Sharpe, 1988) and Browne’s study (Browne, 1997). Another way to determine these constants is applying genetic algorithms (GAs) to find a set of good parameter values based on historical data. Our statistical test results show that the GA generated piecewise linear GDCPPI strategies outperform the Brownian ones under two performance measures, and the GP generated piecewise nonlinear GDCPPI strategies outperform the GA generated piecewise linear GDCPPI strategies.

The remainder of this paper is organized as follows. Section 2 reviews the GA and GP techniques along with their financial applications. Section 3 formulates the models of our different goal-directed strategies. Section 4 describes our experiments to show the learning effects of genetic algorithms for the piecewise linear GDCPPI strategy and forest genetic programming for the piecewise nonlinear GDCPPI strategy. Section 5 presents our conclusions and suggests directions for future work.

Section snippets

Evolutionary algorithms

Evolutionary algorithms (EAs) are based on Darwin’s theory of evolution: the survival of the fittest. Genetic algorithms (GAs) and genetic programming (GP) are two typical EAs. They have been used to solve various problems in trading (Deboeck, 1994a, Jang and Lai, 1994, Kamijo and Tanigawa, 1990, Kimoto et al., 1990, Tanigawa and Kamijo, 1992). Evolutionary algorithms use an evolutionary process resulting in a fittest solution to solve a problem. The evolutionary process consists of three basic

Portfolio insurance strategy

The formulation and solution of optimal portfolio insurance problem will be described following the work by Grossman and Zhou (1993). Assume there are two assets: a risk-free asset such as a T-bill and a risky asset such as a stock. Let the stock price dynamic be dPtPt=μdt+σdzt, where μ is the mean of return rates, σ is the standard deviation of return rates, and zt is a Brownian motion at time t. The portfolio wealth dynamic is then dWt = rWt dt + xt(μ dt + σ dzt), where r is the risky-free rate of

Experiments and analyses

In this study we apply the Brownian (B), the genetic algorithm (GA) and the genetic programming (GP) techniques to generate GDCPPI strategies. Technique B produces the piecewise linear GDCPPI strategy according to Eqs. (3), (6) under Brownian motion assumption and does not require a learning process as opposed to the GA and GP methods. Technique GA finds the fittest m1 and m2 constants to form the piecewise linear GDCPPI strategy. Technique GP is applied to generate the piecewise nonlinear

Conclusions

Traditional portfolio insurance strategy such as CPPI, a special case of portfolio insurance (PI) strategy, does not consider the goal perspective and may fail an almost reached goal as the result. Although current Browne’s study (Browne, 1997) considers the similar goal-seeking objective, it still does not consider both the objectives of floor protecting and goal seeking. This paper combines the concept of the CPPI strategy and the goal-directed strategy derived from Browne’s study to form a

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