Optimal portfolio positioning within generalized Johnson distributions

Abstract

Many empirical studies have shown that financial asset returns do not always exhibit Gaussian distributions, for example hedge fund returns. The introduction of the family of Johnson distributions allows a better fit to empirical financial data. Additionally, this class can be extended to a quite general family of distributions by considering all possible regular transformations of the standard Gaussian distribution. In this framework, we consider the portfolio optimal positioning problem, which has been first addressed by Brennan and Solanki [J. Financial Quant. Anal., 1981, 16, 279–300], Leland [J. Finance, 1980, 35, 581–594] and further developed by Carr and Madan [Quant. Finance, 2001, 1, 9–37] and Prigent [Generalized option based portfolio insurance. Working Paper, THEMA, University of Cergy-Pontoise, 2006]. As a by-product, we introduce the notion of Johnson stochastic processes. We determine and analyse the optimal portfolio for log return having Johnson distributions. The solution is characterized for arbitrary utility functions and illustrated in particular for a CRRA utility. Our findings show how the profiles of financial structured products must be selected when taking account of non Gaussian log-returns.

Keywords:

• Johnson distributions
• Optimal portfolio positioning
• Structured asset management
• Hedge funds

AMS Subject Classifications:

• C61
• G11

Notes

No potential conflict of interest was reported by the authors.

1 This method is based on a relatively simple strategy to dynamically allocate the portfolio. The investor begins by determining a floor equal to the smallest acceptable portfolio value (discounted), depending on his desired guarantee at maturity. Then he defines the cushion as the difference in excess of the value of the portfolio and the floor. The amount invested in the risky asset is called the exposure. It corresponds to the value of this cushion multiplied by a predetermined factor, called the ‘multiple’. This leads to the fact that when the value of the cushion approaches zero, exposure approaches zero as well. In continuous time and in the absence of jumps in the dynamics of the risky asset reference, this method maintains the portfolio value above the floor, which is exactly the purpose of the guarantee.

2 The comparison between the OBPI and the CPPI has been conducted using various criteria Black and Rouhani (1989), Black and Perold (1992), Bookstaber and Langsam (2000), Bertrand et al. (2001). The hypothesis of lognormal returns is generally used in these studies (see Prigent 2006, 2007 for more details on the portfolio insurance and in particular on the comparison).

3 Different methods can be used to estimate the Johnson parameters. The main ones are developed by Hill et al. (1976), Wheeler and Robert (1980) and Slifker and Shapiro (1980).

4 See Prigent (2006, 2007).

5 See proof in appendix 1.

6 This hedging strategy can be determined by Clark–Haussmann–Ocone formula (see Clark 1970, Aase et al. 2000).

7 We can also examine what happens for the S&P 500 for the same time period, namely from 1994 to 2013, using Johnson processes. We get a log return that is not Gaussian since the function l is not linear. However, the general form of the optimal portfolio value is similar to the corresponding optimal portfolio value when assuming a log-normal distribution for the risky asset log return (the approximated pay-off is still increasing and convex) but with modified parameter values. Nevertheless, fitting with a Johnson distribution allows to take better account of the true historical distribution. As expected, the impact of the non normality is stronger for the hedge fund indices than for standard equity indices.