Asset allocation under shortfall constraints

Abstract

R ver the long term, equity investors have , -5 been richly rewarded for the risks that they have endured. For example, during the 1926-1987 period, the S&P provided an annual return advantage of 6.8%, compared with long-term corporate bonds. Over sh0rte.r periods, by contrast, stocks actually underperformed cash on a surprisingly frequent basis. In particular, stocks have underperformed Treasury bills over the past fifteen years in almost 35% of six-to eighteen-month time periods (see Salomon et al. [1990]).Few professional investors are able to observe callmly and passively while high volatility buffets their portfolio's value over the short run, and most fund sponsors control overall risk by adjusting the extent of their equity position. By adding cash or bonds and thereby lowering equity exposure, fund sponsors reduce portfolio volatility. At the same time, they give up a portion of the risk premium that equity offers; decreased exposure to equity leads to a reduction in expected returns.In this article we focus on the balance between risky and risk-free assets. Although we use equity as the proxy for all the risky assets in a portfolio, our methodology applies equally to any basket of risky assets. We offer a simple model of how to quantify risk tolerance and then use it to determine the maximal equity investment.We measure downside risk by the "shortfall probaldity" relative to a minimum return threshold. By specifymg both this threshold and a shortfall prob-,-ability, we can establish a "shortfall constraint" to determine the maximum allocation to risky assets. (See Leibowitz et al. [1990] for full details.)We also consider the sensitivity of the risky asset allocation to changes in voliltility, equity risk premium, return threshold, and shortfall probability. Finally, we show how this methodology can be applied to multi-year investment horizons. THE EFFICIENT FRONTIER FOR AN EQUITYKASH PORTFOlLIOA portfolio manager with ,a well-established horizon always has a continuum clf choices between risky and riskless assets. For example, over a oneyear investment horizon, the oine-year Treasury STRIP provides a riskless return equal to its yield; that is, this "cash" asset has no return volatility. However, modern theory suggests that a holder of risky assets should be compensated for the associated volatility (risk) by means of a positive increment in expected returnthe so-called risk premium. Current estimates of the equity risk premium for U.S. equities range from a 4% expected return advantage to a 6% expected return advantage.Because cash does not have any return volatility, the volatility in an equitykash portfolio reflects entirely the proportion of equity in that portfolio, and the portfolio manager can control volatility risk by adjusting the equitykash balance. As the percentage of equity increases, so does both portfolio risk and expected return. Figure 1 illustrates the linear rela-MARTIN L. LELBOWITZ is Managing Director, and STANLEY KOGELMAN is Vice President, at S,alomon Brothers Inc in New York (I...

A Note on Optimal Estimation from a Risk-Management Perspective under Possibly Misspecified Tail Behavior

“…. , n. The probabilistic requirement on the portfolio's return in (3) is called a shortfall constraint, see Leibowitz and Kogelman (1991) and Leibowitz et al (1992). As an alternative to (2) and (3), we could use a general utility function in order to characterize the manager's preferences.…”

Betting on Death and Capital Markets in Retirement: A Shortfall Risk Analysis of Life Annuities Versus Phased Withdrawal Plans

“…It was widely applied to investment asset allocation by Leibowitz et al (1996) and used by Leibowitz and Krasker (1988) and Maurer and Schlag (2003) among others to judge the long term risk of stocks and bonds. In addition Libby and Fishburn (1977); Kahneman and Tversky (1979);Laughhuun et al (1980) and March and Shapira (1987) show that in empirical business decisionmaking, many individuals judge the risk of an alternative relative to a reference point.…”

Inflation-Hedging Portfolios in Different Regimes

“…Different methodologies have been developed that move away from the standard mean-variance approach, by changing the risk measure of the portfolio. One branch of the literature considers portfolio selection with value at risk (Agarwal and Naik (2004), Martellini and Ziemann (2007)), or conditional VaR (Rockafellar and Uryasev (2000)); the other branch with shortfall probability (Leibowitz and Henriksson (1989), Leibowitz and Kogelman (1991), Lucas and Klaassen (1998), Billio and Casarin (2007), Smith and Gould (2007)). A useful development of our work would be to reconcile the two approaches and examine shortfall probabilities in the context of non-normal returns.…”

“…Roy defined the shortfall constraint such that the probability of the portfolio's value falling below a specified disaster level is limited to a specified disaster probability. Portfolio optimisations with a shortfall probability risk measure have been conducted before (Leibowitz and Henriksson (1989), Leibowitz and Kogelman (1991), Lucas and Klaassen (1998), Billio (2007), Smith and Gould (2007)), but as far as we know not in the context of an inflation hedging portfolio.…”