A Framework for Constructing Equity-Risk-Mitigation Portfolios

Abstract

The key trade-off among equity-risk-mitigation strategies is their expected return versus their ability to diversify equity risk. In particular, the more reliable a strategy’s equity-hedging properties, the lower its expected return, and vice versa. This article proposes a framework for optimal equity-risk-mitigation portfolio construction. In our model, the investor maximizes the portfolio’s unconditional expected return, subject to a constraint on its conditional equity beta. We show that the return to a risk-mitigation portfolio can be decomposed into hedging and return- generating components. We then demonstrate that optimal risk-mitigation portfolios exhibit better return-defensiveness properties relative to the underlying strategies.

Disclosure: The authors report no conflicts of interest.

Editor’s Note

Submitted 17 June 2019

Accepted 14 April 2020 by Stephen J. Brown

Acknowledgments

Disclaimer: This article contains hypothetical analysis. Results shown may not be attained and should not be construed as the only possibilities that exist. The analysis reflected in this information is based on a set of assumptions believed to be reasonable at the time of creation. Actual returns will vary. Forecasts, estimates, and certain information contained herein are based on proprietary research and should not be considered as investment advice or a recommendation of any particular security, strategy, or investment product.

Hypothetical performance results have many inherent limitations, some of which are described below. No representation is being made that any account will or is likely to achieve profits or losses similar to those shown. In fact, there are frequently sharp differences between hypothetical performance results and the actual results subsequently achieved by any particular trading program. One of the limitations of hypothetical performance results is that they are generally prepared with the benefit of hindsight. In addition, hypothetical trading does not involve financial risk, and no hypothetical trading record can completely account for the impact of financial risk in actual trading. For example, the ability to withstand losses or to adhere to a particular trading program in spite of trading losses are material points that can also adversely affect actual trading results. There are numerous other factors related to the markets in general or to the implementation of any specific trading program that cannot be fully accounted for in the preparation of hypothetical performance results and all of which can adversely affect actual trading results.

This article contains the current opinions of the authors but not necessarily those of PIMCO, and such opinions are subject to change without notice. This article has been distributed for educational purposes only and should not be considered as investment advice or a recommendation of any particular security, strategy, or investment product. Information contained herein has been obtained from sources believed to be reliable but not guaranteed.

Notes

¹ Some examples include currency, fixed-income, and commodity carry strategies, as well as other equity risk premium factors, such as quality and value.

² The term “insurance premium” used herein refers to the lower expected returns due to higher allocations to assets/strategies with better risk-mitigation properties but lower return potential and is not an insurance product backed by the claims-paying ability of an insurance carrier.

³ See also Baz, Granger, Harvey, Le Roux, and Rattray (2015) for a review of trend-following, carry, and value strategies in various asset classes as well as a discussion of their economic rationale.

⁴ When the constraint in Equation 2.2$w^{\prime }{\beta }_c\le {\overline{\beta }}_c,$ (2.2) is not binding, the solution reduces to the standard MVO result.

⁵ Under the capital asset pricing model (CAPM), the expected excess return on the unit-beta portfolio will equal the expected equity excess return $\left({\mu }^{\prime }{w}^B={\mu }_{equity}\right)$; thus, the expected return on the beta-hedging portfolio, ${\overline{\beta }}_c{\mu }_{equity}$, would be negative for ${\overline{\beta }}_c<0$. However, this term could be less negative or even positive when the CAPM does not hold (as we assume in our framework) since the investor can hold assets with both negative betas and positive unconditional expected returns.

⁶ This term can be positive if the unit-beta portfolio loads heavily on negative-beta and positive-expected-return strategies. In general, we should expect the “insurance premium” component to be greater than ${\overline{\beta }}_c{\mu }_{equity}$, but it can be negative if we use such assets as put options. We can think of this term as a cost even if the beta-hedging portfolio has a positive expected return, because the beta-hedging portfolio still has a worse risk–return trade-off than the unconstrained MVO portfolio.

⁷ The value strategy we consider here should not be confused with value investing in the stock market. Our value strategy uses the rationale of “buy cheap, sell rich” for commodities, currencies, and interest rates. The section “Background on Popular Equity-Risk-Mitigation Strategies” provides some examples discussed in the literature.

⁸ As Harvey and Liu (2015) noted, the rule-of-thumb 50% haircut is generally considered industry standard. We applied this broad assumption largely to focus on the intuition of the final result. In practice, investors and portfolio managers would ideally use strategy-specific assumptions. See also Harvey, Liu, and Zhu (2016).

⁹ We made the adjustment to the Sharpe ratio because these strategies can generally be implemented using derivatives; therefore, different volatility targets are relatively easy to achieve. We chose the same volatility target for each strategy to make the final allocations more intuitive.

¹⁰ In this study, we chose conditional beta as our measure of defensiveness. However, any other measure of defensiveness based on statistical estimates would also be subject to estimation error.

¹¹ We first converted the conditional beta to a conditional correlation, and then we constructed the confidence interval of the conditional correlation using Fisher transformation. The (1 – α)% confidence interval is $\left[\tanh \left(\text{art}anh\left(r\right)-\frac{z_{\alpha /2}}{\sqrt{n-3}}\right),\tanh \left(\text{art}anh\left(r\right)+\frac{z_{\alpha /2}}{\sqrt{n-3}}\right)\right],$ where r is the sample correlation, n is the sample size, and tanh and artanh are the hyperbolic tangent and the inverse hyperbolic tangent functions, respectively. Subsequently, we converted the correlation back into a conditional beta value for expositional purposes.

¹² We used excess returns on the S&P 500 as a proxy for equity market returns. The return moments are based on historical values and are reported in Table 1.

¹³ Although these allocations are theoretically correct, using put options to control the equity exposure of the overall portfolio may not be practical for most investors. One way to avoid these short equity positions is to impose a no-short-sale constraint, which we consider in the next section.

¹⁴ For the case with equities in the opportunity set, the maximum return portfolio has a conditional beta of 0.1. That is, the conditional beta constraint does not bind for beta targets greater than 0.1 and all these portfolios have the same optimal allocations.