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Optimizing Risk‑Mitigation Portfolios

Improved mitigation of equity risk may result from combining positive expected return strategies that are also negatively correlated with equities.

With the equity bull market increasingly long in the tooth, many investors are seeking to diversify their equity risk by creating what have become commonly referred to as “risk-mitigation portfolios”1 that seek to cushion their investments during the next downturn. These portfolios are not intended to be low risk but rather to serve as a risk diversifier, and they often combine three diversifying strategies: (1) long Treasuries, (2) trend-following and (3) alternative risk premia. All these portfolio strategies are supported by research showing that improved mitigation of equity risk may result from combining positive expected return strategies that are also negatively correlated with equities.

With the exception of tail risk hedging, none of these strategies is contractually obligated to pay out during a significant equity sell-off; rather, each tends to perform best in different types of market drawdowns, but with varying degrees of uncertainty. Accordingly, combining several strategies (“diversify your diversifiers”) may help make the total portfolio more resilient to varying kinds of equity drawdowns.

But how does one allocate to these strategies to produce an optimal result? We examined this question in detail in “A Theoretical Framework for Equity-Defensive Strategies.” We found that the key trade-off among these strategies is the “cost” of each approach versus the relative certainty that it will work in times of crisis. In other words, they form a theoretical “risk-mitigation frontier” (see Exhibit 1). Investors may be able to achieve positive expected returns on a defensive portfolio if they can allocate a sufficient amount of risk to assets with a positive Sharpe ratio, and they do not “overpay” for the hedging component (tail risk hedging). A higher degree of downside risk mitigation must be “paid for” in the form of lower returns.

Exhibit 1 shows a graph of risk/returns for various defensive strategies. The Y-axis represents expected return, and the X-axis indicates increasing uncertainty of diversification. The strategy on the left, on the Y-axis, is tail risk hedging, and shows a slight negative expected return, with zero increasing uncertainty. From there, other strategies form an arc in the upper right-hand quadrant. Long Treasuries are the next along the arc, with a positive expected return and some increasing uncertainty. Higher up on the curve is trend following, showing a higher expected return and higher uncertainty, followed by alternative risk premia, which has only a slightly higher expected return than trend following, but much higher uncertainty.

Investors, of course, are not seeking to optimize risk-mitigation portfolios for every equity market scenario. Rather, they are seeking to optimize for an equity market sell-off. Accordingly, we use the term “conditional beta” to refer to an asset’s equity beta specifically in down equity markets. (This may be quite different from its beta in normal markets.) This is particularly relevant because the asset classes and styles with the highest unconditional expected returns tend to have the weakest downside equity risk mitigation, and vice versa.

If we make a set of assumptions for the expected return and volatility of the different asset classes (as Exhibit 2 illustrates), we can determine optimal weights. (Carry and value serve as the proxy for alternative risk premia).

We then combine these assets to form a defensive portfolio targeting 10% volatility, as in Exhibit 3. (Note that tail risk hedging is modeled as passive put-buying; we believe investors can improve on this considerably with an active approach.)

Exhibit 2 is a table breaks down assumptions for five different assets into two groups: unconditional moments and conditional moments. The five strategies are S&P 500, long Treasuries, tail risk hedging, trend-following, carry, and value. Variables shown for unconditional moments include expected excess return, volatility, and Sharpe ratio. Variables for conditional moments include volatility, equity covariance, equity beta and equity correlation. Data as of 31 December 2018 are detailed within.

Exhibit 3 is a bar chart of various weights of optimal defensive portfolios in increments of 0.1 negative beta, starting with a conditional beta of zero, out to negative 1.3. On the left-hand side, the portfolio with a zero beta is made up more of higher-returning assets, such as trend following and carry. Moving right on the graph, the importance of equity hedging increases, so as conditional beta becomes increasingly negative, the investor allocates more to long Treasuries and tail risk hedging.

As one would expect, allocations on the left-hand side (conditional beta of zero), are made up of higher-returning assets. In fact, these portfolios actually hold a short position in the tail risk equity hedging portfolio (effectively selling puts) as a means of increasing the unconditional expected return. Moving right on the graph, the importance of equity hedging increases, so the investor allocates to more “reliable” hedging sources, such as long Treasuries and long puts, as the conditional beta becomes increasingly negative. As expected, more negative conditional betas result in portfolios with lower unconditional returns, reflecting the “price” investors must pay to increase the size and effectiveness of the equity hedge.

To see this in further detail, we examine three illustrative portfolios: (1) zero conditional beta, (2) −0.6 conditional beta, and (3) −1 conditional beta. Their weights and risk/return characteristics are shown in Exhibit 4.

Exhibit 4 uses pie charts to show three different portfolios, with different conditional betas. The portfolio on the left, with a conditional beta of zero, has a higher allocation to trend following and put selling, and its expected returns are 7.9%, with a Sharpe ratio of 0.7. By contrast, on the far right, a portfolio with a conditional beta of negative one has higher allocations to long Treasuries, put buying and carry strategies. While it has a lower expected return of 2.8%, its Sharpe ratio is 0.28. More data is within the table below the pie charts.

The zero conditional beta portfolio can be viewed more as a return driver than a diversifier – we would expect it to provide essentially no benefits during periods in which equity markets sell off, but higher expected return (we estimate 7.9%). At the opposite extreme, the −1 conditional beta has a modest (2.8%) expected return, but we would expect it to act more reliably as a diversifier in an equity market sell-off. Accordingly, it relies more heavily on Treasuries and overt tail hedges.

The middle portfolio (−0.6 conditional equity beta), which is perhaps the most useful for investors, consists predominantly of trend-following, Treasuries, and carry/value trades (our proxy for alternative risk premia), while selling puts at the margin to boost expected return. It has an expected return of 6%, and we estimate with 95% confidence that it is likely to produce at least a zero beta in the event of an equity market sell-off.

While this portfolio is compelling, we would caveat these results with a few notes about the assumptions underlying these calculations. First and foremost, this result is driven by the negative equity correlation inherent in long Treasuries and in trend-following – these relationships have held up over time, but are not guaranteed. Should these correlations reverse, these results would be different. For trend-following, in particular, the results depend significantly on how responsive the strategy is to market trends. This will, of course, vary from manager to manager; a shorter reaction window (the time before a response to market trends) leads to more negative equity beta (but potential for a lower Sharpe ratio overall). Additionally, while we believe the return assumptions of these individual strategies are reasonable, they can and will vary, of course, particularly over long periods of time (for example, the lackluster performance of trend-following over the last several years).

PIMCO’s Client Solutions and Analytics team goes into far more detail in “A Theoretical Framework for Equity-Defensive Strategies.” We also encourage interested investors to read “Hedging for Profit: Constructing Robust Risk-Mitigating Portfolios,” which is the basis for the more recent research.

The Author

Josh Davis

Global Head of Risk Management

Steve Sapra

Senior Advisor

Jerry Tsai

Client Solutions and Analytics



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This paper 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 upon a set of assumptions believed to be reasonable at time of creation. Actual returns will vary. Forecasts, estimates, and certain information contained herein are based upon proprietary research and should not be considered as investment advice or a recommendation of any particular security, strategy or investment product.

Hypothetical and simulated examples have many inherent limitations and are generally prepared with the benefit of hindsight. There are frequently sharp differences between simulated results and the actual results. There are numerous factors related to the markets in general or the implementation of any specific investment strategy, which cannot be fully accounted for in the preparation of simulated results and all of which can adversely affect actual results. No guarantee is being made that the stated results will be achieved.

Expected returns are for illustrative purposes only and are not a prediction or a projection of return. Return targets and expectations are an estimate of what investments may earn on average over the long term. Actual returns may be higher or lower than those shown and may vary substantially over shorter time periods.

Figures are provided for illustrative purposes and are not indicative of the past or future performance of any PIMCO product. It is not possible to invest directly in an unmanaged index.

Past performance is not a guarantee or a reliable indicator of future results. All investments contain risk and may lose value. Investing in the bond market is subject to risks, including market, interest rate, issuer, credit, inflation risk, and liquidity risk. The value of most bonds and bond strategies are impacted by changes in interest rates. Bonds and bond strategies with longer durations tend to be more sensitive and volatile than those with shorter durations; bond prices generally fall as interest rates rise, and low interest rate environments increase this risk. Reductions in bond counterparty capacity may contribute to decreased market liquidity and increased price volatility. Bond investments may be worth more or less than the original cost when redeemed. Tail risk hedging may involve entering into financial derivatives that are expected to increase in value during the occurrence of tail events. Investing in a tail event instrument could lose all or a portion of its value even in a period of severe market stress. A tail event is unpredictable; therefore, investments in instruments tied to the occurrence of a tail event are speculative. Derivatives may involve certain costs and risks such as liquidity, interest rate, market, credit, management and the risk that a position could not be closed when most advantageous. Investing in derivatives could lose more than the amount invested. Alternative investments involve a high degree of risk that each prospective investor must carefully consider prior to making such an investment and are suitable only for persons of adequate financial means who have no need for liquidity with respect to their investment and who can bear the economic risk, including the possible complete loss, of their investment. Diversification does not ensure against loss.

There is no guarantee that these investment strategies will work under all market conditions or are suitable for all investors and each investor should evaluate their ability to invest long-term, especially during periods of downturn in the market. Investors should consult their investment professional prior to making an investment decision.

Beta is a measure of price sensitivity to market movements. Market beta is 1. Correlation is a statistical measure of how two securities move in relation to each other. Covariance is a measure of the directional relationship between the returns on two risky assets; a positive covariance means that asset returns move together while a negative covariance means returns move inversely. The Sharpe Ratio measures the risk-adjusted performance. The risk-free rate is subtracted from the rate of return for a portfolio and the result is divided by the standard deviation of the portfolio returns.

This material contains the opinions of the manager and such opinions are subject to change without notice. This material has been distributed for informational 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.

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