**The Taming Of The Skewness by Brendan Hoffman, PhD & Kathryn Kaminski, PhD CAIA – Campbell & Company**

### Executive Summary

Investors are often concerned about the negative skewness, or left-tail asymmetry, of equity returns. In response, they seek risk-mitigating strategies to provide offsetting returns when equity markets fall. Due to their association with positive skewness, trend-following strategies are popular candidates for risk-mitigation or crisis-offset. This paper explores how a trend-following portfolio can achieve positive skewness, and finds that time variation in risk is the primary factor. In fact, any portfolio with a positive Sharpe ratio can achieve positive skewness simply by varying the level of risk taken through time.

To illustrate this point, three different approaches to risk management are applied to trend-following: constant risk targeting (CRT) achieves zero skewness, signal risk targeting (SRT) achieves positive skewness by chance, and equity risk targeting (ERT) achieves positive skewness by design. Each risk targeting approach is studied from 1990 to 2016. The key features are summarized in the table below.

Finally, the paper turns to investor objectives and discusses the distinction between a diversifier and a complement. A diversifier is an investment strategy which has accretive portfolio benefits with the goal to increase the overall long run Sharpe ratio of a relatively diversified portfolio. A complement is an investment strategy which is designed to best improve a concentrated portfolio by exploiting conditional correlation. In this study, the CRT was found to have the best stand-alone performance and was therefore the best diversifier. The ERT portfolio provided the best equity protection with the highest risk-adjusted return during crisis periods. The SRT portfolio achieves the highest skewness, but with less crisis alpha than the ERT and a lower Sharpe ratio than the CRT. This highlights the fact that positive skewness alone is not enough for risk mitigation; timing matters.

Skewness is simply an outcome; the ultimate decision of whether or not to vary risk over time depends on the investor’s objective: to diversify or to complement?

### The Taming Of The Skewness – Introduction

Many investors are interested in return skewness; in fact, certain investors even consider it an explicit objective when selecting an investment. The skewness of a distribution is a measure of asymmetry around the average return. Negatively skewed portfolios usually have most of their returns above the mean, punctuated with fewer, but larger, returns below the mean. 1 In practice, investors often worry about negative skewness in equity markets. In response, they may seek out positively skewed strategies that can mitigate large negative equity movements.

This paper begins by discussing two sources of skewness in portfolio returns: the composition of the underlying assets/strategies (the ingredients) and amount of risk taken (quantity).3 The paper then turns to trend-following to demonstrate how dynamic risk taking can alter return distributions and create positively skewed outcomes. Two illustrative examples are discussed. The first example uses both a contrived heads/tails strategy, as well as a signal risk-targeting strategy, to show how random (or uncontrolled) time-varying risk can lead to portfolio skewness by chance. The second example employs an alternative risk targeting approach to explore how a controlled time-varying risk target might be able to tame the skew and create skewness by design. Finally, the particular objective of creating positive skewness to complement equity portfolios is discussed.

### Portfolio Return Distributions

Portfolio returns depend on two inputs: the composition of assets/strategies and the amount of risk taken. Given these two choices, a stream of portfolio returns which includes positions in *n* assets with returns at each time *t* can be written as

where at time is the risk level for the total portfolio, is the weight in market *i* and is the percent return. The set of is explicitly constrained such that when the portfolio takes unit risk. Take as a simple example a long-only equity strategy using the S&P 500; this portfolio will have the same distribution as the S&P500, which has historically exhibited negative skew.5 As additional assets/strategies are added, what happens to the portfolio’s return distribution?

At this point, it is necessary to differentiate between the point-in-time distribution and the time-series distribution. The point-in-time distribution is the unobservable distribution of potential return outcomes at any point in time. At a given time t, the observed portfolio return will be the weighted sum of observed asset returns (Equation 1), each drawn from potentially time-varying point-in-time distributions of market returns. As the number of markets n gets large and these markets are sufficiently independent, the portfolio’s point-in-time return distribution will approach a normal distribution.

What return distribution does the investor actually observe? Because an investor has only one realization from each point-in-time distribution, the point-in-time distributional properties cannot be measured. Rather, the investor will aggregate portfolio returns over time, and each of these returns will come from a potentially different point-in-time distribution. As a result, the investor actually measures the average distribution across time, which is called the time series distribution.

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