Equity convexity means that a stock outperforms in times of large upward or downward movements of the broad market: its elasticity to the market return is curved upward. Gamma is a measure of that convexity. All else equal, positive gamma is attractive, as a stock would outperform in market rallies and diversify in market stress. However, gamma is not observable, changeable, and needs to be estimated. Only a subset of stocks displays statistically significant gamma. Empirical analysis suggests that convex stocks can mostly be found in the materials, telecom, industrials, and energy sectors. High past volatility and price-to-book ratios have also been indicative of high gamma. Macroeconomic drivers that trigger gamma performance have been interest rates and oil prices. Systematic long-convexity strategies that seek to time convexity exposure have reportedly produced significant investor value.

The below post is based on quotes from the above paper, except where a separate source has been linked next to the quote. Headings, cursive text, and text in brackets have been added. Also, most mathematical expressions have been paraphrased for easier reading.

This post ties in with this site’s summary on macro trends as a basis for systemic trading strategies.

### Understanding equity convexity or gamma

“An investment strategy is __convex if its payoff relative to its benchmark is curved upward__. The image below depicts an investment strategy that exhibits convexity on both the downside and the upside. Convex investment strategies are expected to be highly correlated with the benchmark in typical market environments but diverge to the positive in extreme markets. There are no free lunches though, and convex strategies are expected to lag during quiet markets.” [simplify.us]

“__Time-variant gamma at security level…translates a stock’s exposure to performance in the tails of the benchmark [market portfolio].__ A positive gamma…implies that a stock’s returns are a convex function of market returns, which means that theoretically they should always outperform the benchmark (whether the latter is either in positive or negative territory). A negative gamma instead implies that such a relationship is actually concave, therefore such a stock would systematically underperform the market. __The higher the gamma, the more such a stock would outperform the market, which is a particularly attractive feature__, considering that periods of market stress are generally accompanied by high correlation between stock returns. This is a powerful characteristic in terms of diversification.”

“[For] measuring convexity [one must consider] __time-varying unconditional systematic co-skewness in the traditional capital asset pricing model__. “

*This means that the return of a stock does not just depend linearly on the market return but also on the square of the market return. Convex stocks’ returns increase more than proportionately to the market return and concave stocks’ returns increase less than proportionately to the market return.*

“To illustrate, based on weekly returns against the MSCI World from April 2018 to March 2021, we present [in the figure] below two typical gamma shapes. The first is from Qurate Retail equity, which has been [positively related to squared market returns] for the period of analysis: we witness that it exhibits a convex shape. The second is from Cenovus Energy…which was [ngeatively related to squared market returns] and results in a concave shape when plotted against the MSCI World.”

“In theory, convexity should always pay off, since a convex stock is supposed to outperform the benchmark independently of market conditions. Similarly, concavity should not be of interest for a long-only investor since concave stocks are supposed to consistently underperform the market. However, by definition __we cannot apprehend the stock’s true convexity, and have to estimate it__.”

### How to estimate equity gamma

“We work on the MSCI World universe. For each stock, at each date the market beta and the gamma are estimated together via Ordinary Least Square (OLS), as in [the equation below], using heteroskedasticity robust standard errors, on a rolling window of 3 years, using weekly returns from February 2010 to August 2020 [where R is the return of a stock, rf the risk-free rate, RM the market return, and beta and gamma are the coefficients to be estimated].”

“Gamma is estimated at the end of the months of February, May, August and November. Actually, the beta coefficients derived from this regression are predominantly significant. However, this is not the case for the gamma coefficient in general…__Gamma is not everywhere and only concerns a small subset__ of the stocks within our universe. Therefore, we build a new database, composed of the gamma that are highly significant alongside corresponding stock returns data.”

### What is driving equity convexity or gamma

“Fundamental data is retrieved from the FactSet Fundamentals database on a quarterly basis [and some other sources].”

“From a bottom-up viewpoint….over the last 10 years stocks with convex returns have been mostly found in the Asia-Pacific region and in sectors such as Materials, Telecoms, Industrials, and Energy. Despite varying gamma regimes over the period, we demonstrate that past volatility and the price-to-book ratio have been the most efficient discriminant features of concavity and convexity. Namely, stocks with volatile past returns, from companies that are rather classified as glamorous (as opposed to value) tend to have higher gammas.”

“In a top-down approach, we investigate the macroeconomic drivers of gamma… [i.e.] market environments that trigger gamma performance and subsequently, our ability to forecast this premium…Applying a cointegrating vector framework, we find that it exhibits long-term relationships with the VIX, as expected from gamma’s essence, but also with short-term interest rates, and oil prices.”

### Strategies using equity convexity or gamma

“We propose a systematic long convexity strategy. The rationale in this approach is to time the convexity exposure…We…__evaluate the ability of different models to forecast future convexity premium dynamics__…[and] employ these signals in the design of a systematic long convexity strategy…It leads to significantly improved risk-adjusted returns compared to a capitalization-weighted benchmark, especially in turbulent markets. Convexity exposure appears particularly relevant in a context of monetary policy normalization.”

“The __XMA factor is long convex stocks and short concave ones.__ We propose to embed our XMA forecast in the design of a systematic strategy….The forecast is only a function of lagged macroeconomic data and…analyzing performance exclusively out-of-sample ensures that our model…is not overfitted. In this systematic long convexity strategy, the aim is to time the convexity exposure.”

“Building on [empirical] results, we attempt to forecast the premium [of the] XMA factor (long convexity, short concavity)…Increasing short-term interest rates and market volatility are conducive to the outperformance of the convexity premium in the subsequent period. We use this signal to propose a systematic long convexity strategy.”

“The strategies we propose, [which] aim at timing convexity exposure in a systematic way, deliver strong risk-adjusted returns compared to their benchmark, with a Sharpe ratio close to 2 for our period of analysis. Furthermore, __long convexity strategies, by exposing to convexity during period of market stress, efficiently manage to reduce portfolio volatility__. This authenticates the relevance of such strategy for mitigating equity portfolio losses in turbulent markets, in a defensive manner.”