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This is a summary of "What Is the Expected Return on a Stock?," by Ian W.R. Martin and Christian Wagner, published in the Journal of Finance.

## Overview

The authors derive a formula for calculating a stock’s expected returns using the risk-neutral variances of the stock, the market, and an average stock, which can be computed directly from index and stock option prices. The formula performs well for typical stocks (e.g., those in the S&P 500 Index), both in and out of sample, for periods up to two years.

## What Is the Investment Issue?

The estimation of stocks’ expected returns and volatility has been a central issue for investments in stocks. Most methods, in some form or another, rely on historical data to calculate forward prices. The authors’ formula uses risk-neutral variances for a specific stock, the market, and an average stock. These inputs are directly computable from the prevailing prices of the options on the index and the stock. The authors test the formula for in- and out-of-sample data for 1, 3, 6, 12, and 24 months; explore linkages of the formula with stock characteristics; evaluate the formula’s predictive performance; and compare the formula with competing methodologies for calculating expected stock returns.

## How Did the Authors Conduct This Research?

The authors start with a model of the “growth optimal” portfolio under a risk-neutral expectation. In such a situation, the market itself is growth optimal, and the risk premium of the stock is determined by its risk-neutral covariance with the growth optimal return. The authors derive the risk-neutral variance of the market [SVIXt2=vart*(Rm,t+1/Rf,t+1)], which is similar to the CBOE Volatility Index, or VIX, measure. They further define two new measures. One is the risk-neutral variance of the individual stock [SVIXi,t2=vart*(Ri,t+1/Rf,t+1)], which measures stock-level volatility, and the other is the value-weighted average of stocks’ risk-neutral variance [SVIX¯t2=iwi,tSVIXi,t2], which measures average stock volatility. Using simplifying assumptions, the authors predict stocks’ expected returns in excess of the market as follows:

(EtRi,t+1-R(f,t+1)/Rf,t+1=SVIXt2+12(SVIXi,t2-SVIX¯t2)

Calculating the input variables—the risk-neutral variance of the market, the risk-neutral variance of the individual stock, and the value-weighted average of stocks’ risk-neutral variance—is done directly from option prices. The authors test the formula for daily and monthly data for equity index options on the S&P 500 Index and the S&P 100 between January 1996 and October 2014. Equity index return data are from CRSP, and constituent firm data are supplied by Compustat. The authors use available volatility surfaces for all in-sample index firms provided by OptionMetrics for 1, 3, 6, 12, and 24 months.

## What Are the Findings and Implications for Investors and Investment Professionals?

The return prediction formula performs well both in and out of sample. For out-of-sample studies, the formula does better than a group of competitor predictors across most horizons for both expected returns and expected returns in excess of the market. Also, the formula outperforms when the competing methods know only the in-sample average and the univariate relationship between realized returns and any one of the characteristics. When the competing methods know the in-sample average and the multivariate relationship, the formula outperforms for returns in excess of the market.

The results also show that both size and book-to-market factors are statistically significant predictors of excess returns, though not of returns in excess of the market. In addition, the authors find that stocks have a considerably higher variability of expected returns than normally acknowledged.

The ability to compute the inputs directly from current option prices gives this model certain advantages over other conventional approaches, such as that it can be implemented in real time. Unlike factor models, for which both the factors and factor weights must be estimated, the authors’ formula does not require the estimation of any parameters. It also allows for conditional forecasts on individual stocks, as opposed to unconditional forecasts on portfolios of stocks.

This model is similar to the capital asset pricing model (CAPM), which links expected returns and betas. Like the CAPM, this formula requires a view on conditionally expected market returns. Unlike the CAPM, which estimates forward-looking betas derived from historical data, however, this model works with contemporaneous option prices.

This approach is particularly useful in times of unexpected change, when information—stock specific, market related, macroeconomic, or otherwise—arrives and expectations adjust suddenly. Most conventional models use backward-looking covariances that change with lags; however, option prices change nearly instantaneously. The formula is also useful when only limited historical data are available, as in the case of the dot-com boom during the late 1990s and early 2000s.

When using this model, practitioners need to keep in mind two underlying assumptions. First, the model assumes that the risk-neutral betas of the stock are believed to be close to one another and therefore allow for the linearization of betas. Second, the risk-neutral variances of the residuals are approximated by a time-invariant stock fixed effect. Through empirical tests, the authors demonstrate that these assumptions may be reasonable for select stocks, such as those in the S&P 100 Index, but they may not extend to the average stock or, indeed, other asset classes.