What Is the Expected Return on a Stock?

• Published date: 01 August 2019
• Author: CHRISTIAN WAGNER,IAN W. R. MARTIN
• DOI: http://doi.org/10.1111/jofi.12778
• Date: 01 August 2019

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 4 •AUGUST 2019

What Is the Expected Return on a Stock?

IAN W. R. MARTIN and CHRISTIAN WAGNER∗

ABSTRACT

We derive a formula for the expected return on a stock in terms of the risk-neutral

variance of the market and the stock’s excess risk-neutral variance relative to that

of the average stock. These quantities can be computed from index and stock option

prices; the formula has no free parameters. The theory performs well empirically

both in and out of sample. Our results suggest that there is considerably more vari-

ation in expected returns, over time and across stocks, than has previously been

acknowledged.

IN THIS PAPER,WE DERIVE A NEW formula that expresses the expected return on

a stock in terms of 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. Then we show that the formula performs well empirically.

The inputs to the formula—the three measures of risk-neutral variance—

are computed directly from option prices. As a result, our approach has some

distinctive features that separate it from more conventional approaches to the

cross section.

First, as it is based on current market prices rather than, say, accounting

information, it can in principle be implemented in real time. Nor does it require

that we use any historical information. It thus represents a parsimonious al-

ternative to pooling data on many firm characteristics (as, for instance, in

Lewellen (2015)).

∗Ian Martin is at the London School of Economics. Christian Wagner is at Copenhagen Busi-

ness School. We thank Harjoat Bhamra; John Campbell; Patrick Gagliardini; Christian Julliard;

Marcin Kacperczyk; Binying Liu; Dong Lou; Stefan Nagel (Editor); Christopher Polk; Tarun Ra-

madorai; Tyler Shumway; Paul Schneider; Andrea Tamoni;Fabio Trojani; Dimitri Vayanos; Tuomo

Vuolteenaho; participants at the Western Finance Association Meetings 2017, the 2017 Annual

Meeting of the Society for Economic Dynamics, the 2017 CEPR Spring Symposium, the BI-SHoF

Conference in Asset Pricing, the AP2-CFF Conference on Return Predictability, the 2016 IFSID

Conference on Derivatives, the 4nations Cup 2016; seminar participants at Arrowstreet, AQR,

Banca d’Italia, BlackRock, CEMFI, the European Central Bank, the London School of Economics,

MIT (Sloan), NHH Bergen, Norges Bank Investment Management, the University of Maryland,

the University of Michigan (Ross), and WU Vienna; and two anonymous referees for their com-

ments. Ian Martin is grateful for support from the Paul Woolley Centre and from the ERC under

Starting Grant 639744. Christian Wagner acknowledges support from the FRIC Center for Finan-

cial Frictions, grant no. DNRF102. We have read the Journal of Finance disclosure policy and have

no conflicts of interest to disclose.

This is an open access article under the terms of the Creative Commons Attribution License, which

permits use, distribution and reproduction in any medium, provided the original work is properly

cited.

DOI: 10.1111/jofi.12778

C2019 The Authors. The Journal of Finance published by Wiley Periodicals, Inc. on behalf of

American Finance Association.

1887

1888 The Journal of Finance R

Second, our formula provides conditional forecasts at the level of the individ-

ual stock. Rather than asking, say, what the unconditional average expected

return is on a portfolio of small value stocks, we can ask, what is the expected

return on Apple, today?

Third, the formula makes specific quantitative predictions about the relation-

ship between expected returns and the three measures of risk-neutral variance.

It does not require estimation of any parameters. This can be contrasted with

factor models, in which both factor loadings and the factors themselves are esti-

mated from the data (with all of the associated concerns about data snooping).

There is a closer comparison with the Capital Asset Pricing Model (CAPM),

which makes a specific prediction about the relationship between expected re-

turns and betas, but even the CAPM requires that the forward-looking betas

that come out of theory be estimated based on historical data.

Our approach does not have this deficiency and, as we will show, it performs

better empirically than the CAPM. But, like the CAPM, it requires that we

take a stance on the conditionally expected return on the market. We do so by

applying the results of Martin (2017), who argues that the risk-neutral variance

of the market provides a lower bound on the equity premium. In particular,

we exploit Martin’s more aggressive claim that, empirically, the lower bound

is approximately tight, so that risk-neutral variance directly measures the

equity premium. We also present results that avoid dependence on this claim,

however, by forecasting expected returns in excess of the market.Indoingso,

we isolate the purely cross-sectional predictions of our framework that are

independent from the market-timing issue of forecasting the equity premium.

As these predictions exploit the cross section as well as the time series, the

associated empirical results are stronger in a statistical sense than those of

Martin (2017).

We introduce the theoretical framework in Section I. We then show how to

construct the three risk-neutral variance measures, and discuss some of their

properties, in Section II.

Our main empirical results are presented in Section III. We test the frame-

work for S&P 100 and S&P 500 stocks at forecast horizons ranging from

1 month to 2 years. Papers in the predictability literature typically aim to

uncover variables that are statistically significant in forecasting regressions.

We share this goal, of course, but as our model makes predictions about the

quantitative relationship between expected returns and risk-neutral variances,

we hope also to find that the estimated coefficients on the predictor variables

are close to specific numbers that come out of the theory. For most specifica-

tions, we find that that we do not reject the model, whereas we reject the null

hypothesis of no predictability at the 6-month, 1-year, and 2-year horizons.

In Section IV, we examine how our findings relate to stock characteristics.

Notably, we run panel regressions of realized returns onto beta, size, book-to-

market, and past returns. In our sample, size and book-to-market are statisti-

cally significant forecasters of excess returns, though not of returns in excess

of the market. When we include our predictive variables based on risk-neutral

variance, these characteristics become statistically insignificant, but the

What Is the Expected Return on a Stock? 1889

risk-neutral variance variables themselves are significant predictors (of both

excess returns and excess-of-market returns). Moreover, they enter with coef-

ficients that are insignificantly different from those predicted by our theory. In

a similar vein, we show that returns on portfolios sorted on the characteristics

are consistent with the model.

In Section V, we assess the out-of-sample predictive performance of the for-

mula when its coefficients are constrained to equal the values implied by the

theory. We compute out-of-sample R2coefficients that compare the formula’s

predictions to those of a range of competitors, as in Goyal and Welch (2008).

We start by comparing against competitors that are themselves out-of-sample

predictors (in the sense of being based on a priori considerations, without in-

sample information). The formula outperforms all such competitors at horizons

of 3, 6, 12, and 24 months, both for expected returns and for expected returns

in excess of the market.

More ambitiously, we next compare the formula against competitors that

have in-sample information. At the 6- and 12-month horizons, the only case

in which our model of expected excess returns “loses” is when we allow the

competitor predictor to know both the in-sample average realized return across

stocks and the multivariate in-sample relationship between realized returns

and beta, size, book-to-market, and past returns. When we allow the competitor

to know only the in-sample average and the univariate relationship between

realized returns and any one of the characteristics, our formula outperforms.

Even more strikingly, in the case in which we forecast returns in excess of the

market, the formula outperforms the competitor armed with knowledge of the

in-sample average and of the multivariate relationship.

These empirical successes are particularly notable because the formula

makes some dramatic predictions about stock returns. Figure 1plots the time

series of expected excess returns, relative to the riskless asset and relative to

the market, for Apple and JPMorgan Chase & Co. over the period January

1996 to October 2014. According to our model, expected returns spiked for both

stocks during the depths of the financial crisis of 2008 to 2009. In the case of

Apple, this largely reflected a high market-wide equity premium rather than

an Apple-specific phenomenon, whereas JP Morgan Chase’s expected excess

return was high even relative to the market risk premium. The figure also

plots expected excess returns computed using the CAPM with 1-year rolling

historical betas and the equity premium computed from the SVIX index of Mar-

tin (2017), or fixed at 6%, to illustrate the point (which, as we will show, holds

more generally) that our model generates more volatility in expected returns,

both over time and in the cross section, than does the CAPM.

We conclude in Section VI. The Appendix contains a discussion of the rela-

tionship between our volatility measures and implied correlation, and provides

details of the bootstrap procedure. Finally,f urther empirical results and analy-

sis of the finite-sample properties of our block bootstrap procedure are provided

in an Internet Appendix.1

1The Internet Appendix may be found in the online version of this article.