What is the Expected Return on a Stock? Ian Martin Christian Wagner - PowerPoint PPT Presentation

What is the Expected Return on a Stock? Ian Martin Christian Wagner May, 2018 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 1 / 40

What is the expected return on a stock? In a factor model, E t R i , t + 1 − R f , t + 1 = � K j = 1 β ( j ) i , t λ ( j ) t � � ◮ Eg, in the CAPM, E t R i , t + 1 − R f , t + 1 = β ( m ) E t R m , t + 1 − R f , t + 1 i , t But how to measure factor loadings β ( j ) i , t and factor risk premia λ ( j ) t ? No theoretical or empirical reason to expect either to vary smoothly, given that news sometimes arrives in bursts ◮ Scheduled (or unscheduled) release of firm-specific or macro data, monetary or fiscal policy, LTCM, Lehman, Trump, Brexit, Black Monday, 9/11, war, virus, earthquake, nuclear disaster. . . ◮ Level of concern / market focus associated with different types of events can also vary over time Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 2 / 40

What is the expected return on a stock? Not easy even in the CAPM 20 15 10 5 0 2000 2005 2010 Figure: Martin (2017, QJE , “What is the Expected Return on the Market?”) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 3 / 40

What we do We derive a formula for a stock’s expected excess return: R i , t + 1 − R f , t + 1 t + 1 � � 2 = SVIX 2 SVIX 2 i , t − SVIX E t t R f , t + 1 2 SVIX indices are similar to VIX and measure risk-neutral volatility ◮ market volatility: SVIX t ◮ volatility of stock i : SVIX i , t ◮ average stock volatility: SVIX t Our approach works in real time at the level of the individual stock The formula requires observation of option prices but no estimation The formula performs well empirically in and out of sample Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 4 / 40

What we do We derive a formula for a stock’s expected return in excess of the market: R i , t + 1 − R m , t + 1 = 1 � � 2 SVIX 2 E t i , t − SVIX t R f , t + 1 2 SVIX indices are similar to VIX and measure risk-neutral volatility ◮ market volatility: SVIX t ◮ volatility of stock i : SVIX i , t ◮ average stock volatility: SVIX t Our approach works in real time at the level of the individual stock The formula requires observation of option prices but no estimation The formula performs well empirically in and out of sample Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 4 / 40

What is the expected return on Apple? Expected excess returns Expected returns in excess of the market APPLE INC APPLE INC 0.25 0.20 Expected Return in Excess of the Market 0.20 Expected Excess Return 0.15 0.15 0.10 0.10 0.05 0.05 0.00 0.00 Jan/96 Jan/00 Jan/04 Jan/08 Jan/12 Jan/96 Jan/00 Jan/04 Jan/08 Jan/12 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 5 / 40

What is the expected return on Apple? Expected excess returns Expected returns in excess of the market APPLE INC APPLE INC 0.25 0.20 Model Expected Return in Excess of the Market Model 6% CAPM 6% CAPM 0.20 Expected Excess Return 0.15 0.15 0.10 0.10 0.05 0.05 0.00 0.00 Jan/96 Jan/00 Jan/04 Jan/08 Jan/12 Jan/96 Jan/00 Jan/04 Jan/08 Jan/12 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 5 / 40

Cross-sectional variation in expected returns 90 % − 10 % quantiles of expected returns Model 0.25 6% CAPM Difference in expected returns 0.20 0.15 0.10 0.05 0.00 Jan/96 Jan/00 Jan/04 Jan/08 Jan/12 Expected returns based on our model imply much more cross-sectional variation across stocks than benchmark forecasts Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 6 / 40

Outline Where do the formulas come from? Construction and properties of volatility indices Panel regressions and the relationship with characteristics The factor structure of unexpected stock returns Out-of-sample analysis Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 7 / 40

Theory (1) R g , t + 1 : the gross return with maximal expected log return This growth-optimal return has the special property that 1 / R g , t + 1 is a stochastic discount factor (Roll, 1973; Long, 1990) Write E ∗ t for the associated risk-neutral expectation, � X t + 1 � 1 E ∗ t X t + 1 = E t R f , t + 1 R g , t + 1 Using the fact that E ∗ t R i , t + 1 = R f , t + 1 for any gross return R i , t + 1 , this implies the key property of the growth-optimal return that � R i , t + 1 � R i , t + 1 , R g , t + 1 E t − 1 = cov ∗ t R f , t + 1 R f , t + 1 R f , t + 1 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 8 / 40

Theory (2) For each stock i , we decompose R i , t + 1 R g , t + 1 = α i , t + β i , t + u i , t + 1 (1) R f , t + 1 R f , t + 1 where � � R f , t + 1 , R g , t + 1 R i , t + 1 cov ∗ t R f , t + 1 β i , t = (2) R g , t + 1 var ∗ t R f , t + 1 E ∗ t u i , t + 1 = 0 (3) cov ∗ t ( u i , t + 1 , R g , t + 1 ) = 0 (4) Equations (2) and (3) define β i , t and α i , t ; and (4) follows from (1)–(3) Only assumption so far: first and second moments exist and are finite Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 9 / 40

Theory (3) The key property, and the definition of β i , t , imply that R i , t + 1 R g , t + 1 − 1 = β i , t var ∗ (5) E t t R f , t + 1 R f , t + 1 We also have, from (1) and (4), R i , t + 1 R g , t + 1 = β 2 var ∗ i , t var ∗ + var ∗ t u i , t + 1 (6) t t R f , t + 1 R f , t + 1 We connect the two by linearizing β 2 i , t ≈ 2 β i , t − 1, which is appropriate if β i , t is sufficiently close to one, i.e. replace (6) with R i , t + 1 R g , t + 1 var ∗ = ( 2 β i , t − 1 ) var ∗ + var ∗ t u i , t + 1 (7) t t R f , t + 1 R f , t + 1 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 10 / 40

Theory (4) Using (5) and (7) to eliminate the dependence on β i , t , R i , t + 1 − 1 = 1 R i , t + 1 + 1 R g , t + 1 − 1 2 var ∗ 2 var ∗ 2 var ∗ E t t u i , t + 1 t t R f , t + 1 R f , t + 1 R f , t + 1 Value-weighting, R m , t + 1 − 1 = 1 R j , t + 1 + 1 R g , t + 1 − 1 � � E t w j , t var ∗ 2 var ∗ w j , t var ∗ t u j , t + 1 t t R f , t + 1 2 R f , t + 1 R f , t + 1 2 j j Now take differences. . . Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 11 / 40

Theory (5) Now take differences:     R i , t + 1 − R m , t + 1 1 R i , t + 1 R j , t + 1 1 � �  var ∗ w j , t var ∗  var ∗ w j , t var ∗ = t u i , t + 1 − t u j , t + 1 E t  −  t − t R f , t + 1 2 R f , t + 1 R f , t + 1 2 j j � �� � α i Second term is zero on value-weighted average: we assume it can be captured by a time-invariant stock fixed effect α i Follows immediately if the risk-neutral variances of residuals decompose separably, var ∗ t u i , t + 1 = φ i + ψ t , and value weights are constant over time Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 12 / 40

Theory (6) So, � � R i , t + 1 − R m , t + 1 = 1 R i , t + 1 R j , t + 1 � + α i E t var ∗ − w j , t var ∗ t t R f , t + 1 2 R f , t + 1 R f , t + 1 j � �� � � �� � SVIX 2 i , t 2 SVIX t where fixed effects α i are zero on value-weighted average Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 13 / 40

Theory (6) So, R i , t + 1 − R m , t + 1 = 1 � � 2 SVIX 2 E t i , t − SVIX + α i t R f , t + 1 2 where fixed effects α i are zero on value-weighted average Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 13 / 40

Theory (7) For the expected return on a stock, we must take a view on the expected return on the market Exploit an empirical claim of Martin (2017) that R m , t + 1 − R f , t + 1 R m , t + 1 E t = var ∗ t R f , t + 1 R f , t + 1 Substituting back, � � R i , t + 1 − R f , t + 1 R m , t + 1 + 1 R i , t + 1 R j , t + 1 � E t = var ∗ var ∗ − w j , t var ∗ + α i t t t R f , t + 1 R f , t + 1 2 R f , t + 1 R f , t + 1 j � �� � � �� � � �� � SVIX 2 SVIX 2 2 t i , t SVIX t Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 14 / 40

Theory (7) For the expected return on a stock, we must take a view on the expected return on the market Exploit an empirical claim of Martin (2017) that R m , t + 1 − R f , t + 1 R m , t + 1 = var ∗ E t t R f , t + 1 R f , t + 1 Substituting back, R i , t + 1 − R f , t + 1 t + 1 � � 2 = SVIX 2 SVIX 2 E t i , t − SVIX + α i t R f , t + 1 2 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 14 / 40

Theory (8) Three different variance measures: � � SVIX 2 = var ∗ R m , t + 1 / R f , t + 1 t t � � SVIX 2 var ∗ = R i , t + 1 / R f , t + 1 i , t t � 2 w i , t SVIX 2 SVIX = t i , t i SVIX can be calculated from option prices using the approach of Breeden and Litzenberger (1978) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? May, 2018 15 / 40