What is the Expected Return on a Stock?

What is the Expected Return on a Stock? Ian Martin Christian Wagner November, 2017 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 1 / 38

What is the expected return on a stock? In a factor model, E t R i,t+1 = K j=1 β(j) i,t λ(j) t But how to measure β (j) i,t? And what are the factor risk premia? 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 economy-wide data, major technological innovation, monetary policy, 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? November, 2017 2 / 38

What is the expected return on a stock? Not easy even in the CAPM 20 15 10 5 0 Figure: Martin (2017, QJE, What is the Expected Return on the Market? ) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 3 / 38

What we do We derive a formula for a stock s expected excess return: R i,t+1 E t = SVIX 2 t + 1 ( ) SVIX 2 i,t SVIX 2 t 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: SVIXt 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, both in and out of sample Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 4 / 38

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 E t = 1 ( ) SVIX 2 i,t SVIX 2 t 2 SVIX indices are similar to VIX and measure risk-neutral volatility market volatility: SVIXt volatility of stock i: SVIXi,t average stock volatility: SVIXt 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, both in and out of sample Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 4 / 38

What is the expected excess return on Apple? Expected excess returns Expected returns in excess of the market Expected Excess Return 0.00 0.05 0.10 0.15 0.20 APPLE INC Expected Return in Excess of the Market 0.00 0.05 0.10 0.15 0.20 APPLE INC Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 5 / 38

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? November, 2017 6 / 38

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, ( ) 1 E Xt+1 t X t+1 = E t R g,t+1 Using the fact that E t R i,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 E t 1 = cov Ri,t+1 t, R ) g,t+1 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 7 / 38

Theory (2) For each stock i, we decompose R i,t+1 = α i,t + β i,t R g,t+1 + u i,t+1 (1) where β i,t = cov t ( ) Ri,t+1, R g,t+1 var t R g,t+1 (2) 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? November, 2017 8 / 38

Theory (3) The key property, and the definition of β i,t, imply that We also have, from (1) and (4), var t E t R i,t+1 1 = β i,t var t R i,t+1 = β 2 i,t var t R g,t+1 (5) R g,t+1 + var t u i,t+1 (6) Connect the two by linearizing β 2 i,t 2β i,t 1, which is appropriate if β i,t is sufficiently close to one, i.e. replace (6) with var t R i,t+1 = (2β i,t 1) var t R g,t+1 + var t u i,t+1 (7) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 9 / 38

Theory (4) Using (5) and (7) to eliminate the dependence on β i,t, E t R i,t+1 1 = 1 2 var t R i,t+1 + 1 2 var t R g,t+1 1 2 var t u i,t+1 (8) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 10 / 38

Theory (4) Using (5) and (7) to eliminate the dependence on β i,t, and imposing a further assumption that var t u i,t+1 = φ i + ψ t, E t R i,t+1 1 = 1 2 var t R i,t+1 + 1 2 var t R g,t+1 1 2 (φ i + ψ t ) (8) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 10 / 38

Theory (4) Using (5) and (7) to eliminate the dependence on β i,t, and imposing a further assumption that var t u i,t+1 = φ i + ψ t, E t R i,t+1 1 = 1 2 var t R i,t+1 + 1 2 var t R g,t+1 1 2 (φ i + ψ t ) (8) Without loss of generality, we can ensure that j w j,tφ i = 0 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 10 / 38

Theory (4) Using (5) and (7) to eliminate the dependence on β i,t, and imposing a further assumption that var t u i,t+1 = φ i + ψ t, E t R i,t+1 1 = 1 2 var t R i,t+1 + 1 2 var t R g,t+1 1 2 (φ i + ψ t ) (8) Without loss of generality, we can ensure that j w j,tφ i = 0 Value-weighting, E t R m,t+1 1 = 1 2 w j,t var t j R j,t+1 + 1 2 var t R g,t+1 1 2 ψ t (9) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 10 / 38

Theory (4) Using (5) and (7) to eliminate the dependence on β i,t, and imposing a further assumption that var t u i,t+1 = φ i + ψ t, E t R i,t+1 1 = 1 2 var t R i,t+1 + 1 2 var t R g,t+1 1 2 (φ i + ψ t ) (8) Without loss of generality, we can ensure that j w j,tφ i = 0 Value-weighting, E t R m,t+1 1 = 1 2 w j,t var t j R j,t+1 + 1 2 var t R g,t+1 1 2 ψ t (9) Now take differences... Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 10 / 38

Theory (5) Subtracting (9) from (8), ( R i,t+1 R m,t+1 E t = 1 var R i,t+1 t ) w j,t var R j,t+1 t + α i 2 }{{} j }{{} SVIX 2 i,t SVIX 2 t Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 11 / 38

Theory (5) Subtracting (9) from (8), R i,t+1 R m,t+1 E t = 1 ( ) SVIX 2 i,t SVIX 2 t + α i 2 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 11 / 38

Theory (5) Subtracting (9) from (8), R i,t+1 R m,t+1 E t = 1 ( ) SVIX 2 i,t SVIX 2 t + α i 2 α i = 1 2 φ i, so we have j w j,tα j = 0 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 11 / 38

Theory (6) 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 E t R m,t+1 = var t R m,t+1 Substituting back, E t R i,t+1 ( = var R m,t+1 t + 1 var R i,t+1 t ) w j,t var R j,t+1 t +α i 2 }{{}}{{} j }{{} SVIX 2 t SVIX 2 i,t SVIX 2 t Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 12 / 38

Theory (6) 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 E t R m,t+1 = var t R m,t+1 Substituting back, R i,t+1 E t = SVIX 2 t + 1 ( ) SVIX 2 i,t SVIX 2 t +α i 2 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 12 / 38

Theory (7) option prices call i,t (K) put i,t (K) F i,t K Using Breeden Litzenberger (1978), var t R i,t+1 = 2 F 2 i,t [ Fi,t put i,t (K) dk + 0 F i,t call i,t (K) dk ] Closely related to VIX definition, so call this SVIX 2 i,t F i,t is forward price of stock i, known at time t, spot price For SVIX 2 t, use index options rather than individual stock options Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 13 / 38

Theory: summary Expected return on a stock: E t R i,t+1 = α i + SVIX 2 t + 1 ( ) SVIX 2 i,t 2 SVIX2 t where i w i,tα i = 0 Pure cross-sectional prediction: where i w i,tα i = 0 E t R i,t+1 R m,t+1 = α i + 1 ( ) SVIX 2 i,t 2 SVIX2 t Also consider the possibility that α i = constant = 0 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 14 / 38

Data Prices of index and stock options OptionMetrics data from 01/1996 to 08/2014 Maturities from 1 month to 1 year S&P 100 and S&P 500 Total of 869 firms, average of around 450 firms per day Approx. 2.1m daily observations per maturity Approx. 90,000 to 100,000 monthly observations per maturity Other data: CRSP, Compustat, Fama French library A caveat: American-style vs. European-style options Today: S&P 500 only unless explicitly noted Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 15 / 38

SVIX 2 t and SVIX 2 t One year horizon Stock variance 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 SVIX t 2 SVIX t 2 Jan/96 Jan/98 Jan/00 Jan/02 Jan/04 Jan/06 Jan/08 Jan/10 Jan/12 Jan/14 SVIX t > SVIX t (portfolio of options > option on a portfolio) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 16 / 38

Characteristics and SVIX 2 i 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Panel A. CAPM Beta 0.00 0.05 0.10 0.15 0.20 0.25 Panel B. Size High P.2 P.3 P.4 P.5 P.6 P.7 P.8 P.9 Low Small P.2 P.3 P.4 P.5 P.6 P.7 P.8 P.9 Big 0.00 0.05 0.10 0.15 0.20 0.25 Panel C. Book-to-market Panel D. Momentum Value P.2 P.3 P.4 P.5 P.6 P.7 P.8 P.9 Growth Winner P.2 P.3 P.4 P.5 P.6 P.7 P.8 P.9 Loser 0.00 0.05 0.10 0.15 0.20 0.25 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 17 / 38

Characteristics and SVIX 2 i Panel A. CAPM Beta Panel B. Size Stock variance 0.0 0.1 0.2 0.3 0.4 0.5 0.6 High Medium Low Stock variance 0.0 0.1 0.2 0.3 0.4 0.5 Small Medium Big Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Panel C. Book-to-market Panel D. Momentum Stock variance 0.0 0.1 0.2 0.3 0.4 Value Neutral Growth Stock variance 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Winner Neutral Loser Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 17 / 38

Average excess returns on individual stocks 12-month horizon Excess return 0.0 0.1 0.2 0.3 0.4 Slope: 1.12, R squ: 18.5% (SVIX i 2 SVIX 2 ) deciles 0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.5 (SVIX i 2 SVIX 2 ) Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 18 / 38

Empirical analysis Excess return panel regression: R i,t+1 R ( ) f,t+1 = α i + β SVIX 2 t +γ SVIX 2 i,t R SVIX2 t + ε i,t+1 f,t+1 and we hope to find i w i,tα i = 0, β = 1, and γ = 0.5 Excess-of-market return panel regression: R i,t+1 R m,t+1 ( ) = α i + γ SVIX 2 i,t SVIX2 t + ε i,t+1 and we hope to find i w i,tα i = 0 and γ = 0.5 Pooled and firm-fixed-effects regressions Bootstrap procedure to obtain joint distribution of parameters Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 19 / 38

Expected excess returns Horizon 30 days 91 days 182 days 365 days Firm fixed-effects regressions i w iα i 0.080 0.042-0.009 0.011 [1.11] [0.55] [-0.16] [0.16] β 0.634 1.716 3.189 2.618 [0.28] [0.71] [2.16] [1.75] γ 0.381 0.569 0.845 0.936 [1.29] [1.79] [2.54] [3.06] Panel adj-r 2 (%) 0.609 3.957 10.235 17.157 H 0 : i w iα i = 0, β = 1, γ = 0.5 0.187 0.210 0.160 0.131 H 0 : β = γ = 0 0.396 0.166 0.023 0.008 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 20 / 38

Expected excess returns Horizon 30 days 91 days 182 days 365 days Pooled regressions α 0.060 0.020-0.038-0.021 [0.81] [0.26] [-0.63] [-0.30] β 0.736 1.877 3.485 3.033 [0.32] [0.78] [2.22] [1.89] γ 0.150 0.279 0.448 0.513 [0.55] [1.01] [1.43] [1.61] Pooled adj-r 2 (%) 0.066 0.725 3.176 4.440 H 0 : α = 0, β = 1, γ = 0.5 0.154 0.185 0.158 0.182 H 0 : β = γ = 0 0.860 0.582 0.072 0.092 Theory adj-r 2 (%) -0.198 0.149 1.441 1.981 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 21 / 38

Expected returns in excess of the market Horizon 30 days 91 days 182 days 365 days Firm fixed-effects regressions i w iα i 0.036 0.034 0.033 0.033 [4.55] [4.58] [4.23] [4.00] γ 0.466 0.662 0.896 0.916 [1.64] [2.19] [2.80] [3.16] Panel adj-r 2 (%) 0.343 2.867 7.111 12.649 H 0 : i w iα i = 0, γ = 0.5 0.000 0.000 0.000 0.000 H 0 : γ = 0 0.100 0.029 0.005 0.002 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 22 / 38

Expected returns in excess of the market Horizon 30 days 91 days 182 days 365 days Pooled regressions α 0.018 0.017 0.013 0.014 [1.24] [1.13] [0.82] [0.74] γ 0.247 0.383 0.530 0.554 [0.93] [1.45] [1.75] [1.84] Pooled adj-r 2 (%) 0.098 0.554 1.678 2.917 H 0 : α = 0, γ = 0.5 0.368 0.523 0.635 0.602 H 0 : γ = 0 0.353 0.146 0.080 0.066 Theory adj-r 2 (%) -0.013 0.460 1.580 2.692 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 23 / 38

Conclusions so far * = p-value < 0.1, ** = p-value < 0.05, *** = p-value < 0.01 At 6- and 12-month horizons, we reject β = γ = 0 S&P100 6mo S&P100 12mo S&P500 6mo S&P500 12mo ER, pooled * ** * * ER, FE ** *** ** *** EMR, pooled ** *** * * EMR, FE *** *** *** *** We do not reject β = 1, γ = 1/2 at any horizon in any specification But the FE are significant in the excess-of-market regressions Now we try to break the model by sorting on characteristics known to prove problematic for previous models... Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 24 / 38

Expected excess returns 5x5 [characteristic]-svix i,t double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Portfolio fixed-effects regressions i w iα i -0.015-0.020-0.020-0.009 [-0.22] [-0.27] [-0.29] [-0.14] β 2.807 2.730 2.921 2.763 [1.87] [1.82] [1.87] [1.81] γ 0.684 0.833 0.595 0.778 [1.24] [1.39] [1.06] [1.44] Panel adj-r 2 (%) 21.217 22.495 21.521 22.009 H 0 : i w iα i = 0, β = 1, γ = 0.5 0.244 0.314 0.231 0.250 H 0 : β = γ = 0 0.151 0.129 0.151 0.130 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 25 / 38

Expected excess returns 5x5 [characteristic]-svix i,t double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Pooled regressions α -0.020-0.021-0.022-0.021 [-0.28] [-0.30] [-0.30] [-0.29] β 2.987 2.974 3.037 2.969 [1.86] [1.86] [1.87] [1.85] γ 0.451 0.522 0.447 0.515 [1.38] [1.54] [1.31] [1.54] Pooled adj-r 2 (%) 9.253 9.973 9.255 10.153 H 0 : α = 0, β = 1, γ = 0.5 0.167 0.202 0.158 0.202 H 0 : β = γ = 0 0.117 0.106 0.125 0.109 Theory adj-r 2 (%) 3.511 4.300 3.211 4.451 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 25 / 38

Expected excess returns 5x5 SVIX i,t -[characteristic] double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Portfolio fixed-effects regressions i w iα i -0.014-0.021-0.015-0.008 [-0.21] [-0.29] [-0.23] [-0.12] β 2.826 2.726 2.820 2.711 [1.85] [1.82] [1.83] [1.81] γ 0.664 0.835 0.708 0.844 [1.16] [1.31] [1.21] [1.37] Panel adj-r 2 (%) 20.385 22.022 21.658 21.650 H 0 : i w iα i = 0, β = 1, γ = 0.5 0.239 0.345 0.267 0.253 H 0 : β = γ = 0 0.155 0.138 0.149 0.139 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 26 / 38

Expected excess returns 5x5 SVIX i,t -[characteristic] double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Pooled regressions α -0.020-0.021-0.021-0.021 [-0.29] [-0.30] [-0.30] [-0.29] β 3.034 2.980 3.022 2.980 [1.87] [1.85] [1.86] [1.85] γ 0.399 0.510 0.449 0.501 [1.18] [1.45] [1.32] [1.41] Pooled adj-r 2 (%) 8.270 9.510 9.138 9.608 H 0 : α = 0, β = 1, γ = 0.5 0.141 0.201 0.167 0.187 H 0 : β = γ = 0 0.133 0.112 0.123 0.121 Theory adj-r 2 (%) 2.399 3.795 3.088 3.886 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 26 / 38

Expected returns in excess of the market 5x5 [characteristic]-svix i,t double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Portfolio fixed-effects regressions i w iα i 0.015 0.008 0.014 0.019 [0.87] [1.58] [0.89] [1.10] γ 0.793 0.949 0.714 0.871 [1.61] [1.80] [1.41] [1.78] Panel adj-r 2 (%) 13.073 16.579 12.728 15.240 H 0 : i w iα i = 0, γ = 0.5 0.445 0.069 0.481 0.208 H 0 : γ = 0 0.107 0.073 0.160 0.075 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 27 / 38

Expected returns in excess of the market 5x5 [characteristic]-svix i,t double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Pooled regressions α 0.015 0.013 0.015 0.014 [0.78] [0.68] [0.78] [0.71] γ 0.497 0.575 0.503 0.564 [1.59] [1.78] [1.54] [1.77] Pooled adj-r 2 (%) 8.473 10.032 8.175 10.426 H 0 : α = 0, γ = 0.5 0.634 0.590 0.637 0.601 H 0 : γ = 0 0.111 0.074 0.124 0.076 Theory adj-r 2 (%) 7.682 9.107 7.316 9.514 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 27 / 38

Expected returns in excess of the market 5x5 SVIX i,t -[characteristic] double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Portfolio fixed-effects regressions i w iα i 0.016 0.007 0.015 0.019 [0.96] [0.95] [0.90] [1.07] γ 0.802 0.971 0.850 0.963 [1.49] [1.76] [1.61] [1.74] Panel adj-r 2 (%) 11.225 15.440 13.465 14.605 H 0 : i w iα i = 0, γ = 0.5 0.381 0.254 0.377 0.216 H 0 : γ = 0 0.137 0.078 0.108 0.083 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 28 / 38

Expected returns in excess of the market 5x5 SVIX i,t -[characteristic] double-sorted portfolios, 1-yr horizon Horizon Beta Size B/M Mom Pooled regressions α 0.016 0.014 0.015 0.014 [0.82] [0.69] [0.77] [0.71] γ 0.456 0.568 0.509 0.555 [1.37] [1.70] [1.55] [1.63] Pooled adj-r 2 (%) 6.371 9.072 8.049 9.294 H 0 : α = 0, γ = 0.5 0.648 0.591 0.633 0.614 H 0 : γ = 0 0.170 0.088 0.120 0.103 Theory adj-r 2 (%) 5.516 8.166 7.178 8.416 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 28 / 38

Stock returns and firm characteristics 1-year horizon, excess returns Realized returns Expected returns Unexpected returns estimated theory estimated theory const 0.721 0.453 0.259 0.164 0.462 0.557 [2.12] [1.42] [1.96] [4.70] [1.39] [1.69] Beta i,t 0.038-0.048 0.083 0.097-0.045-0.059 [0.56] [-0.71] [1.30] [5.54] [-0.96] [-0.81] log(size i,t ) -0.030-0.019-0.010-0.009-0.019-0.021 [-2.18] [-1.46] [-1.47] [-5.24] [-1.46] [-1.61] B/M i,t 0.071 0.068 0.003 0.001 0.068 0.069 [2.08] [1.79] [0.27] [0.21] [1.80] [1.90] Ret (12,1) i,t -0.049-0.005-0.046-0.026-0.003-0.023 [-0.77] [-0.09] [-1.09] [-1.66] [-0.06] [-0.40] SVIX 2 t 2.793 [1.90] SVIX 2 i,t SVIX2 t 0.513 [1.44] adj-r 2 (%) 1.926 5.285 17.253 30.373 0.975 1.200 H 0 : b i = 0, c i = 0 0.003 0.001 0.701 0.000 0.188 0.091 H 0 : b i = 0, c 0 = 1, c 1 = 0.5 0.142 Do not reject the model Characteristics related to expected returns but not to residual returns Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 29 / 38

Stock returns and firm characteristics 1-year horizon, excess-of-market returns Realized returns Expected returns Unexpected returns estimated theory estimated theory const 0.426 0.275 0.131 0.107 0.295 0.319 [1.15] [0.73] [1.80] [4.01] [0.81] [0.89] Beta i,t 0.017-0.130 0.114 0.105-0.097-0.088 [0.23] [-2.08] [1.74] [6.37] [-2.09] [-1.12] log(size i,t ) -0.018-0.006-0.009-0.009-0.009-0.010 [-1.31] [-0.44] [-1.63] [-5.42] [-0.62] [-0.75] B/M i,t 0.032 0.031 0.001 0.001 0.031 0.031 [1.28] [1.16] [0.10] [0.11] [1.19] [1.21] Ret (12,1) i,t -0.051-0.029-0.017-0.015-0.034-0.035 [-1.24] [-0.71] [-0.95] [-1.50] [-0.87] [-0.89] SVIX 2 i,t SVIX2 t 0.705 [2.30] adj-r 2 (%) 1.029 3.985 37.617 37.617 1.044 0.964 H 0 : b i = 0, c = 0 0.349 0.019 0.427 0.000 0.161 0.623 H 0 : b i = 0, c = 0.5 0.240 Do not reject the model Characteristics related to expected returns but not to residual returns Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 29 / 38

(And we do not need historical data) Google First IPO on August 19, 2004 OptionMetrics data from August 27, 2004 Included in the S&P 500 from 31, March 2006 GOOGLE INC Expected Excess Return 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 30 / 38

The factor structure of unexpected stock returns What do we miss, relative to an oracle with perfect foresight? Run PCA on unexpected returns-in-excess-of-market, 1yr horizon, S&P 500 firms with complete time-series coverage Residuals from pooled regression Residuals from fixed-effects regression Residuals when coefficients constrained to theoretical values PC1: 25% of variance. PC2: 12%. PC3: 8%. PC4: 7% PC loadings do not show any relationship with market cap, B/M, SVIX i, beta, or industry Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 31 / 38

The factor structure of unexpected stock returns PC1 PC2 4 2 0 2 4 6 pooled (0.251) fixed effects (0.253) theory (0.242) 4 2 0 2 pooled (0.117) fixed effects (0.115) theory (0.125) Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 PC3 PC4 4 2 0 2 pooled (0.084) fixed effects (0.079) theory (0.096) 4 3 2 1 0 1 2 pooled (0.070) fixed effects (0.072) theory (0.067) Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 32 / 38

The formula performs well out-of-sample Out-of-sample R 2 of the model-implied expected returns in excess of the market relative to competing forecasts Horizon 30 days 91 days 182 days 365 days Random walk 0.08 0.67 1.85 3.10 ( β i,t 1) S&P500 t 0.09 0.70 1.90 3.11 ( β i,t 1) CRSP t 0.13 0.79 2.07 3.36 ( β i,t 1) SVIX 2 t 0.03 0.53 1.60 2.82 ( β i,t 1) 6% p.a. 0.11 0.74 1.97 3.21 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 33 / 38

... even against in-sample predictions Out-of-sample R 2 of the model-implied expected returns in excess of the market relative to competing forecasts Horizon 30 days 91 days 182 days 365 days in-sample avg all stocks 0.02 0.49 1.53 2.51 ( β i,t 1) in-sample avg mkt 0.12 0.77 2.03 3.31 Beta i,t 0.02 0.49 1.53 2.48 log(size i,t ) -0.03 0.30 1.20 1.93 B/M i,t -0.01 0.40 1.40 2.34 Ret (12,1) i,t 0.02 0.47 1.39 2.07 All -0.05 0.25 1.04 1.49 We even beat the model that knows the multivariate in-sample relationship between returns and beta, size, B/M, lagged return Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 34 / 38

The formula performs well out-of-sample... (2) Out-of-sample R 2 of the model-implied expected excess returns relative to competing forecasts R 2 OS = 1 SSE model /SSE competitor Horizon 30 days 91 days 182 days 365 days SVIX 2 t 0.00 0.49 1.72 3.10 S&P500 t -0.01 0.70 2.51 3.83 CRSP t -0.18 0.15 1.37 1.71 6% p.a. -0.10 0.38 1.78 2.55 SVIX 2 i,t 1.06 1.99 1.64 2.15 RX i,t 1.31 4.88 11.73 27.06 β i,t S&P500 t 0.00 0.71 2.49 3.77 β i,t CRSP t -0.16 0.20 1.41 1.69 β i,t SVIX 2 t -0.04 0.39 1.54 2.89 β i,t 6% p.a. -0.09 0.39 1.79 2.49 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 35 / 38

... even against in-sample predictions (2) Out-of-sample R 2 of the model-implied expected excess returns relative to competing forecasts R 2 OS = 1 SSE model /SSE competitor Horizon 30 days 91 days 182 days 365 days in-sample avg mkt -0.14 0.23 1.46 1.90 in-sample avg all stocks -0.19 0.09 1.20 1.42 β i,t in-sample avg mkt -0.12 0.26 1.48 1.87 Beta i,t -0.19 0.08 1.17 1.30 log(size i,t ) -0.28-0.25 0.57 0.21 B/M i,t -0.28-0.12 0.83 0.77 Ret (12,1) i,t -0.19 0.07 1.03 1.05 All -0.34-0.39 0.20-0.53 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 36 / 38

Cross-sectional variation in expected returns S&P 100 (top) and S&P 500 (bottom) 75% 25% quantiles of 0.5 SVIX i 2 90% 10% quantiles of 0.5 SVIX i 2 0.05 0.10 0.15 0.1 0.2 0.3 0.4 0.5 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 0.05 0.10 0.15 0.20 0.25 75% 25% quantiles of 0.5 SVIX i 2 0.1 0.2 0.3 0.4 0.5 90% 10% quantiles of 0.5 SVIX i 2 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Jan/96 Jan/99 Jan/02 Jan/05 Jan/08 Jan/11 Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 37 / 38

Summary We derive and test a formula for the expected return on a stock Computable in real time Requires observation of option prices but no estimation Expected returns on individual stocks exhibit substantial time-series and cross-sectional variation Performs well in and out of sample Martin & Wagner (LSE & CBS) What is the Expected Return on a Stock? November, 2017 38 / 38