Common Factors in Return Seasonalities

Common Factors in Return Seasonalities Matti Keloharju, Aalto University Juhani Linnainmaa, University of Chicago and NBER Peter Nyberg, Aalto University AQR Insight Award Presentation 1 / 36

Common factors in return seasonalities 1 The puzzle 2 Systematic risk in seasonalities 3 Theory 4 Seasonalities everywhere Anomalies Segments of U.S. equities Commodities and countries 5 Optimal portfolios and economic magnitudes 2 / 36

The puzzle The puzzle What happens when we estimate a cross-sectional regression of returns this month against returns in month t k? The simplest model of stock returns: The regression slope then equals: ˆb k = cov(r it, r i,t k ) var(r i,t k ) var cs (µ i ) = var cs (µ i ) + σε 2 r it = µ i + ε it, with ε it IID = cov(µ i + ε it, µ i + ε i,t k ) var(r i,t k ) > 0 3 / 36

Seasonalities Some numbers. The average cross-sectional standard deviation of monthly returns is about 15% If the cross-sectional standard deviation of monthly expected returns is 1.5%, then ˆb = (0.015)2 (0.15) 2 = 0.01 Interpreting the regression coefficient Measures the amount of variation in realized returns that emanates from cross-sectional differences in expected returns If expected returns are constant, we get the same slope at every lag Today s returns regressed against returns last month, today s returns regressed against returns 5 years ago,... 4 / 36

A world without seasonalities Estimate cross-sectional regressions: r it = a t + b t r i,t k + e it for lags k up to 10 years Average Fama-MacBeth coefficients ˆb as a function of the lag using simulated data: 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 5 / 36

Regressions using actual data 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 6 / 36

Regressions using actual data 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 7 / 36

Regressions using actual data 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 8 / 36

Regressions using actual data 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 9 / 36

Regressions using actual and simulated data 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 10 / 36

Trading seasonalities Stocks expected returns must vary from month to month Some stocks earn high returns in January, others in April,... Seasonalities are remarkably strong Complete overwhelm unconditional differences in expected returns A seasonality strategy Compute each stock s average same-calendar month return Use up to 20 years of data These average returns are signals of calendar month-specific differences in expected returns, µ i,m(t) Strategy: Buy stocks with high ˆµ i,m(t) s, sell those with low ˆµ i,m(t) s 11 / 36

Estimates Value-weighted return on a 10 1 same-calendar month strategy 1.19% per month (t-value = 6.27) between 1963 and 2011 What if we estimate ˆµ i from other-calendar month returns? 0.96% per month (t = 4.12) Long-term reversals... but far stronger than usual because the signal now cleans out seasonalities! Unconditional exposures against standard risk factors? A same-minus-other strategy: 2.16% per month (t = 7.94) Three-factor model alpha = 1.63% (t = 7.42) 12 / 36

Risk and common factors These seasonalities stem from a set of common factors This is important! Stocks do not have high or low expected returns in certain months for idiosyncratic reasons We see this systematic risk everywhere Additional risk in the seasonality strategy The annualized volatility of the 10 1 strategy is 16.64% A random long-short strategy from the same assets: 7.35% The variance of the true seasonality exceeds that of its randomized counterpart by a factor of five! When we sort stocks into portfolios by ˆµ i,m(t), we therefore group together stocks that are similar in some dimensions 13 / 36

Risk and common factors If the seasonalities were predominantly idiosyncratic, we could capture them without taking almost any risk at all In the data, Sharpe ratios are high but bounded Additional risk through the lens of FMB regressions Fama-MacBeth regression slopes are returns on particular long-short strategies t-values therefore proportional to these strategies Sharpe ratios Simulated data: the regression slope against average same-calendar month return has a t-value > 30! Actual data: t-value = 10.28 14 / 36

Flip the logic around: purge idiosyncratic effects and then trade seasonalities Form well-diversified portfolios by sorting on size, value, momentum, industry,... Sort by: Same Other Difference Assets month month Avg FF3 α Individual stocks 1.19 0.96 2.16 1.63 (6.27) ( 4.12) (7.94) (7.42) Portfolios sorted by Size 1.35 0.94 2.29 2.29 (6.64) ( 3.94) (6.53) (6.78) Value 0.47 0.23 0.24 0.44 (2.76) (1.27) (1.05) (2.31) Momentum 1.83 1.89 0.07 0.34 (5.77) (5.39) ( 0.20) ( 1.21) Industry 0.70 0.81 1.51 1.35 (3.79) ( 4.32) (5.40) (4.71).... Composite 1.30 0.01 1.29 1.18 (8.65) (0.06) (6.31) (5.92) 15 / 36

Cross-sectional regressions using individual stocks 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 16 / 36

Cross-sectional regressions using 58 portfolios 0.10 0.05 ˆbt 0.00-0.05 12 24 36 48 60 72 84 96 108 120 Lag, months 17 / 36

Theory: Seasonalities in common factors A simple and compelling explanation for the data: The seasonalities reside in the risk premia of common factors Theory 1 Any seasonality in factor premia always gets transferred to the cross-section of security returns We only need variation in factor loadings, var cs (β i ) > 0 2 If there are multiple factors security returns aggregate seasonalities in their risk premia 18 / 36

Seasonalities everywhere: Anomalies We almost always measure anomalies unconditional average returns Returns on some anomalies accrue unevenly Small stocks do well and momentum poorly in January because(?) of tax-loss selling and long-term reversals Questions: 1 Is the same true for other anomalies? 2 Is this just about January versus other months? Examine returns on 15 popular anomalies: size, value, momentum, gross profitability,..., and financial distress 19 / 36

All months Excluding January r Jan r Feb = = r Dec, r Dec, # Strategy Mean t p-value Mean t p-value 1 Market 0.439 2.19 0.411 0.384 1.91 0.381 2 Size 0.354 1.17 0.000 0.363 1.24 0.000 3 Value 0.508 2.17 0.000 0.224 0.95 0.007 4 Momentum 1.916 5.38 0.003 2.327 5.79 0.186 5 Gross profitability 0.451 2.69 0.080 0.567 3.31 0.266 6 Dividend to price 0.026 0.11 0.077 0.023 0.10 0.034 7 Earnings to price 0.559 2.61 0.003 0.370 1.59 0.043 8 Investment to assets 0.462 3.25 0.032 0.353 2.42 0.152 9 Return on assets 0.396 1.27 0.000 0.798 2.50 0.576 10 Asset growth 0.454 2.62 0.000 0.236 1.23 0.050 11 Net operating assets 0.672 4.85 0.566 0.695 5.09 0.496 12 Accruals 0.498 2.78 0.974 0.479 2.73 0.954 13 Composite eq. issuance 0.628 3.53 0.524 0.679 3.78 0.519 14 Net issuances 0.845 5.70 0.547 0.874 5.63 0.474 15 Ohlson s O-score 0.205 0.66 0.000 0.655 2.05 0.062 16 Distress 0.663 1.58 0.000 1.599 3.86 0.058 2 15 Joint seasonality test 0.000 0.000 20 / 36

Seasonalities in anomaly returns We can often strongly reject the hypothesis that the risk premia accrue evenly Remarkable given the low power of this test Do all anomalies perform well or poorly at the same time? Or is there seasonal variation in cross-sectional differences in expected returns? An intuitive test Measure anomaly returns using historical returns Learn which anomalies are the most or least profitable Buy anomalies with high ˆµs, sell those with low ˆµs Test: Does it matter whether we extract ˆµ i,m(t) s from sameor other-calendar month data? 21 / 36

Rotating anomalies Long and short positions in top-3 and bottom-3 anomalies: Sample All stocks All-but-microcaps All Excluding All Excluding Meta-strategy months January months January Sort strategies by estimated 1.82 1.20 1.54 1.20 same-calendar month premia (6.39) (4.70) (6.70) (5.99) Sort strategies by estimated 0.39 0.06 0.02 0.30 other-calendar month premia ( 1.57) (0.21) ( 0.09) (1.16) Difference 2.21 1.14 1.57 0.90 (6.08) (3.74) (4.98) (3.22) Remarkable! Historical returns are completely uninformative about which anomalies are above or below average when we study other-calendar month returns Seasonalities completely overwhelm unconditional differences in anomaly returns 22 / 36

Seasonalities everywhere: Subsets of U.S. stock market Partition the U.S. equity market into non-overlapping segments based on size, book-to-market, dividend yield, or credit rating Return on Return on Partition Q 5 Q 1 t-value Partition Q 5 Q 1 t-value Size D/P Micro 0.62 6.44 = 0 1.11 6.61 Small 0.73 6.88 Low 1.14 5.21 Large 0.96 5.66 High 1.23 6.42 Book-to-market Credit rating Growth 1.05 6.07 Low 0.70 2.76 Neutral 1.00 6.00 Medium 0.85 4.63 Value 0.87 4.51 High 1.74 4.23 The credit rating sample begins in 1986. Most anomalies falter in some corners of the market Asset growth, for example, is statistically insignificant among large stocks, growth stocks, and high-dividend yield stocks 23 / 36

Multiple factors: Correlations between seasonality strategies Size Book-to-market Dividend-to-price Partition Micro Small Large Growth Neutral Value = 0 Low High Size Micro 1 Small 0.50 1 Large 0.28 0.41 1 B/M Growth 0.28 0.42 0.85 1 Neutral 0.34 0.46 0.69 0.48 1 Value 0.23 0.36 0.41 0.26 0.39 1 D/P = 0 0.35 0.42 0.59 0.63 0.41 0.26 1 Low 0.20 0.31 0.66 0.65 0.53 0.25 0.33 1 High 0.13 0.17 0.47 0.31 0.50 0.48 0.17 0.27 1 Although seasonalities permeate the entire cross-section of equities, they stem from different factors Seasonalities in high-dividend yield stocks stem from different factors than those in small-cap stocks This the aggregation mechanism at work: Security returns sum up seasonalities across all risk factors no matter what they are 24 / 36

Seasonalities all the time? Period 1963 1973 1983 1993 2003 Strategy 1972 1982 1992 2002 2011 Buy-and-hold strategies ( 99.99) ( 99.99) ( 99.99) ( 99.99) ( 99.99) Net issuances 0.90 0.66 0.84 1.30 0.50 (2.58) (2.36) (3.01) (4.20) (1.31) Momentum 2.08 2.05 2.35 2.98 0.10 (4.38) (3.37) (4.09) (3.03) (0.09) Asset growth 0.05 1.10 0.06 1.34 0.14 ( 0.15) (3.78) ( 0.18) (3.15) ( 0.38) Stambaugh et al. combo 0.55 0.36 0.98 1.10 0.29 (2.80) (1.55) (4.15) (3.47) (0.90) Seasonality strategies Individual stocks 1.13 0.83 1.89 1.65 0.40 (3.32) (1.93) (5.29) (2.87) (1.35) Portfolios Size 0.94 1.33 1.83 1.78 0.77 (1.78) (2.49) (4.30) (3.08) (1.84) Industry 0.68 0.69 0.95 1.57 0.49 (2.66) (1.88) (2.36) (3.77) ( 1.00) Composite 0.98 1.40 1.68 1.77 0.58 (3.76) (3.79) (4.90) (4.53) (1.75) 25 / 36

Seasonalities everywhere: Commodity futures and country portfolios, 1970 2011 Seasonality strategies with commodity futures and country indexes Commodities (N = 24): Aluminium, Copper, Nickel,... Country indexes (N = 15): Austria, Belgium, Canada, Denmark,... All months Excluding January Country Country Commodities indexes Commodities indexes Sort by 0.93 0.48 0.96 0.50 same-month return (1.93) (2.20) (2.03) (2.20) Sort by 0.22 0.36 0.19 0.23 other-month return ( 0.58) ( 1.66) ( 0.47) ( 1.06) Same Other 1.15 0.84 1.15 0.73 (1.97) (2.76) (2.02) (2.28) 26 / 36

Seasonalities everywhere: Higher frequencies There is nothing special about monthly returns Write the return process for different frequencies: daily returns, intraday returns If expected returns vary at these frequencies, we will pick up those seasonalities from past same-period returns Seasonalities in daily stock returns Keim and Stambaugh (1983) and many others: Friday returns are particularly high for small stocks; Monday returns low for stocks overweighted by households,... CS regressions of day-t returns against day-t k returns If expected daily returns have seasonalities r it = µ i,d(t) + ε it then prior same-weekday return is an estimate of µ i,d(t) 27 / 36

Day-of-the-week seasonalities in U.S. stock returns 0.004 0.003 0.002 0.001 ˆbt 0.000-0.001-0.002-0.003 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100105110115120125 Lag, days Economic magnitudes? Form a VW long-short portfolio from average same-weekday returns Average daily return of 0.11% (t-value = 13.3) Other-weekday strategy: 0.05% (t-value = 4.42) 28 / 36

Correlations, once more Monthly. Daily... stocks stocks Countries Commodities Monthly U.S. stocks 1 Daily U.S. stocks 0.05 1 Countries 0.02 0.00 1 Commodities 0.11 0.01 0.09 1 These strategies look the same on the surface But they are dramatically distinct in terms of their risks U.S. stocks are exposed to factors A, B, and C; commodities are exposed to factors X, Y, and Z;... If returns on the factors accrue unevenly, security return always aggregate those seasonalities A very weak requirement! 29 / 36

Optimal portfolios and economic magnitudes Construct HML-style factors for U.S. equity seasonalities Univariate long-short strategies for countries and commodities Compute the ex-post mean-variance efficient portfolio Weights and Sharpe ratios Standard factors Market 100% 22% 8% 8% 7% 3% Size 15% 10% 10% 9% 2% Value 40% 24% 24% 23% 23% Momentum 23% 12% 12% 11% 4% Seasonality factors...... Monthly U.S. stocks 46% 44% 41% 21% Commodities 2% 2% 1% Countries 7% 4% Daily U.S. stocks 41% Sharpe ratio 0.46 1.04 1.67 1.69 1.74 2.75 30 / 36

Conclusions Data dramatically reject the non-seasonal view of the world In all standard models, past returns should significantly predict returns today But after momentum we are left with long-term reversals! The spikes in cross-sectional regressions show that there are persistent differences in expected returns It is just that the unconditional component is negligible in comparison Seasonalities completely overpower unconditional differences in expected returns True everywhere we look: Individual stocks, well-diversified portfolios of stocks, anomalies (except momentum!), country indexes, and commodity futures 31 / 36

The puzzle we should not ignore 0.02 0.01 ˆbt 0.00-0.01 12 24 36 48 60 72 84 96 108 120 Lag, months 32 / 36

Conclusions Seasonality strategies are risky You get the returns by rotating through different factors The returns are just the seasonal component of the risk premium As with any tactical factor, you may suffer greatly if a factor gets a shock when you are exposed to it Seasonalities are not a distinct class of anomalies If we have a theory behind any factor, we can also explain its contribution to seasonalities Tradability of seasonalities You could capture monthly seasonalities using industry ETFs But even if seasonalities are not the primary trading strategy, they assist in trade timing: Just like you avoid trading into the teeth of short-term reversals when doing momentum, you can use past returns differently to decide when to enter and exit positions 33 / 36

Conclusions We dislike seasonalities because they are inconvenient Even though all macroeconomic data are soaked in seasonalities, financial markets should smooth them out But seasonalities are one of the most robust empirical regularities in the data We do not have a theory for most (any?) anomalies But if an anomaly is found everywhere we look, and all the time, we accept it as being real It is the theory that has to give 34 / 36

Seasonalities everywhere: Out of sample Period 1963 1973 1983 1993 2003 Strategy 1972 1982 1992 2002 2011 Buy-and-hold strategies ( 99.99) ( 99.99) ( 99.99) ( 99.99) ( 99.99) Net issuances 0.90 0.66 0.84 1.30 0.50 (2.58) (2.36) (3.01) (4.20) (1.31) Momentum 2.08 2.05 2.35 2.98 0.10 (4.38) (3.37) (4.09) (3.03) (0.09) Asset growth 0.05 1.10 0.06 1.34 0.14 ( 0.15) (3.78) ( 0.18) (3.15) ( 0.38) Stambaugh et al. combo 0.55 0.36 0.98 1.10 0.29 (2.80) (1.55) (4.15) (3.47) (0.90) Seasonality strategies Individual stocks 1.13 0.83 1.89 1.65 0.40 (3.32) (1.93) (5.29) (2.87) (1.35) Portfolios Size 0.94 1.33 1.83 1.78 0.77 (1.78) (2.49) (4.30) (3.08) (1.84) Industry 0.68 0.69 0.95 1.57 0.49 (2.66) (1.88) (2.36) (3.77) ( 1.00) Composite 0.98 1.40 1.68 1.77 0.58 (3.76) (3.79) (4.90) (4.53) (1.75) 35 / 36

Seasonalities everywhere: Out of sample Period 1963 1973 1983 1993 2003 2012 Strategy 1972 1982 1992 2002 2011 2014 Buy-and-hold strategies ( 99.99) ( 99.99) ( 99.99) ( 99.99) ( 99.99) ( 99.99) Net issuances 0.90 0.66 0.84 1.30 0.50 0.23 (2.58) (2.36) (3.01) (4.20) (1.31) (0.50) Momentum 2.08 2.05 2.35 2.98 0.10 0.11 (4.38) (3.37) (4.09) (3.03) (0.09) ( 0.11) Asset growth 0.05 1.10 0.06 1.34 0.14 0.62 ( 0.15) (3.78) ( 0.18) (3.15) ( 0.38) (1.28) Stambaugh et al. combo 0.55 0.36 0.98 1.10 0.29 0.11 (2.80) (1.55) (4.15) (3.47) (0.90) ( 0.34) Seasonality strategies Individual stocks 1.13 0.83 1.89 1.65 0.40 1.43 (3.32) (1.93) (5.29) (2.87) (1.35) (2.55) Portfolios Size 0.94 1.33 1.83 1.78 0.77 0.81 (1.78) (2.49) (4.30) (3.08) (1.84) (2.42) Industry 0.68 0.69 0.95 1.57 0.49 0.27 (2.66) (1.88) (2.36) (3.77) ( 1.00) ( 0.42) Composite 0.98 1.40 1.68 1.77 0.58 0.44 (3.76) (3.79) (4.90) (4.53) (1.75) (0.96) 36 / 36