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Seasonality in Equilibrium Stock Returns: A Dynamic Perspective on SAD Xifeng Diao Maurice Levi U of Calgary UBC

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Motivation Saunders (1993), Hirshleifer and Shumway(2003) Sunshine/Daylight (measured by cloudiness) → Mood → Bias → Returns (low returns on rainy days) Kamstra, Kramer, and Levi (KKL, 2003) Sunshine/Daylight (measured by length of day) → SAD → Risk Aversion → Returns (high returns on short days) The implication of seasonal risk aversion for returns –Static effect: higher risk aversion in fall and winter → higher required returns in fall and winter –Dynamic effect: risk aversion decreases in winter and spring → price ↑ → high realized returns in W and Sp –Which effect dominates? Not clear in previous study

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This Paper’s Contribution Dynamic effect should dominate Our model shows that it is the change, not the level, of daylight or risk aversion that determines the pattern of SAD-induced seasonal returns (Winter, Spring > Summer, Fall) Others: –Concavity (diminishing marginal impact of daylight: W>Sp, Su>F) –Cross-sectional differences (size, beta, STD) –Empirical Evidence

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The Model Assets –Zero net supply of bond, r f normalized to 0. –A single risky asset with a terminal payoff D T,. dD t / D t = μ D dt + σ D dB t Agent –A representative agent with CRRA utility –Objective function: Max E { U(W T ) }

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Equilibrium Returns Return from time t to t+d log (S t+d / S t ) = (γ t -0.5) σ D 2 d +(γ t – γ t+d ) σ D 2 (T–t-d) + σ D (B t+d –B t ) where γ: risk aversion. Concavity: γ t = -g(l t ), where l t is length of day at t, and g(l t ) is concave. Dynamic TermVariance TermStatic Term

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Summary of Implications Winter > Spring > Summer > Fall Cross-sectional: –Risk: riskier firms have larger amplitudes –Size: smaller firms tend to have more ownership by individual investors, who are more susceptible to behavioural factors

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Quarterly Excess Returns ( )

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Quarterly Excess Returns (Size Deciles)

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Comparing Different Hypotheses

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AR-GARCH-M Model

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Size/Risk and Seasonality Amplitude

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Summary Theory –Dynamic Effect Should Dominate: W,Sp>Su,F –Concavity Effect: W>Sp; Su>F Further Inferences –Cross-sectional Differences: Size/Risk Tests –Strongly supportive (size premium is even reversed in fall)

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