Seasonality in Equilibrium Stock Returns: A Dynamic Perspective on SAD

Xifeng Diao Maurice Levi

U of Calgary UBC

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

This Paper’s Contribution

• Dynamic effect should dominateOur 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

The Model

• Assets– Zero net supply of bond, rf normalized to 0.

– A single risky asset with a terminal payoff DT,.

dDt / Dt = μD dt + σD dBt

• Agent– A representative agent with CRRA utility

– Objective function: Max E { U(WT) }

Equilibrium Returns• Return from time t to t+d

log (St+d / St)

= (γt-0.5) σD2d +(γt – γt+d)σD

2(T–t-d) +σD(Bt+d–Bt)

where γ: risk aversion.

• Concavity: γt = -g(lt), where lt is length of day at t, and g(lt) is concave.

Dynamic Term Variance TermStatic Term

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

Quarterly Excess Returns (1962-2000)

WinterWinter

(excl. Jan)Spring Summer Fall

EW Index 12.8341% 8.1675% 2.6118% 1.7378% -0.7695%

(std) 14.1986% 12.4446% 10.8333% 10.4472% 10.2821%

Panel A: Index Returns

Quarterly Excess Returns (Size Deciles)

WinterWinter

(excl. Jan)Spring Summer Fall

Decile 1 20.97% 8.8899% 1.55% -0.60% -4.55%

Decile 2 15.41% 7.1486% 0.42% -0.91% -3.64%

Decile 3 13.06% 6.2572% 0.07% -0.89% -3.34%

Decile 4 12.10% 6.3304% 0.02% -0.60% -2.83%

Decile 5 10.16% 5.4138% 0.34% -0.26% -2.29%

Decile 6 9.50% 5.6837% -0.01% -0.21% -1.57%

Decile 7 8.07% 5.2684% 0.44% 0.42% -1.33%

Decile 8 6.96% 5.1507% 0.86% 0.56% -0.88%

Decile 9 6.09% 5.4059% 0.86% 0.39% -0.36%

Decile 10 3.70% 3.5310% 1.05% 0.04% 0.54%

AR-GARCH-M ModelMK Index (EW) Size Decile 5 STD Decile 5 Beta Decile 5

0.000320 -0.000037 -0.000057 0.000440(1.29) (-0.14) (-0.19) (1.54)

0.000052 0.000042 0.000040 0.000031(4.83***) (3.57***) (3.18***) (2.52***)

-0.001800 -0.001961 -0.001474 -0.001533(-20.09***) (-18.38***) (-12.19***) (-12.71***)

0.004341 0.004428 0.002180 0.003202(13.8***) (12.02***) (5.54***) (8.53***)

-0.000043 -0.000020 0.000595 0.000518(-0.29) (-0.12) (3.21***) (2.83***)

0.160803 0.192898 0.274341 0.216926(3.93***) (5.66***) (6.64***) (4.62***)

Note: * Significant at 0.1, ** 0.05, *** 0.01

Delta

Turn of Month

Alpha (t-statistic)

Dynamic Cycle (Concave)

Monday

Tax Dummy

Size/Risk and Seasonality AmplitudeDCNCS DCCS R-Square

0.0063

(14.18***)

0.0005 0.0011

(0.5) (6.96***)

0.0064

(12.72***)

-0.0016 0.0015

(-1.53) (8.52***)

0.0061

(14.03***)

0.0012 0.0006

(0.48) (1.96*)

Note: 1) * Significant at 0.1, ** 0.05, *** 0.01

2) DCCS: dyanmic concave SAD cycle with cross-sectional effects on its aplitude

DCNCS: no cross-sectional effects

Model I (t-statistics)

14.44%Size

DecilesModel II 17.08%

STD Deciles

Model I 18.02%

Model II 21.76%

Beta Deciles

Model I 19.64%

Model II 19.85%

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)