Daniel Chi-Hsiou Hung

1. Systematic Risks and Nonlinear Market Models in International Size and Momentum Strategies Daniel Chi-Hsiou Hung

2. Research questions asked: Do higher order Capital Asset Pricing Models better describe asset returns than the standard CAPM?Can higher co-moment risks (coskewness and cokurtosis) capture the stylized effects of momentum and size strategies in international stock markets?

3. Why and how are they relevant to finance? • The beta of the Capital Asset Pricing Model (CAPM) may not be sufficient to describe systematic risks • Risk management, where the estimation and control of risk profiles of hedged positions of a company or an investment are critical elements of effective hedging • The estimation of the cost of capital of a company, which is directly related to corporate valuation and capital budgeting

4. Why and how are they relevant to finance? (continued) • Insight from higher co-moments can be beneficial to the formation of portfolio strategies, especially for hedge funds that typically have highly skewed return distributions • Performance of managed funds could be evaluated by comparing average return of a managed portfolio with that of benchmark portfolios that have similar beta, coskewness and cokurtosis

5. Research questions are answered in many aspects: • Higher co-moments are priced in 20 international equity markets • When higher co-moments are included into the two-moment CAPM, model intercepts become insignificant in all cases for examining the two-way sorted, momentum-size portfolios

6. Research questions answered (continued): • Develop and test a cubic-market model, which shows better performance than the linear CAPM in explaining the stylised effects • Market models are found to predict payoffs from momentum strategies that buy the past return winners and sell the past return losers

7. Research questions answered (continued): • When up and down markets are tested separately, the CAPM beta is highly significant in explaining the cross-section of international stock returns • The smallest size decile has positive average returns in both up- and down-markets

8. Higher order systematic risks and evidence • The intuition for the preference of skewness • If a risky asset has a return distribution with a long tail in the negative direction, it is more likely to have more extreme negative returns. • Other things being equal, a rational investor will require a higher mean rate of return on assets that contribute negative skewness to the market

9. Higher order systematic risks and evidence • Kraus and Litzenberger (1976) Non-increasing absolute risk aversion leads to a preference for positive skewness. (U.S. stock data) • Harvey and Siddique (2000) Assets that make the portfolio returns more left-skewed are less desirable and should command higher expected returns. (U.S. stock data)

10. Higher order systematic risks and evidence • The intuition for the preference of kurtosis • Since kurtosis measures the probability of extreme outcomes, a rational investor will prefer short-tailed distributions to long-tailed distributions • Other things being equal, the effect of a risky asset contributing to market leptokurtosis will be to increase the required mean rate of return on the asset

11. Higher order systematic risks and evidence • Fang and Lai (1997) Propose an extended model that incorporates a cokurtosis term which is significant in explaining the cross-section of U.S. stock returns, 1969 to 1988 • Dittmar (2002) Decreasing absolute prudence leads to an aversion for kurtosis

12. Higher order systematic risks and evidence • Christie-David and Chaudhry (2001) for U.S. commodity contracts • Hung, Shackleton and Xu (2004) for U.K. stock data

13. Methodology • Sorts and portfolio formation • Size/Momentum/Country sorts • Equally-weighted deciles are formed for examining the characteristics of portfolios • 36 two-way sorted, size-momentumportfolios • Returns of the two extreme deciles of size and momentumsorts are examined in the time series

14. Data and descriptive Statistics • Monthly U.S. dollar returns, from August 1988 to November 2003 (Datastream) • 44,290 stocks from 20 markets: Canada, U.S., Belgium, Denmark, Finland, France, Germany, Italy, Netherlands, Norway, Spain, Sweden, Switzerland, U.K., Australia, Hong Kong, Japan, Korea, Singapore and Taiwan • Market value of equity and the London Financial Times Euro dollar one-month rate (Datastream)

19. Cross-sectional tests of co-moment pricing 100 size portfolios 100 momentum portfolios 36 two-way sorted, size-momentum portfolios

20. Cross-sectional tests of co-moment pricing • Portfolio beta, gamma (Coskewness) and delta (Cokurtosis) estimation

21. Cross-sectional tests of co-moment pricing The estimates of βpt, γpt and δpt are used in cross-sectional regressions to estimate premia ηβ,η γ andηδ associated with covariance, coskewness and cokurtosis

25. Tests of ex-post beta and return relationships • Pettengill et al.(1995): significance tests of beta premia should be separately conducted according to up-or down-market status • According to the CAPM, the ex ante market premium should be positive and that higher beta portfolios should have higher expected returns than lower beta portfolios

26. Tests of ex-post beta and return relationships Ex post, the market premium can be negative in some periods. Thus, averaging all cross-sectional periods might result in an insignificant market premium Even though the market premium is significant according to Sharp ratio of the market (Eq. 5)

27. Tests of ex-post beta and return relationships And also those portfolios with higher betas can have more negative (lower) realised returns than that of lower beta portfolios. Thus reduce the average realised returns of higher beta portfolios. Consequently, the coefficient for beta might appear insignificantly

31. Time-series tests of nonlinear market models A cubic model (Eq. 6), which is consistent with the four-moment extension of the CAPM is applied to explain the time-series returns of size, momentum and country sorted portfolios

37. Model predicted returns of momentum and size deciles For each portfolio, the intercept and slope coefficients (C0, C1, C2 and C3) of time-series regression model (Eq. 6) are estimated in each month on a rolling basis from the month of portfolio formation to the 5th month following formation for momentum sorts (the 11th month for size sorts)

38. Model predicted returns of momentum and size deciles In the second stage, the estimates of C0, C1, C2 and C3 for each asset are used to predict excess return of the asset in the next period by utilizing realized excess market return in the next period

39. Linear model predicted returns of momentum deciles

44. Summary and conclusion • By using a large international stock data, this paper shows evidence for the pricing of higher order systematic risks in returns of size and momentum portfolios • The inclusion of coskewness and cokurtosis to the standard CAPM can provide incremental explanatory power on stock returns of size and momentum sorts

45. Summary and conclusion (continued) This paper also develops and tests a cubic market model that is consistent with the four-moment CAPM In time-series tests, the benefit of adopting non-linear market models is evidenced for both size and momentum sorts and also for eight international markets