Daniel Chi-Hsiou Hung

Daniel Chi-Hsiou Hung

Systematic Risks and Nonlinear Market Models in International Size and Momentum Strategies

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?

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

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

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

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

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

Research questions answered (continued):

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

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)


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


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


Christie-David and Chaudhry (2001) for U.S. commodity contracts

Hung, Shackleton and Xu (2004) for U.K. stock data


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-momentum portfolios

Returns of the two extreme deciles of size and momentum sorts are examined in the time series

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)

Panel A Value weighted countries (markets) returns, from September 1988 to November 2003

Country N. stocks Mean MV Mean Median Stdev, Max Min Skewness Kurtosis J. B. Test Total 44,250 1,964 0.0072 0.0083 0.0335 0.1005 -0.1027 -0.20 3.49 3.05* Canada 5,350 263 0.0054 0.0074 0.0446 0.2172 -0.1633 0.15 7.56 159*** U.S. 14,040 1,388 0.0096 0.0157 0.0415 0.1089 -0.1300 -0.54 3.63 12.1*** Belgium 699 6,848 0.0092 0.0121 0.0346 0.1158 -0.1168 -0.62 4.51 29.1*** Denmark 382 674 0.0088 0.0120 0.0446 0.1360 -0.1152 -0.37 3.40 5.38* Finland 284 1,137 0.0093 0.0115 0.0727 0.2242 -0.2342 -0.10 3.60 3.03* France 2,121 12,506 0.0110 0.0079 0.0384 0.2202 -0.0971 1.35 8.67 301*** Germany 4,433 2,336 0.0099 0.0066 0.0513 0.1599 -0.1758 0.04 3.79 4.78** Italy 493 1,903 0.0084 0.0096 0.0646 0.1994 -0.1860 0.15 3.45 2.22 Netherlands 521 3,193 0.0078 0.0128 0.0414 0.0866 -0.2802 -2.23 15.25 1296*** Norway 471 425 0.0080 0.0117 0.0530 0.1614 -0.2113 -0.29 3.93 9.26*** Spain 247 4,794 0.0079 0.0079 0.0538 0.1384 -0.1696 -0.38 3.86 10.02*** Sweden 995 1,155 0.0070 0.0134 0.0511 0.1438 -0.1809 -0.45 3.65 9.24*** Switzerland 737 4,395 0.0087 0.0101 0.0335 0.1484 -0.0757 0.43 4.72 28.4*** U.K. 5,159 1,756 0.0039 0.0033 0.0298 0.0822 -0.0782 -0.08 3.42 1.56 Australia 1,955 490 0.0078 0.0093 0.0461 0.1215 -0.1615 -0.37 3.62 7.1** Hong Kong 892 870 0.0121 0.0131 0.0696 0.2094 -0.3348 -0.47 6.10 79.8*** Japan 3,077 1,276 0.0004 -0.0084 0.0613 0.2631 -0.1534 0.70 4.41 30.3*** Korea 601 406 0.0107 0.0007 0.1090 0.4028 -0.3010 0.57 4.39 24.6*** Singapore 767 446 0.0067 0.0072 0.0604 0.2099 -0.1936 0.09 5.06 32.6*** Taiwan 1,066 383 0.0047 -0.0009 0.1121 0.3384 -0.3886 -0.10 3.88 6.25** Euo-$ 1M - - 0.0041 0.0041 0.0017 0.0081 0.0009 -0.001 2.63 1.04

Panel B. Equally weighted size portfolios, from October 1988 to November 2003

N. stocks MV ($ M) Past

return Mean Stdev Skewness Kurtosis J. B. Test Beta Gamma Delta

Small 2551.4 1 -0.0106 0.0310 0.0425 0.65 4.74 35.9*** 0.42 2.24 0.59 2 2551.3 5 0.0300 0.0183 0.0400 0.59 5.81 71*** 0.53 2.01 0.65 3 2551.3 14 0.0469 0.0117 0.0345 0.07 4.19 10.9*** 0.55 1.53 0.67

4 2551.1 28 0.0377 0.0091 0.0325 -0.16 4.13 10.4*** 0.58 1.50 0.69 5 2550.9 51 0.0460 0.0078 0.0332 -0.23 4.14 11.5*** 0.65 1.37 0.75

6 2550.9 93 0.0505 0.0071 0.0341 -0.24 4.37 16.1*** 0.72 1.35 0.83 7 2550.9 172 0.0569 0.0068 0.0357 -0.24 4.32 15*** 0.80 1.29 0.90 8 2550.9 353 0.0635 0.0054 0.0372 -0.26 4.67 23.2*** 0.86 1.28 0.96 9 2550.8 920 0.0682 0.0058 0.0378 -0.36 4.58 23.1*** 0.90 1.30 0.98

Big 2550.8 14,614 0.0640 0.0067 0.0352 -0.44 4.32 19.2*** 0.90 1.25 0.93 S - B - -14,613 -0.0746 0.0242 0.0073 1.09 0.43 - -0.48 0.99 -0.34

Panel C. Equally weighted momentum portfolios, from February 1989 to November 2003

N. stocks MV ($ M) Past

return Mean Stdev Skewness Kurtosis J. B. Test Beta Gamma Delta

Loser 2469.8 365 -0.4511 0.0218 0.0795 0.61 5.90 75.3*** 1.03 2.58 1.24 2 2469.7 935 -0.2154 0.0074 0.0512 -0.06 5.29 40.2*** 0.82 1.90 1.01

3 2469.6 1,760 -0.1150 0.0055 0.0384 -0.36 6.09 76.7*** 0.69 1.58 0.86 4 2469.5 1,870 -0.0503 0.0040 0.0302 -0.34 5.24 41.7*** 0.56 1.19 0.71 5 2469.4 1,824 -0.0109 0.0059 0.0251 -0.21 5.68 56.2*** 0.47 0.87 0.53 6 2469.3 1,798 0.0238 0.0078 0.0214 -0.004 3.98 7.29** 0.39 0.73 0.41 7 2469.2 1,790 0.0697 0.0091 0.0238 0.03 3.18 0.27 0.48 0.46 0.46 8 2469.2 2,590 0.1381 0.0091 0.0317 -0.65 4.24 24.7*** 0.66 1.38 0.71

9 2469.1 2,311 0.2641 0.0127 0.0373 -0.75 5.01 47.9*** 0.80 1.86 0.90 Winner 2468.9 1,548 0.9621 0.0203 0.0545 -0.90 7.24 162*** 1.01 2.62 1.14

W - L - 1,183 1.4131 -0.0015 -0.0249 -1.51 1.35 - -0.02 0.04 -0.10

Panel D. Equally weighted momentum-size, two-way sorts, February 1989 to November 2003

Momentum Size

Number

Stocks

MV

($ M)

RP

(P. 6M)

Rp

(Mean) Stdev Skew Kurt J. B. Test Beta Gamma Delta

1 1016.1 2 -0.4356 0.0509 0.081 1.12 6.79 147*** 0.64 3.98 0.98

2 871.4 14 -0.3758 0.0101 0.067 0.65 5.88 76.1*** 0.83 2.49 1.04

3 711.9 44 -0.3487 0.0024 0.072 0.78 8.22 226*** 0.96 2.04 1.14

4 614.7 120 -0.3334 0.0009 0.077 0.61 6.46 102*** 1.16 1.76 1.35

5 527.5 382 -0.3144 0.0014 0.079 0.42 5.37 48.1*** 1.28 1.45 1.42

Lowest

Past

Return

6 374.5 5,097 -0.2903 0.0021 0.076 0.19 4.71 23.3*** 1.27 1.36 1.28

1 580.6 2 -0.1191 0.0193 0.043 0.32 4.78 27.2*** 0.43 1.99 0.68

2 695.0 15 -0.1190 0.0068 0.036 -0.12 5.06 32.7*** 0.52 1.69 0.71

3 711.0 45 -0.1173 0.0042 0.037 -0.39 5.79 63.9*** 0.61 1.69 0.79

4 707.9 121 -0.1179 0.0026 0.041 -0.20 6.60 100*** 0.74 1.54 0.92

5 705.4 393 -0.1169 0.0022 0.044 -0.22 5.74 58.8*** 0.86 1.29 0.99

2

6 716.0 8,671 -0.1147 0.0029 0.044 -0.43 4.68 27.1*** 0.92 1.35 1.00

1 786.1 2 -0.0199 0.0088 0.026 0.03 3.72 3.93* 0.34 1.17 0.42

2 684.2 15 -0.0232 0.0051 0.025 -0.14 3.95 7.48** 0.38 1.07 0.49

3 668.1 44 -0.0238 0.0044 0.025 0.19 6.26 82.3*** 0.42 0.90 0.54

4 649.1 121 -0.0239 0.0037 0.026 0.13 6.42 89.5*** 0.52 0.96 0.62

5 657.5 391 -0.0241 0.0035 0.030 0.04 6.35 85.4*** 0.64 0.88 0.75

3

6 670.8 10,733 -0.0248 0.0042 0.031 -0.02 4.88 26.9*** 0.71 0.88 0.77

1 795.1 2 0.0347 0.0149 0.025 0.80 3.96 26.5*** 0.24 0.87 0.25

2 690.6 15 0.0373 0.0088 0.021 0.10 3.87 6.11** 0.28 0.57 0.29

3 685.0 44 0.0387 0.0078 0.020 0.08 3.43 1.63 0.34 0.48 0.33

4 669.5 121 0.0394 0.0068 0.021 0.06 4.05 8.47** 0.42 0.53 0.41

5 630.8 390 0.0396 0.0066 0.025 0.14 3.82 5.66** 0.54 0.56 0.51

4

6 644.5 10,752 0.0405 0.0065 0.027 0.08 3.87 6.02** 0.64 0.47 0.56

1 441.3 2 0.1364 0.0214 0.047 1.61 9.39 391*** 0.42 1.45 0.50

2 555.1 15 0.1405 0.0145 0.035 0.12 6.16 76.7*** 0.53 1.54 0.60

3 656.2 45 0.1409 0.0095 0.030 -0.54 4.57 27.7*** 0.58 1.35 0.65

4 730.3 122 0.1416 0.0080 0.030 -0.56 4.08 18.4*** 0.64 1.25 0.71

5 802.2 397 0.1426 0.0058 0.032 -0.55 3.92 15.8*** 0.73 1.32 0.79

5

6 930.3 10,201 0.1449 0.0061 0.035 -0.51 3.74 12.0*** 0.86 1.30 0.90

1 497.2 2 0.9796 0.0363 0.070 0.58 4.59 29.4*** 0.66 3.57 0.86

2 619.3 15 0.8460 0.0216 0.055 -0.09 7.55 158*** 0.73 2.69 0.92

3 683.5 45 0.6964 0.0168 0.048 -0.92 8.15 228*** 0.85 2.18 0.98

4 744.1 122 0.6354 0.0150 0.049 -1.02 7.70 200*** 1.00 2.17 1.12

5 792.1 391 0.6064 0.0113 0.051 -0.64 6.54 108*** 1.11 2.01 1.21

Highest

Past

Return

6 779.2 7,784 0.5083 0.0097 0.049 -0.44 6.18 82.9*** 1.07 1.84 1.14

Cross-sectional tests of co-moment pricing

100 size portfolios

100 momentum portfolios

36 two-way sorted, size-momentum portfolios

Portfolio beta, gamma (Coskewness) and delta (Cokurtosis) estimation

1

60

21

60

/t

t

mtm

t

t

mtmptppt rrrrrr

1

60

31

60

2/

t

t

mtm

t

t

mtmptppt rrrrrr

1

60

41

60

3/

t

t

mtm

t

t

mtmptppt rrrrrr

(2)


The estimates of βpt, γpt and δpt are used in cross-sectional regressions to estimate premia ηβ, η γ and ηδ associated with covariance, coskewness and cokurtosis

ptpttpttptttptr 0 (4)


ptpttpttptttptr 0

Panel A. 100 Size sorted portfolios

Model η0 ηβ ηγ ηδ Adj. R2

1 0.0312 -0.0343 0.2325 5.76 -5.91 2 0.0112 -0.0297 0.0100 0.3028 3.18 -5.25 5.77 3 0.0167 -0.0739 0.0500 0.2911 5.04 -6.55 5.63 4 0.0104 -0.0499 0.0080 0.0228 0.3377 3.40 -4.85 4.45 2.54

ptpttpttptttptr 0

Panel B. 100 Momentum sorted portfolios


1 -0.0072 0.0183 0.2214 -2.88 2.72 2 -0.0085 0.0300 -0.0040 0.2818 -2.96 2.92 -1.67 3 -0.0067 0.0406 -0.0206 0.2708 -2.71 3.10 -1.96 4 -0.0071 0.0343 -0.0048 -0.0071 0.3430 -2.56 2.85 -1.26 -0.51

ptpttpttptttptr 0

Panel C. 36 Momentum and size two-way sorted portfolios


1 0.0085 -0.0054 0.2028 3.26 -1.02 2 0.0015 -0.0185 0.0105 0.3386 0.61 -3.09 4.88 3 0.0031 -0.0515 0.0446 0.3269 1.12 -4.10 3.54 4 0.0007 -0.0363 0.0083 0.0212 0.4134 0.26 -2.75 3.01 1.41

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

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)


m

fmppmfp

RRRR

(5)

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


Mean Monthly Returns of Size Sorted Portfolios

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

1 2 3 4 5 6 7 8 9 10

Portfolios

UpDownMean

Mean Monthly Returns of Momentum Sorted Portfolios

-0.03-0.02

-0.010.00

0.010.02

0.030.04

0.050.06

1 2 3 4 5 6 7 8 9 10

Portfolios

UpDownMean

ptpttpttptttptr 0

36 Momentum and size two-way sorted portfolios Model Status η0 ηβ ηγ ηδ Adj. R2

Up 0.0068 0.0205 0.1935 1 2.73 4.34 Down 0.0109 -0.0427 0.2163 3.95 -10.31 Up -0.0002 0.0113 0.0092 0.3150 2 -0.09 2.24 6.29 Down 0.0040 -0.0614 0.0123 0.3725 1.73 -12.31 4.28 Up 0.0008 -0.0175 0.0393 0.3181 3 0.28 -1.40 2.96 Down 0.0063 -0.1003 0.0523 0.3395 2.52 -8.98 4.47 Up -0.0030 -0.0056 0.0072 0.0223 0.4026 4 -1.02 -0.38 3.48 1.32 Down 0.0061 -0.0805 0.0099 0.0195 0.4290 2.49 -8.70 2.78 1.67

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

tmtmtpmtmtpmtpppt rrCrrCrCCr 3

3

2

210(6)


3

2

210

Panel A. Explaining returns of Small and Big portfolios

Model C0 C1 C2

C3 Adj. R2

1 Small 0.0253 0.4605 0.1251 8.48 5.18 Big -0.0004 0.9643 0.8444 -0.42 31.36 2 Small 0.0315 0.4210 -5.4257 0.1706 9.10 4.82 -3.30 Big 0.0013 0.9531 -1.5423 0.8496 1.10 31.22 -2.68 3 Small 0.0260 0.3116 37.8746 0.1301 8.62 2.28 1.43 Big -0.0001 0.8940 17.8854 0.8468 -0.08 18.96 1.96 4 Small 0.0315 0.3605 -5.1523 15.8891 0.1676 9.08 2.67 -3.01 0.59 Big 0.0013 0.9066 -1.3325 12.1996 0.8501 1.11 19.30 -2.23 1.30

Panel B. Explaining returns of Winner and Loser portfolios

Model C0 C1 C2

C3 Adj. R2

1 Winner 0.0135 1.0164 0.3888 4.18 10.63 Loser 0.0147 1.1193 0.2198 2.78 7.11 2 Winner 0.0199 0.9710 -5.6381 0.4192 5.33 10.29 -3.19 Loser 0.0211 1.0748 -5.5192 0.2306 3.36 6.80 -1.86 3 Winner 0.0140 0.9116 26.4639 0.3883 4.28 6.17 0.93 Loser 0.0159 0.8847 59.2217 0.2225 2.96 3.65 1.27 4 Winner 0.0199 0.9593 -5.5862 3.0399 0.4159 5.32 6.61 -3.04 0.11 Loser 0.0210 0.9262 -4.8553 38.8624 0.2291 3.36 3.81 -1.58 0.81


3

2

210

Model C0 C1 C2

C3 Adj. R2

Canada 1 -0.0011 0.7767 0.3376 -0.42 9.68 2 0.0025 0.7533 -3.2071 0.3502 0.80 9.39 -2.12 3 -0.0012 0.7892 -3.1790 0.3339 -0.44 6.35 -0.13 4 0.0025 0.8226 -3.5200 -18.2135 0.3485 0.80 6.64 -2.24 -0.74 US 1 0.0027 0.8954 0.5248 1.26 14.21 2 0.0057 0.8760 -2.6635 0.5352 2.29 13.93 -2.25 3 0.0025 0.9211 -6.5517 0.5225 1.17 9.44 -0.35 4 0.0057 0.9496 -2.9959 -19.3477 0.5352 2.29 9.79 -2.44 -1.00


3

2

210

Belgium 1 0.0022 0.8916 0.7496 1.74 23.36 2 0.0043 0.8786 -1.7884 0.7567 2.84 23.13 -2.50 3 0.0026 0.8260 16.6999 0.7512 1.97 14.05 1.47 4 0.0043 0.8414 -1.6204 9.7792 0.7562 2.84 14.36 -2.18 0.83 Denmark 1 0.0020 0.8681 0.4266 0.79 11.68 2 0.0051 0.8480 -2.7677 0.4357 1.74 11.39 -1.98 3 0.0022 0.8163 13.1929 0.4245 0.88 7.10 0.59 4 0.0051 0.8423 -2.7423 1.4803 0.4325 1.74 7.32 -1.88 0.06 Netherlands 1 0.0010 0.8498 0.4738 0.45 12.84 2 0.0029 0.8374 -1.7005 0.4762 1.11 12.56 -1.36 3 0.0002 1.0064 -39.8340 0.4827 0.09 9.93 -2.03 4 0.0029 1.0308 -2.5738 -50.8269 0.4912 1.13 10.18 -2.00 -2.51


3

2

210

Australia 1 0.0011 0.8179 0.3470 0.38 9.89 2 0.0054 0.7904 -3.7671 0.3642 1.65 9.59 -2.43 3 0.0001 1.0102 -48.9123 0.3575 0.03 7.97 -1.99

4 0.0054 1.0574 -4.9723 -70.1497 0.3878 1.67 8.48 -3.14 -2.81

Taiwan 1 -0.0024 0.9462 0.0752 -0.30 3.97

2 0.0134 0.8455 -13.8137 0.1179 1.44 3.60 -3.13 3 -0.0001 0.4808 118.3849 0.0842 -0.01 1.31 1.67 4 0.0134 0.6016 -12.7127 64.0875 0.1169 1.44 1.66 -2.77 0.89


3

2

210

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)


3

2

210 (6)

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

31,1,3

21,1,21,101,

^

tmtmptmtmptmpptp rrCrrCrCCr (10)

Model predicted returns of momentum and size deciles

Linear Market Model Predicted Momentum Portfolio Return

-0.01

0.00

0.01

0.02

0.03

0.04

1 2 3 4 5 6

Month following portfolio formation

Pre

dict

ed

Re

turn

Winner

Loser

Linear model predicted returns of momentum deciles

Linear Market Model Predicted Size Portfolio Return

-0.01

0

0.01

0.02

1 2 3 4 5 6 7 8 9 10 11 12

Month following Portfolio Formation

Pre

dict

ed R

etur

n

Small

Big

Deflated Excess Returns of Momentum Deciles

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5Linear Model P redicted Mean Return, % per month

Realis

ed M

ean R

etu

rn, %

Panel A. Predicted versus Realised Returns of the Equally Weighted Size Deciles

Deciles Small D2 D3 D4 D5 D6 D7 D8 D9 Big Mean

Square Error

Realised 0.0211 0.0129 0.0078 0.0049 0.0030 0.0024 0.0009 0.0000 0.0002 0.0016

Linear Model Predicted

0.0109 0.0075 0.0051 0.0042 0.0048 0.0052 0.0058 0.0066 0.0077 0.0083 3.22×10-5

Quadratic Model Predicted

0.0104 0.0074 0.0054 0.0047 0.0052 0.0060 0.0064 0.0070 0.0084 0.0086 3.65×10-5

Cubic Model Predicted

0.0102 0.0073 0.0056 0.0049 0.0054 0.0064 0.0069 0.0074 0.0091 0.0094 4.1×10-5

Panel B. Predicted versus Realised Returns of the Equally Weighted Momentum Deciles

Deciles Small D2 D3 D4 D5 D6 D7 D8 D9 Big Mean

Square Error

Realised 0.0166 0.0036 0.0001 -0.0002 -0.0001 0.0026 0.0040 0.0058 0.0086 0.0147

Linear Model Predicted

0.0064 0.0048 0.0033 0.0016 0.0001 0.0009 0.0033 0.0059 0.0104 0.0235 2.03×10-5

Quadratic Model Predicted

0.0077 0.0057 0.0038 0.0021 0.0002 0.0011 0.0036 0.0063 0.0112 0.0255 2.29×10-5

Cubic Model Predicted

0.0075 0.0065 0.0047 0.0026 0.0001 0.0014 0.0034 0.0064 0.0113 0.0263 2.64×10-5

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

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

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Daniel Chi-Hsiou Hung

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