قیمت گذاری دارایی ها با لحظات بالاتر: شواهدی از بازار سهام استرالیا و ایالات متحده

This paper investigates the importance of higher moments of return distributions in capturing the variation of average stock returns for companies listed in the leading S&P US and Australian indices. We find that Australian stocks are more negatively skewed but less leptokurtic than US stocks. As a result, we find that co-skewness plays a more important role in explaining Australian returns while co-kurtosis is consistently influential for US stock returns. We postulate that the differences in results are related to the underlying firm characteristics of the companies in the two indices, where principally the Australian firms are noticeably smaller than their US counterparts and concentrated in a smaller number industry sectors. This implies that for many smaller exchanges around the world higher moment characteristics displayed by the US market may not be applicable. We also show our results are robust to partly explaining average stock returns in the presence of size, value, and momentum effects.

It has long been well documented that stock returns do not follow a normal distribution. For example, Mandelbrot (1963) and Mandelbrot and Taylor (1967) show that stock returns exhibit excess kurtosis, also commonly referred to as fat tail distributions. Fama (1965) finds that large stock returns tend to be followed by stock returns of similar magnitude but in the opposite direction. This can lead to the volatility clustering effect that is related to how information arrives and is received by the market (see Campell and Hentschel (1992)). This clustering in return volatility has raised a fundamental question on whether a mean and variance asset pricing model using only the first two moments of the return distribution is adequate in capturing variation in average stock returns. Subsequent voluminous empirical tests on Sharpe's CAPM (1964) have largely rejected the validity of the model which assumes that an investor's utility function is quadratic and that the co-movement with the market return is the only important factor in pricing stocks (see Campbell et al. (1995) for a comprehensive review). Given that the empirical stock return distribution is observed to be asymmetric and leptokurtic, a natural extension of the elegant but oversimplified two-moment asset pricing model is to incorporate the co-skewness (third moment) and co-kurtosis (fourth moment) factors. An investor whose utility is non-quadratic and is described by non-increasing absolute risk aversion may prefer positive skewness and less kurtosis in the return distribution. Stocks of negative co-skewness and of larger co-kurtosis with the market should therefore be related to higher risk premia. Therefore, movement of higher co-moments unfavourable to the investors’ risk preferences requires compensation in the form of additional returns. This particular approach of characterizing stock pricing behaviour not only can be intuitively appealing but may also improve the explanatory power of a model on the expected stock returns. In this paper, we examine the importance of co-skewness and co-kurtosis for average stock returns, along with the well documented Fama and French (1993) 3 common risk factors (namely firm size, book-to-market equity (BV/MV), and market returns) and the Jegadeesh and Titman (1993) momentum effect. In particular, we test the presence of higher co-moment effects in the Australian stock market and compare them with those in the US market. Our interest in the behaviour of Australian stocks rests with the glaring absence of any direct studies on the pricing of higher co-moments in the current Australian literature despite some evidence of skewness and kurtosis in the stock return distribution. For instance, Beedles (1986) and Alles and Spowart (1995) find that Australian stocks exhibit significant skewness. Furthermore, Bird and Gallagher (2002) and Brands and Gallagher (2004) document that Australian mutual funds are characterized by a leptokurtic distribution. In particular, they noticed that portfolio returns of larger funds had more negative skewness and larger kurtosis relative to smaller mutual funds. Although they suggest that the non-normal distribution may have implications for diversification benefits, they did not pursue the analysis to directly measure this through these higher moments. Even for studies on the US, direct examination of higher moments is usually quite limited, and approaches to examining it can be varied. Fang and Lai (1997) examine the importance of co-skewness and co-kurtosis within the four-moment CAPM framework. Dittmar (2002) tests the four moment factors with non-linear pricing kernels to improve the pricing kernel's ability to describe the cross-section of returns. His methodology is linked to the nonparametric models of Bansal and Viswanathan (1993) and Chapman (1997) in which the pricing kernel is non-linear in the market return. On the other hand, Kan and Zhou (2003) and Ando and Hodoshima (2006) examine the robustness of the asymptotic covariance matrix of least square errors (LSE) of alphas and betas in a linear asset pricing model when the joint distribution of the factors and error terms may not be normal or conditionally homoskedastic. In contrast, our approach is more consistent with the spirit of Ross’ APT (1976) or Merton's ICAPM (1973) in which additional factors such as size, BV/MV, and momentum may also capture variation in average stock returns. Our approach can therefore be viewed as a more direct test on the presence of higher co-moments. We draw a comparison of return behaviour between stocks listed as part of the Australian S&P ASX 300 index and the US S&P 500 to highlight the potential different roles that co-skewness and co-kurtosis perform in each market. Since an average Australian firm tends to be smaller and less volatile than in other developed markets, negative skewness could be more dominant than kurtosis in pricing stocks. On the other hand, an average US firm is larger but more volatile (also shown in the descriptive statistics in Table 1 and Table 2) such that its variance risk or kurtosis could be a more influential factor. Despite this casual observation on the different stock market characteristics, no studies to our knowledge have addressed the potential differential pricing effect of co-skewness and co-kurtosis. Most studies, especially those in the US, rather focus simply on the skewness of the return distribution, when kurtosis could be equally or more important. Earlier works including Arditti (1967), Kraus and Litzenberger (1976), Friend and Westerfield (1980), Lim (1989), Harvey and Siddique, 1999 and Harvey and Siddique, 2000 and Smith (2007) have examined the return distribution that only includes skewness. Our study intends to fill the gap in the literature by addressing the relative importance of higher co-moments in markets of different characteristics. Examining these two different markets also provides some robustness checks on the importance of each pricing factor. Table 1. Summary statistics of the returns of 25 US portfolios formed by size and BV/MV: January 1992–July 2007. Size BV/MV Mean SD Unconditional skewness Excess unconditional kurtosis Normality test Observations Jarque–Bera Probability Large Low Portfolio 1–1 −0.00027 0.01193 −0.55992 7.05873 8617.58 0 4049 2 Portfolio 1–2 0.00005 0.01076 0.03064 2.81503 1337.54 0 4049 3 Portfolio 1–3 0.00005 0.01115 −0.21915 5.51658 5166.66 0 4049 4 Portfolio 1–4 0.00003 0.01213 0.21107 5.52309 5176.44 0 4049 High Portfolio 1–5 0.00013 0.01593 0.21826 4.01917 2757.41 0 4049 2 Low Portfolio 2–1 0.00019 0.01103 −0.04318 4.08743 2819.88 0 4049 2 Portfolio 2–2 0.00034 0.00920 −0.07564 3.28607 1825.62 0 4049 3 Portfolio 2–3 0.00026 0.01060 0.01666 4.92388 4090.45 0 4049 4 Portfolio 2–4 0.00025 0.01168 0.03459 5.16342 4498.72 0 4049 High Portfolio 2–5 0.00044 0.01191 −0.04082 3.03562 1555.77 0 4049 3 Low Portfolio 3–1 0.00042 0.00941 0.14778 3.93310 2624.53 0 4049 2 Portfolio 3–2 0.00046 0.00963 0.03205 4.58425 3546.16 0 4049 3 Portfolio 3–3 0.00037 0.00979 0.10882 3.77506 2412.26 0 4049 4 Portfolio 3–4 0.00038 0.01137 −0.10549 3.36970 1923.17 0 4049 High Portfolio 3–5 0.00056 0.01074 −0.04084 7.05465 8397.42 0 4049 4 Low Portfolio 4–1 0.00027 0.01047 −0.04773 5.97045 6015.36 0 4049 2 Portfolio 4–2 0.00042 0.01001 0.21201 6.50191 7162.45 0 4049 3 Portfolio 4–3 0.00040 0.01024 −0.19771 3.86752 2549.87 0 4049 4 Portfolio 4–4 0.00045 0.01039 −0.01058 3.95522 2639.32 0 4049 High Portfolio 4–5 0.00046 0.01109 0.02718 4.59737 3566.29 0 4049 Small Low Portfolio 5–1 0.00069 0.01381 −0.01686 6.95935 8171.18 0 4049 2 Portfolio 5–2 0.00044 0.01244 −0.01017 3.38753 1936.06 0 4049 3 Portfolio 5–3 0.00051 0.01097 0.05701 3.74465 2367.89 0 4049 4 Portfolio 5–4 0.00045 0.01024 −0.12154 4.44389 3341.65 0 4049 High Portfolio 5–5 0.00048 0.01131 −0.09959 5.63357 5361.01 0 4049 Index 0.000106 0.004235 −0.124432 4.543455 3493.092 0 4049 The sample consists of all stocks listed at any point in time during the sample period that were part of the S&P 500. Each portfolio comprises of 4049 observations and is constructed by the intersection of 5 size and 5 BV/MV groups. Portfolio 1–1 contains large-cap and low BV/MV stocks while portfolio 5–5 contains small-cap and high BV/MV stocks. Co-higher moments are based on the direct method. The daily returns of each portfolio are the value-weighted returns of stocks in the portfolio. Unconditional skewness and kurtosis are the third and the fourth moment of the daily returns. The Jarque–Bera normality test is a test of whether the stock returns are normally distributed. Table options Table 2. Summary statistics of 25 Australian portfolios formed by size and book-to-market value: January 2001–July 2007. Size BV/MV Mean SD Unconditional skewness Excess unconditional kurtosis Normality test Observations Jarque–Bera Probability Large Low Portfolio 1–1 −0.001078 0.015451 −0.4141 28.01132 55659.54 0 1701 2 Portfolio 1–2 −0.000641 0.011673 −0.89338 8.17112 4958.399 0 1701 3 Portfolio 1–3 −0.00097 0.013862 −0.14476 5.442801 2105.548 0 1701 4 Portfolio 1–4 −0.000314 0.013093 −1.37125 22.56568 36623.33 0 1701 High Portfolio 1–5 −0.000682 0.014213 −0.61091 5.383741 2160.095 0 1701 2 Low Portfolio 2–1 0.0000503 0.009885 −1.1804 11.55367 9855.93 0 1701 2 Portfolio 2–2 0.000167 0.008852 −0.54744 3.587539 997.1531 0 1701 3 Portfolio 2–3 0.000248 0.010429 −0.47329 3.539279 951.3193 0 1701 4 Portfolio 2–4 0.00003 0.011205 −0.7352 4.796208 1783.619 0 1701 High Portfolio 2–5 0.000373 0.010618 −0.72905 5.276479 2123.931 0 1701 3 Low Portfolio 3–1 0.000369 0.008598 −0.26806 3.737123 1010.218 0 1701 2 Portfolio 3–2 0.000331 0.009714 −0.53622 6.153249 2765.017 0 1701 3 Portfolio 3–3 0.00045 0.009091 −0.40085 3.661214 995.5952 0 1701 4 Portfolio 3–4 0.00082 0.009157 −0.1679 2.891133 600.4111 0 1701 High Portfolio 3–5 0.000275 0.012259 −0.55098 3.001116 724.4126 0 1701 4 Low Portfolio 4–1 0.000477 0.007567 −0.04241 2.27273 366.6006 0 1701 2 Portfolio 4–2 0.000363 0.007203 −0.12133 2.1406 328.9343 0 1701 3 Portfolio 4–3 0.000313 0.008484 −0.57527 4.228345 1360.986 0 1701 4 Portfolio 4–4 0.000315 0.010089 −2.64937 41.52602 124207.5 0 1701 High Portfolio 4–5 0.000828 0.008785 −0.40689 3.543715 936.9783 0 1701 Small Low Portfolio 5–1 0.0005 0.009136 0.507958 6.132786 2738.833 0 1701 2 Portfolio 5–2 0.000445 0.00817 −1.5602 16.25734 19422.44 0 1701 3 Portfolio 5–3 0.000599 0.00726 −0.43624 2.173984 388.9225 0 1701 4 Portfolio 5–4 0.000637 0.007411 −0.41203 2.219143 397.1606 0 1701 High Portfolio 5–5 0.000404 0.008728 −0.35203 1.771397 257.527 0 1701 Index 0.000154 0.002988 −0.488384 3.058635 725.1418 0 1701 The sample consists of all stocks listed at any point in time during the sample period that were part of the S&P ASX 300. Each portfolio comprises of 1701 observations and is constructed by the intersection of 5 size and 5 BV/MV groups. Portfolio 1–1 contains large-cap and low BV/MV stocks while portfolio 5–5 contains small-cap and high BV/MV stocks. Co-higher moments are based on the direct method. The daily returns of each portfolio are the value-weighted returns of stocks in the portfolio. Unconditional skewness and kurtosis are the third and the fourth moment of the daily returns. The Jarque–Bera normality test is a test of whether the stock returns are normally distributed. Table options We find strong evidence for the higher co-moment factors in the US stocks and co-skewness in Australian stocks. Consistent with the investor preference theory discussed earlier, average stock returns are negatively related to co-skewness but positively related to co-kurtosis. These two factors remain robust when we regress them along with excess market returns, size, BV/MV, and momentum in our data. Our results therefore suggest that both co-skewness and co-kurtosis explain part of the return variation that is not captured by these other well known factors. Our findings do not support Chung et al. (2006) who argue that Fama and French factors are proxies for the pricing of higher order co-moments, but are more consistent with Smith (2007) who finds that adding co-skewness to the Fama and French 3 factor model improves the explanatory power of the model. Our analysis also shows that although co-skewness is important in both the Australian and US markets, its influence varies in degree. The co-skewness effect is stronger for the Australian stocks compared to the co-kurtosis effect for US stocks. The importance of the co-skewness effect can perhaps be partly explained by the positive relationship between size and skewness which is found to be more pronounced in Australia than in the US. Since the average size of the sampled Australian firm is smaller than the US firm,1 it follows that Australian stocks may appear to be more sensitive to downside risk given that the return distribution is more negatively skewed. Co-skewness may therefore play a larger role in the Australian market. On the other hand, the return distribution in the US appears to be more leptokurtic as the stock returns tend to be more volatile. Subsequently, the significance of the co-kurtosis effect is more noticeable in the US data. The source of the larger return volatility in the US market may be traced to the specific characteristics of the US firms. Contrary to Australian firms which are more closely related to primary industries in commodity and mining,2 US firms (at least from our sample of S&P 500 firms) are more represented by technology related firms or high growth firms. Their returns therefore display more extreme values at both tails, leading to co-kurtosis being more influential for US stocks. The remainder of the paper is as follows. Section 2 describes the data and our methodology while Section 3 presents the empirical results. Section 4 concludes the study.

This paper suggests that co-skewness and co-kurtosis are important in pricing stocks. The degree of the importance, however, depends on the firm characteristics of the stocks and the risk preference of investors. As Australian stocks tend to be more skewed and less leptokurtic, we find that co-skewness plays a more significant role in explaining average stock returns. For US stocks, co-kurtosis is more influential as returns exhibit higher excess kurtosis. We believe the differences in results between the US and Australian markets come from the fact that the Australian stocks are relatively small to begin with, when compared to their counterparts in the US. The implication being that for many medium to small sized exchanges, co-skewness may be a more relevant factor than co-kurtosis. Our results are also robust to a number of model specifications. Although size, BV/MV, and momentum are correlated with co-skewness and co-kurtosis, the importance of co-skewness and co-kurtosis remain largely unchanged in their presence. It implies that the higher co-moments capture parts of variation in average stock returns that are not explained by their effects. Adding co-skewness and co-kurtosis also improves the explanatory power of the Carhart (1997) four-factor model that includes market, size, BV/MV, and the momentum factors. This is despite the fact that we believe our sample is bias against us finding higher co-moment effects, as we focus on only analyzing the larger capitalized firms in the US and Australia. If our study included smaller cap stocks we would expect even further evidence of the presence of higher moments. We therefore believe that our findings do support the need to incorporate higher co-moments into asset pricing models and developing a theoretical asset pricing model that addresses the non-normality of the return distribution should remain important for future research.