1. Title, authors, journals
Title: Conditional Skewness in Asset Pricing Tests
Authors: CAMPBELL R. HARVEY and AKHTAR SIDDIQUE
Journal: THE JOURNAL OF FINANCE
2. Research Question + underlying intuition
Research Question: Everything else being equal, do investors prefer portfolios that are right-skewed
to portfolios that are left-skewed?
Underlying intuition: In his 2000 paper in the Journal of Finance with Siddique, Harvey presents a
two-part argument in favour of incorporating skewness. First, asset returns are not normally
distributed. Second, investors like positive skew (big profits) and dislike negative skew (big losses);
Harvey argues these preferences need to be taken into account in both portfolio management and
risk management. Harvey also asserts estimates are imprecise and this uncertainty needs to be taken
into account when making investment decisions.
The results show that conditional skewness helps explain the cross-sectional variation of expected
returns across assets and is significant even when factors based on size and book-to-market are
included.
The results suggest that the momentum effect is related to systematic skewness.
The low expected return momentum portfolios have higher skewness than high expected return
portfolios.
In probability theory and statistics, skewness is a measure of the asymmetry of the probability
distribution of a real-valued random variable about its mean. The skewness value can be positive or
negative, or undefined. The qualitative interpretation of the skew is complicated and unintuitive
One clue that pushed us in the direction of skewness is the fact that some of the empirical
shortcomings of the standard CAPM stem from failures in explaining the returns of specific securities
or groups of securities such as the smallest market-capitalized deciles and returns from specific
strategies such as ones based on momentum.
A positively skewed investment return means there were frequent small losses and a few
large gains.
Negatively skewed means there were frequent small gains and a few large losses.
, - Managers may prefer portfolios with high positive skewness.
3. Main methodology
- Our work focuses primarily on monthly U.S. equity returns from CRSP.
- We form portfolios of equities on various criteria such as industry, size, book-to-market
ratios, coskewness with the market portfolio (where we define coskewness as the
component of an asset's skewness related to the market portfolio's skewness), and
momentum using both monthly holding periods as well as longer holding periods.
- Additionally, we also examine individual equity returns.
- We analyze the ability of conditional coskewness to explain the crosssectional variation of
asset returns in comparison with other factors.
- We find that coskewness can explain some of the apparent nonsystematic components in
cross-sectional variation in expected returns even for portfolios where previous studies have
been unsuccessful.
- The pricing errors in portfolio returns using other asset pricing models can also be partly
explained using skewness.
- Our results, however, show that the asset pricing puzzle is quite complex and the success of a
given multifactor model depends substantially on the methodology and data used to
empirically test the model.
- We also find that an important role is played by the degree of precision involved in
computing the asset betas with respect to the factors-that is, what may be a proxy for
estimation risk.
Methodology from “II. Does Skewness Exist in the Returns Data?- Portfolio Formation and
Summary Statistics”
- The first group represents 32 value-weighted industry portfolios.
- The second set are the 25 portfolios sorted on size and book-to-market value used by Fama
and French (1995, 1996).
- Third, we investigate 10 momentum portfolios formed by sorting on past return over t - 12 to
t - 2 months and holding the stock for six months.
, - The fourth group are size (market capitalization) deciles used in a number of empirical
studies.
- Finally, we look at the three-way classification based on book-to-market value, size, and
momentum detailed in Carhart (1997).
4. Data
II. Does Skewness Exist in the Returns Data?- Portfolio Formation and Summary Statistics
- For the empirical work, we use monthly U.S. equity returns from CRSP NYSE/AMEX and
Nasdaq files.
- We form portfolios from the equities as well as analyze individual equity returns.
- Most of our work focuses on the period July 1963 to December 1993.
- We use a longer sample to investigate the interactions of momentum and skewness.
- As factors capable of explaining cross-sectional variations in excess returns, we use the CRSP
NYSE/AMEX value-weighted index as the market portfolio.
- To capture the effects of size and book-to-market value, we use the SMB and HML hedge
portfolios formed by Fama and French.
,5. Results
, - Figure 1, Panel A, presents a mean-variance-skewness surface.
- Slicing the surface at any level of skewness, we get the familiar positively sloping portion of
the mean-variance frontier.
- Skewness adds the following possibility: at any level of variance, there is a negative trade-off
of mean return and skewness.
- That is, to get investors to hold low or negatively skewed portfolios, the expected return
needs to be higher.
- This is evident in the graph.
Exam Question: Explain why expected return is higher for lower levels of skewness.
Investors dislike negative skewness (larger probability of negative return). Therefore, the demand for
this types of stocks will be lower, so prices will be lower. Given a certain expected price, this will
cause higher expected returns.
Other way of saying this: Investors want to be compensated for holding stocks with low skewness, so
demand a higher expected return.
, - Panel B of Figure 1 introduces the risk-free rate.
- The capital market "line" starts out at zero variance-zero skewness.
- Think of a ray from the risk-free rate (at zero variance) that is tangent to the surface at a
particular variance skewness combination.
- For that level of variance, there are many possible portfolios with different skewnesses.
- The tangency point is the one with the highest skewness.
- Now add another ray from the risk-free rate that is tangent to a different variance-skewness
point.
- In the usual mean variance analysis, there is a single efficient risky-asset portfolio.
- In the mean-variance-skewness analysis, however, there are multiple efficient portfolios.
- The optimal portfolio for the investor is chosen as the tangency of the investor's indifference
surface to the capital market plane.
