Higher Moments
Four-Moment Index Model
Skewness, kurtosis, tail measures and distribution tests are statistical approaches to describe the characteristics of a given return distribution.
A more economic approach is the Four-Moment Index Model (also called the cubic model, four-moment CAPM, depending on the context), which extends the single-index model model by including squared and cubic unexpected index returns as additional factors.
The cubic model is defined as follows...
r(i,t) - r(f,t) = a1 + a2*{r(m,t)-r(f,t)} + a3*{r(m,t)-mean[r(m)]}^2 + a4*{r(m,t)-mean[r(m)]}^3 + e(t)
with...
r(i,t)... return of instrument i at point in time t
r(f,t)... riskfree rate in t
r(m,t)... benchmark return in t
mean[r(m)]... expected indexc return calculated as the arithmetic mean return
r(m,t)-mean[r(m)]... unexpected index return
e(t).... white noise
The most important goal of the cubic model is to test which of the parameters a1, a2, a3 are significantly different from zero (=t-tests). The parameters can be interpreted as sensitivities relative to the benchmark skewness and kurtosis (co-skewness and co-kurtosis).
Cornish-Fisher Exansion
Risk measures quantifying certain characteristics of the return distribution can also be affected by the presence of higher moments. Some workarounds exist, see for example see calculating VaR with the Cornish-Fisher expansion.
Higher Moments and Investor Preferences
The relevance of higher moments depends very much on the functioncal form of the utility functions of investors, respectively their "risk tolerance". Investors can be assumed to like positive skewness and dislike fat tails (kurtosis) as well as "long tails".
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