Market commentators frequently use the word “volatility” to mean that the market is moving downwards. To a financial economist, this is an error. Volatility means the stochastic element to price changes and it is not a synonym for the proportion of the distribution below the mean. In his Nobel Prize lecture, Harry Markowitz suggested the use of what he termed semi-variance to assess risk, a definition of risk that concentrates solely on the downsize and which is more in line with the usage of the commentator than the statistician.

In our prior two posts, discussing Asymmetric GARCH and the VOLVIX process, we found evidence of an option-like non-linear response of the variance to quadratic stimuli. In this post we look for a similar structure in a less esoteric series — the daily returns of the S&P 500 Index.

Applying a AGARCH(1,1) GJR model to the daily returns of the S&P 500 Index, but assuming a fundamentally leptokurtotic driving process, in this instance draws from Student's t Distribution, we find a model that provides an acceptibly good description of the data (the results of which are presented below).

Fit of AGARCH(1,1) Model to S&P 500 Index - Estimation by BFGS
Convergence in    31 Iterations. Final criterion was  0.0000028 <=  0.0000100
Daily(5) Data From 2000:01:03 To 2010:05:24
Usable Observations   2333
Log Likelihood                    7299.94328188

   Variable           Coeff       Std Error      T-Stat     Signif
*********************************************************************
1.  Mean             0.00008427   0.00017596      0.47891  0.63200177 ← mean daily return
2.  C                0.00000095   0.00000029      3.27083  0.00107232 ← static variance term
3.  A               -0.01428908   0.00888489     -1.60825  0.10778138 ← ARCH MA(1) parameter
4.  B                0.93586512   0.01186219     78.89480  0.00000000 ← GARCH AR(1) parameter
5.  D                0.14418349   0.01892191      7.61992  0.00000000 ← asymmetry parameter
6.  Shape           10.98934650   2.22903583      4.93009  0.00000082 ← distributional kurtosis 

This model strongly likes the downside response term in the variance and almost completely rejects its symmetric counterpart. The MLR test statistic for the inclusion of the asymmetry term in the model is χ²(1) = 78.2, which is very significant.

So, in conclusion, we can state that yes, as far as the variance response is concerned, the usage of the term “volatility” as a synonym for “downside risk” seems to be confirmed.