skewness_and_kurtosis

[liordp]

Hi suzanne'

i am confused
http://www.bionicturtle.com/learn/article/skewness_and_kurtosis_quantitative/

you say "Financial asset returns are typically considered (i.e., heavy or fat- tailed)"

in the reading i find that return distribution i s normal but price distribution is lognormal

tnx

 

[krose]

Hi Convexity

Normality is only an assumption which makes a lot of calculations easier (option pricing, VaR, etc). If you take a look at an empirical distribution you often see that it is not normally distributed . Compare an empirical distribution to the normal distribution and you will notice a central tendency in the returns. The returns are most of the time making small "jumps" around the mean but every now and then the market "blows up" and makes huge leaps. This is basically why the empirical distributions have excess kurtosis and are skewed. As far as i remember from Davids screencasts, the more statistical reason why the empirical distributions are not normal in the moments is because the variance and the mean are not stable over time.

I have uploaded an Excel sheet that compares the normal distribution to the empirical distribution from the S&P 500. You will notice a spike in the frequency of zero in the empirical distribution but this is because it didn't clean the data from the holidays where there market is closed but for some reason shows a return of zero. All the 0's will also affect the results of the other calculations but the spreadsheet is just to show you the basic difference between the normal and the empirical distribution.

The skew and kurtosis is calculated based on the setup of one of Davids spreadsheets (hope this is alright?).

/Kenneth

 

[David Harper CFA FRM]

I think Kenneth's answer is great! (@Kenneth: of course you can adopt XLS, I am glad it is useful). It speaks to the thematic difference between MODEL vs REALITY which is related to parametric/analytical versus empirical. Notice when we use historical data to compute a volatility, then even the simple act of scaling the volatility to (e.g.) 95% normal VaR = 1.645*volatility is to "impose normality" (Linda Allen's term) on the historical data. The data itself isn't normally distributed; but our 1.645 normal deviate (after all, we get it from an artificial standard normal) "superficially" overlays a clean, false (parametric) normal on messy, empirical non-normal data. (I find Kenneth's other reference even more relevant: even if returns are perfectly normal from period-to-period, they won't be stable over time and the time-varying dynamic creates fat tails. This is GARCH: conditional normal but unconditionally non-normal, even in the purely parametric form). So, I do think this amount to our *practical* reliance on an assumption that underlying risk factors are normal (an expedient) compared to our knowledge that real returns are not so well-behaved such that the question becomes, are we bending the rules too much, or can the false normality help us generate sufficiently useful *approximate* measures.

The other point you raise re: lognormal is not empirical. Under GBM/BSM, we assume log returns are normal: LN(S1/S0) ~ N(0,1). This implies LN(St) ~ N(.) and this is the definition of a lognormal distribution. So, in this regard, the difference is between log RETURNS (normal) versus PRICE LEVELS (lognormal). And since the lognormal has heavy-tails, we could fairly say (although you don't see it much) that the price levels (i.e., prices rather than returns) under a normal return process are heavy-tailed. Hope that helps, David

 

[liordp]

Hi David
Your answer made it clear