Normal vs Log-normal   a continuation of Part I

We continue with our study of the S&P 500 to determine whether the monthly returns are Normal or Log-normal or ... whatever. However, now we'll look not at the fractional changes in price (as we did in Part I) but at the price itself.

Recall that we used r and g=1+r to denote fractional changes in price and gain factor (respectively). That is, if a stock (our S&P500) goes from price P1 (this month) to P2 (next month), then

r = P2/P1 - 1      and      g = P2/P1

In Fig. 1 there are two graphs: the S&P500 and its (natural, to-the-base-e) logarithm. If, at some time t, the logarithm has a value Q and the S&P has the value P, then

Q = log(P)
and P = eQ

Fig. 1
If we were to consider a monthly change in the stock price, say P2 - P1, in 1950, then (staring at the blue S&P graph) we see that the same change would be insignificant if it happened in the nineties. However, that insignificance would not be the case for a change in the logarithm.

Fig. 2
 Indeed:   log(P2) - log(P1) = log(P2/P1) = log(g)

Fig. 2 shows the set {g} of monthly gain factors for the S&P500; the distribution doesn't change much, eh?

We'll consider N months and let P1, P2, P3, ... PN denote the N end-of-month stock prices, and use P0 as the starting price (9:30 am ET on Jan 1, 1950, for our S&P example, where N=600). We can also use Q1, Q2, etc. to denote their logarithms.

>Can you show the distribution of log(g)?

Here it is

If we now calculate the total change in log(P) we get:

log(PN) - log(P0) = log(PN/P0)

and we may recognize PN/P0 as the N-month gain factor and {PN/P0}1/N as the equivalent per-month gain factor and {PN/P0}12/N as the equivalent per-year (or annualized) gain factor ... over the N-month period.

>Example?
  • P600/P0 = 1469.25/16.66 = 88.19 (the S&P grew by a factor 88.19)
  • {88.19}1/600 = 1.0075 (the equivalent per-month gain factor was 1.0075)
  • {1.0075}12 = 1.094 (the equivalent annualized gain factor was 1.094)
  • so the annualized gain was 9.4%
And, just to be obstreperous (so you don't fall asleep), I'll remind you of the chart with the distribution of the monthly S&P500 returns (since 1926) so you can compare them with Normal and Log-normal distributions with the same Mean and Standard Deviation ... like so

>Why are you telling me all this? Are you going to suggest a Normal or a Log-normal distribution? Are you ...?
Me? Suggest Normal or Log-normal? Are you kidding! Look at the comparison. Is it good?

Actually, I'd like to introduce you to ..

Ta DUM

>zzz ZZZ

for part III