27-March-2006
A Few Abnormal Things, by Kim Zussman

The mean and stdev were calculated for 14146 daily returns of SP500 since 1950 to check for normality. These values were used to generate random daily returns from a normal distribution with same mean and stdev. Comparing the worst 20 daily returns for the real series and the normally-distributed simulation:

```t-Test: Two-Sample Assuming Equal Variances

daily ret SP500        Sim
Mean    -0.0617       -0.0286
Variance 0.001293822 1.62355E-06
Observations 20 20
Pooled Variance 0.000647723
Hypothesized Mean Difference 0
df 38
t Stat -4.113703144
P(T<=t) one-tail 0.000100616
t Critical one-tail 1.685954461
P(T<=t) two-tail 0.000201232
t Critical two-tail 2.024394147```

i.e., significantly greater mean decline in 20 worst actual SP500 days as opposed to 20 worst in the normally distributed simulation. The same exercise was done for the 20 best returns of both series, and again the actual returns were significantly higher (t=8).

Taking the 10 best and 10 worst daily returns of the SP500 series, here is a list of how many standard deviations from the mean they were (negatives are declines):

```10.18 -22.90
6.41   -9.26
6.05   -7.68
5.97   -7.61
5.72   -7.57
5.70   -7.47
5.62   -7.41
5.61   -6.84
5.51   -6.52
5.33   -6.02```

The drops were more severe than the gains, and yes that 22 is in Oct 1987.

So what if daily returns are not normally distributed, especially if both top and bottom tails are heavy? Besides influencing choice of statistical tests, there is the question of ruin-especially for those high enough on the food chain to leverage their IQ's.

To check this, located the greatest rolling 20-day declines for each of the 14 non-overlapping 1000 day periods for the last 55 years in SP500. The table below lists the year at the end of each 1000 day period, and the % drop for the single greatest 20 day decline in each period:

```2006 -19.65
2002 -16.88
1998 -8.01
1994 -13.73
1990 -29.56
1986 -7.35
1982 -12.98
1978 -14.69
1974 -13.90
1970 -13.76
1966 -8.67
1962 -15.47
1958 -9.31
1954 -7.76
```