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N=250 (config):
N=5000 (config):
N=20,000,000 (config - warning:
long runtime, 85m on my Macbook Air M1):
In comparison to the normal distribution, the same points look very different
for the t-distribution with a=3:
- Even within 250 samples we are seeing a couple of samples near 6 MADs away from the mean; at 5K samples about 50 (1%) lie more than 5 MADs away, some of which reach over 12 MADs (which are already practically unreachable for the normal distribution), and at 20M there are samples more than 200 MADs away.
- Mean and MAD still converge fairly well, even if their 99% CIs are about twice
as wide as for the normal distribution, but the same
sigma's CI is 3x wider even for 20M samples, all the while having occasional jumps a whole order of magnitude away from its expected value forN=250and about 3x its value forN=5K.
In other words, the standard deviation converges a lot slower and is a lot
noisier than MAD, and hence, I've decided to use MAD rather than sigma.
Additionally, MAD is a more intuitive measure of volatility.
Recall that the market on average (as judged e.g. by the NASDAQ Composite index) has a daily mean of about 3-5% of its MAD. Let's optimistically take the 5%. This implies that the mean's CI must be no more than +-0.05 (5% of MAD) just to give an indication whether the price series is rising or falling on average, and a lot narrower to estimate its actual value.
For a 1-year period, the 99% mean's CI is +-20% of MAD for the normal, and 25% for the t-distribution, which is too wide to estimate anything useful about the mean. In fact, the required +-5% width can only be obtained for a CI with a confidence level of 45% for the normal, and 40% for the t-distribution, which is a dubious confidence level to say it nicely.
For a 20-year period, the normal's 99% mean's CI is now a bit below +-5%, but the t-distribution is closer to +-6%, and the required 5% is obtained by the 98% confidence level. Once again, this is just to be sure with 98% confidence that the stock or index actually grows over time, but not to tell how fast.
For reference, a 10-year period achieves the 5% accuracy with 90% confidence for
t-distribution, and the same 90% confidence yields 3.7% accuracy over 20 years,
which might in fact be passable for a practical investor. Note, that assuming 5%
mean / MAD ratio for a typical 1% volatility is 0.05% daily growth, which
translates to 3.5% annualized growth, and +-3.7% accuracy is a 90% confidence
that the stock grows between 1.4% and 8.9% annually. I'd say, that's still a
pretty lousy accuracy, even if somewhat reassuring.
Since the normal distribution has comparable ranges for the mean, the precision and the confidence are not much higher, even if we take a wild chance and decide to use it as a model.
Next: The Tale of Fat Tails