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## Monday, October 7, 2013

### iid assumption in cumulative return

Time diversification? First look at asset diversification. Split \$200k into 2 uncorrelated investments so when one is down, the other might be up. Time-div assumes we could add up the log returns of perod1 and period2. Since the 2 values are two N@Ts and very likely non-perfectly-correlated (i.e. corr < 1.0), one of them might cushion the other.

Background -- the end-to-end (log) return over 30 years is (by construction) sum of 30 annual returns --

r_0to1 is a N@T from noisgen1 with mu and sigma
r_1to2 is a N@T from noisgen2.
...
r_29to30 is a N@T

So the sum r_0to30 (denoted r) is also a random var with a distribution. Without assuming normality of noisegen1, if the 30 random variables are IID, then the sum would follow a normal distribution with E(r) = 30mu and stdev(r) = sigma * sqrt(30)

This is a very important and widely used result, at the heart of a lot of quizzes, a lot of financial data. However, the underlying IID assumption is controversial.

* The indep assumption is not too wrong. Stock return today is not highly correlated with yesterday's. Still AR(1) models include preceding period's return .... Harmless.
* The ident assumption is more problematic. We can't go back in time to run the noisegen1 again, but there are data to prove that the ident assumption is not supported by real data.

Here's my suggestion to estimate noisegen1's sigma. Look at log return. r_day1to252 = r_day1to2 + r_day2to3 + ... + r_day251to252. Assuming the 252 daily return values are a sample of a single noisegenD, we can estimate noisegenD's mean and stdev, then derive the stdev of r_day1to252. This stdev is the stdev of noisegen1.