# Volatility Estimation

Let say I ran two strategies and got its weights at each rebalance and equity curves. I would like to combine these systems to get the performance if I were to trade them concurrently from a portfolio level.

The first way to do it would be to just take the equity curves of the respective strategies and treat them as asset class return streams and allocate weights at desired rebalance. To get the performance, I simply do w'R, where w = weights at each rebalance and R being the returns of the strategies. If I were to calculate portfolio volatility given weights w, I would simple employ p.risk = w'E w.

The second way to do it, is to first calculate the get the desired weights for each strategies. From there I can drill deeper and get the weights of the asset classes for each strategy. (assume they trade the same universe) But the problem here is that when I get the aggregate exposure for each asset class and calculate portfolio level volatility (using w'Ew (of asset classes)), I get a number that is less than the first way of doing it. Why is that? Should both methods be the same, all else equal?

example of second way

Each strategy trades Asset1 and Asset2

at first rebalance:

Strategy 1 allocated 30%

Strategy 2 allocated 70%

Within each strategy:

Strategy 1 allocates 10% to Asset1 and 90% to Asset2

Strategy 2 allocates 60% to Asset1 and 40% to Asset2

To get portfolio level asset exposure simply:

Strategy1: x= asset 1 (30%*10%), asset2 (30% * 90%) = 3% , 27%
Strategy2: y= asset 1 (70%*60%), asset2 (70% * 40%) = 42%, 28%

Aggregate Asset 1 exposure = 3% + 42% = 45%
Aggregate Asset 2 exposure = 27% + 28% = 55%
for a total of 100%

So given asset1 = 45% weight and asset2 of 55% weight, I can simply get portfolio risk by w'Ew where E = covariance between the two assets.

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Is E in the first case the covariance between strategies? Are these two E-s coherent/how is it ensured? –  Quartz Jun 11 '13 at 10:08
Sorry for the vagueness. The first E is the covariance for the two strategies while the second one is the covariance of the asset classes, instruments traded by each strategy. (assume they are the same universe)...in terms of coherence, I am assuming they as intuitively the covariance of the two strategies are made up of exposure to the correlation and variances of the asset classes held at each rebalance –  user1234440 Jun 11 '13 at 13:37
Yes, you can definitely use the classic $\sigma^2=w' \Sigma w$ formula to compute the volatility.
In fact, you first compute your consolidated portfolio $w_p$ from your two strategies portfolios $w_A$ and $w_B$ by giving them weights weights $\alpha_A$ and $\alpha_B$ as follows:
$$w_p = \alpha_A \cdot w_A + \alpha_B \cdot w_b$$
Then the variance is calculated for $w_p$ as it would for any portfolio.