You combine two uncorrelated return streams on equal capital, one running 12% annualised volatility and the other 16%. What is the volatility of the sum, and what changes at a correlation of 0.5?

You combine two uncorrelated return streams on equal capital, one running 12% annualised volatility and the other 16%. What is the volatility of the sum, and what changes at a correlation of 0.5?

Approach: Add the variances rather than the volatilities, take the square root, then reintroduce the covariance term for the correlated case.

20%. Variance adds for uncorrelated streams, so the combined variance is 12^2 + 16^2 = 144 + 256 = 400 and the volatility is sqrt(400), which is 20%. Volatility itself does not add, since 12 + 16 = 28 overstates the risk by 40%, because the two streams offset each other part of the time. This is a Pythagorean sum, so the right triples are worth memorising: 3-4-5 covers 12 and 16, and 5-12-13 and 8-15-17 cover most interview numbers. At a correlation of 0.5 the cross term returns and the variance becomes 144 + 256 + 2 * 0.5 * 12 * 16 = 592, so the volatility is 24.3%, which is 21% above the uncorrelated case and still under the naive 28.

Follow-up: What correlation would make the combined volatility exactly 16%, and what does that imply about the second stream as a hedge?

Key concepts: variance adds, correlation, Pythagorean sum.