# How to calculate standard deviation cone around expected returns?

I would like to evaluate the returns of an investment manager who has given me their return and volatility expectations for their fund. I would like to calculate both 1 and 2 standard deviations from what is expected. The chart below by Bridgewater Associates is what I am trying to replicate.

• @noob2 Would you convert the comments to an answer so that I can properly upvote =P – Helin Sep 6 '19 at 1:22

## 1 Answer

The returns (or rather alphas, i.e. returns relative to the benchmark) plotted are logarithmic returns, not the simple returns usually reported by investment managers. This makes them additive over time.

The green line is a straight line with slope 18% (the expected annual alpha). The thin purple curve is $$\sigma \sqrt{t}$$ above and below the green line, where t is the time in years since inception. The thick purple curve is twice as far from the green line, i.e. $$\pm 2 \sigma \sqrt{t}$$ from the line. And $$\sigma$$, the annual standard deviation, is also 18%.

[The chart itself looks quite impressive, I did not know that Bridgewater had such good performance over this period. I wonder how it has done since then].

• It is impressive. I know they've struggled over the past few years but with over \$100B it becomes quite difficult. – Joe Sep 6 '19 at 2:56