Building Return Cones from Expected Alpha and Volatility

Summary

The document describes how to plot a range of cumulative outcomes around an investment manager’s expected return. It uses logarithmic returns, framed as alpha relative to a benchmark, so returns add over time. The expected cumulative alpha is drawn as a straight line whose slope is the annual expected alpha. Bands around that line widen with the square root of elapsed time, using annual volatility for a one-standard-deviation range and twice that distance for a two-standard-deviation range.

This is a concise construction for visualizing uncertainty around a stated return and volatility forecast. The example sets expected annual alpha and annual standard deviation to the same stated value, but it does not provide a historical performance test or discuss the assumptions behind the bands. In particular, the chart does not establish that future returns follow the distribution implied by the standard-deviation ranges, and its use of relative logarithmic returns differs from the simple returns commonly reported by managers.

Key ideas

• Use logarithmic returns relative to a benchmark so cumulative alpha adds over time.
• Plot expected cumulative alpha as a straight line with annual expected alpha as its slope.
• Scale the one-standard-deviation band by annual volatility times the square root of elapsed years.
• Place the two-standard-deviation band at twice the one-standard-deviation distance from the expected path.
• The construction visualizes a forecast and does not establish that future returns will fit the bands.

Tags

# How to calculate standard deviation cone around expected returns?


# 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.

## Answer by nbbo2 (score 2, accepted)

https://quant.stackexchange.com/a/47521

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].

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.