Annualizing Sharpe, Information, Sortino, and Treynor Ratios
Summary
The document explains how annualization differs across common performance ratios calculated from monthly returns. Under an independent and identically distributed monthly-return assumption, Sharpe and Information ratios scale by the square root of twelve: their return numerator grows with the number of months, while standard deviation grows with its square root. The Sortino ratio is treated similarly because its downside deviation is annualized by the square root of twelve.
The Treynor ratio uses average excess return divided by market beta. Since beta is unchanged by expressing returns annually, the response scales this ratio by twelve. These rules follow from the described ratio definitions and assumptions, rather than from a sample performance calculation. The key limitation is the IID assumption behind the volatility scaling; serial correlation or changing return distributions can make simple annualization inaccurate. The answer does not give adjustments for those cases.
Key ideas
- Sharpe and Information ratios scale by the square root of twelve under IID monthly returns.
- Sortino annualization uses the square-root scaling of downside deviation.
- Treynor scales by twelve because its numerator is annualized while beta remains unchanged.
- Serial dependence or changing return distributions can undermine these simple scaling rules.
Tags
Full text
# Annualising Data
# Annualising Data
I have a 3 year performance track record of monthly returns. I am trying to calculate the Sortino Ratio, Information Ratio, Treynor index etc.
In calculating the Sharpe Ratio I have multiplied the formula by (sqrt12) to annualise. Does this apply to the other performance measures, ie. multiply by (sqrt12) to annualise it?
Thanks.
## Answer by Tim Wilding (score 3)
https://quant.stackexchange.com/a/41006
It depends on the ratio you are looking at. Most of them are scaled by $\sqrt{12}$, but the Treynor index is a bit different and is scaled by $12$.
Sharpe and Information ratios are both ratios of average returns to standard deviations. They are annualized by assuming that the monthly returns are IID. Hence, average monthly return is scaled up by 12 and standard deviation is scaled up by $\sqrt{12}$. Hence, the final scale factor is $\sqrt{12}$. A full description is available at How to annualize Sharpe Ratio?
The Sortino ratio is the ratio of average excess return to downside standard deviation. So, annualizing the Sortino ratio involves annualizing the downside standard deviation, but this is very similar to annualizing the standard deviation. The annual downside standard deviation is $\sqrt{12} \times$ the monthly value, and this again leaves us to scale up the monthly Sortino ratio by $\sqrt{12}$.
The Treynor ratio is the ratio of the average excess return to the market beta. The market beta does not change when considering annual returns. Hence, the Treynor ratio is scaled up by 12 because the average return is the only element that needs to be annualized.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.