# Sharpe ratio

This ratio compares the return on a portfolio to the risk on a benchmark. It is:

d/σd where d = rp - rb

where rp is the return on a portfolio,
rb is the return on a benchmark,
d is called the differential return, and,
σd is the tracking error.

This may be calculated either ex post (historical) or ex ante using forecast data. When using forecast data one uses a point prediction of returns over a single period, together with the standard deviation as a measure of the uncertainty of that prediction. When using historical data the differential return is the average over several periods.

The Sortino ratio is a variant of the Sharpe ratio which only penalises downside risk, which is more intuitively correct. The Treynor index only adjusts for non-diversifiable risk.

## Sharpe ratio vs information ratio

A very simple case of this is where the benchmark is a risk free investment, in which case the Sharpe ratio is the excess return on the portfolio divided by the standard deviation of the return on the portfolio.

This is the original Sharpe ratio, and it is usual to call this the Sharpe ratio, and call the more generalised comparison to a benchmark the information ratio. However, some authorities, including Sharpe himself, refer to them as the “original” and “generalised” Sharpe ratio.

## Uses of the Sharpe ratio/information ratio

The Sharpe ratio is interesting because it is a measure of the relationship between risk and return, a concept that is central to financial theory. It can, as explained above, be applied to both ex-ante (expected) returns (to assess an investment) and to ex-post (historical) returns (to test the relationship between risk and reward).

One useful property of the Sharpe ratio is that the Sharpe ratio of a portfolio does not depend on the time over which it is measured. It will change with time period depending on the actual historic data, but there is no correlation between the Sharpe ratio and the length of time period. This is because the return and the standard deviation both increase with time. Sharpe ratios calculated over different periods of time are directly comparable.