## Risk and Return

Many investors recognize the importance of risk but often do not have the tools to measure it. For example, look at the mean (average) annual returns from two strategies.

Strategy A: -10% and 30%: Mean Return: 10%
Strategy B: 5% and 15%: Mean Return: 10%

Both have the same average but that is hiding the fact that one strategy had much greater variation or risk. Clearly given the same return, any investor would rather have lower risk. A common statistical measure for this variation is called Standard Deviation, which we will refer to as Risk. Think of this as the average distance from the mean. The standard deviation of Strategy A is 28% and Strategy B is 7%, quantifying that the average risk of Strategy A is four times Strategy B.

William Sharpe, Nobel Prize winner in Economics, devised a single measure of Return versus Risk, called the Sharpe Ratio . This is calculated simply by dividing Excess Return by Risk. This measure allows you to compare any two strategies on both risk and return, so that a strategy that has twice the risk, needs twice the return to compensate. Below shows that the Sharpe ratio of Strategy B is four times A, which is 1.4 units of return for each unit of risk you take.

Strategy A: Risk: 28%, Sharpe Ratio: 0.36
Strategy B: Risk: 7%, Sharpe Ratio: 1.42

## Confidence Interval So would you put your money to work in strategy B? We definitely would not because two measurements is not enough to gain confidence these values are not just chance. To quantify this confidence we can calculate the standard error, which allows us to state the average return is not really a single number but that we are 95% confident that it falls within a certain range. Essentially we adjust our confidence in the estimates based on number of observations as well as how much they vary. The confidence levels calculated below suggest there is a lot of uncertainty in these estimates and therefore we need more data. The graph on the right shows top Sharpe Ratio strategies from a scan of 10,000+ strategy combinations for the S&P 500 index. This also demonstrates how much more fluctuation there is in measurements when dealing with a small number of trades or samples.

Strategy A: Average Return is 10% +/- 40% (with 95% confidence)
Strategy B: Average Return is 10% +/- 10% (with 95% confidence)

Continue...