### Sharpe Ratios Made Easy

In today’s near-zero interest rate economy, the reward versus risk of an investment portfolio can be measured using the ** Sharpe ratio**. Like a batting average, higher numbers are better, and 0.400 is very good.

If portfolio Z has a forward-looking Sharpe ratio of 0.400, and an expected return of 8%, there is a 68% chance its 1-year return will be between -12% and +28%.

The math is surprisingly easy. Because the Sharpe ratio is a return/risk ratio it can be transformed into a risk/return ratio by finding its inverse (using the “1/x” button on a calculator). The inverse of 0.400 is 2.5. The return is 8%, so the “risk” is 2.5 times 8% which is 20%.

For the Sharpe ratio, the downside risk and the upside “risk” are the same. So the downside is 8% -20%, or -12%. The upside risk is 8%+20%, or 28%. Easy!

### Sharpe Ratios and Risk (more detail)

Where did the “68% chance” come from? The answer is a bit more complicated, but still fairly easy to understand.

It comes from the 3-sigma1 rule of statistics. The range of -12% to +28% comes from 1 standard deviations of the mean (or plus or minus one sigma). The 3-sigma rule also says that 95% of outcomes will fall within two standard deviations. Double the deviation means two times the upside and downside risk, so the 95% confidence range becomes -32% to 48%. Finally the 3-sigma rule means triple the upside and downside risk, meaning outcomes from -52% to +68% will occur 99.7 percent of the time.

Almost every investor will be be pleased with a positive sigma event, where the return is above 8%. For example a +1 sigma (+1σ) occurrence has a +28% return — quite nice.

A downside event is potentially quite troublesome. Even a -1σ event means a 12% loss. A -2σ is a much worse 32% loss.

### Ex Ante and Ex Post Sharpe Ratios

Forward-looking (ex ante) Sharpe ratios are *predictions* “prior to the event(s)”. They are always positive, because no rational investor would invest in a negative expected return. The assumptions baked into an ex ante Sharpe ratio predictions are 1) expected standard deviation of total return, σ, 2) expected future return.

Backward-looking, or after the fact, (ex post) Sharpe ratios can be negative or positive. In fact, assuming “normal distributions of return”, there is a reasonable (but less than 50%) chance of a negative ex post Sharpe ratio.

Sigma1 HAL0 software optimizes for Sharpe ratios by optimizing for return and standard deviation. It also optimizes for semivariance. More “plain English” on that advantage later.