The Appraisal Ratio and Sortino Ratio

You don't get to talk about performance evaluation without mentioning the Sharpe ratio.

It's an industry standard. Any meaningful measurement of returns must include a component of risk, and a good starting point is zero. The Sharpe ratio does this by measuring only returns beyond what you would get with zero risk and dividing these returns by the most common variability measure: standard deviation. $$\displaystyle \mbox{Sharpe ratio} = \frac{\mbox{Portfolio return}~ - ~\mbox{Risk-free rate}}{\mbox{Standard deviation of portfolio return}} $$ To finish the starting point, what would the Sharpe ratio be for an asset earning the risk-free rate?

No, it's 0. Notice that the returns in the numerator would be equal.

Right.

Then it obviously goes up from there. This linear measure isn't best for all investors, but it's a popular starting point.

There's also the **appraisal ratio** which is the alpha earned divided by the annualized residual risk, also known as the standard error of regression: >$$ AR = \frac{\alpha}{\sigma_{\epsilon}} $$ You can think of this as the benefit of active management divided by the risk of that active management. How is this similar to the Sharpe ratio?

Yes. It's still a measure of return over risk, so a higher measure is better, whether you're comparing to another asset, a benchmark, or whatever.

That's not it. The risk-free rate is removed from returns in the Sharpe ratio, so it does matter. It will also affect the alpha calculation here.

Not quite; that's just for the appraisal ratio where you're looking at relative risk on the bottom. The Sharpe ratio is a standalone measure.

Suppose an investment firm had a manager who outperformed the benchmark every year. Every single time. But, those returns had a lot of variance. Every one was a winner, but you could never know by how much. Does that sound like something you'd want to punish the manager for?

Absolutely. There's a researcher named Frank Sortino who agrees with you.

Probably not. At least, many would disagree with you, including a researcher named Frank Sortino.

The __Sortino ratio__ is once again return divided by risk, but both of these include a benchmark return. This benchmark is called the minimum acceptable return (MAR), and the return measure is return in excess of this. The denominator is a target semideviation calculation, or __downside deviation__, using just those returns below the MAR. What does this mean for the risk calculation?

Exactly. $$\displaystyle \mbox{Sortino ratio} = \frac{\mbox{Portfolio return}~ - ~\mbox{MAR}}{\mbox{Downside deviation}} $$ Compare this to the Sharpe ratio for a moment. If you set the MAR at the risk-free rate, and all returns were below this, then the two would be identical. But that probably won't be the case. If you set the MAR that low, then it should be easy enough to have some returns in excess of this. But the key is that the Sortino ratio wouldn't include those values as part of the risk calculation, which would be a bit odd. The MAR would be set higher, and the manager would get a "free pass" on whatever deviations were earned above that level. So is that the answer? No, there is no perfect metric. Just recognize the differences, and experience will lead you to best practices over time.

No, not necessarily. It would depend on the values.

Not quite. These observations aren't used in a target semideviation calculation, so they wouldn't reduce measured risk.

To summarize: [[summary]]

0

1

Higher is better

The risk-free rate doesn't matter

It's based on a benchmark portfolio

No

Yes

Risk related to high returns is ignored

Using more observations reduces measured risk

High returns cause the risk measure to be reduced

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