# Evaluation of portfolio other than Sharpe Ratio

Is Sharpe ratio always the best way to evaluate a portfolio?

I'm not really sure what this potential interview question wants me to answer. I have read that Sharpe ratio essentially explains how much the return on our asset will change with a change in its volatility (i.e. becoming more or less certain about the price at maturity) and is defined as $S(X) = \frac{r_X-R_f}{\sigma}$ where $r_X$ is average return, $R_f$ is risk free (volatility=0) return and $\sigma$ is the standard deviation (measure of volatility).

The only thing I can think about saying is that $\sigma$ isn't the only measure of volatility. There is also e.g. $\beta$ which measures volatility relative to overall market (in contrast to $\sigma$ which just compares with previous performance of $X$ and nothing else). Therefore there might be a better way that uses $\beta$ instead of $\sigma$.

Is there an alternative to Sharpe ratio? Why is it better? How should one approach a question like this?

Thanks.

• The main problem with the Sharpe ratio is that it can give misleading results when the portfolio includes options (which can alter the prob dist of returns away from the normal dist). – Alex C Feb 13 '16 at 21:34
• @AlexC Thanks. How can we see that explicitly? – user11128 Feb 13 '16 at 21:35
• Skim an article by Goetzmann called Sharpening Sharpe Ratios. – Alex C Feb 13 '16 at 21:36
• The ratio that uses $\beta$ instead of $\sigma$ is called the Treynor Ratio, but it is not much used and does not address the non-normality issue. – Alex C Feb 14 '16 at 4:20

Go through the research paper by Tripathi & Bhandari(2015). In this paper, authors compared the performance of various funds using various risk adjusted measure like Sharpe Ratio, Treynor ratio, Jensen's Alpha, and information ratio. Authors have carefully examined the limitation of each and every ratio and also suggested for an alternatives measures.

Beside this you can use other measures like VaR, expected shortfall etc.

For a more theoretical view, you could also check out the Gibbons, Ross and Shanken paper (1989). In short, sharpe ratio is a measure of absolute performance. Choosing the portfolio with the maximum sharpe ratio is equivalent to maximizing end of period wealth.

In contrast, Jensen's alpha is a measure of relative performance. It measures the marginal contribution of an asset if this asset is added to a portfolio.

For example, consider a classical mean-variance diagram with the usual tangency line. The slope of the tangency line determines the maximum achievable sharpe ratio. Now suppose you add an additional asset. That will increase the slope of the tangency line. GRS point out, besides other stuff, that the relationship between Jensen's alpha and the sharpe ratio is:

SR_old^2 + A^2 = SR_new^2


where A is called the appraisal ratio and is defined as (alpha/sigma_e)^2. Alpha is the intercept from a time-series regression of excess returns on the market, sigma_e is the standard deviation of the residuals from that regression.

I don't think this is a proper answer but I cannot comment (no rep), so I just wanted to give you that reference. Maybe it helps.

The Sharpe ratio is only useful to the extent that variance is an appropriate measure of risk. However, even if that condition is fulfilled, the Sharpe ratio remains hard to interpret.

Additionally, portfolios that realized a negative return in the past period have a negative Sharpe ratio. Possibly even inflated (when the standard deviation is very low). So even when the portfolio in itself may have a good risk-return profile, it may not look such a good portfolio based on the Sharpe ratio.

One alternative measure, developed specifically to tackle this issue of interpretation, is the M2-measure of Modigliani & Modigliani. It is closely related to the Sharpe ratio but offers the advantage being able to rank the portfolios relatively based on this metric.

The only thing I can think about saying is that σ isn't the only measure of volatility.

Guessing you meant "...isn't the only measure of risk." not volatility. You're correct, and to your point there are a host of alternatives all in service of establishing risk-adjusted return.

There is also e.g. β which measures volatility relative to overall market (in contrast to σ which just compares with previous performance of X and nothing else). Therefore there might be a better way that uses β instead of σ.

Dividing by beta is another option called the Treynor ratio. Sortino is also used (return divided by semi- or downside deviation), where semi-deviation is SD calculated over only down moves. We also have the Sterling ratio which is calculated by taking return and dividing by max drawdown or average drawdown.

Alternatively, I've also seen a measure called omega used to deal with the absence of skew and kurtosis from most or all of the previous. It's calculated by taking the sum of up moves by the magnitude of the sum of down moves. Percentage win/loss as well as average win/loss are also used.

In short, there isn't a single stat that's going to give you all the information you want or need to make an investment decision. Commonly a combination of any or all of these are used, while also considering things like capacity, trade size/days-to-trade, etc.