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How good is your portfolio? The Sharpe Ratio

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When you want to examine the performance of an investment, one of the most common used measures is the Sharpe Ratio, developed by Nobel Laureate William Sharpe. It measures the excess return (or risk premium) per unit of deviation in an investment asset or a trading strategy. For a monthly calculated Sharpe Ratio (as Morningstar uses) the formula is as follows:

Sharpe ratio_M

The numerator is the average of the monthly – the M stands for monthly – excess return (the difference between the asset return and the “risk-free rate” return), while the denominator is the standard deviation of these excess returns.

The higher the difference between the returns of your selected asset and an adequate risk-free rate (the Treasury bill if the asset is located in the US, for example), the better.

Numerador_Sharpe

Where

legenda_numerador Sharpe

The standard deviation of the excess returns is then given by

Denominador_Sharpe

Quick note: If you don’t know or don’t remember why it is used the unbiased standard deviation (why it is divided by n-1), or want to review the main concepts of mean and standard deviation, I suggest a quick look at one of the many great websites that teach this for free, like the Khan Academy (take it from here).

Hence

Sharpe ratio_M2

Normally it is presented the annualized Sharpe Ratio, which is obtained by multiplying the numerator by 12 and the denominator by the square root of 12, resulting in

Sharpe ratio_M3

The Sharpe ratio tells the investor how well he/she is compensated compared to a “riskless” investment, taking into account its volatility. You can easily compute this ratio as long as you have these two things: an observed series of returns and the benchmark returns. Easy to calculate and to use for comparison.

The simplicity of the formula, however, encompasses both quality and weakness. Since it uses the standard deviation, it relies on the fact that the results are normally distributed, and in many investments this is not the case. Abnormalities like kurtosisfatter tails, higher peaks, or skewness on the distribution can be problematic for the ratio interpretation, as standard deviation doesn’t have the same effectiveness when these problems exist. I suggest a look at your results before calculating the Sharpe ratio, to see if the returns are close to be normally distributed. You have several methods to check this, and computer software that can instantly do this as well. It also relies on the comparison of the returns with a “risk-free rate”, or a benchmark rate of some sort, which is always controversial since it is hard to indicate with precision a rate which is truly free of risk. Nevertheless, examples like the US T-bills could be adequate enough. Another usually referred weakness is the fact that inflation is not taken into account.

This way, it is understandable that investors rely on other indicators beside the Sharpe ratio. It is still, however, one of the most simple and practical when examining the performance of a portfolio.

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