The (annualized) Sharpe ratio
1. The (annualized) Sharpe ratio
We have now seen how to compute the mean and standard deviation of the monthly returns on a risky portfolio. To interpret these numbers2. Benchmarking performance
we need a benchmark to compare with. The standard choice of benchmark3. Benchmarking performance
is the investment in a risk-free asset, such as a Treasury Bill issued by the US government. Because there is no risk, the volatility of its return is zero, and the return itself is called the risk free rate. I find it useful to consider this comparison between the risk-free asset and the risky portfolio in a scatter plot,4. Risk-return trade-off
with, on the x-axis, the volatility of the portfolio return,5. Risk-return trade-off
and, on the y-axis, the mean return.6. Risk-return trade-off
We thus have two points on the graph. The first one is the risk-free asset, for which the volatility is of course zero, and for which the expected return is called7. Risk-return trade-off
the risk-free rate.8. Risk-return trade-off
The second point on the graph is the risky portfolio for which the expected return is higher, in compensation for the risk taken.9. Risk-return trade-off
The vertical two-sided arrow indicates the difference between the mean return on the risky portfolio and the risk-free rate. This difference is called the portfolio’s excess return. It tells you how much additional return you can expect on the risky portfolio compared to the risk-free rate.10. Capital allocation line
Let's now look at the line11. Capital allocation line
connecting the two points. This is called the capital allocation line. It connects the portfolio that is fully invested in the risk-free asset with the risky portfolio. Each point in between those two portfolios is another portfolio that12. Capital allocation line
is invested in both the risk-free asset and the risky portfolio. Going from left to right, the allocation to the risky portfolio increases. When we go13. Capital allocation line
beyond the risky portfolio, the investor takes leverage. She is borrowing money to invest more than she has in the risky portfolio.14. The Sharpe ratio
The capital allocation line is also15. The Sharpe ratio
important because of its slope. As you can see, the slope equals the mean excess return of the risky portfolio, divided by the portfolio volatility. The slope is thus a measure for the risk-adjusted return of the portfolio: It shows the reward per unit of risk taken. Investors call this the portfolio Sharpe ratio and thus compute it as the excess portfolio mean return, divided by the portfolio volatility.16. The Sharpe ratio
17. The Sharpe ratio
Altogether, we have seen three statistics to describe the portfolio performance based on a sample of historical returns, namely the mean return (both arithmetic and geometric), the volatility, and the Sharpe ratio.18. Performance statistics in action
As an illustration, suppose now that we have the sample of eight monthly portfolio returns, as listed on the slide. Straightforward calculations then lead to the following numbers.19. Performance statistics in action
The arithmetic mean is 1.5%.20. Performance statistics in action
The geometric mean is 1.46%.21. Performance statistics in action
Their standard deviation is 2.7%.22. Performance statistics in action
If we assume a risk-free rate of 0.4%, then the Sharpe ratio is approximately 0.4. There is one shortcoming of the previous table. It shows the performance over one month, while in professional investment reports, the performance is often reported in terms of annualized numbers to match with the performance over a one-year investment horizon.23. Annualize monthly performance
We thus need to annualize the performance measures. For the simple average, the convention is to do this by multiplying with 12. For the geometric mean approach, we obtain the annualized return by raising the product of total returns to the power 12 divided by the number of observations. And for annualizing the volatility, the convention is to use the square root of time rule. It consists of multiplying the monthly volatility with the square root of 12.24. Performance statistics in action
If we apply this to our sample of eight monthly returns, we then obtain a table in which not only the mean and volatility have increased, but also the Sharpe ratio.25. Performance statistics in action
This is easy to understand since we annualized the mean by multiplying it with 1226. Performance statistics in action
, while we annualized the volatility by27. Performance statistics in action
multiplying it with the square root of 12.28. Performance statistics in action
The Sharpe ratio, being the ratio of both, thus increases with a factor equal to the square root of 12.29. Let's practice!
That’s it for now. Let’s go back to our sample of S&P 500 returns to put the theory into practice.Create Your Free Account
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