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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 numbers

2. Benchmarking performance

we need a benchmark to compare with. The standard choice of benchmark

3. 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 called

7. 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 line

11. 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 that

12. 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 go

13. 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 also

15. 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 12

26. Performance statistics in action

, while we annualized the volatility by

27. 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.

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