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Calculating Return Risk and the Sharpe Ratio from Sample Returns

Article Quant Q&A · Author: Veronica
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Summary

The worked example explains how to calculate return risk and a Sharpe ratio when no risk-free investment is specified. In that case, it uses the mean return as the numerator and the square root of the population variance of returns as the risk denominator. Variance is computed as the average squared deviation from the mean, and risk is its standard deviation.

The answer works through two separate years and then pools the returns across both periods. It reports that each individual year has the same Sharpe ratio, while the ratio for the combined observations differs, illustrating that aggregation changes both the mean and dispersion. The calculation treats all listed observations equally and divides by the number of observations; it does not explain annualization, a risk-free rate adjustment, or whether a sample-variance convention using a different denominator is appropriate. Its figures therefore illustrate the stated exercise rather than a general performance-analysis protocol.

Key ideas

  • When no risk-free investment is given, the example divides mean return by return standard deviation.
  • The standard deviation is obtained by taking the square root of the average squared deviations from the mean.
  • The answer uses population variance, dividing by the number of observations.
  • Combining periods can change the overall Sharpe ratio even when separate periods share a ratio.
  • The example does not address annualization or adjustments for a risk-free return.

Tags

StatisticsRisk management
Full text
# Calculate total risk


# Calculate total risk












I have a question regarding how the risk is calculated, if I have only the returns. I think the risk premium (rp) is just the average of the returns and the sharpe ratio is the risk premium divided by the total risk. Let me know if I am mistaken.

But how do they calculate the risk? Thanks in advance!

PS:The exercise is in the attached pictures.

## Answer by stochazesthai (score 1)

https://quant.stackexchange.com/a/24603

Notice that the problem does not give you a risk-free investment, so the computation of the Sharpe ratio becomes:

$$SR = \frac{E(r)}{\sqrt{VAR(r)}}$$

Year 1:

$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{4}(-2 + 6 - 2 + 6) = \frac{1}{4}(8) = 2$

$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{4}((-4)^{2} + 4^{2} + (-4)^{2} + 4^{2})} = \sqrt{\frac{1}{4}(16 + 16 + 16 + 16)} = \sqrt{\frac{1}{4}(64)} = \sqrt{16} = 4$

$SR = \frac{2}{4} = 0.5$

Year 2:

$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{4}(-6 + 18 - 6 + 18) = \frac{1}{4}(24) = 6$

$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{4}((-12)^{2} + 12^{2} + (-12)^{2} + 12^{2})} = \sqrt{\frac{1}{4}(144 + 144 + 144 + 144)} = \sqrt{\frac{1}{4}(576)} = \sqrt{144} = 12$

$SR = \frac{6}{12} = 0.5$

Year 1+2:

$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{8}(-2 + 6 - 2 + 6 - 6 + 18 - 6 + 18) = \frac{1}{8}(32) = 4$

$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{2}((-6)^{2} + (-2)^{2} + (-6)^{2} + (-2)^{2} + (-10)^{2} + 14^{2} + (-10)^{2} + 14^{2})} = \sqrt{\frac{1}{8}(36 + 4 + 36 + 4 + 100 + 196 + 100 + 196)} = \sqrt{\frac{1}{8}(672)} = \sqrt{84} = 9.165$

$SR = \frac{4}{9.165} = 0.436$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.