Calculate total risk [closed]

Question

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 1

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:

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

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

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

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Year 2:

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

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

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

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Year 1+2:

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

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

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