Obsérvese que el problema no da una inversión sin riesgo, por lo que el cálculo del ratio de Sharpe se convierte en
$$SR = \frac{E(r)}{\sqrt{VAR(r)}}$$
Año 1:
$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{4}(-2 + 6 - 2 + 6) = \frac{1}{4}(8) = 2$
$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{4}((-4)^{2} + 4^{2} + (-4)^{2} + 4^{2})} = \sqrt{\frac{1}{4}(16 + 16 + 16 + 16)} = \sqrt{\frac{1}{4}(64)} = \sqrt{16} = 4$
$SR = \frac{2}{4} = 0.5$
Segundo año:
$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{4}(-6 + 18 - 6 + 18) = \frac{1}{4}(24) = 6$
$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{4}((-12)^{2} + 12^{2} + (-12)^{2} + 12^{2})} = \sqrt{\frac{1}{4}(144 + 144 + 144 + 144)} = \sqrt{\frac{1}{4}(576)} = \sqrt{144} = 12$
$SR = \frac{6}{12} = 0.5$
Año 1+2:
$r_{p} = E(r) = \frac{1}{n}\sum_{i = 1}^{n}{r_{i}} = \frac{1}{8}(-2 + 6 - 2 + 6 - 6 + 18 - 6 + 18) = \frac{1}{8}(32) = 4$
$\sigma(r_{p}) = \sqrt{VAR(r)} = \sqrt{\frac{1}{n}\sum_{i = 1}^{n}{(r_{i} - r_{p})^{2}}} = \sqrt{\frac{1}{2}((-6)^{2} + (-2)^{2} + (-6)^{2} + (-2)^{2} + (-10)^{2} + 14^{2} + (-10)^{2} + 14^{2})} = \sqrt{\frac{1}{8}(36 + 4 + 36 + 4 + 100 + 196 + 100 + 196)} = \sqrt{\frac{1}{8}(672)} = \sqrt{84} = 9.165$
$SR = \frac{4}{9.165} = 0.436$