Risk-adjusted metrics

Financial Analytics in Google Sheets

David Ardia

Professor in Quantitative Methods for Finance

Sharpe ratio

$$Sharpe\,\,ratio = \frac{m_G - r_f}{sd}$$

Effective rate of return: $m_G$

Risk-free rate: $r_f$

Volatility of the returns: $sd$

$$Sharpe\,\,ratio = \frac{m_G - r_f}{sd}$$

Effective rate of return: $m_G$

Risk-free rate: $r_f$ → Interest rate of the US Treasury Bill

Volatility of the returns: $sd$

$R_1=50\%,R_2=-20\%,R_3=5\%,R_4=-3\%$

$$Sharpe\,\,ratio = \frac{m_G - r_f}{sd} = \frac{5\% - 1\%}{30\%} = 0.13$$

The semideviation is a measurement of dispersion of the returns which are below the average return:

$$smd=\sqrt{\frac{(R_1^\star-m_A)^2+(R_2^\star-m_A)^2+\cdots+(R_L^\star-m_A)^2}{L}}$$

where $R_1^\star, R_2^\star,\ldots,R_L^\star$ are the $L$ historical returns which are below $m_A$

$$Sortino\,\,ratio = \frac{m_G - r_f}{ smd}$$

$R_1=50\%,R_2=-20\%,R_3=5\%,R_4=-3\%$

$$smd=\sqrt{\frac{(-20\%-8\%)^2+(5\%-8\%)^2+(-3\%-8\%)^2}{3}}=17\%$$

$R_1=50\%,R_2=-20\%,R_3=5\%,R_4=-3\%$

$$Sortino\,\,ratio = \frac{m_G - r_f}{smd} = \frac{5\% - 1\%}{17\%} = 0.23 $$

Financial Analytics in Google Sheets