Investment Performance Measures¶
1. Sharpe's Ratio¶
Sharpe's ratio is a tool used to help investors understand the return of an investment compared to its risk. The idea is to find out how much excess return you are receiving for the extra volatility that you endure for holding a riskier asset.
Formula¶
The Sharpe ratio is calculated by subtracting the risk-free rate of return from the mean rate of return of the asset, and then dividing the result by the standard deviation of the asset's returns:
\(Sharpe\ Ratio (S_i) = \frac{\bar{R_i} - R_f}{\sigma_i}\)
Where: - \( R_i \) = Mean rate of return of the asset - \( R_f \) = Risk-free rate of return - \( sigma_i \) = Standard deviation of the asset’s returns (risk)
Interpretation¶
A higher Sharpe ratio is preferable as it indicates that the investment is providing a better return for its level of risk. Essentially, it measures the performance of an investment adjusted for its risk.
2. Jensen's Alpha¶
Jensen's Alpha is a performance measure that represents the average return on a portfolio over and above that predicted by the Capital Asset Pricing Model (CAPM), given the portfolio's beta and the average market return. This alpha is a gauge of a manager's ability to generate excess returns.
Formula¶
Jensen's Alpha is calculated as follows:
\(Jensen's\ Alpha = \bar{R_i} - (R_f + \beta_i (R_m - R_f))\)
Where: - \(R_i \) = Actual portfolio return - \( R_f \) = Risk-free rate - \( beta_i \) = Portfolio’s beta - \( R_m \) = Average market return
Interpretation¶
A positive Jensen's Alpha indicates a manager's performance has added value to a fund on a risk-adjusted basis. Conversely, a negative alpha suggests underperformance.
3. Treynor's Ratio¶
Treynor's Ratio is similar to the Sharpe ratio, but instead of using total risk (standard deviation), it uses beta, which represents the risk from exposure to general market movements as opposed to idiosyncratic factors.
Formula¶
Treynor's Ratio is calculated by dividing the excess return over the risk-free rate by the portfolio's beta:
\(Treynor's\ Ratio = \frac{\bar{R_i} - R_f}{\beta_i}\)
Where: - \( R_i \) = Mean rate of return of the portfolio - \( R_f \) = Risk-free rate - \( beta_i \) = Beta of the portfolio
Interpretation¶
A higher Treynor's Ratio indicates a more favorable risk-adjusted return, particularly relevant when comparing portfolios with similar market exposure. It is especially useful for evaluating the performance of portfolios that are diversified across different types of assets.
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