Portfolio Evaluation¶
- Evaluation of portfolio as whole, without examining the individual securities.
- However, for portfolio revision, you need to examine the individual securities.
- Always perform evaluation using synthetic dataset (Gaussian Copula, Cholesky decomposition) to ensure portfolio will perform well in different possible scenarios
Metrics¶
| Type | Metric | Comment | ||
|---|---|---|---|---|
| Excess | Inflation Adjusted Return | \(\dfrac{1 + R_{P_{t_1, t_2}}}{1+\pi_{t_1, t_2}} - 1\) \(\pi_{t_1, t_2} = \dfrac{\text{CPI}_{t_2} - \text{CPI}_{t_1}}{\text{CPI}_{t_1}}\) | Return corrected for inflation | Corrects for past, not necessarily indicative of future |
| Jensen \(\alpha\) | \(R_p - R_\min\) | Excess return more than required | ||
| Ratio | Sharpe | \(\dfrac{R_P - R_f}{\sigma_p}\) | Price premium per unit risk | |
| Sortino | \(\dfrac{R_P - R_f}{{\sigma_\text{semi}}_p}\) | Price premium per unit of downside risk | ||
| Treynor | \(\dfrac{R_P-R_f}{\beta_P}\) | Price premium per unit \(\beta\) | ||
| Calmar | \(\dfrac{R_p}{\text{Max Drawdown}}\) | |||
| Sterling | \(\dfrac{R_p}{\text{Max Drawdown} - 10 \%}\) | |||
| Omega | \(\dfrac{P(\text{gain}) \times \mu_\text{gain}}{P(\text{loss}) \times \mu_\text{loss}}\) \(\dfrac{R_p - R_f}{\sum \min \{ w R_{pt} - R_f, 0 \}}\) |
Drawdown¶
Percentage peak-to-trough decline during a specific time period
Measured once a new high is reached, because a minimum cannot be measured yet since the value could decrease further
Sharpe Ratio¶
Limitations¶
Non-normality leads to under-estimating the variance in sharpe ratio estimate
Selection bias of strategies results in false-positives regarding the success of a strategy
Non-Normality Adjusted Sharpe Ratio¶
\[ \hat \sigma^2 (\widehat {\text{SR}}) = \dfrac{1}{n-1} \left( 1 - \hat \gamma_3 \widehat {\text{SR}} + \dfrac{\hat \gamma_4 - 1}{4} \widehat {\text{SR}}^2 \right) \]
Deflated Sharpe Ratio¶
\[ \begin{aligned} \text{DSR} &= P(\text{SR}^* \le \widehat {\text{SR}} ) \\ &= \Phi \left( \dfrac{ \text{SR}^* - \widehat {\text{SR}} }{\hat \sigma(\widehat {\text{SR}})} \right) \end{aligned} \]
- \(\text{SR}^* =\) benchmark sharpe ratio
- \(\widehat{\text{SR}} =\) estimated sharpe ratio of portfolio
- \(\phi=\) cdf of normal distribution
Probability that SR is statistically-significant, after controlling for inflationary effect of
- No of independent trials with the strategy \(k\)
- List all the returns of all strategies
-
Find the independent series
-
Data Dredging \(V \left[ \widehat{\text{SR}}_k \right]\)
- Non-normality of returns: \(\hat y_3, \hat y_4\)
- Length of time series \(T\)
Can help identify if the benefits is due to chance