Regression Statistics
Regression statistics provide insights into the relationship between the portfolio returns and benchmark returns. These metrics help traders understand how their portfolio is performing relative to the benchmark and assess the portfolio's risk-adjusted performance.
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Key Metrics
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Regression Alpha
The intercept of the regression equation with the vertical axis. It represents the excess return of the portfolio over the expected return predicted by the regression model.
\[
\text{Regression Alpha} \; \alpha_{R} = \bar{r} - \beta_{R} * \bar{b}
\]
Where:
- \(\bar{r}\) is the portfolio mean return.
- \(\bar{b}\) is the benchmark mean return.
- \(\beta_{R}\) is the regression beta.
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Regression Beta
The slope or gradient of the regression equation, indicating the portfolio's sensitivity to the benchmark.
\[
\text{Regression Beta} \; \beta_{R} = \frac{\text{Covariance}}{\sigma_{b}^{2}}
\]
Where:
- \(\text{Covariance}\) is the covariance between portfolio returns and benchmark returns.
- \(\sigma_{b}^{2}\) is the variance of the benchmark return.
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CAPM Beta
The beta coefficient in the Capital Asset Pricing Model (CAPM), reflecting the systematic risk of the portfolio relative to the benchmark.
\[
\text{CAPM Beta} \; \beta = \frac{\sum_{i=1}^{N} [((r_{i} - r_{F_{i}}) - (\bar{r} - \bar{r}_{F})) * ((b_{i} - r_{F_{i}}) - (\bar{b} - \bar{r}_{F}))]}{\sum_{i=1}^{N} \left( (b_{i} - r_{F_{i}}) - (\bar{b} - \bar{r}_{F}) \right)^{2}}
\]
Where:
- \(r_{i}\) is the portfolio return in period \(i\).
- \(r_{F_{i}}\) is the risk-free rate in period \(i\).
- \(\bar{r}\) is the mean portfolio return.
- \(\bar{r}_{F}\) is the mean risk-free rate.
- \(b_{i}\) is the benchmark return in period \(i\).
- \(\bar{b}\) is the mean benchmark return.
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Jensen's Alpha
The intercept of the regression equation in the CAPM, representing the excess return adjusted for systematic risk.
\[
\text{Jensen's Alpha} \; \alpha = \bar{r} - \bar{r}_{F} - \beta * (\bar{b} - \bar{r}_{F})
\]
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Annualized Jensen's Alpha
Jensen's Alpha calculated based on annualized data.
\[
\text{Annualized Jensen's Alpha} \; \tilde{\alpha} = \tilde{r} - \tilde{r}_{F} - \beta * (\tilde{b} - \tilde{r}_{F})
\]
Where:
- \(\tilde{r}\) is the annualized mean return.
- \(\tilde{r}_{F}\) is the annualized risk-free rate.
- \(\tilde{b}\) is the annualized mean benchmark return.
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Bull Beta (\(\beta^{+}\))
Coefficient Beta calculated for only positive benchmark returns, indicating the portfolio's sensitivity during positive market periods.
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Bear Beta (\(\beta^{-}\))
Coefficient Beta calculated for only negative benchmark returns, indicating the portfolio's sensitivity during negative market periods.
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Beta Timing Ratio
The ratio of Bull Beta to Bear Beta, assessing the portfolio's performance in different market conditions.
\[
\text{Beta Timing Ratio} = \frac{\beta^{+}}{\beta^{-}}
\]
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R Squared (\(R^2\))
The proportion of variance in fund returns that is related to the variance of benchmark returns, indicating the degree of diversification.
\[
R^2 = \frac{\sigma_{s}^{2}}{\sigma^{2}} = \frac{\beta_{R}^{2} * \sigma_{b}^{2}}{\sigma^{2}} = \rho_{r,b}^{2}
\]
Where:
- \(\sigma_{s} = \beta_{R} * \sigma_{b}\) is the systematic risk.
- \(\sigma\) is the total risk.
- \(\rho_{r,b}\) is the correlation between portfolio and benchmark returns.
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Treynor Ratio
Similar to the Sharpe ratio, but the denominator uses systematic risk instead of total risk.
\[
\text{Treynor Ratio} \; TR = \frac{\tilde{r} - \tilde{r}_{F}}{\beta}
\]
Where:
- \(\tilde{r}\) is the annualized portfolio return.
- \(\tilde{r}_{F}\) is the annualized risk-free rate.
- \(\beta\) is the CAPM Beta.
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Modified Treynor Ratio
A variation of the Treynor Ratio that uses annualized systematic risk.
\[
\text{Modified Treynor Ratio} \; MTR = \frac{\tilde{r} - \tilde{r}_{F}}{\tilde{\sigma}_{s}}
\]
Where:
- \(\tilde{\sigma}_{s}\) is the annualized systematic risk.
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Appraisal Ratio
Similar to the information ratio but uses Jensen’s alpha (excess return adjusted for systematic risk) in the numerator and specific risk (standard deviation of the error term) in the denominator.
\[
\text{Appraisal Ratio} = \frac{\tilde{\alpha}}{\tilde{\sigma_{e}}}
\]
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K-Ratio
A reward-to-risk measure based on the slope of cumulative portfolio returns through time.
\[
\text{K Ratio} = \frac{\beta_{c}}{\sigma_{c}}
\]
Where:
- \(\beta_{c}\) is the slope of the portfolio’s cumulative return through time.
- \(\sigma_{c}\) is the standard deviation of the resultant regression error term.
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Modified K-Ratio
The K ratio standardized over different time periods.
\[
\text{Modified K Ratio} = \frac{\beta_{c}}{\sigma_{c} * n}
\]
Where:
- \(n\) is the number of periods.
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Variable Explanations
- \(\bar{r}\): Portfolio mean return.
- \(\bar{b}\): Benchmark mean return.
- \(\beta_{R}\): Regression beta.
- \(\text{Covariance}\): Covariance between portfolio returns and benchmark returns.
- \(\sigma_{b}^{2}\): Variance of the benchmark return.
- \(r_{i}\): Portfolio return in period \(i\).
- \(r_{F_{i}}\): Risk-free rate in period \(i\).
- \(\bar{r}_{F}\): Mean risk-free rate.
- \(b_{i}\): Benchmark return in period \(i\).
- \(\tilde{r}\): Annualized mean return.
- \(\tilde{r}_{F}\): Annualized risk-free rate.
- \(\tilde{b}\): Annualized mean benchmark return.
- \(\beta\): CAPM Beta.
- \(\beta^{+}\): Bull Beta.
- \(\beta^{-}\): Bear Beta.
- \(\sigma_{s}\): Systematic risk.
- \(\sigma\): Total risk.
- \(\rho_{r,b}\): Correlation between portfolio and benchmark returns.
- \(\tilde{\sigma}_{s}\): Annualized systematic risk.
- \(\tilde{\sigma_{e}}\): Specific risk (standard deviation of the error term).
- \(\tilde{\alpha}\): Annualized Jensen's Alpha.
- \(\beta_{c}\): Slope of the portfolio’s cumulative return through time.
- \(\sigma_{c}\): Standard deviation of the resultant regression error term.
- \(n\): Number of periods.
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These regression statistics provide a comprehensive view of the relationship between the portfolio and the benchmark, helping traders to assess performance and risk in a detailed manner.