Performance Attribution from Bacon

Abstract

This vignette gives a brief overview of the functions developed in Bacon(2008) to evaluate the performance and risk of portfolios that are included in PerformanceAnalytics and how to use them. There are some tables at the end which give a quick overview of similar functions. The page number next to each function is the location of the function in Bacon (2008)

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Risk Measure

Mean absolute deviation (p.62)

To calculate Mean absolute deviation we take the sum of the absolute value of the difference between the returns and the mean of the returns and we divide it by the number of returns.

${MeanAbsoluteDeviation = \frac{\sum^{n}_{i=1}\mid r_i - \overline{r}\mid}{n}}$

where n is the number of observations of the entire series, $r_i$ is the return in month $i$ and $\overline{r}$ is the mean return

data(portfolio_bacon)
print(MeanAbsoluteDeviation(portfolio_bacon[,1])) #expected 0.0310

## [1] 0.03108333

Frequency (p.64)

Gives the period of the return distribution (i.e. 12 if monthly return, 4 if quarterly return)

data(portfolio_bacon)
print(Frequency(portfolio_bacon[,1])) #expected 12

## [1] 12

Sharpe Ratio (p.64)

The Sharpe ratio is simply the return per unit of risk (represented by variability). In the classic case, the unit of risk is the standard deviation of the returns.

$\frac{\overline{(R_{a}-R_{f})}}{\sqrt{\sigma_{(R_{a}-R_{f})}}}$

data(managers)
SharpeRatio(managers[,1,drop=FALSE], Rf=.035/12, FUN="StdDev")

##                                      HAM1
## StdDev Sharpe (Rf=0.3%, p=95%): 0.3201889

Risk-adjusted return: MSquared (p.67)

$M^2$ is a risk adjusted return useful to judge the size of relative performance between different portfolios. With it you can compare portfolios with different levels of risk.

$M^2 = r_P + SR * (\sigma_M - \sigma_P) = (r_P - r_F) * \frac{\sigma_M}{\sigma_P} + r_F$

where $r_P$ is the portfolio return annualized, $\sigma_M$ is the market risk and $\sigma_P$ is the portfolio risk

data(portfolio_bacon)
print(MSquared(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 0.1068

##                      benchmark.return....
## benchmark.return....              0.10062

MSquared Excess (p.68)

excess is the quantity above the standard M. There is a geometric excess return which is better for Bacon and an arithmetic excess return

$M^2 excess (geometric) = \frac{1 + M^2}{1 + b} - 1$ $M^2 excess (arithmetic) = M^2 - b$

where $M^2$ is MSquared and $b$ is the benchmark annualized return (normally denoted as $r_a$ in most other texts).

data(portfolio_bacon)
print(MSquaredExcess(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.00998

##                      benchmark.return....
## benchmark.return....          -0.01553103

print(MSquaredExcess(portfolio_bacon[,1], portfolio_bacon[,2], Method="arithmetic")) #expected -0.011

##                      benchmark.return....
## benchmark.return....          -0.01736344

Regression analysis

Regression equation (p.71)

$r_P = \alpha + \beta * b + \epsilon$ ### Regression alpha (p.71)

“Alpha” purports to be a measure of a manager’s skill by measuring the portion of the managers returns that are not attributable to “Beta”, or the portion of performance attributable to a benchmark.

data(managers)
print(CAPM.alpha(managers[,1,drop=FALSE], managers[,8,drop=FALSE], Rf=.035/12))

## [1] 0.005960609

Regression beta (p.71)

CAPM Beta is the beta of an asset to the variance and covariance of an initial portfolio. Used to determine diversification potential.

data(managers)
CAPM.beta(managers[, "HAM2", drop=FALSE], managers[, "SP500 TR", drop=FALSE], Rf = managers[, "US 3m TR", drop=FALSE])

## [1] 0.3383942

Regression epsilon (p.71)

The regression epsilon is an error term measuring the vertical distance between the return predicted by the equation and the real result.

$\epsilon_r = r_p - \alpha_r - \beta_r * b$

where is $\alpha_r$ the regression alpha, $\beta_r$ is the regression beta, $r_p$ is the portfolio return and $b$ is the benchmark return.

data(managers)
print(CAPM.epsilon(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.013

## [1] -0.01313932

Jensen’s alpha (p.72)

The Jensen’s alpha is the intercept of the regression equation in the Capital Asset Pricing Model and is in effect the excess return adjusted for systematic risk.

$\alpha = r_p - r_f - \beta_p * (b - r_f)$

where $r_f$ is the risk free rate, $\beta_r$ is the regression beta, $r_p$ is the portfolio return and $b$ is the benchmark return

data(portfolio_bacon)
print(CAPM.jensenAlpha(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.014

## [1] -0.01416944

Systematic Risk (p.75)

Systematic risk as defined by Bacon(2008) is the product of beta by market risk. Be careful ! It’s not the same definition as the one given by Michael Jensen. Market risk is the standard deviation of the benchmark. The systematic risk is annualized

$\sigma_s = \beta * \sigma_m$

where $\sigma_s$ is the systematic risk, $\beta$ is the regression beta, and $\sigma_m$ is the market risk

data(portfolio_bacon)
print(SystematicRisk(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 0.013

## [1] 0.132806

Specific Risk (p.75)

Specific risk is the standard deviation of the error term in the regression equation.

data(portfolio_bacon)
print(SpecificRisk(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 0.0329

## [1] 0.03293109

Total Risk (p.75)

The square of total risk is the sum of the square of systematic risk and the square of specific risk. Specific risk is the standard deviation of the error term in the regression equation. Both terms are annualized to calculate total risk.

$Total Risk = \sqrt{Systematic Risk^2 + Specific Risk^2}$

data(portfolio_bacon)
print(TotalRisk(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 0.0134

## [1] 0.136828

Treynor ratio (p.75)

The Treynor ratio is similar to the Sharpe Ratio, except it uses beta as the volatility measure (to divide the investment’s excess return over the beta).

$TreynorRatio = \frac{\overline{(R_{a}-R_{f})}}{\beta_{a,b}}$

data(managers)
print(round(TreynorRatio(managers[,1,drop=FALSE], managers[,8,drop=FALSE], Rf=.035/12),4))

## [1] 0.2528

Modified Treynor ratio (p.77)

To calculate modified Treynor ratio, we divide the numerator by the systematic risk instead of the beta.

data(portfolio_bacon) 
print(TreynorRatio(portfolio_bacon[,1], portfolio_bacon[,2], modified = TRUE)) #expected 1.677

## [1] 0.7806747

Appraisal ratio (or Treynor-Black ratio) (p.77)

Appraisal ratio is the Jensen’s alpha adjusted for specific risk. The numerator is divided by specific risk instead of total risk.

$Appraisal ratio = \frac{\alpha}{\sigma_{\epsilon}}$

where $\alpha$ is the Jensen’s alpha, $\sigma_{\epsilon}$ is the specific risk.

data(portfolio_bacon)
print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="appraisal")) #expected -0.430

## [1] -0.4302756

Modified Jensen (p.77)

Modified Jensen’s alpha is Jensen’s alpha divided by beta.

$Modified Jensen's alpha = \frac{\alpha}{\beta}$

where $\alpha$ is the Jensen’s alpha and $\beta$ is the regression beta

data(portfolio_bacon)
print(AppraisalRatio(portfolio_bacon[,1], portfolio_bacon[,2], method="modified"))

## [1] -0.01418576

Fama decomposition (p.77)

Fama beta is a beta used to calculate the loss of diversification. It is made so that the systematic risk is equivalent to the total portfolio risk.

$\beta_F = \frac{\sigma_P}{\sigma_M}$

where $\sigma_P$ is the portfolio standard deviation and $\sigma_M$ is the market risk

data(portfolio_bacon)
print(FamaBeta(portfolio_bacon[,1], portfolio_bacon[,2])) #expected 1.03

##                              portfolio.monthly.return....
## portfolio.monthly.return....                     1.030395

Selectivity (p.78)

Selectivity is the same as Jensen’s alpha

$Selectivity = r_p - r_f - \beta_p * (b - r_f)$

where $r_f$ is the risk free rate, $\beta_r$ is the regression beta, $r_p$ is the portfolio return and $b$ is the benchmark return

data(portfolio_bacon)
print(Selectivity(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.0141

## [1] -0.01416944

Net selectivity (p.78)

Net selectivity is the remaining selectivity after deducting the amount of return require to justify not being fully diversified

$Net selectivity = \alpha - d$

where $\alpha$ is the selectivity and $d$ is the diversification

If net selectivity is negative the portfolio manager has not justified the loss of diversification

data(portfolio_bacon)
print(NetSelectivity(portfolio_bacon[,1], portfolio_bacon[,2])) #expected -0.017

##                              portfolio.monthly.return....
## portfolio.monthly.return....                   -0.0178912

Relative Risk

Tracking error (p.78)

A measure of the unexplained portion of performance relative to a benchmark.

Tracking error is calculated by taking the square root of the average of the squared deviations between the investment’s returns and the benchmark’s returns, then multiplying the result by the square root of the scale of the returns.

$TrackingError = \sqrt{\sum\frac{(R_{a}-R_{b})^{2}}{len(R_{a})\sqrt{scale}}}$

where $R_{a}$ is the investment’s return, $R_{b}$ is the benchmark’s return and $scale$ is the number of observations of the entire series

data(managers)
TrackingError(managers[,1,drop=FALSE], managers[,8,drop=FALSE])

## [1] 0.1131667

Information ratio (p.80)

The Active Premium divided by the Tracking Error.

$InformationRatio = \frac{ActivePremium}{TrackingError}$

This relates the degree to which an investment has beaten the benchmark to the consistency with which the investment has beaten the benchmark.

data(managers)
InformationRatio(managers[,"HAM1",drop=FALSE], managers[, "SP500 TR", drop=FALSE])

## [1] 0.3604125

Return Distribution

Skewness (p.83)

measures the deformation from a normal deformation

$Skewness = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^3$

where . is the number of return, . is the mean of the return distribution, . is its standard deviation and is its sample standard deviation

data(managers)
skewness(managers)

##                HAM1    HAM2      HAM3       HAM4       HAM5       HAM6
## Skewness -0.6588445 1.45804 0.7908285 -0.4310631 0.07380869 -0.2799993
##          EDHEC LS EQ   SP500 TR  US 10Y TR  US 3m TR
## Skewness  0.01773013 -0.5531032 -0.4048722 -0.328171