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Performance Attribution from Bacon
In PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis
::: article
``` {r include=FALSE}
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library(PerformanceAnalytics)
## 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
``` {r }
data(portfolio_bacon)
print(MeanAbsoluteDeviation(portfolio_bacon[,1])) #expected 0.0310
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
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")
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
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
print(MSquaredExcess(portfolio_bacon[,1], portfolio_bacon[,2], Method="arithmetic")) #expected -0.011
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))
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])
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
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
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
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
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
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))
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
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
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"))
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
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
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
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])
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])
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)
Sample skewness (p.84)
$$Sample Skewness = \frac{n}{(n-1)(n-2)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^3 $$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return
distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample
standard deviation
data(portfolio_bacon)
print(skewness(portfolio_bacon[,1], method="sample")) #expected -0.09
Kurtosis (p.84)
Kurtosis measures the weight or returns in the tails of the distribution
relative to standard deviation.
$$Kurtosis(moment) = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return
distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard
deviation
data(portfolio_bacon)
print(kurtosis(portfolio_bacon[,1], method="moment")) #expected 2.43
Excess kurtosis (p.85)
$$Excess Kurtosis = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4 - 3$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return
distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample
standard deviation
data(portfolio_bacon)
print(kurtosis(portfolio_bacon[,1], method="excess")) #expected -0.57
Sample kurtosis (p.85)
$$Sample kurtosis = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^4$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return
distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample
standard deviation
data(portfolio_bacon)
print(kurtosis(portfolio_bacon[,1], method="sample")) #expected 3.03
Sample excess kurtosis (p.85)
$$Sample excess kurtosis = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^4 - \frac{3(n-1)^2}{(n-2)(n-3)}$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return
distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample
standard deviation
data(portfolio_bacon)
print(kurtosis(portfolio_bacon[,1], method="sample_excess")) #expected -0.41
Drawdown
Pain index (p.89)
The pain index is the mean value of the drawdowns over the entire
analysis period. The measure is similar to the Ulcer index except that
the drawdowns are not squared. Also, it's different than the average
drawdown, in that the numerator is the total number of observations
rather than the number of drawdowns. Visually, the pain index is the
area of the region that is enclosed by the horizontal line at zero
percent and the drawdown line in the Drawdown chart.
$$Pain index = \sum^{n}_{i=1} \frac{\mid D'_i \mid}{n}$$
where $n$ is the number of observations of the entire series, $D'_i$ is the
drawdown since previous peak in period $i$
data(portfolio_bacon)
print(PainIndex(portfolio_bacon[,1])) #expected 0.04
Calmar ratio (p.89)
Calmar ratio is another method of creating a risk-adjusted measure for
ranking investments similar to the Sharpe ratio.
data(managers)
CalmarRatio(managers[,1,drop=FALSE])
Sterling ratio (p.89)
Sterling ratio is another method of creating a risk-adjusted measure for
ranking investments similar to the Sharpe ratio.
data(managers)
SterlingRatio(managers[,1,drop=FALSE])
Burke ratio (p.90)
To calculate Burke ratio we take the difference between the portfolio
return and the risk free rate and we divide it by the square root of the
sum of the square of the drawdowns.
$$Burke Ratio = \frac{r_P - r_F}{\sqrt{\sum^{d}_{t=1}{D_t}^2}}$$
where $d$ is the number of drawdowns, $r_P$ is the portfolio return, $r_F$ is
the risk free rate and $D_t$ the $t^{th}$ drawdown.
data(portfolio_bacon)
print(BurkeRatio(portfolio_bacon[,1])) #expected 0.74
Modified Burke ratio (p.91)
To calculate the modified Burke ratio we just multiply the Burke ratio
by the square root of the number of data points.
$$Modified Burke Ratio = \frac{r_P - r_F}{\sqrt{\sum^{d}_{t=1}\frac{{D_t}^2}{n}}}$$
where $n$ is the number of observations of the entire series, $d$ is number of
drawdowns, $r_P$ is the portfolio return, $r_F$ is the risk free rate and $D_t$
the $t^{th}$ drawdown.
data(portfolio_bacon)
print(BurkeRatio(portfolio_bacon[,1], modified = TRUE)) #expected 3.65
Martin ratio (p.91)
To calculate Martin ratio we divide the difference of the portfolio
return and the risk free rate by the Ulcer index
$$Martin Ratio = \frac{r_P - r_F}{\sqrt{\sum^{n}_{i=1}\frac{{D'_i}^2}{n}}}$$
where $r_P$ is the portfolio return, $r_F$ is the risk free rate, $n$ is the
number of observations of the entire series, $D'_i$ is the drawdown since
previous peak in period $i$.
data(portfolio_bacon)
print(MartinRatio(portfolio_bacon[,1])) #expected 1.70
Pain ratio (p.91)
To calculate Pain ratio we divide the difference of the portfolio return
and the risk free rate by the Pain index
$$Pain ratio = \frac{r_P - r_F}{\sum^{n}_{i=1} \frac{\mid D'_i \mid}{n}}$$
where $r_P$ is the portfolio return, $r_F$ is the risk free rate, $n$ is the
number of observations of the entire series, $D'_i$ is the drawdown since
previous peak in period $i$.
data(portfolio_bacon)
print(PainRatio(portfolio_bacon[,1])) #expected 2.66
Downside risk
Downside risk (p.92)
Downside deviation, similar to semi deviation, eliminates positive
returns when calculating risk. Instead of using the mean return or zero,
it uses the Minimum Acceptable Return as proposed by Sharpe (which may
be the mean historical return or zero). It measures the variability of
underperformance below a minimum target rate. The downside variance is
the square of the downside potential.
$$DownsideDeviation(R , MAR) = \delta_{MAR} = \sqrt{\sum^{n}{t=1}\frac{min[(R{t} - MAR), 0]^2}{n}}$$
$$DownsideVariance(R, MAR) = \sum^{n}{t=1} \frac{min[(R{t} - MAR), 0]^2}{n}$$
$$DownsidePotential(R, MAR) = \sum^{n}{t=1}{\frac{min[(R{t} - MAR), 0]}{n}}$$
where $n$ is either the number of observations of the entire series or the
number of observations in the subset of the series falling below the MAR.
data(portfolio_bacon)
MAR = 0.5
DownsideDeviation(portfolio_bacon[,1], MAR) #expected 0.493
DownsidePotential(portfolio_bacon[,1], MAR) #expected 0.491
UpsideRisk (p.92)
Upside Risk is the similar to semideviation taking the return above the
Minimum Acceptable Return instead of using the mean return or zero.
$$UpsideRisk(R , MAR) = \sqrt{\sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]^2}{n}}$$
$$UpsideVariance(R, MAR) = \sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]^2}{n}$$
$$UpsidePotential(R, MAR) = \sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]}{n}$$
where $n$ is either the number of observations of the entire series or the
number of observations in the subset of the series falling above the MAR.
data(portfolio_bacon)
MAR = 0.005
print(UpsideRisk(portfolio_bacon[,1], MAR, stat="risk")) #expected 0.02937
print(UpsideRisk(portfolio_bacon[,1], MAR, stat="variance")) #expected 0.08628
print(UpsideRisk(portfolio_bacon[,1], MAR, stat="potential")) #expected 0.01771
Downside frequency (p.94)
To calculate Downside Frequency, we take the subset of returns that are
less than the target (or Minimum Acceptable Returns (MAR)) returns and
divide the length of this subset by the total number of returns.
$$DownsideFrequency(R , MAR) = \sum^{n}{t=1}\frac{min[(R{t} - MAR),0]}{R_{t}*n}$$
where $n$ is the number of observations of the entire series.
data(portfolio_bacon)
MAR = 0.005
print(DownsideFrequency(portfolio_bacon[,1], MAR)) #expected 0.458
Bernardo and Ledoit ratio (p.95)
To calculate Bernardo and Ledoit ratio we take the sum of the subset of
returns that are above 0 and we divide it by the opposite of the sum of
the subset of returns that are below 0
$$BernardoLedoitRatio(R) = \frac{\frac{1}{n}\sum^{n}{t=1}{max(R{t},0)}}{\frac{1}{n}\sum^{n}{t=1}{max(-R{t},0)}}$$
where $n$ is the number of observations of the entire series
data(portfolio_bacon)
print(BernardoLedoitRatio(portfolio_bacon[,1])) #expected 1.78
d ratio (p.95)
The d ratio is similar to the Bernardo Ledoit ratio but inverted and
taking into account the frequency of positive and negative returns.
It has values between zero and infinity. It can be used to rank the
performance of portfolios. The lower the d ratio the better the
performance, a value of zero indicating there are no returns less than
zero and a value of infinity indicating there are no returns greater
than zero.
$$DRatio(R) = \frac{n_{d}\sum^{n}{t=1}{max(-R{t},0)}}{n_{u}\sum^{n}{t=1}{max(R{t},0)}}$$
where $n$ is the number of observations of the entire series, $n_{d}$ is the
number of observations less than zero and $n_{u}$ is the number of
observations greater than zero
data(portfolio_bacon)
print(DRatio(portfolio_bacon[,1])) #expected 0.401
Omega-Sharpe ratio (p.95)
The Omega-Sharpe ratio is a conversion of the omega ratio to a ranking
statistic in familiar form to the Sharpe ratio.
To calculate the Omega-Sharpe ration we subtract the target (or Minimum
Acceptable Returns (MAR)) return from the portfolio return and we divide
it by the opposite of the Downside Deviation.
$$OmegaSharpeRatio(R,MAR) = \frac{r_p - r_t}{\sum^n_{t=1}\frac{max(r_t - r_i, 0)}{n}}$$
where $n$ is the number of observations of the entire series
data(portfolio_bacon)
MAR = 0.005
print(OmegaSharpeRatio(portfolio_bacon[,1], MAR)) #expected 0.29
Sortino ratio (p.96)
Sortino proposed an improvement on the Sharpe Ratio to better account
for skill and excess performance by using only downside semivariance as
the measure of risk.
$$SortinoRatio=\frac{(\overline{R_{a} - MAR})}{\delta_{MAR}}$$
where $\delta_{MAR}$ is the DownsideDeviation.
data(managers)
round(SortinoRatio(managers[, 1]),4)
Kappa (p.96)
Introduced by Kaplan and Knowles (2004), Kappa is a generalized downside
risk-adjusted performance measure.
To calculate it, we take the difference of the mean of the distribution
to the target and we divide it by the l-root of the lth lower partial
moment. To calculate the lth lower partial moment we take the subset of
returns below the target and we sum the differences of the target to
these returns. We then return return this sum divided by the length of
the whole distribution.
data(portfolio_bacon)
MAR = 0.005
l = 2
print(Kappa(portfolio_bacon[,1], MAR, l)) #expected 0.157
Upside potential ratio (p.97)
Sortino proposed an improvement on the Sharpe Ratio to better account
for skill and excess performance by using only downside semivariance as
the measure of risk. That measure is the Sortino ratio. This function,
Upside Potential Ratio, was a further improvement, extending the
measurement of only upside on the numerator, and only downside of the
denominator of the ratio equation.
$$UPR=\frac{ \sum^{n}{t=1} (R{t} - MAR) }{ \delta_{MAR} }$$
where $\delta_{MAR}$ is the DownsideDeviation.
data(edhec)
UpsidePotentialRatio(edhec[, 6], MAR=.05/12) #5 percent/yr MAR
Volatility skewness (p.97)
Volatility skewness is a similar measure to omega but using the second
partial moment. It's the ratio of the upside variance compared to the
downside variance.
$$VolatilitySkewness(R , MAR) = \frac{\sigma_U^2}{\sigma_D^2}$$
where $\sigma_U$ is the Upside risk and $\sigma_D$ is the Downside Risk
data(portfolio_bacon)
MAR = 0.005
print(VolatilitySkewness(portfolio_bacon[,1], MAR, stat="volatility")) #expected 1.32
Variability skewness (p.98)
Variability skewness is the ratio of the upside risk compared to the
downside risk.
$$VariabilitySkewness(R , MAR) = \frac{\sigma_U}{\sigma_D}$$
where $\sigma_U$ is the Upside risk and $\sigma_D$ is the Downside Risk
data(portfolio_bacon)
MAR = 0.005
print(VolatilitySkewness(portfolio_bacon[,1], MAR, stat="variability")) #expected 1.15
Adjusted Sharpe ratio (p.99)
Adjusted Sharpe ratio was introduced by Pezier and White (2006) to
adjusts for skewness and kurtosis by incorporating a penalty factor for
negative skewness and excess kurtosis.
$$Adjusted Sharpe Ratio = SR * [1 + (\frac{S}{6}) * SR - (\frac{K - 3}{24}) * SR^2]$$
where $SR$ is the Sharpe Ratio with data annualized,$S$ is the skewness and
$K$ is the kurtosis
data(portfolio_bacon)
print(AdjustedSharpeRatio(portfolio_bacon[,1])) #expected 0.81
Skewness-kurtosis ratio (p.99)
Skewness-Kurtosis ratio is the division of Skewness by Kurtosis.' It is
used in conjunction with the Sharpe ratio to rank portfolios. The higher
the rate the better.
$$SkewnessKurtosisRatio(r,MAR) = \frac{S}{K}$$
where $S$ is the skewness and $K$ is the Kurtosis
data(portfolio_bacon)
print(SkewnessKurtosisRatio(portfolio_bacon[,1])) #expected -0.034
Prospect ratio (p.100)
Prospect ratio is a ratio used to penalize loss since most people feel
loss greater than gain
$$ProspectRatio(R) = \frac{\frac{1}{n}\sum^{n}_{i=1}(Max(r_i,0)+2.25Min(r_i,0) - MAR)}{\sigma_D}$$
where $n$ is the number of observations of the entire series, $MAR$ is the
minimum acceptable return and $\sigma_D$ is the downside risk
data(portfolio_bacon)
MAR = 0.05
print(ProspectRatio(portfolio_bacon[,1], MAR)) #expected -0.134
Return adjusted for downside risk
M Squared for Sortino (p.102)
M squared for Sortino is a calculated for Downside risk instead of Total
Risk
$$M^2_S = r_P + Sortino ratio * (\sigma_{DM} - \sigma_D)$$
where $M^2_S$ is MSquared for Sortino, $r_P$ is the annualized portfolio return,
$\sigma_DM$ is the benchmark annualized downside risk and $D$ is the portfolio
annualized downside risk
data(portfolio_bacon)
MAR = 0.005
print(M2Sortino(portfolio_bacon[,1], portfolio_bacon[,2], MAR)) #expected 0.1035
Omega excess return (p.103)
Omega excess return is another form of downside risk-adjusted return. It
is calculated by multiplying the downside variance of the style
benchmark by 3 times the style beta.
$$\omega = r_P - 3\beta_S\sigma_{MD}^2$$
where $\omega$ is omega excess return, $\beta_S$ is style beta, $\sigma_D$ is
the portfolio annualized downside risk and $\sigma_MD$ is the benchmark annualized
downside risk.
data(portfolio_bacon)
MAR = 0.005
print(OmegaExcessReturn(portfolio_bacon[,1], portfolio_bacon[,2], MAR)) #expected 0.0805
Tables
Variability risk
Table of Mean absolute difference, Monthly standard deviation and
annualized standard deviation
data(managers)
table.Variability(managers[,1:8])
Specific risk
Table of specific risk, systematic risk and total risk
data(managers)
table.SpecificRisk(managers[,1:8], managers[,8])
Information risk
Table of Tracking error, Annualized tracking error and Information ratio
data(managers)
table.InformationRatio(managers[,1:8], managers[,8])
Distributions
Table of Monthly standard deviation, Skewness, Sample standard
deviation, Kurtosis, Excess kurtosis, Sample Skewness and Sample excess
kurtosis
data(managers)
table.Distributions(managers[,1:8])
Drawdowns
Table of Calmar ratio, Sterling ratio, Burke ratio, Pain index, Ulcer
index, Pain ratio and Martin ratio
data(managers)
table.DrawdownsRatio(managers[,1:8])
Downside risk
Table of Monthly downside risk, Annualized downside risk, Downside
potential, Omega, Sortino ratio, Upside potential, Upside potential
ratio and Omega-Sharpe ratio
data(managers)
table.DownsideRiskRatio(managers[,1:8])
Sharpe ratio
Table of Annualized Return, Annualized Std Dev, and Annualized Sharpe
data(managers)
table.AnnualizedReturns(managers[,1:8])
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::: article ``` {r include=FALSE} library(knitr) opts_chunk$set( engine='R' )
``` {r include=FALSE}
library(knitr)
opts_chunk$set(
concordance=TRUE
)
``` {r echo = FALSE} library(PerformanceAnalytics)
## 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
``` {r }
data(portfolio_bacon)
print(MeanAbsoluteDeviation(portfolio_bacon[,1])) #expected 0.0310
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
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")
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
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 print(MSquaredExcess(portfolio_bacon[,1], portfolio_bacon[,2], Method="arithmetic")) #expected -0.011
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))
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])
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
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
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
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
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
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))
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
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
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"))
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
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
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
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])
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])
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)
Sample skewness (p.84)
$$Sample Skewness = \frac{n}{(n-1)(n-2)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^3 $$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard deviation
data(portfolio_bacon) print(skewness(portfolio_bacon[,1], method="sample")) #expected -0.09
Kurtosis (p.84)
Kurtosis measures the weight or returns in the tails of the distribution relative to standard deviation.
$$Kurtosis(moment) = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard deviation
data(portfolio_bacon) print(kurtosis(portfolio_bacon[,1], method="moment")) #expected 2.43
Excess kurtosis (p.85)
$$Excess Kurtosis = \frac{1}{n}*\sum^{n}_{i=1}(\frac{r_i - \overline{r}}{\sigma_P})^4 - 3$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard deviation
data(portfolio_bacon) print(kurtosis(portfolio_bacon[,1], method="excess")) #expected -0.57
Sample kurtosis (p.85)
$$Sample kurtosis = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^4$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard deviation
data(portfolio_bacon) print(kurtosis(portfolio_bacon[,1], method="sample")) #expected 3.03
Sample excess kurtosis (p.85)
$$Sample excess kurtosis = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum^{n}{i=1}(\frac{r_i - \overline{r}}{\sigma{S_P}})^4 - \frac{3(n-1)^2}{(n-2)(n-3)}$$
where $n$ is the number of returns, $\overline{r}$ is the mean of the return distribution, $\sigma_P$ is its standard deviation and $\sigma_P$ is its sample standard deviation
data(portfolio_bacon) print(kurtosis(portfolio_bacon[,1], method="sample_excess")) #expected -0.41
Drawdown
Pain index (p.89)
The pain index is the mean value of the drawdowns over the entire analysis period. The measure is similar to the Ulcer index except that the drawdowns are not squared. Also, it's different than the average drawdown, in that the numerator is the total number of observations rather than the number of drawdowns. Visually, the pain index is the area of the region that is enclosed by the horizontal line at zero percent and the drawdown line in the Drawdown chart.
$$Pain index = \sum^{n}_{i=1} \frac{\mid D'_i \mid}{n}$$
where $n$ is the number of observations of the entire series, $D'_i$ is the drawdown since previous peak in period $i$
data(portfolio_bacon) print(PainIndex(portfolio_bacon[,1])) #expected 0.04
Calmar ratio (p.89)
Calmar ratio is another method of creating a risk-adjusted measure for ranking investments similar to the Sharpe ratio.
data(managers) CalmarRatio(managers[,1,drop=FALSE])
Sterling ratio (p.89)
Sterling ratio is another method of creating a risk-adjusted measure for ranking investments similar to the Sharpe ratio.
data(managers) SterlingRatio(managers[,1,drop=FALSE])
Burke ratio (p.90)
To calculate Burke ratio we take the difference between the portfolio return and the risk free rate and we divide it by the square root of the sum of the square of the drawdowns.
$$Burke Ratio = \frac{r_P - r_F}{\sqrt{\sum^{d}_{t=1}{D_t}^2}}$$
where $d$ is the number of drawdowns, $r_P$ is the portfolio return, $r_F$ is the risk free rate and $D_t$ the $t^{th}$ drawdown.
data(portfolio_bacon) print(BurkeRatio(portfolio_bacon[,1])) #expected 0.74
Modified Burke ratio (p.91)
To calculate the modified Burke ratio we just multiply the Burke ratio by the square root of the number of data points.
$$Modified Burke Ratio = \frac{r_P - r_F}{\sqrt{\sum^{d}_{t=1}\frac{{D_t}^2}{n}}}$$
where $n$ is the number of observations of the entire series, $d$ is number of drawdowns, $r_P$ is the portfolio return, $r_F$ is the risk free rate and $D_t$ the $t^{th}$ drawdown.
data(portfolio_bacon) print(BurkeRatio(portfolio_bacon[,1], modified = TRUE)) #expected 3.65
Martin ratio (p.91)
To calculate Martin ratio we divide the difference of the portfolio return and the risk free rate by the Ulcer index
$$Martin Ratio = \frac{r_P - r_F}{\sqrt{\sum^{n}_{i=1}\frac{{D'_i}^2}{n}}}$$
where $r_P$ is the portfolio return, $r_F$ is the risk free rate, $n$ is the number of observations of the entire series, $D'_i$ is the drawdown since previous peak in period $i$.
data(portfolio_bacon) print(MartinRatio(portfolio_bacon[,1])) #expected 1.70
Pain ratio (p.91)
To calculate Pain ratio we divide the difference of the portfolio return and the risk free rate by the Pain index
$$Pain ratio = \frac{r_P - r_F}{\sum^{n}_{i=1} \frac{\mid D'_i \mid}{n}}$$
where $r_P$ is the portfolio return, $r_F$ is the risk free rate, $n$ is the number of observations of the entire series, $D'_i$ is the drawdown since previous peak in period $i$.
data(portfolio_bacon) print(PainRatio(portfolio_bacon[,1])) #expected 2.66
Downside risk
Downside risk (p.92)
Downside deviation, similar to semi deviation, eliminates positive returns when calculating risk. Instead of using the mean return or zero, it uses the Minimum Acceptable Return as proposed by Sharpe (which may be the mean historical return or zero). It measures the variability of underperformance below a minimum target rate. The downside variance is the square of the downside potential.
$$DownsideDeviation(R , MAR) = \delta_{MAR} = \sqrt{\sum^{n}{t=1}\frac{min[(R{t} - MAR), 0]^2}{n}}$$ $$DownsideVariance(R, MAR) = \sum^{n}{t=1} \frac{min[(R{t} - MAR), 0]^2}{n}$$
$$DownsidePotential(R, MAR) = \sum^{n}{t=1}{\frac{min[(R{t} - MAR), 0]}{n}}$$
where $n$ is either the number of observations of the entire series or the number of observations in the subset of the series falling below the MAR.
data(portfolio_bacon) MAR = 0.5 DownsideDeviation(portfolio_bacon[,1], MAR) #expected 0.493 DownsidePotential(portfolio_bacon[,1], MAR) #expected 0.491
UpsideRisk (p.92)
Upside Risk is the similar to semideviation taking the return above the Minimum Acceptable Return instead of using the mean return or zero.
$$UpsideRisk(R , MAR) = \sqrt{\sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]^2}{n}}$$
$$UpsideVariance(R, MAR) = \sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]^2}{n}$$
$$UpsidePotential(R, MAR) = \sum^{n}{t=1}\frac{max[(R{t} - MAR), 0]}{n}$$
where $n$ is either the number of observations of the entire series or the number of observations in the subset of the series falling above the MAR.
data(portfolio_bacon) MAR = 0.005 print(UpsideRisk(portfolio_bacon[,1], MAR, stat="risk")) #expected 0.02937 print(UpsideRisk(portfolio_bacon[,1], MAR, stat="variance")) #expected 0.08628 print(UpsideRisk(portfolio_bacon[,1], MAR, stat="potential")) #expected 0.01771
Downside frequency (p.94)
To calculate Downside Frequency, we take the subset of returns that are less than the target (or Minimum Acceptable Returns (MAR)) returns and divide the length of this subset by the total number of returns.
$$DownsideFrequency(R , MAR) = \sum^{n}{t=1}\frac{min[(R{t} - MAR),0]}{R_{t}*n}$$
where $n$ is the number of observations of the entire series.
data(portfolio_bacon) MAR = 0.005 print(DownsideFrequency(portfolio_bacon[,1], MAR)) #expected 0.458
Bernardo and Ledoit ratio (p.95)
To calculate Bernardo and Ledoit ratio we take the sum of the subset of returns that are above 0 and we divide it by the opposite of the sum of the subset of returns that are below 0
$$BernardoLedoitRatio(R) = \frac{\frac{1}{n}\sum^{n}{t=1}{max(R{t},0)}}{\frac{1}{n}\sum^{n}{t=1}{max(-R{t},0)}}$$
where $n$ is the number of observations of the entire series
data(portfolio_bacon) print(BernardoLedoitRatio(portfolio_bacon[,1])) #expected 1.78
d ratio (p.95)
The d ratio is similar to the Bernardo Ledoit ratio but inverted and taking into account the frequency of positive and negative returns.
It has values between zero and infinity. It can be used to rank the performance of portfolios. The lower the d ratio the better the performance, a value of zero indicating there are no returns less than zero and a value of infinity indicating there are no returns greater than zero.
$$DRatio(R) = \frac{n_{d}\sum^{n}{t=1}{max(-R{t},0)}}{n_{u}\sum^{n}{t=1}{max(R{t},0)}}$$
where $n$ is the number of observations of the entire series, $n_{d}$ is the number of observations less than zero and $n_{u}$ is the number of observations greater than zero
data(portfolio_bacon) print(DRatio(portfolio_bacon[,1])) #expected 0.401
Omega-Sharpe ratio (p.95)
The Omega-Sharpe ratio is a conversion of the omega ratio to a ranking statistic in familiar form to the Sharpe ratio.
To calculate the Omega-Sharpe ration we subtract the target (or Minimum Acceptable Returns (MAR)) return from the portfolio return and we divide it by the opposite of the Downside Deviation.
$$OmegaSharpeRatio(R,MAR) = \frac{r_p - r_t}{\sum^n_{t=1}\frac{max(r_t - r_i, 0)}{n}}$$
where $n$ is the number of observations of the entire series
data(portfolio_bacon) MAR = 0.005 print(OmegaSharpeRatio(portfolio_bacon[,1], MAR)) #expected 0.29
Sortino ratio (p.96)
Sortino proposed an improvement on the Sharpe Ratio to better account for skill and excess performance by using only downside semivariance as the measure of risk.
$$SortinoRatio=\frac{(\overline{R_{a} - MAR})}{\delta_{MAR}}$$
where $\delta_{MAR}$ is the DownsideDeviation.
data(managers) round(SortinoRatio(managers[, 1]),4)
Kappa (p.96)
Introduced by Kaplan and Knowles (2004), Kappa is a generalized downside risk-adjusted performance measure.
To calculate it, we take the difference of the mean of the distribution to the target and we divide it by the l-root of the lth lower partial moment. To calculate the lth lower partial moment we take the subset of returns below the target and we sum the differences of the target to these returns. We then return return this sum divided by the length of the whole distribution.
data(portfolio_bacon) MAR = 0.005 l = 2 print(Kappa(portfolio_bacon[,1], MAR, l)) #expected 0.157
Upside potential ratio (p.97)
Sortino proposed an improvement on the Sharpe Ratio to better account for skill and excess performance by using only downside semivariance as the measure of risk. That measure is the Sortino ratio. This function, Upside Potential Ratio, was a further improvement, extending the measurement of only upside on the numerator, and only downside of the denominator of the ratio equation.
$$UPR=\frac{ \sum^{n}{t=1} (R{t} - MAR) }{ \delta_{MAR} }$$
where $\delta_{MAR}$ is the DownsideDeviation.
data(edhec) UpsidePotentialRatio(edhec[, 6], MAR=.05/12) #5 percent/yr MAR
Volatility skewness (p.97)
Volatility skewness is a similar measure to omega but using the second partial moment. It's the ratio of the upside variance compared to the downside variance.
$$VolatilitySkewness(R , MAR) = \frac{\sigma_U^2}{\sigma_D^2}$$
where $\sigma_U$ is the Upside risk and $\sigma_D$ is the Downside Risk
data(portfolio_bacon) MAR = 0.005 print(VolatilitySkewness(portfolio_bacon[,1], MAR, stat="volatility")) #expected 1.32
Variability skewness (p.98)
Variability skewness is the ratio of the upside risk compared to the downside risk.
$$VariabilitySkewness(R , MAR) = \frac{\sigma_U}{\sigma_D}$$
where $\sigma_U$ is the Upside risk and $\sigma_D$ is the Downside Risk
data(portfolio_bacon) MAR = 0.005 print(VolatilitySkewness(portfolio_bacon[,1], MAR, stat="variability")) #expected 1.15
Adjusted Sharpe ratio (p.99)
Adjusted Sharpe ratio was introduced by Pezier and White (2006) to adjusts for skewness and kurtosis by incorporating a penalty factor for negative skewness and excess kurtosis.
$$Adjusted Sharpe Ratio = SR * [1 + (\frac{S}{6}) * SR - (\frac{K - 3}{24}) * SR^2]$$
where $SR$ is the Sharpe Ratio with data annualized,$S$ is the skewness and $K$ is the kurtosis
data(portfolio_bacon) print(AdjustedSharpeRatio(portfolio_bacon[,1])) #expected 0.81
Skewness-kurtosis ratio (p.99)
Skewness-Kurtosis ratio is the division of Skewness by Kurtosis.' It is used in conjunction with the Sharpe ratio to rank portfolios. The higher the rate the better.
$$SkewnessKurtosisRatio(r,MAR) = \frac{S}{K}$$
where $S$ is the skewness and $K$ is the Kurtosis
data(portfolio_bacon) print(SkewnessKurtosisRatio(portfolio_bacon[,1])) #expected -0.034
Prospect ratio (p.100)
Prospect ratio is a ratio used to penalize loss since most people feel loss greater than gain
$$ProspectRatio(R) = \frac{\frac{1}{n}\sum^{n}_{i=1}(Max(r_i,0)+2.25Min(r_i,0) - MAR)}{\sigma_D}$$
where $n$ is the number of observations of the entire series, $MAR$ is the minimum acceptable return and $\sigma_D$ is the downside risk
data(portfolio_bacon) MAR = 0.05 print(ProspectRatio(portfolio_bacon[,1], MAR)) #expected -0.134
Return adjusted for downside risk
M Squared for Sortino (p.102)
M squared for Sortino is a calculated for Downside risk instead of Total Risk
$$M^2_S = r_P + Sortino ratio * (\sigma_{DM} - \sigma_D)$$
where $M^2_S$ is MSquared for Sortino, $r_P$ is the annualized portfolio return, $\sigma_DM$ is the benchmark annualized downside risk and $D$ is the portfolio annualized downside risk
data(portfolio_bacon) MAR = 0.005 print(M2Sortino(portfolio_bacon[,1], portfolio_bacon[,2], MAR)) #expected 0.1035
Omega excess return (p.103)
Omega excess return is another form of downside risk-adjusted return. It is calculated by multiplying the downside variance of the style benchmark by 3 times the style beta.
$$\omega = r_P - 3\beta_S\sigma_{MD}^2$$
where $\omega$ is omega excess return, $\beta_S$ is style beta, $\sigma_D$ is the portfolio annualized downside risk and $\sigma_MD$ is the benchmark annualized downside risk.
data(portfolio_bacon) MAR = 0.005 print(OmegaExcessReturn(portfolio_bacon[,1], portfolio_bacon[,2], MAR)) #expected 0.0805
Tables
Variability risk
Table of Mean absolute difference, Monthly standard deviation and annualized standard deviation
data(managers) table.Variability(managers[,1:8])
Specific risk
Table of specific risk, systematic risk and total risk
data(managers) table.SpecificRisk(managers[,1:8], managers[,8])
Information risk
Table of Tracking error, Annualized tracking error and Information ratio
data(managers) table.InformationRatio(managers[,1:8], managers[,8])
Distributions
Table of Monthly standard deviation, Skewness, Sample standard deviation, Kurtosis, Excess kurtosis, Sample Skewness and Sample excess kurtosis
data(managers) table.Distributions(managers[,1:8])
Drawdowns
Table of Calmar ratio, Sterling ratio, Burke ratio, Pain index, Ulcer index, Pain ratio and Martin ratio
data(managers) table.DrawdownsRatio(managers[,1:8])
Downside risk
Table of Monthly downside risk, Annualized downside risk, Downside potential, Omega, Sortino ratio, Upside potential, Upside potential ratio and Omega-Sharpe ratio
data(managers) table.DownsideRiskRatio(managers[,1:8])
Sharpe ratio
Table of Annualized Return, Annualized Std Dev, and Annualized Sharpe
data(managers) table.AnnualizedReturns(managers[,1:8])
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