PerformanceAnalytics (version 0.9.7.1)

chart.Style: calculate and display effective style weights

Description

Functions that calculate effective style weights and display the results in a bar chart. chart.Style calculates and displays style weights calculated over a single period. chart.RollingStyle calculates and displays those weights in rolling windows through time. style.fit manages the calculation of the weights by method. style.QPfit calculates the specific constraint case that requires quadratic programming.

Usage

chart.Style(R.fund, R.style, method = c("constrained", "unconstrained", "normalized"), leverage = FALSE, main = NULL, ylim = NULL, unstacked=TRUE, ...)
chart.RollingStyle(R.fund, R.style, method = c("constrained","unconstrained","normalized"), leverage = FALSE, width = 12, main = paste(colnames(R.fund)[1]," Rolling ", width ,"-Month Style Weights", sep=""), space = 0, ...)
style.fit(R.fund, R.style, model = FALSE, method = c("constrained", "unconstrained", "normalized"), leverage = FALSE, ...)
style.QPfit(R.fund, R.style, model = FALSE, leverage = FALSE, ...)

Arguments

R.fund

matrix, data frame, or zoo object with fund returns to be analyzed

R.style

matrix, data frame, or zoo object with style index returns. Data object must be of the same length and time-aligned with R.fund

method

specify the method of calculation of style weights as "constrained", "unconstrained", or "normalized". For more information, see style.fit

leverage

logical, defaults to 'FALSE'. If 'TRUE', the calculation of weights assumes that leverage may be used. For more information, see style.fit

model

logical. If 'model' = TRUE in style.QPfit, the full result set is shown from the output of solve.QP.

unstacked

logical. If set to 'TRUE' and only one row of data is submitted in 'w', then the chart creates a normal column chart. If more than one row is submitted, then this is ignored. See examples below.

space

the amount of space (as a fraction of the average bar width) left before each bar, as in barplot. Default for chart.RollingStyle is 0; for chart.Style the default is 0.2.

main

set the chart title, same as in plot

width

number of periods or window to apply rolling style analysis over

ylim

set the y-axis limit, same as in plot

...

for the charting functions, these are arguments to be passed to barplot. These can include further arguments (such as 'axes', 'asp' and 'main') and graphical parameters (see 'par') which are passed to 'plot

Value

• The chart functions plot a column chart or stacked column chart of the resulting style weights to the current device. Both style.fit and style.QPfit produce a list of data frames containing 'weights' and 'R.squared' results. If 'model' = TRUE in style.QPfit, the full result set is shown from the output of solve.QP.

Details

These functions calculate style weights using an asset class style model as described in detail in Sharpe (1992). The use of quadratic programming to determine a fund's exposures to the changes in returns of major asset classes is usually refered to as "style analysis".

The "unconstrained" method implements a simple factor model for style analysis, as in: $$Ri = bi1*F1+bi2*F2+...+bin*Fn+ei$$ where $Ri$ represents the return on asset i, $Fj$ represents each factor, and $ei$ represents the "non-factor" component of the return on i. This is simply a multiple regression analysis with fund returns as the dependent variable and asset class returns as the independent variables. The resulting slope coefficients are then interpreted as the fund's historic exposures to asset class returns. In this case, coefficients do not sum to 1.

The "normalized" method reports the results of a multiple regression analysis similar to the first, but with one constraint: the coefficients are required to add to 1. Coefficients may be negative, indicating short exposures. To enforce the constraint, coefficients are normalized.

The "constrained" method includes the constraint that the coefficients sum to 1, but adds that the coefficients must lie between 0 and 1. These inequality constraints require a quadratic programming algorithm using solve.QP from the 'quadprog' package, and the implementation is discussed under style.QPfit. If set to TRUE, "leverage" allows the sum of the coefficients to exceed 1.

According to Sharpe (1992), the calculation for the constrained case is represented as: $$min var(Rf - sum[wi * R.si]) = min var(F - w*S)$$ $$s.t. sum[wi] = 1; wi > 0$$

Remembering that:

$$\sigma(aX + bY) = a^2 \sigma(X) + b^2 \sigma(Y) + 2ab cov(X,Y) = \sigma(R.f) + w'*V*w - 2*w'*cov(R.f,R.s)$$

we can drop $var(Rf)$ as it isn't a function of weights, multiply both sides by 1/2:

$$= min (1/2) w'*V*w - C'w$$ $$s.t. w'*e = 1, w_i > 0$$

Which allows us to use solve.QP, which is specified as: $$min(-d' b + 1/2 b' D b)$$ and the constraints $$A' b >= b.0$$

so: b is the weight vector, D is the variance-covariance matrix of the styles d is the covariance vector between the fund and the styles

The chart functions then provide a graphical summary of the results. The underlying function, style.fit, provides the outputs of the analysis and more information about fit, including an R-squared value.

Styles identified in this analysis may be interpreted as an average of potentially changing exposures over the period covered. The function chart.RollingStyle may be useful for examining the behavior of a manager's average exposures to asset classes over time, using a rolling-window analysis.

References

Sharpe, W. Asset Allocation: Management Style and Performance Measurement Journal of Portfolio Management, 1992, 7-19. See http://www.stanford.edu/~wfsharpe/art/sa/sa.htm

See Also

barplot, par

Examples

data(edhec)
data(managers)
style.fit(managers[97:132,2,drop=FALSE],edhec[85:120,], method="constrained", leverage=FALSE)
chart.Style(managers[97:132,2,drop=FALSE],edhec[85:120,], method="constrained", leverage=FALSE, unstack=TRUE)
chart.RollingStyle(managers[,2,drop=FALSE],edhec[,1:11], method="constrained", leverage=FALSE, width=36, cex.legend = .7, colorset=rainbow12equal, las=1)