# portfolio-riskPfolio: Risk and Related Measures for Portfolios In fPortfolio: Rmetrics - Portfolio Selection and Optimization

## Description

Computes Value-at-Risk and related measures for a portfolio of assets.

The functions are:

 `pfolioVaR` computes Value-at-Risk for a portfolio of assets, `pfolioCVaRplus` computes Value-at-Risk+ for a portfolio of assets, `pfolioCVaR` computes Conditional Value-at-Risk for a PF of assets, `lambdaCVaR` computes CVaR's atomic split value lambda, `pfolioCVaRoptim` computes Conditional VaR from mean-CVaR optimization, `pfolioMaxLoss` computes Maximum Loss for a portfolio of assets, `pfolioReturn` computes return values of a portfolio, `pfolioTargetReturn` computes the target return of a portfolio, `pfolioTargetRisk` computes the target risk of a portfolio, `pfolioHist` plots a histogram of the returns of a portfolio.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```pfolioVaR(x, weights = NULL, alpha = 0.05) pfolioCVaRplus(x, weights = NULL, alpha = 0.05) pfolioCVaR(x, weights = NULL, alpha = 0.05) lambdaCVaR(n, alpha = 0.05) pfolioCVaRoptim(x, weights = NULL, alpha = 0.05) pfolioMaxLoss(x, weights = NULL) pfolioReturn(x, weights = NULL, geometric = FALSE) pfolioTargetReturn(x, weights = NULL) pfolioTargetRisk(x, weights = NULL) pfolioHist(x, weights = NULL, alpha = 0.05, range = NULL, details = TRUE, ...) ```

## Arguments

 `x` a 'timeSeries' object, data frame or any other rectangular object which can be expressed as a matrix. The first dimension is the number of observations, we call it `n`, and the second is the number of assets in the data set, we call it `dim`. `weights` usually a numeric vector which has the length of the number of assets. The weights measures the normalized weights of the individual assets. By default `NULL`, then an equally weighted set of assets is assumed. `geometric` a logical flag, should geometric returns be used, by default FALSE `alpha` a numeric value, the confidence interval, by default 0.05. `details` a logical value, should details be printed? `n` the number of observation from which the CVaR's atomic split value `lambda=1-floor(alpha*n)/(alpha*n)` will be evaluated. `range` a numeric vector of two elements limiting the plot range of the histogram. This is quite useful if one likes to compare several plots on the same scale. If `range=NULL`, the default value, then the range will be selected automatically. `...` optional arguments to be passet to the function `hist`.

## Details

The percentile measures of loss (or reward) are defined in the following way: Let f(x ,y) be a loss functions depending upon a decision vector x = (x_1, ..., x_n ) and a random vector y = (y_1, ..., y_m), then

pfolioVaR is the alpha-percentile of the loss distribution, a smallest value such that the probability that losses exceed or are equal to this value is greater or equal to alpha.

pfolioCVaRplus or "CVaR+" or the "upper CVaR" are the expected losses strictly exceeding VaR. This is also also called "Mean Excess Loss" and "Expected Shortfall".

pfolioCVaR is a weighted average of VaR and CVaRplus defined as CVaR = lambda*VaR + (1-lambda) CVaRplus, for 0 <= lambda <= 1.

Note, CVaR is convex, but VaR and CVaRplus may be non-convex. The following inequalities are valid: VaR <= CVaR <= CVaRplus.

## Value

`pfolioVaR`
returns the value of risk, VaR, for a portfolio of assets, a numeric value.

`pfolioCVaRplus`
returns the conditional value of risk plus, CVaRplus, for a portfolio of assets, a numeric value.

`pfolioCVaR`
returns the conditional value of risk, CVaR, for a portfolio of assets, a numeric value.

`lambdaCVaR`
returns CVaR's atomic split value `lambda`, a numeric value.

`pfolioMaxLoss`
returns the maximum loss value of the portfolio, a numeric value.

`pfolioReturn`
returns the total portfolio return computed from the set of assets `x`, a numeric vector.

`pfolioTargetReturn`
returns the total return or target return computed from the set of assets `x` and weights `weights`, a numeric value.

`pfolioTargetRisk`
returns the total risk (Sigma) or target risk computed from the set of assets `x` and `weights` via the formual `sqrt(weights %*% cov(x) %*% weights)`, a numeric value.

`pfolioHist`
plots a histogram of portfolio returns and adds the values for the VaR (blue), for the CVaRplus (red), and for the maximum loss (green) to the histogram plot. The function invisibly returns a list with the following elements: VaR, VaRplus, maxLoss, mean, and sd. If `details` is `TRUE`, then the result is printed.

## References

Uryasev S. (2000); Conditional Value-at-Risk (CVaR): Algorithms and Applications, Risk Management and Financial Engineering Lab, University of Florida

Wuertz, D., Chalabi, Y., Chen W., Ellis A. (2009); Portfolio Optimization with R/Rmetrics, Rmetrics eBook, Rmetrics Association and Finance Online, Zurich.

fPortfolio documentation built on March 26, 2020, 9:17 p.m.