Package 'Dowd'

Description

Dowd Kevin Dowd's book "Measuring Market Risk" gives overview of risk measurement procedures with focus on Value at Risk (VaR) and Expected Shortfall (ES).

Acknowledgments

Without Kevin Dowd's book Measuring Market Risk and accompanying MATLAB toolbox, this project would not have been possible.

Peter Carl and Brian G. Peterson deserve special acknowledgement for mentoring me on this project.

Author(s)

Dinesh Acharya

Maintainer: Dinesh Acharya [email protected]

References

Dowd, K. Measuring Market Risk. Wiley. 2005.

Description

Estimates the ES hotspots (or vector of incremental ESs) for a portfolio with portfolio return adjusted for non-normality by Cornish-Fisher corerction, for specified confidence level and holding period.

Usage

AdjustedNormalESHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Hotspots for ES for randomly generated portfolio
   vc.matrix <- matrix(rnorm(16),4,4)
   mu <- rnorm(4)
   skew <- .5
   kurtosis <- 1.2
   positions <- c(5,2,6,10)
   cl <- .95
   hp <- 280
   AdjustedNormalESHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Description

Estimates the VaR hotspots (or vector of incremental VaRs) for a portfolio with portfolio return adjusted for non-normality by Cornish-Fisher corerction, for specified confidence level and holding period.

Usage

AdjustedNormalVaRHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Hotspots for ES for randomly generated portfolio
   vc.matrix <- matrix(rnorm(16),4,4)
   mu <- rnorm(4)
   skew <- .5
   kurtosis <- 1.2
   positions <- c(5,2,6,10)
   cl <- .95
   hp <- 280
   AdjustedNormalVaRHotspots(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Description

Function estimates the Variance-Covariance ES of a multi-asset portfolio using the Cornish - Fisher adjustment for portfolio return non-normality, for specified confidence level and holding period.

Usage

AdjustedVarianceCovarianceES(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Variance-covariance ES for randomly generated portfolio
   vc.matrix <- matrix(rnorm(16), 4, 4)
   mu <- rnorm(4)
   skew <- .5
   kurtosis <- 1.2
   positions <- c(5, 2, 6, 10)
   cl <- .95
   hp <- 280
   AdjustedVarianceCovarianceES(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Description

Estimates the variance-covariance VaR of a multi-asset portfolio using the Cornish-Fisher adjustment for portfolio-return non-normality, for specified confidence level and holding period.

Usage

AdjustedVarianceCovarianceVaR(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Variance-covariance for randomly generated portfolio
   vc.matrix <- matrix(rnorm(16),4,4)
   mu <- rnorm(4)
   skew <- .5
   kurtosis <- 1.2
   positions <- c(5,2,6,10)
   cl <- .95
   hp <- 280
   AdjustedVarianceCovarianceVaR(vc.matrix, mu, skew, kurtosis, positions, cl, hp)

Description

Anderson-Darling(AD) test can be used to carry out distribution equality test and is similar to Kolmogorov-Smirnov test. AD test statistic is defined as:

$A^2=n\int_{-\infty}^{\infty}\frac{[\hat{F}(x)-F(x)]^2}{F(x)[1-F(x)]}dF(x)$

which is equivalent to

$=-n-\frac{1}{n}\sum_{i=1}^n(2i-1)[\ln F(X_i)+\ln(1-F(X_{n+1-i}))]$

Usage

ADTestStat(number.trials, sample.size, confidence.interval)

Arguments

Value

Confidence Interval for AD test statistic

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Anderson, T.W. and Darling, D.A. Asymptotic Theory of Certain Goodness of Fit Criteria Based on Stochastic Processes, The Annals of Mathematical Statistics, 23(2), 1952, p. 193-212.

Kvam, P.H. and Vidakovic, B. Nonparametric Statistics with Applications to Science and Engineering, Wiley, 2007.

Examples

# Probability that the VaR model is correct for 3 failures, 100 number
   # observations and  95% confidence level
   ADTestStat(1000, 100, 0.95)

Description

Estimates ES of American Put Option using binomial tree to price the option and historical method to compute the VaR.

Usage

AmericanPutESBinomial(amountInvested, stockPrice, strike, r, volatility,
  maturity, numberSteps, cl, hp)

Arguments

Value

ES of the American Put Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Market Risk of American Put with given parameters.
   AmericanPutESBinomial(0.20, 27.2, 25, .16, .05, 60, 20, .95, 30)

Description

Estimates ES of American Put Option using binomial tree to price the option valuation tree and Monte Carlo simulation with a binomial option valuation tree nested within the MCS. Historical method to compute the VaR.

Usage

AmericanPutESSim(amountInvested, stockPrice, strike, r, mu, sigma, maturity,
  numberTrials, numberSteps, cl, hp)

Arguments

Value

Monte Carlo Simulation VaR estimate and the bounds of the 95 confidence interval for the VaR, based on an order-statistics analysis of the P/L distribution

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Market Risk of American Put with given parameters.
   AmericanPutESSim(0.20, 27.2, 25, .16, .2, .05, 60, 30, 20, .95, 30)

Description

Estimates the price of an American Put, using the binomial approach.

Usage

AmericanPutPriceBinomial(stockPrice, strike, r, sigma, maturity, numberSteps)

Arguments

Value

Binomial American put price

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Estimates the price of an American Put
   AmericanPutPriceBinomial(27.2, 25, .03, .2, 60, 30)

Description

Estimates VaR of American Put Option using binomial tree to price the option and historical method to compute the VaR.

Usage

AmericanPutVaRBinomial(amountInvested, stockPrice, strike, r, volatility,
  maturity, numberSteps, cl, hp)

Arguments

Value

VaR of the American Put Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Market Risk of American Put with given parameters.
   AmericanPutVaRBinomial(0.20, 27.2, 25, .16, .05, 60, 20, .95, 30)

Description

The basic idea behind binomial backtest (also called basic frequency test) is to test whether the observed frequency of losses that exceed VaR is consistent with the frequency of tail losses predicted by the mode. Binomial Backtest carries out the binomial backtest for a VaR risk measurement model for specified VaR confidence level and for a one-sided alternative hypothesis (H1).

Usage

BinomialBacktest(x, n, cl)

Arguments

Value

Probability that the VaR model is correct

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Kupiec, Paul. Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, Winter 1995, p. 79.

Examples

# Probability that the VaR model is correct for 3 failures, 100 number
   # observations and  95% confidence level
   BinomialBacktest(55, 1000, 0.95)

Description

Estimates ES of Black-Scholes call Option using Monte Carlo simulation

Usage

BlackScholesCallESSim(amountInvested, stockPrice, strike, r, mu, sigma,
  maturity, numberTrials, cl, hp)

Arguments

Value

ES

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Market Risk of American call with given parameters.
   BlackScholesCallESSim(0.20, 27.2, 25, .16, .2, .05, 60, 30, .95, 30)

Description

Derives the price of European call option using the Black-Scholes approach

Usage

BlackScholesCallPrice(stockPrice, strike, rf, sigma, t)

Arguments

Value

Price of European Call Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Hull, John C.. Options, Futures, and Other Derivatives. 5th ed., p. 246.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Estimates the price of an American Put
   BlackScholesCallPrice(27.2, 25, .03, .2, 60)

Description

Estimates ES of Black-Scholes Put Option using Monte Carlo simulation

Usage

BlackScholesPutESSim(amountInvested, stockPrice, strike, r, mu, sigma, maturity,
  numberTrials, cl, hp)

Arguments

Value

ES

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Market Risk of American Put with given parameters.
   BlackScholesPutESSim(0.20, 27.2, 25, .03, .12, .05, 60, 1000, .95, 30)

Description

Derives the price of European call option using the Black-Scholes approach

Usage

BlackScholesPutPrice(stockPrice, strike, rf, sigma, t)

Arguments

Value

Price of European Call Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Hull, John C.. Options, Futures, and Other Derivatives. 5th ed., p. 246.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Estimates the price of an American Put
   BlackScholesPutPrice(27.2, 25, .03, .2, 60)

Description

Derives the Blanco-Ihle forecast evaluation loss measure for a VaR risk measurement model.

Usage

BlancoIhleBacktest(Ra, Rb, Rc, cl)

Arguments

Value

First Blanco-Ihle score measure.

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Blanco, C. and Ihle, G. How Good is Your Var? Using Backtesting to Assess System Performance. Financial Engineering News, 1999.

Examples

# Blanco-Ihle Backtest For Independence for given confidence level.
   # The VaR and ES are randomly generated.
   a <- rnorm(1*100)
   b <- abs(rnorm(1*100))+2
   c <- abs(rnorm(1*100))+2
   BlancoIhleBacktest(a, b, c, 0.95)

Description

Estimates the bootstrapped ES for confidence level and holding period implied by data frequency.

Usage

BootstrapES(Ra, number.resamples, cl)

Arguments

Value

Bootstrapped Expected Shortfall

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates bootstrapped ES for given parameters
   a <- rnorm(100) # generate a random profit/loss vector
   BootstrapVaR(a, 50, 0.95)

Description

Estimates the 90 level and holding period implied by data frequency.

Usage

BootstrapESConfInterval(Ra, number.resamples, cl)

Arguments

Value

90

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# To be modified with appropriate data.
   # Estimates 90% confidence interval for bootstrapped ES for 95%
   # confidence interval
   Ra <- rnorm(1000)
   BootstrapESConfInterval(Ra, 50, 0.95)

Description

Plots figure for the bootstrapped ES, for confidence level and holding period implied by data frequency.

Usage

BootstrapESFigure(Ra, number.resamples, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# To be modified with appropriate data.
   # Estimates 90% confidence interval for bootstrapped ES for 95%
   # confidence interval
   Ra <- rnorm(1000)
   BootstrapESFigure(Ra, 500, 0.95)

Description

Estimates the bootstrapped VaR for confidence level and holding period implied by data frequency.

Usage

BootstrapVaR(Ra, number.resamples, cl)

Arguments

Value

Bootstrapped VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates bootstrapped VaR for given parameters
   a <- rnorm(100) # generate a random profit/loss vector
   BootstrapES(a, 50, 0.95)

Description

Estimates the 90 level and holding period implied by data frequency.

Usage

BootstrapVaRConfInterval(Ra, number.resamples, cl)

Arguments

Value

90

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# To be modified with appropriate data.
   # Estimates 90% confidence interval for bootstrapped Var for 95%
   # confidence interval
   Ra <- rnorm(1000)
   BootstrapVaRConfInterval(Ra, 500, 0.95)

Description

Plots figure for the bootstrapped VaR, for confidence level and holding period implied by data frequency.

Usage

BootstrapVaRFigure(Ra, number.resamples, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# To be modified with appropriate data.
   # Estimates 90% confidence interval for bootstrapped VaR for 95%
   # confidence interval
   Ra <- rnorm(1000)
   BootstrapESFigure(Ra, 500, 0.95)

Description

Function estimates the ES of a portfolio assuming P and L data set transformed using the BoxCox transformation to make it as near normal as possible, for specified confidence level and holding period implied by data frequency.

Usage

BoxCoxES(loss.data, cl)

Arguments

Value

Estimated Box-Cox ES. Its dimension is same as that of cl

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Hamilton, S. A. and Taylor, M. G. A Comparision of the Box-Cox transformation method and nonparametric methods for estimating quantiles in clinical data with repeated measures. J. Statist. Comput. Simul., vol. 45, 1993, pp. 185 - 201.

Examples

# Estimates Box-Cox VaR
   a<-rnorm(200)
   BoxCoxES(a,.95)

Description

Function estimates the VaR of a portfolio assuming P and L data set transformed using the BoxCox transformation to make it as near normal as possible, for specified confidence level and holding period implied by data frequency.

Usage

BoxCoxVaR(PandLdata, cl)

Arguments

Value

Estimated Box-Cox VaR. Its dimension is same as that of cl

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Hamilton, S. A. and Taylor, M. G. A Comparision of the Box-Cox transformation method and nonparametric methods for estimating quantiles in clinical data with repeated measures. J. Statist. Comput. Simul., vol. 45, 1993, pp. 185 - 201.

Examples

# Estimates Box-Cox VaR
   a<-rnorm(100)
   BoxCoxVaR(a,.95)

Description

If X and Y are position P/Ls, then the VaR is equal to minus quantile. In such cases, we insert the negative of the VaR as the quantile, and the function gives us the value of 1 minus VaR confidence level. In other words, if X and Y are position P/Ls, the quantile is the negative of the VaR, and the output is 1 minus the VaR confidence level.

Usage

CdfOfSumUsingGaussianCopula(quantile, mu1, mu2, sigma1, sigma2, rho,
  number.steps.in.copula)

Arguments

Value

Probability of X + Y being less than quantile

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Dowd, K. and Fackler, P. Estimating VaR with copulas. Financial Engineering News, 2004.

Examples

# Prob ( X + Y < q ) using Gaussian Copula for X with mean 2.3 and std. .2
   # and Y with mean 4.5 and std. 1.5 with beta 1.2 at 0.9 quantile
   CdfOfSumUsingGaussianCopula(0.9, 2.3, 4.5, 1.2, 1.5, 0.6, 15)

Description

If X and Y are position P/Ls, then the VaR is equal to minus quantile. In such cases, we insert the negative of the VaR as the quantile, and the function gives us the value of 1 minus VaR confidence level. In other words, if X and Y are position P/Ls, the quantile is the negative of the VaR, and the output is 1 minus the VaR confidence level.

Usage

CdfOfSumUsingGumbelCopula(quantile, mu1, mu2, sigma1, sigma2, beta)

Arguments

Value

Probability of X + Y being less than quantile

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Dowd, K. and Fackler, P. Estimating VaR with copulas. Financial Engineering News, 2004.

Examples

# Prob ( X + Y < q ) using Gumbel Copula for X with mean 2.3 and std. .2
   # and Y with mean 4.5 and std. 1.5 with beta 1.2 at 0.9 quantile
   CdfOfSumUsingGumbelCopula(0.9, 2.3, 4.5, 1.2, 1.5, 1.2)

Description

If X and Y are position P/Ls, then the VaR is equal to minus quantile. In such cases, we insert the negative of the VaR as the quantile, and the function gives us the value of 1 minus VaR confidence level. In other words, if X and Y are position P/Ls, the quantile is the negative of the VaR, and the output is 1 minus the VaR confidence level.

Usage

CdfOfSumUsingProductCopula(quantile, mu1, mu2, sigma1, sigma2)

Arguments

Value

Probability of X + Y being less than quantile

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Dowd, K. and Fackler, P. Estimating VaR with copulas. Financial Engineering News, 2004.

Examples

# Prob ( X + Y < q ) using Product Copula for X with mean 2.3 and std. .2
   # and Y with mean 4.5 and std. 1.5 with beta 1.2 at 0.9 quantile
   CdfOfSumUsingProductCopula(0.9, 2.3, 4.5, 1.2, 1.5)

Description

Carries out the Christoffersen backtest of independence for a VaR risk measurement model, for specified VaR confidence level.

Usage

ChristoffersenBacktestForIndependence(Ra, Rb, cl)

Arguments

Value

Probability that given the data set, the null hypothesis (i.e. independence) is correct.

Author(s)

Dinesh Acharya

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Christoffersen, P. Evaluating Interval Forecasts. International Economic Review, 39(4), 1992, 841-862.

Examples

# Has to be modified with appropriate data:
   # Christoffersen Backtest For Independence for given parameters
   a <- rnorm(1*100)
   b <- abs(rnorm(1*100))+2
   ChristoffersenBacktestForIndependence(a, b, 0.95)

Description

Carries out the Christiffersen backtest for unconditional coverage for a VaR risk measurement model, for specified VaR confidence level.

Usage

ChristoffersenBacktestForUnconditionalCoverage(Ra, Rb, cl)

Arguments

Value

Probability, given the data set, that the null hypothesis (i.e. a correct model) is correct.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Christoffersen, P. Evaluating interval forecasts. International Economic Review, 39(4), 1998, 841-862.

Examples

# Has to be modified with appropriate data:
   # Christoffersen Backtest For Unconditional Coverage for given parameters
   a <- rnorm(1*100)
   b <- abs(rnorm(1*100))+2
   ChristoffersenBacktestForUnconditionalCoverage(a, b, 0.95)

Description

Function estimates the ES for near-normal P/L using the Cornish-Fisher adjustment for non-normality, for specified confidence level.

Usage

CornishFisherES(mu, sigma, skew, kurtosis, cl)

Arguments

Value

Expected Shortfall

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Zangri, P. A VaR methodology for portfolios that include options. RiskMetrics Monitor, First quarter, 1996, p. 4-12.

Examples

# Estimates Cornish-Fisher ES for given parameters
   CornishFisherES(3.2, 5.6, 2, 3, .9)

Description

Function estimates the VaR for near-normal P/L using the Cornish-Fisher adjustment for non-normality, for specified confidence level.

Usage

CornishFisherVaR(mu, sigma, skew, kurtosis, cl)

Arguments

Value

Value at Risk

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Zangri, P. A VaR methodology for portfolios that include options. RiskMetrics Monitor, First quarter, 1996, p. 4-12.

Examples

# Estimates Cornish-Fisher VaR for given parameters
   CornishFisherVaR(3.2, 5.6, 2, 3, .9)

Description

Generates Monte Carlo VaR for DB pension in Chapter 6.7.

Usage

DBPensionVaR(mu, sigma, p, life.expectancy, number.trials, cl)

Arguments

Value

VaR for DB pension

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates the price of an American Put
   DBPensionVaR(.06, .2, .05, 80, 100, .95)

Description

Generates Monte Carlo VaR for DC pension in Chapter 6.7.

Usage

DCPensionVaR(mu, sigma, p, life.expectancy, number.trials, cl)

Arguments

Value

VaR for DC pension

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates the price of an American Put
   DCPensionVaR(.06, .2, .05, 80, 100, .95)

Description

Generates Monte Carlo VaR for default risky bond portfolio in Chapter 6.4

Usage

DefaultRiskyBondVaR(r, rf, coupon, sigma, amount.invested, recovery.rate, p,
  number.trials, hp, cl)

Arguments

Value

Monte Carlo VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for default risky bond portfolio for given parameters
   DefaultRiskyBondVaR(.01, .01, .1, .01, 1, .1, .2, 100, 100, .95)

Description

Generates Monte Carlo lognormal VaR with filter portfolio strategy

Usage

FilterStrategyLogNormalVaR(mu, sigma, number.trials, alpha, cl, hp)

Arguments

Value

Lognormal VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates standard error of normal quantile estimate
   FilterStrategyLogNormalVaR(0, .2, 100, 1.2, .95, 10)

Description

Estimates the ES of a portfolio assuming extreme losses are Frechet distributed, for specified confidence level and a given holding period.

Usage

FrechetES(mu, sigma, tail.index, n, cl, hp)

Arguments

Details

Note that the long-right-hand tail is fitted to losses, not profits.

Value

Estimated ES. If cl and hp are scalars, it returns scalar VaR. If cl is vector and hp is a scalar, or viceversa, returns vector of VaRs. If both cl and hp are vectors, returns a matrix of VaRs.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Embrechts, P., Kluppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997, p. 324.

Reiss, R. D. and Thomas, M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, Birkhaueser, Basel, 1997, 15-18.

Examples

# Computes ES assuming Frechet Distribution for given parameters
   FrechetES(3.5, 2.3, 1.6, 10, .95, 30)

Description

Plots the ES of a portfolio against confidence level assuming extreme losses are Frechet distributed, for specified confidence level and a given holding period.

Usage

FrechetESPlot2DCl(mu, sigma, tail.index, n, cl, hp)

Arguments

Details

Note that the long-right-hand tail is fitted to losses, not profits.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Embrechts, P., Kluppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997, p. 324.

Reiss, R. D. and Thomas, M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, Birkhaueser, Basel, 1997, 15-18.

Examples

# Plots ES against vector of cl assuming Frechet Distribution for given parameters
   cl <- seq(0.9,0.99,0.01)
   FrechetESPlot2DCl(3.5, 2.3, 1.6, 10, cl, 30)

Description

Estimates the VaR of a portfolio assuming extreme losses are Frechet distributed, for specified range of confidence level and a given holding period.

Usage

FrechetVaR(mu, sigma, tail.index, n, cl, hp)

Arguments

Details

Note that the long-right-hand tail is fitted to losses, not profits.

Value

Value at Risk. If cl and hp are scalars, it returns scalar VaR. If cl is vector and hp is a scalar, or viceversa, returns vector of VaRs. If both cl and hp are vectors, returns a matrix of VaRs.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Embrechts, P., Kluppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997, p. 324.

Reiss, R. D. and Thomas, M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, Birkhaueser, Basel, 1997, 15-18.

Examples

# Computes VaR assuming Frechet Distribution for given parameters
   FrechetVaR(3.5, 2.3, 1.6, 10, .95, 30)

Description

Plots the VaR of a portfolio against confidence level assuming extreme losses are Frechet distributed, for specified range of confidence level and a given holding period.

Usage

FrechetVaRPlot2DCl(mu, sigma, tail.index, n, cl, hp)

Arguments

Details

Note that the long-right-hand tail is fitted to losses, not profits.

Author(s)

Dinesh Acharya

References

Dowd, K. Measurh ing Market Risk, Wiley, 2007.

Embrechts, P., Kluppelberg, C. and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997, p. 324.

Reiss, R. D. and Thomas, M. Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields, Birkhaueser, Basel, 1997, 15-18.

Examples

# Plots VaR against vector of cl assuming Frechet Distribution for given parameters
   cl <- seq(0.9, .99, .01)
   FrechetVaRPlot2DCl(3.5, 2.3, 1.6, 10, cl, 30)

Description

Derives VaR using bivariate Gaussian copula with specified inputs for normal marginals.

Usage

GaussianCopulaVaR(mu1, mu2, sigma1, sigma2, rho, number.steps.in.copula, cl)

Arguments

Value

Copula based VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Dowd, K. and Fackler, P. Estimating VaR with copulas. Financial Engineering News, 2004.

Examples

# VaR using bivariate Gaussian for X and Y with given parameters:
   GaussianCopulaVaR(2.3, 4.1, 1.2, 1.5, .6, 10, .95)

Description

Estimates the ES of a portfolio assuming losses are distributed as a generalised Pareto.

Usage

GParetoES(Ra, beta, zeta, threshold.prob, cl)

Arguments

Value

Expected Shortfall

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

McNeil, A., Extreme value theory for risk managers. Mimeo, ETHZ, 1999.

Examples

# Computes ES assuming generalised Pareto for following parameters
   Ra <- 5 * rnorm(100)
   beta <- 1.2
   zeta <- 1.6
   threshold.prob <- .85
   cl <- .99
   GParetoES(Ra, beta, zeta, threshold.prob, cl)

Description

Plots of emperical mean excess function and Generalized mean excess function.

Usage

GParetoMEFPlot(Ra, mu, beta, zeta)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES assuming generalised Pareto for following parameters
   Ra <- 5 * rnorm(100)
   mu <- 0
   beta <- 1.2
   zeta <- 1.6
   GParetoMEFPlot(Ra, mu, beta, zeta)

Description

Plots of emperical mean excess function and two generalized pareto mean excess functions which differ in their tail-index value.

Usage

GParetoMultipleMEFPlot(Ra, mu, beta, zeta1, zeta2)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES assuming generalised Pareto for following parameters
   Ra <- 5 * rnorm(100)
   mu <- 1
   beta <- 1.2
   zeta1 <- 1.6
   zeta2 <- 2.2
   GParetoMultipleMEFPlot(Ra, mu, beta, zeta1, zeta2)

Description

Estimates the Value at Risk of a portfolio assuming losses are distributed as a generalised Pareto.

Usage

GParetoVaR(Ra, beta, zeta, threshold.prob, cl)

Arguments

Value

Expected Shortfall

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

McNeil, A., Extreme value theory for risk managers. Mimeo, ETHZ, 1999.

Examples

# Computes ES assuming generalised Pareto for following parameters
   Ra <- 5 * rnorm(100)
   beta <- 1.2
   zeta <- 1.6
   threshold.prob <- .85
   cl <- .99
   GParetoVaR(Ra, beta, zeta, threshold.prob, cl)

Description

Derives VaR using bivariate Gumbel or logistic copula with specified inputs for normal marginals.

Usage

GumbelCopulaVaR(mu1, mu2, sigma1, sigma2, beta, cl)

Arguments

Value

Copula based VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Dowd, K. and Fackler, P. Estimating VaR with copulas. Financial Engineering News, 2004.

Examples

# VaR using bivariate Gumbel for X and Y with given parameters:
   GumbelCopulaVaR(1.1, 3.1, 1.2, 1.5, 1.1, .95)

Description

Estimates the ES of a portfolio assuming extreme losses are Gumbel distributed, for specified confidence level and holding period. Note that the long-right-hand tail is fitted to losses, not profits.

Usage

GumbelES(mu, sigma, n, cl, hp)

Arguments

Value

Estimated ES. If cl and hp are scalars, it returns scalar VaR. If cl is vector and hp is a scalar, or viceversa, returns vector of VaRs. If both cl and hp are vectors, returns a matrix of VaRs.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

National Institute of Standards and Technology, Dataplot Reference Manual. Volume 1: Commands. NIST: Washington, DC, 1997, p. 8-67.

Examples

# Gumber ES Plot
   GumbelES(0, 1.2, 100, c(.9,.88, .85, .8), 280)

Description

Estimates the EV VaR of a portfolio assuming extreme losses are Gumbel distributed, for specified confidence level and holding period.

Usage

GumbelESPlot2DCl(mu, sigma, n, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots ES against Cl
   GumbelESPlot2DCl(0, 1.2, 100, seq(0.8,0.99,0.02), 280)

Description

Estimates the EV VaR of a portfolio assuming extreme losses are Gumbel distributed, for specified confidence level and holding period.

Usage

GumbelVaR(mu, sigma, n, cl, hp)

Arguments

Value

Estimated VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Gumbel VaR
   GumbelVaR(0, 1.2, 100, c(.9,.88, .85, .8), 280)

Description

Estimates the EV VaR of a portfolio assuming extreme losses are Gumbel distributed, for specified confidence level and holding period.

Usage

GumbelVaRPlot2DCl(mu, sigma, n, cl, hp)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against Cl
   GumbelVaRPlot2DCl(0, 1.2, 100, c(.9,.88, .85, .8), 280)

Description

Estimates the value of the Hill Estimator for a given specified data set and chosen tail size. Notes: 1) We estimate Hill Estimator by looking at the upper tail. 2) If the specified tail size is such that any included observations are negative, the tail is truncated at the point before observations become negative. 3) The tail size must be a scalar.

Usage

HillEstimator(Ra, tail.size)

Arguments

Value

Estimated value of Hill Estimator

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Hill Estimator of
   Ra <- rnorm(15)
   HillEstimator(Ra, 10)

Description

Displays a plot of the Hill Estimator against tail sample size.

Usage

HillPlot(Ra, maximum.tail.size)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Hill Estimator - Tail Sample Size Plot for random normal dataset
   Ra <- rnorm(1000)
   HillPlot(Ra, 180)

Description

Estimates value of Hill Quantile Estimator for a specified data set, tail index, in-sample probability and confidence level.

Usage

HillQuantileEstimator(Ra, tail.index, in.sample.prob, cl)

Arguments

Value

Value of Hill Quantile Estimator

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Next reference

Examples

# Computes estimates value of hill estimator for a specified data set
   Ra <- rnorm(1000)
   HillQuantileEstimator(Ra, 40, .5, .9)

Description

Estimates the Expected Shortfall (aka. Average Value at Risk or Conditional Value at Risk) using historical estimator approach for the specified confidence level and the holding period implies by data frequency.

Usage

HSES(Ra, cl)

Arguments

Value

Expected Shortfall of the portfolio

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Cont, R., Deguest, R. and Scandolo, G. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10(6), 2010, 593-606.

Acerbi, C. and Tasche, D. On the coherence of Expected Shortfall. Journal of Banking and Finance, 26(7), 2002, 1487-1503

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. Coherent Risk Measures of Risk. Mathematical Finance 9(3), 1999, 203.

Foellmer, H. and Scheid, A. Stochastic Finance: An Introduction in Discrete Time. De Gryuter, 2011.

Examples

# Computes Historical Expected Shortfall for a given profit/loss
   # distribution and confidence level
   a <- rnorm(100) # generate a random profit/loss vector
   HSES(a, 0.95)

Description

Estimates percentiles of historical simulation ES distribution function, using theory of order statistics, for specified confidence level.

Usage

HSESDFPerc(Ra, perc, cl)

Arguments

Value

Value of percentile of VaR distribution function

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles for random standard normal returns and given perc
   # and cl
   Ra <- rnorm(100)
   HSESDFPerc(Ra, .75, .95)

Description

Plots figure showing the historical simulation VaR and ES and histogram of L/P for specified confidence level and holding period implied by data frequency.

Usage

HSESFigure(Ra, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots figure showing VaR and histogram of P/L data
   Ra <- rnorm(100)
   HSESFigure(Ra, .95)

Description

Function plots the historical simulation ES of a portfolio against confidence level, for specified range of confidence level and holding period implied by data frequency.

Usage

HSESPlot2DCl(Ra, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots historical simulation ES against confidence level
   Ra <- rnorm(100)
   cl <- seq(.90, .99, .01)
   HSESPlot2DCl(Ra, cl)

Description

Estimates the Value at Risk (VaR) using historical estimator approach for the specified range of confidence levels and the holding period implies by data frequency.

Usage

HSVaR(Ra, Rb)

Arguments

Value

Value at Risk of the portfolio

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill, 2006

Cont, R., Deguest, R. and Scandolo, G. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance, 10(6), 2010, 593-606.

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. Coherent Risk Measures of Risk. Mathematical Finance 9(3), 1999, 203.

Foellmer, H. and Scheid, A. Stochastic Finance: An Introduction in Discrete Time. De Gryuter, 2011.

Examples

# To be added
   a <- rnorm(1000) # Payoffs of random portfolio
   HSVaR(a, .95)

Description

Estimates percentiles of historical simulation VaR distribution function, using theory of order statistics, for specified confidence level.

Usage

HSVaRDFPerc(Ra, perc, cl)

Arguments

Value

Value of percentile of VaR distribution function

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles for random standard normal returns and given perc
   # and cl
   Ra <- rnorm(100)
   HSVaRDFPerc(Ra, .75, .95)

Description

Function plots the historical simulation VaR and ES of a portfolio against confidence level, for specified range of confidence level and holding period implied by data frequency.

Usage

HSVaRESPlot2DCl(Ra, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots historical simulation VaR and ES against confidence level
   Ra <- rnorm(100)
   cl <- seq(.90, .99, .01)
   HSVaRESPlot2DCl(Ra, cl)

Description

Plots figure showing the historical simulation VaR and histogram of L/P for specified confidence level and holding period implied by data frequency.

Usage

HSVaRFigure(Ra, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots figure showing VaR and histogram of P/L data
   Ra <- rnorm(100)
   HSVaRFigure(Ra, .95)

Description

Function plots the historical simulation VaR of a portfolio against confidence level, for specified range of confidence level and holding period implied by data frequency.

Usage

HSVaRPlot2DCl(Ra, cl)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots historical simulation VaR against confidence level
   Ra <- rnorm(100)
   cl <- seq(.90, .99, .01)
   HSVaRPlot2DCl(Ra, cl)

Description

Generates Monte Carlo VaR for insurance portfolio in Chapter 6.5

Usage

InsuranceVaR(mu, sigma, n, p, theta, deductible, number.trials, cl)

Arguments

Value

VaR of the specified portfolio

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates VaR of Insurance portfolio with given parameters
   InsuranceVaR(.8, 1.3, 100, .6, 21,  12, 50, .95)

Description

Generates Monte Carlo VaR and ES for insurance portfolio.

Usage

InsuranceVaRES(mu, sigma, n, p, theta, deductible, number.trials, cl)

Arguments

Value

A list with "VaR" and "ES" of the specified portfolio

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates VaR and ES of Insurance portfolio with given parameters
   y<-InsuranceVaRES(.8, 1.3, 100, .6, 21,  12, 50, .95)

Description

Jarque-Bera (JB) is a backtest to test whether the skewness and kurtosis of a given sample matches that of normal distribution. JB test statistic is defined as

$JB=\frac{n}{6}\left(s^2+\frac{(k-3)^2}{4}\right)$

where $n$ is sample size, $s$ and $k$ are coefficients of sample skewness and kurtosis.

Usage

JarqueBeraBacktest(sample.skewness, sample.kurtosis, n)

Arguments

Value

Probability of null hypothesis H0

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Jarque, C. M. and Bera, A. K. A test for normality of observations and regression residuals, International Statistical Review, 55(2): 163-172.

Examples

# JB test statistic for sample with 500 observations with sample
   # skewness and kurtosis of -0.075 and 2.888
   JarqueBeraBacktest(-0.075,2.888,500)

Description

The output consists of a scalar ES for specified confidence level.

Usage

KernelESBoxKernel(Ra, cl)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using box kernel approach
   Ra <- rnorm(30)
   KernelESBoxKernel(Ra, .95)

Description

The output consists of a scalar ES for specified confidence level.

Usage

KernelESEpanechinikovKernel(Ra, cl, plot = TRUE)

Arguments

Value

Scalar ES

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# ES for specified confidence level using Epanechinikov kernel approach
   Ra <- rnorm(30)
   KernelESEpanechinikovKernel(Ra, .95)

Description

The output consists of a scalar ES for specified confidence level.

Usage

KernelESNormalKernel(Ra, cl)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# ES for specified confidence level using normal kernel approach
   Ra <- rnorm(30)
   KernelESNormalKernel(Ra, .95)

Description

The output consists of a scalar ES for specified confidence level.

Usage

KernelESTriangleKernel(Ra, cl)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using triangle kernel approach
   Ra <- rnorm(30)
   KernelESTriangleKernel(Ra, .95)

Description

The output consists of a scalar VaR for specified confidence level.

Usage

KernelVaRBoxKernel(Ra, cl, plot = TRUE)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using box kernel approach
   Ra <- rnorm(30)
   KernelVaRBoxKernel(Ra, .95)

Description

The output consists of a scalar VaR for specified confidence level.

Usage

KernelVaREpanechinikovKernel(Ra, cl, plot = TRUE)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using epanechinikov kernel approach
   Ra <- rnorm(30)
   KernelVaREpanechinikovKernel(Ra, .95)

Description

The output consists of a scalar VaR for specified confidence level.

Usage

KernelVaRNormalKernel(Ra, cl, plot = TRUE)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using normal kernel approach
   Ra <- rnorm(30)
   KernelVaRNormalKernel(Ra, .95)

Description

The output consists of a scalar VaR for specified confidence level.

Usage

KernelVaRTriangleKernel(Ra, cl, plot = TRUE)

Arguments

Value

Scalar VaR

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# VaR for specified confidence level using triangle kernel approach
   Ra <- rnorm(30)
   KernelVaRTriangleKernel(Ra, .95)

Description

Kolmogorov-Smirnov (KS) test statistic is a non parametric test for distribution equality and measures the maximum distance between two cdfs. Formally, the KS test statistic is :

$D=\max_i|F(X_i)-\hat{F}(X_i)|$

Usage

KSTestStat(number.trials, sample.size, confidence.interval)

Arguments

Value

Confidence Interval for KS test stat

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Chakravarti, I. M., Laha, R. G. and Roy, J. Handbook of Methods of #' Applied Statistics, Volume 1, Wiley, 1967.

Examples

# Plots the cdf for KS Test statistic and returns KS confidence interval
   # for 100 trials with 1000 sample size and 0.95 confidence interval
   KSTestStat(100, 1000, 0.95)

Description

Kuiper test statistic is a non parametric test for distribution equality and is closely related to KS test. Formally, the Kuiper test statistic is :

$D*=\max_i\{F(X_i)-\hat{F(x_i)}+\max_i\{\hat{F}(X_i)-F(X_i)\}$

Usage

KuiperTestStat(number.trials, sample.size, confidence.interval)

Arguments

Value

Confidence Interval for KS test stat

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots the cdf for Kuiper Test statistic and returns Kuiper confidence
   # interval for 100 trials with 1000 sample size and 0.95 confidence
   # interval.
   KuiperTestStat(100, 1000, 0.95)

Description

Estimates the ES of a portfolio assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalES(...)

Arguments

Value

Matrix of ES whose dimension depends on dimension of hp and cl. If cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, the matrix is column matrix and if both cl and hp are vectors, the matrix has dimension length of cl * length of hp.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalES(returns = data, investment = 5, cl = .95, hp = 90)

   # Computes ES given mean and standard deviation of return data
   LogNormalES(mu = .012, sigma = .03, investment = 5, cl = .95, hp = 90)

Description

Estimates the percentiles of ES distribution for normally distributed geometric returns, for specified confidence level and holding period using the theory of order statistics.

Usage

LogNormalESDFPerc(...)

Arguments

Value

Percentiles of ES distribution function

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles of ES distribution
   data <- runif(5, min = 0, max = .2)
   LogNormalESDFPerc(returns = data, investment = 5, perc = .7, cl = .95, hp = 60)

   # Estimates Percentiles given mean, standard deviation and number of sambles of return data
   LogNormalESDFPerc(mu = .012, sigma = .03, n= 10, investment = 5, perc = .8, cl = .99, hp = 40)

Description

Gives figure showing the VaR and ES and probability distribution function against L/P of a portfolio assuming geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalESFigure(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots lognormal VaR, ES and pdf against L/P data for given returns data
   data <- runif(5, min = 0, max = .2)
   LogNormalESFigure(returns = data, investment = 5, cl = .95, hp = 90)

   # Plots lognormal VaR, ES and pdf against L/P data with given parameters
   LogNormalESFigure(mu = .012, sigma = .03, investment = 5, cl = .95, hp = 90)

Description

Plots the ES of a portfolio against confidence level assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalESPlot2DCL(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots ES against confidence level
   data <- runif(5, min = 0, max = .2)
   LogNormalESPlot2DCL(returns = data, investment = 5,
                       cl = seq(.9,.99,.01), hp = 60)

   # Plots ES against confidence level
   LogNormalESPlot2DCL(mu = .012, sigma = .03, investment = 5,
                       cl = seq(.9,.99,.01), hp = 40)

Description

Plots the ES of a portfolio against holding period assuming that geometric returns are normal distributed, for specified confidence level and holding period.

Usage

LogNormalESPlot2DHP(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalESPlot2DHP(returns = data, investment = 5, cl = .95, hp = 60:90)

   # Computes v given mean and standard deviation of return data
   LogNormalESPlot2DHP(mu = .012, sigma = .03, investment = 5, cl = .99, hp = 40:80)

Description

Plots the ES of a portfolio against confidence level and holding period assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalESPlot3D(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against confidene level given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalESPlot3D(returns = data, investment = 5, cl = seq(.9,.99,.01), hp = 1:100)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogNormalESPlot3D(mu = .012, sigma = .03, investment = 5, cl = seq(.9,.99,.01), hp = 1:100)

Description

Estimates the VaR of a portfolio assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalVaR(...)

Arguments

Value

Matrix of VaR whose dimension depends on dimension of hp and cl. If cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, the matrix is column matrix and if both cl and hp are vectors, the matrix has dimension length of cl * length of hp.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes VaR given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalVaR(returns = data, investment = 5, cl = .95, hp = 90)

   # Computes VaR given mean and standard deviation of return data
   LogNormalVaR(mu = .012, sigma = .03, investment = 5, cl = .95, hp = 90)

Description

Estimates the percentile of VaR distribution function for normally distributed geometric returns, using the theory of order statistics.

Usage

LogNormalVaRDFPerc(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles of VaR distribution
   data <- runif(5, min = 0, max = .2)
   LogNormalVaRDFPerc(returns = data, investment = 5, perc = .7, cl = .95, hp = 60)

   # Computes v given mean and standard deviation of return data
   LogNormalVaRDFPerc(mu = .012, sigma = .03, n= 10, investment = 5, perc = .8, cl = .99, hp = 40)

Description

Plots the VaR and ETL of a portfolio against confidence level assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalVaRETLPlot2DCL(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR and ETL against confidene level given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalVaRETLPlot2DCL(returns = data, investment = 5, cl = seq(.85,.99,.01), hp = 60)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogNormalVaRETLPlot2DCL(mu = .012, sigma = .03, investment = 5, cl = seq(.85,.99,.01), hp = 40)

Description

Gives figure showing the VaR and probability distribution function against L/P of a portfolio assuming geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalVaRFigure(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots lognormal VaR and pdf against L/P data for given returns data
   data <- runif(5, min = 0, max = .2)
   LogNormalVaRFigure(returns = data, investment = 5, cl = .95, hp = 90)

   # Plots lognormal VaR and pdf against L/P data with given parameters
   LogNormalVaRFigure(mu = .012, sigma = .03, investment = 5, cl = .95, hp = 90)

Description

Plots the VaR of a portfolio against confidence level assuming that geometric returns are normally distributed, for specified confidence level and holding period.

Usage

LogNormalVaRPlot2DCL(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against confidene level given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalVaRPlot2DCL(returns = data, investment = 5, cl = seq(.85,.99,.01), hp = 60)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogNormalVaRPlot2DCL(mu = .012, sigma = .03, investment = 5, cl = seq(.85,.99,.01), hp = 40)

Description

Plots the VaR of a portfolio against holding period assuming that geometric returns are normal distributed, for specified confidence level and holding period.

Usage

LogNormalVaRPlot2DHP(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes VaR given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogNormalVaRPlot2DHP(returns = data, investment = 5, cl = .95, hp = 60:90)

   # Computes VaR given mean and standard deviation of return data
   LogNormalVaRPlot2DHP(mu = .012, sigma = .03, investment = 5, cl = .99, hp = 40:80)

Description

Plots the VaR of a portfolio against confidence level and holding period assuming that geometric returns are normal distributed, for specified confidence level and holding period.

Usage

LogNormalVaRPlot3D(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against confidene level given geometric return data
   data <- rnorm(5, .09, .03)
   LogNormalVaRPlot3D(returns = data, investment = 5, cl = seq(.9,.99,.01), hp = 1:100)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogNormalVaRPlot3D(mu = .012, sigma = .03, investment = 5, cl = seq(.9,.99,.01), hp = 1:100)

Description

Estimates the ES of a portfolio assuming that geometric returns are Student-t distributed, for specified confidence level and holding period.

Usage

LogtES(...)

Arguments

Value

Matrix of ES whose dimension depends on dimension of hp and cl. If cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, the matrix is column matrix and if both cl and hp are vectors, the matrix has dimension length of cl * length of hp.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtES(returns = data, investment = 5, df = 6, cl = .95, hp = 90)

   # Computes ES given mean and standard deviation of return data
   LogtES(mu = .012, sigma = .03, investment = 5, df = 6, cl = .95, hp = 90)

Description

Plots the ES of a portfolio against confidence level assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtESDFPerc(...)

Arguments

Value

Percentiles of ES distribution function

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles of ES distribution
   data <- runif(5, min = 0, max = .2)
   LogtESDFPerc(returns = data, investment = 5, perc = .7, df = 6, cl = .95, hp = 60)

   # Computes v given mean and standard deviation of return data
   LogtESDFPerc(mu = .012, sigma = .03, n= 10, investment = 5, perc = .8, df = 6, cl = .99, hp = 40)

Description

Plots the ES of a portfolio against confidence level assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtESPlot2DCL(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtESPlot2DCL(returns = data, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 60)

   # Computes v given mean and standard deviation of return data
   LogtESPlot2DCL(mu = .012, sigma = .03, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 40)

Description

Plots the ES of a portfolio against holding period assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtESPlot2DHP(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes ES given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtESPlot2DHP(returns = data, investment = 5, df = 6, cl = .95, hp = 60:90)

   # Computes v given mean and standard deviation of return data
   LogtESPlot2DHP(mu = .012, sigma = .03, investment = 5, df = 6, cl = .99, hp = 40:80)

Description

Plots the ES of a portfolio against confidence level and holding period assuming that geometric returns are Student-t distributed, for specified confidence level and holding period.

Usage

LogtESPlot3D(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots ES against confidene level given geometric return data
   data <- rnorm(5, .09, .03)
   LogtESPlot3D(returns = data, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 1:100)

   # Computes ES against confidence level given mean and standard deviation of return data
   LogtESPlot3D(mu = .012, sigma = .03, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 1:100)

Description

Estimates the VaR of a portfolio assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtVaR(...)

Arguments

Value

Matrix of VaRs whose dimension depends on dimension of hp and cl. If cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, the matrix is column matrix and if both cl and hp are vectors, the matrix has dimension length of cl * length of hp.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes VaR given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtVaR(returns = data, investment = 5, df = 6, cl = .95, hp = 90)

   # Computes VaR given mean and standard deviation of return data
   LogtVaR(mu = .012, sigma = .03, investment = 5, df = 6, cl = .95, hp = 90)

Description

Plots the VaR of a portfolio against confidence level assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtVaRDFPerc(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Estimates Percentiles of VaR distribution
   data <- runif(5, min = 0, max = .2)
   LogtVaRDFPerc(returns = data, investment = 5, perc = .7,
                 df = 6, cl = .95, hp = 60)

   # Computes v given mean and standard deviation of return data
   LogtVaRDFPerc(mu = .012, sigma = .03, n= 10, investment = 5,
                 perc = .8, df = 6, cl = .99, hp = 40)

Description

Plots the VaR of a portfolio against confidence level assuming that geometric returns are Student-t distributed, for specified confidence level and holding period.

Usage

LogtVaRPlot2DCL(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against confidene level given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtVaRPlot2DCL(returns = data, investment = 5, df = 6, cl = seq(.85,.99,.01), hp = 60)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogtVaRPlot2DCL(mu = .012, sigma = .03, investment = 5, df = 6, cl = seq(.85,.99,.01), hp = 40)

Description

Plots the VaR of a portfolio against holding period assuming that geometric returns are Student t distributed, for specified confidence level and holding period.

Usage

LogtVaRPlot2DHP(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes VaR given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtVaRPlot2DHP(returns = data, investment = 5, df = 6, cl = .95, hp = 60:90)

   # Computes VaR given mean and standard deviation of return data
   LogtVaRPlot2DHP(mu = .012, sigma = .03, investment = 5, df = 6, cl = .99, hp = 40:80)

Description

Plots the VaR of a portfolio against confidence level and holding period assuming that geometric returns are Student-t distributed, for specified confidence level and holding period.

Usage

LogtVaRPlot3D(...)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots VaR against confidene level given geometric return data
   data <- runif(5, min = 0, max = .2)
   LogtVaRPlot3D(returns = data, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 1:100)

   # Computes VaR against confidence level given mean and standard deviation of return data
   LogtVaRPlot3D(mu = .012, sigma = .03, investment = 5, df = 6, cl = seq(.9,.99,.01), hp = 1:100)

Description

Function derives the VaR of a long Black Scholes call for specified confidence level and holding period, using analytical solution.

Usage

LongBlackScholesCallVaR(stockPrice, strike, r, mu, sigma, maturity, cl, hp)

Arguments

Value

Price of European Call Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Hull, John C.. Options, Futures, and Other Derivatives. 4th ed., Upper Saddle River, NJ: Prentice Hall, 200, ch. 11.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Estimates the price of an American Put
   LongBlackScholesCallVaR(27.2, 25, .03, .12, .2, 60, .95, 40)

Description

Function derives the VaR of a long Black Scholes put for specified confidence level and holding period, using analytical solution.

Usage

LongBlackScholesPutVaR(stockPrice, strike, r, mu, sigma, maturity, cl, hp)

Arguments

Value

Price of European put Option

Author(s)

Dinesh Acharya

References

Dowd, Kevin. Measuring Market Risk, Wiley, 2007.

Hull, John C.. Options, Futures, and Other Derivatives. 4th ed., Upper Saddle River, NJ: Prentice Hall, 200, ch. 11.

Lyuu, Yuh-Dauh. Financial Engineering & Computation: Principles, Mathematics, Algorithms, Cambridge University Press, 2002.

Examples

# Estimates the price of an American Put
   LongBlackScholesPutVaR(27.2, 25, .03, .12, .2, 60, .95, 40)

Description

Derives the first Lopez (i.e. binomial) forecast evaluation score for a VaR risk measurement model.

Usage

LopezBacktest(Ra, Rb, cl)

Arguments

Value

Something

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Lopez, J. A. Methods for Evaluating Value-at-Risk Estimates. Federal Reserve Bank of New York Economic Policy Review, 1998, p. 121.

Lopez, J. A. Regulatory Evaluations of Value-at-Risk Models. Journal of Risk 1999, 37-64.

Examples

# Has to be modified with appropriate data:
   # LopezBacktest for given parameters
   a <- rnorm(1*100)
   b <- abs(rnorm(1*100))+2
   LopezBacktest(a, b, 0.95)

Description

Plots mean-excess function values of the data set.

Usage

MEFPlot(Ra)

Arguments

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Plots mean-excess function values
   Ra <- rnorm(1000)
   MEFPlot(Ra)

Description

Estimates the ES of a portfolio assuming that P/L is normally distributed, for specified confidence level and holding period.

Usage

NormalES(...)

Arguments

Value

Matrix of ES whose dimension depends on dimension of hp and cl. If cl and hp are both scalars, the matrix is 1 by 1. If cl is a vector and hp is a scalar, the matrix is row matrix, if cl is a scalar and hp is a vector, the matrix is column matrix and if both cl and hp are vectors, the matrix has dimension length of cl * length of hp.

Author(s)

Dinesh Acharya

References

Dowd, K. Measuring Market Risk, Wiley, 2007.

Examples

# Computes VaR given P/L
   data <- runif(5, min = 0, max = .2)
   NormalES(returns = data, cl = .95, hp = 90)

   # Computes VaR given mean and standard deviation of P/L data
   NormalES(mu = .012, sigma = .03, cl = .95, hp = 90)