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QRM

Credit

Credit Risk Modelling

Description

Functions for modelling credit risk:

Bernoulli mixture model with prescribed default and joint default probabilities
Bernoulli mixture model with Clayton copula dependencies of default.
Probitnormal Mixture of Bernoullis
Beta-Binomial Distribution
Logitnormal-Binomial Distribution
Probitnormal-Binomial Distribution

Usage

cal.beta(pi1, pi2)
cal.claytonmix(pi1, pi2)
cal.probitnorm(pi1, pi2)
dclaytonmix(x, pi, theta) 
pclaytonmix(q, pi, theta) 
rclaytonmix(n, pi, theta)
rtcopulamix(n, pi, rho.asset, df)
dprobitnorm(x, mu, sigma) 
pprobitnorm(q, mu, sigma) 
rprobitnorm(n, mu, sigma)
rbinomial.mixture(n = 1000, m = 100,
                  model = c("probitnorm", "logitnorm", "beta"), ...)
rlogitnorm(n, mu, sigma)
fit.binomial(M, m)
fit.binomialBeta(M, m, startvals = c(2, 2), ses = FALSE, ...)
fit.binomialLogitnorm(M, m, startvals = c(-1, 0.5), ...)
fit.binomialProbitnorm(M, m, startvals = c(-1, 0.5), ...)
momest(data, trials, limit = 10)

Arguments

`data`	`vector`, numbers of defaults in each time period.
`df`	`numeric`, degree of freedom.
`limit`	`intgeger`, maximum order of joint default probability to estimate.
`M`	`vector`, count of successes.
`m`	`vector`, count of trials.
`model`	`character`, name of mixing distribution.
`mu`	`numeric`, location parameter.
`n`	`integer`, count of random variates.
`pi`	`numeric`, default probability.
`pi1`	`numeric`, default probability.
`pi2`	`numeric`, joint default probability.
`q`	`numeric`, values at which CDF should be evaluated.
`sigma`	`numeric`, scale parameter.
`ses`	`logical`, whether standard errors should be returned.
`startvals`	`numeric`, starting values.
`theta`	`numeric`, parameter of distribution.
`trials`	`vector`, group sizes in each time period.
`x`	`numeric`, values at which density should be evaluated.
`rho.asset`	`numeric`, asset correlation parameter.
`...`	ellipsis, arguments are passed down to either mixing distribution or `nlminb()`.

Details

cal.beta(): calibrates a beta mixture distribution on unit interval to give an exchangeable Bernoulli mixture model with prescribed default and joint default probabilities (see pages 354-355 in QRM).
cal.claytonmix(): calibrates a mixture distribution on unit interval to give an exchangeable Bernoulli mixture model with prescribed default and joint default probabilities. The mixture distribution is the one implied by a Clayton copula model of default (see page 362 in QRM).
cal.probitnorm(): calibrates a probitnormal mixture distribution on unit interval to give an exchangeable Bernoulli mixture model with prescribed default and joint default probabilities (see page 354 in QRM).
dclaytonmix(), pclaytonmix(), rclaytonmix(): density, cumulative probability, and random generation for a mixture distribution on the unit interval which gives an exchangeable Bernoulli mixture model equivalent to a Clayton copula model (see page 362 in QRM).
fit.binomial(): fits binomial distribution by maximum likelihood.
dprobitnorm(), pprobitnorm(), rprobitnorm(): density, cumulative probability and random number generation for distribution of random variable Q on unit interval such that the probit transform of Q has a normal distribution with parameters mu and sigma (see pages 353-354 in QRM).
fit.binomialBeta(): fit a beta-binomial distribution by maximum likelihood.
fit.binomialLogitnorm(): fits a mixed binomial distribution where success probability has a logitnormal distribution. Lower and upper bounds for the input parameters M and m can be specified by means of the arguments lower and upper, which are passed to nlminb(). If convergence occurs at an endpoint of either limit, one need to reset lower and upper parameter estimators and run the function again.
fit.binomialProbitnorm(): Fits a mixed binomial distribution where success probability has a probitnormal distribution. Lower and upper bounds for the input parameters M and m can be specified by means of the arguments lower and upper, which are passed to nlminb(). If convergence occurs at an endpoint of either limit, one need to reset lower and upper parameter estimators and run the function again.
momest(): calculates moment estimator of default probabilities and joint default probabilities for a homogeneous group. First returned value is default probability estimate; second value is estimate of joint default probability for two firms; and so on (see pages 375-376 in QRM).
rbinomial.mixture(): random variates from mixed binomial distribution (see pages 354-355 and pages 375-377 of QRM).
rlogitnorm(): Random number generation for distribution of random variable Q on unit interval such that the probit transform of Q has a normal distribution with parameters mu and sigma (see pages 353-354 in QRM).
rtcopulamix(): random generation for mixing distribution on unit interval yielding Student's t copula model (see page 361 in QRM, exchangeable case of this model is considered).

Examples

## calibrating models
pi.B <- 0.2
pi2.B <- 0.05 
probitnorm.pars <- cal.probitnorm(pi.B, pi2.B) 
probitnorm.pars 
beta.pars <- cal.beta(pi.B, pi2.B) 
beta.pars 
claytonmix.pars <- cal.claytonmix(pi.B, pi2.B) 
claytonmix.pars 
q <- (1:1000) / 1001 
q <- q[q < 0.25] 
p.probitnorm <- pprobitnorm(q, probitnorm.pars[1],
                            probitnorm.pars[2]) 
p.beta <- pbeta(q, beta.pars[1], beta.pars[2]) 
p.claytonmix <- pclaytonmix(q, claytonmix.pars[1],
                            claytonmix.pars[2]) 
scale <- range((1 - p.probitnorm), (1 - p.beta), (1 - p.claytonmix)) 
plot(q, (1 - p.probitnorm), type = "l", log = "y", xlab = "q", 
           ylab = "P(Q > q)",ylim=scale) 
lines(q, (1 - p.beta), col = 2) 
lines(q, (1 - p.claytonmix), col = 3) 
legend("topright", c("Probit-normal", "Beta", "Clayton-Mixture"), 
          lty=rep(1,3),col = (1:3))
## Clayton Mix
pi.B <- 0.0489603 
pi2.B <- 0.003126529 
claytonmix.pars <- cal.claytonmix(pi.B, pi2.B)
claytonmix.pars
q <- (1:1000) / 1001
q <- q[q < 0.25]
d.claytonmix <- dclaytonmix(q, claytonmix.pars[1], claytonmix.pars[2])
head(d.claytonmix)
## SP Data
data(spdata.raw) 
attach(spdata.raw) 
BdefaultRate <- Bdefaults / Bobligors 
## Binomial Model
mod1a <- fit.binomial(Bdefaults, Bobligors)
## Binomial Logitnorm Model
mod1b <- fit.binomialLogitnorm(Bdefaults, Bobligors) 
## Binomial Probitnorm Model
mod1c <- fit.binomialProbitnorm(Bdefaults, Bobligors)
## Binomial Beta Model
mod1d <- fit.binomialBeta(Bdefaults, Bobligors); 
## Moment estimates for default probabilities
momest(Bdefaults, Bobligors)
pi.B <- momest(Bdefaults, Bobligors)[1]
pi2.B <- momest(Bdefaults, Bobligors)[2]
## Probitnorm
probitnorm.pars <- cal.probitnorm(pi.B, pi2.B) 
q <- (1:1000)/1001
q <- q[ q < 0.25]
d.probitnorm <- dprobitnorm(q, probitnorm.pars[1], probitnorm.pars[2])
p <- c(0.90,0.95,0.975,0.99,0.995,0.999,0.9999,0.99999,0.999999)
sigma <- 0.2 * 10000 / sqrt(250)
VaR.t4 <- qst(p, df = 4, sd = sigma, scale = TRUE)
VaR.t4
detach(spdata.raw)
## Binomial Mixture Models
pi <- 0.04896 
pi2 <- 0.00321 
beta.pars <- cal.beta(pi, pi2)
probitnorm.pars <- cal.probitnorm(pi, pi2) 
n <- 1000 
m <- rep(500, n) 
mod2a <- rbinomial.mixture(n, m, "beta", shape1 = beta.pars[1],
                          shape2 = beta.pars[2]) 
mod2b <- rbinomial.mixture(n, m, "probitnorm",
                          mu = probitnorm.pars[1],
                          sigma = probitnorm.pars[2])

QRM

Provides R-Language Code to Examine Quantitative Risk Management Concepts

v0.4-31

GPL (>= 2)

Authors

Bernhard Pfaff [aut, cre], Marius Hofert [ctb], Alexander McNeil [aut] (S-Plus original (QRMlib)), Scott Ulmann [trl] (First R port as package QRMlib)

Initial release

2020-02-15