Extended Vasicek Credit Loss Model with Macroeconomic Factors

Description

Fits the extended Vasicek single-factor credit loss model where the probability of default depends on macroeconomic covariates. Maximum likelihood estimates of all parameters, including asset value correlation, are obtained via closed-form probit-transformed OLS regression. Also provides density, distribution, quantile, and random generation functions for the Vasicek loss distribution.

Author(s)

Dmitriy Mayorov

References

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 160–162.

Yang, B. H. (2014). Estimating Long-Run PD, Asset Correlation, and Portfolio Level PD by Vasicek Models. MPRA Paper No. 57244.

See Also

vasicekfit, predict.vasicekfit, dvasicek

The Vasicek Loss Distribution

Description

Density, distribution function, quantile function, and random generation for the Vasicek credit loss distribution.

Usage

dvasicek(x, p, rho, kappa = NULL, u = NULL, log = FALSE)
pvasicek(q, p, rho, kappa = NULL, u = NULL, lower.tail = TRUE, log.p = FALSE)
qvasicek(prob, p, rho, kappa = NULL, u = NULL, lower.tail = TRUE, log.p = FALSE)
rvasicek(n, p, rho, kappa = NULL, u = NULL)

Arguments

Details

The Vasicek loss distribution arises from the single-factor Gaussian copula model. Its probability density function is

f(x) = \sqrt{\frac{1-\rho}{\rho}} \exp\left\{\frac{1}{2}\left(\Phi^{-1}(x)^2 - \left[\frac{\sqrt{1-\rho}\,\Phi^{-1}(x) - \sum \kappa_j u_j - \Phi^{-1}(p)}{\sqrt{\rho}}\right]^2\right)\right\}

When kappa and u are NULL, this reduces to the standard two-parameter Vasicek distribution.

Value

dvasicek gives the density, pvasicek gives the distribution function, qvasicek gives the quantile function, and rvasicek generates random deviates.

References

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 160–162.

See Also

vasicekfit

Examples

# Standard Vasicek density
curve(dvasicek(x, p = 0.03, rho = 0.10), from = 0.001, to = 0.20,
      ylab = "Density", main = "Vasicek loss distribution")

# 99th percentile (Basel-style)
qvasicek(0.99, p = 0.03, rho = 0.10)

# Check CDF and quantile are inverses
qvasicek(pvasicek(0.05, p = 0.03, rho = 0.10), p = 0.03, rho = 0.10)

# Random sample
set.seed(1)
hist(rvasicek(10000, p = 0.03, rho = 0.10), breaks = 100,
     main = "Vasicek loss distribution samples")

Predict Method for vasicekfit Objects

Description

Obtain predictions from a fitted Vasicek model, optionally at specified confidence levels.

Usage

## S3 method for class 'vasicekfit'
predict(object, newdata = NULL,
        type = c("link", "response"), alpha = NULL, ...)

Arguments

Value

If alpha is NULL, a numeric vector of predictions. If alpha is supplied, a matrix with rows corresponding to observations and columns to confidence levels. When alpha is a scalar, the result is a vector.

See Also

vasicekfit

Examples

set.seed(42)
n <- 100
u <- rnorm(n)
x <- pnorm((qnorm(0.03) + 0.1 * u + sqrt(0.02) * rnorm(n)) / sqrt(1 - 0.02))
d <- data.frame(default_rate = x, macro = u)
fit <- vasicekfit(default_rate ~ macro, data = d)

# Conditional mean PD
predict(fit, type = "response")

# 99th percentile loss rate under stress
predict(fit, newdata = data.frame(macro = 2), alpha = 0.99,
        type = "response")

# Multiple confidence levels
predict(fit, newdata = data.frame(macro = c(0, 1, 2)),
        alpha = c(0.95, 0.99, 0.999), type = "response")

Fit an Extended Vasicek Credit Loss Model

Description

Fits the extended Vasicek single-factor credit loss model where the probability of default depends on macroeconomic covariates. All parameters, including asset value correlation, are estimated via closed-form probit-transformed OLS.

Usage

vasicekfit(formula, data, bias_correct = FALSE, portfolio_size = NULL)

Arguments

Value

An object of class "vasicekfit", which is a list containing:

p

estimated probability of default (baseline PD).

rho

estimated asset value correlation.

kappa

named numeric vector of macro-factor sensitivities.

sigma2

MLE variance estimate in probit space.

lm_fit

the underlying lm object.

fitted.values

fitted values in probit (Y) space.

residuals

residuals in probit (Y) space.

formula

the model formula.

call

the matched call.

terms

the terms object.

model

the model frame.

bias_correct

logical flag used.

portfolio_size

portfolio size used, or NULL.

References

Vasicek, O. A. (2002). The distribution of loan portfolio value. Risk, 15(12), 160–162.

Yang, B. H. (2014). Estimating Long-Run PD, Asset Correlation, and Portfolio Level PD by Vasicek Models. MPRA Paper No. 57244.

See Also

predict.vasicekfit, dvasicek

Examples

set.seed(42)
n <- 100
u1 <- rnorm(n)
u2 <- rnorm(n)
p_true <- 0.03; rho_true <- 0.02
kappa_true <- c(0.13, -0.07)
z <- rnorm(n)
x <- pnorm((qnorm(p_true) + kappa_true[1] * u1 + kappa_true[2] * u2 +
  sqrt(rho_true) * z) / sqrt(1 - rho_true))
d <- data.frame(default_rate = x, unemp = u1, hpi = u2)
fit <- vasicekfit(default_rate ~ unemp + hpi, data = d)
fit

Methods for vasicekfit Objects

Description

Print, summary, coefficient extraction, covariance, confidence interval, and accessor methods for "vasicekfit" objects.

Usage

## S3 method for class 'vasicekfit'
print(x, ...)
## S3 method for class 'vasicekfit'
summary(object, ...)
## S3 method for class 'vasicekfit'
coef(object, ...)
## S3 method for class 'vasicekfit'
fitted(object, ...)
## S3 method for class 'vasicekfit'
residuals(object, ...)
## S3 method for class 'vasicekfit'
vcov(object, ...)
## S3 method for class 'vasicekfit'
confint(object, parm, level = 0.95, ...)

Arguments

Details

coef returns the recovered Vasicek parameters (p, rho, and kappas) in the original parameter space.

fitted and residuals return values in probit (Y) space, consistent with the underlying OLS regression.

vcov returns the delta-method covariance matrix for the original-space parameters (p, \rho, \kappa_1, \ldots, \kappa_m), obtained by propagating the joint covariance of (\hat\beta, \hat\sigma^2) through the closed-form recovery transform. The (\hat\beta, \hat\sigma^2) block is treated as block-diagonal by the independence of sample mean and variance under normality. The variance of \hat\sigma^2 uses 2 \sigma^4 (N - m - 1)/N^2 for the MLE and 2 \sigma^4 / (N - m - 1) when bias_correct = TRUE.

confint returns Wald intervals based on vcov.

Value

print and summary return object invisibly.

coef returns a named numeric vector.

fitted and residuals return numeric vectors.

vcov returns a (m+2) \times (m+2) covariance matrix.

confint returns a two-column matrix of lower and upper bounds.

See Also

vasicekfit, predict.vasicekfit