Estimating Portfolio Credit Losses in Downturns

• Published date: 01 December 2015
• DOI: http://doi.org/10.1111/fmii.12033
• Date: 01 December 2015
• Author: Fernando F. Moreira

Estimating Portfolio Credit Losses in Downturns

BYFERNANDO F. MOREIRA

This paper suggests formulas able to capture potential strong connection among credit

losses in downturns without assuming any specific distribution for the variables involved.

We first show that the current model adopted by regulators (Basel) is equivalent to a

conditional distribution derived from the Gaussian Copula (which does not identify tail

dependence). We then use conditional distributions derivedfrom copulas that express tail

dependence (stronger dependence across higher losses) to estimate the probability of credit

losses in extreme scenarios (crises). Next, we use data on historical credit losses incurred

in American banks to compare the suggested approach to the Basel formula with respect

to their performance when predicting the extreme losses observed in 2009 and 2010. Our

results indicate that, in general, the copula approach outperforms the Basel method in

two of the three credit segments investigated. The proposed method is extendable to other

differentiable copula families and this gives flexibility to future practical applications of

the model.

Keywords: Credit risk, downturns, Basel Accords, conditional distributions, copulas.

JEL Classification: G28, G21, G32, C46, C49.

I. INTRODUCTION

The model (Basel Accord) adopted by regulators in many countries to calculate the

capital to cover unexpectedcredit losses in financial institutions assumes normally-

distributed variables and uses the linear correlation to measure dependence among

losses. However, these assumptions do not allow the identification of possible

asymmetric dependence across losses in extreme scenarios (which seems to be the

case for several financial assets, loans included) and, therefore, the Basel method

may misestimate joint credit losses in periods of crisis.

Albeit the formula currently used in Basel Accords has a derivation not asso-

ciated to copula functions, we show that it turns out to be equivalent to the first

derivative of the Gaussian Copula (which denotes symmetric association without

tail dependence). Moreover, the distributionof one variable conditional on another

variable can be calculated as the first derivative of the copula that represents the

dependence between the considered variables with respect to the conditioning

variable. In other words, the Basel formula can be interpreted as the cumulative

distribution of a latent variable (asset returns of obligors, for instance) conditional

on the economic status. Based on this interpretation of the Basel model, we propose

the use of copulas that capture stronger dependence among high losses (stronger

dependence among low values of debtors’ asset returns) to generate alternative

conditional distributions. So, we keep the basic intuition of the traditional approach

but change the dependence structure so that we can, for example, identify higher

probability of default in adverse scenarios. The alternative model is basically set

Corresponding author: Fernando F. Moreira, University of Edinburgh, Phone: +44(0)131–651–5312. Email:

Fernando.Moreira@ed.ac.uk.

C2015 New YorkUniversity Salomon Center and Wiley Periodicals, Inc.

392 Fernando F. Moreira

as the first derivativeof the copula chosen to represent the relationship between the

latent variable and the economic factor with respect to the latter variable. At this

point, we face a challenge pertaining to the copula parameter that measures the

dependence intensity. For some copulas, this parameter can be directly deduced

from the rank correlation (Kendall’s tau) between the variables. Thus we need to

find the rank correlation between the latent variable of each loan and the economic

factor but we cannot calculate it since we do not have enough information about

the second variable. To overcome this problem, we show that the rank correlation

between the latent variable of each debtor and the economic factor is related to the

rank correlation between two latent variables (e.g., asset returns of two obligors)

which can be presumed from past losses (default rates). Once we have an estimate

for the former rank correlation, we will have all necessary information to calculate

the conditional probability by means of the first derivativeof a copula with a given

confidence (unfavorable economic level).

As examples, we present two formulas originated from the Clayton and the

Student t Copulas that are able to detect stronger connection (tail dependence)

among low values of latent variables (which is equivalent to identify higher de-

pendence among high credit losses). These formulas do not assume any kind of

distribution for the variables considered and therefore such approach overcomes

the limitations of the existing method with regard to the assumption of normality

and the use of the linear correlation.

We use aggregate data on losses in American banks to check the performance

of the suggested approach and our analyses show that, for two of the three credit

segments considered, the copula formulas typically outperform the Basel formula

regarding the estimation of unusually high losses.

In short, our contributions are threefold: (i) we present an alternative derivation

of the Basel formula and show that it corresponds to the first derivative of the

Gaussian Copula; (ii) we set up a model able to capture stronger dependence among

credit losses in unfavorable scenarios which results in more efficient estimations

of potential extreme losses; and (iii) we propose a way to derive the dependence

between a latent variable of each loan and an economic factor from the dependence

observed across loans’ default rates.

II. COPULAS AND CONDITIONAL DISTRIBUTIONS

Copulas are multivariate distribution functions with uniformly distributed mar-

gins in (0,1) that link marginal (individual) distributions of variables to their joint

distributions:

F1...n(x1,...,xn)=C(F1(x1),...,Fn(xn))

where F(.) denotes a cumulative distribution function and Cstands for a cop-

ula. Thus, Cis an expression (function) with ninputs and, when evaluated at

F1(x1),...,Fn(xn), returns the joint cumulative distribution of the nvariables