[Paper Review] Incorporating exchange rate risk into PDs and asset correlations

Dirk Tasche|ArXiv.org|Dec 20, 2007

Credit Risk and Financial Regulations2 references3 citations

TL;DR

This paper develops a framework to adjust default probabilities (PDs) and asset correlations for exchange rate risk by integrating Merton’s structural model, Garman-Kohlhagen foreign exchange pricing, and Vasicek’s correlation model. The key contribution is a consistency condition (Equation 12) that links adjusted PDs and correlations without requiring estimates of hard-to-estimate parameters like asset volatility.

ABSTRACT

Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well-known models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we develop simple representations of PDs and asset correlations that take into account exchange rate risk. From these results, consistency conditions can be derived that link the changes in PD and asset correlation and do not require knowledge of hard-to-estimate parameters like asset value volatility.

Motivation & Objective

• To model how exchange rate risk increases the probability of default (PD) and asset correlation for borrowers with mismatched currency denominations.
• To derive adjustments to PDs and asset correlations when assets and debt are in different currencies, using established financial models.
• To identify a consistency condition linking adjusted PDs and correlations that does not require knowledge of asset value volatility or other hard-to-estimate parameters.
• To provide a practical tool for credit risk modeling under currency mismatch, especially in regulatory and internal risk management contexts.

Proposed method

• Combines Merton’s structural default model with Garman-Kohlhagen’s foreign exchange model to represent the default condition under currency mismatch.
• Models the logarithmic change in exchange rates and firm asset values as correlated geometric Brownian motions with a shared correlation parameter.
• Derives explicit formulas for adjusted PDs (Equation 11a) and asset correlations (Equation 11b) under the assumption of independent exchange rate and asset value processes.
• Derives a consistency condition (Equation 12) by eliminating asset volatility and exchange rate volatility parameters under the assumptions of zero mean exchange rate change and uncorrelated processes.
• Applies the Vasicek (2002) framework for default correlation to model dependence between borrowers under currency risk.
• Uses the standard normal inverse cumulative distribution function to express PDs and correlations in terms of standardized normal variables.

Experimental results

Research questions

• RQ1 How does exchange rate risk affect the probability of default for a borrower whose assets and debt are denominated in different currencies?
• RQ2 How does exchange rate risk influence the asset correlation between two borrowers with currency mismatches?
• RQ3 Can a consistency condition be derived that links adjusted PDs and asset correlations without requiring estimates of asset volatility or exchange rate volatility?
• RQ4 What is the impact of exchange rate risk on the joint default behavior of borrowers in a credit portfolio?
• RQ5 How do the adjusted PDs and correlations behave under the homogeneous portfolio assumption?

Key findings

• When assets and debt are in different currencies, the PD increases due to exchange rate risk, even if the mean exchange rate change is zero.
• The asset correlation between borrowers increases under exchange rate risk, especially when their asset values and exchange rates are correlated.
• Under the assumption of uncorrelated asset and exchange rate processes and zero mean exchange rate change, a consistency condition (Equation 12) links adjusted PDs and correlations without requiring asset volatility estimates.
• For homogeneous borrowers, the consistency condition simplifies to $\frac{1-\varrho^{\ast}}{1-\varrho} = \frac{\Phi^{-1}(p^{\ast})^2}{\Phi^{-1}(p)^2}$, showing that correlations grow over-proportionally as adjusted PDs decrease.
• The adjusted asset correlation approaches 1 as the original PD approaches 0.5, indicating increasing tail dependence under currency mismatch.
• The curves relating adjusted PD and adjusted correlation are strictly concave, implying that the impact of exchange rate risk is most pronounced at low default probabilities.