Modeling Credit Risk (1): Credit Risk Assessment and Empirical Analysis

Abstract

In recent years, credit risk has become the most important risk borne by banks and other credit institutions. Consequently, credit risk analysis and management methods have generated considerable interest among academics, practitioners, and regulators. In addition, in the aftermath of the last financial crises, the control and monitoring of counterparty risk (distinct from credit risk) has also become a key factor in the performance, and sometimes survival, of these institutions. This chapter addresses the analysis and valuation of securities subject to credit and/or counterparty risk. Section 28.1 focuses on empirical analysis and evaluation tools, while Sect. 28.2 presents the main valuation models and some applications.

We thank Riadh Belhaj and Andras Fulop for their helpful comments on this chapter.

Appendix

Notation and calculation rules for jumps and default events in intensity models.

Basic Definitions

N(t) = number of jumps between 0 and t; τ = date of the first jump or default date.

Events

$$ \left\{ jump\kern0.17em between\ t\;\mathrm{and}\;t+ dt\right\}\equiv \left\{ dN=1\right\}; $$

$$ \left\{ default\ between\ t\ \mathrm{and}\ t+ dt\right\}\equiv \left\{\tau \in \left[t,t+ dt\right]\right\}\equiv \left\{ dN=1\;\mathrm{and}\;N(t)=0\right\}\equiv \left\{ dN=1\;\mathrm{and}\;\tau >t\right\}; $$

$$ \left\{ default\ between\;t\;\mathrm{and}\;t+ dt| survival\ until\ t\right\}\equiv \left\{\tau \in \left[t,t+ dt\right]\ |\tau >t\right\}. $$

Probabilities

Proba{dN = 1} = λ(t) dt (= Proba{dN = 1 | N(t) = n} for all n).

Proba{Ν(t) = 0} = Proba{τ > t} = γ(t) = probability of survival at time t;

γ(t) = ${e}^{\int_0^t-\lambda (u) du}$ = e^{–t Λt}; if λ(t) = λ = constant: γ(t) = e^–λt .

Proba{τ ≤ t} = 1 − γ(t) (distribution law of τ) = ϕ_t;

Proba{τ ∈ [t, t + dt]} = γ(t) λ(t) dt = − dγ.

Proba {τ ∈ [t, t + dt] | τ > t} = λ(t) dt (hence λ(t) = hazard rate).