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### Credit Risk Models

Question: What is an appropriate modeling approach to value defaultable debt (bonds and loans)?

Gieseke “Credit Risk Modeling and Valuation: An Introduction,” October 2004

Three main approaches to modeling credit risk in the finance literature

- Structural approach: Assumptions are made about the dynamics of a firm’s assets, its capital structure, and its debt and share holders. A firm defaults if the assets are insufficient according to some measure. A liability is characterized as an option on the firm’s assets.
- Reduced form approach: No assumptions are made concerning why a default occurs. Rather, the dynamics of default are exogenously given by the default rate (or intensity). Prices of credit sensitive securities can be calculated as if they were default free using the risk free rate adjusted by the level of intensity.
- Incomplete information approach: Combines the structural and reduced form approaches.

1. We want to use the structural approach to incorporate bond default risk in bond valuation

The value of the firm’s assets are assumed to follow the process,

where μ is the instantaneous expected rate of return on assets, and σ is the standard deviation of the return on assets.

Let D(t,T) be the date t market value of debt with promised payment B at date t.

The second line in (18.2) says that the payoff to the creditors equals the promised payment (B) minus the payoff on a European put option written on the firm’s assets with exercise price B.

Market value of firm debt, D(t)

Let P(t,T) represent the current date t price of a default-free, zero-coupon bond that pays $1 at date T, where the bond conforms with the Vasicek model in Ch. 9.

Pennacchi asserts that using results for pricing options (Ch. 9.3) when interest rates are random (as in 9.58), we can write

Market value of firm equity, E(t)

Shareholder equity is similar to a call option on the firm’s assets, since at maturity the payoff to equity holders is max [A(t) – B, 0].

However, shareholder equity is different from a European option if the firm pay dividends to shareholders prior to maturity as reflected in the first term of the last line in (18.4) where δ denotes the dividend rate.

Critique of the Merton model

The Merton model assumption is that the firm has a single issue of zero-coupon debt. That is unrealistic. Modeling multiple issues with different maturities and seniorities complicates default.

In response some models have suggested that default occurs when the firm’s assets hit a lower boundary. That boundary has a monotonic relation to the firm’s total outstanding debt. The first passage time is when the value of the firm’s assets crosses through the lower boundary.

First passage model - - bond indenture provisions often include safety covenants that give bond holders the right to reorganize the firm if the value falls below a given barrier.

The first passage model defines the survival probability as p(t,T) that the distance to default does not reach zero at any date τ between t and T. The distance to default is often measured in terms of standard deviations.

Structural approach: default in the first passage model (Black & Cox, 1976)

2. We want to use the reduced form approach to incorporate bond default risk in bond valuation

The reduced form model was developed to overcome the nontradeability and nonobservability of the firm’s asset value process (Jarrow &Turnbull, 1992).

Default is not tied to the dynamics of asset prices and this breaks the link between the firm’s balance sheet and the likelihood of default.

Rather, default is based on an exogenous Poisson process, so it may be better able to capture the effects of default due to additional unobserved factors.

Reduced form models can also be used to value defaultable bonds using the techniques used for default-free bonds.

In the reduced form framework, we assume that the default event depends on a “reduced form process,” that may depend on the firm’s assets and capital structure, but also on other macroeconomic factors that influence default.

The default event for a firm’s bond is modeled as a Poisson process with a time-varying “default intensity.”

Conditional on no default occurring up to date t, the instantaneous probability of default at (t, t+dt) is denoted as λ(t) dt, where λ(t) is the physical default intensity, or “hazard rate,” where it is assumed that λ(t) ≥ 0. Since λ(t) is time-varying, it may be linked to changes in underlying state variables.

We can compute the physical probability that a bond will not default from date t to date τ where t ≤ τ ≤ T. This physical survival probability is written

Zero recovery bond (bond holders receive nothing in the event of a default)

With zero recovery the bondholder payoff at date T is either D(T,T) = B if no default occurs, or D(T,T) = 0 if default occurred during (t,T).

If we apply risk-neutral pricing, the date t value of a zero-recovery bond can be written

where r is the instantaneous “default-free” interest rate, which gives us the risk-neutral default intensity rather than the physical default intensity in (18.5).

The risk-neutral default intensity accounts for the market price of risk due to the Poisson arrival of the default event.

Value of the zero-recovery defaultable bond

Using the calculated survival probability in (18.5) we get

So, (18.9) indicates that valuing a zero-recovery defaultable bond is similar to valuing a default-free bond, except that we use the discount rate, r(u) + λ(t), rather than just r(u).

Default depends on both Brownian motion vector (dz) for the state variables and the Poisson process (dq) for arrival of default) - - the default process is “doubly stochastic”

3. We want to extend the structural and reduced form models for bonds to the case of bank loans

The link between loans and optionality can be illustrated by a payoff function to a bank lender. Here repayment of the loan requires amount 0B. But the market value of project assets can be AL or AH. At AL the borrow would have an incentive to default on the loan contract by forfeiting the assets to the bank. Above 0B the bank earns a fixed return on the loan.

This is analogous to the payoff to a put option writer on a stock with exercise price B.

Structural model (KMV)

The value of a put option on a stock can be written as,

F(S, X, r, σ, T)

The value of a default option on a loan can be written as,

G(A, B, r, σA, T)

where A is the value of the firm’s assets and B is the repayment at maturity. We note that the values for A and σA are not directly observable.

The KMV Credit Monitor Model turns the bank’s lending problem around and considers it from the perspective of the borrower.

To solve for the two unknowns, A and σA , the model uses

- the structural relationship between market value of equity and market value of assets, and
- the relationship between volatility of assets and volatility of equity.

Loan repayment from the perspective of the borrower (equity holder)

The payoff function of the equity holder is a call option on the assets of the firm, H(A, σA, r, B, T).

KMV solves the unobservables problem by assuming that σE = g(σA) where σE is the observable volatility of firm equity and with two equations in two unknowns, we can solve for A and σA . Once these values are derived, KMV calculates the expected default frequency (EDF).

Calculating the theoretical EDF

If A = 100, σA = 10, and B = 80, the distance to default = (A-B)/σA = 2 standard deviations. The value of assets would have to decline by 2 standard deviations in order to enter default.

Based on a sufficiently large sample of firms, we can map the distance to default into EDF

Critique of the KMV model

It is difficult to construct the theoretical EDF curves without the assumption of normality of asset returns

Private firm EDFs can only be constructed by using accounting data and other observable characteristics of the borrower

The KMV approach does not distinguish between different types of debt (bonds that vary by seniority, collateral, covenants, convertibility, etc.)

The KMV model is static - - once the debt is in place the firm does not change it. The default behavior of firms that manage their leverage positions is not captured.

Reduced-form model (CreditRisk+)

The Credit Risk+ model is based on an insurance approach where default is an event that resembles other insurable events (casualty losses, death, injury, etc.). These are generally referred to as mortality models which involve actuarial estimate of the events occurring.

- Default is modeled as a continuous variable with an underlying probability distribution.
- Default uncertainty is one type of uncertainty, there is also uncertainty surrounding the size or severity of the loss.
- Loss severities are distributed into “bands,” and the number of bands is adjusted to get greater accuracy in the estimation.
- The frequency of losses and the severity of losses produce a distribution of losses for each band. Summing across these bands we construct the loss distribution for a portfolio of loans.

Constructing the loss distribution in the CreditRisk+ model

Using the formula for the Poisson distribution

The loss distribution for a single loan portfolio (severity rate = $20,000 per $100,000 of loan)

Critique of the CreditRisk+ model

The observed distribution of losses may have a larger variance than the model shows. This would tend to underestimate the true economic capital requirement.

- This may be due to an assumption that the mean default rate is constant within each band. So, increase the number of bands for more accuracy.
- Default rates across bands may be correlated due to underlying state variables that have broader impact on borrowers.
- The predictive usefulness of the approach depends on the size of the sample of loans.
- The model is not a “full VaR model” because it concentrates on loss rates, not on loan value changes. It is a default model, not a mark-to-market model.

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