Credit Risk Models
Question: What is an appropriate
modeling approach to value
defaultable debt (bonds and loans)?
“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
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.
Structural approach: default in the classical Merton
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, zerocoupon 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
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
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
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
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,
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
• Default is modeled as a continuous variable with an underlying
• 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
Using the formula for the Poisson distribution
The calculated probability of default in band 1
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
• 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 markto-market model.