Credit Risk, Credit Derivatives and Firm Value Based

Models

Willi Semmler∗and Michael Robert

October 2006

Abstract

∗
Schwartz Center for Economic Policy Analysis, New School, New York, and Center for Empirical
Macroeconomics, Bielefeld

1
1 Introduction
Nowadays there have been developed many instruments to transfer credit risk. These
instruments are called credit derivatives. There have also been developed many model
and methods to evaluate credit risk. They range from practical market methods to theory
guided methods relying on firm value. In this note, first some well known instruments
for transferring credit risk are discussed and then, second, firm value based models on
evaluating credit risk are studied. Of course, there are other evaluation methods of credit
risk, for example, intensity based models or credit rating models but here we want to
focus on firm value based models. Those ones have a sound theoretical foundation and,
they are based on the theoretical development of the 1970, put forward by Black and
Scholes (1973) and Merton (1974). Further theoretical foundations of this approach can
be found in Schönbucher (2003), Grüne and Semmler (2005) and Grüne, Semmler and
Bernard (2006).

2 The Relevance of Credit Derivatives

The market for credit derivatives was created in the early 1990s in London and New
York and it is fastest growing derivative market at the moment. Considering only the
period between June 2001 and June 2004, the notional amounts outstanding in billions of
US dollars were 695 and 4.477 respectively according to a recent survey of the Bank for
International Settlements, Switzerland (see table 3 in the appendix). That is a growth of
more than 500 per cent in only three years.
Participants in the market for credit derivatives can be divided into five major groups.
Banks form the largest group with a fraction of about 47 per cent. The second largest
group consists of insurances and re-insurances which cover about 23 per cent of the mar-
ket’s notional outstanding. Other groups are hedge funds (8 per cent) and investment
funds (5 per cent) as well as industrials (4 per cent) of different branches.

2
Table 1 Market share by instrument type (rounded numbers)

Instrument Share (%)

Credit default swaps (including FtDs) 67
Synthetic balance sheet CDOs 12
Tranched portfolio default swaps 9
Credit-linked notes, asset repackaging, asset swaps 7
Credit spread options 2
Managed synthetic CDOs 2
Total return swaps 1
Hybrid credit derivatives 0,2
Source: Risk (Patel, 2002).

When one takes a look at the derivative market with respect to instrument types, one
can see that credit default swaps (CDS) represent about 67 per cent of all transactions
made in that field (see table 1). A reason for this may be the standards for ”plain vanilla”
CDSs developed by the International Swaps and Derivatives Association (ISDA), leading
to lower transaction costs and simplifying the whole business. Further types are discussed
later in this paper.
Purposes for using credit derivatives are, as the types of instruments themselves, man-
ifold. One can think of using credit derivatives as investments, for the credit risk manage-
ment of bond portfolios, for hedging counterparty or country risk in isolated cases, as a
funding opportunity for banks through the securitisation of loan portfolios or for portfolio
optimization for bond and loan portfolio managers. Referring to former times, a bank
could only manage its credit risk at origination. During the whole lifetime of a loan the
risk remained on the books until the loan was paid off or the obligor defaulted. With the
possibilities of these instruments, however, a bank and all the other previously mentioned
institutions are able to conduct active risk management. Due to these features and the
fact that credit is now a trading asset, the market of credit derivatives is growing and
should keep growing in the future.

3
 After giving a short introduction about the important role credit derivatives play in the
financial world today, the terminology of the general credit derivative is described. Next
we provide an overview of different types of credit derivatives, and give an example to show
how they are used to conduct active risk management. In the second part of the paper,
we talk about the idea of firm’s value models and their connection to credit derivatives.
In particular the Black/Scholes-Merton model and Moody’s KMV are discussed.

3 Terminology
A useful definition of credit derivatives is formulated by Phillip Schönbucher (2003):
”A credit derivative is a derivative security that has a payoff which is conditioned
on the occurrence of a credit event. The credit event is defined with respect to a refer-
ence credit (or several reference credits), and the reference credit asset(s) issued by the
reference credit. If the credit event has occurred, the default payment has to be made
by one of the counterparties. Besides the default payment a credit derivative can have
further payoffs that are not default contingent. This definition can be extended to include
derivative securities whose payoffs are materially affected by credit events and derivatives
on defaultable underlying securities.”

For most derivatives, one can use the following definitions:

• A is the counterparty which receives a payment in the event of a default

• B is the counterparty which has to make the payment in the event of a default

• C is the reference credit

• Reference entity/reference credit is the issuer of the reference obligation/reference

credit asset whose default triggers the credit event

• Reference obligations/reference credit asset is a set of assets issued by the

reference entity

4
 • Credit event/default event occurs e.g. for the following reasons:

– Bankruptcy

– Failure to pay with certain requirements

– Obligation default

– Ratings downgrade below given thresholds (only for ratings-triggered credit

derivatives)

• Default payment is the payment which has to be made by B if a credit event

occurs

4 Some Types of Credit Derivatives

4.1 Total Return Swaps (TRS)

In a total return swap (or total rate of return swap), A wants to change its entire payoff
from a defaultable investment (e.g. a bond, denoted by C with the entire payoff B receives
from its default-free Libor investment.
There are several effects appearing from this contract. First, B is long the C-bond
without having paid for this investment. Therefore B normally has to put collateral (this
can be the C-bond, which legally still belongs to A), depending on its creditworthiness.
Second, A has hedged its exposure to the C-bond and bears a certain counterparty risk
now, but which should be minimized because of the collateral.
Concerning the purpose of credit derivatives, A transmits the credit AND market risk
of the reference credit C to B and ensures a risk-free Libor interest rate plus a certain
spread, reflecting the creditworthiness of B.

4.2 Credit Default Swaps (CDS)

The most important difference between a TRS and a CDS is the matter of isolating credit
risk. While a TRS transfers both credit AND market risk (whereas a certain risk remains

5
for counterparty A because only the risk of one of the reference credit is transferred,
not the whole default risk), the default risk of this type of credit derivative is completely
isolated.
In a credit default swap (or credit swap), B takes the default risk of A’s defaultable
asset and has to make a default payment of a credit event occurs. In exchange for
this service, A pays a fee for the default protection.
With respect to the default payment, there are several possibilities. A physical
delivery requires the delivery of the reference assets against a repayment at par. When a
cash settlement is arranged, B has to pay the difference between the post-default market
value and the face value of the asset. A default digital swap, in contrast, demands a fixed
amount of money, agreed to at the time of the contract.
Since A and B can declare any asset of C they want, they are able to widen the range
of assets so that the default risk of C is completely transferred.

According to the International Swaps and Derivatives Association (ISDA), the following
information should be part of a CDS contract:

• The reference obligor and his reference assets

• The definition of a credit event that is to be insured

• The notional of the CDS

• The start of the CDS

• The maturity date

• The credit default swap spread

• The frequency and day count convention for the spread payments

• The payment of the credit event and its settlement

6
4.3 Collateralized Debt Obligations (CDO)

Collateralized debt obligations belong to the group of exotic credit derivatives as their
construction is very special. The aim of a CDO is to securitize a complete portfolio of
defaultable assets like a basket of bonds or loans in order to sell these securities and the
credit risk of the assets with them.
The way a CDO is born looks like this: first, a portfolio of defaultable assets is set
up and then sold to a company, exclusively created for this aim and denoted by special
purpose vehicle (SPV). The second step is to divide the portfolio into several tranches in
a way that every single tranche can be securitized and sold to investors with different risk
aversions and different demands for the yield, respectively. The obligations sold by the
SPV are collateralized by the underlying debt portfolio.

Bond 1 Tranche 1
Bond 2 1st 5% of loss
Bond 3 Yield=35%

Tranche 2
2nd 10% of loss
Yield=15%
Bond n Trust

Tranche 3
3rd 10% of loss
Average yield
Yield=7.5%
3,5%

Tranche 4
Residual loss
Yield=6%

Figure 1: Collateralized debt obligation

According to the tranche an investor owns, he or she is confronted with more or less
risk. Assuming the investor has obligations of the first tranche, in the example given in
figure 1 he or she suffers already from the first 5 per cent of losses the portfolio gains. Since
the risk of losing money is very high in this case, the yield one gets os correspondingly
very high, too. Normally, it is a multiple of the average yield of the assets of the portfolio.

7
An investor of the forth tranche, in contrast, is only burdened with a loss when already
more than 25 per cent of the assets of the portfolio defaulted. Of course, people investing
in this tranche have a lower expected yield than the average expected portfolio yield.

4.4 Example of a CDS with Real Quotes

The following example should give an idea how a plain vanilla credit default swap looks in
practice. Given the bid/offer quotes of a market maker in table 2, one can think through
several cases.

Table 2 Credit default swap quotes (basis points)

Maturity
Company Rating 3 years 5 years 7 years 10 years
Toyota Motor Corp AaI/AAA 16/24 20/30 26/37 32/53
Merrill Lynch Aa3/AA- 21/41 40/55 41/83 56/96
Ford Motor Company A+/A 59/80 85/100 95/136 118/159
Enron BaaI/BBB+ 105/125 115/135 117/158 182/233
Nissan Motor Co.Ltd. BaI/BB+ 115/145 125/155 200/230 244/274

Looking at Toyota, the market maker is prepared to buy three-year default protection
for 16 basis points per year and sell three-year default protection for 24 basis points per
year and so on (Hull 2002).
Supposing that a bank had several hundred million dollars of loans outstanding to
Enron and was concerned about its exposure. It could buy a $100 million five-year CDS
on Enron from the market maker for 135 basis points or $1.35 million per year. This
would shift part of the bank’s Enron credit exposure to the market maker (Hull 2002).
Another possibility could be an exchange in the bank’s credit risk. If the bank is
interested in shifting part of its credit risk to another industry, it could, for example, sell
a five-year $100 million CDS on Nissan for $1.25 million per year while buying a similar
CDS on Enron at the same time. The net cost of this strategy would be 10 basis points

8
or $100,000 per year. So the bank had changed part of its credit risk from Enron for a
certain credit risk of Nissan. Due to the differences in these industries, one can say that
the bank has diversified its credit exposure (Hull 2002).

5 Firm Value Based Models and Black and Scholes

So far we have talked about the characteristics of credit derivatives in general and how to
use them as tools for active risk management. Now we will focus on the pricing of credit
derivatives using a specific modeling approach: the approach of firm’s value models.
To be able to price credit derivatives, we have to know something about the default
risk (credit risk) of the underlying asset. Modeling the default risk is the aim of credit
derivatives pricing models such as intensity and spread-based models. Compared to those,
firm’s value models use a much more fundamental approach to valuing defaultable debt
and in addition try to provide a link between the values of equity and debt of the firm.
Firm’s value models assume a fundamental process V , denoting the total value of the
assets of the firm that has issued the bonds in question. V is described as a stochastic
process, influenced by the prices of all securities issued by the firm. A very important
point of this type of model is that all claims on the firm’s value are modelled as derivative
securities with the firm’s value as underlying.
Black and Scholes 1973) and Merton (1974) were the first people modeling credit risk
with what we know today as a firm’s value model. Modeling credit risk means modeling
default probability. In their consideration a default could only occur at maturity of the
debt, i.e. if the difference firm value V minus outstanding debt at maturity is negative, a
default happens, otherwise the firm continuous to exist. Mertion (1974) explicitly treated
the corporate liability from the perspective of derivative pricing. We will come to another
and more realistic view later. For further theoretical development see, Schönbucher (2003),
Grüne and Semmler (2005) and Grüne, Semmler and Bernard (2006).
As already mentioned above, the value V of the firm’s assets is described as a stochastic
process. Fischer Black, Myron Scholes and Robert C. Merton set up for V the following

9
geometric Brownian motion:

dV = µV dt + σV dW (1)

or

dV
= µdt + σdW (2)
V
where the variable σ is the volatility of firm value, the variable µ is the expected rate
of return and DW as a Wiener process (for the derivation of this equation see Hull 2002,
11.3).
From now on in this model, the prices of both debt B(V, t) and shares S(V, t) are
functions of the firm’s value V and the time t. What Black and Scholes (1973) and
Merton (1974) did was a breakthrough. They showed that both equity and debt of the
firm can be seen as derivative securities on the value V of the firm’s assets. The payoff
structure of these derivative securities looks like this (D is the exercise price):

B(V, t) = min(D, V ) (3)

S(V, t) = max(V − D, 0) (4)

10
 80 .
70 .
60 . . . . . . . . .
50 . .
Payoff 40 . B (V , t ) .. S(V,t)
30 .. .
20 . .
10 . .
0
150
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
Firm‘s value

Figure 2: Payoffs of shares and bonds at t = T for D = 60

As we are interested in pricing equity and debts of the firm and credit derivatives,
respectively, we set up a risk-neutral portfolio by hedging one bond with ∆-shares. The
value of the portfolio is:

Π = B(V, t) + ∆S(V, t) (5)

The change in value can be derived from Ito’s lemma (see appendix 11A in Hull 2002)
and is:

dΠ = dB∆dS (6)
∂B 1 ∂ 2 B ∂S ′ 1 ∂ 2 S ′
 
= + +∆ + ∆ dt
∂t 2 ∂V 2 ∂t 2 ∂V 2
 
∂B ∂S
+ +∆ dV
∂V ∂V

To be fully hedged and to have a predictable return, the number of shares must be:

11
 ∂B/∂V
∆=− (7)
∂S/∂V
This leads to the well known Black-Scholes partial differential equation:

∂ 1 ∂2 ∂
S + σ2V 2 2
S + rV S − ΓS = 0 (8)
∂t 2 ∂V ∂V
Now we can compute the value of a share with the Black-Scholes formula C BS for a
European call option on V . The expiry date is denoted by T , the exercise price by D, the
underlying volatility by σ and the interest rate by rf :

S(V, t) = C BS (V, t; D, σ, rf ) (9)

= V N (d1 ) − e−rf (T −t) DN (d2 ) (10)

where

ln(V /D) + rf − 12 σ 2 (T − t)


d1 = √ (11)
σ T −t
and

√
d2 = d1 − σ T − t (12)

Note that in the risk neutral case the V in equ. (10) refers to the current value of the
firm, but of course it is determined by the discounted future income stream of the firm.
Yet in the risk free case we can have

S(V, t) = C BS (V, t; D, σ, rf )

= e−rf (T −t) (V N (d1 )erf (T −t) − DN (d2 ))

A Gauss computer program for the above evaluation of corporate debt from the per-
spective of derivative pricing is available.1 Schönbucher (2003, ch.) extends the model
1
available upon request

12
by also taking into account a safety covenant acting as a default barrier. He also intro-
duces bankruptcy cost, and a time varying interest rate, following a Brownian motion, for
example, co-varying with stock market shocks. The firm value approach tries to model
the whole obligor at once through linking the debt and equity with a hedge. A large and
important disadvantage of the model is that one does not observe the process V with its
driving factors.

6 Computing Firm Value and Creditworthiness

In Grüne and Semmler (2005) the firm value is derived from an intertemporal behavior
of firms. There, however, only for the deterministic case. Yet, in Grüne, Semmler and
Bernard (2006) the stochastic case is also considered.
We give a formal presentation of the deterministic model. We can say in the bilaterial
contract between a creditor and debtor there are two problems involved. The first pertains
to the computation of debt and the second to the computation of the debt ceiling. The
first problem is usually answered by employing an equation of the form

Ḃ(t) = θB(t) − f (t), B(0) = B0

where B(t) is the level of debt2 at time t, θ the interest rate determining the credit
cost and f (t) the net income of the agent. The second problem can be settled by defining
a debt ceiling such as

B(t) ≤ C, (t > 0)

or less restrictively by

supB(t) < ∞
t≥0

2
Note that all subsequent state variables are written in terms of efficiency labor along the line of
Blanchard (1983).

13
or even less restrictively by the transversality condition

lim e−θt B(t) = 0. (13)

t→∞

The ability of an obligator to service the debt, i.e. the feasibility of a contract, will
depend on the obligator’s source of income. Along the lines of intertemporal models of
borrowing and lending3 we model this source of income as arising from a stock of capital
k(t), at time t, which changes with the investment rate j(t) at time t through

k̇(t) = j(t) − σ (k(t)) , k(0) = k0 . (14)

In our general model both the capital stock and the investment are allowed to be
multivariate. As debt service we take the net income from the investment rate j(t) at
capital stock level k(t) minus some minimal rate of consumption.4 Hence

Ḃ(t) = θB(t)) − f (k(t), j(t)) , B(0) = B0 (15)

where θB(t) is the credit cost. Note that the credit cost is not necessarily a constant
factor (a constant interest rate). We call B ∗ (k) the creditworthiness of the capital stock
k. The problem to be solved is how to compute B ∗ .
H(B,k)
If there is a constant credit cost factor (interest rate), θ = B
, then, it is easy to
see, B ∗ (k) is the present value of k or the asset price of k:

Z ∞
∗
B (k) = M ax e−θt f (k(t), j(t)) dt − B(0) (16)
j 0

3
Prototype models used as basis for our further presentation can be found in Blanchard (1983),
Blanchard and Fischer (1989) or Turnovsky (1995).
4
In the subsequent analysis of creditworthiness we can set consumption equal to zero. Any positive
consumption will move down the creditworthiness curve. Note also that public debt for which the Ricar-
dian equivalence theorem holds , i.e. where debt is serviced by a non-distortionary tax, would cause the
creditworthiness curve to shift down. In computing the ”present value” of the future net surpluses we
do not have to assume a particular interest rate. Yet, in the following study we neither elaborate on the
problem of the price level nor on the exchange rate and its effect on net debt and creditworthiness.

14
s.t.

k̇(t) = j(t) − σ (k(t)) , k(0) = ks (17)

Ḃ(t) = θB(t) − f (k(t), j(t)) , B(0) = B0 . (18)

The more general case is, however, that θ is not a constant. As in the theory of credit
market imperfections we generically may let θ depend on k and B, see below.5 Employing
a dynamic model of the firm6 we can use the following net income function that takes
account of adjustment investment and adjustment cost of capital.

f (k, j) = k α − j − j β k −γ (19)

where σ > 0, α > 0, γ > 0 are constants.7 In the above model σ > 0 captures both
a constant growth rate of productivity as well as a capital depreciation rate. Blanchard
(1983) used β = 2, γ = 1 to analyze the optimal indebtedness of a firm (see also Blanchard
and Fischer 1989, Chap. 2).
The maximization problem (16)-(18) can be solved by using the necessary conditions
of the Hamiltonian for (16)-(17). Thus we maximize
Z ∞
M ax e−θt f (k(t), j(t))dt
j 0

s.t. (17).

The Hamiltonian for this problem is

5
The more general theory of creditworthiness with state dependent credit cost is provided in Grüne,
Semmler and Sieveking (2004). Note that instead of relating the credit cost inversely to net worth, as in
Bernanke, Gertler and Gilchrist (1998), one could use the two arguments, k and B, explicitly.
6
The subsequent model can be viewed as a standard RBC model where the stochastic process for
technology shocks is shut down and technical change is exogenously occurring at a constant rate.
7
Note that the production function k α may have to be multiplied by a scaling factor. For the analytics
we leave it aside here.

15
 H(k, x, j, λ) = maxH(k, x, j, λ)
j

H(k, x, j, λ) = λf (k, j) + x(j − σk)

. −∂H
x= + θx = (σ + θ) x − λfk (k, j).
∂k
We denote x as the co-state variable in the Hamiltonian equations and λ is equal to
1.8 The function f (k, j.) is strictly concave by assumption. Therefore, there is a function
j(k, x) which satisfies the first order condition of the Hamiltonian

fj (k, j) + x = 0 (20)
x − 1 β−1
1
j = j(k, x) = ( −γ
) (21)
k ·β
and j is uniquely determined thereby. It follows that (k, x) satisfy
·
k = j(k, x) − σk (22)
·
x = (σ + θ)x − fk (k, j(k, x)) (23)
·
The isoclines can be obtained by the points in the (k, x) space for β = 2 where k =0
satisfies
x = 1 + 2σk 1−γ (24)
·
and where x = 0 satisfies
p
x± = 1 + ϑk 1−γ ± ϑ2 k 2−2γ + 2ϑk 1−γ − 4αγ −1 k α−γ (25)

where ϑ = 2γ −1 (σ + θ). Note that the latter isocline has two branches.
If the parameters are given, the steady state – or steady states, if there are multiple
ones – can be computed and then the local and global dynamics studied. We scale the
9
production function by α.
8
For details of the computation of the equilibria in the case when one can apply the Hamiltonian, see
Semmler and Sieveking (1998), appendix.
9
We have multiplied the production function by a = 0.30 in order to obtain sufficiently separated
equilibria, and take c = 0. We employ the following parameters: α = 1.1, γ = 0.3, σ = 0.15, θ = 0.1.

16
 There is another solution technique which allows one to solve for firm value by using
a dynamic programming approach. The alternative solution method uses the Hamilton-
Jacobi-Bellman (HJB) equation.
In this appendix we present the solution technique of how to find the solution of the
HJB-equation. We describe an algorithm which enable us to compute the asset price
of the firm for the HJB equation of a type such as (1) which will give us the present
value borrowing constraint. We show of how one can explicitly compute firm value using
modern dynamic decision theory.
The HJB-equation for our problem reads

θV = max[k α − j − j 2 k −γ + V ′ (k)(j − σk)] (26)

j

Using the HJB equation we also can compute the steady state equilibria.
For the steady state, for which 0 = j − σk holds, we obtain:

f (k, j)
V (k) = (27)
θ

∂
f ′ (k, j) (k α − σk − σ 2 k 2−γ )
V ′ (k) = = ∂k
(28)
θ θ
Using the information of (27)-(28) in (26) gives, ater taking the derivatives of (26)
with respect to j, the steady states for the stationary HJB equation:

αk α−1 − σ − σ 2 (2 − γ)k 1−γ

−1 − 2jk −γ + =0 (29)
θ
Note that hereby j = σk 10 . Given our parameters the equation may admit multiple
steady states.

We specify the company’s technology parameters to be σ = 0.15, A = 0.29, αi =

0.7, βi = 2, γ = 0.3 and θ = 0.1. The remaining parameters are specified below.
10
Note that this gives us the same equilibria as using the Hamiltonian approach.

17
 As for the numerical procedure an example was computed for different k ′ s in the
compact interval [0.2], using dynamic programming with control range j ∈ [0, 0.25]. The
dynamic programming algorithm (DP) used here is builton the HJB equation and is
explained in Grüne and Semmler (2004). From this algorithm we obtain the figure below
which approximates the present value curve V (k) representing firm value.
We have considered our deterministic formulation above. In this case, debt is issued,
but with no default premium. Thus, the credit cost is given by H(k, B) = θB. We
have used the above mentioned DP algorithm in order to solve the discounted infinite
horizon problem (16)-(18). Figure 3 shows the corresponding value function representing
the present value curve, V (k). The present value curve represents the asset value of the
company for initial conditions k(0) and thus its creditworthiness.

V
2.523

1.898

1.474

0.949

0.425

-0.100 k
0.000 0.400 0.800 1.200 1.600 2.000

k=0.996

Figure 3: Present Value of Company’s Capital Assets

The debt control problem is solved whenever debt is below the firm’s asset value, so
that we have V − B ≥ 0. The optimal investment strategy is not constrained and thus the
asset value which represents the maximum debt capacity V , is obtained by a solution for

18
an unconstrained optimal investment strategy, represented by the present value curve in
Figure 3. For initial values of the capital assets above or below k ∗ , the optimal trajectories
tend to the domain of attraction k ∗ = 0.996. For all initial conditions, the debt dynamics
remain bounded as long V −B ≥ 0, thus allowing the company’s equity holders to exercise
the option of retiring the debt. Any initial debt above the present value curve will be
explosive and the company will lose its creditworthiness, since it will not be able to pay
its obligations.
For the more general case where a default premium is to be paid we can use the
following function to represent risk premia:

α1
H(k(t), B(t)) =  µ θB(t)
N (t)
α2 + k(t)

For the model (16)-(18) with a risk premium included in the company’s borrowing cost,
it is not possible to transform the model into a standard infinite horizon optimal control
problem. This results because debt is now an additional constraint on the optimization
problem. Hence, we need to use another method firm value and one can undertake ex-
periments for different shapes of the credit cost function representing different alternative
functions for the risk premium. An important class o functions for risk premia is defined
by the steepness of the slope defined by the parameter α2 , for details, see Grüne, Semmler
and Bernard (2006). There are also results reported as to what extent the value of this
company is affected by a default premium.

7 Moody’s KMV
Due to the difficulties in computing the present value for firm value models11 a practical
implementation has been developed which comes with solutions to this problem. The
KMV model, named after the founders Kealhover, McQuown and Vasicek (2001), models
credit risk and the default probability of a firm as follows.
11
A more practical methodic of computing firm value is proposed in Benninga (1998, chs. 2-3).

19
7.1 The Distance-to-Default

The model states that there are three main elements determining the default probability
of a firm:

• Value of assets is the market value of the firm’s assets.

• Asset risk is the uncertainty or risk of the asset value. This is a measure of the
firm’s business and industry risk.

• Leverage is the extent of the firm’s contractual liabilities. It is the book value of
liabilities relative to the market value of assets.

As in equs (3) and (10) the default risk of the firm increases when the value of the
assets approaches the book value of the liabilities. The firm defaults when the market
value of the assets is smaller than the book value of the liabilities.
According to Peter Crosbie and Jeff Bohn (2003) who wrote the paper Modelling
Default Risk for Moody’s, their studies do not confirm this thesis in general. Not all
the firms which reach the point where the asset value goes below the book value of their
liabilities default. There are many which continue and serve their debt. The reason for this
can be found in the long-term liabilities which enable the firms to continue their business
until the debt becomes due. The firms may also have credit lines at their disposal.
Crosbie and Bohn draw the conclusion that the default point, the asset value at which
the firm will default, generally lies somewhere between total liabilities and short-term
liabilities. The relevant net worth of the firm is therefore defined as:

[Market Net Worth] = [Market Value of Assets] - [Default Point] (30)

If the market net worth of a firm is zero, the firm is assumed to default. To measure
the default risk, one can combine all three elements determining the default probability
in a single measure of default risk, the distance-to-default:

20
 (31)
[Market Net Worth]
[Distance-to-Default] =
[Size of One Standard Deviation of the Asset Value]
[Market Value of Assets]-[Default Point]
= (32)
[Asset Volatility]

The distance-to-default is the number of standard deviations the asset value is way
from default The default probability can then be computed directly from the distance-to-
default if the probability distribution of the asset value is known.

7.2 The Probability of Default

Crosbie and Bohn (2003) give 6 variables that determine the default probability of a firm
over some horizon, from now until time H (see figure 4):

1. The current asset value

2. The distribution of the asset value at time H

3. The volatility of the future assets value at time H

4. The level of the default point, the book value of the liabilities

5. The expected rate of growth in the asset value over the horizon

6. The length of the horizon, H

21
 Market Distribution
Value of asset value 2
Assets at the horizon
Possible
asset value
path

1 V0
5 DD

4 Default
Point
EDF

0 H6 Time

Figure 4:

The probability of default (expected default frequency or EDFvalue) can be com-

puted with the aid of the measure we calculated above and data on historical default and
bankruptcy frequencies. The database that Mody’s usses consists of more than 400,000
company-years of data and more than 4,900 incidents of default or bankruptcy (see figure
4). From this data, a frequency table can be generated which relates the likelihood of
default to the distance-to-default measure.
For example, a firm that is 7 standard deviations away from default has an expected
default frequency (EDF value) of 5 per cent which leads to a rating of AA. In this case,
Moody’s analysis the default history of the fraction of firms which were 7 standard devia-
tions away from the default point and defaulted over the next year. According to Crosbie
and Bohn (2003), Moody’s tested the relationship between distance-to-default and default
frequency for industry, size, time and other effects and has found that the relationship is
constant across all of these variables.

22
Those relationships can be developed in mathematical terms. According to the Black-
Scholes model and as above in equ. (1) presumed, the market value of the firm’s underlying
assets is described by the following stochastic process:

dVA = µVA dt + σA VA dz (33)

where

VA , dVA are the firm’s asset value and change in asset value
µ, σA are the firm’s asset value drift rate and volatility
dz is a Wiener process

The probability of default that the market value of the firm’s assets will be less than the
book value of the firm’s liabilities by the time the debt matures:

pt = P r[VAt ≤ Xt | VA0 = VA ]

= P r[lnVAt ≤ lnXt | VA0 = VA ] (34)

where

pt is the probability of default by time t

VAt is the market value of the firm’s assets at time t
Xt is the book value of the firm’s liabilities due at time t

The change in the value of the firm’s assets is described by (16), so the value at time
t, VAt , given that the value at time 0 is VA , is:

σ2 √
 
lnVAt = lnVA + µ − A t + σa tε (35)
2
where

µ is the expected return on the firm’s asset

ε is the random component of the firm’s return

23
Equation (18) describes the asset value path shown in figure 3. Combining (17) and (18),
one can write the probability of default as:

σ2 √
   
pt = P r lnVA + µ − A t + σA tε ≤ lnXt (36)
2
or
 2

σA
 
ln VXAt + µ − 2
t
p t = P r − √ ≥ ε (37)
σA t

Since the Black-Scholes model assumes that ε is normally distributed, one can write
the default probability as:
 2
 
σA

ln VXAt + µ − 2
t
pt = N − √  (38)
σA t

Since the distance-to-default measure is nothing else than the number of standard
deviations that the firm is away from default, one can write this measure with the Black-
Scholes notation as:
 2

σA
ln VXAt
+ µ− 2 t
[Distance-to-Default] = √ (39)
σA t
Given an example that we compute a distance-to-default from equation (22) that
equals 3.0, the probability of default using equation (21) will then be 13 basis points or
13 per cent. In practice, this distance-to-default measure is adjusted to include several
other factors which play a role in measuring the default probability.

8 Empirical Evidence for Firm Value Based Models

There are several advantages and disadvantages that firm value based models have in
practice. The predictions of firms value based models on the dynamics of share and debt
prices of firms, are discussed briefly in this section. After a few empirical papers are
discussed the general importance of these models will be evaluated.

24
 While the majority of firm value based models predict a hilly shape for the term
structure of credit spreads, Litterman and Iben (1991) showed that this is only true for
rating classes of firms with bad rating. For other classes, like investment-grade rating
classes, they observed increasing credit spreads rather than hilly ones.
The aim of another empirical work the one by Lardic and Rouzeau (1999), was to
reproduce the risk ranking of obligors using firm value models. The test was designed not
to study the real market value of the firms but to derive the risk level of firms in such a
way that allowed to differentiate between riskier and less risky assets. The results however
showed that the models were not able to reproduce the risk ranking of obligors. Instead,
they were only able to recognize changes in the credit quality of the same obligor.
Longstaff and Schwartz (1995) investigated credit spread movements. With their
tests using Moody’s corporate bond yield averages, they found that there is a negative
correlation between spreads and rates, meaning that firm value based models cannot be
used for hedging purposes.
Concerning the pricing accuracy of firm value based models, Eom et al. (2000) run a
test where they priced corporate bonds using the current share prices and balance sheet
data of firms that issued the bonds. According to this test where the dynamics of the
spreads were not included, it was found that there are pricing errors in all models.
Approximating data on fundamental is an essential strength of firm value based mod-
els, but defining the actual firm value can be really an complex issue. The problems can
quickly become too complex to be handled by empirical tests. Despite all the compli-
cations one has to deal with when using firm value models, a more practical approach
like Moody’s KMV shows that one can obtain acceptable results and a better pricing
performance with some pragmatic approach (see section 7).

25
Appendix 1

Table A1
OTC derivatives market1
Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity
Amounts outstanding in billions of US dollars

Notional amounts Gross market values

End-June End-June End-June End-June
2001 2004 2001 2004
GRAND TOTAL 99,659 220,058 3.045 6,394

A. Foreign exchange contracts 20,434 31,510 967 1,118

Outright forwards
and forex swaps 13,275 16,764 548 483
Currency swaps 4,302 7,939 339 506
Options 2,824 6,806 80 150
Other 33 7 0 0

B. Interest rate contracts2 75,813 177,432 1,748 4,581

FRAs 7,678 14,399 32 211
Swaps 57,220 137,277 1,531 3,978
Options 10,913 25,756 185 393
Other 2 0 0 0

C. Equity-linked contracts 2,039 5,094 220 321

Forwards and swaps 373 774 55 72
Options 1,666 4,320 164 249

D. Commodity contracts3 674 1,354 88 177

Gold 278 360 25 47
Other 396 995 63 130
Forwards and swaps 235 541 – –
Options 162 453 – –
1
All figures are adjusted for double counting. Notional amounts outstanding have been adjusted by

halving positions vis-a-vis other reporting dealers. Gross market values have been calculated as the sum

of the total gross positive market value of contracts and the absolute value of the gross negative market

value of contracts with non-reporting counterparties.

2
Single currency contracts only.
3
Adjustments for double-counting partly estimated.
4
Gross market values after taking into account legally enforcable bilateral netting agreements.
5
Sources: FlOW TRADEdata, Future industry Association; various futures and options exchanges.

27
Appendix 2: The Numerical Solution of the Model
We here briefly describe the dynamic programming algorithm as applied in Grüne and
Semmler (2004) that enables us to numerically solve the dynamic model as proposed in
section 3. The feature of the dynamic programming algorithm is an adaptive discretization
of the state space which leads to high numerical accuracy with moderate use of memory.
Such algorithm is applied to discounted infinite horizon optimal control problems of
the type introduced in section 3. In our model variants we have to numerically compute
V (x) for

Z ∞
V (x) = max e−r f (x, u)dt
u 0

s.t. ẋ = g(x, u)

where u represents the control variable and x a vector of state variables.

In the first step, the continuous time optimal control problem has to be replaced by a
first order discrete time approximation given by

∞
X
Vh (x) = max Jh (x, u), Jh (x, u) = h (1 − θh)U f (xh (i), ui ) (A1)
j
i=0
where xu is defined by the discrete dynamics

xh (0) = x, xh (i + 1) = xh (i) + hg(xi , ui ) (A2)

and h > 0 is the discretization time step. Note that j = (ji )i∈N0 here denotes a discrete
control sequence.
The optimal value function is the unique solution of a discrete Hamilton-Jacobi-
Bellman equation such as

Vh (x) = max{hf (x, uo ) + (1 + θh)Vh (xh (1))} (A3)

j

where xh (1) denotes the discrete solution corresponding to the control and initial value
x after one time step h. Abbreviating

28
 Th (Vh )(x) = max{hf (x, uo ) + (1 − θh)Vh (xh (1))} (A4)
j

the second step of the algorithm now approximates the solution on grid Γ covering a
compact subset of the state space, i.e. a compact interval [0, K] in our setup. Denoting
the nodes of Γ by xi , i = 1, ..., P , we are now looking for an approximation VhΓ satisfying

VhΓ (X i ) = Th (VhΓ )(X i ) (A5)

for each node xi of the grid, where the value of VhΓ for points x which are not grid
points (these are needed for the evaluation of Th ) is determined by linear interpolation.
We refer to the paper cited above for the description of iterative methods for the solution
of (A5). Note that an approximately optimal control law (in feedback form for the discrete
dynamics) can be obtained from this approximation by taking the value j ∗ (x) = j for j
realizing the maximum in (A3), where Vh is replaced by VhΓ . This procedure in particular
allows the numerical computation of approximately optimal trajectories.
In order the distribute the nodes of the grid efficiently, we make use of a posteriori
error estimation. For each cell Cl of the grid Γ we compute

ηl := max | Th (VhΓ )(k) − VhΓ (k) |

k∈cl

More precisely we approximate this value by evaluating the right hand side in a number
of test points. It can be shown that the error estimators ηl give upper and lower bounds for
the real error (i.e., the difference between Vj and VhΓ ) and hence serve as an indicator for
a possible local refinement of the grid Γ. It should be noted that this adaptive refinement
of the grid is very effective for computing steep value functions and models with multiple
equilibria, see Grüne and Semmler (2004).

29
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[12] Crosbie, P. and J. Bohn (2003), ”Modeling Default Risk”, Moody’s KMV Com-
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30
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31