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308

Risk Management and Shareholders’ Value in Banking

place within each grade, and this percentage (empirical default rate) is used as a PD

estimate for all customers belonging to a given class.25

When credit-scoring models are used, attention must be paid to some problems. This

is especially true if the models are used not only to decide which companies to lend to,

but also to estimate their PD. Those problems are illustrated brieﬂy below.

• A ﬁrst important problem relates to the deﬁnition of an “abnormal” or “insolvent”

company. Different “degrees of insolvency” exist, ranging from a simple delay in payment of interest to the compulsory liquidation of the company, with many intermediate

stages. The deﬁnition used to break down the estimation sample obviously affects the

results of the model, even when it is applied to a new company. Thus, for example,

if a very broad deﬁnition of insolvency is used (e.g., a simple delay in the payment

of interest), a model will be obtained which classiﬁes a large number of companies as

insolvent and assigns higher PDs.

• A second problem is associated with the fact that the meaningfulness of the independent

variables used by the scoring model may vary over time, due to the effect of the

economic cycle, ﬁnancial market variables and other factors; there is no economic

reason why the weight of the individual economic/ﬁnancial indicators used to explain

default should remain unchanged.

• A third problem is due to the fact that credit-scoring models ignore numerous qualitative

factors, which can be highly signiﬁcant in determining the insolvency of a company26 .

They include the company’s reputation (which inﬂuences its ability to access alternative

sources of credit), the stage of the economic cycle, the quality of management and the

outlook for the industry to which the company belongs.

• A fourth problem is due to the fact that the companies in the estimation sample

should belong, as far as possible, to the same industry. There are two reasons for

this: (i) because the individual economic/ﬁnancial indicators take very different mean

values from industry to industry; and (ii) because the same indicator may have a different importance in determining default in the various industries. In practice, however,

the difﬁculty of obtaining data from a large number of insolvent companies often forces

banks to work with samples that include companies in different industries.

• Precisely because complete data often exists only for a few defaulting companies

(whereas it is advisable to work with large samples in order to estimate the models with sufﬁcient accuracy), there is often a risk that the estimation samples will be

“unbalanced”, and include an excessively high percentage of healthy companies. Paradoxically, from this standpoint, a bank with poor historical portfolio quality is in a

better position than a bank which has suffered few defaults in the past (and therefore

is forced to work with estimate samples which are either too small or signiﬁcantly

unbalanced).

25

See Chapter 13 for further details.

Also, a number of subjective passages are present in credit-scoring models: the choice of the sample of

companies used to generate the scoring function, the deﬁnition of insolvency used to distinguish a priori

between the two sub-samples, the choice of the starting independent variables, together with more technical,

statistical choices (e.g., whether or not to perform an initial transformation of the input variables based on

principal component analysis so as to eliminate the problem of correlation between independent variables).

However, these subjective decisions are only involved in the construction of the model. Once constructed, the

model is applied objectively to all companies.

26

Credit-Scoring Models

309

SELECTED QUESTIONS AND EXERCISES

1. A bank has analysed the ﬁnancial statements of “good” and “bad” (i.e., defaulted)

customers and found that:

– the ratio of equity to total assets was on average 50 % for “goods” and 20 % for

“bads”;

– the ratio of liquid assets to short-term liabilities was on average 2 for “goods” and

0.4 for “bads”.

– the variance/covariance matrix between the two ratios is the following:

=

– its inverse is

−1

=

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0.04 0.07

0.07 0.51

32.9 −4.52

−4.52 2.58

The bank wants to use this information in a discriminant analysis model.

Compute:

– the coefﬁcients of the discriminant function;

– the centroids;

– the threshold to be used to separate good and bad customers, if the ex ante probability

(prior) of having a bad customer is 10 % and error costs are not available;

– the threshold (based on the same priors as above) for a customer with error costs

of 20,000 euros (loan granted to a customer that will default) and 1,800 euros (loan

refused to a good customer).

2. In a discriminant analysis model, if a borrower offers collateral that reduces the

expected LGD. . .

(A) . . .his/her score remains the same, while his/her PD gets reduced;

(B) . . .his/her PD remains constant, while the cut-off point decreases as the costs

associated to a wrong classiﬁcation change;

(C) . . .his/her score remains constant, while the cut-off point increases as the costs

associated to a wrong classiﬁcation decrease;

(D) . . .his/her score, PD and cut-off point remain all unchanged.

3. A customer has applied for a loan of 500,000 euros, providing a cash collateral of

100,000 (so that, in the event of default, the loss to the bank would be equal to 80 %

of the loan). The rate applied would be 12 %; the cost of funds and all other operating

expenses for the bank would amount to 10 %, leaving a net proﬁt margin of 2 %.

The customer’s score, based on a discriminant analysis model, is 6.1. This is below

the minimum threshold for a loan to be issued, which (based on the error costs shown

above and on a prior “bad” probability of 10 %) would be 7.

How much more cash collateral should the customer provide, leaving the 12 % rate

unchanged, for his loan request to be approved?

Based on the information in this exercise, could you also estimate the customer’s PD?

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310

Risk Management and Shareholders’ Value in Banking

4. Consider the following statements on linear probability models and logit models:

(i) Linear probability models need to be truncated between 0 and 1, for their results

to be equal to those of a logit model;

(ii) Linear probability models need to be truncated between 0 and 1, for their output

ranges to be equal to those of a logit model;

(iii) Linear probability models and logit models have the same coefﬁcients, but the

estimated PDs are different because logit models involve a non-linear ﬁlter;

(iv) Linear probability models, unlike logits, generate biased estimates.

Which ones would you agree with?

(A)

(B)

(C)

(D)

ii) and iii);

iv) only;

ii) and iv);

all of them.

5. A scoring model has been modiﬁed by adding a new input, which is 95 % correlated

with an input that already was in the model and that was considered highly signiﬁcant

by the bank’s experts. Which of the following changes in the model’s performance

would you expect?

(A) A strong improvement in the model, since the new input will also be highly

signiﬁcant;

(B) A strong decrease in performance, because we are duplicating information;

(C) This cannot be told in advance, as it depends on the speciﬁc data sample;

(D) Almost no change in performance.

Appendix 10A

The Estimation of the Gamma Coefﬁcients

in Linear Discriminant Analysis

Equation (10.3):

−1

γ =

(x1 − x2 )

was used in this chapter to compute the γ coefﬁcients of the linear discriminant analysis

model. We now demonstrate how this formula was obtained.

We wish to construct our discriminant function

n

z=

γj xj = γ x

j =1

by choosing γ in such a way as to maximise the (standardised) distance between the

centroids, namely

|γ xA − γ xB |

|zA − zB |

Max

=

(10A.1)

γ

σz

σz

To avoid working with the absolute value (a non-differentiable function), we can work

on the square q of the expression in [10A.1] and rewrite our problem as follows:

Max

γ

(γ xA − γ xB )2

(γ xA − γ xB )2

≡ q(γ )

=

σz2

γ γ

(10A.2)

Before proceeding, note that function q(.) is homogeneous of degree zero in γ . In other

words: q(kγ )= q(γ ) for any real scalar k. In fact:

q(kγ ) =

k 2 (γ xA − γ xB )2

(kγ xA − kγ xB )2

=

= q(γ )

kγ kγ

k2γ γ

This remark will be useful very soon.

Next, to identify γ which maximizes q, we will calculate the gradient of q and require

it to be equal to zero:

∂q

2(γ xA − γ xB )(xA − xB )γ γ − 2 γ (γ xA − γ xB )2

=0

=

∂γ

(γ γ )2

(10A.3)

i.e. require its numerator–ignoring the scalar2(γ xB − γ xB )- to be nil:

(xA − xA )γ

γ =

−1

γ−

(xA − xB )

γ (γ xB − γ xB ) = 0

γ γ

(γ xA − γ xB )

(10A.4)

(10A.5)

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Risk Management and Shareholders’ Value in Banking

γ γ

in (10A.5) it is a scalar. As q(γ ) is homo(γ xA − γ xB )

geneous of grade 0, this scalar can be eliminated from the solution without altering the

value of q(.). Thus, (10A.5) simply becomes

Now, consider the quantity

γ =

−1

(x1 − x2 )

that is, equation (10.3) in the body of the chapter.

11

Capital Market Models

11.1 INTRODUCTION

In recent decades, the development of international capital markets (both stocks and bonds)

has been accompanied by development in the mathematical asset pricing models being

used. In many areas of ﬁnance, this has led to securities prices – which summarize all

available information, as well as of the expectations of investors – being used as input in

estimating the value of other market variables. Take, for example, the use of spot rates

in order to get to the forward rates, or the use of options prices in order to estimate the

volatility of the underlying asset.

This would also include models that use the price of stocks and bonds as an input,

in order to estimate the likelihood of default by the issuing company (i.e. capital market

approaches). In this chapter, we will look at the characteristics of these models, their

basic assumptions, and their potential applications.

In section 11.2, we present a number of examples of models based on bond prices

or, to be more precise, on the term structure of spreads between corporate bonds (which

include an element of default risk), and risk-free government bonds. Section 11.3, then,

analyzes structural models based on the contingent claims approach, which was originally

developed by Nobel Prize winner Robert Merton in the early 70’s, and shows how these

models can be applied using stock prices as an input.

11.2 THE APPROACH BASED ON CORPORATE BOND

SPREADS

The starting point for the approach based on the term structure of bond spreads is relatively

simple and intuitive. The higher yields demanded by investors for “risky” bonds (as

compared with securities of the same maturity that are free from default risk) reﬂect market

expectations as to the likelihood of issuer default. These spreads, therefore, represent a

summary of all available information on the factors (both speciﬁc and systemic) that

inﬂuence the probability of default (PD).

Speciﬁcally, the data needed as input for these models is as follows:

• the curve of the spreads between the yields of the zero-coupon corporate bonds of a

given company and the zero-coupon yields of risk-free securities (essentially, Treasury

bonds);

• an estimate of the expected recovery rate on corporate bonds in the event of default.

Based on this data, we can then calculate the data related to expected default rates for

each future period. In particular, we have two ways in which we can estimate PD over a

time horizon of more than one year, and these ways will be analyzed in sequence below.

We may, in fact, use either the spreads on long-term bonds or the spreads implied by

forward rates.

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Risk Management and Shareholders’ Value in Banking

11.2.1 Foreword: continuously compounded interest rates

In this chapter, interest rates (r) will be expressed as continuously compounded rates.

Practically speaking, the value (V ) of a debt at year-end will be equal to the initial capital

(C) multiplied by an exponential function as follows:

V = Cer

rather than using the standard equation for simple or compound interest:

V = C(1 + r)

The reason for this will become clear later in this section. For now, it is sufﬁcient to

note that it is always possible to switch from a simple or compound rate (rs ) to the

corresponding continuously compounded rate (rc ) by simply imposing that both lead to

the same ﬁnal value, given the same initial capital. From:

Cerc = V = C(1 + rs )

we get:

rs = erc − 1;

rc = ln(1 + rs )

For example, a continuously compounded rate of 4 % equals a simple (or annually compounded) rate of approximately 3.92 %.

11.2.2 Estimating the one-year probability of default

Let us assume that the PD of a company that has issued a bond is equal to p. Let us further

assume that, in the event of default, investors will lose all of their capital (LGD = 100 %),

i.e. they will be unable to recover anything. Finally, let r be the one-year risk-free rate

(given by the yield on one-year treasury bills), and let r ∗ = r + d be the yield on risky

one-year corporate bond (where d represents the spread between the risk-free security

and the corporate bond).

A risk-neutral investor1 should be indifferent to the two investment alternatives (i.e.

the risky bond and the risk-free government bond) when the value of one euro invested

in the risk-free security is equal to the value of one euro invested in the corporate bond,

weighted by the probability that the bond will be redeemed as scheduled:

er = (1 − p)er+d

from which we get:

(11.1)

p = 1 − e−d

(11.2)

1

We will return to the effects of assuming risk-neutrality later in this chapter. This assumption requires that

an investor be indifferent to either a risk-free investment or a risky one, as long as the expected ﬁnal value

of both are the same. For example, such an investor would be indifferent to a certain future payment of one

million euros or a lottery that has a 1 % chance of paying 100 million euros and a 99 % probability of paying

nothing.

Capital Market Models

315

From (11.2), we see that p is an increasing function of d: the greater the spread (d)

required by the market, the greater the probability of default. We also see that the PD

implied in bond rates does not depend on the level of the rates (i.e. on r and r ∗ ), but

solely on the spread between them.2

Let us assume that r ∗ is equal to 5 % and that r is equal to 4 %. The spread d would

then be 1 % and we would have:

p = 1 − e−0,01 = 0,995 %

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This, therefore, is the value of PD that leads investors to demand a one percent risk

premium (5 %–4 %) over the risk-free rate.

At this point, let us assume that, more realistically, in case of default creditors are

able to recover a portion k of the capital invested, plus related interest at the rate r ∗ , by

participating in the liquidation of the business after the default.

In this case, a risk-neutral investor would be indifferent to the two investments (the

risk-free security and the corporate bond) if:

er = [(1 − p) + pk]er+d = [1 − p(1 − k)]er+d

from which we get:

p=

1 − e−d

1−k

(11.3)

(11.4)

Note that 1 − k represents the expected loss given default (LGD) on the bond, expressed

as a percentage of the initial capital.

If the spread d is still equal to 1 % and the recovery rate k is 50 %, from (11.4) we get:

p=

1 − e−0,01

= 1,99 %

1 − 0,5

This is twice as much as the probability of default obtained above (assuming a zero

recovery rate). This is quite logical: if investors continue to demand the same risk premium

(d = 1%) despite an expected recovery in the event of default of 50 %, this means that

they are estimating a PD that is signiﬁcantly higher than in our previous example based

on the assumption of a zero recovery.

11.2.3 Probabilities of default beyond one year

Thus far, we have limited ourselves to the simpler case of interest rates for one year and,

using the spread, we have calculated the implied probability of default for the issuer.

We must now extend our analysis to the more complex, and realistic, case of longer

maturities. As we will see, using the spreads for various maturities, we are able to estimate

the PDs for various time horizons.

Let us take the curve of zero-coupon risk-free rates at various maturities, as well as

the curve for zero-coupon rates on the corporate bonds of a given issuer and the related

2

This simpliﬁcation is made possible by the use of continuously compounded rates. If we were to rewrite (11.1)

and (11.2) using traditional simple or compound rates, we would see that it is not possible to write p as a

function of the spread alone, but the absolute level of interest rates must be speciﬁed, as well.

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Risk Management and Shareholders’ Value in Banking

Table 11.1 Continuously compounded zero-coupon rates

Maturity

(years)

Return on Return on Spread

risk-free

risky

(dT )

bonds (rT ) corporate

∗

bonds (rT )

pT

pT

conditional on

no prior

default

1

5.00 %

1.00 %

2.49 %

2.49 %

2

4.10 %

5.20 %

1.10 %

5.44 %

3.03 %

3

4.20 %

5.50 %

1.30 %

9.56 %

4.36 %

4

4.30 %

5.80 %

1.50 % 14.56 %

5.52 %

5

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4.00 %

4.50 %

6.20 %

1.70 % 20.37 %

6.80 %

7%

6%

5%

4%

3%

2%

Rate on risk-free bonds

Rate con corporate bonds

1%

Spread (d)

0%

1

2

3

Maturity

4

5

Figure 11.1 Zero-coupon rates on corporate bonds and risk-free bonds

spreads (see the ﬁrst four columns of Table 11.1 and Figure 11.1 below). In this example,

we can see that the spread increases as the maturity increases.

We will now use pT to indicate the cumulative probability of default for a period of T

years, i.e. the probability that the issuer will default between today and the end of theT th

year.

If investors in T -year corporate bonds are risk neutral, they will expect the ﬁnal value

of one euro invested in the corporate bond (redemption of the principal weighted for

the probability that the bond will be redeemed as scheduled, plus the recovery value k

weighted by the probability of default) to be equal to the value of one euro invested in

Capital Market Models

317

the risk-free security. Generalizing equation (11.3) above, we therefore impose that:

erT T = [1 − pT + pT k]e(rT +dT )T = [1 − pT (1 − k)]e(rT +dT )T

(11.5)

from which we get:

pT =

1 − e−dT T

1−k

(11.6)

Applying (11.6), we get the cumulative probabilities of default associated with the various

maturities.3 An example is shown in column ﬁve of Table 11.1, where we have assumed

a value for k of 60 %: we can see that, as the time horizon increases, cumulative PDs

also increase, since each one incorporates the risk of the previous periods plus the risk

of default in year T .

Let us indicate by sT ≡ 1 − pT the probability that the debtor survives (i.e. does not

default) between now and the end of year T . Also, let us use sT to indicate the marginal

survival probability during year T , i.e. the probability (conditional on no default through

the end of year T − 1) that the debtor will not default during year T . Accordingly, for

any T , we have:

sT = sT −1 · sT

11.7

In other words, the probability of survival from 0 to T is given by the product of the

probability of survival from 0 to T − 1 and the (marginal) probability of survival for year

T . It follows, then, that the marginal probability of survival can be expressed as follows:

sT =

sT

sT −1

(11.8)

The marginal probability of default during year T (pT ) will then be one minus the related

marginal probabilities of survival:

pT = 1 − sT = 1 −

sT

1 − pT

= 1−

sT −1

1 − pT −1

(11.9)

Applying (11.9), we can use cumulative probabilities of default in order to estimate the

marginal default probabilities associated with the spreads in Table 11.1. For example, the

marginal probability of default for year two (again assuming that k is equal to 60 %)

would be equal to:

p2 = 1 −

1 − p2

1 − 5.44 % ∼

=1−

= 3.03 %

1 − p1

1 − 2.49 %

This value and those for the subsequent years are shown in the last column of Table 1.

3

As noted by Hull (2005), this formula provides precise results only if the recovery rate k refers to the bond’s

no-default value. In practice, this may not be true, and in such cases, the formula is only an approximation.

318

Risk Management and Shareholders’ Value in Banking

11.2.4 An alternative approach

An alternative approach4 to computing marginal PDs uses the zero-coupon forward rates,5

i.e. the forward rates implied by the curves shown in the table above.6 Based on the theory

of expectations,7 these rates represent the rates expected by the market on investments

starting in the future.

Given the spot rates at years T and T − 1, the forward rate for a one-year transaction

starting on T − 1 can be expressed as follows:

T −1 r1

= rT T − rT −1 (T − 1)

(11.10)

Based on this relationship, Table 11.2 shows the one-year forward rates for investments

starting from years 0, 1, 2, 3 and 4 (with the ﬁrst, actually being spot rates). As before,

these are continuously compounded rates.

Table 11.2 One-year forward rates

Starting

date

Maturity

(years)

Forward

Forward

Forward

rate on

rate on risky

spread

risk-free bonds corporate bonds (T−1 d1 )

(T−1 r1 )

(T−1 r ∗ 1 )

pT

conditional on

no prior default

pT

0

1

2

3

1

2

3

4

4.00 %

4.20 %

4.40 %

4.60 %

5.00 %

5.40 %

6.10 %

6.70 %

1.00 %

1.20 %

1.70 %

2.10 %

2.49 %

2.98 %

4.21 %

5.20 %

2.49 %

5.40 %

9.38 %

14.09 %

4

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5

5.30 %

7.80 %

2.50 %

6.17 %

19.39 %

It is interesting to note that the spreads between the spot rates for risky and risk-free

securities happen to be lower than the spreads between the corresponding forward rates.

This is due to the fact that the curve of the spot spreads is positively sloping and, therefore,

assumes that spreads are expected to increase in the future. From such expectations it

follows, then, that the curve of forward spreads is higher than the curve of spot spreads

(Figure 11.2).

Now that we have the spreads for subsequent years, we can estimate the probability of

default for years beyond the ﬁrst using the same criteria used to calculate the probability

of default for year one.

In particular, given that pT represents the probability of default during the T th year,

assuming that the debtor survives through the end of year T − 1, we can rewrite equation

(11.3) as follows:

eT −1 r1 = [(1 − pT −1 ) + pT −1 k]eT −1 r1 +T −1 d1 = [1 − pT −1 (1 − k)]eT −1 r1 +T −1 d1

(11.11)

where the left-hand side is the ﬁnal amount of a risk-free forward investment and the

right-hand side denotes the expected ﬁnal amount of a forward investment in the corporate

4

The relationship between PDs and forward rates is used by Elton, et al. (2001), for example.

See Appendix 3A of Chapter 3.

6

For the economic signiﬁcance and the calculation criteria of forward rates, see Appendix 1B .

7

See Appendix 1A of Chapter 1.

5

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