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3 Pillar One: The Standard Approach to Credit Risk

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20%

20%



Banks



Banks, depending on

the country of

incorporation



A



BBB+

50%



BBB

50 %



BBB−

100%



BB+

100%



BB−



B



BB



A−



A+



AA−



35%

From 100% to 50%, upon discretion of the national supervisory authorities



Non-residential real

estate mortgages



Below

B−

150%



150%



150%



150%



Residential real estate

mortganges



100%



100%



B+



Retail



50%



20%



50%



B−



75%



0%



AAA



Sovereign entities



AAA−

20%



AA+



Corporates



AA



Table 20.1 Risk Weights in the standard approach

Unrated

100%



50%



100%



100%



150%



100%



150%



Past-due



594

Risk Management and Shareholders’ Value in Banking



The New Basel Accord



595



sovereign entities, banks, small enterprises, and private parties grouped in the “retail

portfolios” category) plus some specific types of loans; columns show the different ratings

that might be assigned to a counterpart. By combining rows and columns, for instance, a

loan of ¤ 100 to a non-financial company with a AAA rating translates into ¤ 20 of riskweighted assets, thus leading to a capital requirement of 20 × 8 % = ¤ 1.6 (or, in other

words, 1.6 % of the non-weighted exposure). Similarly, a loan of ¤ 100 to a sovereign

State with a rating lower than B- produces a non-weighted exposure of ¤ 150, thus leading

to a capital requirement of 150 × 8 % = ¤ 12 (12 % of the nominal value).5

The last two columns need some brief clarifications. First, exposures to unrated companies (i.e. where no rating has been assigned by a qualified ECAI) are subject to a 100 %

weight (as in the 1988 Accord). This is likely to be the case also with a high percentage

of European non-financial companies (although many of them, as illustrated below, have

access to a favorable treatment within the category of the retail portfolio). Secondly, pastdue loans (i.e. loans whose interest or principal repayment is overdue by over 90 days)

are subject to a 150 % risk weight (as is the case with the worst rating buckets), since the

delay in repayment is likely to stem from difficulties on the part of the borrower.6

A few comments on some of the Table rows are also appropriate. First, exposures to

banks might be weighted in two different ways:7 They may be classified based on the

rating assigned to the borrowing bank, or based on the rating of the country where the

bank is headquartered. In the latter case, all credit institutions having their registered

office in the same State are given the same weight (for instance, 20 % if they are based

in a State with a rating of at least AA−). Credit with banks may also be subject to more

favorable terms8 in case of loans with a maximum initial maturity of three months.

Secondly, retail loans (which, for simplicity, may be defined as loans of no more than

Euro 1 million granted to individuals and small enterprises9 ) fall into a separate category

other than loans to corporates. Such loans almost invariably have no rating, but, because

they are highly fractioned (and thus guarantee a good risk diversification), are nonetheless

subject to a reduced weighting coefficient of 75 %. Also, as in the 1988 Accord, loans

backed by a mortgage on the borrower’s home are given a reduced risk weight (down from

today’s 50 % to 35 %), while for mortgages on other types of real estates the supervisors

of each individual country has the option of reducing the weight down to 50 %, provided

that the assets involved are subject to a small price-fluctuation risk.

Finally, the standard approach also offers a specific weighting system (not showed in

Table 20.1) for securitization transactions (see Chapter 15). This system imposes considerable capital requirements onto banks investing in junior or equity tranches (which are

often subscribed by the originating bank in order to facilitate placement of the remaining

5

With reference to sovereign States, note that being an OECD country is no longer one of the criteria for

calculating risk-weights.

6

Weights below 150 % might still be used for mortgages (see Table 20.1) or loans that are covered sufficiently

by analytical write-offs.

7

National supervisors will select one of the two available options and their choice will be implemented consistently to all regulated entities.

8

The risk weight associated with the next higher rating class is assigned, still subject to the minimum requirement of 20 %.

9

Specifically, the retail portfolio might include exposures to individuals or small enterprises for no more

than Euro 1 million, represented by specific types of products (revolving credits, such as credit cards and bank

overdrafts, personal loans and leases, credit facilities and credit lines to small enterprises, specifically excluding

securities) and sufficiently fractioned (for instance, supervisory authorities might require that no exposure to a

single counterparty exceeds 0.2 % of the total retail portfolio).



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Risk Management and Shareholders’ Value in Banking



securities of the special purpose vehicle), and is aimed at avoiding an uncontrolled

development of a high-risk junk loan market.

20.3.2 Collateral and Guarantees

Except for the last two rows, the capital coefficients set forth in Table 20.1 refer to unsecured loans. As a matter of fact, the standard approach offers the option of reducing the

capital requirement by obtaining suitable collaterals. Two approaches, with an increasing

degree of complexity, are suggested:

(1) The “simple approach”, which applies to a specific list of financial collaterals (cash,

gold, debt securities, some types of listed stocks and units of mutual funds investing

only in the above-listed assets), and

(2) The “comprehensive approach”, which also applies to all other listed stocks.

With the simple approach, the exposure portion covered with a valid collateral is weighted

by the coefficient established for the collateral (for instance, the applicable coefficient for

sovereign States if the collateral is made of government bonds) instead of using the

debtor’s coefficient (although, as a general rule, a minimum weight of 20 % is required).

Table 20.2 Haircuts established for different types of collaterals

Collateral



Rating



Maturity



Cash (in the same currency)

Government bonds



From AAA to AA−



Haircut

0.0 %

0.5 %



From 1 to 5 years



2.0 %



Over 5 years



4.0 %



Within 1 year



1.0 %



From 1 to 5 years



3.0 %



Over 5 years



From A+ to BBB−



Within 1 year



6.0 %



From BB+ to BB−

From AAA to AA−



Within 1 year



1.0 %



From 1 to 5 years



4.0 %



Over 5 years



8.0 %



From A+ to BBB− and



Within 1 year



2.0 %



bank bonds with no rating



Non-government bonds



Any



From 1 to 5 years



6.0 %



Over 5 years



15.0 %



12.0 %



Stocks included in the major indexes and gold



15.0 %



Other listed stocks



25.0 %



The New Basel Accord



597



With the comprehensive method, no capital requirement is applied on the exposure

portion backed by a valid collateral. However, in the calculation of such portion, the

value of the collateral must be reduced by a haircut reflecting the risk that the market

value of the financial instrument provided by the debtor may fall during the loan term.

Therefore, haircuts reflect the collaterals’ market risk and have been assessed by the

Committee by using VaR models similar to those illustrated in the second part of this

book. So, they are obviously stricter for securities such as stocks or bonds with a long

duration (see Table 20.2), as they are more exposed to the effects of volatility in market

factors. They are also increased as the number of days that the bank needs to recalculate

the collateral’s market value or to request the debtor to provide for an addition increases.

Such an increase reflects the principles illustrated in Chapter 6 and equals the square root

of the number of days.

Among risk-mitigating instruments, the Accord also includes guarantees and credit

derivatives (provided that they are issued by States or other public authorities, banks

and other financial institutions subject to supervision, non-financial companies with a

rating of at least A−). With such guarantees, the debtor’s risk weight is replaced by the

guarantor’s one, which usually implies a lower capital requirement. Incidentally, a similar

solution (the so called “guaranteed by guarantor replacement”) represents a simplified and

prudential approach: indeed, with a loan backed by a guarantee the bank only risks losses

if the guarantor’s default occurs together with the main debtor’s default (an event that,

as such, is more rare than a default of the guarantor only). So, the weighting associated

with the loan should somehow take into account the low risk associated with the event of

a joint default. However, the possibility to take this effect (“double default effect”) into

account is offered only to those banks that use the internal ratings method.



20.4 THE INTERNAL RATINGS-BASED APPROACH

20.4.1 Risk Factors

Banks applying for the internal ratings approach (whose risk measurement systems must

be approved by the national supervisory authorities) are fully or partly responsible for

assessing the degree of risk associated with each individual loan and to the credit portfolio

as a whole.

In this respect, the 2004 Accord – explicitly or implicitly – identifies six major “risk

drivers” that are likely to define the extent of possible future losses on a credit exposure.

In short (see also Table 20.3), such risk drivers are:

(1) Default risk, measured through the one-year probability of default (PD) as captured

by the bank’s rating system (see Chapter 13);

(2) Recovery risk, measured though the loss given default or LGD. Such loss must include

the costs incurred in the recovery process and the financial value of the time between

default and (partial) recovery;

(3) Exposure risk, due to the fact that the exposure at default (EAD) might differ (also

largely) from the current one.

(4) The three profiles mentioned above refer to losses incurred into by a bank in case

of the debtor’s default. As shown in Chapter 14, another factor needs to be taken

into account. Indeed, loans with longer maturities are also subject to a downgrading



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Risk Management and Shareholders’ Value in Banking



risk.10 Such a downgrading risk increases with the maturity (M) of a loan11 ; hence,

maturity is the fourth risk profile to be measured.

(5) Moving from the analysis of a stand-alone exposure to the assessment of risk at a

portfolio level, two further factors come into play. The former is the granularity of

the exposures (i.e. the tendency to grant few large credit lines versus a high number

of loans for smaller amounts). The formulas for calculating the minimum capital

requirement set forth in the Accord are based on the assumption – to be analyzed in

detail later in this chapter – that the portfolio has an infinite granularity, i.e. a null

concentration12 (in other words, it is made up of an extremely large number of small

exposures).

(6) The second relevant parameter at a credit portfolio level is the correlation among

borrowers. It is higher if the bank grants loans to borrowers that are concentrated in

few geographical areas or industries, i.e. borrowers exposed to common risk factors,

while it is lower if the bank’s portfolio is highly diversified to borrowers whose

conditions appear to be relatively independent. The Committee opted for a simplified

framework, specifying the values for correlation among borrowers through a system

based on a few large, standard categories. Therefore, banks are not required to check

the actual degree of diversification of their portfolios

Factors from 1 to 4 (PD, LGD, EAD, and maturity) are the fundamental parameters to

be adequately measured by a rating system.

Depending on the degree of sophistication of their models and on their historical data,

banks might be allowed to use two different approaches:13

– A foundation approach, allowing to estimate only the debtors’ PD using their own internal methods, while requiring to refer to pre-established values set by the Authorities

for LGD, EAD, and maturity;

– An advanced approach that allows to measure all four risk profiles with the banks’

internal methodologies (whose effectiveness and robustness, however, is to be demonstrated clearly).14

10

Let us take a 10-year loan as an example. The borrower creditworthiness is assessed with a specific rating

reflecting its PD in the following 12 months. After one year, the debtor does not default, but has badly worsened

so that its rating is downgraded to a much higher risk level. If the bank were free to renegotiate the loan terms,

it would now require a larger rate spread in order to compensate for the higher risk level. However, it cannot

do so because the loan was granted for 10 years upon pre-defined terms. Hence, the economic value of the

loan has decreased and, in fact, this lower value is a loss (even if it is not recorded on any secondary market).

The longer the loan maturity, i.e. the number of years in which the transaction will continue to operate with

an inadequate spread, the larger the loss.

11

Furthermore, the risk is higher for high-rating loans since those with a medium-low rating, in addition to

being downgraded, may also be upgraded.

12

This is obviously an unrealistic assumption. The individual national Supervisors may require corrections,

under the provisions of pillar two (see section 20.5).

13

Following a so-called “evolutionary” approach, banks will first be prompted to adopt the foundation approach:

only as they grow more confident about their internal estimates (and can show the regulators the databases on

which those estimates are based) will they be allowed to move to the advanced approach.

14

Considering the highly diversified and heterogeneous characteristics of retail loans granted by individual

banks, which cannot be summarized in a unique set of parameters defined by the Authorities, the foundation

approach cannot be adopted for this portfolio. In other words, all banks wishing to apply an internal ratings

system will directly use the advanced approach, thus internally estimating not only PD, but also LGD and

EAD (indeed, no adjustment for maturity is envisaged for the retail portfolio). However, such parameters do

not need to be estimated for each loan or each counterpart (it would be too time-consuming, considering the



The New Basel Accord



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Table 20.3 Risk factors included in the Revised Accord

Factor



Meaning



Features



Notes



Entity qualified

for assessment



PD



Probability that

the counterpart

becomes

insolvent



Calculated on a time

horizon of 12 months,

but keeping into

account any possible

economic downturn



A default occurs

when the debtor is

“unable or

unwilling to pay in

full” and in any

case after a delay

of over 90/180 days



The bank,

provided that it

has an internal

ratings system

validated by the

Supervisors.



LGD



Loss unit rate

in case of

insolvency



Calculated

considering loan

recovery costs and

the financial value of

time



Affected by the

technical form and

by the provision of

collaterals



EAD



Bank’s

Calculated

exposure at the considering the

time of default available margins on

credit lines by cash

and by signature



Maturity (M)



Loan maturity



Calculated as a

duration, i.e.

considering

repayments expected

before the final expiry

date



Granularity



Tendency to

grant few, large

loans or several

small loans



Pre-established and

not calculated (it is

assumed to be

infinite)



Tendency of

different

debtors to

“default

simultaneously”



Pre-established and

not calculated

(different values for

different types of

clients)



Correlation



Constant for

technical forms

with a

pre-established

sinking plan



The

Supervisors or

the bank,

provided that

the latter has

an advanced

rating system

validated by

the Supervisors



Possible correction

within the

framework of

“pillar two”



The

Supervisors



As mentioned above, banks are in no case allowed to measure the granularity and correlation of their loan portfolio as they are fixed at “standard” levels by the Authorities and

identical for all banks subject to the Accord.

This means that banks, despite their ability to estimate the inputs in the credit risk

assessment model, are not authorized to replace the model designed by the Authorities

large number of positions included in the retail portfolio) and banks will simply group retail loans in pools

having homogeneous quality and technical form, and measure their characteristics (past default rate, unit loss

rate, degree of utilization of available margins, etc.) at the level of the entire pool and not of each individual

loan. This approach based on large categories should make it easier (and more cost effective) to manage the

rating system for retail customers.



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Risk Management and Shareholders’ Value in Banking



(see section 20.4.3) with their own internal ones. Indeed, the Committee considered the

latter models to be too “young” to be fully reliable. Specifically, the Committee justified

its rejection of internal credit risk models with two main arguments:

– Lack and poor reliability of input data necessary for such models and in particular of

data on correlations among borrowers;

– Inability to validate the output of the model on an ex post basis, by means of statistically

significant back tests, due to the long time horizon – usually one year – adopted by

the credit risk models.15

20.4.2 Minimum Requirements of the Internal Ratings System

If a bank wants to be authorized to use its own internal ratings system for calculating the

minimum capital requirement, it must comply with a set of minimum requirements. The

general principle underlying such requirements is that the risk estimate procedures must

be able to correctly differentiate among different risk levels, providing a correct, accurate,

and consistent assessment in line with the bank’s past experience.

The first requirement is that the risk measurement for a client be kept rigorously separate

from the risk measurements of the individual loans granted to such client.16 Indeed,

the former can be measured in terms of a specific probability of default (PD), while,

in the case of individual loans, different and further issues are at stake, such as the

expected recovery rate (which depends on LGD) and the exposure risk (EAD, which

increases with the available undrawn margins). Consequently, a bank would be wrong to

upgrade the customer’s rating if such customer provides collaterals on the loans. In fact,

such collaterals would not change the default probability of the counterpart, but only the

consequences that any default would cause on the actual amount of losses.

As to the customers’ PD, banks are required to document in writing the key characteristics of the adopted measurement systems. It is therefore necessary to specify the

definitions given to the different “creditworthiness grade” (seven as a minimum) of the

rating scale, and especially the “plausible and intuitive” criteria used for granting a specific creditworthiness grade to a given counterpart. Such specifications must be detailed

so as to allow the bank’s analysts (even if they are based in different areas or facilities)

to work homogeneously and consistently, and to allow any internal auditors (in addition

to the Supervisory Authority) to easily understand its structure and logics.

The New Accord does not elaborate on how the PD rating system of a bank should

be built; so it does not identify which indicators (for instance which balance-sheet ratios)

should be used for granting a rating, nor does it oblige banks to adopt automatic systems,

based on statistical scoring techniques.17 In this respect, although the Accord provides for

the use of statistical algorithms, it only accepts them as a primary or partial basis for rating

15

Backtesting of market risk models (see Chapter 8) is carried out by comparing the daily VaR for the past

250 days with the corresponding economic results (P&L) of the trading activity (see the previous chapter).

Given the time horizon of credit risk portfolio models – typically one year – , a similar comparison would

require using past data relating to 250 years!

16

However, an exception to this principle is envisaged for retail loans whose evaluation is carried out within

large homogeneous categories (and not for each individual counterpart or loan) and can be done directly by

referring to the “expected loss unit rate”, which includes the effects of PD, LGD, and EAD.

17

See Chapter 10. See also Resti (2002a) for an introduction to the differences between scoring algorithms and

the approach followed by rating agencies.



The New Basel Accord



601



assignments, a sort of preliminary tool subject to the supervision of human experts, aimed

at ensuring that all relevant information are taken into account (including information that

do not fall within the scope of the algorithm). The Accord actually states that, in assigning

ratings, banks should take into account all relevant available information, making sure

that it is sufficiently up-to-date.18

The Committee gave a definition of default to be referred to for estimating PD. A

debtor is insolvent if one of the following two conditions occurs:

– a subjective condition: the bank deems unlikely that the borrower fulfills its obligations in full. This evaluation may result from the bank having partly written off the

original exposures, set up specific provisions, or granted a credit restructuring (waiving

or deferring part of the scheduled payments), or from the borrower having filed for

bankruptcy or for a procedure aimed at protection from creditors;

– an objective condition: the counterpart is more than 90 days late in fulfilling at least

one of its obligations. This term can be extended to 180 days for loans to private

parties and families (retail portfolio) or public administrations, since these types of

debtors are often late in paying, without this implying necessarily their unwillingness

or inability to fulfill their obligations.

As to the measurement of LGD, EAD and maturity a distinction should be made, as

mentioned above, between banks that are authorized to use the foundation or the advanced

approach. For the former, LGD, EAD and maturity are measured based on the criteria

required by the Regulators.

Foundation approach – LGD is fixed at 45 % for all unsubordinated and unsecured

exposures. Such value is increased to 75 % for subordinated loans subscribed by the bank,

but can be reduced for loans assisted by adequate collateral. If, for instance, collateral

represented by financial instruments similar to those admitted in the standard approach

(see section 20.3.2) is provided, LGD can be reduced down to 0 %, depending on the

security value as corrected by the haircuts set forth in Table 20.2. Three other types of

non-financial securities are also accepted: trade receivables, real estate properties (both

residential and commercial), and other collaterals (including equipment or machinery,

but excluding any asset acquired by the bank following the debtor’s default). If such

securities are provided, LGD may drop to 40 % (35 % for trade receivables and real estate

properties).

On the other hand, EAD is equal to 100 % of the current exposure, plus 75 % of any

available margin on credit lines that are not immediately and unconditionally revocable.

Off-balance sheet exposures are transformed into loan equivalents based on the credit

conversion factors illustrated in Chapter 18. By convention, maturity is 2.5 years for all

credits.

Advanced approach – Banks are authorized to use their internal estimates for LGD and

EAD, provided that they satisfy the Regulators that such models are conceptually sound

and consistent with the past experience. Estimates must represent a long-term defaultweighted average, and be based on data relating to similar exposures and counterparts.

18

Such information may include ratings granted by external agencies, provided that they appear consistent

with the other information available to the bank. The Accord specifically states that different categories of

counterparts (for instance, large corporations, SMEs, private parties, banks) might require different algorithms

and rating processes.



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Risk Management and Shareholders’ Value in Banking



Banks are required to use historical data concerning an entire economic cycle and in any

case no less than seven years19 (five years for retail loans).

As described in Chapter 12, LGD has to be measured in an economic and not merely

in an accounting manner, correcting the face value of recoveries by the effect of all

factors that are likely to reduce its present value (specifically the discounting effect due

to the time necessary for recovery, as well as administrative, direct and indirect, costs

associated with recovery). The Basel Committee also provided that, in estimating LGD,

banks should take into account the state of the economic cycle. This allows to account

for the correlation between LGD and PD that resulted from some empirical studies (see

Chapter 12), due to the fact that recovery rates tend to decrease in recession phases, when

default rates increase.

Finally, maturity must be estimated taking into account the impact of any intermediate payments during the loan life (more specifically, the duration formula set forth

in Chapter 2 shall be used, taking zero as the interest rate). The value thus calculated

should be truncated, if necessary, to five years, while maturities of less than one year are

authorized only in few specific cases.

Next to the above-mentioned requirements, which relate to the technical characteristics of the rating system, there are others – just as important – concerning the way the

instruments described above must be transferred into the operational reality of the bank,

i.e. the interaction between the rating system and the process for assessing, granting and

managing loans.

The revised Accord expressly provides that a rating system featuring all the abovelisted requirements is not acceptable – for the purpose of capital requirements – unless it

plays “an essential role in authorizing credit lines, in risk management, internal capital

allocation, and corporate governance functions”.

The goal thus is for the rating system, notwithstanding the necessary exceptions, to

gradually become the decision-making infrastructure to be placed at the core of the loan

granting decisions, the calculation of provisions against future losses, the estimation of

the economic capital allocated to the individual business areas relating to the bank’s credit

portfolio, and finally (although this is not expressly required by the Accord), the definition

of fair lending rates covering expected and unexpected losses.20

This obligation to make effective and large use of the rating system looks as an appropriate and far-sighted requirement, as the gradual fine-tuning of a risk assessment model

can only be achieved through its daily implementation, i.e. through a comparison between

its results and the operational and commercial know-how of those in charge of customer

relationships.

As to the corporate functions affected by the rating process, the Accord provides that

banks set up independent credit risk control units responsible for designing or selecting,

implementing, reviewing rating systems, and overlooking their ex post performance. In

order to avoid any “conflict of interests”, such units must be functionally independent of

the staff that, in any respect, is responsible for the origination of loans. Furthermore, the

assignment of ratings to individual clients and their periodic review are to be carried out

and/or approved by entities that do not directly benefit from credit granting. This seems

to draw a rather sharp distinction between customer relationship managers (who “sell”

19

Pursuant to Directive 2006/49, EU banks may be granted up to three years’ discount when the New Accord

is implemented for the first time.

20

See Chapter 15.



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603



loans to the clients and meet their own budgets based on the results of such sales) and

risk managers (who assess the creditworthiness of borrowers and loans).

The Accord further provides that the internal auditing function or another independent

entity review the bank’s rating system and its operation once a year. Finally, all material

aspects of the rating process shall be approved by the Board of Directors (or its executive committee) and by the management of the bank. Such bodies must have a general

knowledge of the system and an in-depth knowledge of the reports to be submitted to the

management.

20.4.3 From the Rating System to the Minimum Capital Requirements

With the standard approach, the minimum capital associated with an exposure is simply

8 % of risk-weighted assets, where weighting is to be carried out subject to the system described in Table 20.1 (subject to adjustments due to any securities as described in

section 20.3). The IRB approach, on the other hand, relies on a more complex mechanism for transforming the characteristics of a loan (PD, LGD, EAD, maturity) and of its

portfolio (granularity and correlation) into a capital requirement..

Such a mechanism is based on a simple credit risk model, whose structure21 and

parameters shall be analyzed in this section. Also, some adjustments to the basic model

will be illustrated, introduced in order to account for the distinction between expected

and unexpected losses, and for the effect that a longer maturity has on credit risk.

The reference model – Consider a credit portfolio made up of a large number of small

loans (i.e. an “infinitely granular” portfolio). Suppose, in line with Merton’s model,22 that

every borrower defaults if and only if the value of its assets drops below a specific

threshold (e.g. the value of debt) at the end of a given time horizon. Let us also assume

that the percentage change that shall occur next year in the asset value of the i-th borrower

(asset value return, AVR, see Chapter 14) can be expressed as:

Zi = w · Z +



1 − w 2 · εi



(20.1)



i.e. as a linear combination23 of two components: factor Z, which is correlated to the

macroeconomic cycle (and thus impacts on all borrowers in the same way), and factor εi ,

which only depends on the individual (idiosyncratic) risk of the borrower.

Depending on the weights used in the formula, a borrower may be more or less exposed

to the cycle: as w increases, all borrowers tend to be more and more correlated to one

another, while a decrease in w means that the idiosyncratic characteristics prevail and

that the individual borrowers are more independent.

Note that this representation of the effect of the macroeconomic variables on a company’s asset value is a simplification of the multifactor models (e.g. CreditMetrics, Creditrisk+) described in Chapter 14, where two or more random variables represent different

industries, geographical areas, or macroeconomic factors. However, such models would

21

For a short introduction see Finger (2001), while for a more detailed and rigorous description of the model

see Gordy (2001).

22

See Chapter 11.

23

Note that, since Z and εi have a unit variance, and the variance of the sum of two random independent

values, var(αx1 + βx2 ), is always equal to α 2 var(x1 ) + β 2 var(x2 ), in order for Zi to follow a standard normal

distribution we have to impose that α 2 + β 2 = 1. In our case, this was done by making the second weight

equal to the square root of 1 − w 2 , where w is the first weight.



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Risk Management and Shareholders’ Value in Banking



hardly be “manageable” for regulatory purposes.24 For this reason, the Basel Committee

decided to adopt a framework based on a single factor such as the model presented in

this section.

If we assume that Z and εi follow a standard normal distribution, then equation (20.1)

implies that Zi also is standard-normally distributed. For each pair of i and j borrowers, the correlation between asset value returns ( “asset correlation”, see Chapter 14) is

given by:

(20.2)

ρ(Zi , Zj ) = w2

Quite logically, the higher the dependence (w) of each company’s assets on the macroeconomic cycle, the higher the correlation (ρ) between the asset returns of the two companies.

We know that borrower i becomes insolvent if and only if Zi < α, where α represents

its default point. If pi = PD is the unconditional probability of default (independent

of the value of factor Z) of such borrower, then N (α) = pi (see Figure 20.1), where

N (.) indicates the standard normal cumulative probability distribution.

f(Zl)



pi = N(a)



a = N −1(pi)



Zi



Figure 20.1 Asset value returns of company i and its default point α



Let us now assume that we know the trend of the macroeconomic variable in the next

year (obviously, this is an unrealistic assumption, which will be removed later on). This

is like assuming we know the specific value (let us say Z ∗ ) that macroeconomic factor Z

shall take. Then,

Zi = w · Z ∗ + 1 − w2 · εi

(20.3)

and company i shall become insolvent if and only if

Zi = w · Z ∗ +

i.e. if



1 − w 2 · εi < α



α − w · Z∗

N −1 (pi ) − w · Z ∗

=

εi < √



1 − w2

1 − w2



(20.4)



(20.5)



24

A multifactor model would make calculations more complex and, above all, would make the capital requirements on a new loan dependent on the portfolio composition of each bank.



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