The Differentiations in Loan Valuation with Retail and
Corporate Exposure under Basel II
By
Jiun-Fei Chiu
Associate Professor in Department of Finance,
Yuan-Ze University
00-886-34638800 Ext 2666
fnjfchiu@saturn.yzu.edu.tw
Chia-Tien Chang
PhD student in Department of Business Management,
Ji-Nan University
00-886-932834261
hcj.alex@msa.hinet.net
Abstract
The explicit differences in asset characteristics and risk management practices
between retail and corporate exposures lead to differentiation in loan valuation.
For corporate exposure, structured rating system assigning a specific rating to
each borrower or loan is based on a combination of objective and subjective
criteria. The judgment plays a significant role for banks that employ statistical
rating models. While for retail exposure, portfolio of loans are divided into
“segments” made up of exposures with similar risk characteristics assumed to
exhibit homogeneous default characteristics. The loss performance will follow
predictable patterns over the forecast periods. In this paper we focus on loan
valuation in continuous time framework under the IRB approach. We extend the
framework Merton (1974) combined with the theory of portfolio to characterize
the dynamics of the value of loan portfolio for both retail and corporate exposures.
By considering the specific differences in risk components inherent in between,
both exposures exhibit quite different valuation of loan portfolio. The effects of
asset return correlation and granularity in the loan portfolio on valuation are also
examined. The main results are as follows: first, the differentiations in risk
components will ultimately result in quite different risk profile in the whole
0
portfolio of loans; second, both the VaR and the value of loan portfolio move in
tandem over time, which contrast to the single time period setting in literature;
third, the value of bank’s loan portfolio depends on the instantaneous correlation
coefficient between the instantaneous rate of return on the loans; fourth, the
instantaneous idiosyncratic risk of the loan portfolio would wane if the number of
loans in the portfolio increases, especially for retail exposures; fifth, when the
average instantaneous coefficient of correlation decreases, the instantaneous
systematic risk of the loan portfolio and the extent to which the value of the loan
portfolio impacted by the business cycle would become less; sixth, when the level
of concentration , granularity, increases over some critical level, the variance of
the value of loan portfolio would increase with it. This would incur additional
charged capital for the bank.
Introduction
As the new Basel capital accord (henceforth, Basel Ⅱ) has been finalized by the Basel
Committee on Banking Supervision (henceforth, BCBS), along with it comes the new
era for banks all over the world not only in the financial risk regulation but in the
valuation of the portfolio of loans as well. The well-known three pillars in Basel Ⅱ
including minimum capital requirements, supervisory review, and market discipline
are proposed to increase substantially the risk sensitivity of the capital required. That
is, the supreme purpose of the new financial risk regulation is to arrange capital
adequacy assessment more closely with the key components of banking risk. To
clarify the relationships between risk management and valuation of loans for banks,
the impact of risk components, having been considered in capital charge, need to be
further examined. Since in general the corporate loans constitute the major
components of loan portfolio for commercial banks, it spontaneously incurs an
important problem of how the loans to corporate obligors are valued over time in
order to meet the requirements for risk management.
One of the BCBS’s goals in setting forward an IRB approach is to align more
accurately capital requirements with the intrinsic amount of credit risk to which a
bank is exposed. The orientation of the IRB approach is to assess internally both their
credit risk profile and their capital adequacy. There are three main elements to the
IRB approach for corporate exposures:
(1) Risk components: a bank must provide, using either its own estimates or
standardized parameters;
(2) Risk-weight function: which provides risk weights and hence capital requirements
for given sets of these risk components;
(3) A set of minimum requirements: a bank must meet in order to be eligible for IRB
1
treatment.
Under the framework of internal ratings-based (IRB) approaches to credit risk
constructed in Pillar I of Basel II, banks are allowed to compute the capital charges
for each exposure based on the estimates of the four risk components, i.e. the
probability of default (PD) of an obligor, the loss given default (LGD), exposure at
default (EAD), and maturity (M) of the transaction. As the IRB approaches include
two variants: the foundation approach and the advanced approach. In the former one,
banks input their own assessment of the risk of default, i.e. PD, of the obligor, but
estimates of additional risk factors, i.e. LGD, EAD and M are derived through the
application of standardized supervisory rules. In the latter one, banks are allowed to
use their own internal assessments of these components, subject to supervisory
minimum requirements. In addition, banks may use their estimates of LGD, EAD,
and/or the treatment of guarantees and credit derivatives subject to meeting additional
minimum requirements specific to each risk component. Credit risk mitigation in the
form of collateral, credit derivatives and guarantees and on-balance sheet netting, can
materially impact upon a bank’s estimation of PD, LGD or EAD. In the advanced IRB
approach, banks are permitted to use their own estimates for the effect of credit risk
mitigation techniques on their estimates of PD, LGD and EAD.
LGD is influenced by key transaction characteristics such as the presence of collateral
and the degree of subordination. In most cases EAD will equal the nominal amount of
the facility, but for certain facilities it will include an estimate of future lending prior
to default. Thus a bank might be able to differentiate EAD and LGD values on the
basis of a wider set of transaction characteristics (e.g. product type, wider range of
collateral types) as well as borrower characteristics. The estimates of PD, LGD and in
some cases M (M) associated with an exposure combine to map into a schedule of
regulatory capital risk weights. The risk weights reflect the full spectrum of credit
quality through use of a continuous function of risk weights. All banks take into
account facility characteristics such as third-party guarantees, collateral, and
seniority/subordination of the obligation in making lending decisions and in their
credit risk mitigation processes. Moreover, facility characteristics are also explicitly
considered in assessing the credit quality of an exposure and/or analyzing internal
profitability or capital allocations.
The intention to require banks to adopt IRB approach is to secure two key objectives:
the first is additional risk sensitivity in that a capital requirement based on internal
ratings can be more sensitive to the drivers of credit risk and economic loss in a
bank’s portfolio; the second is incentive compatibility in that an appropriately
structured IRB approach can encourage banks to continue to improve their internal
risk management practices.
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As the four risk components are inherent in the loans to corporate obligors, upon
which the capital charges for banks depend, the estimates of these four risk
components would inevitably exert impact upon the value of the loans. In this paper,
we focus on corporate exposures and intend to value the corporate loans in continuous
time framework under the IRB approach. A corporate exposure, in Basel II, is defined
as a debt obligation of a corporation, partnership or proprietorship. In other words, a
loan is an asset for a bank, meanwhile a debt for an obligor. Therefore, under certain
circumstances the methodologies used to value a risky debt can be extended to price a
loan.
The valuation of risky debt is such an important issue that there have been
plethoric studies in the financial literature. To value the corporate debt, we can trace
back to Merton（1974）, which initially utilizes the option pricing model to develop a
structural model for valuing risky corporate debt. In his model, the value of a firm’s
risky debt is contingent on the market value of its assets. By applying Black and
Scholes’ (1973) option pricing formulae and define the market price of a risky
zero-coupon bond, Merton derives the credit spread to price the corporate debt on
which the issuer may default. Following the valuation model in Merton（1974）, Black
and Cox（1976） further analyze the impact of various types of provisions in corporate
bond indenture upon the corporate debt value . Black and Cox（1976）indicate that
some specific provisions do increase the value of corporate bonds. Other extensions
from Merton（1974） include Brennen and Schwartz (1977, 1978, 1980), Geske
(1977), Ingersoll (1976, 1977), Leland (1994), Leland and Toft (1996), Longstaff and
Scvhwartz (1995), and Zou (1997), etc.
Another stream of studying loan pricing is based on Gordy (2002) which releases
a simple way to calculate economic capital charges with a portfolio model of credit
value-at-risk. This stream stems from the concept of well-known traditional pricing
theory, Capital Asset Pricing Model (CAPM). Gordy (2002) treats the marginal VaR
contribution of an additional loan to a portfolio of loans as portfolio-invariant, and
derives a single systematic risk-factor model. The marginal capital requirement for the
portfolio depends on the properties it holds, but not the characteristics of the
instruments. In order to reach portfolio-invariant, two conditions must be satisfied.
First, the loan portfolio is well fine-grained, to diversify specific instrument risk.
Second, there is only a single systematic risk factor. Gordy (2002) indicates that
performance of the portfolio is affected by systematic risk factor, such as
macroeconomic variables, and idiosyncratic risk factor to the individual instrument.
By employing Gordy’s (2002) model, Dietsch and Petey（2002）devote to the risk
modeling issues about small and medium-size enterprise (SME) loans portfolios.
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Dietsch and Petey notice some features of retail loans portfolios. First, SME won’t be
repaid like corporate loan, there is no market value for SME, and there is no transition
process. So the existing models can’t be used to value retail loans. Second, the
number of SME is much larger than that of corporate loan portfolios. It doesn’t
behoove pricing retail loans with corporate models. It consumes too much time to
simulate the parameters at the individual segments and levels. Finally, there are either
rating agencies or financial market price for the models of corporate portfolios. By
using Probit and Gamma distribution models to measure VaR in SME loan portfolios,
Dietsch and Petey model the PD and use two types of information：（1）default score
（2）the balance sheet amount of the firm bank debt to compute the risk of each small
business and rank borrowers risk classes. Similar to Gordy (2002), the default rate is
driven by a single systematic risk factor and the specific risk factor. The results show
that capital requirements derived from an internal model are significantly lower than
those derived by the standard capital ratio and the IRB approach as well. This
demonstrates the interest of taking into account the correlation among the exposures
in internal credit risk models explicitly even in the case of retail portfolios. Dietsch
and Petey also verify that one of the main advantages of an internal credit risk model
is to lead to a better allocation of capital and to better loan pricing. Linde and
Roszbach (2004) employ data from two Swedish banks’ to display that SME and
retail loan portfolios are not affected by systematic risk factors as much as corporate
loan portfolios.
Repullo and Suarez (2004) assess loans of portfolio with Gordy’s model under
Basel capital requirement. It builds a model with a perfectly competitive market of
business lending under the IRB approach. Repullo and Suarez show that low risk
borrower will concentrate borrowing from banks that adopt the IRB approach and
high-risk borrowers prefer borrowing from banks that adopt the standardized
approach. The simulations show that IRB approach compare to Basel I may decrease
the loan rates of 0.65% with a PD of 0.10%, and for loans with a PD of 10% rise
about 1.25%. Repullo and Suarez also argue that to estimate the PD correctly is the
most important in IRB approach. The supervisory should build an incentive system
with penalties and rewords not only to avoid moral hazard in banking but to ensure
the appropriate estimate and true report the risk of their loan portfolios.
To value the corporate loans in continuous time framework under the IRB
approach, we extend the pricing framework of Merton (1974) by embedding the
characteristics of corporate exposures into the loan portfolio of a bank. That is, the
four risk components PD, LGD, EAD, and M are taken as inputs for loan pricing in
our model. As Bohn (2000) asserts that the structural model requires characterization
of issuer’s asset value process, issuer’s capital structure, LGD, default-risk-free
4
interest rate, PD, correlation between the default-free interest rate and the asset price,
and the terms and conditions of the debt issue. In this paper we assume LGD is a
function of firm’s value, debt’s collateralization, and priority and is stochastic. In
addition, we also utilize the framework of portfolio theory to model the dynamics of
the value of loan portfolio. Notably, the effect of granularity to loan portfolio would
be examined in this framework. Recall that both the transition probability matrics
based on credit ratings in CreditMetrics and the one based on EDF in KMV approach
reflect the fact that the value of portfolio of loans would fluctuate with the changes in
credit quality of obligors over time. Hence, we consider the value of loan portfolio
changes in continuous time framework, which would incur changes in the value of the
capital charge through the linkage of the risk components between them. This
apparently contrast to the single time period models posed in Dietsch and Petey
（2002）, Repullo and Suarez (2004), etc.
As the purpose of the new financial risk regulation is to arrange capital adequacy
assessment more closely with the banking risk in order to prevent the bank from being
insolvent, we consider a bank as a levered firm with the probability of insolvency
which would be supervised by regulatory agency. Further, we take the loans to
corporate obligors into account to reflect the decision of risky investment made by the
bank management.
The main results in this study are as follows: first, the capital charge considered
in terms of VaR in the risk management is liked to the dynamics of the value of loan
portfolio; second, both the VaR and the value of loan portfolio move in tandem over
time, which contrast to the single time period setting in literature.
The results show that the credit spread of the loan increases with PD, LGD, M,
and debt ratio of the obligors.
The structure of this paper is as follows: section 2 is the valuation framework of the
loans; section 3 demonstrates the dynamics of the value of loan portfolio; the effect of
granularity is discussed in section 4, and section 5 concludes.
The Valuation Framework
Consider a time set [0, T] as a credit horizon for an economy , where T  (0,
∞). A filtered probability space (  , F, F, P) is set with respect to this credit horizon,
where F= {Ft : t  [0, T]} satisfies increasing, right-continuous and augmented usual
conditions and is the standard filtration of standard Brownian motion B. Let B=
( B1 , B2 ,..., Bd ) be the standard Brownian motion in R d and Ft a Martingale with
5
respect to the filtered probability space. Therefore, B represents the continuous
resolution generated by the d sources of uncertainty over time that would determine
the risks in the economy by influencing all external economic activities.
Suppose that in this economy there exists one bank with its transaction
counterparties including depositors and owners as its claimholders, and N corporate
borrowers as its debt obligors. Further, suppose that each corporate obligor which
may default during the credit horizon [0, T] with the probability of default, PDi , i= 1,
2, …,N, borrows from the bank one loan so that the asset portfolio of the bank is
constituted by N loans of different credit qualities. For IRB purposes, the estimate of
M will be based on a standard supervisory definition of effective M, which
emphasizes the profile of a loan’s contractual principal payments over time.
The credit quality of individual loan would be reflected in the credit spread
charged by the bank. In specific, the bank finances its N risky projects of loan with
two sources of funds: the first is a given amount of deposit which will be reimbursed
to the depositors at the loan maturity date of time T; the second is the capital charged
which is required by the banking supervisory agent and hence in turn depends on the
value of the loan portfolio.
By extending the assumptions in Merton (1974), we make the following ones:
A.1 there are no transactions cost, taxes, or problems with indivisibilities of assets and
restrictions of short sales.
A.2 the assets are traded at the market price continuously in the credit horizon.
A.3 the risk-free rate of return is r.
A.4 prior to the maturity date, T, of the deposit and the N loans, any cash outflows
incurred by cash dividends and share repurchase, or cash inflows incurred by
issuing new securities are not allowed for both the bank and the N corporate
obligors.
A.5 without buying policies of deposit insurance, the bank has to handle its asset and
liability management well enough to pay for two classes of claims. That is, the
bank is required by the supervisory agent to reimburse D dollars to the depositors
on the specified date T, and to charge adequate capital E to cover the loss in even
the worst case faced by the bank during the credit horizon.
A.6 all the N corporate obligors also have simple capital structure with two classes of
claims: a single loan borrowed from the bank as its debt, and the equity.
A.7 the value of the bank, V0t , and those of the corporate obligor i, Vi , i= 1, 2, …, N,
through time follow Geometric Brownian motions with stochastic differential
equations:
d Vit / Vit = i dt+  i d Bit
i= 0, 1, 2, …, N, and
6
t  [0, T].
where i is the instantaneous expected rate of return on the Vi ;
 i is the instantaneous standard deviation of the rate of return on the Vi ;
i and i are Borel measurable and satisfy the Lipschits and growth
conditions;
d Bi is a standard Brownian motion;
d Bi d B j =  ij dt, where  ij is the instantaneous correlation coefficient
between d Bi and d B j .
For each loan i with the market value of Ait , the loan rate is Rit = r+ CSi , where CSi
is the credit spread charged by the bank and is the function of the probability of
default, PDit and the loss given default of loan i, LGDti , i= 1, 2, …,N, and t  [0,
T]. As usual, LGDti depends on obligor i’s value, debt’s collateralization, and
priority and is stochastic. LGD reflects differences in lending standards, the type of
collateral is taken, policies and procedures in pursuing recoveries from defaulted
borrowers, and other areas. LGD can also be dependent on the economic cycle. The
Banks see a range of borrower and transaction specific characteristics as having an
impact on LGD. Borrower characteristics include asset size, country of incorporation,
industry sector and whether the corporate is a holding or operating company.
Transaction specific characteristics include the seniority of the transaction, the
amount and nature of any collateral taken and loan covenants. In using estimates of
LGD, the bank may takes any form of collateral and it can demonstrate that this
serves to reduce its experience of loss, and then this should be recognized in the
bank’s own internal estimates.
For simplicity, all loans are lent to the N obligors at time 0 and would be
reimbursed at time T to the bank without being collateralized or be guaranteed so that
the exposure at default (EAD) to the obligor i is also Ait . The three risk components
( PDit , LGDti , Ait ) inherent in loan i combine to provide a measure of expected
intrinsic, or economic loss, i= 1, 2, …,N.
As for the deposit of bank, suppose that the deposit rate is R0 t = r+ CS0 , where
CS0 is the credit spread to the depositors to reflect the insolvency risk of the bank,
t  [0, T]. Denote the deposit reimbursed at time T by D, so that at time 0 the market
value of deposit is D e  R0 0T . The bank would utilize D e  R0 0T and VE 0 the initial value
of equity at time zero to invest in N risky projects as loans to N obligors. That is,
N
A
i 1
i0
＝D e  R0 0T ＋ VE 0
(1)
7
N
where Ai 0 is the initial value of loan i lent to the obligor i and
A
i 1
i0
is the
initial value of loan portfolio at time zero. During the credit horizon [0, T], the value
of the asset portfolio of the bank at any time t is equal to the sum of those of the N
loans, i.e.:
N
V0t ＝  Ait ＝D e  R0 t (T  t ) ＋ VEt ,
t  [0, T]
(2)
i 1
By applying the European call option pricing formula of Black and Scholes (1973),
the value of equity of the bank at any time t, VEt , can be expressed as
VEt ＝ V0t  ( x01 )－D e  r  ( x02 ),
t  [0, T]
(3)
where  ＝T－t, t  (0, T);
x01 ＝{ ln[ V0t /D e  R0 t (T  t ) ]＋( r＋
1 2
 )  }/  0  ;
2 0
1 2
 )  }/  0  ;
2 0
is the standard deviation of the instantaneous rate of return on the assets of
x02 ＝{ ln[ V0t /D e  R0 t (T  t ) ]＋( r－
0
the bank, d V0t / V0t ;
V0t is the bank i’s asset;
 (.) is the cumulative Normal distribution function
Note that the value of equity VEt is the target to be supervised. As the credit qualities
of the corporate obligors may change over time which would be reflected in PDit , this
will lead the value of loan portfolio of the bank V0t to change in response and then in
turn would impact upon the equity value of the bank VEt . For the purpose of
preventing the bank from being insolvent, the bank supervisor would require the bank
to charge adequate capital to cover the potential loss incurred in even the worst case
where most obligors default during the credit horizon [0, T]. In the perfectly
supervisory reviewing situation that the value of equity VEt is regulated to the extent
to which the capital charged is raised to an exactly adequate enough level, and the
probability of insolvency of the bank goes down to zero: i.e.,
VEt ＞0 and VEt ↓ 0  ,
t  [0, T]
(4)
However, it would be extremely difficult and may cost too much for the bank and the
supervisor to meet such a strictly accurate requirement. Instead, a specified low level
of the probability of insolvency of the bank is allowed in general. Consider that if V0t
≦ D e  R0 t (T  t ) , then VEt ≦ 0 which means that the bank will be insolent, and this
8
implies from the option pricing formula that
V0t  ( x01 )≦ D e  r  ( x02 )
(5)
By assumption A.7, the value of the bank follows Geometric Brownian motions
through time, V0t would be equal to
V0t ＝ V00 exp{ ( 0  1  0 2 )t   0 t Bt }
(6)
2
with Bt ~N(0, 1), 0 and  0 being the mean and standard deviation of the
instantaneous rate of return on the assets of the bank, d V0t / V0t , respectively.
Substituting (6) into (5), we obtain
V00 exp{ (  0  1  0 2 )   0 t Bt }exp{r(T－r)} / D≦  ( x02 )/  ( x01 )
2
(7)
Then the probability of insolvency of the bank can be expressed as
ln
Pr obins ( Bt ≦－
( x01 )
V
1 2
 ln 00  r  ( 0   0 )t
( x02 )
D
2
)
 t
( x01 )
V
1 2
 ln 00  r  ( 0   0 )t
( x02 )
D
2
＝ ( －
)
 t
since Bt ~N0, 1).
ln
(8)
The probability of insolvency of the bank can be also considered as
Pr obins ( V0t ≦D e  R0 t (T  t ) )
(9)
where D e  R0 t (T  t ) is viewed as default point. Equation (9) implies that
ln
Pr ob ( Bt ≦ －
＝ ( －
ln
V00
1 2
 R0t  ( 0   0 )t
D
2
)
 t
V00
1 2
 R0t  ( 0   0 )t
D
2
)
 t
(10)
where  ＝(T－t). If equations (8) and (10) are required to be equal, then
ln
( x01 )
V
1 2
V
1 2
 ln 00  r  ( 0   0 )t
ln 00  R0t  ( 0   0 )t
( x02 )
D
2
2
＝ D
 t
 t
which can be further simplified to
1  ( x01 )
1
R0 －r＝ ln
＝－ { ln  ( x02 )－ ln  ( x01 )}


 ( x02 )
9
(11)
(12)
Recall that the deposit rate is R0 t = r+ CS0 , then
1
CS0 ＝－ { ln  ( x02 )－ ln  ( x01 )}

is the credit spread to the depositors. Under certain conditions, the credit spread in (12)
would be equal to that in to the equation (14) in Merton (1974) as shown below:
CSi ＝ Ri (  )－r＝ －
＝－
1
ln{  ( x02 )+( V0t / A0 t )  (－ x01 )}

1
{ ln  ( x02 )－ ln  ( x01 )}

Then
 ( x02 ) + ( V0t / A0 t )  (－ x01 )＝  ( x02 )/  ( x01 )
(13)
(14)
where A0 t ＝D e  r is the market value of deposit at time t.
Equation (14) can be reduced to
V0t / A0 t ＝  ( x02 )/  ( x01 )＜ 1
(15)
It shows that when the deteriorating credit qualities of the obligors lead to the ratio of
asset value of the bank over the deposit value decreases to a specified
level  ( x02 )/  ( x01 ) which is less than one, the bank will be insolvent even prior to
the maturity time T.
The dynamics of the value of loan portfolio
The minimum capital required for an exposure will depend both on the risk of
individual exposures and on the relationships between a bank’s exposures. The total
minimum capital required for credit risk of the bank will be determined by the mean
and variance of market value of the bank’s whole portfolio of loans. Since the market
value of the bank’s loan portfolio would change with the credit qualities of all
obligors in terms of transition in PD and EAD given the LGD and M in each loan
unchanged, the dynamics of the value of loan portfolio would reveal the ongoing
impact of the lending policies or asset allocation decisions made at time zero by the
bank management. According to the equation (14) in Merton (1974), we know that
the credit spread can be expressed as:
CSi ＝ Ri (  )－r＝－
1
ln{  ( xi 2 )+( Vit / Ait )  (－ xi1 )}

where  ＝T－t, t  (0, T);
xi1 ＝{ ln[ Vit / Ait ]＋( r＋
1 2
 )  }/  i  ;
2 i
10
(13)
1 2
 )  }/  i  ;
2 i
is the value of obligor i’s asset;
xi 2 ＝{ ln[ Vit / Ait ]＋( r－
Vit
Ait is the value of obligor i’s debt
 (.) is the cumulative Normal distribution function.
By equation (13), it can be seen that the credit spread depends on the obligor’s firm
value Vit , debt Ait , the volatility of the firm value  i , time to maturity  , and risk-free
rate r. The yield-to-maturity Ri (  ) of loan i in the portfolio would change with the
credit quality of obligor i which may transit over time during the credit horizon. Thus,
at any time t the correlation between Ri (  ) and R j (  ) will depends on the credit
quality of obligor I and credit quality of obligor j.
Since the focus of this paper is to determine the relationship between loan
pricing and the capital charged given the deposit D reimbursed at time T for the bank,
we now proceed to examine the dynamics of the value of the loan portfolio. Recall
from equation (1) that
N
A
i 1
i0
＝D e  R0 0T ＋ VE 0
where Ai 0 is the initial value of loan i at time zero which would mature at time T.
The market value of loan i at any time t during the credit horizon would be Ait ＝
Ai 0 e Rit t , t  [0, T], where Rit ≡ Ri (  ) is defined in equation (13). Let AiT ≡
Ai 0 e RiT T be the payment of loan i reimbursed to the bank. By equation (13) in Merton
(1974), the market value of loan i at any time t Ait can be expressed as
Ait ＝ AiT e  r  ( xi 2 )+ Vit  (－ xi1 )
(14)
By the yield to maturity of loan i at time t, Ri (  ) in equation (13), and the market
value of loan i at time t, Ait in equation (14), the rate of return RPt , and the variance
of the loan portfolio Var ( RP ) at time t can be derived.
As the value of loan i would change over time, the weight of loan i at time t in the
loan portfolio is
N
Wit ＝ Ait /  Ait ,
t  [0, T]
(15)
i 1
Note that the weight of loan i at time zero Wi 0 is determined by the bank through the
allocation of fund from deposit and equity to loan i, while the weight of loan i at time
11
t  (0, T] in the loan portfolio Wit would change over time as the results of changing
market values of the all loans.
From equations (13) and (15), the rate of return of the loan portfolio at time t is
N
RPt ＝ Wit Rit ,
t  [0, T]
(16)
i 1
where  ＝T－t ,and RPt ≡ RP (  ).
Rit is the yield to maturity on the loan i lent to the obligor i, i＝1, 2, …, N.
To grasp the characteristics of the loan portfolio in a continuous time framework, we
need to examine the instantaneous rate of return of loan i, Rit . As in equation (1) in
Merton (1974), the dynamics of the value of loan i’ can be expressed as
d Ait / Ait ＝  Ai dt+  Ai d B Ait
i= 0, 1, 2, …, N, and t  [0, T]
(17).
where  Ai is the instantaneous expected rate of return on the Ait ;
 Ai is the instantaneous standard deviation of the rate of return on the Ait
 Ai and  Ai are Borel measurable and satisfy the Lipschits and growth
conditions;
d B Ait is a standard Brownian motion.
Assume that the asset value of obligor i, Vit and that of loan i, Ait are perfectly
correlated, then d B Ait ≡d Bi , and d B Ait d B Ajt =  ij dt, where  ij is defined as the
instantaneous correlation coefficient between d Bi and d B j in Assumption 7.
By equation (17), Rit ＝  Ai dt+  Ai d B Ait . Therefore the instantaneous correlation
coefficient  ij between d Bi and d B j would be equal to the one between d Ait / Ait
and d A jt / A jt since Rit ＝d Ait / Ait ＝  Ai dt+  Ai d B Ait is the instantaneous rate of
return on the Ait , i= 0, 1, 2, …, N, and t  [0, T]. Then
Cov ( Rit , R jt )＝ Cov (  Ai dt+  Ai d B Ait ,  Aj dt+  Aj d B Ajt )
＝ E{[(  Ai dt+  Ai d B Ait ) － E(  Ai dt+  Ai d B Ait )][(  Aj dt+  Aj d B Ajt ) －
E(  Aj dt+  Aj d B Ajt )]}
＝  Ai  Aj E{ d B Ait d B Ajt }
＝  ij  Ai  Aj dt
(18)
12
By equation (18), the instantaneous coefficient of correlation between Rit and R jt is
 ij ＝[ Cov ( Rit , R jt )/  ( Rit )  ( R jt )]/dt.
(19)
The instantaneous variance of the rate of return of the loan portfolio at time t is
Var ( RPt )＝  2 ( RPt )
N
N
i 1
i 1
j i
＝  Wit2  2 ( Rit )＋  WitW jt Cov ( Rit , R jt ), t  [0, T]
(20)
where Cov ( Rit , R jt ) is the covariance between the yields to maturity, Rit and R jt ,
on the loan i and loan j per unit time.
 2 ( Rit ) is the variance of the yield to maturity Rit on the loan i per unit time.
Accordingly, on the right hand side of the second equality in equation (20) it
can be seen that at any time t, the first term is the instantaneous idiosyncratic risk of
the whole loan portfolio, and the second one represents the instantaneous systematic
risk, t  [ 0, T]. The instantaneous coefficient of correlation  ij  [－1, 1] would
depend on the yield-to-maturity of loan i and loan j at time t, Rit and R jt , which in
turn are influenced by the credit spreads of obligors i, and j. The lower is the  ij , the
smaller the  2 ( RPt ) is. In addition, although the weight of loan i at time t, Wit , would
continuously fluctuate during the credit horizon, the instantaneous idiosyncratic risk
of the loan portfolio would wane if the number of loans in the portfolio increases.
Further, if the average instantaneous coefficient of correlation  shifts down, other
things being equal, the instantaneous systematic risk of the loan portfolio will
decrease. This implies that the extent to which the value of the loan portfolio
impacted by the business cycle would become less. In brief, if the number of loans N
in the portfolio remains constant, the dynamics of the value of the loan portfolio
would depend on the average instantaneous coefficient of correlation  which is
incurred by the changes in transition pattern in credit quality of obligors over time.
13
The Effect of Granularity
The minimum capital required by the supervisor depends on the risk-weighted assets
which result from multiplying the risk weights by the estimates of EAD, and then
summing up the products amounts across the portfolio. The granularity adjustment is
applied to the entire portfolio as a whole after the sum of all Baseline RWA for all
portfolio exposures is computed. The granularity of a bank portfolio describes the
extent to which there remain significant single borrower concentrations. The more
thoroughly has the bank diversified away the idiosyncratic component of credit risk
associated with individual positions, the lower is the loan portfolio’s economic capital
requirement. In the limit of a perfectly-diversified loan portfolio, the idiosyncratic risk
would wane down to zero. As there exists never a loan portfolio which is infinitely
finely-granular, there is still a residual of undiversified idiosyncratic risk in the loan
portfolio.
Consider that the N loans in the asset portfolio are split into two heterogeneous
components: one is a set consisting of n homogeneous loans with exposures of dollar
amount ASit , i＝1, 2, …, n, hereafter called portfolio S; the other called portfolio L is
composed of m homogeneous loans with exposures of dollar amount ALjt , j＝1, 2, …,
m. The two portfolios are supposed to behave discrepantly in that n is much larger
than m, but ASit is much less than ALjt . Thus, N＝n＋m, and the value of loan
portfolio at time 0 is
n
m
i 1
j 1
 ASi0 ＋  ALj 0 ＝D e  R0 0T ＋ VE 0
(21)
During the credit horizon [0, T], the value of the loan portfolio of the bank at any time
t becomes
n
m
i 1
j 1
 ASit ＋  ALjt ＝ V0t ,
t  [0, T]
(22)
N
Recall that V0t ＝  Ait ＝D e  R0 t (T  t ) ＋ VEt defined in equation (2).
i 1
By equation (22), the loan portfolio is in effect a combination of two heterogeneous
portfolios with weights WSt , and WLt , respectively:
n
m
i 1
j 1
WSt ＝  ASit / V0t and WLt ＝  ALjt / V0t
t  [0, T]
By equation (23), the rate of return of the loan portfolio at time t is
14
(23)
RPt ＝ WSt RSt ＋ WLt RLt
(24)
n
where RSt ＝  WSit RSit is the rate of return of the portfolio S at time t;
i 1
N
RLt ＝ WLjt RLjt is the rate of return of the portfolio L at time t.
i 1
The mean and variance of the rate of return of the loan portfolio at time t are
E( RPt )＝ WSt E( RSt )＋ WLt E( RLt )
(25)
 2 ( RPt )＝ WSt2  2 ( RSt )＋ WLt2  2 ( RLt )＋2 WSt WLt  SL  ( RSt )  ( RLt )dt
＝ (1  WLt ) 2  2 ( RSt )＋ WLt2  2 ( RLt )＋2 (1  WLt ) WLt  SL dt  ( RSt )  ( RLt )
(26)
since Cov ( RSt , RLt )＝  SL  ( RSt )  ( RLt )dt and WSt ＋ WLt ＝1.
By equation (26), the weight on portfolio L, WLt , and the coefficient of correlation
between loan portfolios S and L,  SL will contribute to the volatility of value of the
bank’s loan portfolio. Besides, since m ＜ n and N＝n＋m, the portfolio L would be
expected to be less well-diversified than portfolio S. That is,  ( RLt ) is expected to
be larger than  ( RSt ). The effect of increasing WLt on  2 ( RPt ) is
d 2 ( RPt )
＝2{( WLt －1)  2 ( RSt )＋ WLt  2 ( RLt )＋(1－2 WLt )  SL dt  ( RSt )  ( RLt )}
dWLt
(27)
From equation (27), we obtain that if
WLt ＞ ( ＝ )( ＜ )[  2 ( RSt ) －  SL dt  ( RSt )  ( RLt )]/[  2 ( RSt ) ＋  2 ( RLt ) －
2  SL dt  ( RSt )  ( RLt )]
(28)
d 2 ( RPt )
＞(＝)(＜)0
dWLt
For a specified level of  SL , such as  SL ＝1, 0, or －1, equation (28) can be reduced
then
further to a simpler form. For a given  SL , let
WLt* ≡ [  2 ( RSt ) －  SL dt  ( RSt )  ( RLt )]/[  2 ( RSt ) ＋  2 ( RLt ) －
2  SL dt  ( RSt )  ( RLt )].
Apparently, when WLt is larger than (equal to)(less than) WLt* , the effect of
increasing WLt on  2 ( RPt ) is positive(zero)(negative). WLt* is a critical point on
which the risk of the whole loan portfolio will depend. If the concentration on
15
portfolio L is raised to a level higher than WLt* , the increased variance of  2 ( RPt )
would incur additional charged capital for the bank.
Value at risk and the Capital Adequacy
The term "Value at risk" is the expected loss from an adverse market movement with
a specified probability over a particular period of time. In specific, the Value at Risk
(VaR) for a bank to meet the requirements of capital adequacy is defined as: The
worst loss expected from holding a loan portfolio over a given credit horizon, given a
specified level of probability. For example,
VaR = Expected Value of the loan portfolio– Worst Case Value at the 99% confidence
level.
That is, VaR is the maximum loss at the 99% confidence level, measured relative to
the expected value of the loan portfolio and is the distance of the first percentile from
the mean of the distribution. Expected Loss (EL) is the mean of the loss distribution,
while Unexpected Loss (UL) is the chosen percentile cutoff of extreme losses. By
Basel II, there is 0.1% likelihood that losses will exceed UL. The UL is the measure
of VaR.
According to the discussion in preceding sections, the value of bank’s loan
portfolio depends on the instantaneous correlation coefficient  ij between the
instantaneous rate of return on the loans i, j, i= 0, 1, 2, …, N, and i≠j. The lower is
the  ij , the smaller the  2 ( RPt ) is. Besides, the instantaneous idiosyncratic risk of
the loan portfolio would wane if the number of loans N in the portfolio increases.
Further, when the average instantaneous coefficient of correlation decreases, the
instantaneous systematic risk of the loan portfolio and the extent to which the value of
the loan portfolio impacted by the business cycle would become less. As for the effect
of granularity, the level of concentration is critical the value of loan portfolio. When it
increases to a higher than some specific level, the variance of the value of loan
portfolio would increase with it. Then, this would incur additional charged capital for
the bank. The results show that given LGD and M, the impact of risk components PD,
EAD on the yield to maturity of loan i will materially determine the risk of the
individual loan and the instantaneous correlation coefficient between loans, which in
16
turn would influence the risk of the whole loan portfolio, and hence the capital
charged.
Conclusion
In this paper, we focus on corporate exposures and intend to value the corporate
loans in continuous time framework under the IRB approach. We extend the
framework Merton (1974) combined with the theory of portfolio to characterize the
dynamics of the value of loan portfolio of the bank and the effect of granularity. By
taking the risk components inherent in the loan to obligors into account, the effects of
asset return correlation and granularity are examined. The main results are as follows:
first, the capital charge considered in terms of VaR in the risk management is linked to
the dynamics of the value of loan portfolio; second, both the VaR and the value of
loan portfolio move in tandem over time, which contrast to the single time period
setting in literature; third, the value of bank’s loan portfolio depends on the
instantaneous correlation coefficient between the instantaneous rate of return on the
loans;fourth, the instantaneous idiosyncratic risk of the loan portfolio would wane if
the number of loans N in the portfolio increases; Fifth, when the average
instantaneous coefficient of correlation decreases, the instantaneous systematic risk of
the loan portfolio and the extent to which the value of the loan portfolio impacted by
the business cycle would become less; Sixth, granularity, when the level of
concentration increases to a higher than some critical level, the variance of the value
of loan portfolio would increase with it. This would incur additional charged capital
for the bank; Seventh, the impact of risk components on the yield to maturity of loans
will ultimately determine the risk of the whole loan portfolio, and hence the capital
charged.
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