This document discusses homogeneous debt portfolios. A homogeneous portfolio contains at least 30 debts with identical exposure amounts, loss given default rates, probability of default rates, and default dependency among borrowers. Treating all risks as maturing after one year, a homogeneous portfolio allows for an analytical approach to calculating credit risk that incorporates diversification effects from lending to multiple borrowers. The portfolio one-year expected loss, which simply sums individual debt expected losses, does not account for diversification benefits of a multi-debt portfolio.

1. Managing Credit Risk Under The Basel III Framework 35
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Homogeneous Debt Portfolios
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KEY CONCEPTS
• Debt portfolio
• Independent homogeneous portfolio
• Extreme case loss
• Gaussian copula
• Finite homogeneous portfolio
• Infinite homogeneous portfolio
3 Homogeneous debt portfolios
3.1 Debt portfolio
In a real lending business, a professional lender lends to many borrowers. When the
number of borrowers grows beyond hundred, the effort of managing the credit risk on an
individual debt basis becomes large. In addition, the diversification effect arising from
the investments in many debts also changes the risk characteristics of the debt
investments as a whole. Therefore, the concept of debt portfolio emerges.
A debt portfolio is a collection of debts lent to many borrowers from the same lender.
The default loss of a debt portfolio is essentially a result of the joint defaults of individual
debts in the portfolio. Similar to other financial investments, as an effect of
diversification, the overall credit risk is reduced if the same total amount is divided and
lent to two different borrowers instead of just one borrower. The overall credit risk is
further reduced if the same total amount is divided and lent to many different borrowers
instead of just two borrowers. As such, for a debt portfolio, the diversification effect,
which is characterized by: (i) the concentration of debts; and (ii) the default dependency
among borrowers, essentially introduces a third dimension to the characterization of
credit risk.
Figure 3.1 Credit risk factors of a debt portfolio

2. 36 Managing Credit Risk Under The Basel III Framework
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3.2 Portfolio one-year expected loss
The sum of the 1-year ELs of all debts in a portfolio is referred to as the portfolio 1-year
EL which is a natural extension of the 1-year EL to a debt portfolio from a single debt.6
For a debt portfolio comprising NOB debts from NOB different borrowers, the portfolio
1-year EL is:
( ){ }
[ ]( )
k
NOB
k
k=1
NOB
RM
k k k k
k=1
NOB
k k k k
k=1
Portfolio 1-year EL = 1-year EL
= EAD × LGD × Min PD , 1 - 1 - PD
EAD × LGD × PD × Min 1, RM
 
 
≈
∑
∑
∑
Consider a debt portfolio comprising only one debt with credit risk factors EAD USD
10,000, LGD 90 percent, PD 3 percent and RM one year. The portfolio 1-year EL is
simply:
10,000 × 90% × 3% × 1 = USD 270
Consider another debt portfolio comprising two debts lent to two different borrowers,
each with credit risk factors EAD USD 5,000, LGD 90 percent, PD 3 percent and RM
one year. Again, the portfolio EAD is:
5,000 + 5,000 = USD 10,000
and the portfolio 1-year EL is also:
5,000 × 90% × 3% × 1 + 5,000 × 90% × 3% × 1 = USD 270
Apparently, the portfolio 1-year EL has yet to capture the reduction in credit risk arising
from the diversification effect and is not an effective credit risk measure for a debt
portfolio. In fact, the formula of the portfolio 1-year EL is constructed without any factor
of the diversification effect incorporated.
3.3 Homogeneous portfolio ★★★★★★★★★★★★
A homogeneous portfolio is a debt portfolio comprising at least thirty debts with identical
characteristics in terms of EAD, LGD, PD and default dependency among borrowers.
The RMs are unified artificially to one year across all debts, subject to three criteria that:
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In contrast to the 1-year EL, the concept of the EL cannot be extended to a debt portfolio.