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Capital Structure in Banks 179
UL i
=
predetermined period of time (usually, again, between
time 0 and H [one] year)
Unexpected Loss of the i-th credit asset as defined above
in Equation (5.5).
Therefore, considering a loan at the portfolio level, the contribution of
a single UL i
to the overall portfolio risk is a function of:
■
■
■
The loan’s expected loss (EL), because default probability (PD), loss
rate (LR), and exposure amount (EA) all enter the UL-equation
The loan’s exposure amount (i.e., the weight of the loan in the portfolio)
The correlation of the exposure to the rest of the portfolio
To calculate the unexpected loss contribution 211 ULC i
of a single loan
i analytically, we first need to determine the marginal impact of the inclusion
of this loan on the overall credit portfolio risk. This is done by taking the
first partial derivative of the portfolio UL with respect to UL i
(for loan i):
ULMC
i
UL
≡ ∂ ∂UL
P
∂( ULP
)
=
∂UL
⎛
∂
= ⎛ ⎜
⎝ ⎜ 1 ⎞ ⎝
⎟ ⋅
2UL
⎠
i
P
n
2 12 /
= ⎛ ULP
i ⎝ ⎜ 1⎞
2
−12
/
⎟ ⋅( ) ⋅
2⎠
n
∑∑
⎞
UL ⋅UL
ρ ⎟
j k jk
j= 1 k=
1
⎠ j=
1
∂UL
i
=
n
∑
2
∂( ULP
)
∂UL
UL ρ
UL
P
j
i
ij
(5.10)
where ULMC i
is the marginal contribution of loan i to the overall
portfolio unexpected loss.
Note that in the above formula, the marginal contribution only depends
on the (UL-) weights of the different loans in the portfolio, not on the size
of the portfolio itself. In order to calculate the portfolio volatility attributable
to loan i, we use the following property for a marginal change in portfolio
volatility:
dUL
Port
≡
n
n
∂ULP
⋅ dULi = ULMCi dULi
∂UL ∑ ⋅
i= 1 i
i=
1
∑
(5.11)
211 Note that we follow the argument made by Ong (1999), p. 133, in this discussion
and ignore the weights w i
in the derivation of ULC. We can do so if we assume that
UL i
is measured in dollar terms rather than as a percentage of the overall portfolio.