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Capital Structure in Banks 181
n
∑ ∑ ∑ ∑
2
ULP
= ULi
+ cov
i j= UL 2
,
i
+ 2 ρULiUL
≈ nUL ⋅ +
i
n
j, i< j i
j, i<
j
nn ( −1) 2 ⋅ ρ ⋅ UL = [ n + ρn( n − 1)
]⋅UL
2
2 2 2
i i i
n
n
j
(5.16)
and hence:
2
UL = UL n + ρ( n −n)
(5.17)
P
i
Using the assumption of similar credits within the portfolio previously
described, we can now rewrite:
( ) = + −
ULC UL P
1
n n UL n n n UL 1 ⎛ ⎞
i
= =
i
+ −
i ⎜1 1
2
ρ ρ ⎟ (5.18)
n ⎝ n⎠
which reduces for large n to:
ULC
i
= UL ρ (5.19)
Combining Equation (5.13) with (5.19) and rearranging the terms, we
can arrive at:
n
i
∑ UL jρ
ij
j
ρ = = 1 (5.20)
UL
which clearly shows that ρ is the (weighted) average correlation between
loans in the portfolio (as was assumed above).
This derivation provides some important insights:
■ If one tried to estimate the portfolio UL by using Equation (5.8),
one would need to estimate [n(n – 1)]/2 pairwise default correlations.
214 Given that typical loan portfolios contain many thousand
credits, this is impossible to do. Additionally, one needs to consider
the fact that default correlations are very difficult, if not impossible,
to observe. 215
P
214 As indicated above, one would also need to estimate the correlation of a joint
movement in credit quality.
215 However, they can be estimated from observable asset correlations. See e.g., Gupton
et al. (1997), Ong (1999), pp. 143–145, Pfingsten and Schröck (2000), pp. 14–15.