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Capital Structure in Banks 229
since asset volatilities are very low for banks compared to the volatility of
its traded equity 450 reflecting the high leverage (that magnifies the low σ A
).
We then proceed by iterating on Equations (5.52) and (5.53) until the
V E
inferred from the option-pricing formula deviates from the observed input
value V E
, and σ E
inferred from the other simultaneous Equation (5.53) deviates
from the observed value σ E
less than ε (reasonably small), each by
changing V A
and σ A
.
Hence, we have now derived a realistic estimate for V A
and σ A
and have
thus been able to use the information implied in the stock market to derive
forward-looking measures instead of backward-looking accounting information.
We will now use the publicly available (long-term [senior debt] bank)
rating and the results from Brand and Bahar 451 to map ratings to (one-year)
default probabilities (PDs). 452
As we indicated previously, the following relationship between the probability
of default (PD) and the distance to default (DTD) exists:
and therefore:
PD = N −DTD
( ) (5.54)
−1
DTD N PD
(5.55)
where N -1 = The inverse of the standard normal cumulative distribution
=− ( )
As we have also mentioned, we are now applying actual instead of riskneutral
default probabilities. Therefore, we need to use µ instead of the riskfree
rate to derive the normalized distance to default (DTD):
DTD =
⎛ VA
⎞
⎜
⎝ DP
⎟
⎠
+ ⎛
ln ⎜
⎝
− 1
µ σ
2
,0 2
A
σ
A
⋅ T
⎞
⎟T
⎠
(5.56) 453
where µ = Expected return on the bank’s assets
450 No adjustment is necessary to directly observed equity volatility because it already
reflects the higher leverage of banks.
451 See Brand and Bahar (1999), p. 15.
452 For more details see example below.
453 See Crouhy et al. (1999), p. 10.