Risk Management and Value Creation in ... - Arabictrader.com

Capital Structure in Banks 223
where
d
1
⎛
2
VA
,
⎞
0
A
⎜ r T t
⎝ VDT
⎠
⎟ + ⎛
+ σ ⎞
ln ⎜
⎝ 2
⎟
⎠
⋅ ( − )
,
=
σ ⋅ T −t
A
and
d
2
⎛
2
VA
,
⎞
0
A
⎜ r T t
V
⎟
⎝ DT ⎠
+
⎛
− σ ⎞
ln ⎜
2
⎟
⎝ ⎠
⋅ ( − )
,
=
= d1
−σ
A
⋅ T −t
σ ⋅ T −t
A
where N(⋅) = Cumulative standard normal probability distribution function
T = Time of maturity
r = Risk-free rate
We use the risk-free rate as the expected rate of return here, because we
are in a world of risk-neutral evaluation. 425 As can be easily seen, the Black-
Scholes equation does not include any variables that are affected by the risk
preferences of the investors.
We can diagrammatically summarize the above as follows (see Figure
5.11).
As we can see in Figure 5.11, r represents the expected return on the
bank’s assets in such a risk-neutral world 426 and σ A
the volatility of the bank’s
assets, which drive the probability that V A,T
is smaller than (or equal to)
V D,T
and hence the probability that the bank is in default at time T.
Specific Theoretical Foundations As was already indicated, the purpose of the
suggested approach is to apply the previously depicted approach to banks
and turn it upside-down by using the default probability implied in publicly
available agency ratings. Since these ratings try to estimate the probability
of a company’s specific debt issue being in default in one year’s time, we set
our horizon T equal to 1. 427 However, this complicates the analysis: Usually,
we do not know for certain how much of the total debt is due in one year’s
time. 428 The only reasonable proxy we can use is the balance sheet information
on debt due in one year. 429 However, companies typically have the
possibility of refinancing some of their short-term debt as long-term liabili-
425 See Hull (1997), pp. 239–240.
426 It will later be replaced by µ, the actual return on assets.
427 Other reasons are, for example, that the reporting cycle in most European countries
is one year and the previously mentioned arguments in the horizon discussion.
428 Deposits can be withdrawn at any time without prior notice.
429 This will typically be the book value and not the required market value (that is
unobservable in most of the cases).