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

248 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS
which is similar to an option payoff. Therefore V E,t
and V D,t
can be expressed
as:
V D,t
=
F
– p[V
( 1 + ) ( ) A,t
; F] (6.4)
T−t
R f
and
V E,t
= c[V A,t
; F] (6.5)
where p and c are the values of a European call and put option on the
underlying asset A with strike price F. 51
Note that, as shown in Equation (6.4), the value of debt at time t is the
value of a default-free bond with face value F discounted at the risk-free rate
R f
less an insurance premium for the default risk, which is quantified by the
value of the put option. 52
We use the same assumptions for the market value of A as used previously
in the top-down estimation of economic capital, and additionally
assume:
V A,t
= V E,t
+ V D,t
(6.6)
since capital structure is irrelevant.
Therefore, the probability of default of the bank is:
p(default) = p[V A,T
≤ F] = N(–d 2
), (6.7)
where
d
2
⎛
2
VA
,
⎞
0
A
⎜
A
T
⎝ F
⎟
⎠
+
⎛
− σ ⎞
ln ⎜µ
⎝ 2
⎟
⎠
⋅
=
σ ⋅ T
A
(6.8)
and N(⋅) is the cumulative standard normal probability distribution function,
estimating the actual default probability. 53
One of the underlying premises of RAROC is that the required economic
capital to support a transaction (or the overall bank) is calculated to
keep the probability that the bank will go into default constant (at the chosen
confidence level). Therefore, d 2
in the above equation needs to be fixed.
51 Both options can be priced according to the Black-Scholes option-pricing theory
(OPT). However, we do not show the details of their determination here. For a
discussion see, for example, Hull (1997), pp. 240–242.
52 See Merton and Perold (1993), p. 20, and their references to the literature.
53 As shown in the previous chapter, we therefore use µ A
instead of R f
(as the OPT
would do to estimate risk-neutral probabilities).