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

228 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS
and
d
2
2
⎛ VA′
⎞ ⎛ σ ⎞
A
ln⎜
⎟ + r T
⎝ DP′
⎜ −
⎠ 2
⎟
⎝ ⎠
⋅
=
= d1
−σ
A
⋅
σ ⋅ T
A
where N(⋅) = Cumulative standard normal probability distribution
function
r = Risk-free rate (note that we are still in a risk-neutral evaluation
world 444 for deriving the value of V E
)
T = Time to maturity (= 1 [see horizon discussion above])
Whereas in Equation (5.52) V A
′ is only a first proxy of what the true
value of the assets of a bank is, σ A
is unknown. Yet, both input parameters
are unobservable. However, since we know that equity is a call option on
the firm’s value, equity can be also defined as a portfolio that consists of ∆
units 445 of firm value and a short position in the risk-free asset. Therefore,
we can infer that the return on equity is perfectly correlated with the return
on the value of the firm for small changes in the value of the firm 446 and can
show that the volatility of the rate of return (σ E
) of the option V E
is:
VA
σE
= ∆ ⋅σA
(5.53)
VE
This means that the volatility of equity is equal to ∆V A
/V E
times the
volatility of the firm σ A
and the ∆ is equal to the option delta N(d 1
). 447
Given that we can observe the volatility of the rate of return σ E
in the
stock market, we have now two unknowns and two equations: Equations
(5.52) and (5.53). Therefore, we can now determine V A
and σ A
. However,
as when trying to determine implied volatilities for quoted option prices, we
cannot invert the option-pricing formula and need to apply an iterative search
procedure 448 to solve both equations simultaneously. 449
We use as observable input V E
, DP′, r, and T, and estimate σ E
as the
annualized volatility of the stock market returns in the year prior to the
estimation point in time (i.e., year end). As defined previously, we use as a
starting point for the iterative procedure:
V A
′ = DP′ + V E
(see Equation [5.51] above) and σ A
= σ E
/4
T
444 See Hull (1997), pp. 239–240.
445 As we will see shortly, ∆ is the option delta N(d 1
).
446 A small change in the value of the firm d changes the value of the equity by ∆dV A
,
see Stulz (2000), p. 18-9.
447 See Cordell and King (1995), p. 538.
448 See Hull (1997), p. 246.
449 See Stulz (2000), p. 18-12.