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

222 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS
We now assume for simplicity that there is only one class of equity and
one class of debt in the bank. At least for the debt this assumption does not
seem unrealistic, because in the case when the bank approaches default,
basically all debt holders will want to reclaim their liabilities at the same
time (bank run), making different classes of seniority and maturity less relevant.
422
We further assume an M&M world with respect to the value additivity
423 of these two classes of debt and equity so that the financing decision
has (initially) no impact on the total value of assets. Therefore:
V = V + V
(5.45)
At , Et , Dt ,
where V E,t
= Market value of equity at time t
V D,t
= Market value of debt at time t
V D,t
= V D,T
⋅ e-r' (T – t)
assuming that the debt matures at time T and that V D,t
is the value of a zerocoupon
bond with face value V D,T
(which is equal to the book value of
debt) discounted at interest rate r' (which is typically not the risk-free rate).
From that we can infer—given that shareholders have to repay the debt
holders and can then claim the remainder of the asset value—the payoff for
the shareholders at time T is:
V E,T
= max[(V A,T
– V D,T
); 0] (5.46)
which is identical to the payoff structure of a call option. Thus, in
the simplified case, where we allow for only one class of equity and one
class of outstanding debt, equity can be viewed as a call option on the
underlying assets of the firm with a strike price equal to the value of the
firm’s liabilities.
Since we evaluate the value of the shareholders’ claim at the time of
maturity T of the debt, we can use the Black-Scholes formula for pricing
European call options to price V E
at any time t before T: 424
−⋅ r( T −t)
V = V
0
⋅Nd (
1) −V ⋅e ⋅Nd
(
2
)
(5.47)
Et , A, DT ,
422 See discussion above.
423 Note that this assumption only relates to value additivity and not the overall topdown
approach.
424 See Hull (1997), p. 241. Of course this assumes that hedging using the underlying
is possible and that all assets are fully tradable. Additionally, all other assumptions
of the Black-Scholes world need to be met.