Question

The following facts about a company are known: 1. At present, its total capital is £3 million. 2. It has just sold zero-coupon bonds with the total nominal value of £2 million which it promises to repay in 18 months from now. 3. The total capital F(t) of the company follows the geometric Brownian motion with parameters µ = 0.15 and σ = 0.2. The continuously compounded annual interest rate r = 6%. Within the framework of the Merton model, establish the following.

(a) What is the total value of the shares of this company? [5]

(b) How much money has the company raised from the sale of the bonds? [2]

(c)What is the probability that the company would default on its promise to bond holders? [5]

Answer 1

(a)£3 million

(b)

$3+(2×18)=39$ million

(c)

So we can say that the bond will default if $F(1.5)<200000$ therefore the probability of default is equal to the probability $P[F(1.5)<200000].$

 

We know that F(1.5) follows a geometric Brownian motion which means

$F(1.5)~N[ln(3000000)+0.15×1.5, 0.2×1.5] lnF(1.5)~N[15.139, 0.3]$

$P[F(1.5)<2000000]=P[ln{F(1.5)}<ln{2000000}$

$P[Z<\frac{ln{2000000}−15.139}{0.3}]$

$P[Z<−2.101]=1.78$

So there is 1.78% probability that the company will default on bond payments.