Naga Sai Shivanee
Member
In (iii) of the question we are given the nominal value and the issue price of the bonds A & B. When calculating the default rates, in place of B(t,T) they divided bond prices by nominal value. Why is that so? Is issue price different from bond price?
Here is the question:
Two companies have zero-coupon defaultable bonds in issue. Bond A has
£2m nominal in issue. Bond B has £3m nominal in issue. Both bonds
redeem in exactly 2 years' time.
Under a risk-neutral measure, each bond defaults (not necessarily
independently) at a constant rate. Both bonds have a 60% recovery rate.
Assume:
• a continuously compounded risk-free rate of interest of 3% pa
• the issue of Bond A is priced at £1.6m
• the issue of Bond B is priced at £2.2m
(iii) Evaluate the two default rates (under a risk-neutral measure).
Solution:
The ZCB formula is:
B(t, T) = e^-r(T-t) [ 1-(1- d){1-e^-L(T-t) }]
where,
d is delta
L is lambda
Rearranging gives:
L = (-1/ T-t ) log [ 1- { 1-B(t,T)e^r(T-t) / 1-d } ]
So for Bond A,
L,A = ( -1/2 ) log [ 1- { 1-(1.6/2)e^0.03(2) / 1-0.6 } ] = 0.23606
For Bond B,
L,B = ( -1/2 ) log [ 1- { 1-(2.2/3)e^0.03(2) / 1-0.6 } ] = 0.40293
Doubt:
So here in place of B(t,T) (1.6/2) and (2.2/3) were used instead of 1.6 and 2.2. Why are the issue prices being divided by the nominal value instead of being used directly? Is issue price different from bond price? What is the difference between issue price and bond price?
Thankyou in advance.
Here is the question:
Two companies have zero-coupon defaultable bonds in issue. Bond A has
£2m nominal in issue. Bond B has £3m nominal in issue. Both bonds
redeem in exactly 2 years' time.
Under a risk-neutral measure, each bond defaults (not necessarily
independently) at a constant rate. Both bonds have a 60% recovery rate.
Assume:
• a continuously compounded risk-free rate of interest of 3% pa
• the issue of Bond A is priced at £1.6m
• the issue of Bond B is priced at £2.2m
(iii) Evaluate the two default rates (under a risk-neutral measure).
Solution:
The ZCB formula is:
B(t, T) = e^-r(T-t) [ 1-(1- d){1-e^-L(T-t) }]
where,
d is delta
L is lambda
Rearranging gives:
L = (-1/ T-t ) log [ 1- { 1-B(t,T)e^r(T-t) / 1-d } ]
So for Bond A,
L,A = ( -1/2 ) log [ 1- { 1-(1.6/2)e^0.03(2) / 1-0.6 } ] = 0.23606
For Bond B,
L,B = ( -1/2 ) log [ 1- { 1-(2.2/3)e^0.03(2) / 1-0.6 } ] = 0.40293
Doubt:
So here in place of B(t,T) (1.6/2) and (2.2/3) were used instead of 1.6 and 2.2. Why are the issue prices being divided by the nominal value instead of being used directly? Is issue price different from bond price? What is the difference between issue price and bond price?
Thankyou in advance.