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.
Hi Thanks for your question. The "nominal in issue" is the amount which the bond promises to pay at maturity. This is a value (ie size) of the bond but it's not the "price" of the bond. The "bond price" is the price of the bond now, ie the amount investors are prepared to pay now to enter into the promise of the nominal amount being paid to them in the future. The ZCB formula you've quoted is for a zero-coupon bond which pays a unit amount at time T. In order to use the formula in this form to solve Q4(iii) we need to scale the bond price by the nominal in issue. Hope that helps.
