Question

Consider a mortgage with an initial principal value of $1,000,000. It will make yearly payments for 3 years after which it will be paid off. The annual interest rate is 5%. Take the annual payment to amortize from the loan (you must compute this) and construct a CMO. That is, construct three annual coupon bonds paying an annual coupon rate of 5% with maturities of one, two and three years. What will be the face value of each bond (hint, each will be different).

Answer 1

The annual Payment (A) for the mortgage is given by

A/0.05*(1-1/1.05^3) = 1000000

=> A = $367208.56

So , the mortgage pays $367208.56 at the end of each year

For the CMO, the 3 year bond will pay 5% + 100% of Face value at the end of year 3 i.e. 105% at the end of year 3

So, 105% of Face value of 3 year bond =$367208.56

Face value of 3 year bond = 367208.56/1.05 = $349722.44

At the end of 2 years , one gets 5% of FV of 3 year bond + 105% of FV of 2 year bond

=> 349722.44*0.05+105%*FV of 2 year bond =367208.56

FV of 2 year bond = $333068.99

At the end of 1 year , one gets 5% of FV of 3 year bond + 5% of FV of 2 year bond+105% of FV of 1 year bond

=> 349722.44*0.05+333068.99*0.05+ 105%*FV of 1 year bond =367208.56

FV of 1 year bond = $317208.56