Approximating the price of long-term bonds
The present value of an infinite stream of dollar payments of $z (that starts next year) is $z/i when the nominal interest rate, i, is constant. This formula gives the price of a consol-a bond paying a fixed nominal payment each year, forever. It is also a good approximation for the present discounted value of a stream of constant payments over long but not infinite periods, as long as i is constant. Let's examine how close the approximation is.
a. Suppose that i = 10%. Let $z = 100. What is the present value of the consol?
b. If i = 10%, what is the expected present discounted value of a bond that pays $z over the next 10 years? 20 years? 30 years? 60 years? (Hint: Use the formula from the chapter but remember to adjust for the first payment.)
c. Repeat the calculations in (a) and (b) for i = 2% and i = 5%.