Approximating the price of long-term bonds The present value of an infinite stream of dollar payments of

\(\$ z\) (that starts next year) is \(\$ \mathrm{z} / \mathrm{i}\) when the nominal interest rate, \(\mathrm{i}\), 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 \(\mathrm{i}\) 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 \(i=5 \%\)