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 \(\mathrm{i}\) 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? (Use the formula from the chapter but remember to adjust for the first payment.)

c. Repeat the calculations in parts a and b for \(i=2 \%\) and \(i=5 \%\).