$4$ Suppose that the prices today of zero$-$coupon bonds with various maturities are given in the following table. The face value of each bond is $$1,000.$

You may assume annual compounding for this question.

Maturity in years Price YTM Forward

$1952\mathrm{.}3809524$ y$1$

$2889\mathrm{.}99644$ y$2$

$3816\mathrm{.}2978769$ y$3$

$4735\mathrm{.}0298528$ y$4$

$5649\mathrm{.}9313863$ y$5$

a Calculate the yield to matruity for each zero bond $($in col E$)$ and the one$-$year forward rate of interest for each year $($in col G$).$ The forward rates are denoted f$2,$ f$3,$ f$4,$ and f$5.$

The yield to maturities are denoted as y$1,$ y$2,$ y$3,$ y$4,$ and y$5.$

Please provide your answers in Column E $($for YTM$)$ and column G $($for Forward$)$

b Suppose that today $($i$.$e$.,$ t$=0)$ you buy one $3-$year maturity zero coupon bond. How many $5-$year maturity zeros would you have to sell to make

your initial cash flow equal to zero?

c What are the cash flows from the strategy in part $($b$)$ in each year?

d What is the effective $2-$year interest rate on the effective $3-$year$-$ahead forward loan?

e Confirm that the effective $2-$year interest rate equals:

You therefore can interpret the $2-$year loan rate as a $2-$year forward rate for the last $2$ years.

Alternatively, show that the effective $2-$year forward rate equals: