The bond with six months left to maturity has a semiannual discount rate of 0.025/2 = 0.0125 or 2.5% on an annual bond equivalent yield (BEY) basis. Since the bond will only make a single payment of 101.25 in six months, the YTM is the spot rate for cash flows to be received six months from now.
The one-year bond will make two payments, one in six months of 1.75 and one in one year of 101.75. We can solve for the one-year spot rate in the equation:
where S1.0 is the annualized 1-year spot rate.
Using the 6-month and 1-year spot rates, we can use the same approach to find the 18-month spot rate from the equation:
where S1.5 is the annualized 18-month spot rate.
Solving we get:
S1.5 = 4.53047%.
