FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 97
respect to the seller. What is the market value of the swap with respective to the
fixed-rate payer at that time?
Solution: For the interest rate swap contract in Example 3.12, at the beginning of
the fourth year, there are two remaining expected cash flows before the end of the
contract, namely, m 4 (R S − i F 1 ) and m 5(R S − i F 2 ) payable at the end of the first
and second year, respectively, after the market maker has closed his position. For
the expected values of these two cash flows, i F 1 and iF 2 should be computed using
the prevailing spot rates of interest at the time of the swap termination. We have
i F 1 =4.5% and iF 2 = 1.0472
1.045
− 1=4.9%. From the solutions of Example 3.12,
we have R S =5.1145% and m 4 = m 5 =$1million. The numerical values of
these two expected cash flows are $1,000,000 × (5.1145% – 4.5%) = $6,145 and
$1,000,000 × (5.1145% – 4.9%) = $2,145. Hence, the market value of the swap
for the market maker is
6,145
1.045 + 2,145
(1.047) 2 =$7,837.12.
The market value of the swap to the fixed-rate payer is –$7,837.12.
3.6 Summary
1. Investment outlays at time 0 earn the spot rates of interest, which may vary
according to the time horizon of the investment. Investments that are committed
at time 0 but due in the future earn the forward rates of interest, which
may also vary with the time horizon of the investment.
2. A plot of the spot rates of interest is called the yield curve. The mathematical
relationship between the spot rates of interest and time to maturity is called
the term structure of interest rates.
3. Yield curves may take various shapes. They may be flat, upward sloping,
download sloping or humped. Yield curves have important implications for
future movements of interest rates and the real economy.
4. Spot and forward rates of interest are related through no-arbitrage conditions.
Forward rates of interest that satisfy such conditions are called the implicit
forward rates, which may differ from the forward rates quoted in the market
in practice.