Term Structure of Interest Rates: Spot and Forward Rates

Spot Rates

The spot rate s_t is the annual effective yield on a zero-coupon bond maturing in t years. The present value of $1 due in t years is (1 + s_t)^(-t). Spot rates define the term structure of interest rates. When spot rates differ across maturities, the yield curve is not flat, and using a single discount rate for all cash flows is an approximation.

To price a coupon bond with the term structure: P = Fr/(1+s_1) + Fr/(1+s_2)^2 + ... + (Fr+C)/(1+s_n)^n. Each cash flow is discounted at the spot rate matching its timing.

Forward Rates

The forward rate f(t, t+1) is the rate agreed upon today for borrowing from time t to time t+1. The no-arbitrage relationship between spot and forward rates is: (1 + s_n)^n = (1 + s_1)(1 + f(1,2))(1 + f(2,3))...(1 + f(n-1,n)). Equivalently, (1 + f(t, t+1)) = (1 + s_{t+1})^(t+1) / (1 + s_t)^t.

For multi-period forward rates: (1 + f(t, t+k))^k = (1 + s_{t+k})^(t+k) / (1 + s_t)^t, where f(t, t+k) is the annual rate for the k-year period starting at time t.

Exam FM Applications

Exam FM problems typically give a set of spot rates and ask you to compute forward rates, or give some combination of spot and forward rates and ask for missing values. Use the fundamental relationship: accumulation from 0 to n at spot rates equals accumulation from 0 to 1 at s_1, then 1 to 2 at f(1,2), and so on. The term structure also determines whether forward rates are above or below spot rates. When the yield curve is upward-sloping, forward rates exceed spot rates.