Term Structure of Interest Rates: Spot and Forward Rates

Spot Rates and the Term Structure

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, which describes how yields vary with maturity. 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.

The yield curve can take several shapes. An upward-sloping (normal) curve means longer maturities have higher spot rates. A downward-sloping (inverted) curve means shorter maturities have higher rates. A flat curve means all spot rates are equal. The shape of the yield curve has implications for forward rates, bond pricing, and investment decisions.

Pricing Bonds with Spot Rates

To price a coupon bond with the term structure, discount each cash flow at the spot rate matching its timing: P = Fr/(1+s_1) + Fr/(1+s_2)^2 + ... + (Fr+C)/(1+s_n)^n. This is more accurate than using a single yield to maturity, especially when the yield curve is steeply sloped.

Example: A 3-year bond with face value $1,000 and annual coupons of $50 is priced using spot rates s_1 = 3%, s_2 = 4%, s_3 = 5%. The price is P = 50/(1.03) + 50/(1.04)^2 + 1050/(1.05)^3 = 48.54 + 46.23 + 907.03 = $1,001.80. Notice that each cash flow uses a different discount rate, reflecting the term structure.

The yield to maturity of this bond is the single rate y that satisfies P = 50*a-angle-3(y) + 1000*v^3(y) = 1001.80. This rate is a weighted average of the spot rates, where the weights depend on the timing and size of the cash flows.

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)). This means accumulating at spot rates from time 0 to n gives the same result as accumulating period by period at the one-year forward rates.

Equivalently, (1 + f(t, t+1)) = (1 + s_{t+1})^(t+1) / (1 + s_t)^t. This formula lets you extract forward rates from spot rates. 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.

Computing Forward Rates from Spot Rates

Example: Given s_1 = 3%, s_2 = 4%, s_3 = 5%, find the one-year forward rates. f(1,2) = (1.04)^2 / (1.03) - 1 = 1.0816/1.03 - 1 = 0.0501 = 5.01%. f(2,3) = (1.05)^3 / (1.04)^2 - 1 = 1.1576/1.0816 - 1 = 0.0702 = 7.02%.

Notice that when the yield curve is upward-sloping (s_1 < s_2 < s_3), the forward rates exceed the spot rates and are themselves increasing. The forward rate f(2,3) = 7.02% is much higher than s_3 = 5% because it must compensate for the lower rates in the earlier years. This relationship is a mathematical consequence of the no-arbitrage condition.

For the two-year forward rate starting at year 1: (1 + f(1,3))^2 = (1.05)^3 / (1.03) = 1.1576/1.03 = 1.1239, so f(1,3) = (1.1239)^(1/2) - 1 = 0.0601 = 6.01%. This is the geometric average of f(1,2) and f(2,3).

Extracting Spot Rates from Forward Rates

The process also works in reverse. Given forward rates, you can build spot rates: (1 + s_n)^n = (1 + f(0,1))(1 + f(1,2))...(1 + f(n-1,n)), where f(0,1) = s_1 is the first spot rate. Solve for s_n by taking the n-th root.

Example: Given f(0,1) = 2%, f(1,2) = 3%, f(2,3) = 4%, find the spot rates. s_1 = 2%. (1 + s_2)^2 = (1.02)(1.03) = 1.0506, so s_2 = (1.0506)^(1/2) - 1 = 2.499%. (1 + s_3)^3 = (1.02)(1.03)(1.04) = 1.0926, so s_3 = (1.0926)^(1/3) - 1 = 2.997%.

Study Tips and Common Pitfalls

The most frequent error is confusing the notation. Make sure you clearly distinguish between s_t (the t-year spot rate) and f(t, t+1) (the one-year forward rate from time t to t+1). Another common mistake is forgetting to raise to the correct power. When computing (1 + s_t)^t, the exponent t matches the maturity, not the period number.

Always verify your forward rates by checking that the accumulation relationship holds: (1 + s_1)(1 + f(1,2))...(1 + f(n-1,n)) should equal (1 + s_n)^n. If it does not, you have a computation error. Practice at least 6 problems converting between spot and forward rates, and 3 problems pricing bonds with a non-flat term structure.