The Term Structure and Interest Rate Dynamics

Two managers face the same upward-sloping curve. One buys a five-year bond and holds it; the other buys a six-year bond and sells it in five years. If yields never move, the second manager wins. The reason is the entire content of this reading.

The spot rate $z_N$ is the yield on a default-risk-free zero-coupon bond maturing in $N$ years. Because a zero has one cash flow, its stated yield equals its realized return if held to maturity, with no reinvestment assumption. The discount factor $D{F}_N$ is the price today of one unit received at $N$.

$$D{F}_N=\frac{1}{(1+z_N{)}^N}$$

A forward rate $f_{A,B-A}$ is a rate agreed today for a loan starting at $A$ and ending at $B$. No-arbitrage links it to spot rates through the forward rate model.

$$(1+z_B{)}^B=(1+z_A{)}^A(1+f_{A,B-A}{)}^{B-A}$$

Rearranged as a geometric mean, the T-year spot rate is the compounded chain of the one-year spot rate and the one-year forwards $f_{1,1},f_{2,1},\dots ,f_{T-1,1}$.

Common mistakes

• Reading forwards as forecasts. Implied forwards are the best accessible proxy for market expectations, but liquidity preference and preferred habitat both say they contain a risk premium, so a forward is not an unbiased expected spot rate.
• Assuming YTM equals expected return. With the spot curve at 5%, 6%, 7%, 8%, 9%, a 10% five-year bond prices at 105.43 with a YTM of 8.62%, yet reinvesting at the implied forwards gives an expected annualized return of 9.00%. They agree only when the curve is flat.
• Bootstrapping from the wrong curve. Bootstrapping runs off the par curve, not off yields on seasoned discount or premium bonds, and each step must discount earlier coupons at previously solved spot rates, not at the par rate.

Bottom line

• Spot rate is a geometric average of the one-year rate and successive one-year forwards; forwards are the marginal rates
• Upward-sloping spot curve puts forwards above spot; downward-sloping puts them below; flat makes them equal
• If forwards are realized, every bond earns the one-period spot rate; active management bets they are not realized
• Bootstrapping: solve zero rates from par rates one maturity at a time, discounting earlier coupons at solved spot rates

Exam shortcut

Read the slope first. Sloping up means forwards above spot, rolldown works, and long-bond return beats the short rate if nothing moves. Sloping down means the mirror image. When a vignette gives two spot rates and asks for a forward, always divide the longer compounded factor by the shorter and take the root of the maturity difference.

The full lesson (about 2,945 words, 20 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• term structure

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