Spot Rates, Forward Rates, and Yield-Curve Pricing

A trader can lock in a one-year loan rate that starts a year from now, today, with no view on where rates will go. The forward rate that makes it arbitrage-free is already baked into the spot curve.

A spot rate $s_t$ is the annual yield on a zero-coupon bond maturing at time $t$. Each cash flow is discounted at the spot rate for its own maturity:

$$PV=\sum _t\frac{C{F}_t}{(1+s_t{)}^t}$$

When the curve is flat (all $s_t$ equal), this collapses to ordinary discounting at one rate. FM defaults to a flat curve unless a problem hands you spot or forward rates.

You rarely get spot rates directly; you back them out of coupon-bond prices, shortest maturity first. Price a 1-year bond to get $s_1$. Then price a 2-year bond, plug in the known $s_1$ for its first coupon, and solve the remainder for $s_2$. Continue outward. Each step has exactly one unknown because every earlier spot rate is already pinned.

Common mistakes

• Bootstrapping out of order. You cannot solve $s_3$ without $s_1$ and $s_2$. Each step relies on every earlier spot already being known.
• Flipping the forward-rate fraction. The larger accumulation factor goes on top: $(1+s_{t+1}{)}^{t+1}/(1+s_t{)}^t$. Inverting it gives a negative forward on a normal curve, a clear signal you flipped it.
• Discounting every cash flow at one spot rate. Off a non-flat curve, the year-1 coupon uses $s_1$, the year-2 coupon uses $s_2$, and so on. Using $s_3$ for all three understates the early coupons.

Bottom line

• Spot rate $s_t$ is the yield on a zero maturing at $t$: a cash flow at $t$ is discounted by $(1+s_t{)}^t$.
• Bootstrap spot rates shortest-first: solve $s_1$ from a 1-year bond, then use it to solve $s_2$ from a 2-year bond, and so on.
• Forward rate: $(1+s_t{)}^t(1+f_{t,t+m}{)}^m=(1+s_{t+m}{)}^{t+m}$. The larger accumulation factor's exponent is always on top.
• No-arbitrage lock-in: borrow short, invest long, and the implied gap return is exactly the forward, which is lockable today, not a forecast.

Exam shortcut

Build the accumulation-factor column $(1+s_t{)}^t$ first; every forward is a ratio of adjacent entries minus 1, and every present value is a cash flow divided by an entry. Bootstrap in order, one unknown at a time. "Big over small, minus one" for forwards. "Short to long" for bootstrapping. "Each flow at its own spot" for pricing off the curve.

The full lesson (about 2,404 words, 16 min read) adds 4 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 5a
• 5b
• 5c

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