The Term Structure of Interest Rates: Spot, Par, and Forward Curves

A 10-year Treasury yield is one number, but the term structure behind it is three separate curves: spot, par, and forward. Confuse them at your peril.

Define a spot rate $s_t$ as the yield to maturity on a default-free zero-coupon bond maturing in $t$ years. Each spot rate is tied to one specific maturity. The spot curve plots $s_1,s_2,s_3,\dots$ against maturity. Every other term-structure curve can be derived from it.

KEY: Spot rates are pure discount rates with no reinvestment assumption. A 5-year spot rate is the single rate that grows $1 today into the maturity value of a 5-year zero.

To calculate the price of a coupon bond, discount each cash flow at the spot rate matching its date:

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

A par rate is the coupon rate that sets the price of an annual-pay bond exactly equal to par (100), given today's spot curve. Solve for the coupon $c$:

$$100=\sum _{t=1}^n\frac{c}{(1+s_t{)}^t}+\frac{100}{(1+s_n{)}^n}$$

Common mistakes

• Discounting all cash flows at the same spot rate. Each cash flow must be discounted at the spot rate matching its own maturity. Using $s_3$ for the year-1 coupon is a YTM shortcut. Trap: applying $s_3=5\%$ to all three years and getting $1,000 exactly.
• Inverting the forward formula. The numerator is the longer compounded spot, the denominator is the shorter. Trap: computing $(1.03)/(1.04{)}^2-1$ and getting a negative forward when the curve is rising.
• Confusing par rate with the coupon of an existing bond. Par rate is the coupon that would price a new bond at par today given the current spot curve. The coupon on an older bond is irrelevant. Trap: "the bond pays a 6% coupon, so the par rate is 6%."

Bottom line

• Spot rate $s_t$ = yield on a zero-coupon bond maturing at time $t$; the spot curve plots $s_t$ across maturities and is the foundation of term structure
• A bond's price equals the sum of its cash flows each discounted at the spot rate matching its own maturity; a single YTM only compresses that curve into a shortcut
• Par rate = coupon rate that makes a new bond's price equal par (100) given today's spot curve; the coupon on an existing bond is irrelevant
• Forward rate $f_{a,b}$ = rate implied today for borrowing from year $a$ for $b$ more years, satisfying $(1+s_{a+b}{)}^{a+b}=(1+s_a{)}^a(1+f_{a,b}{)}^b$

Exam shortcut

For pricing on an upward-sloping curve, the spot-rate price is below the flat-YTM price because long cash flows get discounted harder. For forwards, numerator longer, denominator shorter, root equal to the difference in maturities. For curve comparison, forwards are the marginal rate pulling the average spot up, par is the coupon-weighted average pulled down by the principal. Memorize F-S-P for upward-sloping and flip for downward.

The full lesson (about 1,995 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• term structure spot par forward

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