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.

Picture a default-free zero-coupon bond that pays $100 in exactly 3 years. Its yield is the 3-year spot rate, so you define spot rates as zero-coupon yields, one per maturity. Write the $t$-year spot rate as $s_t$. The spot curve plots $s_1, s_2, s_3, \ldots$ 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.

A coupon bond is just a bundle of zeros: one zero for each coupon and one for the final coupon plus principal. So you calculate the price of a bond by discounting each cash flow at the spot rate for its own date:

$$P = \sum_{t=1}^{n} \frac{CF_t}{(1+s_t)^t}$$

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 (−4.77%) when the curve is rising.
• Skipping the forward root. The spot-factor ratio for $f_{1,2}$ covers two years of growth. Trap: reporting 1.12391 − 1 = 12.39% instead of taking the square root to get 6.01%.

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.
• 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 $(1+s_{a+b})^{a+b} = (1+s_a)^a (1+f_{a,b})^b$
• Upward-sloping curve: forward > spot > par. Downward: forward < spot < par. Flat: all three equal

Exam shortcut

For pricing on an upward-sloping curve, the spot-rate price is above the price you get by discounting every cash flow at the longest spot, because the early cash flows get discounted at lower rates ($1,001.80 versus $1,000 in Example 1). For forwards, numerator longer, denominator shorter, root equal to the difference in maturities. For bootstrapping, work maturities in order and never skip the root on the last discount factor.

The full lesson (about 3,777 words, 25 min read) adds 11 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• term structure spot par forward

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