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CFA® Level I · Fixed Income · free lesson
The Term Structure of Interest Rates: Spot, Par, and Forward Curves
A single yield-to-maturity discounts every cash flow at the same rate, but the market charges a different rate for each maturity. This module introduces the three ways of describing that term structure: spot rates for pricing each cash flow, par rates for setting coupons, and forward rates for the future periods the curve implies. You will price bonds with spot and forward rates, move between the three curves, and interpret their shapes.
What you will learn
1. Define spot rates and the spot curve, and calculate the price of a bond by discounting each cash flow at its own spot rate.
- Spot rate = yield on a zero-coupon bond for that maturity; the spot curve plots them.
- Arbitrage-free price: discount each cash flow at its own spot rate.
- Stripping and reconstitution keep coupon bond prices consistent with the spot curve.
- YTM is a weighted average of spot rates, weighted heavily toward the final one.
- Same maturity but different coupons can mean different YTMs from the same spot curve.
2. Define par rates and the par curve, calculate par rates from spot rates, and derive spot rates from par rates by bootstrapping.
- Par rate = coupon rate at which a bond of that maturity is priced at par, given the spot curve.
- Par rate formula: c = (1 - DF_N) / (DF_1 + ... + DF_N).
- On-the-run government yields approximate the par curve; official curves are usually par curves.
- Bootstrapping derives spot rates one maturity at a time from par rates.
- Upward-sloping curve: par rate < spot rate at the same maturity; downward-sloping: par > spot.
3. Define forward rates, calculate forward rates from spot rates and spot rates from forward rates, and price a bond using forward rates.
- AyBy = B-year rate starting A years from now (1y1y, 2y1y, 1y2y).
- No-arbitrage: (1 + z_A)^A x (1 + IFR)^(B-A) = (1 + z_B)^B.
- Spot rates are geometric means of one-period forward rates.
- Pricing with forward rates gives the same price as pricing with spot rates.
- A forward rate is the breakeven reinvestment rate between a short and a long investment.
4. Compare the spot, par and forward curves and interpret the shape of the yield curve.
- Spot rates average forward rates; par rates average spot rates.
- Upward-sloping: forward > spot > par. Inverted: forward < spot < par. Flat: all equal.
- All three curves share the same one-period starting point.
- The forward curve is the most volatile and magnifies kinks in the spot curve.
- Inverted curves usually signal expected rate cuts; forwards include risk premiums, so they are not pure forecasts.
Module at a glance
- Spot rates are zero-coupon yields; discount each cash flow at its own spot rate for the arbitrage-free price.
- YTM is a weighted average of spot rates, dominated by the final one.
- Par rate = coupon that prices a bond at par: c = (1 - DF_N) / sum of DF_t; bootstrap spot rates from par rates.
- Forward rates follow from (1 + z_A)^A (1 + IFR)^(B-A) = (1 + z_B)^B; annualize multi-year forwards.
- Pricing with chained forward rates gives the same price as pricing with spot rates.
- Upward-sloping: forward > spot > par; inverted: forward < spot < par.
- Inverted curves imply forward rates below current short rates, consistent with expected rate cuts.
This free module has 4 lessons, 13 practice questions with full explanations.