Library/CFA (Chartered Financial Analyst)/FIXED INCOME: CFA® Program Curriculum, 2026 • Level I • Volume 6/Learning Module 9 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Learning Module 9 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

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Overview and Key Concepts5 min

This chapter introduces the term structure of interest rates and three interconnected yield curves used in fixed-income analysis: the spot (zero) curve, the par curve, and the forward curve. Spot rates are the yields on default-risk-free zero-coupon bonds for each maturity; using the sequence of spot rates to discount a bond's cash flows produces a no-arbitrage price. In practice, spot curves are estimated from recently issued government coupon bonds and interpolated across maturities because pure zero-coupon data are limited. The bond pricing formula with spot rates is PV = sum_{i=1..N} PMT_i / (1 + Z_i)^i + FV / (1 + Z_N)^N, where Z_i is the i-period spot rate. A par rate is the coupon/yield that makes a hypothetical bond priced at par when its cash flows are discounted using spot rates; par rates are derived by setting PV = 100 and solving for the constant coupon (PMT). The forward rate IFRA,B-A (the implied rate from time A to B) relates to spot rates via (1 + z_A)^A * (1 + IFR)^{B-A} = (1 + z_B)^B and can be interpreted as the breakeven reinvestment rate for extending maturity. Forward rates are calculated from spot rates and spot rates may be recovered by compounding (geometric average) of successive one-period forward rates. The chapter includes numeric examples: computing bond prices using spot rates; computing par rates from spot rates; solving 2y1y, 3y1y forward rates; deriving spot from a sequence of one-year forward rates; and pricing a coupon bond using forwards or spot rates (both yield identical results). It explains relationships among the curve shapes: if the spot curve is upward sloping, par rates are slightly below spot rates and forward rates exceed spot rates; in a flat spot curve, par and forward curves coincide with spot; in a downward-sloping spot curve, par rates exceed spot and forward rates lie below spot.

Key Points

Spot curve: default-risk-free zero rates used to discount each cash flow to get no-arbitrage prices.
Par rate: coupon/yield that prices a hypothetical bond at par; derived from spot rates.
Forward rate: implied breakeven reinvestment rate for a future period; calculated from spot rates.
Pricing with spot or forward rates yields the same bond price.
Curve shapes (upward/flat/inverted) determine relative positions of spot, par, and forward curves.

Pricing Bonds with Spot Rates and Deriving Par Rates6 min

This section develops the bond pricing formula using a sequence of spot rates: PV = sum_{k=1..N} PMT / (1 + Z_k)^k + FV / (1 + Z_N)^N. An example: a 3-year 1% coupon bond priced using one-, two-, three-year spot rates where each coupon is discounted by the corresponding spot rate. The result is a no-arbitrage price. Par rates are derived by setting PV = 100 and solving for the constant coupon PMT: 100 = PMT/(1+z1) + PMT/(1+z2)^2 + ... + (PMT+100)/(1+zN)^N; PMT/100 is the par rate. Par rates are often published (e.g., Treasury par curve) and are useful as benchmark yields for coupon-paying securities. The section shows numeric examples using Canadian and Australian spot rate tables to compute five-year bond prices and yields-to-maturity based on spot-based prices.

Key Points

To compute a bond price with spot rates, discount each cash flow by the spot rate matching its payment date.
A par rate is solved by setting PV = 100 and computing the coupon that satisfies the equality.
Spot-derived par rates control for distortions (tax, liquidity) in actual coupon issues.
Spot-based bond pricing yields the same total PV as discounting by a single YTM but gives different PVs for individual cash flows.

Forward Rates: Calculation and Interpretation6 min

Implied forward rates link short- and long-term spot rates and represent the marginal return for extending investment from A to B. The formula is (1 + z_A)^A * (1 + IFR_{A,B-A})^{B-A} = (1 + z_B)^B; solve for IFR. Forward rates are often named in the format '2y1y' (one-year rate two years from now). Economically, the forward rate is the breakeven reinvestment rate at which an investor is indifferent between buying a longer-term zero or rolling over a shorter-term investment. The forward curve can be used to compute spot rates by geometric compounding: (1 + z_N)^N = product_{i=0..N-1} (1 + IFR_{i,1}). Examples show deriving 2y1y and 3y1y and pricing bonds using forward rates equivalently to spot rates.

Key Points

Forward rates computed from spot rates are breakeven reinvestment rates.
Notation: AxBy refers to a y-year forward starting at A years (e.g., 2y1y).
Spot rates are the geometric average of successive one-period forward rates.
Bond pricing using forward rates (discounted by cumulative forwards) equals pricing with spot rates.

Curve Shapes and Their Relationships5 min

The shape of the spot curve (normal/upward, flat, inverted/downward) governs the relative positions of par and forward curves. In an upward-sloping spot curve: par rates are slightly below spot rates (especially at the long end) and forward rates lie above spot rates. In a flat spot curve: spot, par, and forward curves coincide. In a downward-sloping (inverted) spot curve: par rates exceed spot rates and forward rates lie below spot rates. Real-world data examples (Canada, Australia, Germany, Switzerland) illustrate these relationships, including handling of negative spot rates—forward rates can still become positive when the spot curve is upward sloping even though spot rates are negative. Key practical point: par curves are widely used as benchmarks (e.g., Treasury par curve).

Key Points

Upward spot curve => par < spot and forward > spot.
Flat spot curve => spot = par = forward.
Inverted spot curve => par > spot and forward < spot.
Negative spot rates do not preclude positive forward rates if the curve is upward sloping.

Practical Calculations and Applications6 min

The final section walks through numerous numerical practice problems: calculating bond prices, YTMs, G-spread/I-spread/Z-spread distinctions (from earlier modules), par rates, implied forwards, converting money market quotes to bond-equivalent yields, and using forward and spot rates interchangeably to price coupon bonds. It emphasizes Excel/financial calculator use (RATE, PRICE, and Solver) for solving non-closed-form equations such as finding Z-spreads or discount margins. Interpretive problems include reading curve charts, matching spot/forward maturities, and analyzing how curve movements affect pricing and reinvestment breakevens. The chapter underlines that forward rates are implied rates (not predictions) and that spot curve construction requires interpolation and care with day-count conventions.

Key Points

Use spreadsheet functions (RATE, PRICE, Solver) to compute yields, Z-spreads, and discount margins.
Convert money market discount/add-on quotations to comparable bond-equivalent yields before comparisons.
Forward rates are useful for hedging and for decisions about buy-and-hold versus reinvest strategies.
When constructing curves, be consistent with periodicity and day-count conventions.

Questions

Question 1

Which of the following best defines a spot rate?

Learning Module 9 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

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