CFA Level I · CFA Level I Exam

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

Full chapter guide

Key formulas

Discount factor: DF_t = 1 ÷ (1 + S_t)^t; S_t is the annual spot rate for maturity t years, with annual compounding. For semiannual compounding, use S_t ÷ 2 per period and 2t periods.
Bond price from spot rates: PV = Σ [CF_t ÷ (1 + S_t)^t] = Σ [CF_t × DF_t]; Each cash flow is discounted at the spot rate for its own maturity. The final cash flow includes principal.
Spot rate from discount factor: S_t = (1 ÷ DF_t)^(1/t) − 1; Use this to move back from a discount factor to the spot rate.
Spot rate from a zero-coupon price: S_t = (Face value ÷ Price)^(1/t) − 1; For a zero-coupon bond the spot rate equals its YTM.
Price with a single YTM: PV = Σ [CF_t ÷ (1 + YTM)^t]; Same cash flows, but one rate for all dates. The YTM is the single rate that sets this present value equal to the market price. The spot-based price equals the market price only if the bond is priced with no arbitrage.
Spot-forward relationship: (1 + z_B)^B = (1 + z_A)^A × (1 + f(A, B−A))^(B−A); z is the annual spot rate. A is the start of the forward period and B−A is its length in years. Works for annual compounding.
Implied forward rate: f(A, B−A) = [ (1 + z_B)^B ÷ (1 + z_A)^A ]^(1 ÷ (B−A)) − 1; Always take the root of the ratio over the loan length, B − A, not over B.
One-year forward from adjacent spots: f(n−1, 1) = (1 + z_n)^n ÷ (1 + z_(n−1))^(n−1) − 1; Use this to build the one-year forward curve step by step.
Spot rate from forward rates: (1 + z_n)^n = (1 + f(0,1)) × (1 + f(1,1)) × … × (1 + f(n−1,1)); The spot rate is the geometric average of the one-year forward rates. f(0,1) equals z_1.
Bond value using forward rates: PV = CF_1 ÷ (1 + f(0,1)) + CF_2 ÷ [(1 + f(0,1))(1 + f(1,1))] + …; Gives the same value as discounting each cash flow at its spot rate.
Forward price of a zero-coupon bond: F = DF_B ÷ DF_A, where DF_t = 1 ÷ (1 + z_t)^t; Price per 1 of par, for a contract made today to buy at time A a bond that matures at B (A < B).
Par bond pricing identity (annual pay): 1 = c × (DF₁ + DF₂ + … + DFₙ₋₁) + (1 + c) × DFₙ; c is the par rate for maturity n, written as a decimal. DF is the discount factor per 1 of face value.
Bootstrapped discount factor: DFₙ = [1 − c × (DF₁ + … + DFₙ₋₁)] ÷ (1 + c); Use the par rate cₙ for maturity n and every earlier discount factor.
Spot rate from discount factor: zₙ = (1 ÷ DFₙ)^(1/n) − 1; On the exam this is usually annual compounding.
Discount factor from spot rate: DFₙ = 1 ÷ (1 + zₙ)ⁿ; Use this to go back and check a result.
First step: z₁ = c₁; The 1-year par rate equals the 1-year spot rate when the bond pays once a year.
Bond price using YTM: PV = Σ CF_t ÷ (1 + y)^t; One rate y for all cash flows. Solve y with the calculator.
Bond price using spot rates: PV = Σ CF_t ÷ (1 + z_t)^t; Each cash flow uses the spot rate z_t for its own maturity.
G-spread: G-spread = bond YTM − government bond YTM (same maturity); Interpolate between two government bonds if no exact match exists.
I-spread: I-spread = bond YTM − swap rate (same maturity); Benchmark is the interest rate swap curve, not government bonds.
Z-spread: PV = Σ CF_t ÷ (1 + z_t + Z)^t; Z is constant across all maturities. Found by trial and error or a solver.
OAS for an option bond: OAS = Z-spread − option value (in spread terms); Callable: OAS < Z-spread. Putable: OAS > Z-spread.
Forward rate from spot rates: (1 + z_B)^B = (1 + z_A)^A × (1 + f(A, B−A))^(B−A); f(A, B−A) is the rate for a loan of B−A years starting in A years.
Rolling-down return (unchanged curve): Return = P(horizon) ÷ P(today) − 1; P(horizon) uses the spot rate for the bond's remaining maturity from today's curve.

Quick revision

Spot rate is the yield on a zero-coupon bond for a given maturity.
Discount factor = 1 ÷ (1 + spot rate)^t, with annual compounding.
Bond price = sum of each cash flow times its own discount factor.
Forward rate is the rate for a future period implied by today's spot rates through no-arbitrage.
Forward rate link: (1 + z_B)^B = (1 + z_A)^A × (1 + f)^(B − A).
When the spot curve slopes upward, forward rates lie above spot rates.
Par rate is the coupon rate at which a bond prices at par.
Bootstrapping solves for spot rates one maturity at a time from par rates.
YTM is the single discount rate that equates a bond's price to its cash flows, assuming it is held to maturity and reinvestment at that rate.
YTM is a blend of spot rates and generally differs from any single spot rate.
Yield spread is the difference between yields of two bonds, often quoted in basis points.
A parallel shift moves all rates by the same amount; steepening and flattening change the gap between long and short rates.

Common mistakes

Discounting all cash flows at one yield when the question gives spot rates. Fix: Use the spot rate matching each cash flow date. One rate for all dates is the YTM approach.
Treating spot rates as YTMs of coupon bonds. Fix: Spot rates come from zero-coupon bonds. A coupon bond YTM blends several spot rates and depends on the coupon.
Averaging spot rates instead of using growth factors Fix: Always compound. Use (1 + z_B)^B ÷ (1 + z_A)^A. A simple difference understates the forward rate in an upward-sloping curve.
Taking the root over B instead of B − A Fix: The root order equals the length of the forward loan. For a 2y3y forward, take the 3rd root.
Treating the par rate as the spot rate for every maturity Fix: Only the 1-year (single cash flow) par rate equals the spot rate. Longer par rates discount all coupons at one rate, so bootstrap them.
Discounting the coupons at the new par rate Fix: Use the already-found discount factors for the earlier coupons. The par rate only sets the coupon size.
Using the wrong benchmark for a spread, such as subtracting a swap rate for a G-spread. Fix: Link the letter to the benchmark: G for government, I for interest rate swap, Z for the zero-volatility (spot) curve.
Discounting at one YTM when the question gives spot rates. Fix: Whenever you see a spot curve, discount each cash flow at its own spot rate.

Exam tips

Questions are standalone with three options. If the coupon is above all spot rates, the bond must price above par. Use that to remove an option before calculating.
Watch for the words zero-coupon or spot. They signal that each cash flow gets its own rate.
A discount factor above 1, or one that rises with maturity under positive rates, signals an error.
Check compounding in the stem. Semiannual is a common trap.
Know that spot-based pricing is arbitrage-free pricing. If market price differs, an arbitrage exists through stripping or reconstituting.
Most questions give two spot rates and ask for a forward rate. Write A and B − A before touching the calculator.
Use direction to eliminate options. With an upward-sloping spot curve, the forward rate is above the longer spot rate. With a downward-sloping curve, it is below.
Expect the forward-rate bond valuation to match the spot-rate valuation. If a question asks you to choose between the two, the prices are equal.