CFA Level I · CFA Level I Exam

Yield and Yield Spread Measures for Fixed-Rate Bonds: formula sheet

Full chapter guide

Key formulas

Bond price: PV = PMT ÷ (1+r) + PMT ÷ (1+r)² + … + (PMT + FV) ÷ (1+r)^N; r is the periodic yield (annual yield ÷ m). N is years × m. PMT is the coupon per period.
Yield to maturity: Find r such that price = Σ PMT ÷ (1+r)^t + FV ÷ (1+r)^N; It is the IRR of the bond's cash flows. Solve with a calculator, not by algebra.
Annualised (street convention) yield: Quoted yield = periodic rate × m; m = 2 for semiannual, 4 for quarterly, 12 for monthly. No compounding within the year.
Effective annual yield: EAY = (1 + quoted yield ÷ m)^m − 1; Use this to compare bonds with different payment frequencies.
Convert between frequencies: (1 + y₁ ÷ m₁)^m₁ = (1 + y₂ ÷ m₂)^m₂; Both sides equal 1 + EAY. Solve for the unknown yield.
Current yield: Current yield = annual coupon ÷ bond price; Use the annual coupon in currency terms. For a semiannual bond, add both coupons.
Yield ordering: Premium: coupon rate > current yield > YTM. Discount: coupon rate < current yield < YTM. Par: all equal.; A quick check that your answer is sensible.
Yield to call pricing equation: Price = Σ PMT ÷ (1 + r)^t, t = 1 to n, + Call price ÷ (1 + r)^n; n is the number of periods to the call date and r is the periodic yield. Use the call price, not par, as the final redemption value.
Annualizing the periodic yield: Annual yield (bond-equivalent) = r × number of periods per year; For semiannual coupons, double the periodic rate. Do not compound it unless the question asks for an effective annual yield.
Yield to worst: YTW = minimum of (YTM, YTC at each call date); Include every call date in the schedule. For a putable bond, compute the yield to put separately when put exercise is expected.
Cash flow yield: Price + accrued interest = Σ CFt ÷ (1 + y)^t; CFt are projected interest and principal, including prepayments under an assumed speed. y is the periodic (usually monthly) rate, then annualized.
TVM keys for yield to call: N = periods to call; PMT = coupon per period; FV = call price; PV = –price; CPT I/Y; The price entered as PV must be negative if PMT and FV are positive.
FRN price (periodic): PV = Σ (t = 1 to N) {[(Index + QM) ÷ m × FV] ÷ (1 + (Index + DM) ÷ m)^t} + FV ÷ (1 + (Index + DM) ÷ m)^N; m = payments per year, N = number of periods, t = 1 to N. Use today's index for all projected coupons unless told otherwise.
QM versus DM and price: DM = QM → price = par; DM > QM → price < par; DM < QM → price > par; This holds on a reset date with a flat projected index, and lets you eliminate options without calculating.
Discount rate pricing: PV = FV × (1 − (Days ÷ Year) × DR); DR is quoted on face value. Year is 360 or 365 depending on the market.
Add-on rate pricing: FV = PV × (1 + (Days ÷ Year) × AOR); AOR is quoted on the price paid, so it is higher than the discount rate for the same instrument.
Discount rate to add-on rate: AOR = (Year × DR) ÷ (Year − Days × DR); Same as ((FV ÷ PV) − 1) × (Year ÷ Days).
Bond equivalent yield: BEY = ((FV ÷ PV) − 1) × (365 ÷ Days); For a 360-day add-on rate, BEY = AOR × 365 ÷ 360.
Bond price using spot rates: PV = CF₁ ÷ (1 + S₁) + CF₂ ÷ (1 + S₂)² + … + (CFₙ + Par) ÷ (1 + Sₙ)ⁿ; Each cash flow uses the spot rate for its own date. Annual coupons assumed here.
Discount factor: DFₜ = 1 ÷ (1 + Sₜ)ᵗ; Price = Σ (cash flow × discount factor).
Par bond bootstrapping (2-year example): 100 = C₂ ÷ (1 + S₁) + (100 + C₂) ÷ (1 + S₂)²; Use the par yield as the coupon rate C₂. Known spots go in first. Solve for the one unknown spot.
Implied forward rate: (1 + S_(A+B))^(A+B) = (1 + S_A)^A × (1 + f(A,B))^B; f(A,B) is the B-year rate starting A years ahead.
Forward rate solved: f(A,B) = [ (1 + S_(A+B))^(A+B) ÷ (1 + S_A)^A ]^(1/B) − 1; Take the B-th root. For B = 1 there is no root.
Forward rate from discount factors: 1 + f(A,1) = DF_A ÷ DF_(A+1); A quick route when the question gives discount factors.
G-spread: G-spread = YTM of bond − YTM of government bond with same maturity; Interpolate between two government yields if no exact maturity match exists. Quote in bps.
I-spread: I-spread = YTM of bond − swap rate of same maturity; Uses the swap curve as the benchmark, so it reflects the credit and liquidity of the interbank swap market, not the government.
Linear interpolation: Benchmark = Y1 + (T − T1) ÷ (T2 − T1) × (Y2 − Y1); Use it when the bond's maturity lies between two quoted benchmark maturities.
Z-spread pricing equation: Price = Σ CFt ÷ (1 + St + Z)^t; St is the benchmark spot rate for period t. Z is the same constant for every period. Solve for Z.
OAS for a callable bond: OAS = Z-spread − option value (in spread terms); OAS is lower than the Z-spread because the call option benefits the issuer.
OAS for a putable bond: OAS = Z-spread + option value (in spread terms); OAS is higher than the Z-spread because the put option benefits the investor.

Quick revision

Current yield = annual coupon ÷ price. It ignores capital gains and reinvestment.
YTM is the discount rate that equates price to promised cash flows; it assumes the bond is held to maturity and coupons are reinvested at the YTM.
The annualised YTM is the periodic rate × the number of periods per year. For a semiannual bond this means periodic rate × 2, which gives the bond-equivalent yield. Do not double for other frequencies.
Price below par means YTM above the coupon rate, and price above par means YTM below the coupon rate.
Yield to worst is the lowest of the yields to each call date and to maturity.
A callable bond's required yield or coupon is typically higher than that of an otherwise identical non-callable bond, to compensate the investor for call risk.
Floating-rate notes use a quoted margin and discount margin; the discount margin is the spread that makes the FRN price equal to its cash flows.
Spot rates discount single cash flows. Forward rates are implied by spot rates through no-arbitrage.
G-spread is the bond's yield minus the government yield at the same maturity (interpolated if needed); I-spread is the yield minus the swap fixed rate at the same maturity (interpolated if needed). Each is measured against a single point on the government or swap curve.
Z-spread is the constant spread added to every spot rate on the whole benchmark spot curve so that the discounted cash flows equal the price.
Z-spread = OAS + option cost (in spread terms). For a callable bond OAS is lower than the Z-spread; for a putable bond OAS is higher than the Z-spread.

Common mistakes

Using the annual yield as the periodic rate in a semiannual bond. Fix: Divide the annual yield by m and multiply the years by m before you enter anything. Both N and I/Y must be per period.
Calling the periodic rate times m an effective annual yield. Fix: The street convention yield does not compound. Only (1 + periodic)^m − 1 is the effective annual yield, and it is always higher when m > 1 and the rate is positive.
Using par as the final payment when calculating yield to call. Fix: Read the call price in the stem. Put the call price in FV and the number of periods to the call date in N.
Forgetting to double or halve for semiannual coupons. Fix: Convert N to periods and PMT to the coupon per period. Then multiply the result by the frequency to annualize.
Treating the quoted margin as the required return on an FRN Fix: QM is a contract term and never changes. DM is the market's required spread and changes with credit risk. The coupon uses QM, and the discounting uses DM.
Forgetting to divide the annual rates by m Fix: Divide (Index + QM) and (Index + DM) by m before using them as the coupon rate and discount rate per period.
Using YTM to discount every cash flow when asked to price with spot rates. Fix: Each cash flow uses the spot rate for its own date. Write the rate beside each cash flow before computing.
Treating a par yield as a spot rate for maturities beyond one year. Fix: Only the first maturity is identical. Bootstrap every later maturity.
Subtracting a spot rate from the YTM and calling it the G-spread. Fix: G-spread and I-spread use yields to maturity of the benchmark. Z-spread is the only one of the four tied to spot rates.
Adding the option value to the Z-spread for a callable bond. Fix: The issuer owns the call, so investors demand extra yield in the Z-spread. Remove it: OAS = Z-spread − option value. For a putable bond the investor owns the option, so add it.

Exam tips

Wrong options often reflect common mistakes, such as forgetting to divide by m or reporting the effective annual yield instead of the quoted yield. Work out what each option could represent before you calculate, but do not assume any fixed pattern.
Use the premium/discount order (coupon, current yield, YTM) to remove two options quickly. This works with no calculator at all.
Always check whether the question says semiannual bond basis, annual-pay basis or effective annual. They are different numbers for the same bond.
Practise the TVM keystrokes until they are automatic: N, I/Y, PMT, FV, CPT PV, and the reverse for I/Y. Each question should take about 90 seconds.
If a question asks what YTM assumes, remember: held to maturity, all payments made as promised, and coupons reinvested at the YTM.
Read the stem for the call price, the call date and the coupon frequency before you touch the calculator. Most errors come from the wrong FV or N.
If a question says the bond is trading at a premium to the call price, expect the answer to be a yield to call. If it trades at a discount, test YTM first.
Use the coupon rate as a check, with care. A premium bond's yield should be below the coupon rate and a discount bond's above it for YTM, and for YTC when the call price is par or the bond trades above the call price. If the call price is above par, YTC can exceed the coupon even at par, as in the second worked example.