Fixed-Income Bond Valuation: Prices and Yields: formula sheet

Key formulas

Bond price (on a coupon date)

PV = PMT ÷ (1+r)¹ + PMT ÷ (1+r)² + … + (PMT + FV) ÷ (1+r)ᴺ

r is the yield per period, PMT the coupon per period, FV the principal, N the number of periods remaining.

Annuity form

PV = PMT × [1 − (1+r)⁻ᴺ] ÷ r + FV ÷ (1+r)ᴺ

Same value as the sum above. Use it for hand calculation or to check your calculator.

Periodic inputs

r = YTM ÷ m; N = years × m; PMT = annual coupon ÷ m

m is the number of coupon payments per year (1 for annual, 2 for semiannual).

Zero-coupon bond

PV = FV ÷ (1+r)ᴺ

No coupons, so only the principal is discounted.

Full price between coupon dates

Full price = PV at last coupon date × (1+r)^(t ÷ T)

t is days since the last coupon, T is days in the coupon period. Equivalent to discounting all remaining flows to the settlement date.

Accrued interest

AI = (t ÷ T) × PMT

Share of the current coupon earned by the seller.

Flat price

Flat price = Full price − AI

The quoted (clean) price.

Bond price

PV = Σ [PMT ÷ (1 + r)^t] + FV ÷ (1 + r)^N

r is the yield per period. Higher r means lower PV, which is the inverse relationship.

Coupon vs yield rule

Coupon rate > YTM → premium; coupon rate = YTM → par; coupon rate < YTM → discount

Compare the coupon rate with the yield per period on the same basis.

Duration-only price change

%ΔPV ≈ −ModDur × ΔYield

Linear estimate. It understates the price rise for a yield fall and overstates the price fall for a yield rise on an option-free bond.

Duration plus convexity

%ΔPV ≈ −ModDur × ΔYield + ½ × Convexity × (ΔYield)²

Enter ΔYield as a decimal, e.g. 0.01 for 100 bp. The convexity term is always positive when convexity is positive.

Sensitivity rules (other things equal)

Longer maturity ↑ sensitivity; lower coupon ↑ sensitivity; lower yield level ↑ sensitivity

Use these for ranking questions. They are comparisons, so keep the 'other things equal' condition.

Bond price at any time

PV = Σ C ÷ (1 + r)^t + FV ÷ (1 + r)^N

Use r as the yield per period and N as the periods remaining. For the trajectory, keep r fixed and reduce N.

Price roll-forward (just after a coupon)

P(t+1) = P(t) × (1 + r) − C

Use per-period r. The price grows at the yield, then the coupon is paid out. It is a quick check on any trajectory.

Premium amortization per period

Amortization = C − r × P(t)

Positive for a premium bond. The price falls by this amount.

Discount accretion per period

Accretion = r × P(t) − C

Positive for a discount bond. The price rises by this amount.

Direction rule at constant yield

Coupon rate < YTM: price rises. Coupon rate > YTM: price falls. Equal: price stays at par.

Price equals par at maturity in every case, ignoring the final coupon.

Bond price using spot rates

PV = PMT ÷ (1 + Z1)^1 + PMT ÷ (1 + Z2)^2 + ... + (PMT + FV) ÷ (1 + Zn)^n

Zt is the spot rate for maturity t. For semiannual bonds, use periods and the per-period spot rate (annual rate ÷ 2).

Discount factor

DFt = 1 ÷ (1 + Zt)^t

Price = Σ (cash flow t × DFt). Useful when the question gives discount factors directly.

Price using YTM

PV = Σ PMT ÷ (1 + y)^t + FV ÷ (1 + y)^N

One rate y for all cash flows. Solve with the calculator's N, I/Y, PMT, FV keys.

Linear interpolation of yield (matrix pricing)

y = y1 + [(T − T1) ÷ (T2 − T1)] × (y2 − y1)

T1 and T2 are the maturities of the two comparable bonds with yields y1 and y2. T is the subject bond's maturity. Comparables must have the same credit quality.

Current yield

Current yield = annual coupon ÷ flat price

For a semiannual bond, annual coupon = 2 × the semiannual coupon. Use the flat price, not the full price.

Bond pricing equation (YTM)

PV = PMT/(1+r) + PMT/(1+r)² + … + (PMT + FV)/(1+r)ᴺ

Solve for r, the periodic yield. N is the number of periods and PMT is the coupon per period.

Annualizing a periodic yield

Annual yield (APR) = periodic yield × m

m = periods per year. This is the bond-equivalent yield for m = 2. No compounding is applied.

Periodicity conversion

(1 + APR_m ÷ m)ᵐ = (1 + APR_n ÷ n)ⁿ

Use it to restate a yield on another compounding basis. For semiannual to annual, EAR = (1 + APR ÷ 2)² − 1.

Yield to call

PV = Σ PMT/(1+r)ᵗ + call price/(1+r)ᴺ, with N = periods to the call date

Same equation as YTM, but N and the redemption amount change.

Yield to worst

YTW = lowest of YTM and the YTC for each call date

Compute all candidates, then take the minimum.

Price from discount rate

PV = FV × (1 − (Days ÷ Year) × DR)

Year is usually 360 for the discount rate. The discount is FV × (Days ÷ Year) × DR.

Price from add-on rate

PV = FV ÷ (1 + (Days ÷ Year) × AOR)

The add-on rate is measured against the amount invested, not the face value.

Discount rate to add-on rate (360-day year)

AOR = (360 × DR) ÷ (360 − Days × DR)

The add-on rate is always higher than the discount rate for the same instrument.

Bond equivalent yield

BEY = AOR(360-day) × 365 ÷ 360

Equivalently, BEY = (FV ÷ PV − 1) × 365 ÷ Days for instruments of one year or less.

FRN price

PV = Σ(t=1 to N) [(Index + QM) ÷ m × FV] ÷ [1 + (Index + DM) ÷ m]^t + FV ÷ [1 + (Index + DM) ÷ m]^N

m is payments per year and N is the number of periods. The standard exam assumption is that the index stays at its current level.

Margin rule

DM = QM → price = par; DM > QM → price < par; DM < QM → price > par

This holds on a reset date, when the index in the coupon equals the index in the discount rate.

Forward rate from spot rates

(1 + z_B)^B = (1 + z_A)^A × (1 + f_(A,B−A))^(B−A)

z is the spot rate for the maturity in years. f is the forward rate for a loan of B−A years starting at A. Solve for f. Use annual compounding unless told otherwise.

One-year forward rate starting in one year

f(1,1) = (1 + z2)² ÷ (1 + z1) − 1

Special case of the formula above, with A = 1 and B = 2.

Bond price using spot rates and Z-spread

PV = CF1 ÷ (1 + z1 + Z)¹ + CF2 ÷ (1 + z2 + Z)² + … + CFn ÷ (1 + zn + Z)ⁿ

Z is the constant Z-spread. It is found by trial and error until PV equals the market price.

G-spread

G-spread = bond YTM − government yield at same maturity

Interpolate between two government bonds if no exact match exists.

I-spread

I-spread = bond YTM − swap rate at same maturity

Benchmark is the interest rate swap curve, not government bonds.

OAS

OAS = Z-spread − option value (in bp)

The Z-spread includes the effect of the option: higher for callable, lower for putable. Option value is positive for a call held by the issuer and negative for a put held by the investor. Callable: OAS < Z-spread. Putable: OAS = Z-spread + |put value|, so OAS > Z-spread.

Quick revision

• Bond price = Σ cash flows discounted at the yield per period, plus principal discounted at the same rate.
• Price and yield move in opposite directions.
• Coupon rate above yield gives a premium price; below yield gives a discount; equal gives par.
• For a semiannual bond, halve the annual yield and coupon and double the years to get N.
• The price-yield curve is convex, so a given yield fall raises price more than the same yield rise lowers it.
• With a constant yield, a discount bond's price rises toward par and a premium bond's price falls toward par.
• A spot rate is the yield on a zero-coupon bond for a given maturity; price a bond by discounting each cash flow at its own spot rate.
• Matrix pricing estimates the yield of an illiquid bond from comparable bonds with similar credit quality and maturity.
• Know which yield is which: yield to maturity, current yield, yield to call and yield to worst each have different assumptions.
• A yield spread is a bond's yield minus a benchmark yield, and it compensates for credit, liquidity and other risks.
• Always check whether a question gives a quoted annual rate with periodic compounding before you set N and the rate.
• Use the direction of change to remove one or two options before you calculate.

Common mistakes

• Using the annual yield with semiannual cash flows Fix: Divide the yield and the coupon by m and multiply N by m before keying anything.
• Using the coupon rate as the discount rate Fix: The coupon only sets the cash flow. Discount at the YTM. The coupon rate is used for PMT only.
• Treating the price-yield relationship as a straight line Fix: Remember that duration is the tangent and the curve lies above it for option-free bonds. Convexity adds back the gap.
• Subtracting the convexity term instead of adding it Fix: Write −ModDur × ΔY + ½ × Conv × (ΔY)². The squared yield change makes the second term positive for either direction of change.
• Saying every bond's price rises as maturity approaches. Fix: Compare the coupon rate with the YTM first. Below YTM, price rises. Above YTM, price falls. At par, it stays.
• Treating the price path as a straight line. Fix: At a constant yield the change each period is C − r × P(t), which differs every period. The path is curved. A straight line is not the effective interest method.
• Discounting all cash flows at one spot rate, such as the final-maturity rate. Fix: Spot-rate pricing needs a different rate for each cash flow date. Write Z1, Z2, ... Zn beside each payment before calculating.
• Calling the YTM the same as the spot rate for that maturity. Fix: A spot rate applies to one single payment at one date. YTM is one blended rate for a whole coupon bond. They are equal only when the curve is flat.
• Forgetting to double N or halve the coupon for a semiannual bond. Fix: Convert first: N = years × 2, PMT = annual coupon ÷ 2. Your answer is a semiannual rate until you multiply by 2.
• Reporting the periodic rate from the calculator as the annual yield. Fix: Multiply by m for the bond-equivalent yield, or compound it for the effective annual yield. Check which one the question wants.

Exam tips

• Write the per-period inputs (PMT, r, N) on your scratch sheet before touching the calculator. This is where most marks are lost.
• Use the par check to eliminate options. Numerical options are in ascending order, so a premium bond must be in the option above 1,000 and a discount bond below.
• Watch for distractors built from the classic errors: annual inputs on a semiannual bond, or the coupon rate used as the discount rate.
• If the question says 'quoted' or 'flat' price, subtract accrued interest from the full price. If it says 'amount paid' or 'full price', do not.
• With roughly 90 seconds per question, trust the calculator and avoid the long-hand annuity formula unless asked for the structure of the equation.
• Expect three-option conceptual items asking which bond is most or least sensitive. Check that only one feature (coupon, maturity or yield) differs before you apply a rule.
• If an option says price falls by the same amount for equal rises and falls in yield, treat it as wrong for an option-free bond.
• In estimate questions, the options are listed from smallest to largest. Compute the duration-only figure first. The correct option is usually just beyond it.