Yield and Yield Spread Measures for Floating-Rate Instruments: formula sheet

Key formulas

FRN coupon rate

Coupon rate = Reference rate + Quoted margin

Annual rate. Quoted margin is fixed at issue and is stated in basis points or percent.

Coupon payment per period

Coupon payment = Coupon rate × Period fraction × Par value

Quarterly reset: period fraction ≈ 0.25. Use the annual rate, not the rate per period, as the base.

Cap and floor

Coupon = min(cap, max(floor, reference rate + margin))

Apply the cap and floor to the total coupon rate unless the question says otherwise.

Inverse floater coupon

Coupon rate = Fixed rate − (Leverage factor × Reference rate)

Coupon rises when the reference rate falls. Often has a floor of 0%.

FRN coupon per period

Coupon = (Reference rate + QM) ÷ m × Par

m is the number of payments per year. QM is fixed at issue.

FRN price

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

Assumes the index stays constant. Sum t from 1 to N, where N = years × m. Per 100 of par.

Shortcut price

Price ≈ 100 + [(QM − DM) ÷ m] × Annuity factor at (Index + DM) ÷ m for N periods

Valid on a reset date with the index held constant. Gives the same result as the full formula.

Pricing rule

DM = QM → par; DM > QM → discount; DM < QM → premium

Required margin above quoted margin means the price is below par.

FRN price

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

Sum from t = 1 to N. Index is the reference rate, QM the quoted margin, DM the discount (required) margin, m the payments per year, N the total periods. Assumes the index stays constant.

Periodic coupon

Coupon = FV × (Index + QM) ÷ m

Use the index rate that applies to the period. Margins are annual, so divide by m.

Periodic discount rate

r = (Index + DM) ÷ m

Use this as I/Y on the calculator.

Par, premium or discount rule

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

Exact par holds on a reset date with the index held constant and a flat projection. Between reset dates, accrued interest complicates the quoted price.

Price gap from par

Price − FV = FV × (QM − DM) ÷ m × annuity factor at r for N periods

Useful shortcut because the reference rate drops out of the numerator.

Price from discount rate

PV = FV × (1 − (Days/360) × DR)

Use for T-bills and commercial paper quoted on a discount basis. Days is the actual days to maturity.

Bank discount yield

DR = (360/Days) × (FV − PV) ÷ FV

Based on face value and a 360-day year. It understates the return.

Price from add-on rate

PV = FV ÷ (1 + (Days/Year) × AOR)

Year is 360 for most instruments but 365 for some currencies such as GBP.

Holding period yield

HPY = (FV − PV) ÷ PV

Unannualized return over the life of the instrument.

Money market yield

MMY = HPY × (360/Days)

Equals the add-on rate on a 360-day basis. Also MMY = (360 × DR) ÷ (360 − Days × DR).

Bond equivalent yield

BEY = HPY × (365/Days)

Simple annualization on 365 days. Also BEY = MMY × (365/360).

Effective annual yield

EAY = (1 + HPY)^(365/Days) − 1

Compounded version. Do not confuse with BEY, which is not compounded.

Quick revision

• FRN coupon = reference rate + quoted margin, reset at each coupon date.
• Quoted margin is fixed in the contract; required margin changes with credit risk and liquidity.
• At a reset date, required margin above quoted margin: FRN prices below par.
• At a reset date, required margin below quoted margin: FRN prices above par.
• At a reset date, required margin equal to quoted margin: FRN prices at par.
• Between reset dates, price also depends on the current reference rate versus the rate already set for the current coupon, so the par rule is not exact.
• Discount margin is the spread over the reference rate that equates discounted cash flows to price.
• Discount each FRN cash flow at reference rate plus required margin, adjusted for the period.
• A cap limits the coupon from above and helps the issuer; a floor limits it from below and helps the investor.
• Money market yields use different day counts and bases, so identify the convention before calculating.
• A discount rate is quoted on face value, so for the same instrument it understates the yield on the amount invested (the add-on rate).
• Money market instruments are quoted on a discount rate or add-on rate basis; the bond equivalent yield is a conversion for comparison with bonds, not a quote basis.
• Always check whether a question wants a price, a margin, or a yield, and whether the period is annual or per coupon.

Common mistakes

• Forgetting to apply the cap or floor Fix: Always compare the total coupon rate with the cap and floor before paying out.
• Using the annual coupon rate as the period payment Fix: Multiply by the period fraction, such as 0.25 for quarterly payments.
• Using the quoted margin as the discount margin when pricing. Fix: QM sets the coupon. RM or DM sets the discount rate. Always put each in its own place.
• Forgetting to divide the annual rate by m. Fix: Divide the coupon rate and discount rate by m, and set N = years × m.
• Using the annual margin or annual rate as the periodic rate Fix: Divide both the coupon rate and the discount rate by m, and set N = years × m.
• Choosing a premium when the required margin is above the quoted margin Fix: A higher required margin means a higher discount rate relative to the coupon, so the price falls below par. Remember: DM > QM gives a discount.
• Treating the discount rate as a return on the price paid Fix: Remember that DR is applied to face value. Convert to price first, then compute the yield on price.
• Using 365 instead of 360 for money market yield Fix: MMY always uses 360. Use 365 only for BEY, or when a question says the add-on rate uses 365 days.

Exam tips

• Questions often hide the answer in the cap or floor. Check whether the total coupon rate breaches the limit.
• Know who benefits: cap helps the issuer, floor helps the investor, collar combines both.
• Match the period fraction to the reset frequency before multiplying.
• Know that LIBOR has been replaced by overnight risk-free rates such as SOFR, SONIA and €STR, and that these are transaction-based.
• An inverse floater's coupon moves opposite to the reference rate. Its price falls when rates rise, and it typically has a longer effective duration than a comparable fixed-rate bond.
• Most questions are three-option MCQs. Use the margin comparison to eliminate options on the wrong side of par before calculating.
• Watch the wording: required margin and discount margin are used for the same measure. Quoted margin is the contractual one.
• Check the payment frequency. Annual margins divided by the wrong m cause wrong answers that appear as options.