FI 8 Yield and Yield Spread Measures for Floating-Rate Instruments
Earlier lessons priced bonds that carry a fixed coupon and a time-to-maturity of one year or more. This lesson widens the field to two related families. Floating-rate instruments pay a variable rather than a fixed coupon, and money market instruments mature within a year of issue. Both matter to issuers and investors alike.
Because a floater resets its cash flows as interest rates move, it carries less price risk than an otherwise similar fixed-rate instrument. That property makes floaters useful for hedging exposures and for matching the cash flows of assets against those of liabilities. Most loans are, in fact, floating-rate instruments. Money market instruments, for their part, are a large source of short-term funding: a short maturity lets investors reinvest and issuers refinance quickly, which trims interest rate risk on both sides.
The vocabulary of a floater
A floating-rate instrument derives its interest from an observed market reference rate (MRR) to which a quoted margin is added. The interest payment resets on scheduled dates, so each reset captures whatever change has occurred in the MRR. The quoted margin is a fixed spread set above or below the reference rate. The required margin (its other name is the discount margin) is the spread investors truly demand.
The required margin reflects bottom-up, issuer-specific and security-specific risk, so it plays the same role for a floater that a yield spread plays for a fixed-rate bond. One consequence is worth committing to memory: when a floater trades at par value, its quoted margin and its required margin are equal.
| Margin relationship | Price on the reset date |
|---|---|
| Required margin greater than quoted margin | Discount |
| Required margin equal to quoted margin | Par |
| Required margin less than quoted margin | Premium |
How money market instruments are quoted
Money market instruments follow different conventions from longer-dated securities. Their yields are quoted on either a discount rate basis or an add-on rate basis. A money market discount rate divides interest income by the face value, or maturity value, of the instrument. That construction understates the return an investor truly earns whenever the purchase price sits below the face value, and it overstates it in the opposite case. To make money market quotes comparable with each other and with longer-term securities, they can be converted onto a common basis, a task the second half of this lesson works through in detail.
Three floating-rate notes (FRNs) were issued at par value, each using the three-month MRR as its reference rate.
| FRN | Quoted margin | Discount margin |
|---|---|---|
| FRN 1 | 140 bps | 128 bps |
| FRN 2 | 145 bps | 145 bps |
| FRN 3 | 150 bps | 165 bps |
Floating-rate notes and most loans differ from fixed-rate bonds in one central respect: their coupons change from period to period with the level of a reference interest rate. That design does two things at once. It keeps a borrower’s base rate aligned with market conditions, and it hands the lender less price risk when rates move. In principle a floater holds a stable price even through volatile rates, because its cash flows adjust; a fixed-rate security, with cash flows locked in, must instead reprice.
On a floater the reference rate is usually set at the start of a period and the interest is paid at the end, a structure described as paying in arrears. Accrued interest is most often computed on an actual/360 or actual/365 day count. Consider the four-year notes issued by Antelas AG.
| Term | Detail |
|---|---|
| Issuer | Antelas AG |
| Principal amount | EUR250 million |
| Maturity | Four years from settlement |
| Interest | MRR plus 250 bps per year |
| Reset and payment | MRR reset quarterly, interest paid quarterly in arrears |
| Seniority | Secured, unsubordinated, ranking pari passu with like debt |
| Business days | Frankfurt |
The 250 bps added to the MRR is the quoted margin. Its purpose is to compensate the investor for the gap between the issuer’s credit risk and the risk implied by the reference rate itself.
Yes. A firm whose credit risk is lower than that embedded in the reference rate may be able to borrow at a negative quoted margin, paying the MRR minus a spread rather than plus one.
The required margin and the pull to par
The required margin is the spread over or under the reference rate that would price the floater at par value on a reset date, and the market determines it. Suppose a floater is issued at par paying MRR plus 0.50 percent, a quoted margin of 50 bps. If the issuer’s credit risk does not change, the required margin stays at 50 bps and the floater prices at par on every reset date. Between dates its flat price drifts to a premium or a discount as the MRR falls or rises, but as long as the required margin keeps matching the quoted margin, that flat price gets drawn back to par as the next reset nears. At the reset, any change in the MRR flows into the next period’s interest.
Shifts in the required margin usually trace back to changes in the issuer’s credit risk, though changes in liquidity or tax status can move it too. These are the same forces that drive the yield spread on a fixed-rate bond. If a downgrade lifts the required margin to 75 bps while the quoted margin stays at 50 bps, the issuer now pays a deficient interest amount, and the floater trades at a discount. That discount is the present value of a 25 bps per period annuity, the gap between the required and quoted margins, over the remaining life. Were the required margin instead to drop to 40 bps, the floater would trade at a premium equal to the present value of a 10 bps per period excess payment.
Fixed-rate and floating-rate bonds are alike when credit risk changes. On a fixed-rate bond the premium or discount reflects the distance between its coupon rate, which is fixed, and the yield-to-maturity investors require; on a floater it reflects the distance between the quoted margin, which is fixed, and the required margin. The two diverge sharply, however, when benchmark interest rates move, because the floater’s coupon adjusts and the fixed-rate bond’s does not.
A simplified FRN pricing model
Valuing a floater needs a pricing model. Recall that a fixed-rate bond with market discount rate r and a level payment per period, PMT, is priced by discounting each cash flow. For a floater the payment depends on the MRR together with the quoted margin, while the discount rate depends on the MRR together with the discount margin. Combining those ideas gives a simplified FRN pricing model.
Here PV is the price of the floater; MRR is the market reference rate, expressed as an annual percentage rate (sometimes labelled the Index); QM is the annual quoted margin; FV is the par value returned at maturity; m is the periodicity, meaning how many payment periods fall in a year; DM is the discount, or required, margin stated as an annual rate; and N is the count of evenly spaced periods left until maturity. Because MRR, QM, and DM are annual figures, each is divided by m. In compact form the numerator and the periodic rate are:
The model is deliberately simple. It prices as of a reset date with N evenly spaced periods left, so there is no accrued interest and the flat price equals the full price. It assumes a 30/360 day count so that m is a whole number, even though most floaters use actual/360 in practice. Most important, it applies the same MRR to every cash flow, whereas fuller models place projected forward rates on top and spot rates underneath. The discount margin it produces is therefore an estimate that depends on these assumptions.
Price a two-year, semiannual FRN paying MRR plus 0.50 percent. The MRR is 1.25 percent and investors require a spread of 40 bps. So MRR is 0.0125, QM is 0.0050, FV is 100, m is 2, DM is 0.0040, and N is 4.
Backing out the discount margin from a price
Given a market price, the same model can be inverted to estimate the discount margin. The practical route is to solve for the periodic discount rate first, using a financial calculator or a rate function with the periodic payment as PMT, then read DM out of the relationship r equals (MRR plus DM) divided by m.
Take a two-year FRN paying MRR plus 0.75 percent semiannually, with MRR at 1.10 percent and a price of 95.50, a discount driven by worsening credit. Setting the sum of the present values equal to 95.50 and solving for DM returns 3.12 percent, or 312 bps. At issue investors required only 75 bps; after the downgrade they require an estimated 312 bps, so the fixed 75 bps quoted margin is deficient by 237 bps per period. As the model is simplified, the 312 bps is an estimate.
The Antelas AG four-year note uses a three-month MRR held constant at minus 0.55 percent and a quoted margin of 250 bps. It trades at 97 for each 100 of par. Use a 30/360 day count and evenly spaced periods.
Reset frequency and price sensitivity
A floater’s interest rate sensitivity comes chiefly from the time left until its next reset. A longer gap between resets makes a floater resemble a short-dated fixed-rate security more closely, so its price can swing further, while a shorter gap between resets tames that movement. This is why a five-year floater does not reprice like an otherwise identical fixed-rate bond when rates shift: the answer depends on the coupon reset frequency, not on duration in the fixed-rate sense.
A money market instrument is a debt security whose original maturity does not exceed one year. The category is broad, spanning overnight repurchase agreements (repos), Treasury bills that mature within a year, negotiable bank certificates of deposit, commercial paper, bankers’ acceptances, and time deposits tied to reference rates. Money market mutual funds, which hold only eligible short-term securities, are sometimes treated as an alternative to bank deposits.
Three ways money market yields differ from bond yields
First, a bond yield-to-maturity is annualized and also compounded, whereas money market yields are annualized but left uncompounded; a money market return is stated on a simple interest basis. Second, bond yields-to-maturity are usually quoted for one common periodicity across all maturities, while money market instruments of different maturities carry different periodicities. Third, bond yields-to-maturity come from standard time-value-of-money analysis, but money market instruments are often quoted with non-standard rates that call for their own pricing formulas.
In practice these quotes take one of two forms, a discount rate or an add-on rate. Commercial paper, Treasury bills, and bankers’ acceptances are typically priced off a discount rate, whereas bank certificates of deposit, repos, and reference-rate indexes rely on an add-on rate. In the money market a discount rate applies to an instrument whose interest is folded into its face value, whereas an add-on rate applies interest on top of the principal or investment amount.
Pricing on a discount rate basis
The price of an instrument quoted on a discount rate basis is:
PV is the price; FV is the face value paid at maturity; Days is the count of days from settlement to maturity; Year is the day count of the year; and DR is the annual discount rate.
A 91-day Indian Treasury bill carrying INR10 million of face value is quoted at a 3.45 percent discount rate on a 360-day year.
Isolating the discount rate exposes its quirk:
The first term, Year divided by Days, is the periodicity. The second term is the odd part: it divides the interest earned, FV minus PV, by FV rather than by PV. In theory a rate of return divides earnings by the amount invested (PV), not by the maturity value (FV), which already contains those earnings. So whenever DR is above zero, PV is below FV, and by construction the discount rate understates both the investor’s return and the issuer’s cost of funds.
Pricing on an add-on rate basis
For an add-on quote, the price and the rate are:
Here FV is the redemption amount, principal plus interest, and AOR is the annual add-on rate. Note that the add-on rate divides earnings by PV, the amount actually invested, which is why it states the return honestly.
CFP Bank issues a 90-day certificate of deposit at a quoted add-on rate of 0.12 percent on a 365-day year, with an initial principal of EUR20 million.
| Convention | Discount rate (DR) | Add-on rate (AOR) |
|---|---|---|
| Rate formula | (Year / Days) × (FV − PV) / FV | (Year / Days) × (FV − PV) / PV |
| Quoted amount | Face value at maturity (FV) | Price at issuance (PV) |
| Typical instruments | Commercial paper, Treasury bills, bankers’ acceptances | Bank CDs, repos, market reference rates |
Comparing instruments: the bond equivalent yield
Two frictions make money market comparison awkward: some instruments quote a discount rate and others an add-on rate, and some assume a 360-day year while others use 365. The fix is to convert every quote to a bond equivalent yield, defined as a money market rate expressed on a 365-day add-on basis. The recipe for a discount-rate instrument is to price it, then recompute the rate as a 365-day add-on rate.
An investor compares two 90-day instruments with equal credit risk: commercial paper from Bright Wheel Automotive at a discount rate of 0.100 percent on a 360-day year, and a CFP Bank certificate of deposit quoted at a 0.120 percent add-on rate on a 365-day year.
Four 180-day instruments carry the quotes below. Converting each to a bond equivalent yield lets the investor rank them.
| Instrument | A | B | C | D |
|---|---|---|---|---|
| Quotation basis | Discount | Discount | Add-on | Add-on |
| Days in the year | 360 | 365 | 360 | 365 |
| Quoted rate | 4.33% | 4.36% | 4.35% | 4.45% |
| PV / FV | 97.835 | 97.850 | 102.175 | 102.195 |
| Add-on rate | 0.04487 | 0.04456 | 0.04410 | 0.04450 |
| Bond equivalent yield | 4.487% | 4.456% | 4.410% | 4.450% |
Periodicity and converting to a bond basis
Because bond yields-to-maturity compound, they carry a well-defined periodicity; a semiannual yield, for instance, has a periodicity of two. Money market rates use simple interest, so their periodicity is simply the year’s day count divided by the days left to maturity, and it shifts with maturity. To line a money market rate up with a semiannually compounding bond, use the periodicity conversion:
A 91-day Indian rupee Treasury bill quoted at a bond equivalent yield of 3.50 percent has a periodicity of 365 divided by 91. Converting to a periodicity of two:
So 3.50 percent for a periodicity of 365 divided by 91 corresponds to 3.515 percent on a semiannual bond basis, a difference of minus 1.5 bps. The size of that gap depends on the level of rates: the lower the rate, the smaller the difference between any two periodicities.