Fixed Income · Reading 58

Yield and Yield Spread Measures for Floating-Rate Instruments

CFA Level I · Fixed Income · Reading 58 · about 20 min

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What you'll learn

LOS 58.a Calculate and interpret the quoted margin and discount (required) margin of a floating-rate note and estimate its value on a reset date.
LOS 58.b Calculate and interpret money market yields on discount and add-on bases, 360- and 365-day years, and the bond equivalent yield.

Module 58.1

Yield and Yield Spread Measures for Floating-Rate Instruments

This reading covers the spread measures used for floating-rate notes and the yield conventions used for money market instruments. A candidate must be able to value an FRN on a reset date from its quoted and discount margins, solve for the discount margin from a price, and convert money market quotes among discount yields, add-on yields, the bond equivalent yield and a semiannual bond basis.

LOS 58.a — Quoted margin, discount margin and FRN value

How a floating-rate note pays

A floating-rate note (FRN) pays a coupon that is reset each period to the current market reference rate (MRR) plus a fixed spread. The MRR (typically an interbank or overnight-based rate) is set at the start of the period and paid at the end, so interest is paid in arrears. Because the coupon keeps catching up with market rates, an FRN's price is more stable than the price of a fixed-rate bond with the same maturity.

The fixed spread written into the indenture is the quoted margin (QM). It reflects the issuer's credit risk at issuance compared with that of the banks behind the MRR (liquidity and tax treatment can also affect it).
An issuer riskier than those banks pays MRR plus a margin; an issuer with less credit risk than those banks pays MRR minus a margin, so its coupon rate is below the MRR.
The spread investors currently require over the MRR is the required margin, also called the discount margin (DM). It is the margin that would make the FRN price equal to par on a reset date.

QM versus DM: the price rule

Key concept

Situation since issuance	Relation	Price on a reset date
Credit quality unchanged	QM = DM	At par
Credit quality worsened	DM > QM	Below par (discount)
Credit quality improved	DM < QM	Above par (premium)

FRNs are normally issued at par, with QM = DM. As long as the issuer's credit quality is unchanged, QM stays equal to DM throughout the note's life. The passage of time does not open a gap between them. A change in the MRR itself also does not create a gap, because both the coupon (MRR + QM) and the required return (MRR + DM) move with it. The analogy with fixed-rate bonds: coupon rate < required yield means a discount price; for an FRN, MRR + QM < MRR + DM means the QM is deficient and the note trades below par.

Simplified valuation on a reset date

Project every future coupon using today's MRR plus the QM, and discount at today's MRR plus the DM, with the rates divided by the number of periods per year:

V = t = 1 \sum N \frac{( MRR + QM ) \times F V / m}{( 1 + \frac{MRR + DM}{m} ) ^{t}} + \frac{F V}{( 1 + \frac{MRR + DM}{m} ) ^{N}}

Example. A $1 million FRN pays annual coupons of MRR + 60 bp and has three years left. On a reset date the MRR is 2.80% and investors now require a DM of 90 bp.
Coupon $= (2.80% + 0.60%) \times 1, 000, 000 = $34, 000$ ; discount rate $= 2.80% + 0.90% = 3.70%$ .
TI BA II Plus (P/Y = C/Y = 1, END mode): N = 3, I/Y = 3.70, PMT = 34,000, FV = 1,000,000, CPT PV = −991,627. The value is about $991,627, below par because DM > QM. This shortcut ignores expected changes in the MRR; more complex models give better estimates of value.

Solving for the DM. When the price is known, the same formula is run backward:

Coupon per period $= (MRR + QM) \times F V / m$ .
Solve for the periodic discount rate (I/Y) with N, PMT, FV and PV = −price.
Multiply the periodic rate by $m$ to annualize it.
Subtract the MRR. What remains is the DM.

Example. Suppose the same note were priced at $988,000. With $m = 1$ : N = 3, PMT = 34,000, FV = 1,000,000, PV = −988,000, CPT I/Y = 3.831, so $D M = 3.831% - 2.80% = 1.03%$ , about 103 bp. The DM is above the 60 bp QM, as it must be for a note priced below par on a reset date.

Common exam traps

Swapping the roles. The coupon uses the QM; the discount rate uses the DM. In the $1 million example, swapping them values the note at about $1,008,421 instead of $991,627.
Subtracting the MRR from a quarterly periodic rate. Multiply the periodic rate by 4 first, then subtract the MRR.
Assuming an FRN always trades at par on a reset date. This holds only when QM = DM.
Expecting the approach of maturity or a move in the MRR to open a gap between the QM and the DM.
Treating the MRR as a floor on the coupon rate. A lower-risk issuer pays less than the MRR.

LOS 58.b — Money market yield conventions

Money market securities mature in one year or less. Their yields are quoted under two sets of choices:

Choice	Option 1	Option 2
What is annualized	Add-on yield: interest earned ÷ amount invested today	Discount yield: discount from face value ÷ face value
Day-count year	360 days	365 days

Key concept

Annualized add-on rates are the quoting convention for bank CDs, repos and market reference rates.
Annualized discount yields on a 360-day year are the convention for US Treasury bills and commercial paper.
The bond equivalent yield (BEY) of a money market instrument is an add-on yield based on a 365-day year.

Key formulas: money market yields

Holding period yield: $H P Y = \frac{amount received at maturity}{price paid} - 1$

Key concept

Add-on yield (365-day) = H P Y \times \frac{365}{days} Discount yield = \frac{F V - P}{F V} \times \frac{360}{days}

The 360 in the discount-yield formula becomes 365 when an instrument is quoted as a discount yield on a 365-day year.

Rearranged for pricing, the two conventions run in opposite directions. A discount yield starts from the face value and works back to today's price. An add-on yield starts from the amount invested today and works forward to the amount received at maturity:

P = F V \times (1 - discount yield \times \frac{days}{year}) F V = P V \times (1 + add-on yield \times \frac{days}{year})

Here "year" is 360 or 365 days, whichever basis the quote uses. Converting an add-on yield from a 360-day to a 365-day basis: multiply by $365/360$ .

Because a discount yield divides by face value (larger than the price) and uses a 360-day year, the BEY of a discount instrument priced below face value is higher than its quoted discount yield.

Example 1. A 60-day T-bill is quoted at a 3.00% discount yield.
Discount $= 3.00% \times 60/360 = 0.50%$ , so the price is 99.50 per 100 of face value.
$H P Y = 100/99.50 - 1 = 0.5025%$ ; $B E Y = 0.5025% \times 365/60 = 3.057%$ , above the 3.00% quote.

Example 2. A 45-day interbank deposit quoted at a 2.00% add-on yield on a 360-day basis has a BEY of $2.00% \times 365/360 = 2.028%$ .

Comparing with a semiannual-pay bond

To compare a money market yield with the YTM of a semiannual-pay bond (a semiannual bond basis yield): find the HPY, compound it to an effective annual yield, convert to an effective semiannual rate and double it:

E A Y = (1 + H P Y)^{365/ days} - 1, semiannual bond basis = 2 [(1 + E A Y)^{1/2} - 1]

Example 3. A 200-day deposit quoted at 3.60% add-on (365-day basis): $H P Y = 3.60% \times 200/365 = 1.9726%$ ; $E A Y = 1.01972 6^{365/200} - 1 = 3.6293%$ ; semiannual bond basis $= 2 (1.03629 3^{0.5} - 1) = 3.597%$ . With a holding period longer than half a year (periodicity below 2), the semiannual bond basis yield is slightly below a simple 365-day add-on quote; with a holding period shorter than half a year it is slightly above.

Common exam traps

Annualizing an HPY by $days /365$ instead of $365/ days$ . For the 60-day T-bill in Example 1, that gives 0.083% instead of a BEY of 3.057%.
Using 360 instead of 365 when the question asks for a BEY. For the T-bill in Example 1, this gives 3.015% instead of 3.057%.
Treating a discount yield as a return on the price: a discount yield is computed on face value.
Compounding when the BEY is required. The BEY is a simple (non-compounded) annualization. For the T-bill in Example 1, compounding gives 3.096% instead of 3.057%.

Exam shortcuts

An FRN priced below par on a reset date must have DM > QM, and one priced above par must have DM < QM, so answer choices on the wrong side of the quoted margin can be eliminated without solving for I/Y.
An add-on yield quoted on a 360-day basis converts to a 365-day basis, the BEY, by multiplying by 365/360; the holding period yield is not needed.

Bottom line

An FRN's coupon rate is reset each period to the market reference rate set at the start of the period plus the quoted margin, and the coupon is paid at the end of the period, in arrears.
The quoted margin is fixed in the indenture and reflects the issuer's credit risk at issuance, while the discount (required) margin is the spread investors currently require, the margin that would make the FRN worth par on a reset date.
On a reset date an FRN trades at par when QM = DM, below par when DM > QM because credit quality has worsened, and above par when DM < QM because credit quality has improved.
While the issuer's credit quality is unchanged, the QM stays equal to the DM; neither the passage of time nor a change in the MRR opens a gap between them.
The simplified FRN valuation projects each coupon at today's MRR plus the QM and discounts at today's MRR plus the DM, both divided by the periods per year, and it ignores expected changes in the MRR.
US Treasury bills and commercial paper are quoted as discount yields on a 360-day year, computed on face value, while the convention for bank CDs, repos and market reference rates is an annualized add-on rate.
A money market instrument's BEY annualizes the add-on return over a 365-day year, $H P Y \times 365/ days$ , and for a discount instrument priced below face value it is higher than the quoted discount yield.
To compare a money market yield with a semiannual-pay bond's YTM, compound the HPY to an effective annual yield, $(1 + H P Y)^{365/ days} - 1$ , and restate it as $2 [(1 + E A Y)^{1/2} - 1]$ .

Quick check

Question 1Core

Carrow Logistics has a $5 million floating-rate note outstanding that pays an annual coupon equal to the market reference rate (MRR) plus a quoted margin of 110 basis points. Today is a reset date with exactly four years remaining to maturity. The MRR is 4.20%, and the discount margin the market currently requires on Carrow's credit is 70 basis points. Using the simplified approach that holds the MRR constant, the estimated value of the note is closest to:

Show answer and explanation

Correct answer: A

Coupons are projected at MRR + QM and discounted at MRR + DM. The QM (110 bp) exceeds the DM (70 bp), so the note is worth more than par.

Coupon rate $= 4.20% + 1.10% = 5.30%$ , so the annual coupon $= 0.053 \times $5, 000, 000 = $265, 000$ .

Discount rate $= 4.20% + 0.70% = 4.90%$ .

V = t = 1 \sum 4 \frac{265 , 000}{1.04 9 ^{t}} + \frac{5 , 000 , 000}{1.04 9 ^{4}} = $5, 071, 084

TI BA II Plus (P/Y = C/Y = 1, END mode): N = 4, I/Y = 4.90, PMT = 265,000, FV = 5,000,000, CPT PV = −5,071,084.

Why the other options are wrong

B. $5,000,000 assumes the FRN must trade at par on a reset date. That holds only when the quoted margin equals the discount margin; here QM > DM.
C. $4,929,572 swaps the two margins: coupons at 4.20% + 0.70% = 4.90% discounted at 4.20% + 1.10% = 5.30%, which produces a discount price.

Key takeaway Coupons come from the QM and the discount rate from the DM. If QM > DM the note trades at a premium; if QM < DM, at a discount.

Practice Questions

Question 2Core

Tenby Savings Bank issues a 91-day negotiable certificate of deposit. An investor who deposits $1,000 today receives $1,006.50 at maturity. The certificate's bond equivalent yield is closest to:

Show answer and explanation

Correct answer: B

A CD is an add-on instrument: the investor deposits the full amount and receives it back with interest. The BEY annualizes the add-on return for the 91 days using a 365-day year, without compounding.

H P Y = \frac{1 , 006.50}{1 , 000} - 1 = 0.65%

B E Y = 0.65% \times \frac{365}{91} = 2.607%

Why the other options are wrong

A. 2.57% annualizes the 0.65% return on a 360-day year ( $0.65% \times 360/91$ ); a BEY uses 365 days.
C. 2.63% compounds the period return into an effective annual yield ( $1.006 5^{365/91} - 1$ ). The BEY is a simple annualization with no compounding.

Key takeaway The BEY is a simple add-on annualization on a 365-day year, with no compounding and no 360-day year.

Question 3Core

A floating-rate note issued by Penhale Chemicals pays the market reference rate plus 85 basis points. Dealers now price the note at a discount margin of 55 basis points. Which of the following is the most likely explanation for the difference?

Show answer and explanation

Correct answer: B

The quoted margin (QM) is the fixed spread, set in the indenture, that is combined with the MRR to determine each coupon (it can be positive or negative). The discount (required) margin (DM) is the spread over the MRR at which the note would be priced at par on a reset date. Because the QM was fixed at issuance for the credit risk investors saw then, a DM below the QM means investors now demand less compensation, most likely because the issuer's credit quality has improved. With QM > DM, the note trades at a premium.

Why the other options are wrong

A. A change in the reference rate flows into both the coupon (MRR + QM) and the required return (MRR + DM), so it does not open a gap between the two margins.
C. When the discount margin is below the quoted margin, the coupon exceeds the required return and the note trades above par.

Key takeaway DM below QM points to improved credit and a premium price; DM above QM points to deteriorated credit and a discount.

This reading has 15 questions in the full bank. Practice all of them.

Key Takeaways

Coupons come from the QM and the discount rate from the DM. If QM > DM the note trades at a premium; if QM < DM, at a discount.
The BEY is a simple add-on annualization on a 365-day year, with no compounding and no 360-day year.
DM below QM points to improved credit and a premium price; DM above QM points to deteriorated credit and a discount.