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Yield and Yield Spread Measures for Floating-Rate Instruments

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Topic 8: Yield and Yield Spread Measures for Floating-Rate Instruments

Learning Objectives Coverage

LO1: Calculate and interpret yield spread measures for floating-rate instruments

Core Concept

  • Definition: Floating-rate notes (FRNs) have coupons that reset periodically based on a reference rate plus a quoted margin. The key spread measure is the discount margin (DM), which is the spread required by the market to price the FRN at its current value.
  • Why it matters: FRNs protect investors from interest rate risk while still exposing them to credit risk. Understanding spread measures helps evaluate whether an FRN is fairly priced relative to its credit risk.
  • Key components:
    • Market Reference Rate (MRR): Usually SOFR, Euribor, or similar
    • Quoted Margin (QM): Fixed spread set at issuance
    • Discount Margin (DM): Required spread by market
    • Reset frequency: Typically quarterly

Formulas & Calculations

  • FRN Coupon Rate:

    Coupon = Market Reference Rate + Quoted Margin

  • FRN Pricing Formula:

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

    Where: MRR = reference rate, QM = quoted margin, DM = discount margin, m = payments per year

  • Pricing Relationships:

    If QM > DM → Price > Par (Premium)
If QM = DM → Price = Par
If QM < DM → Price < Par (Discount)

  • HP 12C Steps (Finding Discount Margin):

    Example: 4-year FRN, quarterly reset, MRR = -0.55%, QM = 250 bps, Price = 97

Step 1: Calculate coupon payment
(-0.55 + 2.50)/4 = 0.4875% per quarter

Step 2: Use TVM keys
97 [CHS] [PV]    (current price)
100 [FV]         (par value)
0.4875 [PMT]     (quarterly payment)
16 [n]           (quarters to maturity)
[i]              (solve for rate = 0.8225%)

Step 3: Annualize and find DM
0.8225 × 4 = 3.29%
DM = 3.29% - (-0.55%) = 3.84% or 384 bps

Practical Examples

  • Traditional Finance Example: Antelas AG 4-year FRN

    • Three-month MRR: -0.55%
    • Quoted margin: 250 bps
    • Price: 97 per 100 par
    • Calculated discount margin: 329 bps
    • Interpretation: Credit deterioration since issuance (DM > QM)

  • Credit Quality Changes:

    • Issuer upgrade: DM falls below QM, FRN trades at premium
    • Issuer downgrade: DM rises above QM, FRN trades at discount
    • No change: DM = QM, FRN trades at par

DeFi Application

  • Protocol example: Aave V3 variable rate borrowing
  • Implementation: Interest rates adjust every block based on utilization formula, similar to FRN resets but continuous
  • Advantages/Challenges:
    • Advantages: Real-time rate adjustment, transparent utilization-based pricing
    • Challenges: Rate volatility, no fixed spread concept like traditional FRNs
    • Key insight: Utilization rate acts like reference rate, protocol risk premium like quoted margin

LO2: Calculate and interpret yield measures for money market instruments

Core Concept

  • Definition: Money market instruments are short-term debt securities (≤1 year maturity) quoted using either discount rates or add-on rates, requiring special calculation methods different from bonds.
  • Why it matters: These instruments form the foundation of short-term funding markets and require conversion to common basis for comparison.
  • Key components:
    • Discount rate instruments (T-bills, commercial paper)
    • Add-on rate instruments (CDs, repos)
    • Day count conventions (360 vs 365)
    • Bond equivalent yield for comparison

Formulas & Calculations

  • Discount Rate Formula:

    PV = FV × (1 - Days/Year × DR)
DR = (Year/Days) × (FV - PV)/FV

  • Add-On Rate Formula:

    PV = FV / (1 + Days/Year × AOR)
AOR = (Year/Days) × (FV - PV)/PV

  • Bond Equivalent Yield (365-day add-on basis):

    BEY = (365/Days) × (FV - PV)/PV

  • Conversion from Discount to Add-On:

    AOR = DR / (1 - Days/Year × DR)

  • HP 12C Steps (Money Market Calculations):

    Example: 90-day T-bill, 0.10% discount rate, 360-day basis

100 [ENTER]      (face value)
90 [ENTER]       
360 [÷]          (days/year fraction)
0.001 [×]        (discount rate)
[-]              (subtract from FV)
Result: 99.975 (price)

For bond equivalent yield:
100 [ENTER] 99.975 [-]  (gain)
99.975 [÷]              (divide by price)
365 [×] 90 [÷]          (annualize)
Result: 0.1014% BEY

Practical Examples

  • Commercial Paper vs CD Comparison:

    90-day commercial paper: 0.100% discount rate (360-day)
Price = 100 × (1 - 90/360 × 0.001) = 99.975
BEY = (365/90) × 0.025/99.975 = 0.1014%

90-day CD: 0.120% add-on rate (365-day)
BEY = 0.120% (already on 365-day basis)

CD offers 1.86 bps more yield

  • CFP Bank CD Example:

    Principal: EUR 20 million
90-day CD at 0.12% (365-day basis)
Redemption = 20M × (1 + 90/365 × 0.0012) = EUR 20,005,918

DeFi Application

  • Protocol example: Compound cTokens and money market rates
  • Implementation: cTokens accumulate interest continuously, similar to add-on rate instruments but with per-block compounding
  • Advantages/Challenges:
    • Advantages: Transparent rates, instant liquidity, no maturity constraints
    • Challenges: Rate volatility, smart contract risk, gas costs for small positions
    • Key difference: DeFi uses continuous compounding vs simple interest in TradFi money markets

Core Concepts Summary (80/20 Principle)

Essential Knowledge (20% that delivers 80% value)

  1. FRN Spread Measures

    • Quoted Margin (QM): Fixed spread set at issuance
    • Discount Margin (DM): Market-required spread
    • Price relationship: QM > DM → Premium; QM < DM → Discount

  2. Money Market Quotations

    • Discount rate: Based on face value (T-bills, CP)
    • Add-on rate: Based on price/principal (CDs, repos)
    • Always convert to bond equivalent yield for comparison

  3. Key Calculations

    • FRN value: Discount at MRR + DM, pay MRR + QM
    • Money market: No compounding, use simple interest
    • Day count matters: 360 vs 365 significantly affects yields

  4. Duration Concepts

    • FRNs: Duration ≈ time to next reset (very low)
    • Money market: Duration < 1 year by definition
    • Interest rate risk minimal for both

Comprehensive Formula Sheet formula

Floating-Rate Notes

Coupon Rate = MRR + QM
Required Return = MRR + DM

FRN Price Formula:
PV = Σ [(MRR + QM) × FV/m] / [1 + (MRR + DM)/m]^t

Pricing Rules:
QM > DM → Price > Par
QM = DM → Price = Par  
QM < DM → Price < Par

Discount Margin Calculation:
Solve for DM using PV, FV, PMT, and N

Money Market Instruments

Discount Rate Instruments:
PV = FV × (1 - Days/Year × DR)
DR = (Year/Days) × (FV - PV)/FV

Add-On Rate Instruments:
PV = FV / (1 + Days/Year × AOR)
AOR = (Year/Days) × (FV - PV)/PV

Bond Equivalent Yield (365-day add-on):
BEY = (365/Days) × (FV - PV)/PV

Conversions:
AOR = DR / (1 - Days/Year × DR)
DR = AOR / (1 + Days/Year × AOR)

Day Count Adjustments

360 to 365 day conversion:
Rate365 = Rate360 × (365/360)

Actual/360 to Actual/365:
Rate(A/365) = Rate(A/360) × (360/365)

HP 12C Calculator Sequences hp12c

Calculate FRN Discount Margin

Example: 2-year FRN, semiannual, MRR = 2%, QM = 150 bps, Price = 98.5

Step 1: Calculate payment
2 [ENTER] 1.5 [+]     (total rate = 3.5%)
2 [÷]                 (semiannual = 1.75%)

Step 2: Find discount rate
98.5 [CHS] [PV]       (current price)
100 [FV]              (par value)
1.75 [PMT]            (payment)
4 [n]                 (periods)
[i]                   (solve = 2.13%)

Step 3: Annualize and find DM
2.13 [ENTER] 2 [×]    (annualize = 4.26%)
4.26 [ENTER] 2 [-]    (DM = 2.26% or 226 bps)

Money Market Discount to Price

Example: 180-day T-bill, 1.5% discount rate, 360-day basis

100 [ENTER]           (face value)
180 [ENTER] 360 [÷]   (time fraction = 0.5)
1.5 [ENTER] 100 [÷]   (discount rate decimal)
[×]                   (discount amount = 0.0075)
[-]                   (price = 99.25)

Money Market Add-On to Future Value

Example: EUR 10M CD, 270 days, 2.4% add-on, 365-day basis

10 [ENTER]            (principal in millions)
270 [ENTER] 365 [÷]   (time fraction)
2.4 [ENTER] 100 [÷]   (rate decimal)
[×]                   (interest rate × time)
1 [+]                 (growth factor)
[×]                   (FV = 10.178M)

Bond Equivalent Yield

Example: 90-day CP at 99.5 price

100 [ENTER] 99.5 [-]  (gain = 0.5)
99.5 [÷]              (return = 0.00503)
365 [ENTER] 90 [÷]    (annualization factor)
[×]                   (BEY = 2.04%)

Practice Problems

Basic Level

  1. FRN Coupon Calculation

    • Q: FRN with 3-month SOFR + 200 bps. SOFR = 3.5%. Find quarterly coupon.
    • A: (3.5% + 2.0%) / 4 = 1.375% quarterly

  2. Money Market Discount Price

    • Q: 91-day T-bill, 0.5% discount rate, 360-day basis. Find price.
    • A: PV = 100 × (1 - 91/360 × 0.005) = 99.874

  3. Add-On Rate Calculation

    • Q: 180-day CD, price 98, par 100, 365-day basis. Find add-on rate.
    • A: AOR = (365/180) × (2/98) = 4.14%

Intermediate Level

  1. FRN Valuation

    • Q: 3-year quarterly FRN, MRR = 1%, QM = 180 bps, DM = 220 bps. Find price.
    • A: Payment = (1% + 1.8%)/4 = 0.7%; Required = (1% + 2.2%)/4 = 0.8% Using calculator: PV = 98.79 (discount since DM > QM)

  2. Money Market Comparison

    • Q: Compare 182-day CP at 1.2% discount (360) vs 182-day CD at 1.25% add-on (365).
    • A: CP BEY = (365/182) × (1.21/98.79) = 2.46% CD BEY = 1.25% (already 365-day) CP offers 121 bps more yield

  3. Discount Margin Change

    • Q: FRN issued at par with QM = 150 bps, now trades at 102. Did DM increase or decrease?
    • A: DM decreased below 150 bps (premium price means DM < QM)

Advanced Level

  1. Complex FRN Analysis

    • Q: 5-year FRN, quarterly reset, MRR = -0.25%, QM = 300 bps, price = 95. Calculate DM and explain pricing.
    • A: Payment = 2.75%/4 = 0.6875% Using N=20, PV=-95, FV=100, PMT=0.6875 Quarterly rate = 1.05%, Annual DM = 4.45% DM (445 bps) > QM (300 bps), indicating credit deterioration

  2. Money Market Arbitrage

    • Q: 270-day T-bill at 2% discount (360) vs 270-day repo at 2.1% add-on (360). Which is better?
    • A: T-bill price = 98.5, BEY = (365/270) × (1.5/98.5) = 2.06% Repo BEY = 2.1% × (365/360) = 2.13% Repo offers 7 bps advantage

  3. FRN Credit Migration

    • Q: FRN with QM = 200 bps. If issuer downgraded, DM rises to 350 bps. MRR = 2%, 2 years remaining. Find new price.
    • A: Coupon = 4%/4 = 1% quarterly Required = 5.5%/4 = 1.375% quarterly N=8, PMT=1, FV=100, i=1.375 PV = 97.04 (2.96% discount)

DeFi Applications & Real-World Examples

Traditional Finance Examples

  1. Corporate FRN Issuance

    • Microsoft FRN: 3-month SOFR + 40 bps
    • Trading at par → DM = QM = 40 bps
    • Strong credit maintains spread stability

  2. Bank Funding Markets

    • JP Morgan 90-day CP: 0.15% discount rate
    • Bank of America 90-day CD: 0.18% add-on rate
    • Reflects slight credit differentiation

  3. Government Money Markets

    • 4-week T-bill: 0.05% discount
    • 13-week T-bill: 0.08% discount
    • Yield curve shape in short-term markets

DeFi Protocol Comparisons

  1. Variable Rate Lending Spreads

    Aave USDC Supply: 3.2% APY
Aave USDC Borrow: 4.8% APR
Spread: 160 bps (protocol revenue)

Similar to FRN where:
- Supply rate = MRR equivalent
- Borrow rate = MRR + spread

  2. Utilization-Based Pricing

    Compound DAI Market:
0-80% utilization: Rate = 2% + 10% × U
80-100% utilization: Rate = 10% + 100% × (U - 0.8)

At 85% utilization:
Rate = 10% + 100% × 0.05 = 15%

  3. Money Market Protocol Yields

    Curve 3pool (DAI/USDC/USDT):
Base APY: 0.5%
CRV rewards: 2.1%
Total APY: 2.6%

Comparable to enhanced money market funds

Innovative DeFi Structures

  1. Frax Finance FRAX/USDC Pool

    • Algorithmic stablecoin paired with fiat-backed
    • Yield varies with FRAX demand
    • Similar to FRN with dynamic reference rate

  2. Liquity LUSD Stability Pool

    • Earns liquidation premiums + LQTY rewards
    • Variable yield based on liquidation frequency
    • Acts like FRN with event-driven margin

  3. MakerDAO DSR (Dai Savings Rate)

    • Set by governance, not market
    • Currently 5% fixed until changed
    • Hybrid between fixed and floating rate

Common Pitfalls & Exam Tips

Frequent Mistakes

  1. Day Count Confusion

    • Error: Using 360-day basis for 365-day instrument
    • Fix: Always identify convention before calculating
    • Example: 2% on 360-day ≠ 2% on 365-day (actually 2.03%)

  2. Discount vs Add-On Mix-up

    • Error: Using discount rate formula for CD
    • Fix: Discount = from FV; Add-on = from PV
    • Remember: T-bills/CP = discount; CDs/repos = add-on

  3. FRN Spread Misunderstanding

    • Error: Confusing QM with DM
    • Fix: QM is fixed at issue, DM is market-required
    • Key: Price deviation from par indicates DM ≠ QM

Exam Strategies

  1. Quick Identifications

    • FRN at premium → DM < QM (credit improved)
    • FRN at discount → DM > QM (credit deteriorated)
    • Money market < 1 year maturity always

  2. Calculator Shortcuts

    • Store 365/360 = 1.0139 for conversions
    • For small rates: AOR ≈ DR × (1 + DR × Days/Year)
    • BEY always uses 365-day add-on basis

  3. Time Savers

    • If comparing money markets, convert all to BEY
    • For FRNs, if price = par, then DM = QM
    • Negative rates: Same formulas, just negative inputs

Conceptual Traps

  1. Simple vs Compound Interest

    • Money markets: Simple interest only
    • Bonds/FRNs: Compound interest
    • Never mix methodologies

  2. Reset vs Payment Frequency

    • Can differ (quarterly reset, semiannual payment)
    • Use reset frequency for duration estimate
    • Use payment frequency for cash flow timing

  3. Spread Stability Assumptions

    • DM changes with credit quality
    • QM never changes (fixed at issuance)
    • Market conditions affect DM, not QM

Key Takeaways

Must-Know Concepts

  1. FRN pricing depends on relationship between QM and DM
  2. Money market instruments use simple interest, not compound
  3. Day count conventions significantly impact yields
  4. Bond equivalent yield enables comparison across instruments
  5. Duration of FRNs approximately equals time to next reset

Critical Formulas

  • FRN Coupon: MRR + QM
  • Discount Rate: DR = (Year/Days) × (FV-PV)/FV
  • Add-On Rate: AOR = (Year/Days) × (FV-PV)/PV
  • BEY: (365/Days) × (FV-PV)/PV

DeFi Applications

  • Variable lending rates act like continuously resetting FRNs
  • Utilization models replace traditional reference rates
  • Protocol spreads similar to bank net interest margins
  • No standard money market conventions yet in DeFi

Cross-References & Additional Resources

DeFi Resources

Practice Platforms

  • Bloomberg Terminal: FRN function for floating-rate analysis
  • Refinitiv Eikon: Money market monitors
  • DeFi Pulse: Protocol rate tracking
  • Dune Analytics: Utilization and rate dashboards

Review Checklist

Conceptual Understanding

  • Can explain difference between quoted and discount margin
  • Understand FRN pricing relationships (QM vs DM)
  • Know money market quotation conventions
  • Can identify when to use simple vs compound interest

Calculation Proficiency

  • Calculate FRN price given margins
  • Find discount margin from market price
  • Convert between discount and add-on rates
  • Calculate bond equivalent yields
  • Price money market instruments

Application Skills

  • Compare different money market instruments
  • Interpret FRN price changes
  • Apply concepts to DeFi variable rates
  • Recognize impact of day count conventions

DeFi Integration

  • Understand utilization-based rate models
  • Can compare DeFi lending to FRNs
  • Know how protocol spreads work
  • Can evaluate cross-protocol rate differences

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