Bond Math Cheat Sheet

A one-page reference for the formulas, conversions, and decision trees in this chapter.

Key takeaways

• Print or bookmark this page for quick formula lookup during research or portfolio management
• Spreadsheet functions (PRICE, YIELD, DURATION) automate most calculations
• Conversion from 32nds to decimals is a common practical need
• Tax-adjusted yield and spread analysis are the most frequent professional calculations
• Rounding errors compound; use precise interest-day counts in production work

Price and Yield Formulas

Present Value (Bond Price)

Price = Σ [C / (1 + y)^t] + FV / (1 + y)^n

Where:
C = coupon per period
y = yield per period (annual yield / 2 for semi-annual)
t = period (1, 2, ... n)
FV = face value
n = total number of periods

Yield-to-Maturity (YTM)

Solve the above equation for y given price and coupon. Use spreadsheet function: =YIELD(settlement, maturity, coupon_rate, price, face_value, frequency)

Current Yield

Current Yield = Annual Coupon / Current Price
Example: $50 coupon / $980 price = 5.10% current yield

Accrued Interest (at settlement)

Accrued Interest = Coupon × (Days Since Last Coupon / Days in Coupon Period)
Example: $50 coupon × (30 days / 181 days) = $8.29

Tax and Return Adjustments

After-Tax Yield

After-Tax Yield = Nominal Yield × (1 − Marginal Tax Rate)
Example: 5% yield × (1 − 0.37 federal − 0.038 NIIT) = 5% × 0.592 = 2.96%

Real Yield (inflation-adjusted)

Real Yield ≈ Nominal Yield − Inflation Rate
(More precise: Real rate = (1 + nominal) / (1 + inflation) − 1)
Example: 4% nominal − 2.5% inflation = 1.5% real yield

Tax-Equivalent Yield (for municipal bonds)

Tax-Equivalent Yield = Tax-Free Yield / (1 − Marginal Tax Rate)
Example: 3.5% muni yield / (1 − 0.458 marginal rate) = 6.46% taxable equivalent

Spreads and Credit Metrics

Yield Spread (simple)

Spread = Bond Yield − Benchmark Yield (usually Treasury of same maturity)
Example: Corporate yielding 5% − Treasury 3.5% = 1.5% (150 basis points) spread

Option-Adjusted Spread (OAS)

Requires pricing model; spreadsheets don't compute this. Use Bloomberg, Tradeweb, or bond analytics platforms.

Break-Even Inflation Rate (TIPS vs. Nominal)

Implied Inflation = Nominal Yield − TIPS Yield
Example: 10-year Treasury at 3.5% − 10-year TIPS at 0.8% = 2.7% implied inflation

Duration and Interest-Rate Sensitivity

Duration (Macaulay Duration)

Weighted-average time to receive cash flows. Spreadsheet: =DURATION(settlement, maturity, coupon_rate, yield, frequency)

Modified Duration

Modified Duration = Macaulay Duration / (1 + Yield per Period)
Price sensitivity: % price change ≈ −Modified Duration × Yield Change %
Example: 6-year modified duration × 1% rate rise = ~6% price decline

Convexity (advanced)

Adjusts for non-linear price-yield relationship. Typically small for most bonds; ignore in rough calculations.

32nds Conversion (Treasury and Corporate Bonds)

32nds Fraction	Decimal	% of Par	$ per $1,000 Par
0/32	0.000	0.000%	$0.00
2/32	0.0625	0.0625%	$0.625
4/32	0.125	0.125%	$1.25
8/32	0.250	0.250%	$2.50
16/32	0.500	0.500%	$5.00
24/32	0.750	0.750%	$7.50

Quick conversion: price = whole number % + (32nds / 32) × 1%

Example: 102-16 = 102% + (16/32)% = 102.5% = $1,025 per $1,000 par

Spreadsheet Functions Quick Reference

Function	Purpose	Syntax
PRICE	Calculate bond price from yield	=PRICE(settlement, maturity, coupon, yield, redemption, frequency)
YIELD	Calculate yield from price	=YIELD(settlement, maturity, coupon, price, redemption, frequency)
DURATION	Calculate Macaulay duration	=DURATION(settlement, maturity, coupon, yield, frequency)
MDURATION	Calculate modified duration	=MDURATION(settlement, maturity, coupon, yield, frequency)
ACCRINT	Calculate accrued interest	=ACCRINT(issue, first_interest, settlement, coupon_rate, par, frequency)
PV	Present value of cash flows	=PV(rate, nper, pmt, [fv])

Parameters:

• settlement: today's date (or your analysis date)
• maturity: the bond's maturity date
• coupon: annual coupon rate as a decimal (3.5% = 0.035)
• yield / price: annual yield or price (decimal or $)
• redemption: face value (usually 1,000)
• frequency: 1 (annual), 2 (semi-annual), 4 (quarterly)

Bond Type Snapshot

Type	Coupon	Price	Key Risk	Tax Treatment	Best Account Type
Treasury	Fixed	Market	Interest-rate risk	Ordinary income	Taxable (safe; no default)
Investment-Grade Corporate (AAA–BBB)	Fixed	Market	Credit + rate risk	Ordinary income	Tax-advantaged (avoid 40%+ tax)
High-Yield (BB–C)	High	Variable	Default risk	Ordinary income	Tax-advantaged or taxable w/ diversification
Municipal (tax-exempt)	Fixed	Market	Credit + liquidity risk	Federal tax-exempt	Taxable (high-bracket investors)
TIPS	Fixed (real)	Adjusted	Inflation	Real return	Taxable (phantom income on adjustment)
STRIPS (zero-coupon)	0%	Deep discount	Interest-rate risk	Phantom income (OID)	Tax-advantaged (Roth, 401k)
Floating-rate	Variable	Stable	Reinvestment risk	Ordinary income	Taxable (immune to rate risk)

Decision Tree: Which Bond to Buy?

1. Asset allocation: How much bonds vs. stocks?
2. Duration target: How much interest-rate sensitivity?
3. Credit quality: AAA, A, BBB, or higher-yielding?
4. Account type: Taxable, Roth, 401k, 529?
5. Tax bracket: Low (<22%), medium (22–32%), high (35%+)?
6. Time horizon: 1–5 years, 5–10 years, 10+ years?
7. Income need: Current income, reinvestment, or growth?

Example pathway:

• 40% bond allocation → Target 5-year average duration.
• Taxable account, 35% marginal rate → Use tax-advantaged account for high-coupon bonds if possible.
• High-bracket earner → Consider municipal bonds (break-even: 3.5% muni = ~5.4% taxable).
• 20-year horizon, retirement account → Buy low-coupon or zero-coupon; tax drag irrelevant.

Common Mistakes to Avoid

1. Comparing yields without adjusting for risk: A 7% high-yield bond isn't 75% better than a 4% Treasury — the risk is different.
2. Ignoring taxes: A 5% taxable bond yielding 3.25% after-tax is worse than a 3.5% tax-free municipal for a high-bracket investor.
3. Chasing yield without understanding the bond: Buy-and-hold requires you to trust the credit quality and understand why the coupon is high.
4. Confusing OID and market discount: The tax treatment differs; check Form 1099-OID to know which you own.
5. Forgetting accrued interest: The settlement price includes accrued interest; the quote doesn't.
6. Overweighting duration: Long duration means high interest-rate sensitivity; understand your risk tolerance before buying 10+ year bonds.

Typical Range Values (2024–2025 Market Context)

Metric	Typical Range	Notes
Treasury 10-year yield	3.5–4.5%	Policy-rate dependent
Investment-grade corporate spread	1.0–2.5%	Tightens in strong economies
High-yield spread	3.0–6.0%	Widens in recessions
High-yield default rate	2–5% per year	Varies with credit cycle
TIPS real yield (10-year)	0.5–2.0%	Compensates for inflation uncertainty
Municipal bonds (AAA) spread vs. Treasury	0–0.5%	Tight due to tax exemption value
Treasury bid-ask spread (10-year on-the-run)	2–4 basis points	Varies with Fed operations

Glossary of Terms

• Basis point (bps): 0.01%. 100 bps = 1%.
• On-the-run: The most recently issued security of a given maturity; most liquid.
• Off-the-run: Older securities; less liquid, wider spreads.
• YTM: Yield-to-Maturity. The annualized return if held to maturity.
• YTC: Yield-to-Call. The yield if the bond is called before maturity.
• OAS: Option-Adjusted Spread. Spread accounting for embedded call/put options.
• Par: Face value, usually $1,000.
• Clean price: Price excluding accrued interest (what's quoted).
• Dirty price: Price including accrued interest (what you pay at settlement).
• Convexity: Curvature of the price-yield relationship. Most bonds have positive convexity (beneficial).
• Callability: The issuer's right to repay the bond before maturity (usually if rates fall).
• Subordination: Priority in bankruptcy; subordinated bonds have lower recovery rates.

Scenario-Based Quick Answers

"I want current income." → High-coupon bonds (4%+ coupon). Check credit quality. Place in tax-advantaged account if possible.

"I expect rates to fall." → Long-duration bonds (8+ years) or zeros. Higher duration amplifies price gains. Avoid callable bonds.

"I expect rates to rise." → Short-duration bonds (2–4 years) or floating-rate bonds. Minimize price risk.

"I'm in a high tax bracket." → Tax-advantaged accounts for regular bonds. Taxable account for municipal bonds or Treasury STRIPS (phantom income).

"I want to compare a muni to a corporate." → Compute the muni's tax-equivalent yield. Compare to the corporate's after-tax yield.

"The bond is trading cheap (wide spread)." → Understand why. Credit stress? Liquidity drought? Opportunity (if temporary) or value trap (if structural)?

Print-Friendly Summary

One-line formulas for quick reference:

• Price = PV of all cash flows at the yield rate
• YTM = the discount rate that makes price = PV of coupons + principal
• Current Yield = annual coupon / price
• After-tax yield = nominal yield × (1 − tax rate)
• Real yield = nominal yield − inflation
• Duration = weighted avg time to get cash back
• Spread = bond yield − Treasury yield
• Price ≈ −Duration × Yield Change %

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Related concepts

• ./13-pricing-a-bond-with-pv.md
• ./14-spreadsheet-bond-math.md
• ./19-yield-spread-basics.md

How it flows

Next

This chapter has covered the four numbers on a bond ticket (coupon, face value, maturity, yield) and the mathematics and spreadsheet tools used to evaluate them. You're now equipped to read bond markets, compare bonds fairly, and understand the trade-offs in your portfolio. The next chapter dives deeper into duration and convexity — the tools that professional bond investors use to quantify and manage interest-rate risk.