Yield measures and day count — Bonds and Interest Rates (the part your calculator pretends is easy)

If bonds were people, yield measures would be their resumes and day count conventions would be the weird small-print font nobody reads until it's a disaster.

You're coming in hot from "Bond cash flows and conventions" and the Bayesian stew of "Risk, Return, and Probability." So: you already know how coupons flow and how to price cash flows. Now we make those cash flows speak fluent yield — and we teach them to keep time like a sensible calendar.

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What is "yield" here, and why does day count matter?

Yield is a way to express a bond's expected return as an annualized rate. But there are different flavors depending on assumptions: current yield, yield to maturity (YTM), yield to call, money-market yields, spot/zero yields, etc. Each answers a different question (short-term snapshot, whole-life internal rate of return, best-case call scenario, etc.).

Day count conventions determine how we compute fractions of coupon periods and accrual — they turn the messy reality of calendars into the clean inputs yields need. Different markets use different rules; get them wrong and your accrued interest, dirty price, and yield estimates are all off.

Why this matters for you as an investor/manager: yield measures feed directly into expected return estimates used in portfolio optimization, risk-adjusted performance comparisons, and utility computations from earlier modules. If you feed baked-in calendar mistakes into mean-variance calculations, your efficient frontier politely lies to you.

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Key yield measures (and what question each answers)

• Current yield — quick and dirty: annual coupon / price. Good for a snapshot of coupon income, bad at showing total return.

  • Formula: Current yield = Annual coupon / Price

• Yield to Maturity (YTM) — the IRR assuming you hold to maturity and reinvest coupons at the YTM. The workhorse metric for fixed-income comparisons.

  • Bond price equation (solve for y):

Price = Σ_{t=1}^{N} (C_t / (1 + y)^t) + (Face / (1 + y)^N)

• In practice you solve this numerically (financial calculator or root-finding).

• Quick approximation (useful for intuition):

Approx YTM ≈ (Annual coupon + (Face - Price) / n) / ((Face + Price) / 2)

• Yield to Call / Yield to Worst — like YTM but assume issuer calls the bond at the earliest call date. Use when callable features exist.

• Spot/Zero yields — yields on zero-coupon instruments (bootstrapped from coupon bonds). Crucial for discounting cash flows consistently and for forward-rate curve construction.

• Holding Period Return / Realized Compound Yield — actual ex-post return when you sell or reinvest coupons at realized rates. Important for comparing realized performance with ex-ante YTM assumptions.

• Money market vs Bank Discount yields — used for T-bills and short-term instruments. They look similar but are computed differently (T-Bill bank discount rate uses face value as base; money market yield annualizes based on actual days and price).

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Day count conventions: the boring rules that change money math

Different markets use different conventions. Here are the ones you’ll see most often and what they do to accruals.

Convention	Use cases	How it counts	Effect on accrual
Actual/Actual (ISMA, ACT/ACT)	Government & many corporate bonds	Actual days between dates / actual days in coupon period (or actual days in year depending on spec)	Most accurate for long-dated bonds
30/360 (US)	Many corporate and muni markets	Assumes 30 days per month, 360 days per year	Smooths odd-months; widely used in practice
30E/360 (European)	Some Euro corporates	Slight day-roll rule variations	Slightly different accruals for month-ends
Actual/360	Money markets (e.g., LIBOR conventions)	Actual days / 360	Produces slightly higher annualized rates (denominator smaller)
Actual/365 (Fixed)	Some retail products	Actual days / 365	Slightly lower annualized rates versus 360 base

Quick numeric example (semiannual coupon)

• Par = 100, annual coupon = 5% => semiannual coupon = 2.5
• Days since last coupon = 120
• Coupon period length: Actual/Actual: 182 days; 30/360: 180 days

Accrued interest:

• Actual/Actual: 2.5 * 120 / 182 ≈ 1.648
• 30/360: 2.5 * 120 / 180 = 1.667

Not huge? Multiply that small difference across millions or across many trades and you get meaningful P&L — and incorrect YTM if you use dirty/clean price inconsistently.

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Clean price vs Dirty price (and why day count is part of the argument)

• Dirty price = full price = present value of future cash flows (includes accrued interest).
• Clean price = quoted market price = dirty price − accrued interest.

If you calculate accrued interest with the wrong day-count, you'll mis-state the clean price and thus misinterpret the market quote. Many desks quote clean but settle on dirty; don't be the person who confuses the two during a repo or settlement window.

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Practical tips, common mistakes, and diagnostic checklist

• Always confirm the market's day-count convention before computing accrual or quoting yields.
• Be explicit about coupon frequency: semiannual vs annual changes YTM compounding assumption.
• When comparing yields, compare apples to apples: YTM vs YTM, money-market yield vs money-market yield.
• Watch out for settlement dates: the day count uses actual settlement-to-coupon intervals, not trade date arithmetic.
• For callable bonds, compute yield to worst: take min(YTM, YTCs).

Diagnostic checklist:

1. Have I used the correct day-count for accrual? (Yes / No)
2. Is my price clean or dirty? (If clean, have I added accrued interest for YTM calc?)
3. Is coupon reinvestment rate assumed equal to YTM? (If not, expected realized yield differs.)
4. If comparing instruments, are compounding conventions consistent?

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Mini worked example: Why YTM ≠ realized return (again)

Suppose you buy a 2-year semiannual 5% coupon bond at par (100). YTM = 5% obviously. But if you reinvest coupons at a lower rate (say 2%), your realized compound yield will be lower. This ties back to our earlier module on expected returns and utility: the YTM is an ex-ante IRR with a reinvestment assumption — your subjective utility and risk aversion matter when you judge realized welfare from the investment.

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Closing (keep this on your cheat-sheet)

• Yield measures answer different practical questions — pick the one aligned with your decision (snapshot income vs whole-life IRR vs callable considerations).
• Day count conventions are small rules with big consequences; get them right or your yield, accrued interest, and dirty/clean prices lie to you.

Final one-liner you can tattoo on your risk model: A yield is only as honest as the day count under its feet.

Tie-back to previous modules: use correct yield inputs when you compute expected returns for portfolio optimization and when feeding returns into utility-based decisions. Mis-specified yields produce biased expected return estimates and risk-adjusted performance metrics — and that’s how small accounting errors end up changing allocation decisions.

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If you want, I can:

• Give you a ready-to-run Excel layout for price → YTM (with correct day-count handling), or
• Show how to bootstrap a zero curve from coupon bonds step-by-step (because spot rates are the real MVPs for discounting cashflows).

Choose your next chaos.