Fixed Income Mechanics

Start with the core pricing idea: a bond price equals the present value of all its future payments - coupons and redemption - discounted back to today using a yield you specify. The yield plays the role of a discount rate: when yields rise, the discount rate gets higher, and the present value drops; when yields fall, the present value rises.

For a plain-vanilla fixed-rate bond with annual or semiannual coupons, you can write the price mechanically as a sum. If coupon payments occur at times \(t_1, t_2, \dots, t_n\) and the bond redeems face value \(F\) at maturity \(t_n\), then the dirty price (the full value including accrued interest) is:

\[ \text{Dirty Price} = \sum_{i=1}^{n} \frac{C_i}{(1+y)^{t_i}} + \frac{F}{(1+y)^{t_n}} \]

Here \(y\) represents the yield you use for discounting, and \(C_i\) represents the coupon amount paid at each date. In practice, market conventions define how you interpret \(y\) (compounding frequency, day count, and whether you use a spot curve or a single yield). The mechanics stay the same: discount each dated cash flow.

Now bring in the clean vs dirty split. Most trading screens quote a clean price: it excludes the interest that has already accrued since the last coupon date. The market still pays that accrued interest to the seller when you buy, so the dirty price equals:

\[ \text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest} \]

Accrued interest depends on the fraction of the coupon period that has elapsed. If you buy right after a coupon date, accrued interest sits near zero; if you buy near the next coupon date, accrued interest sits near the full coupon amount for that period. That timing difference can easily move the screen quote even when the underlying discounted value hasn’t changed.

To make this concrete, use a semiannual coupon bond with face value \(F = 100\) and coupon rate 6%. That means each semiannual coupon is \(C = 3\). Suppose the bond pays on March 1 and September 1, and today is May 15. The next coupon is September 1, so you have accrued some portion of the March - September coupon. If the market quotes clean price at 101.20, you compute accrued interest (a fraction of 3 based on the day count convention) and add it to get dirty price. Only the dirty price matches the discounted cash-flow value you compute from the yield.

Discount-Flow Ladder is the workflow that keeps this straight. You ladder the cash flows by date, discount each rung, sum to get dirty price, then peel off accrued interest to report clean price - or reverse the steps when you start from a clean quote.

Putting it into practice: build the ladder and reconcile a quote

Use this workflow when you want to price from a yield or reconcile a screen quote. The steps work whether you build in Excel, a pricing tool, or your own scratch model.

1. Write the dated cash flows and amounts - List each coupon payment date and the coupon amount for that bond, plus the maturity redemption (face value). - Example: Face value \(F=100\), 6% annual coupon paid semiannually → coupon per period \(C=3\). If maturity has four remaining coupon dates, you’ll list four coupons of 3 and a final redemption of 100 at the last date.

2. Convert “today” to a time grid using the bond’s day count and coupon schedule - Compute the accrual fraction for accrued interest and the year fractions \(t_i\) you need for discounting. - Example: If today is May 15 and your coupon period runs March 1 to September 1, you compute the fraction elapsed of that period using the relevant day count convention (for many markets, Actual/Actual or Actual/360/365; you follow the bond’s spec). - If you use a simple “per period” approach, the discounting time increments match the coupon frequency; if you use exact year fractions, you use the day-count-based \(t_i\).

3. Discount each cash flow at the chosen yield to get present values, then sum - Compute PV for each coupon: \(\text{PV}_i = \frac{C_i}{(1+y)^{t_i}}\). - Compute PV for redemption at maturity and add it. - Sum all PVs to get dirty price in the same price basis as your yield (per 100 of face, typically).

4. Reconcile with market quoting by adding or removing accrued interest - If the market gives clean price, compute accrued interest and add it to get dirty price before comparing to your discounted sum. - If the market gives dirty price, subtract accrued interest to report clean. - Example logic: if accrued interest equals 0.90 (meaning 0.90 of the next coupon has already “earned” since the last coupon date), then dirty = clean + 0.90.

Here’s a realistic scenario with numbers that match how desks check quotes. Assume a semiannual 6% coupon bond with face value 100. Coupons are 3 each period. Today lies in the current coupon period such that accrued interest equals 0.90. The bond has two coupon dates remaining: one in 3 months and maturity in 9 months (so you will discount two coupon payments and the redemption).

You see a screen quote: clean price = 101.20. You want to know what yield that implies, or at least whether your internal model agrees.

1. Start from the clean quote and compute dirty - Dirty price = 101.20 + 0.90 = 102.10.

2. Build the remaining cash flows - At the first remaining date: coupon = 3. - At maturity: coupon = 3 and redemption = 100 → total at maturity = 103.

3. Choose the yield you want to test - Start with a guess yield \(y\) consistent with your convention. If you assume semiannual compounding, you can treat the discount factor per half-year. If your yield is quoted with semiannual compounding, keep it consistent with your ladder timing.

4. Discount and sum to match dirty - Compute PV of first coupon: \(\frac{3}{(1+y)^{t_1}}\) where \(t_1\) corresponds to 3 months. - Compute PV of maturity cash flow: \(\frac{103}{(1+y)^{t_2}}\) where \(t_2\) corresponds to 9 months. - Adjust \(y\) until PV sum equals 102.10 (or until the model’s clean price matches 101.20 once you subtract accrued interest).

5. Expected outcome - If your yield guess produces dirty PV near 102.10, the quote aligns with the discounted cash-flow mechanics. - If it doesn’t, you likely misapplied accrued interest, used the wrong coupon schedule, or mismatched the yield convention (compounding frequency or day count).

Quick checklist you can run in minutes: - Confirm the coupon schedule and remaining coupon dates. - Compute accrued interest from the spec (day count and period fraction). - Convert clean to dirty (clean + accrued interest) before comparing to discounted PV. - Discount each remaining cash flow at the yield using consistent time fractions. - Sum PVs to get dirty; subtract accrued interest if you need clean for comparison.

The key differentiator in this workflow is that you never compare a discounted PV (which naturally includes “value as of settlement,” i.e., dirty) directly to a clean quote without reconciling accrued interest.

What to watch for: common breakdowns that break the price

Once you can build a Discount-Flow Ladder, you can also spot why your numbers drift. Most mispricings come from convention mismatches or timing errors, not from “mysterious market behavior.”

Accrued interest off by a coupon fraction If you compute accrued interest using the wrong day count or the wrong coupon period boundaries, you will shift clean vs dirty by a noticeable amount. Even a small error in accrued interest can make a two-basis-point yield move look like a much bigger mispricing when you reconcile prices.

Do this: compute accrued interest from the bond’s declared coupon schedule and day count convention, and confirm the accrual fraction lies between 0 and 1 for the current coupon period. Then convert clean ↔ dirty before you compare to a discounted PV sum. Not this: discount cash flows to a dirty value and then compare it directly to a clean quote on the screen without adding or subtracting accrued interest.

Yield convention mismatch (compounding and time grid) A yield quoted with semiannual compounding does not behave the same way as a yield you discount using annual compounding or the wrong exponent schedule. If you plug the yield into the wrong compounding basis, the PV sum won’t hit your target clean or dirty price, and you’ll “solve” for a yield that looks different only because your discounting math changed.

Do this: match the yield convention in your ladder to the bond’s market convention (compounding frequency and how you interpret the yield with respect to the time grid). Keep the time exponents consistent with how your tool defines \(t_i\). Not this: treat every yield as if it discounts with the same compounding frequency regardless of the bond’s spec, or mix exact year fractions with a period-based compounding assumption.

Schedule and settlement timing errors Coupon dates and settlement date matter. If you use the wrong “next coupon” relative to settlement, you’ll shift which cash flows you include and when you discount them. That error often shows up as a systematic bias: your model always prices too high (you included a coupon too early) or too low (you skipped a coupon date).

Do this: verify the next coupon date and the last coupon date relative to settlement, then build the ladder from the first cash flow strictly after settlement (unless your convention includes payment-on-settlement handling). Not this: reuse a schedule from an earlier trade or assume “maturity month = last coupon month” without checking whether the bond has stub periods or nonstandard first/last coupon dates.

The takeaway is simple and practical: bond pricing becomes predictable once you treat the price as a sum of discounted dated cash flows and you respect the clean/dirty split through accrued interest. When you keep those two pieces aligned, you stop arguing about “cheap vs rich” and you start measuring whether the yield you’re trading actually reproduces the market quote. That discipline carries straight into how you move from a single yield to term structures and how intermediaries quote, hedge, and pass through risk.

End of chapter one. 7 more chapters in the full book.