Fixed Income Mechanics

A bond deal can look “cheap” on a screen and still lose money the moment you try to explain it. The missing link usually sits in one place: the cash flows and the discounting. When you know how bond prices come from discounted cash flows, you can translate a quoted yield into a price, sanity-check a trade, and convert between market conventions like clean and dirty prices without getting surprised.

Nadia, 34, a rates analyst at a bank, learned that the hard way. She once compared two bonds with the same coupon rate and similar maturities; one looked richer in price terms, and the other looked cheaper. The trade never made sense until she rebuilt the price from first principles: coupon and principal cash flows discounted at the yield consistent with the deal. The difference turned out not to be “the bond’s quality,” but the timing of accrual and how the market reports the number.

After this chapter, you will be able to: (1) break a bond into its dated cash flows, (2) discount each cash flow at a chosen yield to produce a theoretical price, and (3) move between clean price and dirty price using accrual intuition so your quotes line up with how desks actually trade.

Bond prices as discounted cash flows (and where clean vs dirty fits)

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)....