Using Clean Prices and Accrued Interest

Why Clean Prices Need an Adjustment

A bond quote may look like the amount an investor pays, but it often excludes the coupon interest that has accumulated since the last coupon date. That quoted amount is the clean price. The cash consideration at settlement includes both the clean price and accrued interest; it is the dirty price. In a bootstrap, the distinction matters because the price in the present-value equation must represent the actual settlement consideration.

The calculation is conceptually short: determine accrued interest under the bond’s stated coupon and day-count conventions, add it to the clean price, and use the resulting dirty price in the bootstrap equation. The details that commonly cause errors are the price basis (per 100 of face value or dollars), the elapsed fraction of the coupon period, and whether accrued interest has already been included.

Definition: For a conventional coupon bond between coupon dates, the clean price excludes accrued interest. The accrued interest is the coupon interest attributed to the elapsed part of the current coupon period under the specified day-count convention. The dirty price, also called the invoice price, is the clean price plus accrued interest: $$ \text{Dirty price}=\text{Clean price}+\text{Accrued interest}. $$

Accrued interest is usually calculated as the coupon payment for the period multiplied by the fraction of that coupon period that has elapsed. The required day-count convention is part of the bond’s terms or the problem data. Do not assume that every bond uses the same convention: a problem may specify, for example, 30/360 or Actual/Actual. The examples below state their conventions explicitly.

From Accrued Interest to a Bootstrap Equation

Suppose a bond has remaining cash flows $C_1,C_2,\ldots,C_n$, occurring at times $t_1,t_2,\ldots,t_n$ from the valuation or settlement date. Let $D(t_j)$ be the discount factor for a dollar payable at time $t_j$. If the bond is valued on the same basis as the quoted settlement price, its dirty price satisfies

P_{\text{dirty}}=\sum_{j=1}^{n} C_jD(t_j).

This is a present-value equation: discount each remaining contractual payment to the valuation date and add the results. When bootstrapping, earlier discount factors may already be known, while the factor for the final cash-flow date is the unknown. The price on the left side must be the dirty price, not the clean quote.

Accrued interest is not an additional future payment by the bond issuer. It is an adjustment between the buyer and seller in the settlement amount. The bond’s remaining coupons and redemption payment are the cash flows on the right side of the bootstrap equation; accrued interest is reflected through the dirty price on the left.

Key result: If a quote is clean, first calculate accrued interest and set $P_{\text{dirty}}=P_{\text{clean}}+\text{AI}$. Then equate that dirty price to the present value of the bond’s remaining cash flows. Keep the face-value basis and currency units consistent on both sides.

Why the Adjustment Works

Between coupon dates, a coupon has been earned over part of its accrual period, even though the issuer will pay the full coupon on the next coupon date. In a standard settlement between coupon dates, the buyer acquires the right to receive that full upcoming coupon. The buyer therefore compensates the seller for the part attributed to the seller’s holding period. That compensation is accrued interest.

The clean quote separates this accrued amount from the market quotation of the bond. But a present-value equation is an equation for the full amount exchanged for the remaining cash flows. Adding accrued interest restores that full amount. In symbols, if the quoted clean price is $P_{\text{clean}}$ and accrued interest is $\text{AI}$, then

P_{\text{clean}}+\text{AI} =\sum_{j=1}^{n}C_jD(t_j).

This also explains the direction of the common error. If accrued interest is positive and the clean price is used as though it were the settlement amount, the left side is too small by exactly the accrued interest. Any discount factor solved from that equation will generally be understated when the unknown factor multiplies a positive final cash flow.

Worked Examples

Worked Example: Convert a Clean Quote Before Solving a Discount Factor

A hypothetical bond has face value $1,000 and pays a 6% annual coupon semiannually. Its clean price is $987.40. Under the specified 30/360 convention, 75 of the 180 days in the current coupon period have elapsed. The next coupon is payable in 105 days, and the bond’s final coupon and $1,000 redemption payment are payable in 285 days. The discount factor for the first payment date, 105 days from settlement, has already been bootstrapped as 0.985. Find the dirty price and the discount factor for the final payment date. Use 360 days per year for the stated time coordinates.

Method and accrued interest. The semiannual coupon is $$ 1000(0.06)/2=\$30. $$ The accrued fraction of the coupon period is $75/180$, so accrued interest is $$ \text{AI}=30\left(\frac{75}{180}\right)=\$12.50. $$ Thus the dirty price is $$ P_{\text{dirty}}=987.40+12.50=\$999.90. $$

Set up the cash flows. The next payment is the $30 coupon. At the final date, the bond pays the $30 coupon plus $1,000 principal, or $1,030. The given time coordinates are $105/360$ and $285/360$ years from settlement. The bootstrap equation is $$ 999.90=30(0.985)+1030D(285/360). $$

Solve. The present value of the first coupon is $$ 30(0.985)=29.55. $$ The amount left for the final cash flow is $$ 999.90-29.55=970.35. $$ Therefore, $$ D(285/360)=\frac{970.35}{1030}=0.942087\ldots. $$ To six decimal places, the final discount factor is $0.942087$. This method avoids the common mistake of inserting the clean quote of $987.40 into the present-value equation. The dirty price is the settlement amount represented by the remaining cash flows.

Check. Repricing with the rounded factor gives $29.55+1030(0.942087)=999.89961$, which rounds to $999.90.

Worked Example: Quantify the Error from Mixing Price Conventions

A hypothetical bond has face value $100, an 8% annual coupon paid semiannually, and a clean price of $101.20. Under 30/360, 45 of the 180 days in the coupon period have elapsed. The next $4 coupon is payable in 135 days, and the final $104 payment is payable in 315 days. Use 360 days per year. The discount factor for the next coupon date is 0.980. Find the correct factor for the final date, then find the factor produced by incorrectly using the clean price directly. State the error.

Method and correct calculation. Each semiannual coupon is $$ 100(0.08)/2=\$4. $$ Accrued interest is $$ \text{AI}=4\left(\frac{45}{180}\right)=\$1.00, $$ so the dirty price is $101.20+1.00=\$102.20$. The next coupon has present value $4(0.980)=\$3.92$. The correct bootstrap equation is $$ 102.20=3.92+104D(315/360). $$ The amount attributable to the final cash flow is $102.20-3.92=\$98.28$, giving $$ D(315/360)=\frac{98.28}{104}=0.945. $$

Incorrect calculation. Using the clean price in place of the dirty price gives $$ 101.20=3.92+104D_{\text{wrong}}(315/360), $$ so $$ D_{\text{wrong}}(315/360)=\frac{101.20-3.92}{104} =\frac{97.28}{104}=0.9353846\ldots. $$ The discount-factor error is $0.945-0.9353846\ldots=0.0096154\ldots$, an understatement. In dollar terms, using the clean price makes the left side $1.00 too small, exactly the accrued interest omitted. This is the exam trap: a clean quote cannot be used as the full price in a cash-flow present-value equation.

Check. The correct equation reprices to $3.92+104(0.945)=\$102.20$. The incorrect equation reprices to $3.92+104(0.9353846\ldots)=\$101.20$, which is the clean quote, not the settlement amount.

Worked Example: Apply the Price Basis Consistently Across Several Cash Flows

A hypothetical bond has face value $500 and pays a 5.2% annual coupon semiannually. Its clean price is $496.75. Under 30/360, 40 of the 180 days in the coupon period have elapsed. The remaining payments are $13 in 140 days, $13 in 320 days, and $513 in 500 days. Use 360 days per year. The previously bootstrapped discount factors for the first two payment dates are 0.989 and 0.967. Find the dirty price and the discount factor for the final date.

Method and accrued interest. The coupon per half-year is $$ 500(0.052)/2=\$13. $$ Thus accrued interest is $$ \text{AI}=13\left(\frac{40}{180}\right)=\$2.888888\ldots. $$ The dirty price is $496.75+2.888888\ldots=\$499.638888\ldots$.

Form the equation. The first two cash flows have present values $$ 13(0.989)=\$12.857,\qquad 13(0.967)=\$12.571. $$ Their combined present value is $12.857+12.571=\$25.428$. The final cash flow is $513, so $$ 499.638888\ldots=25.428+513D(500/360). $$

Solve and check. The amount assigned to the final cash flow is $$ 499.638888\ldots-25.428=474.210888\ldots, $$ and hence $$ D(500/360)=\frac{474.210888\ldots}{513}=0.9243877\ldots. $$ To six decimal places, the factor is $0.924388$. Repricing with this rounded factor gives $25.428+513(0.924388)=499.639044$, which rounds to the dirty price of $499.64. The calculation avoids both omitting accrued interest and treating it as a separate issuer cash flow.

Conventions and Common Traps

The method is reliable when the quote basis, accrued-interest basis, and cash-flow units are aligned. In practice, check these items before solving:

Price per 100 versus dollars. If a clean quote is stated per $100 face value, calculate accrued interest on the same $100 basis before adding. If converting to dollars, multiply both price and accrued interest by the same face-value scale.
Coupon frequency. For a stated annual coupon rate paid $m$ times per year, the coupon per period on face amount $F$ is $F i_{\text{coupon}}/m$, provided the stated coupon convention is nominal annual rate convertible $m$ times. Do not accidentally use the full annual coupon as one period’s coupon.
Day-count convention. Use the specified elapsed fraction and coupon-period fraction. A count such as 75 elapsed days does not by itself determine accrued interest unless the relevant convention and period basis are known.
Sign of the adjustment. For a standard coupon bond between coupon dates, accrued interest is added to the clean price. Subtracting it produces an amount below the quoted clean price and reverses the settlement adjustment.
Double counting. Accrued interest belongs in the dirty price. Do not add it to the dirty price again or insert it as another issuer payment in the bond’s future cash-flow list.

When a bond is quoted exactly on a coupon date, accrued interest is ordinarily zero under the usual coupon-period convention. Between coupon dates, however, the clean and dirty prices generally differ. If a problem explicitly gives a dirty price, use it as given rather than adding accrued interest a second time.

Key takeaway: A clean quote is not the full settlement value. Calculate accrued interest under the stated convention, add it to obtain the dirty price, and use that dirty price as the present value of the remaining bond cash flows. This distinction is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated convention and keep the price basis consistent.

A hypothetical bond has face value $1,000, a 4% annual coupon paid semiannually, and a clean price of $992.35. Under 30/360, 60 of the 180 days in the coupon period have elapsed. Find accrued interest and the dirty price, to the nearest cent. (Answer: accrued interest $6.67; dirty price $999.02)
A hypothetical bond has face value $5,000, a 6% annual coupon paid semiannually, and a clean price of $4,980. Under 30/360, 75 of the 180 days in the coupon period have elapsed. Find the dirty price, to the nearest cent. (Answer: $5,042.50)
A hypothetical $100-face bond pays a 5% annual coupon semiannually. Its clean price is $99.10, and 90 of 180 days have elapsed under 30/360. The next $2.50 coupon is due in 90 days and has discount factor 0.990; the final $102.50 payment is due in 270 days. Find the dirty price and the final discount factor, to four decimal places. (Answer: dirty price $100.35; final discount factor 0.9559)
A hypothetical $100-face bond pays a 6% annual coupon semiannually. Its clean price is $103.40, and 30 of 180 days have elapsed under 30/360. The next $3 coupon has present value $2.94; the final payment is $103. Find the final discount factor, to four decimal places. (Answer: 0.9802)
A hypothetical bond has a clean price of $98.60 and accrued interest of $1.20. In its bootstrap equation, all known earlier cash flows have total present value $5.00, and the unknown discount factor multiplies a final cash flow of $104. Find the correct discount factor and the amount by which the factor would be understated if the clean price were used directly, to four decimal places. (Answer: correct factor 0.9115; understatement 0.0115)

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