Bootstrap Workshop with Mixed Coupon Frequencies

Why Mixed Coupon Frequencies Need Careful Standardization

A bond paying coupons once a year and a bond paying coupons quarterly can both help identify discount factors on the same yield curve. The different schedules do not require different curves. They do require care: before combining prices, put every cash flow on the same time axis, identify whether each quoted price is clean or dirty, and account for accrued interest under the stated convention.

The objective here is to recover discount factors from a sequence of hypothetical bond prices. We use a common valuation date, $t=0$, and measure every payment time in years after that date. Coupon rates determine the size and frequency of contractual payments; discount factors determine how those payments are valued. In particular, a coupon rate convertible quarterly is not itself the quarterly discount rate for the curve.

Definition: Let $D(t)$ be the discount factor at time $t$, so that one dollar payable at time $t$ has value $D(t)$ at the valuation date. For known cash flows $C_j$ payable at times $t_j$, the bond's dirty price is $$ P_{\mathrm{dirty}}=\sum_j C_jD(t_j). $$ A clean price excludes accrued interest. Under a stated accrued-interest convention, the corresponding dirty price is the clean price plus accrued interest.

For these examples, cash flows are certain, there are no taxes or transaction costs, and all prices refer to the same valuation date. When a clean price is quoted, accrued interest is calculated using the convention specified in the problem. Coupon dates and time intervals are given exactly; no calendar-day or business-day adjustments are needed. These assumptions matter: a different settlement date, day-count convention, or price convention can change the cash flows' times or the price that must be bootstrapped.

From Settlement and Coupon Schedule to a Bootstrap Equation

A reliable calculation starts by writing down the dates and amounts of all future payments. Measure each payment time from the common valuation date—not from the bond's issue date or from its previous coupon date. If settlement occurs between coupon dates, include only coupons still payable after settlement. Accrued interest is not an additional future cash flow: it is the adjustment that converts the clean quote into the full settlement value.

The reason the pricing equation works is the value-additivity of certain cash flows under a discount curve. A payment of $C_j$ at $t_j$ has present value $C_jD(t_j)$; replacing each future payment by its present value and adding gives the price. When all but one of the discount factors in that sum are already known, the remaining factor can be isolated. For a bond whose final payment is $R$ at $T$, with earlier known payments $C_j$, this gives:

D(T)=\frac{P_{\mathrm{dirty}}-\sum_{t_j<T}C_jD(t_j)}{R}.

This sequential solve requires the earlier payment-date factors to be known and the final payment coefficient $R$ to be nonzero. A bond with an earlier cash flow at a date whose factor has not been established does not, by itself, permit this one-unknown-at-a-time calculation. Coupon frequency affects which dates and amounts appear in the sum; it does not change the form of the equation.

Standardize the valuation date.
Express every future coupon and redemption date as a time measured from the common settlement or valuation date.

Standardize the price.
If the quote is clean, add accrued interest using the specified convention to obtain the dirty price used in present-value equations.

Write the contractual cash flows.
Use the coupon amount implied by the bond's stated frequency and add redemption to the final coupon payment.

Solve in maturity order.
Subtract the present values of known earlier cash flows, then divide by the final cash flow to obtain the new discount factor.

Check by repricing.
Substitute the solved factors into the original price equation and compare the result with the dirty price, allowing for any stated quote rounding.

Worked Examples

Worked Example: Clean Price with Settlement Between Annual Coupons

A hypothetical bond has face value \$100 and annual coupons of \$8. Settlement is halfway through an annual coupon period. The next coupon is due in 0.5 years and maturity is in 1.5 years. Its clean price is \$101.96, accrued interest is calculated linearly, and $D(0.5)=0.97$. Find $D(1.5)$. State the method and the common mistake to avoid.

Method. Convert the clean price to a dirty price, list the remaining cash flows from settlement, and isolate the factor at maturity. The common mistake is to use the clean price as though it already included accrued interest.

Step 1: Accrued interest. Half of the annual coupon period has elapsed, so linear accrual is $$ 8(0.5)=4. $$

Step 2: Dirty price and cash flows. The dirty price is $101.96+4=105.96$. The remaining payments are \$8 at time 0.5 and \$108 at time 1.5; the latter includes the final \$8 coupon and \$100 redemption.

Step 3: Solve. The present-value equation and its solution are $$ 105.96=8(0.97)+108D(1.5), $$ $$ D(1.5)=\frac{105.96-7.76}{108} =\frac{98.20}{108} =0.9092593\ldots\approx 0.9093. $$ The factor is rounded to four decimal places.

Check. Using the unrounded factor gives $8(0.97)+108(0.9092593\ldots)=7.76+98.20=105.96$, the dirty price. Subtracting the \$4 accrued interest returns the quoted clean price of \$101.96.

Trap shown. Treating \$101.96 as the dirty price would give $(101.96-7.76)/108=0.8722$ to four decimal places. That calculation omits the accrued-interest adjustment and does not reprice the stated settlement value.

Worked Example: Sequential Bootstrap with Semiannual and Annual Coupons

At a common valuation date, suppose $D(0.5)=0.97$. A clean-price quote for an annual-coupon bond halfway through its coupon period, with \$8 annual coupons, clean price \$101.96, and the cash flows described in the preceding calculation, establishes $D(1.5)$. Now use two further bonds, both quoted on coupon dates: a \$100-face bond with a 4% annual coupon paid semiannually, maturity in one year, and price \$97.82; and a \$100-face bond with a 5% annual coupon paid annually, maturity in two years, and price \$97.10. Find $D(1)$ and $D(2)$, then check both prices.

Method. Apply the same dirty-price present-value equation to each bond, solving first at the one-year maturity and then at the two-year maturity. The common trap is to use the annual coupon rate as though it were the payment amount at each semiannual date.

Step 1: Establish the one-year factor. A 4% annual coupon paid semiannually means a coupon of $100(0.04)/2=\$2$ every half-year. The payments are \$2 at 0.5 years and \$102 at 1 year. Because settlement is on a coupon date, there is no accrued interest in this example, so the quoted price is the dirty price. $$ 97.82=2(0.97)+102D(1), $$ $$ D(1)=\frac{97.82-1.94}{102} =\frac{95.88}{102} =0.94. $$

Check. Repricing gives $2(0.97)+102(0.94)=1.94+95.88=\$97.82$.

Step 2: Establish the two-year factor. The annual-coupon bond pays \$5 at year 1 and \$105 at year 2. The latter consists of the final coupon and redemption. The one-year factor is now known: $$ 97.10=5(0.94)+105D(2), $$ $$ D(2)=\frac{97.10-4.70}{105} =\frac{92.40}{105} =0.88. $$

Check. Repricing gives $5(0.94)+105(0.88)=4.70+92.40=\$97.10$. The two bonds have different coupon frequencies, but their prices use the same discount-factor curve and the same valuation date.

Worked Example: Quarterly Coupons and a Rounded Market Quote

A hypothetical \$100-face bond pays coupons quarterly at a nominal coupon rate of 8% convertible quarterly and matures in one year. Its clean price, quoted on a coupon date, is \$102.80. Suppose the earlier factors are $D(0.25)=0.995$, $D(0.50)=0.985$, and $D(0.75)=0.972$. Estimate $D(1)$ and check the result to the precision of the quote.

Method. Convert the nominal coupon rate to the contractual quarterly payment, include redemption with the final coupon, and solve for the one-year factor. The common mistake is to apply the annual coupon rate of 8% as the payment at each quarter.

Step 1: Write the payment schedule. The coupon each quarter is $100(0.08)/4=\$2$. Thus the first three cash flows are \$2, and the final cash flow is \$102 at year 1. Settlement is on a coupon date, so the clean price equals the dirty price here.

Step 2: Discount the known payments and solve. $$ 102.80=2(0.995)+2(0.985)+2(0.972)+102D(1), $$ $$ 2(0.995)+2(0.985)+2(0.972)=1.990+1.970+1.944=5.904, $$ $$ D(1)=\frac{102.80-5.904}{102} =\frac{96.896}{102} =0.9499608\ldots\approx 0.9500. $$ The factor is rounded to four decimal places.

Check. Using $D(1)=0.9500$ gives $5.904+102(0.9500)=5.904+96.900=\$102.804$, which rounds to the quoted \$102.80. The small difference before rounding reflects the price being quoted only to the nearest cent.

Interpreting the Results and Avoiding Frequency Confusion

In a mixed-frequency bootstrap, the key consistency condition is not that every bond pays on the same schedule. It is that each bond's cash flows and price are expressed at the same valuation date and valued using the same set of applicable discount factors. A semiannual bond may supply a cash flow at six months, while an annual bond may supply one at one year; both contribute information to the curve if the required earlier factors are available.

Keep three rates or quantities conceptually separate:

Coupon rate: determines the size of each contractual coupon under the bond's payment-frequency convention.
Accrued interest: adjusts a clean quote to a dirty price under the stated convention. It is not another future coupon.
Discount factor: values a payment at its specific time from the common valuation date. It is not inferred by simply dividing a coupon rate by its payment frequency.

A useful final check is to reprice every input instrument, not only the last bond solved. Use unrounded factors during the calculation, then compare repriced amounts with quoted prices at the quotes' stated precision. If a quote is rounded to cents, a factor inferred from it may not reproduce the exact underlying factor; a repriced amount that agrees to the cent is the appropriate check.

Key takeaway: Standardize settlement first, convert clean prices to dirty prices when required, and write each bond's actual dated cash flows. Then solve for discount factors in maturity order and verify the result by repricing. Coupon frequency changes the cash-flow schedule, not the curve's valuation rule.

Check Your Understanding

For each problem, use the stated price and accrued-interest convention. Round discount factors to four decimal places when needed.

A \$100-face bond pays annual coupons of \$12. Settlement is three-quarters of the way through its annual coupon period; the next coupon is due in 0.25 years and maturity is in 1.25 years. The clean price is \$102, accrued interest is linear, and $D(0.25)=0.98$. Find $D(1.25)$. (Answer: $0.8861$)
A \$100-face bond pays a 5% annual coupon semiannually and matures in one year. Its price on a coupon date is \$98.50, and $D(0.5)=0.96$. Find $D(1)$. (Answer: $0.9376$)
A \$100-face bond pays annual coupons of \$4 and matures in two years. Its price on a coupon date is \$98.35, and $D(1)=0.95$. Find $D(2)$. (Answer: $0.9091$)
A \$100-face bond pays coupons quarterly at a nominal annual coupon rate of 12% convertible quarterly and matures in six months. Its price on a coupon date is \$102.88. Given $D(0.25)=0.99$, find $D(0.5)$. (Answer: $0.9700$)
A \$100-face bond pays annual coupons of \$6. Settlement is halfway through the annual coupon period; the next coupon is due in 0.5 years and maturity is in 1.5 years. Its clean price is \$99, accrued interest is linear, and $D(0.5)=0.96$. Find $D(1.5)$. (Answer: $0.9079$)

What You'll Learn