Why Same-Maturity Quotes Need to Agree on the Terminal Factor
Bootstrapping uses observed bond prices to infer discount factors. If earlier discount factors are already known, a bond maturing at time \(T\) supplies an equation for the discount factor at \(T\). A second bond with the same maturity supplies another equation for that same factor. The bonds may have different coupons or redemption values, but under a single consistent pricing model their quotes must imply the same terminal discount factor.
That requirement is a diagnostic, not a reason to force every input to fit. If two quotes imply different factors, at least one input, convention, or modeling assumption needs attention. The discrepancy may reflect a calculation error, differing credit or liquidity characteristics, or a quote that has not been converted to the correct price basis. This comparison is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The setup assumes the bonds are valued on the same date and in the same currency, their stated cash flows and payment dates are correct, and the same discount curve applies to both. It also assumes prices have been expressed on a consistent basis. In particular, a dirty price includes accrued interest; a clean price excludes it. If one bond has material credit risk or a different liquidity profile from the other, a single common curve may not be an appropriate model for both.
Why the Implied-Factor Formula Holds
A bond’s price is the sum of the present values of its promised cash flows. Separate the terminal payment from the earlier payments:
Once the earlier discount factors are known, the earlier cash flows have a known present value. Subtract that value from the price. What remains must be the present value of the terminal cash flow, \(KD(T)\). Dividing by \(K\) isolates the discount factor:
For a coupon bond with redemption value \(R\) and a final coupon \(c_T\), the terminal cash flow is \(K=R+c_T\), not just \(R\). This formula does not determine a unique factor when the terminal cash flow is zero. For ordinary coupon bonds, the terminal payment is positive, so division is valid.
Worked Example: Two Quotes That Imply Different Factors
At a valuation date, the one-year discount factor is \(D(1)=0.95\). Two annual-coupon bonds both mature at year 2 and each has a redemption value of 100. Bond A pays an annual coupon of 6 and has a price of 100. Bond B pays an annual coupon of 10 and has a price of 103. Treat the quotes as exact dirty prices. Find each bond’s implied \(D(2)\), and assess whether one discount curve can fit both exactly.
Method. Write each bond’s price as the present value of its year-1 coupon plus its terminal payment. The year-2 cash flow includes both the final coupon and redemption.
Bond A. Its year-1 coupon has present value \(6(0.95)=5.70\). Its terminal cash flow is \(100+6=106\). Therefore:
Bond B. Its year-1 coupon has present value \(10(0.95)=9.50\). Its terminal cash flow is \(100+10=110\). Therefore:
The factors differ by approximately \(0.889623-0.850000=0.039623\). If all the stated assumptions hold and the prices are exact, there is no single \(D(2)\) that satisfies both price equations. The conflicting inputs cannot both be used as exact observations in this one-curve setup. Common mistake avoided: the final coupon is part of the terminal cash flow; omitting it gives the wrong denominator.
Check the Quote Basis Before Comparing Factors
A clean-price quote is not the price to insert directly into a present-value equation when accrued interest is due. First convert it to a dirty price by adding the accrued interest stated under the bond’s coupon and day-count conventions:
The following example uses a common earlier payment date with known discount factor \(D(t_1)=0.96\), and a shared later maturity date \(T\). Accrued interest is supplied explicitly, so no day-count calculation is needed. The cash flows and prices are in dollars per 100 of face value.
Worked Example: Reconciling a Clean Quote with a Dirty Quote
Bond A pays 5 at the earlier payment date and 105 at maturity. Its clean quote is 98.60, and its accrued interest is 0.70. Bond B pays 8 at the same earlier date and 108 at the same maturity; its quote of 104.88 is a dirty price. Given \(D(t_1)=0.96\), determine the factor each quote implies.
Method. Put both prices on a dirty-price basis, then subtract the present value of the earlier coupon and divide by the terminal cash flow.
For Bond A, the dirty price is \(98.60+0.70=99.30\). The earlier coupon’s present value is \(5(0.96)=4.80\). Thus:
Bond B is already quoted dirty. Its earlier coupon’s present value is \(8(0.96)=7.68\), so:
The factors agree. To see the consequence of misreading Bond A’s quote, using its clean price directly gives \((98.60-4.80)/105=93.80/105\approx0.893333\). That false discrepancy comes from omitting accrued interest, not from inconsistent prices. Common mistake avoided: the quote label “clean” or “dirty” is part of the input and must be checked before solving.
Check Redemption Values and Other Cash-Flow Details
Same maturity does not mean same terminal cash flow. A bond’s terminal amount is the sum of its last coupon and its redemption value. If redemption differs from par, using par instead of the stated redemption value changes the implied factor. The same care applies to coupon amounts, payment dates, and whether a quoted price is per 100 of face value or expressed in another unit.
Worked Example: A Non-Par Redemption Changes the Denominator
Two bonds share a maturity date. The discount factor at their common earlier coupon date is \(D(t_1)=0.97\). Bond A pays a coupon of 4 at \(t_1\), pays a final coupon of 4 at maturity, redeems at 100, and is priced at 95.40. Bond B pays a coupon of 8 at \(t_1\), pays a final coupon of 8 at maturity, redeems at 105, and is priced at 107.20. Treat both prices as dirty prices. Find the implied terminal factors and show the result of incorrectly treating Bond B as redeeming at 100.
Method. Use the stated redemption amount when forming each terminal cash flow. The terminal cash flows are \(104\) for Bond A and \(113\) for Bond B.
For Bond A, the earlier coupon’s present value is \(4(0.97)=3.88\). Hence:
For Bond B, the earlier coupon’s present value is \(8(0.97)=7.76\). Its terminal cash flow is \(105+8=113\), so:
The factors agree. If the stated redemption of 105 is mistakenly replaced by 100, the denominator becomes \(100+8=108\), and the erroneous calculation is \((107.20-7.76)/108=99.44/108\approx0.920741\). Common mistake avoided: do not assume redemption at par merely because the bond’s price is quoted per 100 of face value.
Interpreting a Conflict Rather Than Forcing a Fit
When two correctly assembled equations yield different terminal factors, first check the inputs systematically. A small discrepancy may be due to rounding or bid-ask differences. A larger discrepancy can point to an incorrect clean-to-dirty adjustment, a wrong coupon frequency, an omitted final coupon, an incorrect redemption amount, or cash flows assigned to the wrong dates. It may also indicate that the bonds should not be discounted on the same curve, for example because their credit or liquidity characteristics differ.
Check that both prices refer to the same valuation or settlement date, currency, and price convention.
Confirm coupon amounts and dates, redemption values, and the full terminal payment, including the final coupon.
Discount earlier cash flows with their applicable factors, then solve for the terminal factor separately for each bond.
If the results differ, identify which assumption or input may explain the gap; do not label the inputs an exact fit.
A tempting response is to average the two implied factors and report the average as the exact bootstrap result. That does not satisfy either original price equation in general. A fitted curve can be produced by a separate estimation procedure when quotes are noisy or inconsistent, but its result is an estimate under that procedure, not an exact solution to all the quoted prices. Also, a difference between implied factors is evidence of inconsistency under the stated common-curve model; by itself, it does not establish an executable arbitrage. Trading costs, short-sale constraints, credit risk, liquidity, and quote availability matter to that stronger conclusion.
Check Your Understanding
For each problem, use the stated earlier discount factor and include the final coupon in the terminal cash flow.
- A bond has an earlier coupon of 4, a final coupon of 4, redemption of 100, price 97.48, and known earlier factor \(D(t_1)=0.97\). Find its implied maturity factor. (Answer: \(D(T)=0.900000\))
- A bond pays 7 at the earlier date and 7 at maturity, redeems at 100, and costs 100.81. If \(D(t_1)=0.95\), find its implied maturity factor. (Answer: \(D(T)=0.880000\))
- Two bonds share a maturity date and \(D(t_1)=0.96\). Bond X pays 5 at each coupon date, redeems at 100, and costs 98. Bond Y pays 9 at each coupon date, redeems at 100, and costs 101. Find both implied maturity factors and state whether they match. (Answer: Bond X, \(93.20/105\approx0.887619\); Bond Y, \(92.36/109\approx0.847339\); they do not match.)
- A bond pays 6 at an earlier date and 6 at maturity, redeems at 100, and has a clean quote of 101.89 with accrued interest of 0.45. If \(D(t_1)=0.98\), find the implied maturity factor. (Answer: Dirty price \(102.34\); \(D(T)=(102.34-5.88)/106=0.910000\))
- A bond pays 3 at an earlier date and 3 at maturity, redeems at 110, and costs 98.87. If \(D(t_1)=0.94\), find its implied maturity factor. Also state the factor obtained by incorrectly assuming redemption at 100. (Answer: Correct \(D(T)=(98.87-2.82)/113=0.850000\); incorrect \(D(T)=96.05/103\approx0.932524\))