The Sequential Logic of Curve Bootstrapping

Why Bootstrapping Is Sequential

A coupon bond pays more than one cash flow. Its price reflects the values of payments at several dates, not just the redemption amount at maturity. To infer the discount factor for a later date from that price, the values of the earlier payments must already be known. This is the central logic of curve bootstrapping: establish the value of the earliest cash-flow dates first, then use those values to work out the next maturity.

The method does not assume that one annual interest rate applies to every payment date. Instead, each date has its own discount factor. A one-year payment is valued using the one-year discount factor; a two-year payment is valued using the two-year discount factor. With suitable bond prices at successive maturities, the earlier parts of each bond’s price can be calculated, leaving the new terminal discount factor to be determined.

Definition: Let $D(t)$ be the time-0 price of a certain payment of $1 at time $t$, with no other payments. A payment of $C$ dollars at time $t$ therefore has time-0 value $C D(t)$. Bootstrapping is the sequential process of using market prices of known cash flows to infer discount factors at new payment dates.

The examples use hypothetical default-free bonds, valued immediately after a coupon date. Each bond has annual coupons, a face amount of $100, and a quoted price per $100 face amount. Prices are assumed to be present values at the same valuation date, with no accrued interest or embedded options. All scheduled payments are known and certain. These assumptions let us focus on the discounting logic; if a bond’s cash flows or price convention differ, those details must be accounted for before applying the method.

Why Earlier Maturities Must Be Known First

The price of a bond is the sum of the values of its promised payments. This follows from value additivity: a portfolio of separate claims to the bond’s payments replicates the bond’s cash flows date by date. If each separate claim is worth its payment amount times the discount factor for that date, their total value is the bond’s price.

P=\sum_{j=1}^{n} C_jD(t_j)

Here $P$ is the bond price, $C_j$ is the payment at time $t_j$, and $D(t_j)$ is the discount factor for that date. For a coupon bond with maturity $T$, the last payment includes both the final coupon and redemption. If the discount factors for all earlier coupon dates are known, their contributions can be subtracted from the price. The remaining amount is the value of the terminal payment.

D(T)= \frac{P-\sum_{t_j<T} C_jD(t_j)}{C_T}

This calculation requires $C_T\ne 0$, and the earlier payment dates must have known discount factors. A bond maturing at a new date does not, by itself, reveal that date’s discount factor if some of its earlier cash-flow dates are still unvalued. In that case, the earlier values must first be established using other instruments or additional information.

Worked Examples

Worked Example: Establishing the First Discount Factor

A hypothetical one-year bond has face amount $100, pays a 5% annual coupon, and is priced at $94.50 immediately after its issue date. Find the one-year discount factor. State the method and verify the result.

Method. This is the first maturity in the available sequence, so there are no earlier coupon dates to value. The bond makes one payment at year 1: its $5 coupon plus $100 redemption.

The terminal cash flow is

$$ C_1=0.05(100)+100=5+100=105. $$

A certain $105 payment at year 1 has price $105D(1)$. Equate that value to the observed bond price and solve:

94.50=105D(1), \qquad D(1)=\frac{94.50}{105}=0.90.

Check. Discounting the payment back gives $105(0.90)=94.50$, exactly the quoted price. The common mistake avoided here is to divide the price by the $100 face amount: the bond pays $105 at maturity, not $100.

Worked Example: Using the Known One-Year Value

Suppose the one-year discount factor has been established as $D(1)=0.90$. A hypothetical two-year bond has face amount $100, an 8% annual coupon, and a price of $93.60. Find $D(2)$, and check the bond price.

Method. Value the first coupon using the known one-year discount factor. Then subtract its value from the bond price and divide the remainder by the year-2 payment.

The bond pays $8 at year 1 and $108 at year 2, since the final payment includes the $8 coupon and $100 redemption. The year-1 coupon is worth

$$ 8D(1)=8(0.90)=7.20. $$

The part of the price attributable to the year-2 payment is therefore $93.60-7.20=86.40$. Thus,

108D(2)=86.40, \qquad D(2)=\frac{86.40}{108}=0.80.

Check. Reconstruct the price from both cash flows:

$$ 8(0.90)+108(0.80)=7.20+86.40=93.60. $$

The quoted price is recovered exactly. The mistake to avoid is treating the full $93.60 as the value of the year-2 payment. That would ignore the year-1 coupon, whose value is already determined.

Worked Example: Following a Three-Maturity Sequence

A hypothetical set of annual-coupon bonds has the following prices per $100 face amount: a one-year bond with a 4% coupon costs $99.84; a two-year bond with a 5% coupon costs $99.30; and a three-year bond with a 6% coupon costs $98.08. Find the discount factors at years 1, 2, and 3. Assume all bonds are valued at the same date, immediately after a coupon date.

Method. Work from the shortest maturity to the longest. Each step uses previously determined discount factors to value earlier coupons, leaving the terminal payment to identify the next discount factor.

Year 1. The one-year bond pays $4+100=104$ at year 1. Therefore,

D(1)=\frac{99.84}{104}=0.96.

Year 2. The two-year bond pays $5 at year 1 and $5+100=105$ at year 2. Its year-1 coupon is worth $5(0.96)=4.80$. The remaining price is $99.30-4.80=94.50$, so

D(2)=\frac{94.50}{105}=0.90.

Year 3. The three-year bond pays $6 at year 1, $6 at year 2, and $6+100=106$ at year 3. The first two coupons are worth $6(0.96)=5.76$ and $6(0.90)=5.40$. Together, they are worth $5.76+5.40=11.16$. The remaining price is $98.08-11.16=86.92$, giving

D(3)=\frac{86.92}{106}=0.82.

Check. Reconstruct all three prices in order:

$$ 104(0.96)=99.84, $$ $$ 5(0.96)+105(0.90)=4.80+94.50=99.30, $$ $$ 6(0.96)+6(0.90)+106(0.82)=5.76+5.40+86.92=98.08. $$

Each reconstructed price matches its quote. The common mistake avoided is discounting the first- and second-year coupons using $D(3)$ simply because the bond matures in year 3. Each payment is discounted using the factor for its own date.

A Practical Sequence and Its Limits

The sequential procedure can be organized as a short calculation plan. It is important to keep the payment dates attached to the cash flows throughout: a coupon’s date determines which previously established factor applies.

List the dated payments.
Record each coupon and the final redemption payment, including the date of every cash flow.

Start with the earliest unvalued date.
Use an instrument whose cash flows can be valued from known factors, with the shortest maturity often providing the first factor.

Value earlier payments.
Multiply each known payment by the discount factor for its own date and add these present values.

Isolate the new terminal payment.
Subtract the earlier payments’ present values from the quoted price, then divide by the terminal payment to find its discount factor.

Verify the price.
Use the factors just found to revalue every payment. The sum should reproduce the quoted bond price.

This logic works directly when the available instruments provide prices for the needed dates and the earlier cash flows can be valued. A gap in the available maturities is not automatically filled by the next bond quote: if that bond has coupons at dates whose discount factors are unknown, its price includes those unknown values as well. More information is then required before the terminal factor can be isolated. Similarly, inconsistent prices may prevent one coherent set of discount factors from reproducing every quote.

Common Exam Trap: Discounting Every Payment at the Bond’s Maturity

A frequent error is to use the maturity-date discount factor for every payment in a coupon bond. In the three-maturity example, that would treat all three-year bond payments as if they were paid at year 3. The resulting calculation would be

D(3)\text{ (incorrect)}= \frac{98.08}{6+6+106} =\frac{98.08}{118} \approx 0.8312.

This is incorrect because the $6 coupons arrive at years 1 and 2, not year 3. Their present values use the already established factors $D(1)=0.96$ and $D(2)=0.90$. The correct calculation is

D(3)= \frac{98.08-6(0.96)-6(0.90)}{106} =\frac{98.08-5.76-5.40}{106} =\frac{86.92}{106} =0.82.

A useful check is to label every payment with its time before substituting any discount factor. The maturity of a bond identifies its last payment date; it does not make all of its coupons payments at that date.

Key takeaway: Bootstrapping proceeds one payment date at a time. Value a coupon bond’s earlier cash flows using discount factors already established, subtract those values from the bond price, and use the remainder to infer the discount factor for the terminal payment. Then reprice the bond as a check.

Check Your Understanding

For each hypothetical bond, assume annual coupons, face amount $100, valuation immediately after a coupon date, and no accrued interest. Work from the stated earlier discount factors where provided.

A one-year bond with a 3% coupon is priced at $97.85. Find the one-year discount factor. (Answer: $D(1)=0.95$)
Given $D(1)=0.95$, a two-year bond with a 4% coupon is priced at $97.40. Find $D(2)$, showing which part of the price comes from the year-1 coupon. (Answer: $D(2)=0.90$)
Given $D(1)=0.95$ and $D(2)=0.90$, a three-year bond with a 5% coupon costs $98.50. Find $D(3)$, accounting separately for both earlier coupons. (Answer: $D(3)=0.85$)
For the bond in question 3, a candidate divides $98.50 by the sum of all its payments, $115, to estimate $D(3)$. Explain why this calculation is wrong and give the correctly calculated factor. (Answer: It applies a year-3 factor to earlier coupons; subtract $5(0.95)+5(0.90)$ first, giving $D(3)=(98.50-4.75-4.50)/105=0.85$.)
A three-year annual-coupon bond has a known price, but neither $D(1)$ nor $D(2)$ is available, and there are no other bond prices or instruments. Can its price alone determine $D(3)$? (Answer: No; the price contains unknown values for the year-1 coupon, year-2 coupon, and year-3 payment, so additional information is needed.)

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