Fixed-rate bond contract and cash flows

What you will be able to do

After this lesson, you should be able to:

identify the terms that determine a toy fixed-rate bond’s payments;
calculate the level coupon amount;
construct the complete promised cash-flow schedule.

What a bond is

A bond is a debt claim. The issuer is the borrower; the bondholder is the lender who owns the claim. In the fixed-rate example used here, the issuer promises a sequence of coupon payments and repayment of principal at maturity. The price paid for the bond is a different quantity from the promised cash flows.

The word “promises” is important. A contract can specify an amount without a guarantee that the issuer pays it. This first bond lesson constructs the contractual schedule. The credit-risk lessons introduce default probability and recovery later. Tuckman and Serrat define a coupon bond by its coupon rate, its maturity, and its face amount (also called the par amount or the principal amount). They show the coupons and the principal on the cash-flow timeline[1].

Read the contract before valuing it

Write the simplified contract’s face-value as $\explain{face-value}{F}$ , its annual-coupon-rate as $\explain{annual-coupon-rate}{c}$ , its bond-payment-frequency as $\explain{bond-payment-frequency}{m_{\mathrm B}}$ , and its maturity-time as $\explain{maturity-time}{T}$ . Write the number-of-bond-payments as $\explain{number-of-bond-payments}{n}$ and index those dates with the bond-payment-index $\explain{bond-payment-index}{k}$ . The matching payment-time is $\explain{payment-time}{t_{\explain{bond-payment-index}{k}}}$ , measured from valuation-time $0$ .

Write the level coupon-payment as $\explain{coupon-payment}{C}$ and the promised bond-cash-flow on date $\explain{bond-payment-index}{k}$ as $\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}}$ .

The number of payment dates is the payment frequency multiplied by the maturity:

\explain{number-of-bond-payments}{n}=\explain{bond-payment-frequency}{m_{\mathrm B}}\explain{maturity-time}{T}

(2.1.1)

The bond payment frequency is a different concept from the compounding-frequency $\explain{compounding-frequency}{m}$ of the earlier lessons. So it has a different symbol, with the subscript $\mathrm B$ .

Calculate the coupon

The level coupon payment is the annual coupon rate multiplied by the face value, divided by the payment frequency:

\explain{coupon-payment}{C}=\frac{\explain{annual-coupon-rate}{c}\explain{face-value}{F}}{\explain{bond-payment-frequency}{m_{\mathrm B}}}

(2.1.2)

The coupon rate is applied to the face value, not to the market price of the bond.

Construct every promised payment

For the bondholder, the promised bond cash flow on date $\explain{bond-payment-index}{k}$ is the coupon, plus the face value on the final date:

\explain{bond-cash-flow}{CF_{\explain{bond-payment-index}{k}}^{\mathrm{bond}}} =\explain{coupon-payment}{C}+\mathbf{1}_{\{\explain{bond-payment-index}{k}=\explain{number-of-bond-payments}{n}\}}\explain{face-value}{F}

(2.1.3)

The indicator $\mathbf{1}_{\{\explain{bond-payment-index}{k}=\explain{number-of-bond-payments}{n}\}}$ equals one on the final date and zero on every other date. So the face value is paid only on the final date.

On the regular schedule, the payment time of date $\explain{bond-payment-index}{k}$ , measured from valuation time, is:

\explain{payment-time}{t_{\explain{bond-payment-index}{k}}}=\frac{\explain{bond-payment-index}{k}}{\explain{bond-payment-frequency}{m_{\mathrm B}}}, \qquad 0\text{ is the valuation time}

(2.1.4)

Example 2.1.1 Coupon schedules by payment frequency

Open the set, then choose a payment frequency.

Annual coupons

A two-year bond has a face value of USD 1,000, an annual coupon rate of 5%, and one payment per year. The coupon is:

\explain{coupon-payment}{C}=\frac{0.05\times1000}{1}=\text{USD }50

Payment index $\explain{bond-payment-index}{k}$	Payment time $\explain{payment-time}{t_{\explain{bond-payment-index}{k}}}$ , years	Coupon, USD	Principal, USD	Total, USD
1	1.0	50	0	50
2	2.0	50	1,000	1,050

Semiannual coupons

An 18-month bond has a face value of USD 1,000, an annual coupon rate of 4%, and two payments per year. The coupon and the number of payments are:

\explain{coupon-payment}{C}=\frac{0.04\times1000}{2}=\text{USD }20, \qquad \explain{number-of-bond-payments}{n}=2\times1.5=3

Its payments at 0.5, 1.0, and 1.5 years are USD 20, USD 20, and USD 1,020.

Quarterly coupon amount

For a face value of USD 500, an annual coupon rate of 4.8%, and quarterly payments, the coupon is:

\explain{coupon-payment}{C}=\frac{0.048\times500}{4}=\text{USD }6

The maturity determines the number of payments. Under the level-coupon assumption, the maturity does not change the coupon amount.

Check your understanding

These items cover term interpretation, coupon arithmetic, and schedule construction. Each question stays collapsed until you open it; answers and explanations appear once you check.

Knowledge check 2.1.1 Fixed-rate bond contract and cash flows

Model boundary and review note

This lesson describes promised cash flows only. It does not assign a price, yield, payment probability, or expected recovery.

The contract terms and the level-coupon schedule follow Tuckman & Serrat[2]. The lesson stays draft pending human confirmation of the printed locators.

References

Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). Ch. 1 §1.1, printed p. 50 and Table 1.1. draft ↩
Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). §1.1, “Government Coupon Bonds”: a bond is set by coupon rate, maturity date, and face (par, principal); each semiannual coupon is half the annual rate times face, with principal repaid at maturity (Table 1.1). draft ↩

Notation used on this page (11)

c

Annual coupon ratedraft

Contractual annual rate used to determine a fixed-rate bond's coupon payments; applied to face value, not to market price.

Units: decimal rate per year

CF_k^{\mathrm{bond}}

Bond cash flowdraft

Promised amount paid to the bondholder on one scheduled payment date; a positive receipt for the bondholder, with default excluded in the bond lessons.

Units: stated currency at payment time

m_{\mathrm B}

Bond payment frequencydraft

Number of scheduled coupon payments per year in the simplified regular bond, a positive integer.

Units: scheduled coupon payments per year

k

Bond payment indexdraft

Labels one remaining scheduled bond payment in increasing time order; it selects a payment and is not itself a time or currency amount.

Units: dimensionless schedule index

m

Compounding frequencydraft

Number of equal compounding periods per year under the stated rate convention, a positive integer fixed by the model convention.

Units: compounding periods per year

C

Coupon paymentdraft

Level periodic cash amount promised by the simplified fixed-rate bond; a positive receipt for the bondholder in these lessons.

Units: stated currency per coupon date

F

Face valuedraft

Contractual reference amount used to determine coupons and principal redemption.

Units: stated currency

T

Maturity timedraft

Final scheduled time when principal is redeemed in the simplified bond.

Units: model-years from the valuation date

n

Number of bond paymentsdraft

Counts the remaining regular coupon dates including maturity; a positive integer for the simplified regular bond schedule.

Units: scheduled payment dates

t_k

Payment timedraft

Time from the valuation date to one scheduled cash-flow event; a time coordinate, not a calendar date or payment index.

Units: years from the valuation date

0

Valuation timedraft

Common origin from which later model times and present values are measured.

Units: years from the valuation date