Bootstrapping with Redemption Cash Flows at Coupon Dates

Why the Final Payment Needs Special Care

In a coupon-bond bootstrap, the price is related to the bond’s dated cash flows and the discount factors for those dates. A common situation is that the bond is redeemed on a scheduled coupon date. On that date, the investor receives both the final coupon and the redemption amount. The bootstrap must treat these as one cash flow at one date before solving for the discount factor at that date.

The key issue is not a new pricing principle. It is keeping the cash-flow timeline and the payment amounts precise. A final coupon is not omitted merely because principal is repaid on the same date, and it is not discounted separately at an additional time. If the redemption amount is quoted as principal only, add the coupon to it. If the quoted amount is explicitly the total maturity payment, do not add the coupon again.

Definition: Let $P$ be a bond’s full price at the valuation date, $t_1,\ldots,t_n=T$ its payment dates measured in years from that date, and $D(t)$ the discount factor from the valuation date to time $t$. If coupon $C_k$ is paid at $t_k$, and principal redemption $R$ is paid at the final coupon date $T$, then the final dated cash flow is $C_n+R$. The price equation is $$ P=\sum_{k=1}^{n-1} C_kD(t_k)+(C_n+R)D(T). $$

This equation assumes the bond’s promised payments are fixed, the valuation date and cash-flow dates are known, and the same set of discount factors is used to value each payment. It describes the contractual cash flows; it does not include a separate allowance for default or uncertain payments. The discount factors at dates before $T$ must already be known if this bond is to determine a new terminal factor by itself.

Isolating the Terminal Discount Factor

Once earlier discount factors are available, calculate the present value of every cash flow before $T$. Subtract that amount from the observed price. The remainder is the present value attributable to the combined final coupon and redemption. Dividing by their total amount gives $D(T)$.

D(T)=\frac{P-\displaystyle\sum_{k=1}^{n-1}C_kD(t_k)}{C_n+R}

The denominator is the actual final payment, not just the principal redemption. Before relying on this equation, check that the redemption is in fact paid at the final coupon date and that the stated redemption amount excludes the coupon. If the bond’s terms instead give a single total maturity payment, use that amount directly as the denominator.

Lay out the dates and payments.
List every coupon date, the coupon on each date, and the date and amount of principal redemption. Mark any date on which coupon and redemption coincide.

Combine payments at a shared date.
When the final coupon and redemption are separate components payable on the same date, add their amounts. There is still just one discount factor for that date.

Value known-date cash flows.
Multiply each earlier cash flow by its already bootstrapped discount factor and add the resulting present values.

Solve and check.
Subtract the earlier present value from the bond price, divide by the total final payment, and reprice the bond using the factor found.

Why the Combination Works

The reason for combining the two amounts is visible directly from the pricing equation. Suppose the final coupon $C_n$ and redemption $R$ are both paid at $T$. Discounting each separately gives

$$ C_nD(T)+RD(T)=(C_n+R)D(T). $$

Both payments are multiplied by the same factor because both occur at the same time. Thus the price can be written as the present value of earlier coupons plus one final dated cash flow. Rearranging that price equation yields the terminal-factor formula above. This is an algebraic consequence of the dated-cash-flow model, not an additional assumption about the bond’s yield.

Worked Example: Annual Coupon and Above-Par Redemption

A hypothetical three-year bond has face value $1,000, an annual coupon rate of 6%, and principal redemption of $1,020 at the end of year 3. Its full price at a coupon date is $1,012. The bootstrapped discount factors are $D(1)=0.96$ and $D(2)=0.91$. Find $D(3)$, then check the price. Use the stated annual coupon and fixed redemption cash flows; no default or taxes are assumed.

Method. Write the annual coupon cash flows and combine the year-3 coupon with principal redemption. The coupon is $0.06(1{,}000)=\$60$. The final payment is therefore $\$60+\$1{,}020=\$1{,}080$.

The present value of the earlier coupons is

60(0.96)+60(0.91)=57.60+54.60=\$112.20.

The amount of the price attributable to the final payment is

1{,}012-112.20=\$899.80.

Therefore,

D(3)=\frac{899.80}{1{,}080}=0.833148148\ldots

To check, the price using the unrounded factor is

60(0.96)+60(0.91)+1{,}080(0.833148148\ldots) =57.60+54.60+899.80 =\$1{,}012.00.

Thus $D(3)=0.833148$ to six decimal places. A common mistake would be to divide the residual $899.80 by $1,020, using redemption alone. That would assign the final coupon no present value. The correct denominator is $1,080 because the coupon and redemption are both paid at year 3.

Semiannual Coupons and Different Redemption Amounts

The same logic applies when coupons are paid more than once a year. The coupon frequency affects the coupon amount and the dates, but it does not change the rule that cash flows on one date share one discount factor. Confirm the number of coupon periods and use the discount factor for each exact payment date.

Worked Example: Semiannual Coupon With Below-Par Redemption

A hypothetical two-year bond has face value $1,000, a 5% annual coupon rate paid semiannually, and principal redemption of $975 at year 2. Its full price is $1,000. The known discount factors are $D(0.5)=0.985$, $D(1)=0.965$, and $D(1.5)=0.940$. Find the two-year factor and verify it. Assume the coupon rate is nominal annual with two payments per year, and that $975 is principal redemption only.

Method. There are four half-year coupon periods. Each coupon is $1{,}000(0.05)/2=\$25$. The final coupon coincides with redemption, so the year-2 payment is $\$25+\$975=\$1{,}000$.

The present value of the first three coupons is

25(0.985)+25(0.965)+25(0.940) =24.625+24.125+23.500 =\$72.250.

The residual price for the final payment is

1{,}000-72.250=\$927.750.

It follows that

D(2)=\frac{927.750}{1{,}000}=0.927750.

Repricing gives

72.250+1{,}000(0.927750)=72.250+927.750=\$1{,}000.000.

The answer is $D(2)=0.927750$. The trap here is treating the final principal repayment as if it were the only year-2 payment. In this example the combined payment happens to equal $1,000, but that equality follows from the $975 redemption plus the $25 coupon; it is not an assumption that redemption is always at par.

When the Quoted Maturity Amount Is Already a Total

Bond descriptions do not always state the final amounts in the same way. “Principal is redeemed for $R” normally identifies a principal amount separate from the final coupon. By contrast, a term such as “the total payment at maturity is $M” identifies the entire payment on that date. Read the wording before forming the cash flow: adding a coupon to a total payment counts it twice.

Worked Example: Maturity Payment Quoted as a Total

A hypothetical bond pays annual coupons at years 1 and 2. Its face value is $1,000 and its annual coupon rate is 4%. The contract states that the total payment at year 2, including the final coupon, is $990. The bond’s full price is $950, and $D(1)=0.96$. Find $D(2)$ and check the result. Assume the stated total is the complete year-2 cash flow.

Method. The year-1 coupon is $1{,}000(0.04)=\$40$, with present value $40(0.96)=\$38.40$. The year-2 payment is already stated as a total of $990, so the final coupon must not be added to it.

The residual price for year 2 is

950-38.40=\$911.60.

Thus,

D(2)=\frac{911.60}{990}=0.920808080\ldots

To check using the unrounded factor,

40(0.96)+990(0.920808080\ldots) =38.40+911.60 =\$950.00.

Therefore $D(2)=0.920808$ to six decimal places. The tempting but incorrect approach is to add the $40 coupon to the quoted $990, on the assumption that $990 is principal only. The contract expressly defines $990 as the total payment, so doing so would count the final coupon twice.

Common Mistakes and a Practical Check

At a coupon-date redemption, the most important check is to write the final dated cash flow in words before inserting numbers. State whether the redemption amount is principal alone or a total that already includes coupon. Then confirm that every payment date appears once in the price equation and is paired with its corresponding discount factor.

Leaving out the last coupon: if redemption is quoted as principal alone, the total final payment includes the final coupon.
Counting the last coupon twice: if the quote explicitly gives the total maturity payment, do not add the coupon a second time.
Using a separate factor for same-date payments: the coupon and redemption on one date both use $D(T)$; their amounts can be added first.
Using the wrong date’s factor: a semiannual coupon schedule requires the factor for the actual half-year date, not an annual-date factor selected for convenience.
Failing to reprice: substitute the solved factor into the full cash-flow equation. The result should equal the quoted full price, apart from any stated rounding.

A solved factor should also be interpreted in the context of the input data. If the residual price is negative while the final payment is positive, the algebra produces a negative factor; that is a signal to inspect the quote, cash flows, and earlier factors rather than silently changing the calculation. The price equation determines the factor, while the repricing check confirms that the arithmetic reproduces the input price.

Key takeaway: When redemption occurs on a coupon date, first form the complete cash flow on that date. If redemption is stated separately from coupon, use $C_n+R$; if the quote already gives the total maturity payment, use that total as written. Divide the price remaining after earlier cash flows by this one final payment to obtain the terminal discount factor.

Check Your Understanding

For each problem, assume fixed promised payments, no default, and a full price at the stated valuation date. Use the given earlier discount factors.

A two-year annual-coupon bond has face value $1,000, a 5% annual coupon rate, and principal redemption of $1,000 at year 2. Its price is $987 and $D(1)=0.96$. Find $D(2)$. (Answer: $0.894286$)
A two-year bond pays semiannual coupons. Face value is $1,000, the nominal annual coupon rate convertible semiannually is 5%, principal redemption is $1,020 at year 2, and the price is $1,000. The known factors are $D(0.5)=0.99$, $D(1)=0.975$, and $D(1.5)=0.95$. Find $D(2)$. (Answer: $0.887201$)
A three-year annual-coupon bond has face value $1,000, coupon rate 8%, and principal redemption of $950 at year 3. Its price is $920, with $D(1)=0.97$ and $D(2)=0.92$. Find $D(3)$. (Answer: $0.746408$)
A two-year bond has a $1,000 face value and a 6% annual coupon. Its contract states that the total year-2 payment, including the final coupon, is $1,040. The price is $970 and $D(1)=0.95$. Find $D(2)$, taking care not to add the coupon twice. (Answer: $0.878846$)

What You'll Learn