Solving for a New Terminal Discount Factor

Why the Terminal Discount Factor Can Be Isolated

When a bond has known cash flows at earlier dates and one final payment at maturity, its price equation contains a separate discount factor for each payment date. If the earlier factors are already known, their discounted cash flows can be subtracted from the bond’s full price. The remaining amount is the present value of the maturity payment, which isolates the terminal discount factor.

This is the central algebraic step in extending a discount curve by one maturity. The result is useful only if it is consistent with the pricing model. In particular, with positive cash flows and strictly positive discount factors, the implied terminal factor must be positive. A negative result signals that the stated price and known factors cannot all fit that model.

Definition: A discount factor $D_k$ is the present value, at the valuation date, of one unit payable at time $k$. In this tutorial, a bond has annual coupon dates, is priced at its valuation date, and is quoted at its full price. Its face amount is $F$, its annual coupon rate is $r$, its coupon payment is $C=rF$, and its maturity is $n$ years. The standard par-redemption bond pays $C$ at each coupon date and pays $F+C$ at maturity.

Assume that the discount factors $D_1,\ldots,D_{n-1}$ are known and that $D_n$ is the only unknown. Under this annual-coupon timing assumption, the full-price equation is

P = CD_1+CD_2+\cdots+CD_{n-1}+(F+C)D_n.

The final term includes both the last coupon and the principal. Rearranging gives the terminal factor directly:

D_n=\frac{P-C\displaystyle\sum_{k=1}^{n-1}D_k}{F+C}.

This formula applies when the bond’s stated cash flows and payment dates match the model, the price is a full price at the valuation date, and all factors except $D_n$ are known. It does not directly apply if the quoted price is not the full price, if coupon dates or cash flows differ from those assumed, or if there are other unknown discount factors in the equation.

Why the Rearrangement Works

Each dated payment contributes its amount multiplied by the discount factor for that payment date. The first $n-1$ payments are coupons of $C$, so their combined present value is $C\sum_{k=1}^{n-1}D_k$. At maturity the bond pays $F+C$, so the maturity payment contributes $(F+C)D_n$. Subtracting the known earlier present values from the full price leaves exactly the present value of the maturity payment:

P-C\sum_{k=1}^{n-1}D_k=(F+C)D_n.

Since $F+C$ is a positive cash flow, dividing by it gives the displayed formula. This also explains the sign test: the residual price $P-C\sum_{k=1}^{n-1}D_k$ must be positive if $D_n$ is to be strictly positive.

Write the full-price equation.
List every coupon payment and include both coupon and principal in the maturity cash flow.

Value the earlier coupons.
Multiply each known earlier discount factor by the coupon amount and add the results.

Find the residual price.
Subtract the earlier coupons’ present value from the full bond price.

Divide by the terminal cash flow.
The residual price divided by $F+C$ is the implied $D_n$.

Check admissibility and reprice.
For the positive-discount-factor model, confirm $D_n>0$, then substitute it back into the price equation.

Worked Applications

Worked Example: A Small Residual Price

A three-year bond has face amount $F=\$100$, annual coupon rate $r=5\%$, and full price $P=\$10.00$. The known factors are $D_1=0.97$ and $D_2=0.92$. Find $D_3$, round to four decimal places, and assess admissibility under a strictly positive-discount-factor model.

Method. Subtract the present value of the two known coupons, then divide the residual by the maturity cash flow. The important cash-flow check is that the year-three payment is not just the coupon.

The annual coupon is

C=rF=0.05(100)=\$5.

The maturity payment is principal plus the final coupon:

F+C=100+5=\$105.

The first two coupon present values are

CD_1=5(0.97)=\$4.85, \qquad CD_2=5(0.92)=\$4.60.

Their total is $\$4.85+\$4.60=\$9.45$, so the residual price is

10.00-9.45=\$0.55.

Therefore,

D_3=\frac{0.55}{105}=0.005238095\ldots\approx 0.0052.

The implied factor is positive, so it is admissible under the stated strictly positive-factor condition. It is very small, but that alone does not make it negative or inadmissible under this test. As a check, the maturity cash flow has present value $105(0.005238095\ldots)=\$0.55$; adding the earlier $\$9.45$ gives the quoted $\$10.00$.

Common mistake avoided: The residual is $\$0.55$, not a negative amount. Subtracting the earlier present value in the correct order matters: price minus known present values.

Worked Example: Including the Final Coupon

A two-year bond has face amount $F=\$110$, annual coupon rate $r=4\%$, full price $P=\$104.40$, and $D_1=0.96$. Find $D_2$, to four decimal places, and check the result by repricing.

Method. Use the one-known-coupon version of the terminal-factor equation. The trap is to use only the face amount as the terminal payment.

The annual coupon is

C=0.04(110)=\$4.40.

The year-two payment is $\$110+\$4.40=\$114.40$. The year-one coupon’s present value is

CD_1=4.40(0.96)=\$4.224.

The residual price assigned to the year-two payment is

104.40-4.224=\$100.176.

Thus,

D_2=\frac{100.176}{114.40}=0.8756643357\ldots\approx 0.8757.

The result is positive. To check it before rounding, substitute the unrounded factor: $114.40(0.8756643357\ldots)=\$100.176$, and $\$100.176+\$4.224=\$104.40$. The quoted four-decimal factor is for reporting; the unrounded value is used for the exact check.

Common mistake avoided: Dividing the residual by $\$110$ omits the final coupon. The correct denominator is the full terminal cash flow, $\$114.40$.

Worked Example: Several Earlier Coupons

A four-year bond has face amount $F=\$100$, annual coupon rate $r=6\%$, full price $P=\$101.84$, and known factors $D_1=0.98$, $D_2=0.95$, and $D_3=0.91$. Solve for $D_4$, report four decimal places, and verify the price.

Method. Add the present values of all three earlier coupons before finding the residual. This prevents leaving out a known payment when several terms appear in the price equation.

The annual coupon is $C=0.06(100)=\$6$, and the terminal cash flow is $\$100+\$6=\$106$. The earlier coupon values are

6(0.98)=\$5.88,\qquad 6(0.95)=\$5.70,\qquad 6(0.91)=\$5.46.

Their sum is $\$5.88+\$5.70+\$5.46=\$17.04$. The residual price is

101.84-17.04=\$84.80.

Therefore,

D_4=\frac{84.80}{106}=0.8000.

The implied factor is strictly positive. Repricing confirms the result: the maturity payment contributes $106(0.8000)=\$84.80$, and $\$17.04+\$84.80=\$101.84$, exactly the full price.

Common mistake avoided: Applying the formula after subtracting only one earlier coupon would assign the present value of the omitted coupon to the terminal payment and produce the wrong factor.

Admissibility and Exam Traps

The algebra always produces a quotient when the terminal cash flow is nonzero, but not every quotient is compatible with the model. For a positive terminal payment $F+C$, the sign of $D_n$ is the sign of the residual price. Thus, a positive residual gives a positive implied factor; a zero residual gives $D_n=0$, which fails a strictly positive-factor condition; and a negative residual gives a negative factor, which is inadmissible under that condition.

This check is specific to the stated positivity requirement. Do not impose an additional restriction unless the problem supplies it. For example, the formula itself does not say that the terminal factor must equal the preceding factor, nor does the positivity test require the factors to decrease at every date. The appropriate first diagnostic is whether the solved factor satisfies the assumptions actually stated in the problem.

A frequent sign error is to calculate known present values minus price, then divide by the positive terminal cash flow. That reverses the residual. The equation fixes the correct order:

\text{residual price} = \text{full price} - \text{present value of earlier cash flows}.

A second frequent error is to confuse a coupon with the maturity payment. For a standard par-redemption bond, the maturity cash flow is $F+C$, even though the earlier payments are only $C$. A third is mixing a quoted price with a price basis inconsistent with the cash-flow equation. Before solving, confirm that the supplied price is the full price at the valuation date, as assumed here.

Key takeaway: Subtract all known earlier coupon present values from the full price, then divide by the complete maturity cash flow. A positive implied factor passes the strict positivity check; reprice the bond to verify the arithmetic.

Check Your Understanding

For each problem, use the stated annual-coupon model and round discount factors to four decimal places where needed. Each listed price is a full price at the valuation date.

A two-year bond has face amount $\$100$, annual coupon rate $5\%$, price $\$98.00$, and $D_1=0.96$. Find $D_2$. (Answer: $0.8876$)
A three-year bond has face amount $\$100$, annual coupon rate $4\%$, price $\$99.20$, and known factors $D_1=0.97$ and $D_2=0.94$. Find $D_3$. (Answer: $0.8804$)
A two-year bond has face amount $\$120$, annual coupon rate $3\%$, price $\$118.00$, and $D_1=0.99$. Find $D_2$. (Answer: $0.9259$)
A three-year bond has face amount $\$100$, annual coupon rate $6\%$, price $\$98.00$, and known factors $D_1=0.95$ and $D_2=0.90$. Find $D_3$ and state whether it is admissible under the strictly positive-factor model. (Answer: $0.8198$, admissible)
A two-year bond has face amount $\$100$, annual coupon rate $5\%$, price $\$4.50$, and $D_1=0.96$. Calculate the implied $D_2$ and assess it under the strictly positive-factor model. (Answer: $-0.0029$, inadmissible)

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