Why Conventional Bond Yields Have A Unique Solution

Why Yield Uniqueness Matters

A bond’s yield to maturity is the rate that makes the present value of its promised future payments equal its observed price. A numerical method such as iteration can approximate a solution, but it is useful to know whether the equation can have more than one solution. For a conventional bond with positive coupons and positive redemption, the answer is no: its price function is strictly decreasing over positive yields, so the yield equation has at most one positive solution.

The result depends on the cash flows being fixed and positive. It does not claim that every positive bond price corresponds to a positive yield. A price can be too high for any positive yield to fit; uniqueness and existence are separate questions.

Definition: Consider a bond valued on a coupon date, with $n$ coupon periods remaining. Let $B_k$ be the total promised payment at the end of period $k$, for $k=1,\ldots,n$. For a conventional bond, each coupon is positive, and the final payment includes both the last coupon and the positive redemption amount, so $B_k>0$ at every payment date. If $r$ is the effective yield per coupon period, the bond’s price function is $$ P(r)=\sum_{k=1}^{n}\frac{B_k}{(1+r)^k},\qquad r>0. $$

For annual coupons, $r$ is the effective annual yield $i$. For $m$ coupon periods per year, if the quoted yield is the nominal annual rate $i^{(m)}$, the rate used in each period is $r=i^{(m)}/m$. The cash-flow dates and the rate’s compounding period must match: each exponent $k$ counts coupon periods, not years when coupons are more frequent than annual.

This setup assumes the bond’s promised cash flows are fixed, there is no default or contingent payment, and the yield equation discounts those cash flows at one constant rate per period. It describes a specified bond cash-flow scenario, not a bond whose payments change with interest rates or depend on a future decision.

The Uniqueness Result

Theorem: If every future bond payment $B_k$ is positive, then $P(r)$ is strictly decreasing for $r>0$. Consequently, for any fixed observed price, the yield equation $P(r)=\text{observed price}$ has at most one positive solution.

The key is that increasing a positive yield reduces the present value of each positive payment. Because the bond has at least one payment at a positive future time—and, under the stated conventional-bond assumptions, every listed payment is positive—the total present value falls strictly, not merely weakly.

Compare two candidate yields.
Let $0<r_1<r_2$. Then $1+r_1<1+r_2$, so for every positive period $k$, $(1+r_1)^{-k}>(1+r_2)^{-k}$.

Apply the comparison to every payment.
Since $B_k>0$, multiplying the discount-factor inequality by $B_k$ preserves its direction: $B_k(1+r_1)^{-k}>B_k(1+r_2)^{-k}$.

Add the present values.
Adding the strict inequalities over all payment dates gives $P(r_1)>P(r_2)$. Thus $P$ is strictly decreasing on the positive-yield domain.

Conclude uniqueness.
If two different positive yields both solved the same price equation, the strict inequality would require their prices to differ. They cannot both equal the same observed price, so there is at most one solution.

This is a direct comparison proof: it does not depend on a particular coupon amount, maturity, or numerical root-finding method. It also shows why positivity matters. If some future cash flows were negative, multiplying by them could reverse the inequalities; the argument would no longer establish strict decrease.

Worked Examples

Worked Example: Verify a Yield and Establish Its Uniqueness

A two-year bond pays a coupon of $50 at the end of each year and redeems for $1,000 with its second coupon. Its observed price is $712. Find the annual yield and explain why there is no other positive yield. Use the effective annual yield model.

Method. Write the two future payments, discount them at a candidate yield, and use the uniqueness theorem. The payments are $B_1=50$ and $B_2=50+1{,}000=1{,}050$.

At a candidate yield of $r=25\%=0.25$, the discount factors are $$ (1.25)^{-1}=0.8,\qquad (1.25)^{-2}=0.64. $$ Therefore, $$ P(0.25)=50(0.8)+1{,}050(0.64)=40+672=712. $$ The observed price is exactly $712, so a 25% effective annual yield solves the equation.

Both payments are positive and occur at future dates. The price function is therefore strictly decreasing for positive yields. Since 25% is one solution, it is the only positive solution. As a check on the comparison, at 20% the price is $50/1.20+1{,}050/(1.20)^2=41.6667+729.1667=\$770.83$; at 30% it is $50/1.30+1{,}050/(1.30)^2=38.4615+621.3018=\$659.76$. These values lie on opposite sides of $712, consistent with the 25% solution.

Common mistake avoided: Finding one rate that reproduces the price does not, by itself, rule out another rate. The proof of strict decrease is what establishes that a second positive solution is impossible.

Worked Example: Use the Price Equation to Confirm a Yield

A bond pays $50 annually for two years and redeems for $1,000 with the final coupon. Its observed price is exactly $50/1.05+1{,}050/(1.05)^2$ dollars (approximately $1,000.00). Show that 5% is a yield solution and determine whether there can be another positive solution.

Method. Substitute the proposed effective annual yield into the bond’s present-value equation. The first payment has present value $$ \frac{50}{1.05}=47.6190476\ldots $$ and the second payment has present value $$ \frac{1{,}050}{(1.05)^2} =\frac{1{,}050}{1.1025} =952.3809524\ldots. $$ Adding gives $47.6190476\ldots+952.3809524\ldots=1{,}000$. Thus the price equation is satisfied at an effective annual yield of 5%.

The cash flows at both future dates are positive. The theorem therefore applies to this bond, and no other positive yield can produce the same price. The exact observed price was specified by the present-value expression, so its rounded display as $1,000.00 does not introduce a rounding difference into the equation.

Common mistake avoided: A rounded price is not always interchangeable with an exact price when checking a proposed yield. Keep the unrounded value for the equation, and round only when reporting the displayed dollar amount.

Worked Example: Apply Uniqueness to a Semiannual Yield Quote

A bond has six semiannual periods remaining. It pays a coupon of $24 each half-year and redeems for $1,000 with its final coupon. Its price is set equal to its present value at a nominal annual yield of 8% convertible semiannually. Calculate the price and explain why no other positive quoted yield can produce it.

Method. Convert the nominal annual yield to the period rate before discounting. The semiannual rate is $r=0.08/2=0.04$. The six coupons have total present value $$ 24\left(\frac{1-(1.04)^{-6}}{0.04}\right). $$ Using $(1.04)^{-6}=0.790314526$, the annuity factor is $$ \frac{1-0.790314526}{0.04}=5.24213685. $$ The coupon present value is $24(5.24213685)=125.8112844$. The redemption present value is $1{,}000(0.790314526)=790.314526$. Consequently, the price is $125.8112844+790.314526=916.1258104$, or $916.13 to the nearest cent.

Each coupon and the redemption are positive, so the price is strictly decreasing in the positive semiannual period yield. A nominal annual yield convertible semiannually maps to that period yield by dividing by two; this conversion is one-to-one. Therefore, the quoted yield equation has at most one positive nominal annual solution, and the 8% quote is that solution.

Common mistake avoided: Using 8% as the rate for each half-year would discount at twice the correct period rate. For example, that incorrect calculation gives $24[1-(1.08)^{-6}]/0.08+1{,}000(1.08)^{-6}$, approximately $741.12$, not $916.13. The correct periodic rate is 4%, not 8%.

Uniqueness Is Not the Same as Existence

The theorem says there can be at most one positive yield. It does not say that a positive yield must exist for every positive observed price. The distinction can be made precise for the finite schedule above. At a zero yield, the present value is the sum of the future payments: $$ P(0)=\sum_{k=1}^{n}B_k. $$ For positive yields, each discount factor is less than 1, so $0<P(r)<P(0)$. Also, as $r$ increases without bound, each discounted payment approaches zero and hence $P(r)$ approaches zero. The price function is continuous for $r>0$. Thus, if an observed price lies strictly between zero and the sum of the future payments, continuity gives a positive yield solution, and strict decrease makes that solution unique. If the observed price equals the sum, the yield is zero, not positive; if it is higher, no nonnegative yield solves the equation.

This existence check is useful before applying a numerical method. A numerical search for a positive root should not be expected to succeed when the price is at least the undiscounted total of all future payments. Conversely, a price between zero and that total guarantees one positive solution under the stated fixed-cash-flow model.

Common Traps

Confusing “at most one” with “exactly one.” Strict decrease rules out multiple solutions. A separate existence check is needed to know whether a solution is present.
Using the annual nominal quote as a coupon-period rate. With $m$ coupon periods per year, discount each period using $i^{(m)}/m$, not $i^{(m)}$.
Leaving redemption out of the final payment. At maturity, the cash flow includes both the final coupon and redemption. Omitting redemption changes the price function and the yield equation.
Assuming the argument covers every cash-flow pattern. Positivity is essential to the comparison proof. A cash-flow schedule with negative payments needs a separate analysis.

Key takeaway: For a fixed conventional bond with strictly positive payments at future coupon dates, every increase in the positive yield reduces every discounted payment. The price function is strictly decreasing, so a fixed observed price has at most one positive yield solution. Check existence separately, and match the rate period to the coupon period.

Check Your Understanding

For each problem, use the fixed-cash-flow model and distinguish a verified solution from the reason it is unique.

A two-year annual-coupon bond pays $50 each year and redeems for $1,000. Its price is $50/1.10+1{,}050/(1.10)^2$ dollars. Verify a yield solution and state why it is the only positive solution. (Answer: 10% effective annual yield; it is the only positive solution because the price function is strictly decreasing.)
A bond has five semiannual periods remaining, pays $18 each half-year, and redeems for $1,000 with its last coupon. Its price is set to the value at a 6% nominal annual yield convertible semiannually. What is the periodic yield used in its equation, and can another positive quoted yield give the same price? (Answer: 3% per half-year; no, the 6% nominal annual yield is the only positive quoted yield.)
An annual-coupon bond pays $40 at the end of each of three years and redeems for $1,000 with its last coupon. Its observed price is $960. At 5% the price is approximately $972.77, and at 6% it is approximately $946.54. What can you conclude about its positive yield? (Answer: Exactly one positive yield exists, and it lies between 5% and 6%.)
A one-year bond pays a $40 coupon and redeems for $1,000. Its observed price is $1,040. Is there a positive effective annual yield solving the price equation? (Answer: No; the price equals the undiscounted total of $1,040, so the solution is zero yield, not a positive yield.)
A semiannual bond has four periods remaining, pays $12 each half-year, and redeems for $1,000 with the final coupon. Its price is set at a nominal annual yield of 8% convertible quarterly. What periodic rate belongs in each of the four semiannual discount periods, and does this quoted yield determine a unique price-equation solution? (Answer: 2% per quarter does not match the semiannual payment periods; convert the nominal rate to an effective six-month rate $(1+0.08/4)^2-1=4.04\%$. The price equation has at most one positive semiannual yield solution.)

What You'll Learn