Why a Conventional Bond Price Has One Positive Yield

Why Yield Uniqueness Matters

A bond’s yield is inferred from its price by finding the rate that discounts its scheduled payments to that price. Before trying to calculate the rate, it is useful to know whether the answer can be ambiguous. For a conventional bond, every scheduled coupon and the redemption payment are positive. That sign pattern makes the price strictly decrease as the yield increases, which rules out two different positive yields for the same price.

The result is about uniqueness, not a promise that a positive yield exists for every price. A bond price must also fall below the sum of its undiscounted future payments for the yield to be positive. The distinction between these two claims is an important part of using the yield equation correctly.

Definition: Let a bond pay positive amounts $C_1,C_2,\ldots,C_n$ at the ends of periods $1,2,\ldots,n$. If $y$ is the yield per payment period, its price on a payment date is $$ P(y)=\sum_{k=1}^{n}\frac{C_k}{(1+y)^k}, \qquad y>-1. $$ A conventional bond has positive scheduled coupons and a positive redemption payment, so each $C_k>0$. The yield rate must use the same period as the payment schedule.

For annual coupons, $y$ is an annual effective yield. For semiannual coupons, $y$ is the yield per half-year; if it is quoted as a nominal annual rate convertible semiannually, that nominal rate is $2y$. The payment timing and the rate period must match in the yield equation.

Why the Price Must Decrease

Each cash flow is discounted by a factor that decreases when $y$ rises. The derivative makes the argument precise: it measures the change in price for a small change in yield.

Justification: For $y>-1$, differentiate each term of the finite price sum: $$ \frac{dP}{dy} = \sum_{k=1}^{n} C_k\frac{d}{dy}(1+y)^{-k} = -\sum_{k=1}^{n}\frac{kC_k}{(1+y)^{k+1}}. $$ Every $k$, $C_k$, and $(1+y)^{k+1}$ is positive. Therefore every term in the final sum is negative, and $\frac{dP}{dy}<0$. Thus $P(y)$ is strictly decreasing on $y>-1$, in particular for positive yields.

This conclusion depends on the cash flows being positive and fixed. It does not say that every kind of investment with payments of mixed signs has a unique yield: if some cash flows are negative, the derivative need not remain negative. Nor does it cover a payment schedule that changes when the yield changes. For a conventional bond’s fixed scheduled coupons and redemption, however, the conditions hold.

Theorem: A conventional bond with positive future cash flows has at most one positive yield for any specified price. It has exactly one positive yield if $$ 0<P(0\text{ price})<\sum_{k=1}^{n}C_k. $$ Here $P(0\text{ price})$ means the given market price, not the value of the price function at yield zero.

At a zero yield, no payment is discounted, so the price function has value $\sum C_k$. As $y$ increases without bound, every discounted payment approaches zero, and so does the price. The price function is continuous and strictly decreasing between these limits. Consequently, every positive price below $\sum C_k$ is attained at exactly one positive yield. A price equal to the sum corresponds to a zero yield; a price above the sum cannot be produced by a positive yield.

Check the cash-flow signs.
Confirm that every future coupon and the redemption payment are positive and fixed.

Check the price boundary.
Compute the undiscounted total $\sum C_k$ and compare it with the given positive market price.

State the conclusion carefully.
If the price is below the total, a unique positive yield exists. If it equals the total, the yield is zero; if it exceeds the total, there is no positive yield.

Worked Examples

Worked Example: Confirming a Positive Yield and Its Uniqueness

A two-year bond pays annual coupons of $4$ dollars and redeems at $100$ dollars. Its market price is $100$ dollars, and the yield is measured as an annual effective rate $y$. Determine whether there is a positive yield and, if so, identify it. Assume the scheduled payments are fixed and certain.

Method: check the candidate rate in the yield equation, then use strict monotonicity. The cash flows are $C_1=4$ and $C_2=104$, because the final payment includes both the last coupon and redemption. Their undiscounted total is $$ 4+104=108. $$ The market price $100$ is positive and less than $108$, so the existence condition for a positive yield is satisfied.

Test $y=0.04$. The present value of the first coupon is $$ \frac{4}{1.04}=3.8461538\ldots $$ The present value of the final payment is $$ \frac{104}{(1.04)^2} = \frac{104}{1.0816} = 96.1538461\ldots $$ Therefore, $$ P(0.04)=3.8461538\ldots+96.1538461\ldots=100. $$ The price equation is satisfied at a 4% annual effective yield. Since price is strictly decreasing in yield, no other positive yield can produce the same price. The yield is uniquely $4\%$. The computed price checks against the stated $100$-dollar market price.

Common mistake avoided: Finding one rate that satisfies the equation is not, by itself, a proof that it is the only rate. The strict-decrease result supplies the uniqueness argument.

Worked Example: Showing a Positive Yield Lies Between Two Rates

A bond pays annual coupons of $7$ dollars for two years and redeems for $110$ dollars at the end of year 2. Its market price is $105$ dollars. Determine whether a positive annual effective yield exists, and show that it lies between 5% and 10%. Assume all scheduled payments are fixed and certain.

Method: compare the price function at two positive rates, then use continuity and strict decrease. The cash flows are $7$ at year 1 and $117=7+110$ at year 2. At a zero yield, their total is $$ 7+117=124. $$ Since $0<105<124$, exactly one positive yield exists.

At $y=0.05$, the calculated price is $$ P(0.05) = \frac{7}{1.05}+\frac{117}{(1.05)^2} = 6.6666667\ldots+106.1224490\ldots = 112.7891157\ldots $$ At $y=0.10$, it is $$ P(0.10) = \frac{7}{1.10}+\frac{117}{(1.10)^2} = 6.3636364\ldots+96.6942149\ldots = 103.0578513\ldots $$ The target price $105$ lies between those two prices: $$ 112.7891157\ldots>105>103.0578513\ldots. $$ The price function is continuous, so it takes the value $105$ at a yield between 5% and 10%. Strict decrease ensures that it takes that value only once. The result is checked by the endpoint prices: one is above the target and the other is below it.

Common mistake avoided: A price below the undiscounted total establishes that a positive yield exists, but does not show where it lies. Comparing prices at two rates gives the additional interval conclusion without confusing the existence test with a numerical yield calculation.

Worked Example: Applying Uniqueness with Semiannual Coupons

A hypothetical four-period bond pays $2.50$ dollars at the end of each half-year and returns $100$ dollars with its fourth coupon. Its market price is defined to be the present value of those payments at a yield of $2\%$ per half-year. Find the price to the nearest cent, state the yield per half-year, and express the corresponding annual effective yield. Assume the payments are fixed and certain.

Method: keep the yield period matched to the coupon period, calculate the price, and apply monotonicity. The cash flows per half-year are $2.50, 2.50, 2.50,$ and $102.50$. At $y=0.02$ per half-year, the discount factors are $$ (1.02)^{-1}=0.98039216,\quad (1.02)^{-2}=0.96116878,\quad (1.02)^{-3}=0.94232233,\quad (1.02)^{-4}=0.92384543. $$ The discounted payments are, in order, $$ 2.50(0.98039216)=2.4509804, $$ $$ 2.50(0.96116878)=2.4029220, $$ $$ 2.50(0.94232233)=2.3558058, $$ and $$ 102.50(0.92384543)=94.6941562. $$ Thus the market price, before rounding, is $$ 2.4509804+2.4029220+2.3558058+94.6941562 = 101.9038644\ldots $$ The price rounded to the nearest cent is $\$101.90$. By the definition of the market price in this problem, the yield is $2\%$ per half-year. The corresponding annual effective yield is $$ (1.02)^2-1=1.0404-1=0.0404=4.04\%. $$ The undiscounted cash-flow total is $2.50+2.50+2.50+102.50=110$, which exceeds the positive price, consistent with a positive yield. Strict decrease confirms that the exact, unrounded price has no second positive yield. The cent-rounded price is for reporting; it is not substituted for the exact price definition when identifying the rate.

Common mistake avoided: Treating the 2% half-year yield as a 2% annual effective yield would use the wrong rate period. Compounding two half-year periods gives the annual effective rate of 4.04%.

Common Mistakes and Scope of the Result

The uniqueness result is powerful, but it is easy to state it too broadly. It proves that there cannot be two positive yields for one conventional bond price; it does not mean a positive yield must exist whatever price is supplied. The zero-yield price boundary is the undiscounted sum of all future payments.

Confusing “at most one” with “exactly one.” Strict decrease proves at most one solution. To establish exactly one positive solution, also verify that the price is positive and below the undiscounted total.
Forgetting the final coupon. If redemption is $R$ and the last coupon is $c$, the final cash flow is $R+c$, not just $R$.
Using a mismatched period. With semiannual coupons, discount successive payments using a semiannual yield. Convert to an annual effective rate only after identifying the period yield.
Assuming the result applies to mixed-sign cash flows. The proof uses $C_k>0$ for every scheduled payment. If payments can be negative, the derivative sign must be examined rather than assumed.

The proof is aligned with the SOA Exam FM (Financial Mathematics) syllabus: a candidate should be able to explain why the conventional bond price-yield relationship is strictly decreasing and apply the result without overstating what it guarantees.

Check Your Understanding

For each problem, identify whether a positive yield exists and use the strict-decrease result to state what can be concluded about uniqueness.

A bond pays $5$ dollars at the end of year 1 and $105$ dollars at the end of year 2. Its price is $100$ dollars. Verify that a 5% annual effective yield fits the price equation and state whether another positive yield can fit it. (Answer: The price at 5% is $5/1.05+105/(1.05)^2=100$; it is the unique positive yield.)
A bond has cash flows of $4$ dollars at year 1 and $104$ dollars at year 2, and its price is $102$ dollars. Compare the price with the zero-yield price and state the existence and uniqueness conclusion. (Answer: The zero-yield price is $108$; exactly one positive yield exists.)
A three-year bond pays $6$ dollars annually and redeems for $100$ dollars at year 3. Its price is $118$ dollars. Determine whether its yield is positive, zero, or negative, and whether it has a positive yield solution. (Answer: The undiscounted total is $6+6+106=118$; the yield is zero, so there is no positive yield.)
A three-period bond pays $1.50$ dollars at each of the first two half-year ends and $101.50$ dollars at the third. Its price is $100.50$ dollars. State whether a unique positive yield per half-year exists and explain the boundary comparison. (Answer: The zero-yield price is $104.50$; exactly one positive yield per half-year exists.)
A bond makes positive payments of $3$, $3$, and $103$ dollars at the ends of years 1, 2, and 3, respectively. Its price is $100$ dollars. State whether its positive yield, if any, is unique, and whether the price boundary alone gives its numerical value. (Answer: Exactly one positive yield exists because $0<100<109$; the boundary comparison does not give its numerical value.)

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