How Coupon Amount Affects Yield At A Fixed Price

Introduction

A bond’s coupon amount and yield are connected through the present value of its promised payments. If price, redemption value, and term are held fixed, changing the coupon changes the yield that makes those cash flows worth exactly the stated price. This is a comparative question: it is not enough to observe that a larger coupon produces more cash flow; we must determine how the yield must adjust to preserve the same price.

The result is useful when comparing bonds with the same redemption value and payment dates, or when interpreting how a change in coupon terms affects yield. The key is to keep the comparison conditions explicit. In particular, a comparison that holds yield fixed answers a different question from one that holds price fixed.

Definition: Consider a conventional bond valued on a coupon date, with $n$ remaining coupon periods, level coupon amount $C$ paid at the end of each period, and redemption value $R$ paid with the final coupon. Its yield $i$ per coupon period is the rate that equates the price $P$ to the present value of these contractual payments: $$ P=C a_{\overline{n}|}+Rv^n =C\sum_{k=1}^{n}(1+i)^{-k}+R(1+i)^{-n}, \qquad v=(1+i)^{-1}. $$

This model assumes fixed, known payments and no default, call, or other contingency affecting the cash flows. There is no accrued interest in the comparison because valuation is on a coupon date. The yield is an internal rate of return for these cash flows; it is not a guarantee that an investor will earn that rate if the bond is sold early or payments are not reinvested consistently with the yield interpretation.

Here $C$ is the dollar coupon per period, not the coupon rate. The comparison in this tutorial holds $P$, $R$, and $n$ fixed while changing $C$. The yield $i$ is measured per coupon period. If a bond pays semiannually, for example, the yield variable in this equation is the rate per half-year, not automatically an effective annual rate.

Why a Larger Coupon Raises Yield at a Fixed Price

At any fixed yield, increasing the coupon adds value: each extra dollar of coupon is discounted from one of the payment dates. Its total present-value contribution is positive. Therefore, if the price is not allowed to change, the yield must rise to discount the larger cash flows more heavily.

The argument can be made precise. For $i>-1$, the price function decreases as $i$ increases, provided the bond has positive redemption value and nonnegative coupons. Differentiating the finite present-value sum gives

\frac{\partial P}{\partial i} =-C\sum_{k=1}^{n} k(1+i)^{-k-1} -nR(1+i)^{-n-1}<0.

Each term in the sum is nonnegative, and the redemption term is strictly negative when $R>0$. Thus the price is strictly decreasing in yield. This is the standard price-yield relationship for fixed positive bond cash flows.

Now keep $P$, $R$, and $n$ fixed while allowing both $C$ and its corresponding yield $i$ to vary. Differentiate the price equation with respect to $C$. The direct effect of an extra dollar of coupon is $a_{\overline{n}|}$, and the indirect effect is the change in present value caused by changing the yield. Since the total price must remain unchanged,

0=a_{\overline{n}|} +\frac{\partial P}{\partial i}\frac{di}{dC}, \qquad \frac{di}{dC} = \frac{a_{\overline{n}|}} {C\displaystyle\sum_{k=1}^{n} k(1+i)^{-k-1} +nR(1+i)^{-n-1}}>0.

The numerator is positive, and the denominator is positive under the stated assumptions. Hence a higher coupon corresponds to a higher yield when price, redemption value, and term are fixed. This derivative describes the local change in yield for a small change in coupon; it does not say that yield changes by a constant amount for every coupon increase.

Write the fixed-price equation.
Use the same price, redemption value, number of periods, and payment schedule for both bonds.

Identify the coupon change.
At a fixed yield, a higher coupon increases the present value by the coupon increase multiplied by $a_{\overline{n}|}$.

Adjust the yield in the opposite direction to price.
Because price falls as yield rises, the yield must increase to offset the added coupon value and restore the original price.

Worked Comparisons

The examples use the same conventional-bond model. For the multi-period comparison, all discount factors are computed from their defining powers, and intermediate values are retained at sufficient precision before rounding.

Worked Example: A One-Period Bond at a Fixed Nonpar Price

Two one-period bonds have redemption value $R=\$1{,}000$ and the same price $P=\$8{,}000/7$, or approximately $\$1{,}142.86$. Bond A pays a coupon of $\$200$. Bond B pays a coupon of $\$1{,}800/7$, approximately $\$257.14$. Find each yield per period and compare them.

Method. With one period remaining, the coupon and redemption value are both paid at the end of the period. Thus $P=(C+R)/(1+i)$, so $i=(C+R)/P-1$.

For Bond A, the total payment is $\$200+\$1{,}000=\$1{,}200$. Therefore,

i_A=\frac{1{,}200}{8{,}000/7}-1 =\frac{1{,}200(7)}{8{,}000}-1 =1.05-1 =0.05.

Bond A’s yield is $5\%$ per period. For Bond B, the total payment is $1{,}800/7+1{,}000=8{,}800/7$. Therefore,

i_B=\frac{8{,}800/7}{8{,}000/7}-1 =\frac{8{,}800}{8{,}000}-1 =1.10-1 =0.10.

Bond B’s yield is $10\%$ per period. The price, redemption value, and term are identical, while Bond B’s coupon is larger and so is its yield. As a check, discounting Bond A’s $\$1{,}200$ payment at $5\%$ gives $1{,}200/1.05=8{,}000/7$; discounting Bond B’s $8{,}800/7$ payment at $10\%$ gives $(8{,}800/7)/1.10=8{,}000/7$.

Common mistake avoided: Treating the two yields as equal because the prices are equal. Equal prices do not imply equal yields when the contractual payments differ.

Worked Example: Two Coupons Over Two Periods

A two-period bond has redemption value $\$1{,}000$. A bond paying $\$100$ at the end of each period has yield $4\%$ per period. A second bond has the same price, redemption value, and term, but pays a larger level coupon. Find the coupon amount that makes the second bond’s yield exactly $8\%$ per period, and verify the comparison.

Method. First calculate the common price using the first bond. Then solve the second bond’s present-value equation at the specified yield. For the first bond, the discount factors at $4\%$ are

v=\frac{1}{1.04}=0.961538461538\ldots, \qquad v^2=\frac{1}{1.04^2}=0.924556213018\ldots.

The two-period annuity factor is $a_{\overline{2}|}=v+v^2=1.886094674556\ldots$. The common price is therefore

P=100(1.886094674556\ldots) +1{,}000(0.924556213018\ldots) =1{,}113.165680474\ldots.

At $8\%$, the discount factors are

v=\frac{1}{1.08}=0.925925925926\ldots, \qquad v^2=\frac{1}{1.08^2}=0.857338820302\ldots, \qquad a_{\overline{2}|}=1.783264746228\ldots.

Let $C_2$ be the required coupon. Set the present value at $8\%$ equal to the common price:

C_2(1.783264746228\ldots) +1{,}000(0.857338820302\ldots) =1{,}113.165680474\ldots, $$ $$ C_2 =\frac{1{,}113.165680474\ldots-857.338820302\ldots} {1.783264746228\ldots} =143.4598316\ldots.

Thus the second bond’s coupon is approximately $\$143.46$ per period, and its yield is $8\%$ by construction. Retaining the unrounded coupon expression gives a present value of $1{,}113.165680474\ldots$, the same as the first bond. The first coupon is $\$100$, so the larger coupon is associated with the higher yield.

Common mistake avoided: Rounding the coupon to cents and then claiming the resulting yield is exactly $8\%$. The exact equality uses the unrounded coupon amount; a coupon of $\$143.46$ produces a very slightly different yield.

Worked Example: Interpreting Local Yield Sensitivity

A two-period bond has $C=\$100$, $R=\$1{,}000$, and yield $i=4\%$ per period. Estimate the local yield change for a $\$10$ increase in coupon if price, redemption value, and term remain fixed.

Method. Use the derivative $di/dC$ derived above. At $4\%$, $v^2=0.924556213018\ldots$, $v^3=0.889996359\ldots$, and $a_{\overline{2}|}=1.886094674556\ldots$. The denominator of the derivative is

100\bigl(v^2+2v^3\bigr)+2(1{,}000)v^3 =100(2.704548931\ldots)+2{,}000(0.889996359\ldots) =2{,}050.447611\ldots.

Therefore, the local sensitivity is

\frac{di}{dC} =\frac{1.886094674556\ldots}{2{,}050.447611\ldots} =0.0009198\ldots \quad\text{per dollar of coupon}.

For a $\$10$ increase, the local linear estimate is $\Delta i\approx 10(0.0009198)=0.009198$, or about $0.920$ percentage points per period. This is an approximation around the original coupon and yield, not an exact finite-change calculation. It gives a scale and direction for the yield adjustment; solving the full price equation is required for an exact new yield.

Common mistake avoided: Reporting $0.0009198$ as a percentage-point change. It is a change in the decimal yield per dollar; multiplying by $100$ converts a decimal yield change into percentage points.

What the Comparison Does—and Does Not—Say

The conclusion depends on holding price, redemption value, and term fixed. If instead yield is held fixed, a larger coupon raises the bond’s price by the coupon increase multiplied by $a_{\overline{n}|}$. These are consistent statements about opposite quantities being held constant: at a fixed yield, higher coupons mean a higher price; at a fixed price, higher coupons require a higher yield.

The comparison also assumes the same payment schedule and the same redemption date. If coupon frequency, redemption value, or maturity differs, the cash-flow equations differ in additional ways. Those changes cannot be attributed to coupon amount alone. For a conventional bond with positive redemption value, the price function is strictly decreasing in yield, so any yield that solves the fixed-price equation is unique. In this setting, the comparative result is unambiguous whenever the yields exist.

Common Exam Trap

A frequent error is to calculate the price effect of a larger coupon and mistake it for the yield effect under a fixed-price condition. For example, in the two-period comparison above, increasing the coupon from $\$100$ to approximately $\$143.46$ at a fixed $4\%$ yield would increase price by approximately

(143.4598316\ldots-100)(1.886094674556\ldots) \approx \$81.97.

That calculation is correct only when yield is held fixed. It does not preserve the original price. If price must remain fixed, the yield must rise instead; in the example it rises from $4\%$ to $8\%$. Before calculating, mark which quantities the problem holds fixed and which one it asks you to compare or solve.

Key takeaway: For conventional fixed cash flows, price decreases as yield increases. Consequently, if price, redemption value, and term are fixed, increasing the level coupon amount increases the yield that equates present value to price.

Check Your Understanding

For each question, use the stated per-period cash flows and keep the fixed quantities explicit.

A one-period bond has redemption value $\$1{,}000$, coupon $\$56$, and price $\$960$. Find its yield per period. (Answer: $10\%$)
Two one-period bonds have the same price of $\$1{,}100$ and redemption value of $\$1{,}000$. One pays a coupon of $\$100$; the other pays $\$155$. Find both yields and state which is higher. (Answer: $0\%$ and $5\%$, respectively; the higher-coupon bond has the higher yield.)
A two-period bond has price $\$1{,}000$, redemption value $\$1{,}000$, and level coupon $\$40$. Its yield is $4\%$ per period. A second bond has the same price, redemption value, and term, with coupon $\$60$. Its yield is $6\%$ per period. (Answer: $4\%$ for the first bond and $6\%$ for the second.)
A one-period bond has redemption value $\$1{,}000$, coupon $\$90$, and price $\$950$. Calculate its yield per period and explain how it compares with the yield of a bond with the same price, redemption value, and term but a coupon of $\$50$. (Answer: The $\$90$-coupon bond yields $14.7368\%$; the $\$50$-coupon bond has a lower yield.)
For fixed positive redemption value and term, a bond’s coupon increases while its price remains unchanged. State the direction of the yield change and give the price-yield fact that justifies it. (Answer: Yield increases, because bond price is strictly decreasing in yield for fixed positive cash flows.)

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