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VEE Accounting and Finance · Bond Yields and Interest-Rate Risk · Tutorial 836 of 996

How Coupon Size Affects Bond Price Sensitivity

Learn how coupon size changes a bond’s percentage price response to a yield change, using cash-flow timing and fully worked comparisons.

Intermediate 10 min read

What You'll Learn

  • Define percentage price sensitivity using the initial bond price as the denominator
  • Compare coupon bonds while holding face value, maturity, and yield change fixed
  • Explain why lower coupons shift present value toward the later principal payment
  • Use an exact price-ratio argument to justify the sensitivity comparison
  • Distinguish dollar price changes from percentage changes
  • Recognize when coupon size does not affect percentage sensitivity

Why Coupon Size Matters

Two bonds can have the same face value, maturity, and yield yet react differently when market yields change. The key difference is when their promised cash flows arrive. A bond with a large coupon pays more of its total cash flow early; a low-coupon bond places relatively more value on the principal payment at maturity. Since later cash flows are more affected by a yield change, the low-coupon bond generally has the greater percentage price sensitivity.

The comparison must hold other features fixed. If maturity, face value, yield, payment frequency, or the size of the yield change differs, the observed price responses cannot be attributed to coupon size alone. Also distinguish a dollar price change from a percentage price change: a higher-priced bond can have a larger dollar change even when its percentage change is smaller.

Definition: For an initial bond price \(P_0\) and a later price \(P_1\), the percentage price change is \((P_1-P_0)/P_0\). For a price decline, the percentage decline is \((P_0-P_1)/P_0\). Percentage price sensitivity describes the size of this response relative to the initial price, for a stated change in yield.

Bond Price and the Cash-Flow Timing Effect

Assume a bond has face value \(F\), pays an annual coupon at rate \(c\), and matures in \(n\) years. Its annual coupon is \(C=cF\), and its principal repayment is \(F\) at maturity. Assume the bond is valued just after a coupon date, all cash flows are certain, and \(y\) is its effective annual yield, compounded annually. The price is the present value of the remaining coupons and principal:

$$ P(y)=\sum_{t=1}^{n}\frac{C}{(1+y)^t}+\frac{F}{(1+y)^n} $$

The yield and cash-flow periods must match. For example, with semiannual coupons, the cash flows and yield must both be expressed on a semiannual basis before discounting. The annual formula above does not apply unchanged to mismatched periods or to a bond priced between coupon dates without accrued-interest adjustments.

A coupon bond’s cash flows occur on multiple dates. Reducing the coupon leaves the principal repayment unchanged but reduces the earlier payments. Consequently, a larger share of the bond’s present value comes from the later principal. This is the mechanism behind the general comparison; it is not a claim that every pair of bonds with different terms can be ranked by coupon alone.

Key result: For bonds with the same face value and maturity, valued at the same yield and exposed to the same yield increase, the lower-coupon bond generally has the larger percentage price decline. The comparison assumes fixed promised cash flows and the same payment frequency. It is a comparison of percentage changes, not necessarily dollar changes.

Why Later Cash Flows Increase Sensitivity

The comparison can be justified directly from the present-value formula. Let \(PV_t(y)\) be the present value at yield \(y\) of the cash flow paid at time \(t\). Its share of the bond’s initial price is \(w_t=PV_t(y)/P(y)\). These shares are nonnegative and sum to one. If yield rises by \(\Delta y>0\), the price ratio is:

$$ \frac{P(y+\Delta y)}{P(y)} = \sum_{t=1}^{n} w_t \left(\frac{1+y}{1+y+\Delta y}\right)^t $$

To see why this holds, discount a cash flow \(CF_t\) first at \(y\), giving \(PV_t(y)=CF_t/(1+y)^t\). Discounting that same cash flow at \(y+\Delta y\) gives \(PV_t(y)\big((1+y)/(1+y+\Delta y)\big)^t\). Summing those new present values and dividing by the initial price gives the displayed weighted sum.

Because \(y\) and \(\Delta y\) are positive, the factor \((1+y)/(1+y+\Delta y)\) is less than one. Raising it to a larger power \(t\) makes the factor smaller. Thus, each dollar of initial value assigned to a later cash flow experiences a larger proportional reduction. Reducing the coupon shifts the present-value shares away from the earlier coupon dates and toward the principal at maturity. The weighted price ratio falls, so the percentage decline rises. This reasoning applies to a specified finite yield change and does not require a small-change approximation.

Worked Example: Comparing Two Coupons at the Same Maturity

Two hypothetical bonds each have face value $1,000 and two years remaining. Bond L pays an annual coupon rate of 2%; Bond H pays 8%. Both are valued first at an effective annual yield of 4%, then at 5%. Compare their price declines as percentages of their initial prices. Use annual coupons and round prices to the nearest cent and percentage declines to three decimal places.

Method. Price each bond by discounting its year-1 coupon and its year-2 coupon plus principal at the stated yield. Then divide each dollar decline by that bond’s own initial price. This avoids comparing dollar changes alone.

Bond L, 2% coupon. Its annual coupon is \(0.02(1{,}000)=\$20\). At a 4% yield:

$$ P_L(4\%)=\frac{20}{1.04}+\frac{1{,}020}{1.04^2} =19.230769+943.047337 =\$962.278106 $$

At a 5% yield:

$$ P_L(5\%)=\frac{20}{1.05}+\frac{1{,}020}{1.05^2} =19.047619+925.170068 =\$944.217687 $$

The decline is \(962.278106-944.217687=\$18.060419\). Its percentage decline is \(18.060419/962.278106=0.018768\), or 1.877% to three decimal places. The rounded prices are $962.28 and $944.22.

Bond H, 8% coupon. Its annual coupon is \(0.08(1{,}000)=\$80\). At a 4% yield:

$$ P_H(4\%)=\frac{80}{1.04}+\frac{1{,}080}{1.04^2} =76.923077+998.520710 =\$1{,}075.443787 $$

At a 5% yield:

$$ P_H(5\%)=\frac{80}{1.05}+\frac{1{,}080}{1.05^2} =76.190476+979.591837 =\$1{,}055.782313 $$

The decline is \(1{,}075.443787-1{,}055.782313=\$19.661474\). Its percentage decline is \(19.661474/1{,}075.443787=0.018282\), or 1.828% to three decimal places. The rounded prices are $1,075.44 and $1,055.78. Bond H loses more dollars, but Bond L loses more as a percentage of its initial price. That is the relevant comparison for percentage sensitivity.

Check. Both bonds’ prices fall when yield rises. The low-coupon bond has the greater percentage decline, consistent with its larger share of value arriving at maturity. The common mistake is to compare the $18.06 and $19.66 dollar declines and conclude that the higher-coupon bond is more sensitive.

Worked Comparisons and Boundary Cases

Worked Example: A Zero-Coupon Bond Versus a Coupon Bond

A hypothetical two-year zero-coupon bond and a two-year bond with a 6% annual coupon each have face value $1,000. Their yields increase from 4% to 5%. Compare their percentage price declines, rounding prices to the nearest cent and declines to three decimal places.

Method. The zero-coupon bond has only one cash flow, the principal at year 2. The coupon bond pays $60 at year 1 and $1,060 at year 2. Apply the same yields to both cash-flow patterns.

For the zero-coupon bond:

$$ P_Z(4\%)=\frac{1{,}000}{1.04^2}=\$924.556213,\qquad P_Z(5\%)=\frac{1{,}000}{1.05^2}=\$907.029478 $$

Its decline is \(924.556213-907.029478=\$17.526735\), and its percentage decline is \(17.526735/924.556213=0.018957\), or 1.896%. The rounded prices are $924.56 and $907.03.

For the coupon bond:

$$ P_C(4\%)=\frac{60}{1.04}+\frac{1{,}060}{1.04^2} =57.692308+980.029586 =\$1{,}037.721894 $$
$$ P_C(5\%)=\frac{60}{1.05}+\frac{1{,}060}{1.05^2} =57.142857+961.451247 =\$1{,}018.594104 $$

Its decline is \(1{,}037.721894-1{,}018.594104=\$19.127790\), and its percentage decline is \(19.127790/1{,}037.721894=0.018432\), or 1.843%. The rounded prices are $1,037.72 and $1,018.59. The zero-coupon bond has the greater percentage decline because all of its value is paid at the later date, even though the coupon bond has the larger dollar decline.

Check. The denominator for each percentage is that bond’s own initial price. Using the coupon bond’s dollar decline divided by the zero-coupon bond’s price would not measure either bond’s percentage change.

Worked Example: When Different Coupons Have Equal Percentage Sensitivity

Two hypothetical one-year bonds each have face value $1,000. One has a 3% coupon and the other a 9% coupon. Their effective annual yield rises from 4% to 6%. Find each price change and compare the percentage declines. Round prices to the nearest cent and the percentage to three decimal places.

Method. Since each bond pays only once, at the end of year 1, its entire cash flow is discounted for the same single period. This is an important boundary case: the coupon changes the size of the one payment but not its timing.

The 3% bond pays $1,030. Its prices are:

$$ P_{3\%}(4\%)=\frac{1{,}030}{1.04}=\$990.384615,\qquad P_{3\%}(6\%)=\frac{1{,}030}{1.06}=\$971.698113 $$

Its decline is \(990.384615-971.698113=\$18.686502\). Dividing by its initial price gives \(18.686502/990.384615=0.018868\), or 1.887%.

The 9% bond pays $1,090. Its prices are:

$$ P_{9\%}(4\%)=\frac{1{,}090}{1.04}=\$1{,}048.076923,\qquad P_{9\%}(6\%)=\frac{1{,}090}{1.06}=\$1{,}028.301887 $$

Its dollar decline is \(1{,}048.076923-1{,}028.301887=\$19.775036\), but its percentage decline is \(19.775036/1{,}048.076923=0.018868\), also 1.887%. In fact, for a single payment at time 1, the price ratio is \(1.04/1.06\) for either bond; multiplying the payment by a different coupon amount changes both prices proportionally, not the percentage response.

Check. The higher-coupon bond has the larger dollar decline, while both have the same percentage decline. Coupon size has no timing effect when there is only one payment date.

Common Traps and Practical Interpretation

The coupon comparison is useful because it isolates one feature of bond cash flows, but it does not say that coupon rate alone determines risk. A bond’s response also depends on maturity, yield, payment frequency, and the size and direction of the yield change. The price comparisons here assume promised cash flows remain fixed; they do not account for default, embedded options, or changes in other required yields.

  • Comparing dollars instead of percentages. Divide a price change by the initial price of the same bond. A larger dollar loss does not automatically mean greater percentage sensitivity.
  • Changing more than the coupon. To attribute a difference to coupon size, keep face value, maturity, yield basis, and yield shift the same.
  • Forgetting the principal payment. The final-period cash flow is the last coupon plus face value, not just face value.
  • Mixing annual and periodic rates. Match the yield period to the payment period before discounting. A nominal annual quote is not automatically the effective annual yield used in the formula.
  • Turning a general comparison into an absolute rule. The lower-coupon result is most informative when the bonds are otherwise alike. It does not rank bonds whose other terms differ.

When yields fall rather than rise, fixed-cash-flow bond prices rise. Later cash flows also respond more strongly in that direction. The precise percentage increase should still be calculated from the initial price and the stated new yield; it need not be the same percentage as a yield increase of equal size.

Key takeaway: With maturity, face value, yield, and yield change held fixed, a lower coupon generally means that more present value is concentrated in the later principal payment. That later timing increases percentage price sensitivity. Always verify the comparison using each bond’s own initial price.

Check Your Understanding

For each problem, use the stated cash-flow timing and compare percentage changes using the initial price as the denominator.

  1. A two-year zero-coupon bond and a two-year bond with a 10% annual coupon each have face value $1,000. Yield rises from 0% to 10%. Find both percentage declines and identify the greater one. (Answer: Zero-coupon, 17.355%; coupon bond, 16.667%; the zero-coupon bond has the greater percentage decline.)
  2. Two one-year bonds have face value $1,000 and annual coupon rates of 2% and 11%. Yield rises from 4% to 6%. What are their percentage declines, and why are they equal? (Answer: 1.887% each; each bond has a single payment at the same date, so the payment amount cancels in the price ratio.)
  3. Two bonds have the same face value, maturity, payment frequency, and yield. One has a lower coupon. If yield rises by the same amount for both, which generally has the greater percentage decline, and what cash-flow feature explains it? (Answer: The lower-coupon bond; more of its present value is concentrated in the later principal payment.)
  4. A bond’s price falls from $950 to $930 after a yield change. Calculate its percentage decline, using the initial price as the base. (Answer: \(20/950=2.105\%\).)
  5. A two-year bond has face value $1,000 and a 5% annual coupon. Its effective annual yield changes from 4% to 5%. Find the price at each yield and the percentage decline, rounding prices to cents and the decline to three decimal places. (Answer: At 4%, $1,018.86; at 5%, $1,000.00; decline \(18.860947/1,018.860947=1.851\%\).)

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