Why Maturity Matters
A bond’s price is the present value of its promised cash flows. When market yield changes, a cash flow due later is discounted over more periods than a cash flow due sooner. As a result, postponing repayment generally makes a bond’s percentage price more sensitive to a given yield change, all else equal. The effect is especially clear for a zero-coupon bond, whose only payment is its maturity value.
A fair maturity comparison holds the other important terms fixed: the face value, coupon rate, coupon timing, and starting yield. The bonds should also be valued using the same yield convention. In the examples below, coupons are paid annually, and the annual yield is an effective annual yield applied to every cash flow. Cash flows are assumed fixed and positive, with no default or embedded options; a yield change means the same change at every payment date.
This is a general single-yield present-value model. It does not describe every bond valuation situation: for example, a bond with default risk, embedded options, or yields that differ by payment date may require additional assumptions and methods. The focus here is the maturity effect when otherwise comparable promised cash flows are discounted at a common yield.
The Basic Maturity Comparison
For a zero-coupon bond with face value \(F\), maturity \(n\), and yield \(y\), the price is \(F(1+y)^{-n}\). Extending maturity leaves the face value unchanged but pushes its payment farther away. For a given yield change, the longer-maturity zero therefore generally has a larger percentage price change.
Worked Example: Comparing Two Zero-Coupon Maturities
Two hypothetical zero-coupon bonds each promise $1,000 at maturity. One matures in 2 years and the other in 10 years. The yield is initially 0% and then rises to 10% per year. Find each bond’s initial price, new price, and percentage price change. Use the original price as the percentage-change base.
Method. Discount each maturity payment at the stated yield, then calculate \((\text{new price}-\text{initial price})/\text{initial price}\). This compares percentage effects rather than merely comparing dollar losses.
At a 0% yield, the discount factor is 1 for both maturities. Thus, each initial price is $1,000.
For the 2-year bond, the new price is \(1000/(1.10)^2=1000/1.21=\$826.45\). Its percentage change is \((826.4463-1000)/1000=-0.1735537\), or a decline of 17.36%.
For the 10-year bond, \((1.10)^{10}=2.59374246\), so the new price is \(1000/2.59374246=\$385.54\). Its percentage change is \((385.5433-1000)/1000=-0.6144567\), or a decline of 61.45%.
Check. Both bonds have the same face value and starting yield, and both are discounted at the new 10% yield. The longer-maturity bond has the larger percentage decline: 61.45% versus 17.36%. The mistake to avoid is treating the difference in dollar prices as the sensitivity measure; here the initial prices happen to match, but in many comparisons they will not.
The example uses a large yield move to make the contrast visible. For smaller changes, the same direction generally holds, but exact percentage changes need not be proportional to the size of the yield change. Repricing both bonds at the new yield is the direct way to measure the actual change.
Coupon Bonds: Matching Terms Matters
A coupon bond pays interest before maturity as well as repaying principal at maturity. Extending its maturity postpones principal repayment and adds coupon payments. The price response therefore depends on the timing of the full cash-flow stream, not just on the maturity date in isolation. When face value, coupon rate, and yield are matched, the longer bond generally has greater percentage sensitivity because more value is exposed to later cash flows.
Worked Example: Repricing Matched Coupon Bonds
Two hypothetical bonds have face value $1,000 and an annual coupon rate of 10%, or $100 per year. One matures in 2 years and the other in 6 years. Their initial annual yield is 10%, and it increases to 11%. Find the percentage price change for each bond.
Method. Price each bond from its annual coupons and maturity payment. Since the initial yield equals the coupon rate, each bond initially sells at face value: the coupon stream has present value \(100a_{\overline{n}|}\) and principal has present value \(1000v^n\), where \(v=1/1.10\). Because \(a_{\overline{n}|}=(1-v^n)/0.10\), their sum is \(1000(1-v^n)+1000v^n=\$1,000\).
For the 2-year bond at 11%, the year-1 coupon has present value \(100/1.11=\$90.0901\). The year-2 coupon and principal total $1,100, with present value \(1100/(1.11)^2=1100/1.2321=\$892.7847\). The new price is \(90.0901+892.7847=\$982.8748\), or $982.87 to the nearest cent. Its percentage change is \((982.8748-1000)/1000=-1.7125\%\), a decline of 1.71%.
For the 6-year bond, \(1/(1.11)^6=0.5346408\), and the 6-year annuity factor at 11% is \((1-0.5346408)/0.11=4.230538\). The coupon present value is \(100(4.230538)=\$423.0538\), and the principal present value is \(1000(0.5346408)=\$534.6408\). The new price is \(423.0538+534.6408=\$957.6946\), or $957.69 to the nearest cent. Its percentage change is \((957.6946-1000)/1000=-4.2305\%\), a decline of 4.23%.
Check. At the initial 10% yield, both bonds are at par, as shown above. After the same one-percentage-point increase, the 6-year bond declines by a larger percentage than the 2-year bond. A common mistake is to reprice the final principal but omit the final coupon; the $1,100 payment at maturity includes both.
This result is a comparison under matched terms, not a rule that maturity alone determines sensitivity. If one bond has a different coupon rate, yield, payment frequency, or cash-flow structure, maturity by itself may not be enough to rank the bonds. A longer-maturity bond is not automatically more sensitive than every shorter-maturity bond.
Why Relative Sensitivity Increases with Payment Time
The present-value formula also gives a precise explanation of why later payments matter. Differentiate price with respect to the common yield \(y\). For each fixed payment \(C_t\), the derivative is negative: an increase in yield lowers the present value. Dividing the magnitude of the total derivative by price converts that dollar response into a relative response.
Here \(w_t\) is the fraction of the bond’s price contributed by the payment at time \(t\). The expression \(\sum w_t t\) is the average payment time when each date is weighted by that payment’s share of present value. Thus, relative sensitivity is this weighted average time divided by \(1+y\). Moving present-value weight toward later dates increases the relative sensitivity, at a fixed yield.
For a small yield change \(\Delta y\), the first-order approximation is \(\Delta P/P\approx (dP/dy)\Delta y/P\). The negative sign in the derivative means a positive yield change produces a negative price change. This approximation is useful for understanding the direction and approximate size of a small move; it is not a substitute for exact repricing when the change is large.
Worked Example: Comparing Local Sensitivity from Present-Value Weights
At a 10% annual yield, compare a 2-year and a 6-year bond, each with face value $1,000 and annual coupons of $100. Both prices are $1,000. Calculate the relative sensitivity from the present-value weights, and estimate the percentage price change for a 1-percentage-point yield increase. Use the derivative approximation, not an exact repricing.
Method. Find the present value of each payment, divide each by the $1,000 bond price to obtain its weight, and calculate the weighted average payment time. Then divide by \(1+y=1.10\).
For the 2-year bond, the present values are \(100/1.10=\$90.9091\) at year 1 and \(1100/(1.10)^2=\$909.0909\) at year 2. Their weights are 0.0909091 and 0.9090909. The weighted average time is \(0.0909091(1)+0.9090909(2)=1.909091\) years. Relative sensitivity is \(1.909091/1.10=1.735537\) per unit change in yield.
For the 6-year bond, the present values at years 1 through 5 are, respectively, $90.9091, $82.6446, $75.1315, $68.3013, and $62.0921. The year-6 payment is $1,100, with present value \(1100/(1.10)^6=\$620.9213\). The present values sum to $1,000. Their time-weighted sum is \(1(90.9091)+2(82.6446)+3(75.1315)+4(68.3013)+5(62.0921)+6(620.9213)=\$4,790.7868\). The weighted average time is 4.790787 years, so relative sensitivity is \(4.790787/1.10=4.355261\) per unit change in yield.
A 1-percentage-point increase means \(\Delta y=0.01\). The estimated percentage changes are therefore \(-1.735537(0.01)=-1.7355\%\) for the 2-year bond and \(-4.355261(0.01)=-4.3553\%\) for the 6-year bond. Rounded to two decimals, the estimates are declines of 1.74% and 4.36%.
Check. Each set of present values sums to the $1,000 price, and the longer bond places more of its price in later payments. The mistake to avoid is multiplying the relative-sensitivity figure by 1 rather than 0.01: a one-percentage-point yield change is 0.01 when yield is expressed as a decimal.
Relative Sensitivity Is Not Dollar Sensitivity
The derivative \(dP/dy\) measures an absolute dollar change per unit change in yield. The ratio \(-P^{-1}dP/dy\) measures the corresponding relative change. These are different quantities. In particular, it is the relative sensitivity of a single payment that increases with its time to payment: for a unit payment at time \(t\), present value is \((1+y)^{-t}\), and its relative sensitivity is \(t/(1+y)\).
The absolute derivative for that same unit payment is instead \(-t/(1+y)^{t+1}\). It need not grow with \(t\), because the payment’s present value shrinks as it is moved farther away. For example, at a 10% yield, its absolute derivative is \(1/(1.1)^2\approx0.826\) at year 1, while at year 100 it is \(100/(1.1)^{101}\approx0.00660\). Saying that a later payment always has a larger absolute dollar response would therefore be incorrect.
There is a related comparison trap: a longer-maturity bond can have a smaller dollar price change simply because its initial price is lower. To compare percentage sensitivity, divide the dollar change by the relevant initial price. To isolate maturity, compare bonds with otherwise matched terms and compute their percentage changes, or examine the timing of their present-value weights.
Check Your Understanding
For each problem, compare percentage effects using the stated initial price. Round dollar prices to the nearest cent and percentages to two decimal places where needed.
- A hypothetical $500 zero-coupon bond matures in 3 years. Its yield rises from 4% to 5% per year. Find its initial and new prices and its percentage price change. (Answer: $444.50 initially, $431.92 after the change; a 2.83% decline.)
- A hypothetical 4-year bond has face value $2,000, annual coupons of $100, and an initial annual yield of 5%. The yield rises to 6%. Find the new price and percentage change. (Answer: $1,930.70; a 3.47% decline.)
- A bond priced at $1,000 has present-value weights 0.40 at year 1 and 0.60 at year 3. Its annual yield is 8%. Use the relative-sensitivity formula to estimate the percentage price change for a yield increase of 1 percentage point. (Answer: approximately a 2.04% decline.)
- Two hypothetical zero-coupon bonds each have face value $1,000. Their maturities are 5 and 8 years, and their yield rises from 5% to 6%. Calculate each percentage price change and identify which is more sensitive. (Answer: approximately a 4.63% decline for 5 years and a 7.30% decline for 8 years; the 8-year bond is more sensitive.)