Maturity Effects On Macaulay Duration

Why Maturity Changes Duration

Maturity tells us when a bond’s principal is repaid. Macaulay duration, by contrast, summarizes the timing of all the bond’s payments using their present values as weights. Extending maturity moves the principal payment later, but it also changes that payment’s present value and adds another coupon. The resulting duration change depends on the whole cash-flow pattern, not just on the new maturity date.

A useful comparison holds the face amount, coupon amount, payment frequency, and yield fixed, changing only the maturity. In many ordinary comparisons, the longer bond has a greater Macaulay duration. However, it is important not to turn that common pattern into an unconditional rule: under some assumptions, an extension can reduce duration.

Definition: For a bond with positive cash flows $C_t$ at times $t=1,\ldots,n$, priced at effective yield $i$ per payment period, its Macaulay duration in payment periods is $$ D_n=\frac{\sum_{t=1}^{n} tC_t v^t}{\sum_{t=1}^{n} C_t v^t}, \qquad v=\frac{1}{1+i}. $$ The denominator is the bond price, and each time $t$ is weighted by the present value of the cash flow at that time. With annual payments, duration is in years. With $m$ payments per year, a duration measured in payment periods is converted to years by dividing by $m$.

For an annual level-coupon bond with face amount $F$, annual coupon $C$, and maturity $n$, the cash flow is $C$ at each year-end through year $n$, plus $F$ at year $n$. Thus the last payment is $C+F$. Assuming annual effective yield $i$, its price and duration are

P_n=C\sum_{t=1}^{n}v^t+Fv^n, \qquad D_n=\frac{C\sum_{t=1}^{n}tv^t+nFv^n}{P_n}.

These formulas require a fixed yield, fixed coupon amount, fixed face amount, and the stated payment dates. Comparing bonds whose yields or coupon terms also change does not isolate the effect of maturity.

A Direct Comparison With the Same Bond Terms

The first calculation illustrates the common pattern when the coupon rate equals the yield. The price is at par, so the present-value weights are particularly easy to check.

Worked Example: Extending a Par Bond from Two to Four Years

Two hypothetical bonds each have face amount $F=\$1{,}000$ and annual coupon $C=\$100$. Both have annual effective yield $i=10\%$. Bond A matures in two years; Bond B matures in four years. Compare their Macaulay durations in years.

Method. Discount each annual cash flow at the common annual yield, then divide the present-value-weighted sum of payment times by price. Here $v=1/1.10=0.9090909$.

For Bond A, the cash flows are $\$100$ at year 1 and $\$1{,}100$ at year 2:

P_2=100(0.9090909)+1100(0.8264463) =90.9091+909.0909 =\$1{,}000.0000. $$ $$ D_2=\frac{1(90.9091)+2(909.0909)}{1000} =\frac{1909.0909}{1000} =1.9091\text{ years}.

For Bond B, the cash flows are $\$100$ at years 1, 2, and 3, and $\$1{,}100$ at year 4. The present values, including the separate year-4 coupon and principal, are:

\begin{aligned} PV_1&=100(0.9090909)=90.9091,\\ PV_2&=100(0.8264463)=82.6446,\\ PV_3&=100(0.7513148)=75.1315,\\ PV_4&=1100(0.6830135)=751.3149. \end{aligned}

The price is $90.9091+82.6446+75.1315+751.3149=\$1{,}000.0001$, with the small discrepancy from $\$1{,}000$ due to displayed rounding. Using the unrounded discount factors gives the duration numerator

90.9091+2(82.6446)+3(75.1315)+4(751.3149) \approx 3486.852.

Therefore $D_4\approx3486.852/1000=3.4869$ years. Extending maturity from two to four years raises duration from $1.9091$ to $3.4869$ years, but the four-year duration remains less than the four-year maturity. Common mistake avoided: duration is not simply the maturity; earlier coupon payments pull the weighted average earlier.

Why an Extension Often Raises Duration

For a bond with maturity $n$, extending maturity to $n+1$ leaves the coupons at years 1 through $n$ unchanged. It removes the principal $F$ from year $n$, then pays that principal with the year-$n+1$ coupon at year $n+1$. The duration effect can be derived by treating this as a replacement of one discounted cash flow by another.

Identify the present values being replaced.
At the old maturity, the principal has present value $a=Fv^n$ at time $n$. At the new maturity, the year-$n+1$ coupon and principal together have present value $b=(C+F)v^{n+1}$ at time $n+1$.

Keep the other cash flows fixed.
All coupons through year $n$ remain in place. If their combined price is $P_n$ and their weighted-time numerator is $P_nD_n$, the new price is $P_{n+1}=P_n-a+b$.

Compare the two weighted averages.
The new numerator is $P_nD_n-na+(n+1)b$. Subtracting $D_n$ from the new duration gives the exact change below.

D_{n+1}-D_n =\frac{a(D_n-n)+b(n+1-D_n)}{P_{n+1}}.

To verify the result, write $D_{n+1}=[P_nD_n-na+(n+1)b]/P_{n+1}$, substitute $P_{n+1}=P_n-a+b$, and subtract $D_n$. The $P_nD_n$ terms cancel, leaving the displayed expression. Since the bond has positive cash flows at times before $n$, its old duration is below $n$. The first term in the numerator is therefore negative, while the second is positive. Whether the extension raises duration depends on their relative sizes.

This explains both the usual intuition and its limitation. The later payment at $n+1$ pulls the average later, but its present value $b$ can be much smaller than the principal’s former present value $a$. For common bond terms and maturities, the later payment’s effect often dominates. It is not enough, however, to observe that the maturity date moved later and conclude that duration must rise.

Comparing a Discount Bond at Two Maturities

The next example keeps the annual coupon and yield fixed while extending maturity. It also shows that a bond need not be at par for the comparison to work.

Worked Example: Extending a Low-Coupon Bond

A hypothetical bond has face amount $\$1{,}000$, annual coupon $\$40$, and annual effective yield $i=10\%$. Compare its duration at one-year and three-year maturities.

Method. Use $v=1/1.10=0.9090909$, list each maturity’s cash flows, and calculate the present-value-weighted time. For one year, the only payment is $\$1{,}040$:

P_1=\frac{1040}{1.10}=\$945.4545, \qquad D_1=\frac{1(945.4545)}{945.4545}=1.0000\text{ year}.

For three years, the cash flows are $\$40$, $\$40$, and $\$1{,}040$. Their present values are:

\begin{aligned} PV_1&=40(0.9090909)=36.3636,\\ PV_2&=40(0.8264463)=33.0579,\\ PV_3&=1040(0.7513148)=781.3674. \end{aligned}

Thus $P_3=36.3636+33.0579+781.3674=\$850.7889$. The weighted-time numerator is $36.3636+2(33.0579)+3(781.3674)=\$2{,}446.5816$, so

D_3=\frac{2446.5816}{850.7889}=2.8757\text{ years}.

The three-year bond has greater duration than the one-year bond. Common mistake avoided: the maturity payment is $\$1{,}040$, not just the $\$1{,}000$ face amount; the final coupon must also be included.

The Increase Is Not Guaranteed

The exact change formula shows why a blanket rule would be unsafe. If the coupon is small relative to the face amount and the yield is high, the principal’s present value can shrink substantially when it is moved one period later. At sufficiently long maturities, that reduction can outweigh the time extension.

Worked Example: A Long-Maturity Extension That Reduces Duration

Consider a hypothetical annual-coupon bond with face amount $F=\$100$, annual coupon $C=\$1$, and annual effective yield $i=10\%$. Compare maturity at 100 years with maturity at 101 years. This deliberately extreme comparison tests whether duration must always rise with maturity.

Method. For an $n$-year bond, the present-value sum of the coupons is $C(1-v^n)/i$, and the coupon time numerator is $C[110-10(n+11)v^n]$ at $i=10\%$. The principal contributes $Fv^n$ to price and $nFv^n$ to the numerator. Use $v=1/1.10$, $v^{100}\approx0.000072566$, and $v^{101}\approx0.000065969$.

At 100 years, the coupon present value is $10(1-0.000072566)=9.999274$; the principal present value is $100(0.000072566)=0.007257$. Hence $P_{100}\approx10.006531$. The coupon time numerator is $110-10(111)(0.000072566)\approx109.919452$, and the principal time numerator is $100(100)(0.000072566)\approx0.725660$. Therefore,

D_{100}\approx\frac{109.919452+0.725660}{10.006531} =11.0573\text{ years}.

At 101 years, the coupon present value is $10(1-0.000065969)=9.999340$, and the principal present value is $100(0.000065969)=0.006597$. Thus $P_{101}\approx10.005937$. The coupon time numerator is $110-10(112)(0.000065969)\approx109.926115$, and the principal time numerator is $100(101)(0.000065969)\approx0.666287$. Therefore,

D_{101}\approx\frac{109.926115+0.666287}{10.005937} =11.0527\text{ years}.

Duration falls by about $0.0046$ years even though maturity moves from 100 to 101 years. The change is small, but it refutes the claim that every maturity extension must increase duration. Common mistake avoided: treating later timing as the only effect; the present value of the shifted principal also changes.

Practical Comparison Checks

When comparing two coupon bonds, first check that the comparison isolates maturity. If coupon rates, yields, payment frequencies, or redemption amounts differ, the duration difference cannot be attributed to maturity alone. Keep the yield measured per payment period consistent with the cash-flow dates, and state duration in the corresponding time unit.

A reliable calculation also checks that the result is a weighted average of payment times. With positive cash flows, duration must lie between the first and last payment dates. It will generally be earlier than maturity for a coupon bond because coupons arrive before principal. If a calculated duration exceeds maturity, or is reported in years when it was calculated in semiannual periods without conversion, review the cash flows and units.

Hold coupon, face amount, payment frequency, and yield fixed when isolating maturity.
Include both coupon and principal in the final payment.
Use present values, not undiscounted cash flows, as the duration weights.
Describe a duration increase as a common pattern, not as a universal law.

Key takeaway: Macaulay duration is a present-value-weighted average of payment times. Extending maturity replaces the old principal payment with a later, differently discounted payment that includes another coupon. The usual direction can be checked from the cash flows; maturity alone does not determine it.

This treatment of maturity comparisons is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each item, use the stated annual yield and payment timing. Durations are in years unless stated otherwise.

Two zero-coupon bonds have the same face amount and annual effective yield. One matures in 6 years and the other in 9 years. What are their Macaulay durations? (Answer: 6 years and 9 years, respectively.)
A bond with face amount $\$1{,}000$, annual coupon $\$100$, and annual effective yield $10\%$ matures in either one or two years. Calculate both durations. (Answer: 1.0000 years and 1.9091 years.)
A bond has face amount $\$1{,}000$, annual coupon $\$50$, and annual effective yield $5\%$. Compare maturities of two and three years. (Answer: $D_2=1.9524$ years and $D_3\approx2.8594$ years.)
A bond’s annual coupon is $\$1$ on face amount $\$100$, and its annual effective yield is $10\%$. Using the 100-year and 101-year values in the worked comparison, does duration rise or fall? (Answer: It falls, from approximately 11.0573 to 11.0527 years.)
For a maturity extension, suppose the old duration is $D_n=4$, the old maturity is $n=5$, the present value removed from time 5 is $a=\$20$, the present value added at time 6 is $b=\$18$, and the new price is $\$900$. Find the duration change using the replacement formula. (Answer: $[20(4-5)+18(6-4)]/900=0.0178$ years, an increase.)

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