Coupon Rate Effects On Macaulay Duration

Why Coupon Rate Changes Duration

Two bonds can have the same maturity and yield yet differ in how soon their present value is received. A bond with a larger coupon pays more cash before maturity. Since Macaulay duration is a present-value-weighted average time, those earlier payments can change its duration even though the final repayment date is unchanged.

This tutorial compares fixed-maturity annual-pay bonds, changing only the coupon amount. The comparison is aligned with the SOA Exam FM (Financial Mathematics) syllabus. To isolate the coupon effect, the bonds must have the same face amount, annual effective yield, and maturity.

Definition: For an annual-pay bond with cash flow $CF_t$ at time $t$, annual effective yield $i$, and price $P$, the Macaulay duration $D_M$, measured in years, is

P=\sum_{t=1}^{n} CF_t v^t,\qquad D_M=\frac{\sum_{t=1}^{n} t\,CF_t v^t}{P}, \qquad v=\frac{1}{1+i}.

Here $CF_t v^t$ is the present value of the cash flow at time $t$. The yield must satisfy $i>-1$, so the discount factor is positive. This formula uses annual cash-flow times and an annual effective yield; the time unit and yield period must agree.

For a bond with face amount $F>0$, annual coupon rate $r$, and maturity $n$ years, the annual coupon is $C=Fr$. It pays $C$ at the end of years $1,\ldots,n-1$, and $C+F$ at the end of year $n$. Its price and duration are therefore

P=C a_{\overline{n}|}+Fv^n,\qquad D_M= \frac{C\sum_{t=1}^{n} t v^t+nFv^n} {C a_{\overline{n}|}+Fv^n}, \qquad a_{\overline{n}|}=\sum_{t=1}^{n}v^t.

The coupon at maturity is included in the $C\sum t v^t$ term, while the face amount is included separately at time $n$. This separation makes it possible to see how changing $C$ affects the time-weighted average.

The Coupon-Rate Comparison

Suppose two annual-pay bonds have identical $F$, $n$, and $i$, but different coupon amounts. If $n\geq2$, the bond with the larger coupon has the smaller Macaulay duration. The reason is not simply that its price is higher: its added coupon value is received at times before maturity, pulling the present-value-weighted average time earlier.

Theorem: For fixed $F>0$, annual effective yield $i>-1$, and integer maturity $n\geq2$, Macaulay duration is strictly decreasing as the annual coupon amount $C$ increases. If $n=1$, every such bond has duration exactly one year, regardless of its coupon.

To verify the strict comparison, define

A=\sum_{t=1}^{n}v^t,\qquad B=\sum_{t=1}^{n}t v^t,\qquad Q=Fv^n.

Then the bond price is $CA+Q$, and the time-weighted present-value total is $CB+nQ$. Thus $D_M(C)=(CB+nQ)/(CA+Q)$. Differentiating with respect to the coupon amount gives

\frac{dD_M}{dC} =\frac{B(CA+Q)-A(CB+nQ)}{(CA+Q)^2} =\frac{Q(B-nA)}{(CA+Q)^2}.

The denominator is positive, as is $Q$. Also,

B-nA=\sum_{t=1}^{n}(t-n)v^t.

For $n\geq2$, the terms with $t<n$ are strictly negative and the term with $t=n$ is zero. Therefore $B-nA<0$, so $dD_M/dC<0$. This proves that a larger coupon strictly reduces duration under the stated assumptions. If $n=1$, the only payment is at time 1, so the weighted average time is 1 whatever the payment amount; in the derivative argument, $B-nA=0$.

Worked Examples

Worked Example: Comparing Three-Year Bonds at the Same Yield

Two annual-pay bonds each have face amount $1,000 and mature in three years. The first has a 4% annual coupon rate and the second an 8% rate. Both have an annual effective yield of 5%. Calculate their prices and Macaulay durations, then identify which duration is shorter. Give durations to four decimal places.

Method. Calculate each bond’s present-value total and time-weighted present-value total using the same annual discount factors. The annual coupons are $C_1=1{,}000(0.04)=\$40$ and $C_2=1{,}000(0.08)=\$80$.

At $i=0.05$, the discount factors are $v=1/1.05=0.952380952$, $v^2=0.907029478$, and $v^3=0.863837599$. The sums needed for the coupon stream are

A=v+v^2+v^3=2.723248029,\qquad B=v+2v^2+3v^3=5.357952703.

For the 4% bond, the price is

P_1=40A+1{,}000v^3 =40(2.723248029)+1{,}000(0.863837599) =108.929921+863.837599 =\$972.767520.

Its time-weighted present-value total is

N_1=40B+3(1{,}000)v^3 =40(5.357952703)+3{,}000(0.863837599) =214.318108+2{,}591.512797 =2{,}805.830905.

Therefore $D_{M,1}=N_1/P_1=2{,}805.830905/972.767520=2.8844$ years.

For the 8% bond, the price and time-weighted total are

P_2=80A+1{,}000v^3 =80(2.723248029)+863.837599 =217.859842+863.837599 =\$1{,}081.697441,

N_2=80B+3{,}000v^3 =80(5.357952703)+2{,}591.512797 =428.636216+2{,}591.512797 =3{,}020.149013.

Thus $D_{M,2}=3{,}020.149013/1{,}081.697441=2.7920$ years. The 8% bond has the shorter duration. As a check, both results lie between year 1 and year 3, and the higher-coupon bond has the smaller duration, as the theorem predicts. Round only the final durations to four decimal places; prices above are shown to six decimal places to make the calculation traceable.

Common mistake avoided. The final coupon is paid at year 3 as part of the total $C+F$, but it is still a coupon and belongs in the coupon-stream contribution. Omitting it from the $C A$ and $C B$ terms would understate both price and the time-weighted total.

Worked Example: A Zero-Yield Check on the Direction

Two annual-pay bonds have face amount $1,000, a two-year maturity, and annual effective yield 0%. Their annual coupons are $100 and $300. Calculate both durations and compare them. Give durations to four decimal places.

Method. At a 0% yield, $v=1$, so present values equal the cash flows. The first bond pays $100 at year 1 and $1,100 at year 2. Its price and time-weighted total are

P_1=100+1{,}100=\$1{,}200,\qquad N_1=1(100)+2(1{,}100)=100+2{,}200=\$2{,}300.

Therefore $D_{M,1}=2{,}300/1{,}200=1.9167$ years.

The second bond pays $300 at year 1 and $1,300 at year 2. Hence

P_2=300+1{,}300=\$1{,}600,\qquad N_2=1(300)+2(1{,}300)=300+2{,}600=\$2{,}900.

Therefore $D_{M,2}=2{,}900/1{,}600=1.8125$ years. The $300-coupon bond has the shorter duration. The calculation checks the direction even without discounting: the higher coupon puts a larger share of the total value at year 1. Durations are rounded to four decimal places.

Common mistake avoided. The time-weighted total multiplies the year-1 cash flow by 1 and the year-2 cash flow by 2. For example, the first bond’s total is $100+2(1{,}100)=2{,}300$, not $100+2(100)+2(1{,}000)$ with an extra year-2 coupon counted twice.

Worked Example: The One-Year Boundary Case

Two annual-pay bonds each have face amount $1,000, mature in one year, and have an annual effective yield of 6%. Their coupon rates are 3% and 9%. Find each price and duration, and explain whether the coupon-rate comparison is strict. Report prices to the nearest cent.

Method. With a one-year maturity, each bond makes only one payment, at time 1. The coupons are $1{,}000(0.03)=\$30$ and $1{,}000(0.09)=\$90$, so the maturity payments are $1,030 and $1,090.

The first price is $1{,}030/1.06=\$971.698113$, or $971.70 to the nearest cent. Its duration is $1(1{,}030/1.06)/(1{,}030/1.06)=1.0000$ year. The second price is $1{,}090/1.06=\$1{,}028.301887$, or $1,028.30 to the nearest cent. Its duration is $1(1{,}090/1.06)/(1{,}090/1.06)=1.0000$ year.

The coupon rate changes the payment amount and price, but not the time of the sole payment. Thus there is no strict duration reduction when maturity is one year. The common mistake is to apply the higher-coupon, shorter-duration rule without checking that $n\geq2$.

Interpreting the Result and Avoiding Traps

The weighted-average interpretation gives a useful way to remember the comparison. Each present value $CF_t v^t$ is a positive weight, and duration is the average of the payment times using those weights. Raising the coupon adds present value at each annual coupon date, including the final date, while the face amount remains at maturity. For $n\geq2$, some added coupon value is necessarily assigned to dates earlier than maturity. This pulls the average time earlier.

The result is conditional, not a rule for comparing any two bonds that happen to have different coupons. If yield, maturity, face amount, or payment frequency also changes, this proof does not isolate the coupon effect. In particular, yields determine the discount factors used in the present-value weights. Compare coupon rates only after confirming the other assumptions are held fixed.

Do not confuse coupon rate with yield. The coupon rate determines the scheduled coupon amount; the yield determines the discount factors used to price the cash flows.
Keep the time unit consistent. The formulas here use annual coupons, annual payment times, and an annual effective yield.
Include redemption at maturity. The face amount contributes $Fv^n$ to price and $nFv^n$ to the time-weighted total.
Check the maturity. The strict coupon comparison requires at least one coupon date before maturity. When $n=1$, duration is exactly one year for every coupon.

Key takeaway: With the same positive face amount, annual effective yield, and maturity of at least two years, the higher-coupon annual-pay bond has the lower Macaulay duration. At one-year maturity, both durations equal one year.

Check Your Understanding

For each comparison, use annual coupons and the stated annual effective yield. Answers are given so you can check your calculations.

Two two-year bonds have face amount $500 and yield 0%. Their coupon rates are 5% and 15%. Calculate their durations and identify the shorter one. (Answer: 5% coupon: $1.9545$ years; 15% coupon: $1.8846$ years; the 15% coupon bond is shorter.)
Two three-year bonds have face amount $1,000 and yield 0%. Their annual coupons are $50 and $150. Calculate each duration. (Answer: $50 coupon: $3{,}300/1{,}150=2.8696$ years; $150 coupon: $3{,}900/1{,}450=2.6897$ years.)
Two one-year bonds each have face amount $1,200 and yield 5%, with coupon rates 6% and 11%. Find the prices to the nearest cent and compare durations. (Answer: prices are $1,211.43 and $1,268.57; both durations are $1.0000$ year.)
Two two-year bonds have face amount $1,000 and yield 10%, with annual coupons of $50 and $100. Calculate durations and identify which is shorter. (Answer: $50 coupon: $431/221=1.9502$ years; $100 coupon: $21/11=1.9091$ years; the $100 coupon bond is shorter.)

What You'll Learn