What Macaulay Duration Measures
A bond’s maturity tells you when its final payment is due, but not when the value represented by all its payments is received. A bond that pays coupons returns some value before maturity; a zero-coupon bond does not. Macaulay duration summarizes this timing by averaging the payment dates, with each date weighted by that payment’s share of the bond’s present value.
The weights are based on present values, not on the stated cash amounts alone. A dollar due later is worth less today than a dollar due sooner when the yield is positive. Discounting each payment before assigning its weight therefore makes duration a measure of the timing of the bond’s value as assessed at the valuation date.
For annual payments at the end of years \(1,\ldots,n\), let \(C_t\) be the cash flow at the end of year \(t\). If the bond’s price is calculated using the annual effective yield \(i\), then
Here \(P\) is the value of the promised cash flows at the valuation date. Each weight is nonnegative for nonnegative cash flows and a yield greater than \(-1\); the weights sum to one because their numerators sum to \(P\). Duration is measured in years. These formulas assume fixed cash flows, valuation on a coupon date, and one annual effective yield used to discount every payment. They do not account for default, embedded options that change cash flows, or accrued interest between coupon dates.
Why the Present-Value Weights Form an Average
The weighted-average interpretation follows directly from the price formula. Define the present value of each cash flow as \(PV_t=C_t/(1+i)^t\), and its price weight as \(w_t=PV_t/P\). Then the bond price is the sum of the present values, so the weights add to one:
Substituting those weights into the definition gives \(D_{\mathrm{Mac}}=\sum_{t=1}^{n}t w_t\), which is precisely an average of the payment times. A payment with a larger present value receives a larger share of the weight. Since these weights sum to one, duration lies between the earliest and latest payment times when all cash flows are nonnegative.
Record each cash flow and its payment time in years. Include the final coupon and principal repayment together if they are paid on the same date.
Use the bond’s yield and the correct number of periods to find each payment’s present value.
Divide its present value by the sum of all the present values, which is the bond price under the stated yield.
Multiply each time by its weight and add. If coupons are semiannual, express each period’s time in years.
Annual-Coupon Bonds
For an annual-coupon bond, the cash flow in an ordinary coupon year is the coupon payment. The final cash flow includes both the last coupon and repayment of principal. The yield \(i\) in the formula is the bond’s annual effective yield to maturity, not its coupon rate or current yield. The coupon rate determines the promised coupon amount; the yield is used to discount the payments.
Worked Example: A Three-Year Bond Priced at Par
A hypothetical bond has face value $1,000, pays a $100 coupon annually, and repays principal with its third coupon. Its annual effective yield is 10%. Find its price and Macaulay duration. Use annual payments, fixed promised cash flows, and no default or embedded options.
Method. Discount each annual cash flow at the 10% effective annual yield, then use each present value as a share of the price. The final-year cash flow is $100 + $1,000 = $1,100.
The present values are:
Adding them gives \(P=\$90.9091+\$82.6446+\$826.4463=\$1{,}000.0000\), or $1,000.00 to the cent. The cash-flow times are one, two, and three years. Their present-value-weighted total is:
The price is $1,000.00 and the duration is 2.7355 years. The weights sum to one because the present values sum to the price. Common mistake avoided: using undiscounted payments as weights gives \((1(100)+2(100)+3(1{,}100))/1{,}300=3{,}600/1{,}300=2.7692\) years. That calculation weights payments by their nominal amounts, not by their shares of today’s price, so it is not Macaulay duration.
Worked Example: A Two-Year Bond with a 7% Yield
A hypothetical bond has face value $1,000, pays a $70 coupon at the end of each year, and repays principal with the second coupon. Its annual effective yield is 7%. Find the price and Macaulay duration. Assume the stated cash flows are fixed and the valuation date is a coupon date.
Method. Discount the $70 first-year payment and the $1,070 second-year payment at 7%. Divide each present value by the price to form the weights, then average the two payment times.
The price is \(\$65.4206+\$934.5794=\$1{,}000.0000\), or $1,000.00. The duration calculation is:
Thus the bond’s price is $1,000.00 and its duration is 1.9346 years. The calculation is checked by confirming that the two present values add to the stated price. Common mistake avoided: the second payment occurs at year two, so its present value is multiplied by two in the duration numerator; it is not treated as a year-one payment merely because it includes principal repayment.
Zero-Coupon Bonds
A zero-coupon bond makes no interim coupon payments. Its only cash flow is principal at maturity, so the entire price weight belongs to that one payment. This makes its duration especially direct.
Worked Example: Duration of a Five-Year Zero-Coupon Bond
A hypothetical zero-coupon bond pays $1,000 exactly five years from now. Its annual effective yield is 6%. Find its price and Macaulay duration. Assume there are no other payments, default, or embedded options.
Method. Discount the single payment for five years. Since there is only one payment, its present value is also the bond’s price and its weight is the whole price divided by itself.
The payment’s weight is \(\$747.2582/\$747.2582=1\). Therefore \(D_{\mathrm{Mac}}=5(1)=5.0000\) years. The result checks against the theorem: with no earlier cash flows, the bond’s entire present value is associated with its payment at maturity. Common mistake avoided: discounting a payment changes its present value, but it does not change the date on which the payment is received. Duration is five years, not the bond’s price or a discounted version of its maturity.
Semiannual Coupons and Time Units
When a bond pays coupons semiannually, discounting must use the yield per half-year period, while duration should still be reported in years. If the quoted yield is the nominal annual rate \(i^{(2)}\), convertible semiannually, the rate per period is \(i^{(2)}/2\). Payment \(k\) is made at \(k/2\) years. The duration formula is therefore:
This formula assumes the stated nominal yield is compounded twice per year and the cash flows occur every half-year. Do not use the annual effective yield formula with \(k\) as though each half-year were a full year. If instead a yield is stated as an annual effective rate \(i\), its equivalent half-year rate is \((1+i)^{1/2}-1\); it is not generally \(i/2\).
Worked Example: A Semiannual-Coupon Bond
A hypothetical three-year bond has face value $1,000, a 6% annual coupon rate paid semiannually, and a quoted nominal annual yield \(i^{(2)}=6\%\), convertible semiannually. Find its price and Macaulay duration in years. Assume fixed cash flows and valuation on a coupon date.
Method. Each half-year coupon is \(0.06(\$1{,}000)/2=\$30\), and the final payment is $1,030. There are six periods, the yield per period is \(0.06/2=0.03\), and period \(k\) occurs at \(k/2\) years.
Discounting the six payments gives the following present values, rounded to four decimal places. Calculations retain full precision until the final duration is rounded.
| Period \(k\) | Time (years) | Cash flow | Present value |
|---|---|---|---|
| 1 | 0.5 | $30 | $29.1262 |
| 2 | 1.0 | $30 | $28.2779 |
| 3 | 1.5 | $30 | $27.4542 |
| 4 | 2.0 | $30 | $26.6546 |
| 5 | 2.5 | $30 | $25.8783 |
| 6 | 3.0 | $1,030 | $862.6088 |
For example, the first present value is \(\$30/1.03=\$29.1262\), and the final one is \(\$1{,}030/1.03^6=\$862.6088\). Adding all six present values gives a price of $1,000.0000, or $1,000.00. The time-weighted present values are:
Dividing by the price gives \(D_{\mathrm{Mac}}=\$2{,}789.8537/\$1{,}000=2.7899\) years. The check is that the present values sum to $1,000.00 and the duration is an average of times from 0.5 to 3 years, so it falls within that interval. Common mistake avoided: using period numbers \(1,\ldots,6\) as though they were years would report a duration twice as large. The period times must be converted to years.
Interpretation and Common Traps
Macaulay duration is a timing summary, not the bond’s maturity except in the zero-coupon case. For a coupon bond with positive cash flows, earlier coupons give some of the present-value weight to dates before maturity, so duration is less than maturity. A larger share of value arriving later generally moves the weighted average later, while discounting determines how much weight each payment receives at the stated yield.
- Do not use the coupon rate as the discount rate automatically. Use the stated yield to maturity. The rates coincide only when the problem says so.
- Do not use undiscounted payments as weights. First find each payment’s present value, then divide by the bond price.
- Keep units consistent. With semiannual payments, the discount exponent counts half-year periods, but each time in the duration average is measured in years.
- Combine the final coupon and principal. They are both received on the maturity date and together form the final cash flow.
- Check the weights. Present values divided by price should sum to one, subject only to rounding.
For exam calculations aligned with the SOA VEE Accounting and Finance syllabus, a useful final check is to compare the answer with the cash-flow dates. With nonnegative payments, the duration must lie between the earliest and latest payment times. A result beyond maturity, or a semiannual duration reported in periods when years are requested, signals that the weights or time units need review.
Check Your Understanding
For each problem, use the stated yield basis and report duration in years. Round money to the nearest cent and durations to four decimal places.
- A hypothetical zero-coupon bond pays $1,000 in four years. Its annual effective yield is 5%. Find its price and Macaulay duration. (Answer: price $822.70; duration 4.0000 years)
- A hypothetical three-year bond with face value $1,000 pays a $50 coupon annually and repays principal with its final coupon. Its annual effective yield is 5%. Find its price and duration. (Answer: price $1,000.00; duration 2.8594 years)
- A hypothetical two-year bond with face value $1,000 pays an 8% annual coupon rate semiannually. Its nominal annual yield \(i^{(2)}\) is 8%, convertible semiannually. Find the half-year coupon, price, and Macaulay duration. (Answer: coupon $40 per half-year; price $1,000.00; duration 1.8875 years)
- A hypothetical two-year bond with face value $1,000 pays a $60 annual coupon and repays principal with the second coupon. Its annual effective yield is 5%. Find the price and duration. (Answer: price $1,018.59; duration 1.9439 years)
- A hypothetical two-year bond has cash flows of $40 at 0.5 years and $1,040 at 1 year. Its nominal annual yield \(i^{(2)}\) is 8%, convertible semiannually. Find the price and duration, and state why the duration is below one year. (Answer: price $1,000.00; duration 0.9808 years; the payment at 0.5 years has positive present-value weight)