Zero-Coupon Bonds And Single-Payment Duration

Why a Zero-Coupon Bond Has a Special Duration

A zero-coupon bond makes no coupon payments before maturity. Its only promised cash flow is the redemption payment at the end. This simple cash-flow pattern makes the bond a useful case for understanding the meaning of duration: when there is only one payment date, the present-value-weighted average payment time must be that date.

The key is to keep time measured in years separate from the number of yield periods. A bond with semiannual yield periods and a maturity of 12 years has 24 periods, but its Macaulay duration is measured as 12 years, not 24 years. Modified duration also depends on the stated yield convention.

Definition: A zero-coupon bond with redemption amount $F$ pays no interim coupons and pays $F$ at maturity. If it matures after $n$ periods and the yield per period is $r$, its price is $P=F(1+r)^{-n}$, assuming the redemption payment is certain and the same yield applies in every period. Its Macaulay duration is the present-value-weighted average time, in years, of its promised payments.

For the results below, assume the bond’s single redemption payment is certain, its maturity is an integer number of the stated yield periods, and it has no other cash flows. These are bond cash-flow assumptions, not mortality assumptions; no mortality model or probability distribution is used. If a bond has interim payments or other cash flows, the single-payment formulas do not describe its full duration.

Macaulay Duration: One Payment, One Time

For a bond with cash flows at times $t_k$, measured in years, Macaulay duration is the present-value-weighted average of those times. The weight assigned to a cash flow is its present value divided by the bond price. For a zero-coupon bond, there is just one cash flow, $F$, at maturity $T$. Its present value is the entire price, so its weight is 1.

D_{\mathrm{Mac}} = \frac{T\left(F(1+r)^{-n}\right)} {F(1+r)^{-n}} =T

This is the core reason the result holds: the only payment has all the present-value weight, and it occurs at time $T$. The yield and redemption amount affect the price, but they do not move the payment date. If there are $m$ yield periods per year and the bond matures after $n$ periods, then $T=n/m$ years, so

D_{\mathrm{Mac}}=\frac{n}{m}\text{ years}.

For an annual yield convention, $m=1$. For a semiannual convention, $m=2$, and for a quarterly convention, $m=4$. The period count $n$ is not itself a duration in years unless there is exactly one period per year.

Modified Duration Under a Periodic Yield Convention

Modified duration describes the local sensitivity of price to a change in yield, with the yield convention held fixed. Let $j$ be a nominal annual yield convertible $m$ times per year. The yield per period is $j/m$. For $n$ periods to maturity, the price of a zero-coupon bond is

P(j)=F\left(1+\frac{j}{m}\right)^{-n}.

Differentiate with respect to the annual nominal yield $j$, then divide the negative price derivative by the price. This gives modified duration measured in years per unit change in $j$:

-\frac{1}{P}\frac{dP}{dj} = \frac{n}{m\left(1+j/m\right)} = \frac{D_{\mathrm{Mac}}}{1+j/m}.

The derivative is with respect to the annual nominal yield $j$, not the per-period yield $j/m$. This distinction is why the denominator uses $1+j/m$, while the numerator is Macaulay duration in years. The formula requires $1+j/m>0$, so the periodic accumulation factor is positive.

Theorem: For a zero-coupon bond with $n$ periods remaining, $m$ periods per year, and nominal annual yield $j$ convertible $m$ times per year, $D_{\mathrm{Mac}}=n/m$ years and $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+j/m)$. If yield is stated as an effective annual rate, use $m=1$; if it is stated as a nominal annual rate convertible semiannually, use $m=2$.

The formula is a local sensitivity result. For a small change $\Delta j$, the first-order approximation is $\Delta P/P\approx-D_{\mathrm{mod}}\Delta j$. It is not an exact formula for a finite yield change. The later material on price-change estimates develops how duration is used for such approximations.

Worked Calculations

Worked Example: Annual Effective Yield and a Long Maturity

A hypothetical zero-coupon bond will pay $1,000 in 12 years. Its yield is 4% effective annually. Find its price, Macaulay duration, and modified duration. Report the price to the nearest cent and durations to four decimal places.

Method and assumptions. There is one certain payment, after 12 annual periods. An effective annual yield of 4% means $m=1$, $r=0.04$, and $n=12$. No interim coupons are paid.

Price. Discount the redemption payment for 12 years:

P=\frac{1{,}000}{(1.04)^{12}}, \qquad (1.04)^{12}=1.6010322185\ldots, \qquad \frac{1}{(1.04)^{12}}=0.62459705\ldots

Thus $P=1{,}000(0.62459705\ldots)=\$624.59705\ldots$, which is $624.60 to the nearest cent.

Macaulay duration. The only payment occurs at year 12, so its present-value weight is 1 and $D_{\mathrm{Mac}}=12$ years.

Modified duration. With $m=1$,

D_{\mathrm{mod}}=\frac{12}{1+0.04} =\frac{12}{1.04} =11.5384615\ldots \approx 11.5385\text{ years}.

Check and common mistake. The duration is 12 years, matching the payment date, while modified duration is slightly smaller because the yield-period factor is 1.04. Do not treat the rounded price as an input to the duration calculation; the duration follows from the cash-flow timing and yield convention.

Worked Example: Semiannual Yield and Period Counting

A hypothetical zero-coupon bond pays $1,210 in one year. Its nominal annual yield is 10% convertible semiannually. Find its price and both durations. Give the price to the nearest cent and durations to four decimal places.

Method and assumptions. The yield is nominal and convertible twice per year, so $m=2$. The rate per half-year is $j/m=0.10/2=0.05$. One year contains $n=2$ half-year periods.

Price. Discount for two periods at 5% per period:

P=\frac{1{,}210}{(1.05)^2} =\frac{1{,}210}{1.1025} =1{,}097.5056689\ldots

Therefore the price is $1,097.51 to the nearest cent.

Macaulay duration. The one redemption payment occurs after two half-year periods, or $2/2=1$ year. Therefore $D_{\mathrm{Mac}}=1.0000$ year.

Modified duration. The denominator uses the half-year yield factor:

D_{\mathrm{mod}} = \frac{1}{1+0.10/2} = \frac{1}{1.05} =0.95238095\ldots \approx 0.9524\text{ years}.

Check and common mistake. Two periods divided by two periods per year gives one year of Macaulay duration. Using $n=2$ as though it were years would incorrectly double the duration. Also, using $1.10$ in the modified-duration denominator would mismatch the semiannual yield convention.

Worked Example: Quarterly Yield with a Fraction of a Year to Maturity

A hypothetical zero-coupon bond pays $1,000 after three quarters. Its nominal annual yield is 16% convertible quarterly. Find its price, Macaulay duration, and modified duration. Round the price to the nearest cent and durations to four decimal places.

Method and assumptions. The yield is nominal with $m=4$ periods per year. The quarterly yield is $j/m=0.16/4=0.04$, and the maturity is $n=3$ quarters, or $3/4=0.75$ years.

Price. Discount the payment for three quarterly periods:

P=\frac{1{,}000}{(1.04)^3}, \qquad (1.04)^3=1.124864, \qquad P=888.996\ldots

The price is $889.00 to the nearest cent.

Macaulay duration. Since the only payment is after three quarters, $D_{\mathrm{Mac}}=3/4=0.7500$ year.

Modified duration. Apply the quarterly periodic-yield convention:

D_{\mathrm{mod}} = \frac{0.75}{1+0.16/4} = \frac{0.75}{1.04} =0.7211538\ldots \approx0.7212\text{ years}.

Check and common mistake. Both durations are less than one year because the payment is due after three quarters. Counting the three quarterly periods as three years would be a unit error.

Common Traps and Practical Checks

A zero-coupon bond is simple, but its simplicity can make convention errors especially visible. Before substituting into a formula, identify the yield type, the number of periods per year, the number of periods remaining, and maturity in years.

Confusing periods with years. With semiannual periods, 10 remaining periods represent 5 years. Convert using $T=n/m$.
Using the wrong periodic rate. A nominal annual yield $j$ convertible $m$ times has per-period rate $j/m$. Do not discount each period at $j$.
Mixing modified-duration conventions. Under nominal annual yield convertible $m$ times, use $D_{\mathrm{Mac}}/(1+j/m)$. The yield variable and denominator must match.
Assuming modified duration is the payment date. Macaulay duration equals maturity for a zero-coupon bond. Modified duration is smaller for a positive yield because it divides Macaulay duration by a factor greater than 1.
Applying the single-payment result to a coupon bond. Coupon payments have earlier payment dates and nonzero present-value weights. A bond with coupons requires accounting for all its cash flows.

Key takeaway: For a zero-coupon bond, the single redemption payment has all of the present-value weight. Therefore Macaulay duration is its maturity in years, $n/m$. Under a nominal annual yield $j$ convertible $m$ times per year, modified duration is $(n/m)/(1+j/m)$.

The zero-coupon case is also a useful check when learning duration calculations: if a single-payment bond’s Macaulay duration does not equal its payment time, either the time units or the present-value weights have been mishandled. These definitions and calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the yield period before calculating. Assume each hypothetical bond makes only the stated certain redemption payment and has no interim coupons.

A bond pays $2,000 in five years and yields 6% effective annually. Find its Macaulay duration and modified duration in years. (Answer: $D_{\mathrm{Mac}}=5.0000$ years; $D_{\mathrm{mod}}=5/1.06=4.7170$ years.)
A bond pays $1,000 in six years at a nominal annual yield of 8% convertible semiannually. How many yield periods remain, and what are its two durations? (Answer: 12 periods; $D_{\mathrm{Mac}}=6.0000$ years; $D_{\mathrm{mod}}=6/1.04=5.7692$ years.)
A bond pays $1,000 after two years and yields 12% nominal convertible quarterly. Find its price to the nearest cent and both durations to four decimal places. (Answer: $P=1{,}000/(1.03)^8=\$789.41$; $D_{\mathrm{Mac}}=2.0000$ years; $D_{\mathrm{mod}}=2/1.03=1.9417$ years.)
A bond pays $1,500 after three years at a nominal annual yield of 6% convertible semiannually. Find the number of periods, Macaulay duration, and modified duration. (Answer: 6 periods; $D_{\mathrm{Mac}}=3.0000$ years; $D_{\mathrm{mod}}=3/1.03=2.9126$ years.)
A bond matures in 18 months and yields 8% nominal convertible quarterly. Find its period count, Macaulay duration, and modified duration. (Answer: 6 periods; $D_{\mathrm{Mac}}=1.5000$ years; $D_{\mathrm{mod}}=1.5/1.02=1.4706$ years.)

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