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Duration Bounds And The Meaning Of Weighted Time

Interpret Macaulay duration as a present-value-weighted time, prove its payment-date bounds, and recognize when those bounds do not apply.

Intermediate 9 min read

What You'll Learn

  • Define present-value weights for a stream of positive future payments.
  • Prove that Macaulay duration lies between the earliest and latest payment times.
  • Interpret duration as a weighted time rather than a payment date.
  • Convert weighted time consistently between coupon periods and years.
  • Identify why the positive-cash-flow bound can fail for signed cash flows.

Why Duration Has Payment-Date Bounds

Macaulay duration combines the timing of a security’s payments into one summary measured in units of time. For a stream of positive payments, that summary cannot occur before the first payment or after the last: it is an average of payment times, with each time weighted by that payment’s share of the price. This gives a useful interpretation and a quick check on a calculation.

The bound depends on the weights being positive and summing to one. Those conditions hold for positive future cash flows valued at a yield with a positive discount factor. They need not hold for a stream containing negative cash flows, so the bound is not an unconditional rule for every algebraic cash-flow schedule.

Definition: Suppose a security pays positive amounts \(C_j\) at times \(t_j\), measured in years, and is valued at an effective annual yield \(i>-1\). Let \(v=1/(1+i)\). Its price is \(P=\sum_j C_jv^{t_j}\), and the present-value weight of payment \(j\) is \(w_j=C_jv^{t_j}/P\). Its Macaulay duration is \(D_M=\sum_j t_jw_j\), measured in years.

The price \(P\) is positive because every payment and discount factor is positive. Consequently, each \(w_j\) is positive, and the weights add to one: \(\sum_j w_j=P/P=1\). The discounting is essential: duration weights each payment by its present value, not by its undiscounted amount.

Payment times must be expressed in a consistent unit. If the payments occur at coupon-period numbers, first compute duration in coupon periods; convert to years afterward if required. A result in periods cannot be compared directly with bounds stated in years.

The Bound and Its Justification

Let \(t_{\min}\) be the earliest payment time and \(t_{\max}\) the latest. Every payment time is between these endpoints. Multiplying each such inequality by its positive weight preserves its direction, and adding the weighted inequalities gives the result.

$$ t_{\min}\leq t_j\leq t_{\max} \quad\Longrightarrow\quad t_{\min}w_j\leq t_jw_j\leq t_{\max}w_j $$
$$ \sum_j t_{\min}w_j \leq \sum_j t_jw_j \leq \sum_j t_{\max}w_j \quad\Longrightarrow\quad t_{\min}\leq D_M\leq t_{\max}, $$

because \(\sum_j w_j=1\). This is the weighted-average justification: duration is a convex combination of payment times. If there are positive payments at two or more distinct times, the duration is strictly between the earliest and latest dates. It equals an endpoint only when all the present value is concentrated at that endpoint—for example, for a single-payment zero-coupon security, duration equals its payment time.

Theorem: For a finite stream of strictly positive future payments, valued using positive discount factors, Macaulay duration lies between the first and last payment times. The conclusion follows from positive weights summing to one; it does not require the payment amounts to be equal.

Worked Example: A Positive Cash-Flow Stream

Worked Example: A Positive Cash-Flow Stream

A hypothetical security pays $110 at the end of year 1, $266.20 at the end of year 3, and $146.41 at the end of year 4. Its effective annual yield is 10%. Find its price, its Macaulay duration, and the interval in which the duration must lie.

Method and assumptions. The payments are all positive, occur at annual times, and are discounted at an effective annual yield \(i=0.10\). Thus \(v=1/1.10\), and the present-value weights are suitable for applying the bound.

First calculate each present value:

$$ 110v^1=\frac{110}{1.10}=100,\qquad 266.20v^3=\frac{266.20}{1.10^3}=200,\qquad 146.41v^4=\frac{146.41}{1.10^4}=100. $$

The price is \(P=100+200+100=\$400\). The corresponding weights are \(100/400=0.25\), \(200/400=0.50\), and \(100/400=0.25\); they are positive and sum to 1.

Weight each payment time and add:

$$ D_M=1(0.25)+3(0.50)+4(0.25) =0.25+1.50+1.00 =2.75\text{ years}. $$

The earliest and latest payment times are 1 and 4 years, so the theorem requires \(1\leq2.75\leq4\). The calculated duration satisfies the check. It is closer to year 3 than to year 1 because the payment at year 3 accounts for half of the price.

Common mistake avoided: Averaging the payment dates without weights would give \((1+3+4)/3\), which treats all three dates as equally important. The correct weights come from present values, not from the number of payments.

Coupon-Period Time Versus Years

The bound applies in whichever time unit is used consistently. For a security paying semiannually, it is often simplest to index dates by half-year periods, discount each payment using the yield per half-year, and then convert the duration and endpoints to years. In particular, a nominal annual yield convertible twice per year is not itself the yield for one half-year.

Worked Example: Semiannual Payments and the Bound

A hypothetical security pays $208 at the end of half-year period 1, $112.4864 at the end of period 3, and $350.957568 at the end of period 4. The yield is 8% nominal per year convertible semiannually. Find its price and Macaulay duration in periods and years, then check the payment-date bound.

Method and assumptions. The yield per half-year period is \(0.08/2=0.04\), so the discount factor per period is \(1/1.04\). All three payments are positive, and their times are \(1,3,4\) half-year periods.

Discount the payments period by period:

$$ \frac{208}{1.04}=200,\qquad \frac{112.4864}{1.04^3}=100,\qquad \frac{350.957568}{1.04^4}=300. $$

The price is \(P=200+100+300=\$600\). The present-value weights are \(200/600=1/3\), \(100/600=1/6\), and \(300/600=1/2\). Therefore, duration in half-year periods is

$$ D_M=1\left(\frac13\right)+3\left(\frac16\right)+4\left(\frac12\right) =\frac13+\frac12+2 =\frac{17}{6} =2.8333\text{ half-year periods}. $$

There are two half-year periods per year, so \(D_M=(17/6)/2=17/12=1.4167\) years. The first and last payments occur at \(0.5\) and \(2\) years, respectively, and \(0.5\leq1.4167\leq2\). The bound holds in both units: \(1\leq17/6\leq4\) periods, or \(0.5\leq17/12\leq2\) years.

Common mistake avoided: Treating 8% as the rate for each half-year would discount incorrectly. The stated nominal rate is convertible semiannually, so the per-period rate is 4%. A second trap is reporting 2.8333 periods as 2.8333 years; converting the duration also requires converting the endpoints.

When Signed Cash Flows Break the Bound

The positive-weight argument has a precise limitation. If a cash-flow stream contains a negative amount, its present-value weight is negative (provided the price is positive). The weights may still sum to one algebraically, but multiplying inequalities by a negative weight reverses their directions. The convex-combination proof no longer applies, and the resulting weighted-time ratio can fall outside the earliest-to-latest interval.

Worked Example: A Negative Cash Flow and a Failed Bound

A hypothetical net settlement stream has cash flows of $80 at year 1, \(-\$50\) at year 2, and $10 at year 3. Use an effective annual yield of 0%. Calculate the price and weighted-time ratio, and determine whether the positive-cash-flow bound applies.

Method and assumptions. At \(i=0\), \(v=1\), so each present value equals its stated cash flow. The stream contains a negative amount; therefore, even if its net price is positive, it does not have all-positive present-value weights.

The price is \(P=80-50+10=\$40\). The signed present-value weights are \(80/40=2\), \(-50/40=-1.25\), and \(10/40=0.25\); they sum to 1, but one weight is negative. The weighted-time ratio is

$$ \frac{1(80)+2(-50)+3(10)}{40} =\frac{80-100+30}{40} =\frac{10}{40} =0.25\text{ years}. $$

The ratio \(0.25\) years is earlier than the first cash-flow time, year 1. This does not contradict the theorem: the theorem assumes positive weights, and the year-2 weight here is \(-1.25\). It shows why the bound cannot be applied to this signed stream.

Common mistake and correction: An arithmetic error such as \(80-100+30=30\) would produce \(30/40=0.75\) years. The correct numerator is \(10\), so the ratio is \(0.25\) years. Check the signed time-weighted sum before drawing any conclusion about the bound.

Interpreting the Result

The duration bound is useful both conceptually and as a diagnostic. For a conventional positive-payment bond, duration is a single weighted summary of when its present value is received. It need not equal maturity, and it need not coincide with any coupon date. Earlier payments generally contribute to the summary, while later payments contribute in proportion to their present values.

The bound also helps identify inconsistent inputs or calculations. If a positive-cash-flow security has duration outside its first-to-last payment interval, check whether payment times use a consistent unit, whether the weights are present values divided by the price, and whether each payment has been discounted using the intended yield convention. Confirm the assumptions before treating a bound as a calculation rule: a signed cash-flow stream, a zero or negative net price, or an incorrectly specified discount model can invalidate the positive-weight argument.

Key takeaway: With positive future payments and a positive price, present-value weights are positive and sum to one. Macaulay duration is therefore a weighted-average payment time between the earliest and latest dates. A negative cash flow removes the guarantee because its weight is negative.

Check Your Understanding

For each problem, compute the price and weighted time, keep the time unit consistent, and decide whether the payment-date bound applies.

  1. A security pays $110 at year 1 and $121 at year 2. The effective annual yield is 10%. Find its price and duration, and state the bounds. (Answer: Price $200; duration 1.500 years; bounds 1 to 2 years.)
  2. A security pays $53 at the end of quarter 1, $112.36 at the end of quarter 2, and $178.6524 at the end of quarter 3. The yield is 24% nominal per year convertible quarterly. Find price and duration in quarters and years, and state the bounds. (Answer: Price $300; duration \(7/3=2.3333\) quarters \(=7/12=0.5833\) years; bounds 1 to 3 quarters, or 0.25 to 0.75 years.)
  3. A positive-payment security pays $105 at year 1, $110.25 at year 2, and $243.10125 at year 4. The effective annual yield is 5%. Find its price, duration, and payment-date bounds. (Answer: Price $400; duration 2.750 years; bounds 1 to 4 years.)
  4. At a 0% effective annual yield, a net cash-flow stream pays $60 at year 1, pays \(-\$20\) at year 2, and pays $20 at year 4. Find its net price and weighted-time ratio, and state whether the positive-cash-flow bound applies. (Answer: Price $60; weighted-time ratio \(100/60=1.6667\) years; no, because the cash flows do not all have positive weights.)

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