Yield Effects On Macaulay Duration

How Yield Changes Macaulay Duration

A bond’s cash flows and their payment dates can stay fixed while its yield changes. The price then changes, but so do the proportions of price attributable to each payment. Since Macaulay duration is the present-value-weighted average payment time, those changing proportions affect duration too.

The key comparison in this tutorial holds the bond’s cash flows and payment times fixed and changes only the yield used to discount them. Under that comparison, a higher yield gives relatively more weight to earlier payments and less to later payments. The result is a lower Macaulay duration, provided there are positive cash flows at more than one payment time.

Model and assumptions: Consider a bond with fixed, deterministic cash flows $C_t>0$ at future integer times $t=1,\ldots,n$, measured in years. The same effective annual yield $i>-1$ is used to discount every payment, with annual compounding. Ignore default, taxes, and embedded options; the cash flows do not change when the yield changes.

Under this model, the price is the sum of the discounted cash flows. The present-value weight of a payment is its present value divided by the total price, and Macaulay duration is the weighted average of the payment times.

P(i)=\sum_{t=1}^{n} C_t(1+i)^{-t}, \qquad w_t(i)=\frac{C_t(1+i)^{-t}}{P(i)}, \qquad D_{\mathrm{Mac}}(i)=\sum_{t=1}^{n}t\,w_t(i).

The weights are positive and sum to 1, because the present values sum to the price. To compare durations at different yields, it is not enough to notice that every payment’s present value falls when yield rises. The important question is how quickly each payment’s present value falls relative to the others.

Why the Relative Weights Shift Earlier

Take two payment times $s<t$. Their present-value-weight ratio is

\frac{w_s(i)}{w_t(i)} = \frac{C_s}{C_t}(1+i)^{t-s}.

The fixed cash-flow ratio $C_s/C_t$ does not change with yield. Since $t-s>0$, the factor $(1+i)^{t-s}$ increases as $i$ increases for $i>-1$. Thus the earlier payment gains relative weight compared with the later payment. This is a comparison of relative weights: an individual earlier payment’s weight need not rise in every cash-flow pattern, because all the other payments also affect the total price.

A general derivative makes the duration effect precise. Treat the payment time $T$ as a discrete random variable that takes value $t$ with probability $w_t(i)$. Its mean is $D_{\mathrm{Mac}}(i)$, and its variance is calculated using these same weights.

Theorem: For fixed positive cash flows discounted at one effective annual yield $i>-1$,

\frac{dD_{\mathrm{Mac}}(i)}{di} = -\frac{\operatorname{Var}_{w(i)}(T)}{1+i} = -\frac{\displaystyle\sum_{t=1}^{n}w_t(i)\bigl(t-D_{\mathrm{Mac}}(i)\bigr)^2}{1+i}.

Because $1+i>0$ and a variance cannot be negative, duration cannot increase with yield under these assumptions. If cash flows occur at two or more distinct times, the variance is positive, so duration strictly decreases as yield increases. If there is only one payment time, the variance is zero and duration stays equal to that payment time at every yield.

Deriving the Duration Result

For completeness, the derivative follows directly by differentiating the normalized present-value weights. The price derivative is

P'(i) = -\frac{1}{1+i}\sum_{t=1}^{n}t\,C_t(1+i)^{-t} = -\frac{P(i)D_{\mathrm{Mac}}(i)}{1+i}.

Apply the quotient rule to $w_t(i)=C_t(1+i)^{-t}/P(i)$. The numerator’s logarithmic derivative is $-t/(1+i)$, and the price’s logarithmic derivative is $-D_{\mathrm{Mac}}(i)/(1+i)$. Therefore,

w_t'(i) = -\frac{w_t(i)\bigl(t-D_{\mathrm{Mac}}(i)\bigr)}{1+i}.

Differentiate $D_{\mathrm{Mac}}(i)=\sum_t t\,w_t(i)$, substitute this expression for $w_t'(i)$, and use $\sum_t w_t(i)t=D_{\mathrm{Mac}}(i)$:

\begin{aligned} D_{\mathrm{Mac}}'(i) &= -\frac{1}{1+i}\sum_t w_t(i)t\bigl(t-D_{\mathrm{Mac}}(i)\bigr)\\ &= -\frac{\sum_t w_t(i)t^2-D_{\mathrm{Mac}}(i)^2}{1+i}\\ &= -\frac{\operatorname{Var}_{w(i)}(T)}{1+i}. \end{aligned}

This derivative is specific to fixed cash flows discounted at a common yield. It does not establish the same conclusion if changing the yield also changes the cash flows, or if some cash flows are negative. It is also a statement about Macaulay duration, not a claim that every measure of price sensitivity changes in the same way.

Worked Example: Reweighting a Bond at Two Yields

A hypothetical bond pays $100 at the end of year 1, $100 at the end of year 2, and $300 at the end of year 4. Find its price, present-value weights, and Macaulay duration at effective annual yields of 0% and 100%. Assume these are the bond’s only cash flows.

Method and assumptions. Keep the three positive cash flows and their annual payment times fixed. Discount each at the stated effective annual yield, then divide each present value by the price to obtain its weight.

At $i=0\%$. The discount factor for every payment is 1, so the present values are $100, $100, and $300. The price is

$$ P(0)=100+100+300=500. $$

The weights, in payment-time order, are $100/500=1/5$, $100/500=1/5$, and $300/500=3/5$. Therefore,

D_{\mathrm{Mac}}(0) = 1\left(\frac15\right)+2\left(\frac15\right)+4\left(\frac35\right) = \frac{1+2+12}{5} = 3\text{ years}.

At $i=100\%$. The discount factors at years 1, 2, and 4 are $1/2$, $1/4$, and $1/16$. The present values are $100/2=50$, $100/4=25$, and $300/16=18.75$, giving

$$ P(1)=50+25+18.75=93.75. $$

The corresponding weights are $50/93.75=8/15$, $25/93.75=4/15$, and $18.75/93.75=3/15$. Hence,

D_{\mathrm{Mac}}(1) = 1\left(\frac{8}{15}\right)+2\left(\frac{4}{15}\right)+4\left(\frac{3}{15}\right) = \frac{8+8+12}{15} = \frac{28}{15} \approx 1.8667\text{ years}.

Check and trap avoided. Duration falls from 3 years to about 1.8667 years, consistent with the theorem. A tempting error is to keep the original weights after the yield change: discounting does not scale all payment dates equally, so the weights must be recalculated from the new price.

Worked Example: A Smaller Yield Increase

A hypothetical bond pays $100 at the end of each of years 1 and 2, with no other payments. Calculate its Macaulay duration at effective annual yields of 10% and 20%, and find the percentage decrease in duration as yield rises.

Method and assumptions. Use annual compounding and hold both positive payments fixed. Calculate the present values and normalized weights at each yield; compare the resulting durations.

At $i=10\%$. The present values are

\frac{100}{1.1}=\frac{1000}{11}, \qquad \frac{100}{(1.1)^2}=\frac{10000}{121}.

The price is $11000/121+10000/121=21000/121$. The weights are therefore $11000/21000=11/21$ at year 1 and $10000/21000=10/21$ at year 2. The duration is

D_{\mathrm{Mac}}(0.10) = 1\left(\frac{11}{21}\right)+2\left(\frac{10}{21}\right) = \frac{31}{21} \approx 1.4762\text{ years}.

At $i=20\%$. The present values are

\frac{100}{1.2}=\frac{250}{3}, \qquad \frac{100}{(1.2)^2}=\frac{625}{9}.

Their sum is $750/9+625/9=1375/9$. The weights are $750/1375=6/11$ at year 1 and $625/1375=5/11$ at year 2. Thus,

D_{\mathrm{Mac}}(0.20) = 1\left(\frac{6}{11}\right)+2\left(\frac{5}{11}\right) = \frac{16}{11} \approx 1.4545\text{ years}.

The absolute decrease is $31/21-16/11=5/231$ years. Relative to the original duration, the percentage decrease is

\frac{5/231}{31/21}\times 100\% = \frac{5}{341}\times 100\% \approx 1.47\%.

Check and trap avoided. The year-1 weight rises from $11/21$ to $6/11$, while the year-2 weight falls from $10/21$ to $5/11$. Do not calculate the percentage decrease using the new duration as the base; the stated comparison is the decrease relative to the original duration.

Worked Example: Uneven Cash Flows and a Common-Scale Error

A hypothetical bond pays $60 at the end of year 1, $100 at the end of year 2, and $240 at the end of year 4. Find its duration at effective annual yields of 0% and 50%. Explain why it would be wrong to assume that the present-value weights remain unchanged.

Method and assumptions. The annual cash flows are fixed and positive, and the same effective annual yield discounts each one. Recompute every present value and the total price at each yield.

At $i=0\%$. The price is $60+100+240=400$. The weights are $60/400=3/20$, $100/400=1/4$, and $240/400=3/5$. Therefore,

D_{\mathrm{Mac}}(0) = 1\left(\frac{3}{20}\right) +2\left(\frac14\right) +4\left(\frac35\right) = \frac{3+10+48}{20} = \frac{61}{20} = 3.05\text{ years}.

At $i=50\%$. The year-1 and year-2 present values are $60/1.5=40$ and $100/(1.5)^2=400/9$. The year-4 present value is

\frac{240}{(1.5)^4} = 240\left(\frac{16}{81}\right) = \frac{1280}{27}.

With denominator 27, the three present values are $1080/27$, $1200/27$, and $1280/27$. The price is $3560/27$. The weights are $1080/3560=27/89$, $1200/3560=30/89$, and $1280/3560=32/89$. Consequently,

D_{\mathrm{Mac}}(0.50) = 1\left(\frac{27}{89}\right) +2\left(\frac{30}{89}\right) +4\left(\frac{32}{89}\right) = \frac{27+60+128}{89} = \frac{215}{89} \approx 2.4157\text{ years}.

Check and trap avoided. The duration has decreased, as expected. The incorrect shortcut would be to say that a higher yield multiplies every present value by the same factor and therefore leaves the weights, and duration, unchanged. In fact, the discount factors at years 1, 2, and 4 are $1/1.5$, $1/(1.5)^2$, and $1/(1.5)^4$, which differ. Each yield requires a new set of weights.

When the Comparison Is Useful

Yield comparisons help explain why duration is not determined by payment dates alone: it also depends on the present-value weights assigned to those dates. For a given positive cash-flow schedule, higher yields concentrate the relative value earlier in time, reducing Macaulay duration. This result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

The conclusion depends on the comparison being controlled. If two bonds have different payment dates or cash flows, a yield difference by itself does not determine which bond has the greater duration. The theorem also need not apply if cash flows change with yield, as may occur with options or other contractual features, or if the cash-flow stream includes negative amounts. In those cases, the fixed-positive-cash-flow assumptions used in the derivation are not satisfied.

Hold payment amounts and dates fixed when isolating the effect of yield.
Use the same yield convention and time unit for every payment.
Recalculate the price and every present-value weight after changing yield.
Expect a strict duration decrease only when positive payments occur at multiple distinct times.

Key takeaway: For fixed positive bond cash flows discounted at one effective annual yield, higher yield shifts relative present-value weight toward earlier payments. Macaulay duration therefore decreases strictly when payments occur at more than one time; mathematically, its yield derivative is the negative weighted variance of payment time divided by $1+i$.

Check Your Understanding

For each problem, assume fixed deterministic cash flows, annual compounding, and one effective annual yield unless the question states otherwise.

A bond pays $100 at the end of year 1 and $300 at the end of year 3. Find its Macaulay duration at yields of 0% and 100%. (Answer: 2.5000 years at 0%; 1.8571 years at 100%.)
A bond pays $200 at the end of year 1 and $100 at the end of year 2. Calculate duration at yields of 0% and 50%, and state the decrease. (Answer: $4/3$ years at 0%; $5/4$ years at 50%; decrease $1/12$ year.)
A bond pays $90 at the end of year 1 and $90 at the end of year 3. Find its duration at an effective annual yield of 20%. (Answer: $111/61\approx1.8197$ years.)
For a fixed positive cash-flow schedule with payments at years 1 and 3, explain whether duration rises or falls as the effective annual yield increases, and state the condition that makes the change strict. (Answer: It falls strictly, because the payment-time variance is positive when both payment times have positive weight.)
A bond pays $50, $100, and $150 at the ends of years 1, 2, and 3. Calculate duration at yields of 0% and 100%. (Answer: $7/3\approx2.3333$ years at 0%; $21/11\approx1.9091$ years at 100%.)

What You'll Learn