Avoiding Time-Unit And Compounding Errors

Why Time Units and Compounding Must Agree

A duration calculation can use correct present-value weights and still be wrong if its time and yield conventions do not match. A duration stated in years is not the same quantity as a duration stated in coupon periods. Likewise, a nominal annual yield convertible $m$ times per year is not the same rate as an effective annual yield. These distinctions matter both when reporting Macaulay duration and when estimating sensitivity to a yield change.

The essential check is to keep the cash-flow time unit, the yield period, and the yield variable used for modified duration explicit. This is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: For cash flows $C_k$ paid at times $t_k$ years, and present-value weights $w_k=C_kv_k/P$, Macaulay duration in years is $D_{\mathrm{Mac}}=\sum_k t_kw_k$. Here $v_k$ is the discount factor to time $t_k$, $P=\sum_k C_kv_k$ is the price, and the weights sum to 1. If all cash flows occur at regular intervals of $1/m$ years and $k$ counts those intervals, then $t_k=k/m$: a duration computed first in periods must be divided by $m$ to report years.

Modified duration is tied to a particular yield variable. It describes the local price sensitivity to a change in that specified yield, so its denominator depends on the compounding convention. It is not enough to write “divide by one plus the yield” without saying which yield is meant.

Convention: If $j^{(m)}$ is a nominal annual yield convertible $m$ times per year, its yield per period is $j^{(m)}/m$. For regular $m$-times-per-year cash flows, $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+j^{(m)}/m)$ when sensitivity is measured with respect to the nominal annual yield $j^{(m)}$. If $i$ is an effective annual yield, the matching formula is $D_{\mathrm{mod}}=D_{\mathrm{Mac}}/(1+i)$. The two formulas use different yield variables; use the one matching the yield change being considered.

Why the Modified-Duration Denominator Changes

The denominator follows from differentiating the price with respect to the yield. For regular payments every $1/m$ years discounted at nominal annual yield $j^{(m)}$, write the price as $P=\sum_k C_k(1+j^{(m)}/m)^{-k}$. Differentiating one term gives $\frac{d}{dj^{(m)}}(1+j^{(m)}/m)^{-k}=-(k/m)(1+j^{(m)}/m)^{-k-1}$. Summing these derivatives and dividing by price gives the relative price sensitivity.

-\frac{1}{P}\frac{dP}{dj^{(m)}} = \frac{\sum_k (k/m)C_k(1+j^{(m)}/m)^{-k}} {P(1+j^{(m)}/m)} = \frac{D_{\mathrm{Mac}}}{1+j^{(m)}/m}.

The numerator is Macaulay duration in years because each payment time is $k/m$ years. If instead the yield variable is effective annual $i$, then a cash flow at time $t_k$ has discount factor $(1+i)^{-t_k}$. Differentiating that factor gives $-t_k(1+i)^{-t_k-1}$, so $-P^{-1}dP/di=D_{\mathrm{Mac}}/(1+i)$. These are local sensitivities with respect to the stated annual yield, not guarantees of exact price changes for a finite yield movement.

Worked Examples: Keeping Periods and Years Straight

Worked Example: Quarterly Payments and Nominal Yield

A hypothetical asset pays $100 at the end of each of the next three quarters. Its annual nominal yield is $j^{(4)}=8\%$, convertible quarterly. Find its price, Macaulay duration in years, and modified duration with respect to $j^{(4)}$. State the method and the error it avoids.

Method. Convert the nominal annual yield to the quarterly rate, discount each payment, and compute the present-value-weighted average of payment times. The main error to avoid is reporting a period-based average as years.

The quarterly yield is $0.08/4=0.02$, so $v=1/1.02=50/51$. The three payment times are $1/4$, $2/4$, and $3/4$ years. Their present values are:

\frac{100}{1.02}=98.039216,\qquad \frac{100}{1.02^2}=96.116878,\qquad \frac{100}{1.02^3}=94.232233.

Thus the price is $P=288.388327$, or $288.39 to the nearest cent. The PV-weighted average time, calculated first in quarter periods, is

\frac{1v+2v^2+3v^3}{v+v^2+v^3} = \frac{760050}{382550} = \frac{15201}{7651} \approx 1.986799 \text{ quarters}.

Divide by four to express the result in years:

D_{\mathrm{Mac}} = \frac{1}{4}\left(\frac{15201}{7651}\right) = \frac{15201}{30604} \approx 0.496700\text{ years}.

For the nominal annual yield $j^{(4)}$, the yield per period is $0.08/4=0.02$. Therefore $D_{\mathrm{mod}}=0.4966997778/1.02\approx0.4869605665$, or $0.4870$ to four decimals. As a check, Macaulay duration must lie between the first and last payment times, $0.25$ and $0.75$ years. The result does.

Worked Example: The Same Quarterly Rate Under Two Annual Yield Conventions

A hypothetical asset pays $100 at the end of each of three quarters. The quarterly yield is 10%. Find Macaulay duration and modified duration first with respect to the equivalent nominal annual yield convertible quarterly, then with respect to the equivalent effective annual yield. Give durations to four decimals. This comparison avoids treating two different annual yield measures as interchangeable.

The periodic discount factor is $v=1/1.10=10/11$. The period-weighted average is

\frac{v+2v^2+3v^3}{v+v^2+v^3} = \frac{641}{331} \approx 1.936556\text{ quarters}.

The annual Macaulay duration is therefore $1.936555891/4=0.484138973$, or $0.4841$ years. The nominal annual yield is $j^{(4)}=4(0.10)=40\%$. Its matching modified duration is

D_{\mathrm{mod},\,j^{(4)}} = \frac{0.484138973}{1+0.40/4} = \frac{0.484138973}{1.10} \approx 0.4401.

For the equivalent effective annual yield, compound the quarterly accumulation over four quarters: $1+i=1.10^4=1.4641$, so $i=46.41\%$. The modified duration with respect to this effective annual yield is

D_{\mathrm{mod},\,i} = \frac{0.484138973}{1.4641} \approx 0.3307.

The two modified durations differ because they measure sensitivity to different yield variables, even though both rates imply the same 10% yield per quarter. The result is also consistent with the annual effective yield being 46.41%, not 40%.

Worked Example: A Zero-Coupon Bond and the Period-Count Trap

A hypothetical zero-coupon bond pays $10,000 exactly two years from now. Its yield is $j^{(2)}=6\%$, nominal annual and convertible semiannually. Find its price, Macaulay duration in years, and modified duration with respect to $j^{(2)}$. Identify the time-unit mistake the method prevents.

Method. There are two semiannual periods per year, so the yield per period is $0.06/2=0.03$. Two years contain four such periods. The bond price is

P=\frac{10000}{1.03^4} =\frac{10000}{1.12550881} \approx \$8{,}884.87.

A zero-coupon bond has one cash flow, so its PV weight is 1 and its Macaulay duration equals the time of that payment: $D_{\mathrm{Mac}}=2$ years. Counting four semiannual periods and reporting “4 years” would confuse periods with years; four periods equal two years here. The modified duration is

D_{\mathrm{mod},\,j^{(2)}} = \frac{2}{1+0.06/2} = \frac{2}{1.03} \approx 1.9417.

The duration check is direct: the only payment occurs at two years, so the PV-weighted average payment time must be two years. The price calculation uses four periods, while the duration result is reported in years.

A Reliable Diagnostic Process

When a duration answer appears implausible, check the units before recomputing the present values. The following process separates the decisions that are often accidentally combined.

Name the yield convention.
Record whether the stated rate is nominal annual $j^{(m)}$, effective annual $i$, or a rate per coupon period. Do not infer the convention from the percentage alone.

Convert to the rate used for discounting.
For nominal annual $j^{(m)}$, use $j^{(m)}/m$ per period. For effective annual $i$, the equivalent rate per period is $(1+i)^{1/m}-1$, when payment intervals are $1/m$ years.

Write each payment time in years.
For payment $k$ periods from now at regular frequency $m$, use $t_k=k/m$ years in the Macaulay-duration numerator.

Match modified duration to the yield variable.
Use $1+j^{(m)}/m$ for sensitivity to nominal annual $j^{(m)}$, or $1+i$ for sensitivity to effective annual $i$.

Common Mistakes and Why They Matter

A frequent wrong calculation is to find a weighted average of payment counts $k$, label the result years, and then use it in a duration formula. If the weighted average is 1.986799 quarterly periods, it represents $1.986799/4=0.496700$ years—not 1.986799 years. The latter is four times too large. The correction is to convert each time to years, or divide the period-based weighted average by the number of periods per year.

Another common mistake is applying the annual yield itself as though it were the period yield. With $j^{(4)}=8\%$, quarterly discounting uses 2% per quarter, not 8% per quarter. Using 8% each quarter compounds the yield over four periods per year and prices a different stream.

Finally, the same annual yield percentage cannot be used in both modified-duration denominators without checking its convention. A nominal annual rate of 40% convertible quarterly and an effective annual rate of 40% do not imply the same quarterly return. In the second worked example, the quarterly return of 10% corresponds to nominal $40\%$ but effective annual $46.41\%$. Use the denominator consistent with the yield parameter whose change is being measured.

Key takeaway: Macaulay duration is a time and must be reported in an identified unit. For annual reporting, convert coupon-period times to years. Modified duration is a sensitivity to a specified yield: identify that yield convention first, then use its matching denominator.

Check Your Understanding

For each problem, track the time unit and yield convention before calculating.

An asset pays equal amounts at the ends of three quarters, and its nominal annual yield is $j^{(4)}=12\%$, convertible quarterly. Its Macaulay duration is the PV-weighted average of payment times. Find the duration in years. (Answer: approximately $0.4951$ years.)
A bond has Macaulay duration $2.4$ years and yield $j^{(2)}=5\%$, nominal annual and convertible semiannually. Find modified duration with respect to $j^{(2)}$, to four decimals. (Answer: $2.3415$.)
Find the equivalent quarterly rate for an effective annual yield of 10%, to four decimal places as a percentage. (Answer: approximately $2.4114\%$, since $(1.10)^{1/4}-1\approx0.024114$.)
A zero-coupon bond pays at 18 months and has yield $j^{(4)}=8\%$, nominal annual and convertible quarterly. Find Macaulay duration in years and modified duration with respect to $j^{(4)}$. (Answer: $1.5000$ years and $1.4706$.)
A stream has Macaulay duration $0.4966997778$ years and yield $j^{(4)}=8\%$, nominal annual and convertible quarterly. Find modified duration with respect to that nominal yield to four decimals. (Answer: $0.4870$, using $0.4966997778/1.02$.)

What You'll Learn