Converting Macaulay Duration To Modified Duration

Why the Conversion Depends on Compounding

Macaulay duration and modified duration describe related but different features of a fixed-income cash-flow stream. Macaulay duration is a present-value-weighted average time, measured in years. Modified duration measures the local percentage sensitivity of price to a change in yield. The conversion between them depends on how yield is quoted and compounded.

In this tutorial, assume fixed cash flows occur at regular compounding dates and are discounted using one constant yield. There are $m$ compounding periods per year, and $y$ is the nominal annual yield convertible $m$ times per year. Thus the yield per compounding period is $y/m$. These assumptions are essential: the conversion below is not a general formula for cash flows whose timing and yield-compounding conventions have not been matched.

Definition: For cash flows $C_k$ paid at the end of compounding period $k$, the price at nominal annual yield $y$, convertible $m$ times per year, is $P=\sum_{k=1}^{n} C_k(1+y/m)^{-k}$. Macaulay duration in years is $D_{\text{Mac}}=\frac{1}{P}\sum_{k=1}^{n}\frac{k}{m}C_k(1+y/m)^{-k}$. Modified duration with respect to $y$ is $D_{\text{mod}}=-\frac{1}{P}\frac{dP}{dy}$.

The time $k/m$ in the Macaulay-duration formula is in years, while the exponent $k$ counts compounding periods. Keeping those two units distinct is the key to using the conversion correctly. A cash flow at period 6 under semiannual compounding, for example, is paid after $6/2=3$ years.

Deriving the Conversion

Differentiate the price with respect to the quoted nominal annual yield $y$. The chain rule matters because each period’s discount factor uses $y/m$, not $y$ directly.

Write the price formula.
$P=\sum_{k=1}^{n}C_k(1+y/m)^{-k}$, with $m$ fixed.

Differentiate each discounted cash flow.
The derivative of $(1+y/m)^{-k}$ is $-\frac{k}{m}(1+y/m)^{-k-1}$. Therefore, $\frac{dP}{dy}=-\sum_{k=1}^{n}\frac{k}{m}C_k(1+y/m)^{-k-1}$.

Factor out one discount-base term.
Since $(1+y/m)^{-k-1}=(1+y/m)^{-k}/(1+y/m)$, the derivative becomes $\frac{dP}{dy}=-\frac{1}{1+y/m}\sum_{k=1}^{n}\frac{k}{m}C_k(1+y/m)^{-k}$.

Divide by price and use the duration definition.
The sum in Step 3 is $P D_{\text{Mac}}$. Hence $-\frac{1}{P}\frac{dP}{dy}=\frac{D_{\text{Mac}}}{1+y/m}$.

D_{\text{mod}}=\frac{D_{\text{Mac}}}{1+y/m} \qquad\text{and}\qquad D_{\text{Mac}}=D_{\text{mod}}(1+y/m).

The denominator is one plus the yield per compounding period. The formula uses $y$ as a decimal nominal annual yield convertible $m$ times per year. For example, an 8% nominal annual yield is entered as $y=0.08$, not $8$. The duration values in these formulas are expressed in years.

The derivation also explains the meaning of modified duration. For a small change $\Delta y$ in the nominal annual yield, the first-order price approximation is $\Delta P/P\approx-D_{\text{mod}}\Delta y$. The yield change must be entered as a decimal: a rise of 0.01 is one percentage point. This is a local approximation, not an exact prediction for a large yield change.

Worked Conversions

Worked Example: Annual Compounding

A hypothetical three-payment instrument pays $110 at the end of year 1, $121 at the end of year 2, and $133.10 at the end of year 3. Its yield is 10% effective annually. Calculate its Macaulay and modified durations. Assume the payments are certain, the yield is constant, and annual compounding applies.

Method. Since compounding is annual, $m=1$, and the periodic yield is $0.10/1=0.10$. Discount each payment to time 0:

\frac{110}{1.10}=100,\qquad \frac{121}{(1.10)^2}=100,\qquad \frac{133.10}{(1.10)^3}=100.

The price is $P=100+100+100=300$. These present values are also the weights used to calculate Macaulay duration. The weighted-time numerator is $1(100)+2(100)+3(100)=600$ dollar-years, so $D_{\text{Mac}}=600/300=2$ years.

Convert using $D_{\text{mod}}=D_{\text{Mac}}/(1+y/m)$:

D_{\text{mod}}=\frac{2}{1+0.10/1} =\frac{2}{1.10} =1.8182\text{ years}.

Check. The present-value weights are equal, so their average payment time is $(1+2+3)/3=2$ years, matching the calculated Macaulay duration. The modified duration is smaller because the yield-per-period adjustment divides by 1.10. Both duration results are rounded to four decimal places where needed.

Worked Example: Semiannual Compounding

A hypothetical instrument pays $104 after six months, $108.16 after one year, and $112.4864 after eighteen months. Its yield is 8% nominal per year, convertible semiannually. Find both durations. Assume certain cash flows and a constant yield, with payment dates coinciding with compounding dates.

Method. Here $m=2$, so the periodic yield is $0.08/2=0.04$. The present values of the three payments are:

\frac{104}{1.04}=100,\qquad \frac{108.16}{(1.04)^2}=100,\qquad \frac{112.4864}{(1.04)^3}=100.

Thus $P=300$. The payment times in years are $1/2$, $2/2$, and $3/2$. The weighted-time numerator is $(0.5)(100)+(1)(100)+(1.5)(100)=300$ dollar-years. Therefore $D_{\text{Mac}}=300/300=1$ year.

The conversion uses the semiannual yield $0.04$, or equivalently the nominal rate divided by $m=2$:

D_{\text{mod}}=\frac{1}{1+0.08/2} =\frac{1}{1.04} =0.9615\text{ years}.

Check. Equal present values give an average payment time of $(0.5+1+1.5)/3=1$ year. The conversion denominator is 1.04, not 1.08, because the annual nominal yield is compounded twice per year. Durations are reported to four decimal places where needed.

Worked Example: Recovering Macaulay Duration

A hypothetical fixed-cash-flow portfolio has a modified duration of 3.20 years. Its yield is 9% nominal per year, convertible quarterly. Calculate its Macaulay duration. Assume the quoted modified duration is defined with respect to that nominal annual yield and that the cash flows lie on quarterly dates.

Method. Rearrange the conversion rather than dividing: $D_{\text{Mac}}=D_{\text{mod}}(1+y/m)$. There are $m=4$ periods per year, so the periodic yield is $0.09/4=0.0225$.

D_{\text{Mac}}=3.20(1+0.09/4) =3.20(1.0225) =3.272\text{ years}.

Check. Substituting the result back into the forward conversion gives $3.272/1.0225=3.20$ years. The answer is rounded to four decimal places as $3.2720$ years; the product is exact to the shown precision.

Common Exam Trap: Using the Wrong Yield Factor

A frequent error is to divide by $1+y$ even when $y$ is a nominal annual yield compounded more than once per year. Another is to multiply the annual yield by $m$. Both mistakes use a factor other than the yield per compounding period.

Common mistake: For a Macaulay duration of 4.20 years at 10% nominal annual yield convertible semiannually, the incorrect calculation $4.20/1.10=3.8182$ uses the full annual nominal yield in the denominator. The correct periodic yield is $0.10/2=0.05$, so $D_{\text{mod}}=4.20/1.05=4.0000$ years. The denominator is $1+y/m$, not $1+y$ or $1+my$.

A related unit error occurs when Macaulay duration is supplied in compounding periods rather than years. Convert it to years before applying the formula here: with $m$ periods per year, a duration of $K$ periods is $K/m$ years. Alternatively, use a period-based formula consistently, but do not combine a duration in periods with a yield in annual units.

Before calculating, check three items: Is the yield nominal or effective? How many compounding periods occur per year? Is duration expressed in years? If cash-flow dates do not coincide with the stated compounding periods, first establish a consistent yield and payment-period convention rather than applying this formula mechanically.

Key takeaway: With a nominal annual yield $y$ convertible $m$ times per year, modified duration in years is Macaulay duration in years divided by $1+y/m$. To reverse the conversion, multiply modified duration by that same factor.

Check Your Understanding

For each problem, assume fixed cash flows, a constant yield, and payment dates consistent with the stated compounding convention.

A bond has Macaulay duration 4.20 years and a 6% nominal annual yield convertible semiannually. Find modified duration in years. (Answer: 4.0777 years)
A portfolio has modified duration 2.75 years at a 12% nominal annual yield convertible monthly. Find Macaulay duration. (Answer: 2.7775 years)
A security has Macaulay duration 6.00 years and a 5.6% nominal annual yield convertible quarterly. Find modified duration to four decimal places. (Answer: 5.9172 years)
A liability stream has modified duration 1.92 years at a 7% nominal annual yield convertible semiannually. Find Macaulay duration. (Answer: 1.9872 years)
A cash-flow stream has a Macaulay duration of 8.4 semiannual periods and a 10% nominal annual yield convertible semiannually. Convert the duration to years, then find modified duration in years. (Answer: 4.0000 years)

What You'll Learn