Macaulay And Modified Duration In Portfolio Measurement

Why Duration Is Useful

A bond’s market value depends on both the amount and the timing of its promised cash flows. Two bonds can have the same market value but respond differently when yields change: the bond whose value depends more heavily on distant cash flows is generally more rate-sensitive. Macaulay duration summarizes cash-flow timing, while modified duration translates that timing into a local measure of price sensitivity to yield.

The measures in this tutorial apply to bonds and portfolios with fixed cash flows and a specified yield convention. They are useful for comparing exposures and making small-change estimates. They are not exact forecasts for large yield moves, and the fixed-cash-flow formulas do not account for cash flows that change when interest rates change.

Definition: For a bond with fixed cash flows $C_k$ paid at times $t_k$, present value $P$, and a single annual effective yield $y$, Macaulay duration is the present-value-weighted average payment time: $$ D_{\mathrm{Mac}}=\frac{\sum_k t_k\frac{C_k}{(1+y)^{t_k}}}{P}, \qquad P=\sum_k\frac{C_k}{(1+y)^{t_k}}. $$ The weights are the present values of individual cash flows divided by the bond price. Thus, Macaulay duration is measured in years when each $t_k$ is in years.

For annual coupon payments and annual effective yield $y$, modified duration is Macaulay duration divided by $1+y$. If a bond pays $m$ times per year and its quoted yield $y^{(m)}$ is nominal, convertible $m$-thly, the periodic yield is $y^{(m)}/m$, and modified duration is:

D_{\mathrm{mod}}=\frac{D_{\mathrm{Mac}}}{1+y^{(m)}/m}.

Macaulay duration is a time-weighted summary. Modified duration is a sensitivity measure: for a small change in yield, the approximate proportional price change is $-D_{\mathrm{mod}}$ times the yield change, with the yield change expressed as a decimal in the same annual convention.

From Cash-Flow Timing to Price Sensitivity

The relationship between the two duration measures follows from differentiating the bond-price formula. Assume fixed cash flows, annual payments, and a single annual effective yield $y$. Differentiating each discounted payment gives:

\frac{dP}{dy} = -\sum_k \frac{t_k C_k}{(1+y)^{t_k+1}} = -\frac{1}{1+y}\sum_k t_k\frac{C_k}{(1+y)^{t_k}}.

The final sum is $P D_{\mathrm{Mac}}$, by the definition of Macaulay duration. Dividing by $P$ therefore gives the proportional price sensitivity:

\frac{1}{P}\frac{dP}{dy} = -\frac{D_{\mathrm{Mac}}}{1+y} = -D_{\mathrm{mod}}.

For a sufficiently small yield change $\Delta y$, the first-order approximation is consequently:

\frac{\Delta P}{P}\approx-D_{\mathrm{mod}}\Delta y, \qquad \Delta P\approx-PD_{\mathrm{mod}}\Delta y.

This is a local, linear approximation derived from the price derivative. It is most informative for small changes under the stated fixed-cash-flow assumptions. The sign is important: a positive yield change implies an approximate price decrease, while a negative yield change implies an approximate price increase.

Calculating Duration for Individual Bonds

Worked Example: An Annual-Coupon Bond Priced at Par

A hypothetical bond has a face value of $1,000, pays a 5% annual coupon, and matures in three years. Its annual effective yield is 5%. Calculate its price, Macaulay duration, and modified duration. State the interpretation of the modified duration.

Method. Discount each promised cash flow at the annual effective yield, then use the resulting present values as weights. Because coupon rate and yield are equal, the bond should be priced at par; this provides a check on the calculation.

The cash flows are $50 at the end of year 1, $50 at the end of year 2, and $1,050 at the end of year 3. Their present values are:

PV_1=\frac{50}{1.05}=47.6190,\qquad PV_2=\frac{50}{1.05^2}=45.3515,\qquad PV_3=\frac{1{,}050}{1.05^3}=907.0295.

Adding the present values gives $47.6190+45.3515+907.0295=1{,}000.0000$, as expected for a par bond. The present-value-weighted time numerator is:

$$ 1(47.6190)+2(45.3515)+3(907.0295) =47.6190+90.7030+2{,}721.0885 =2{,}859.4105. $$

Therefore, Macaulay duration is $2{,}859.4105/1{,}000=2.8594$ years. Modified duration is:

D_{\mathrm{mod}}=\frac{2.8594}{1.05}=2.7232.

The modified duration indicates that, for a small change in annual effective yield, the bond’s percentage price change is approximately $-2.7232$ times the yield change. For example, a one-percentage-point increase corresponds to an approximate 2.7232% price decrease. The method avoids treating Macaulay duration itself as the percentage price sensitivity.

Check. The price is $1,000, Macaulay duration is between one and three years, and modified duration is lower than Macaulay duration because the yield is positive.

For payments more frequent than annual, use payment times in years when calculating Macaulay duration. With semiannual payments, for example, the payment times are $0.5,1.0,1.5,\ldots$ years. The modified-duration adjustment uses the yield per payment period, not the full nominal annual yield.

Worked Example: A Semiannual Bond and the Yield-Period Convention

A hypothetical one-year bond has a face value of $1,000, pays coupons semiannually at a 4% nominal annual coupon rate, and has a 4% nominal annual yield convertible semiannually. Calculate its price, Macaulay duration in years, and modified duration.

Method. The coupon per half-year is $1{,}000(0.04)/2=$20$. The yield per half-year is $0.04/2=0.02$. The cash flows are $20 at 0.5 years and $1,020 at 1 year. Discounting each payment:

PV_{0.5}=\frac{20}{1.02}=19.6078,\qquad PV_1=\frac{1{,}020}{1.02^2}=980.3922.

The price is $19.6078+980.3922=$1{,}000.00$. Macaulay duration uses the payment times in years:

D_{\mathrm{Mac}} = \frac{0.5(19.6078)+1(980.3922)}{1{,}000} = \frac{9.8039+980.3922}{1{,}000} =0.9902\text{ years}.

The nominal annual yield is convertible semiannually, so the modified-duration denominator is $1+0.04/2=1.02$, not $1+0.04$:

D_{\mathrm{mod}}=\frac{0.9902}{1.02}=0.9708.

Check. The price is par because the coupon rate and nominal yield agree on the same semiannual basis. Macaulay duration is just under one year because some value is received after six months; modified duration is lower than Macaulay duration. The common mistake this method avoids is using the nominal annual yield as though it were the yield for each half-year.

Measuring Duration for a Portfolio

For a portfolio of fixed-cash-flow bonds, its Macaulay duration is the market-value-weighted average of the bonds’ Macaulay durations, provided the portfolio and component durations use consistent timing and yield conventions. If bond $j$ has market value $V_j$, total market value $V=\sum_jV_j$, and Macaulay duration $D_{\mathrm{Mac},j}$, then:

D_{\mathrm{Mac,port}} = \sum_j\frac{V_j}{V}D_{\mathrm{Mac},j}.

When all bonds use the same annual yield convention, the corresponding portfolio modified duration is the same market-value-weighted average of their modified durations. More generally, portfolio price sensitivity can be aggregated by adding each position’s dollar price sensitivity and dividing by total portfolio value. Do not combine durations expressed on incompatible yield bases.

Worked Example: Portfolio Duration and an Estimated Rate-Driven Loss

A hypothetical insurer holds two annual-coupon bonds. Bond A has a market value of $600,000 and is a three-year, $1,000 par bond with a 5% annual coupon and 5% annual effective yield. Bond B has a market value of $400,000 and is a one-year, $1,000 par bond with the same coupon and yield. Estimate the portfolio’s percentage and dollar price change if the common annual effective yield increases by 20 basis points.

Method. First calculate each bond’s Macaulay duration from its cash flows. Then use the common yield basis to obtain modified durations and market-value-weight the results. Finally, apply the first-order approximation.

For Bond A, the present values of its cash flows are $47.6190, $45.3515, and $907.0295, as obtained by discounting $50, $50, and $1,050 at 5%. The price is $1,000, and the weighted-time numerator is:

$$ 1(47.6190)+2(45.3515)+3(907.0295)=2{,}859.4105. $$

Thus $D_{\mathrm{Mac},A}=2.8594$ years and $D_{\mathrm{mod},A}=2.8594/1.05=2.7232$. Bond B pays $1,050 at year 1; its present value is $1{,}050/1.05=$1{,}000$. Therefore $D_{\mathrm{Mac},B}=1.0000$ year and $D_{\mathrm{mod},B}=1/1.05=0.9524$.

The market-value weights are $600{,}000/1{,}000{,}000=0.6$ and $400{,}000/1{,}000{,}000=0.4$. The portfolio durations are:

D_{\mathrm{Mac,port}} =0.6(2.8594)+0.4(1.0000) =2.1156\text{ years}, $$ $$ D_{\mathrm{mod,port}} =0.6(2.7232)+0.4(0.9524) =2.0149.

A 20-basis-point increase is $\Delta y=0.0020$. Hence the estimated percentage change is $-2.0149(0.0020)=-0.0040298$, or approximately $-0.4030\%$. Applied to the $1,000,000 portfolio value, the estimated dollar change is $-$4,029.80$, so the approximate loss is $4,029.80.

Check. The portfolio Macaulay duration also equals $2.1156/1.05=2.0149$ when converted to modified duration, consistent with the weighted component modified duration. The positive yield change gives a negative price estimate. This calculation avoids the common error of treating 20 basis points as 0.20 rather than 0.0020.

Interpretation, Assumptions, and Exam Traps

Duration is informative because it combines the maturity and size of every cash flow into one interpretable measure. Macaulay duration is the average time at which the bond’s present value is received, with larger present-value cash flows receiving larger weights. Modified duration measures the slope of price with respect to yield, scaled by price. It is therefore suited to a first-order comparison of price sensitivity under a small yield change.

The formulas require a consistent bond-pricing setup. In particular, the cash flows are assumed fixed, the bond is valued using the stated yield convention, and the yield change is applied consistently. For a portfolio approximation, the yield shock must be appropriate to the duration being used; the single-duration calculation represents a parallel change in the relevant yield. The estimate is not an exact repricing result.

A frequent exam trap is to report Macaulay duration as though it were modified duration. For an annual bond with positive yield, the incorrect estimate would be $\Delta P/P\approx-D_{\mathrm{Mac}}\Delta y$. The correction is to divide Macaulay duration by $1+y$ for an annual effective yield, or by $1+y^{(m)}/m$ for a nominal yield convertible $m$-thly. Another trap is mixing units: a 20-basis-point change is $0.0020$, not $20$ or $0.20$. Always state the yield convention, convert basis points to a decimal, and check that an increase in yield produces an estimated price decrease.

Key takeaway: Macaulay duration is a present-value-weighted time in years. Modified duration is the local proportional price sensitivity to yield. For fixed cash flows, a small yield change gives $\Delta P/P\approx-D_{\mathrm{mod}}\Delta y$; portfolio measures require consistent conventions and market-value weights.

Check Your Understanding

For each problem, keep payment timing and yield conventions explicit. The final answer is provided for checking.

A hypothetical two-year bond has face value $1,000, a 4% annual coupon, and a 4% annual effective yield. Calculate its price, Macaulay duration, and modified duration. (Answer: Price $1,000.00; Macaulay duration 1.9615 years; modified duration 1.8861.)
A hypothetical three-year zero-coupon bond has an annual effective yield of 6%. Find both duration measures and estimate its percentage price change for a 25-basis-point yield increase. (Answer: Macaulay duration 3.0000 years; modified duration 2.8302; estimated change $-0.7075\%$.)
A portfolio contains a $250,000 position with modified duration 3.2 and a $750,000 position with modified duration 0.8, both measured on the same yield basis. Calculate portfolio modified duration and the estimated dollar loss for a 15-basis-point yield increase. (Answer: Portfolio modified duration 1.4000; estimated loss $2,100.00.)
A hypothetical three-year zero-coupon bond is priced using a 5% nominal annual yield convertible semiannually. Calculate Macaulay and modified duration, then estimate the percentage price change for a 30-basis-point increase in the quoted nominal annual yield. (Answer: Macaulay duration 3.0000 years; modified duration 2.9268; estimated change $-0.8780\%$.)

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