Market-Value-Weighted Duration For Bond Portfolios

Why Portfolio Duration Uses Market Values

A bond portfolio’s duration summarizes the timing or yield sensitivity of all its holdings together. The key is how much each holding contributes to the portfolio’s current value: a holding worth $300,000 should generally have more influence on the portfolio measure than one worth $30,000. For that reason, component durations are combined using market-value weights, not equal weights or face-value weights.

This calculation assumes the positions are valued at the same valuation date and in the same currency. A position’s market value is its current value in the portfolio. If the data give a unit price and number of units, multiply them to find the position value. Divide each position value by the total portfolio value to find its weight.

Definition: For bond holding $j$, let $V_j$ be its market value and let $D_{M,j}$ be its Macaulay duration. For a portfolio of $k$ holdings, total market value is $V=\sum_{j=1}^{k}V_j$, and the market-value weight of holding $j$ is $w_j=V_j/V$. The portfolio Macaulay duration is $D_M=\sum_{j=1}^{k}w_jD_{M,j}$. The weights are nonnegative and sum to 1 when the holdings have positive market values.

Macaulay duration is measured in time units, usually years. It is the present-value-weighted average time of a bond’s cash flows. Portfolio Macaulay duration is therefore also a present-value-weighted average: the value weight of a holding determines how much its own average cash-flow time contributes.

Why the Weighted-Average Formula Holds

Suppose all component bonds are valued at the same date. For holding $j$, let $PV_{j,t}$ denote the present value of its cash flow payable at time $t$, measured in years. Its market value is the sum of these present values, and its Macaulay duration is their present-value-weighted average time:

V_j=\sum_t PV_{j,t}, \qquad D_{M,j}=\frac{\sum_t t\,PV_{j,t}}{V_j}.

Pooling the holdings means adding their present values at each payment time. The portfolio’s total present value is $\sum_j V_j$, while its time-weighted present-value total is $\sum_j\sum_t t\,PV_{j,t}$. Replace each inner sum with $V_jD_{M,j}$, then divide by the portfolio value:

D_M =\frac{\sum_j\sum_t t\,PV_{j,t}}{\sum_j V_j} =\frac{\sum_j V_jD_{M,j}}{\sum_j V_j} =\sum_j w_jD_{M,j}.

This argument explains why the weighting is by current market value: each $V_j$ is the total present value of that holding’s cash flows. Face value is not the denominator in the duration definition. It can be used to find market value only when the relevant price information is also applied.

Worked Example: Combining Macaulay Durations at a Common Yield

A portfolio has three bond positions with market values of $240,000, $360,000, and $400,000. Their Macaulay durations are 2.4, 5.1, and 7.2 years, respectively. Each bond’s cash flows are valued using the same effective annual yield of 5%. Find the portfolio Macaulay duration and modified duration.

Method. Calculate the portfolio’s total value, convert each position value to a market-value weight, and take the weighted average of the component Macaulay durations.

The total market value is

$$ V=240{,}000+360{,}000+400{,}000=1{,}000{,}000. $$

The weights are $240{,}000/1{,}000{,}000=0.24$, $360{,}000/1{,}000{,}000=0.36$, and $400{,}000/1{,}000{,}000=0.40$. They sum to $1.00$. Thus,

D_M=(0.24)(2.4)+(0.36)(5.1)+(0.40)(7.2) =0.576+1.836+2.880 =5.292\text{ years}.

For an effective annual yield $i$, modified duration is $D_{\text{mod}}=D_M/(1+i)$. Here $i=0.05$, so

D_{\text{mod}}=\frac{5.292}{1.05}=5.04\text{ years}.

Check. The portfolio Macaulay duration lies between the smallest and largest component durations, 2.4 and 7.2 years. The calculation also gives $5.04(1.05)=5.292$, as required by the conversion formula. The common mistake avoided here is taking a simple average of the three durations, which would give $4.9$ years and ignore the differing position values.

Combining Modified Durations

Modified duration measures approximate percentage price sensitivity to a change in yield under a specified compounding convention. For an effective annual yield $i$, it is $D_{\text{mod}}=D_M/(1+i)$. For a nominal annual yield $i^{(m)}$ convertible $m$ times per year, the corresponding formula is $D_{\text{mod}}=D_M/(1+i^{(m)}/m)$, provided Macaulay duration is in years and the yield is quoted with that convention.

Modified durations can be combined as market-value-weighted averages when every component uses the same yield-shock measure and compounding convention. To see why, modified duration is the negative derivative of price with respect to the relevant yield, divided by price. Portfolio price is the sum of component prices. Differentiating that sum and dividing by total portfolio value gives:

D_{\text{mod},P} =-\frac{1}{V}\frac{dV}{dy} =-\frac{1}{\sum_j V_j}\sum_j\frac{dV_j}{dy} =\sum_j \frac{V_j}{V}D_{\text{mod},j}.

Here $y$ denotes the yield variable being shifted, used consistently for all holdings. This result does not mean that modified durations quoted under different compounding conventions can be averaged without adjustment. First express the component measures under one consistent convention. Likewise, if each bond’s duration is calculated using a different yield and the proposed shock is not a common yield shift, the weighted-average calculation may not describe the portfolio’s sensitivity to that shock.

Worked Example: Semiannual Yield Convention

Three bond holdings have market values of $100,000, $150,000, and $250,000. Their Macaulay durations are 3.6, 5.4, and 8.4 years. All are valued at a nominal annual yield of 6% convertible semiannually. Calculate the portfolio Macaulay and modified durations.

Method. Weight Macaulay durations by market value, then apply the semiannual yield conversion. The total value is

$$ V=100{,}000+150{,}000+250{,}000=500{,}000. $$

The market-value weights are $0.20$, $0.30$, and $0.50$. Therefore,

D_M=(0.20)(3.6)+(0.30)(5.4)+(0.50)(8.4) =0.72+1.62+4.20 =6.54\text{ years}.

The yield per half-year is $0.06/2=0.03$. With $m=2$, the modified duration is

D_{\text{mod}}=\frac{6.54}{1+0.06/2} =\frac{6.54}{1.03} =6.349514\ldots\text{ years}.

To three decimal places, the portfolio modified duration is $6.350$ years. Check. Multiplying the unrounded modified duration by $1.03$ returns $6.54$ years. The common mistake avoided here is dividing by $1.06$, which would use an annual effective-rate denominator instead of the semiannual yield-period denominator.

Market Values, Not Face Values

A bond’s face value and market value are different quantities. The market value depends on the bond’s price, which may be above or below par. Duration weights must reflect the market values of the positions at the valuation date. If holdings are specified by face amount and price per $100 of face value, calculate each market value before finding the weights.

The following table summarizes the appropriate calculation for the two common duration measures.

Portfolio measure	Calculation	Important condition
Macaulay duration	Market-value-weighted average of component Macaulay durations	Component durations use the same time unit and valuation date
Modified duration	Market-value-weighted average of component modified durations	Yield shocks and compounding conventions are consistent
Modified duration from portfolio Macaulay duration	Divide by $1+i$, or by $1+i^{(m)}/m$	Use the yield and compounding convention used to value the portfolio

Worked Example: Face Amounts Do Not Determine the Weights

A portfolio holds $100,000 face amount of Bond A, priced at 96% of face, and $200,000 face amount of Bond B, priced at 103% of face. Their Macaulay durations are 4.0 and 6.0 years. Both use a nominal annual yield of 6% convertible semiannually. Find the portfolio Macaulay and modified durations.

Method. Convert face amounts to market values using the quoted prices. Bond A is worth $100{,}000(0.96)=$96{,}000$, and Bond B is worth $200{,}000(1.03)=$206{,}000$. Thus, the total market value is $96{,}000+206{,}000=$302{,}000$.

The market-value weights are $96{,}000/302{,}000=48/151$ and $206{,}000/302{,}000=103/151$. They sum to $1$. The portfolio Macaulay duration is

D_M=\frac{48}{151}(4.0)+\frac{103}{151}(6.0) =\frac{192+618}{151} =\frac{810}{151} =5.364238\ldots\text{ years}.

The semiannual yield per period is $0.06/2=0.03$, so

D_{\text{mod}}=\frac{810/151}{1.03} =5.207998\ldots\text{ years}.

To three decimal places, the portfolio Macaulay duration is $5.364$ years and the modified duration is $5.208$ years. Check. The Macaulay result lies between 4.0 and 6.0 years. Using face amounts as weights instead would give $(100/300)(4)+(200/300)(6)=5.333$ years, which is not the market-value-weighted duration because Bond A is priced below par and Bond B above par.

Common Mistakes and a Practical Checklist

The weighted-average formula is simple, but errors often arise before the multiplication. A reliable calculation separates valuation, weighting, and duration conversion.

Find current position values.
Multiply units by market price, or apply each quoted price to its face amount. Do not use face amounts alone when prices differ from par.

Calculate and check the weights.
Divide each holding’s value by the total portfolio value. Confirm that the weights sum to 1, allowing only for displayed rounding.

Combine like duration measures.
Use Macaulay durations to calculate portfolio Macaulay duration, or modified durations calculated on a consistent basis to calculate portfolio modified duration.

Convert only after checking the yield convention.
For a common effective annual yield use $1+i$; for a common nominal annual yield convertible $m$ times per year use $1+i^{(m)}/m$.

A quick reasonableness check is that a weighted average with nonnegative weights summing to 1 must lie between the smallest and largest component durations. A result outside that range usually indicates a weight error, a unit mismatch, or an arithmetic mistake. Also retain adequate precision in the weights and intermediate calculations; round the final duration only at the requested precision.

Key takeaway: Portfolio duration reflects the market-value share of each holding. Calculate the position values first, verify that the weights sum to 1, and combine durations only when their definitions, time units, and yield conventions are consistent.

Check Your Understanding

For each problem, use market-value weights and keep the stated yield convention in the conversion.

A portfolio has holdings worth $90,000 and $210,000, with Macaulay durations of 2.0 and 5.0 years. Find the portfolio Macaulay duration. (Answer: 4.1 years)
A portfolio holds 50 bonds priced at $1,200 each and 80 bonds priced at $1,125 each. Their Macaulay durations are 2.5 and 6.5 years. Find the portfolio Macaulay duration. (Answer: 4.9 years)
Two holdings are worth $120,000 and $280,000 and have Macaulay durations of 4.0 and 7.0 years. They share a nominal annual yield of 8% convertible semiannually. Find the portfolio modified duration to three decimal places. (Answer: 5.865 years)
A portfolio’s market-value weights are 25%, 35%, and 40%. The component modified durations, calculated under a common yield convention, are 3.2, 5.6, and 7.1 years. Find the portfolio modified duration. If the common yield is an effective annual 4%, also find the portfolio Macaulay duration. (Answer: Modified duration = 5.6 years; Macaulay duration = 5.824 years)
A portfolio holds $100,000 face amount of a bond priced at 96% of face and $200,000 face amount of a bond priced at 103% of face. Their Macaulay durations are 4.0 and 6.0 years. Both use a nominal annual yield of 6% convertible semiannually. Find the portfolio Macaulay and modified durations to three decimal places. (Answer: Macaulay duration = 5.364 years; modified duration = 5.208 years)

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