Combining Bond Macaulay Durations as a Market-Value-Weighted Average

Summary

The document explains how to combine the Macaulay durations of two bonds into a portfolio duration. Its key concept is that the portfolio duration is the market-value-weighted average of the individual bond durations. Each bond’s duration is multiplied by its market value, the products are added, and the result is divided by the total market value.

This relationship lets an investor solve for an unknown component duration when the other bond’s duration, both market values, and the combined duration are known. The response supplies the weighted-average formula but does not carry out the arithmetic for the question’s example. It also does not discuss assumptions or extensions, such as rebalancing, cash flows beyond the stated bonds, or the distinction between Macaulay and other duration measures. The useful lesson is the aggregation rule for a portfolio of bonds under the stated market-value weighting.

Key ideas

• Portfolio Macaulay duration is the market-value-weighted average of component durations.
• Calculate each bond’s contribution by multiplying its duration by its market value.
• Divide the sum of those contributions by the portfolio’s total market value.
• The weighted-average relationship can be rearranged to find an unknown bond duration.

Tags

# Macaulay Duration: Duration for 2 bonds


# Macaulay Duration: Duration for 2 bonds












> Using Macaulay Duration, determine the duration of Bond B if Bond A and B (market value of 600 000 dollars and 400 000 dollars respectively) have a duration of 6.7 years and the duration of A is 8.5 years.

My thinking:

Since duration looks at the time of maturity of the bond prices, Bond A and B's duration will be inter-linked. Hence, I will be able to write µ(ab) ≠ µ(a) or µ(b).

But if that is the case, then I would be unable to draw a link to the two equations between Bond A and B and Bond A?

Is my thinking correct in finding duration of Bond B?

## Answer by amsh (score 3)

https://quant.stackexchange.com/a/20746

Macaulay duration is simply a weighted average.

$MacD(A,B)=\frac{V(A) \cdot MacD(A)+ V(B) \cdot MacD(B)}{V(A)+V(B)}$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.