Portfolio Duration and Risk Aggregation

A portfolio manager holding fifty different bonds — different coupons, maturities, and issuers — needs one number that says how much the whole book will move if interest rates shift by, say, 25 basis points. Computing that directly, bond by bond, cash flow by cash flow, would be exhausting. Fortunately, duration aggregates almost effortlessly: the portfolio's duration is simply the weighted average of each bond's duration, weighted by how much of the portfolio's market value each bond represents.

The aggregation formula

$$D_{portfolio} = \sum_{i=1}^{n} w_i \times D_i, \qquad w_i = \frac{MV_i}{\sum_j MV_j}$$

In words: for each bond $i$, compute its weight $w_i$ as its market value divided by the total portfolio market value, multiply that weight by the bond's own duration $D_i$, and sum across every bond. The result behaves exactly like a portfolio's beta being a weighted average of the betas of its stocks.

The intuition holds because duration measures percentage price sensitivity, and a portfolio's total dollar sensitivity is just the sum of each position's dollar sensitivity — dividing back through by total market value recovers a single weighted-average percentage figure.

Worked example

A $10 million portfolio holds three bonds: $4 million of Bond A (duration 3), $3.5 million of Bond B (duration 7), and $2.5 million of Bond C (duration 12).

1. Weights: $w_A = 4/10 = 0.40$, $w_B = 3.5/10 = 0.35$, $w_C = 2.5/10 = 0.25$.
2. Weighted contributions: $0.40 \times 3 = 1.20$, $0.35 \times 7 = 2.45$, $0.25 \times 12 = 3.00$.
3. Portfolio duration: $1.20 + 2.45 + 3.00 = 6.65$.
4. Interpretation: a 100 bp parallel rise in rates should move the whole $10 million portfolio's value down roughly $6.65\% \times 10{,}000{,}000 = 665{,}000$, i.e. about $665,000, before any convexity adjustment.

What this means in practice

Aggregated duration is the everyday risk figure quoted on any bond fund's factsheet, and it's the number a manager targets when told to run "duration-neutral" relative to a benchmark, or to shorten duration ahead of an expected rate hike. But the weighted average only captures sensitivity to a parallel shift in rates — it says nothing about what happens if short rates and long rates move differently, which is why desks running curve trades also track key-rate durations rather than relying on the single aggregated number alone.