How do you calculate the duration of a portfolio containing multiple bonds?
I understand Macaulay and modified duration for a single bond, but when my FRM practice exam asks for the duration of a portfolio of three or four bonds, I'm not sure how to combine them. Is it just a weighted average? What are the weights — face value, market value, or something else?
Portfolio duration is indeed a weighted average, but the weights must be based on market values, not face values or par amounts. This is a common exam trap.
The Formula
D_portfolio = SUM(w_i x D_i)
Where:
- w_i = MV_i / MV_total (market value weight of bond i)
- D_i = modified duration of bond i
Why Market Value Weights?
Consider two bonds with the same $1,000 face value:
- Bond A: 2% coupon, 10Y maturity, trading at $850 (discount)
- Bond B: 6% coupon, 10Y maturity, trading at $1,150 (premium)
Using par weights would weight them equally, but Bond B contributes more economic exposure ($1,150 vs $850). Market value weights capture this correctly.
Worked Example
Terrastone Fixed Income manages a 3-bond portfolio:
| Bond | Face Value | Market Price | Market Value | Mod Duration |
|---|---|---|---|---|
| Ridgeline Corp 4.5% 2029 | $2,000,000 | 98.50 | $1,970,000 | 3.2 |
| Summerfield Muni 3.8% 2033 | $3,000,000 | 94.20 | $2,826,000 | 5.8 |
| Ironclad Treasury 5.0% 2041 | $1,500,000 | 102.75 | $1,541,250 | 11.4 |
Total market value: $1,970,000 + $2,826,000 + $1,541,250 = $6,337,250
Weights:
- w_Ridgeline = 1,970,000 / 6,337,250 = 0.3109
- w_Summerfield = 2,826,000 / 6,337,250 = 0.4460
- w_Ironclad = 1,541,250 / 6,337,250 = 0.2432
Portfolio duration:
D_p = 0.3109 x 3.2 + 0.4460 x 5.8 + 0.2432 x 11.4
D_p = 0.995 + 2.587 + 2.772
D_p = 6.35 years
Interpretation: If all yields shift up by 100 bps in parallel, the portfolio value will decline by approximately 6.35%.
Dollar Duration
For absolute risk measurement:
Dollar Duration = D_portfolio x MV_total x 0.01
= 6.35 x $6,337,250 x 0.01 = $402,415
This means a 1% parallel rate increase destroys approximately $402K of value.
Limitations
- Assumes a parallel shift in the yield curve — not realistic for non-parallel moves (use key rate duration instead)
- Duration is a linear approximation — for large rate changes, convexity matters
- Different bonds may have different yield curves (corporate vs. treasury), so "parallel shift" is ambiguous
For more fixed income risk analysis, check our FRM Part I course materials.
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