How do I aggregate convexity across a bond portfolio with multiple holdings?

Question

AcadiFi · Accepted Answer

Portfolio convexity is the market-value-weighted average of individual bond convexities:

C_portfolio = Σ w_i × C_i

where w_i is bond i's market value as a fraction of total portfolio value. This is a linear aggregation because convexity (like duration) is defined as a second derivative of price divided by price, and sums of such quantities weighted by market value give the portfolio-level equivalent.

For your portfolio:
- Total MV = $40M + $60M + $25M = $125M
- w₁ = 40/125 = 0.32; w₂ = 60/125 = 0.48; w₃ = 25/125 = 0.20
- C_port = 0.32 × 80 + 0.48 × 150 + 0.20 × 320
- = 25.6 + 72 + 64
- = 161.6

So the portfolio convexity is 161.6.

**Similarly for duration**:
D_port = 0.32 × D₁ + 0.48 × D₂ + 0.20 × D₃

**Portfolio P&L from a yield shift** (parallel shift assumption):
%ΔP_port = −D_port × Δy + ½ × C_port × (Δy)²

For a 50bps rise:
- Duration effect: −D_port × 0.005 (need D_port)
- Convexity effect: 0.5 × 161.6 × (0.005)² = 0.5 × 161.6 × 0.000025 = 0.20%

**Caveats**:

1. **Parallel shift assumption**: The formula works for uniform curve moves. For non-parallel shifts (steepening, flattening), use key rate durations and convexities.

2. **Option-adjusted vs analytical**: If holdings include callables/MBS, use effective convexity from option-adjusted models, not closed-form. A portfolio mixing option-free Treasuries with MBS should use effective convexity for MBS and analytical for Treasuries, then weight-average.

3. **Market value weighting**: Some texts use duration dollars (PVBP × MV) instead, which gives the same answer but is more intuitive for hedging. The formula above is correct for percentage price changes.

4. **Credit convexity**: For corporate bonds, a full treatment separates interest-rate convexity from spread convexity — the price reaction to spread widening also has a second-order term that matters for high-yield portfolios.

Portfolio managers use aggregate convexity to calibrate how much 'second-order protection' they carry in anticipation of large yield moves. Long-convexity portfolios pay more today (via lower yield/higher price) but perform better in volatile regimes.

How do I aggregate convexity across a bond portfolio with multiple holdings?

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