How do I compute effective convexity for a bond with embedded options?

Question

AcadiFi · Accepted Answer

Effective convexity uses the finite-difference method rather than the analytical formula because, for option-embedded bonds, cash flows are not stable as yields change. The formula is:

Effective Convexity = (P− + P+ − 2×P0) / (P0 × (Δy)²)

Where P0 is the base-case price, P+ is the price after a parallel upward shift of Δy, and P− is the price after a parallel downward shift of Δy.

Step-by-step with a realistic MBS example — Larkhaven MBS Pool, current price 101.50.

Step 1: Build or obtain a valuation model (binomial lattice or Monte Carlo) that includes prepayment behavior.

Step 2: Revalue at yields shifted down 25 bp. New cash flow timing reflects faster prepayments. Model produces P− = 102.20.

Step 3: Revalue at yields shifted up 25 bp. Slower prepayments (extension). Model produces P+ = 99.85.

Step 4: Plug in:
Effective Convexity = (102.20 + 99.85 − 2×101.50) / (101.50 × 0.0025²)
                   = (202.05 − 203.00) / (101.50 × 0.00000625)
                   = -0.95 / 0.000634
                   = -1498

The negative value confirms negative convexity.

Compare to a non-callable Treasury with similar duration — it would produce a positive convexity of roughly +85 to +110, a stark ~1600-point spread.

Critical usage points:

1. Choose Δy carefully. Too small and numerical noise dominates. Too large and you average across non-linearities the measure assumes are smooth. Standard practice is 25-50 bp.

2. Use a model that captures all embedded options, not just a yield curve shift. For MBS, prepayment models (PSA-based or econometric) drive cash flow evolution.

3. Repeat at multiple base yield levels. The measure is highly path-dependent — effective convexity at 5% yields differs substantially from effective convexity at 7% yields.

4. Report alongside effective duration; the pair is commonly used in risk reports.

Limitations: effective convexity assumes parallel yield shifts. Real-world curve twists require key-rate or bucket convexity for accuracy.

How do I compute effective convexity for a bond with embedded options?

Master Level II with our CFA Course

Related Questions

Practice Questions