How do you calculate effective convexity and why does it matter for bonds with embedded options?

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AcadiFi · Accepted Answer

Effective convexity measures the second-order curvature effect of yield changes on bond price, estimated numerically using an option-adjusted valuation model rather than...

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**Effective Convexity Formula:**

EffConvexity = [P(-Δy) + P(+Δy) - 2 × P(0)] / [P(0) × (Δy)^2]

Where:
- P(-Δy) = Price if yield decreases by Δy
- P(+Δy) = Price if yield increases by Δy
- P(0) = Current price
- Δy = Yield shift (typically 25-50 bps)

**Worked Example -- Grandview Municipal Bond (fictional)**

Grandview 5.0% coupon, 20-year callable bond at par in 5 years. Using an OAS model with Δy = 50 bps:

| Yield Scenario | Price (per $100 par) |
|---|---|
| Yield down 50 bps | $103.20 |
| Current yield | $100.00 |
| Yield up 50 bps | $96.50 |

EffConvexity = [$103.20 + $96.50 - 2 × $100.00] / [$100.00 × (0.005)^2]
= [-$0.30] / [$100 × 0.000025]
= -$0.30 / $0.0025
= **-120**

**Interpretation of Negative Convexity:**

| Convexity | Bond Type | Behavior |
|---|---|---|
| Positive (e.g., +150) | Option-free bond | Price gains accelerate as yields fall |
| Near zero | At-the-money callable | Gains capped, losses uncapped |
| Negative (e.g., -120) | Callable near call price | Price DECELERATES as yields fall (call caps upside) |
| Very positive (e.g., +300) | Putable bond | Downside protected by put |

The Grandview bond has negative convexity because as yields fall, the probability of the issuer calling the bond increases, capping the price near the call price ($100). But as yields rise, the bond behaves like a straight bond and falls freely. This asymmetry -- limited upside with full downside -- is negative convexity.

Investors in callable bonds demand a yield premium (OAS) to compensate for this unfavorable convexity profile. Portfolio managers use effective convexity alongside effective duration to estimate total price impact: ΔP/P ≈ -EffDur × Δy + 0.5 × EffConvexity × (Δy)^2.

How do you calculate effective convexity and why does it matter for bonds with embedded options?

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