What Is Negative Convexity?
Negative convexity describes a bond price–yield relationship that is concave (curves inward). For these bonds, price gains when yields fall are smaller than price losses when yields rise (i.e., the bond’s price behavior is asymmetrical and worse than implied by duration alone). Negative convexity most commonly appears in mortgage‑backed securities and in callable bonds when yields are low (so exercise/call or prepayment is likely).
Key takeaways
– Convexity is the second derivative of price with respect to yield; it measures how a bond’s duration changes as yields change.
– Positive convexity is desirable: prices rise more when yields fall and fall less when yields rise. Negative convexity is the opposite and is an additional risk.
– Mortgage‑backed securities (prepayment option) and callable bonds (issuer call option) typically show negative convexity in some yield ranges.
– Because duration is a linear approximation, investors add a convexity adjustment to improve price‑change estimates.
– Investors manage negative convexity by reducing exposure, using derivatives (futures, swaps, swaptions), buying non‑callable alternatives, or dynamically hedging.
Understanding negative convexity — intuition and mechanics
– Duration measures first‑order sensitivity: percentage price change ≈ -Duration × change in yield (Δy). This is linear.
– Convexity is the second‑order term that corrects the linear approximation: it reflects curvature in the price‑yield relationship. With positive convexity the curvature is upward; with negative convexity it’s downward (concave).
– Why callable/MBS exhibit negative convexity:
– Callable bond: when yields fall, the issuer is more likely to call (refinance), capping price appreciation. When yields rise, price falls as usual. That asymmetry produces concavity.
– Mortgage‑backed security (MBS): falling rates accelerate prepayments (principal returned early), limiting price upside; rising rates slow prepayments, extending duration and amplifying price decline.
Measuring convexity — formulas and conventions
1) Standard Taylor approximation (common textbook convention)
Percentage price change ≈ -Modified Duration × Δy + 0.5 × Convexity × (Δy)^2
– Here Δy is change in yield in decimal (e.g., 1% = 0.01).
– Convexity in this formula is the “traditional” convexity measure (second derivative scaled appropriately).
2) Practical finite‑difference approximation (easy to compute)
Convexity ≈ [P(+) + P(-) − 2×P(0)] / [2 × P(0) × (Δy)^2]
– P(0) = current price; P(+) = price if yield is decreased by Δy; P(-) = price if yield is increased by Δy.
– This is the formula used in many practical settings to estimate convexity from observed prices.
Example calculation (numbers from Investopedia example)
– P(0) = $1,000
– P(+) (rates −1%) = $1,035
– P(-) (rates +1%) = $970
– Δy = 0.01
Convexity approximation:
Convexity ≈ ($1,035 + $970 − 2×$1,000) / [2 × $1,000 × 0.01^2]= $5 / $0.2 = 25
Convexity adjustment (convention used in the example):
Convexity adjustment = Convexity × 100 × (Δy)^2= 25 × 100 × 0.0001 = 0.25 (percent)
Note on sign and conventions:
– Different sources use slightly different conventions (factor of 0.5, 100 multipliers, sign conventions). The textbook Taylor form (with 0.5×convexity term) is standard; some practical presentations scale convexity so the adjustment is written as Convexity×100×(Δy)^2. Always confirm the unit and sign of duration and convexity before applying formulas. When using the standard formula, note the negative sign on the duration term: price falls when yields rise.
Using duration + convexity to estimate price change (practical)
– Standard percentage price change ≈ -Duration × Δy + 0.5 × Convexity × (Δy)^2.
– Apply Δy with sign: e.g., for a 1% fall in yield use Δy = -0.01.
– If convexity is negative, the second term reduces the price gain for a yield drop, and increases the loss for a yield rise.
Practical steps for investors and portfolio managers
1) Identify where negative convexity exists in the portfolio
– Flag callable issues, MBS, collateralized mortgage obligations, and other structures with optionality.
– Check call schedules and prepayment assumptions (CPR for MBS).
2) Measure sensitivity
– Compute modified duration and approximate convexity (finite difference using small Δy, e.g., 1 or 10 bps).
– Run scenario analysis: simulate price changes for a range of yield moves (±25, ±50, ±100 bps). Include nonlinear effects from option exercise models (e.g., prepayment models).
3) Understand option behavior
– Model the embedded option: for callable securities model the call decision at each possible path; for MBS use prepayment models tied to rate paths.
– Estimate expected cash flows under scenarios—not just static yield shocks—because optionality depends on future rate paths.
4) Set risk limits and decide target convexity profile
– Determine allowed negative convexity per mandate, or target overall portfolio convexity (net across assets and hedges).
– Consider replacing heavily negatively convex exposures with non‑callable or floating instruments if possible.
5) Hedging strategies (practical choices)
– Duration hedge via futures: short Treasury futures to offset interest‑rate exposure (helps when rates rise). This is cheap and fast but linear.
– Use options/swaptions to offset optionlike behavior:
– Buy payer swaptions (right to enter a pay‑fixed receive‑float swap) or buy interest‑rate caps to protect against rising rates.
– These options add positive convexity (nonlinear payoff) and can offset negative convexity from MBS/callables.
– Use swaps and structured overlay:
– Receive floating / pay fixed swaps (or take positions that profit when rates rise) to hedge downside from rate increases.
– Dynamic hedging:
– Rebalance hedges as rates move because option exposure (and effective duration) changes with yield levels (hedge ratios are path‑dependent).
– Diversification and allocation:
– Use floating‑rate notes, T‑bills, inflation‑linked securities, or shorter maturities to reduce overall sensitivity.
6) Implementation checklist
– Calculate P(0), P(+) and P(-) with consistent yield shifts; compute convexity.
– Run scenario P&L and worst‑case analyses.
– Choose hedging instruments based on cost, liquidity, and hedge effectiveness.
– Monitor cash flows (prepayment speeds, call notices) and rebalance hedges frequently when exposures are option‑sensitive.
Limitations, risks and monitoring
– Model risk: Prepayment and call behavior depend on borrower economics and can deviate from models—constant monitoring and model validation are critical.
– Hedging cost: Options are expensive; dynamic hedge costs and basis between hedging instruments and underlying securities can add friction.
– Liquidity risk: Some negatively convex securities are thinly traded; hedge instruments may imperfectly correlate.
– Residual convexity: Hedging duration does not eliminate curvature; explicit convexity hedges (options/swaptions) or dynamic rebalancing are needed.
Summary — what you should remember
– Negative convexity means a bond’s price gains are limited when rates fall and losses may be larger when rates rise; it arises from embedded options (issuer calls, borrower prepayments).
– Duration alone understates the risk for negatively convex securities; convexity and scenario modeling are essential.
– Practical management includes identifying exposures, measuring convexity, running scenarios, and using a mix of linear (futures, swaps) and nonlinear (options, swaptions) hedges plus active monitoring and rebalancing.
Source
– Investopedia: “Negative Convexity” — https://www.investopedia.com/terms/n/negative_convexity.asp
If you’d like, I can:
– Run a worked numeric example applying the standard convexity formula and show the difference depending on convention, or
– Produce a step‑by‑step hedge implementation plan for a sample MBS position with dollar amounts and instrument suggestions. Which would you prefer?