Large Yield Shifts and Non-Linearity
Large Yield Shifts and Non-Linearity
Duration works well for small yield moves, but when rates jump 2, 3, or 4 percent, the relationship between yield and price becomes curved rather than straight. This curvature—called convexity—is why losses exceed the linear duration estimate and why sophisticated bond managers track both metrics.
Key takeaways
- Duration is a linear approximation that underestimates losses when yields rise sharply.
- Convexity measures the curvature of the price-yield relationship.
- Large yield moves expose the non-linearity that duration ignores.
- Positive convexity (typical bonds) means realized losses are smaller than duration suggests.
- Understanding convexity is essential for managing portfolios through volatile rate environments.
The Breakdown of Linear Approximation
The relationship between bond price and yield is fundamentally non-linear. A bond's price is the present value of all future cash flows discounted at the current yield. When you discount at 2%, the price is higher than when you discount at 3%, but the drop from 2% to 3% is not the same as the drop from 3% to 4%, because each cash flow's weight decreases as it moves further into the future.
Duration assumes this relationship is a straight line. If yields rise 1%, the price drops 1× duration. If yields rise 2%, the duration model says the price drops 2× duration. But this is only an approximation. The actual price-yield curve is convex—it bulges—so the actual price loss for a 2% yield rise is less severe than 2× duration suggests.
Consider a 10-year bond with duration 8.5 years and current price 100 at 3% yield:
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Duration predicts: 1% rise → price = 100 − 8.5 = 91.5
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Actual price at 4% yield: 92.3 (convexity adds back 0.8%)
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Duration predicts: 2% rise → price = 100 − 17 = 83
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Actual price at 5% yield: 85.7 (convexity adds back 2.7%)
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Duration predicts: 3% rise → price = 100 − 25.5 = 74.5
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Actual price at 6% yield: 79.8 (convexity adds back 5.3%)
The gap widens as the yield move gets larger. This is convexity at work.
Why Convexity Exists
Convexity arises because bond cash flows are fixed. When yields fall, the present value of far-future cash flows rises a lot, but the present value of near-future cash flows can't fall—they're already being discounted at the current low rate. This creates an asymmetry: price gains from yield falls exceed price losses from equal-sized yield rises.
Imagine two bonds:
Bond A: Pays 100% of cash in year 1.
If yields drop from 3% to 2%, the present value jumps from 97.09 to 98.04 (0.95 gain).
If yields rise from 3% to 4%, the present value drops from 97.09 to 96.15 (0.94 loss).
Very little convexity—the cash flow is too soon to benefit from the curve shape.
Bond B: Pays 0% coupon, 100% in year 10.
If yields drop from 3% to 2%, the present value jumps from 74.41 to 82.03 (7.62 gain).
If yields rise from 3% to 4%, the present value drops from 74.41 to 67.56 (6.85 loss).
Much more convexity—the far-future cash flow benefits enormously from the yield curve shape.
This is why zero-coupon bonds and long-maturity bonds have the most convexity. They have the most cash sitting far in the future, so changes in the discount rate affect them most dramatically.
The 2022 Bond Bear as an Example
In 2022, the 10-year Treasury yield rose from 1.5% to 3.88%—a 2.38% move. A typical long-duration bond fund with 15-year duration would have experienced:
- Duration prediction: 15 × 2.38% = −35.7% loss
- Actual loss: approximately −29% (convexity reduced losses by roughly 6 percentage points)
This wasn't hypothetical. TLT (iShares 20+ Year Treasury ETF) fell 29.7% that year. BND (Vanguard Total Bond Market ETF) fell 13.0%, consistent with its 5.5-year duration predicting 5.5 × 2.38% = −13.1%.
The math worked better for intermediate bonds because convexity's benefit is smallest when you're near par value with moderate duration. It works worst for extreme moves in zero-coupon bonds or very long securities.
Positive vs Negative Convexity
Standard bonds have positive convexity: the price-yield curve bends upward, and realized losses are smaller than linear duration predicts. This is a gift—it means you're overprotected in the duration model. You expected a bigger loss, but convexity cushioned the blow.
But some bonds have negative convexity: mortgage-backed securities (MBS) and callable corporate bonds. These securities contain embedded options. A mortgage-backed security can be prepaid if rates fall (homeowners refinance), capping the upside. When rates fall, you get the price gain only until the MBS is called away. A 30-year MBS with a 3% coupon behaves like a much shorter bond when rates drop. The effective duration changes depending on the rate environment.
For callable corporate bonds, the issuer can call the bond if rates fall. The bondholder benefits if rates rise (capital appreciation) but is capped if rates fall (bond called away at par). This negative convexity is why callable bonds offer higher yields than straight bonds—investors accept the capped upside in exchange for the higher coupon.
A portfolio of callable bonds in a falling-rate environment performs worse than duration would suggest. In a rising-rate environment, they perform about as expected. During periods of rate volatility, negative convexity is costly because the portfolio's effective duration can shorten suddenly when the bond moves near the call price.
Measuring Convexity
Convexity is measured in mathematical terms as the second derivative of the price-yield function. For investors, it's usually presented as a single number—the convexity statistic—in basis points per basis point squared.
A typical 7-year corporate bond might have convexity of 70. This means if yields change by 100 basis points (1%), convexity adds back approximately (100^2 / 10000) × 70 = 0.7% to the price change. For a 2% yield move, convexity adds (200^2 / 10000) × 70 = 2.8% back. For a 3% yield move, it adds 6.3% back.
High-coupon bonds have more convexity than low-coupon bonds. Long-duration bonds have more convexity than short-duration bonds. Zero-coupon bonds have very high convexity. Bond funds and indices publish their convexity, allowing managers to compare portfolios and understand which will hold up best in volatile markets.
Implications for Portfolio Construction
Understanding convexity changes how you size positions. If you expect moderate yield volatility, owning bonds with high convexity (long duration, low coupon, zero-coupon) provides a hedge. If rates spike, convexity cushions losses. If rates plummet, convexity boosts gains.
Conversely, callable bonds look attractive in yield but are unattractive in convexity. You're compensated for accepting negative convexity via a higher coupon. This trade-off made sense in the 2010s when rates were expected to stay low and the probability of rates falling further (triggering calls) was low. It made less sense in 2021–2022, when rates were rising and callable bonds' negative convexity was a significant drag.
A practical rule: In low-yield, rising-rate environments, emphasize positive convexity. In high-yield, falling-rate environments, the cost of negative convexity is lower because rates are less likely to fall further. In uncertain environments, high convexity is insurance.
Non-Linear Risk and Scenario Analysis
Because the price-yield relationship is curved, single-point estimates (like duration) are increasingly unreliable as yields move. Professional bond managers use scenario analysis: they calculate portfolio value at yields 1%, 2%, 3%, and 4% higher and lower than current, then observe the actual curve rather than relying on duration alone.
A portfolio manager holding AGG (the Vanguard Total Bond Market ETF) might ask: "What if 10-year yields jump to 5%? What if they fall to 1%?" The answers give a fuller picture of risk than duration alone. Convexity ensures that the portfolio performs better in both extreme scenarios than a simple duration calculation would suggest, though the gains are asymmetric.
Visual Relationship
Next
Yield curves are rarely flat; different maturities have different yields. The curve can shift in parallel (all rates up or down equally), or it can twist (short rates up while long rates down). Understanding how curves move is essential to managing duration portfolios.