Yield-Based Bond Convexity and Portfolio Properties

Fixed Income. Worth 11 to 14 percent of the exam. One session: the lesson, the rules, the method, then the questions.

Fixed IncomeYield-Based Bond Convexity and Portfolio Properties
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The lesson

Runtime 9 minutes 50 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate the convexity-adjusted percentage price change, describe why convexity is always positive for an option-free bond and what drives it higher, describe when a callable bond or MBS turns negatively convex, and calculate portfolio duration and convexity as market-value-weighted averages.

Duration draws a straight tangent line against the actual price-yield curve, and that line is only ever exactly right at the single point it touches. Convexity is the correction for everything the straight line misses: the full approximation is %change in price is approximately -(modified duration)(change in yield) + 0.5(convexity)(change in yield)^2. The first term is the linear, duration-only estimate; the second term is the convexity adjustment, correcting for the fact that the true price-yield relationship curves rather than running straight.

For an option-free bond, convexity is always a positive number. Because the yield change is squared in the adjustment term, that correction is always added, never subtracted, regardless of whether yields rose or fell. This means duration alone systematically understates an option-free bond's actual behavior in both directions. It understates the real price gain when yields fall, and it overstates the real price loss when yields rise. Convexity, for an option-free bond, always works in the investor's favor.

Convexity rises with the same three drivers that raise duration: longer maturity, lower coupon, lower yield. It scales roughly with duration squared. A zero-coupon bond carries the highest convexity of any bond at a given maturity and yield, for the same reason it carries the highest duration. Every dollar of cash flow arrives at a single future date. Nothing pulls that concentration in earlier the way coupon payments would, which maximizes the curvature of the price-yield relationship rather than minimizing it. The word zero describes the coupon, never the convexity.

Callable bonds and mortgage-backed securities can turn negatively convex, but only once the embedded option sits at or near the money. A callable bond behaves like an ordinary option-free bond, with ordinary positive convexity, whenever yields stay high enough that its call option remains far out of the money. Only once yields fall enough that the bond's price approaches or exceeds its call price does the issuer's incentive to actually call the bond cap further price appreciation. The price-yield curve bends backward into negative convexity. That negative convexity is conditional on where yields currently sit, never a fixed, permanent property of being callable in the first place. An exam answer that states callable bonds always carry negative convexity is a false absolute.

Portfolio duration and portfolio convexity are each computed the same way: a market-value-weighted average of the individual measure across every bond held, weight times duration (or convexity), summed across the whole portfolio. Both figures remain local, linear-plus-quadratic approximations, valid mainly for small-to-moderate parallel shifts in the yield curve; neither one captures a non-parallel shift, where short and long rates move by different amounts at the same time.

Convexity as the gap between the curve and duration's straight line price yield actual price (curved) duration estimate (straight)
Duration draws a straight line tangent to the price-yield curve. The curve bends away from that line, and convexity is a second correction for exactly that gap.

Worked in full

A bond has a modified duration of 6.03 and a convexity of 52. If yields rise by 100 basis points, what is the estimated percentage price change using both duration and convexity? Duration term = -(modified duration) x (change in yield) = -6.03 x 0.01 = -6.03 percent. Convexity adjustment = 0.5 x convexity x (change in yield)^2 = 0.5 x 52 x (0.01)^2 = 0.26 percent. Total estimated price change = -6.03 + 0.26 = -5.77 percent, a smaller loss than the duration-only estimate of -6.03 percent would suggest.

The same problem, one step removed

Same bond: modified duration 6.03, convexity 52, yields rise 100 basis points. Compute the duration term and the convexity adjustment separately, then add them yourself.

The trap

Subtracting the convexity adjustment when yields rise mistakes what squaring the yield change actually does: (change in yield) squared is always positive regardless of direction, so for a positively convex, option-free bond the adjustment is always added, never subtracted, whichever way yields moved.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

The full price change formula adds a convexity term to the duration-only linear estimate to correct for the curvature of the price-yield relationship

The complete approximation is %delta-P is approximately -(Modified Duration)(delta-Y) + 0.5(Convexity)(delta-Y)^2. The first term is the linear, duration-only estimate; the second term is the convexity adjustment, which corrects for the fact that the true price-yield relationship is a curve, not a straight line. The convexity adjustment is always added, and because delta-Y is squared, it is always non-negative for a bond with positive convexity, regardless of whether yields rose or fell.

Convexity is always positive for an option-free bond, and it always works in the investor's favor

For an option-free bond, convexity is a positive number, which means the convexity adjustment always adds to the estimated price change: it reduces the magnitude of an estimated price decline when yields rise, and it increases the magnitude of an estimated price gain when yields fall. Duration alone therefore systematically understates both price gains and price declines for an option-free bond; the larger the yield change, the larger this understatement becomes, which is why the duration-only approximation is least reliable for large yield moves.

Convexity rises with longer maturity, lower coupon, and lower yield, the same directional drivers as duration, and it scales roughly with duration squared

A zero-coupon bond has the highest convexity of any bond at a given maturity and yield, because its entire cash flow sits at a single future date, maximizing the dispersion of cash flows around the duration point; a higher-coupon bond returns more value sooner, reducing both duration and convexity. As yield level rises, convexity falls, because a higher discount rate compresses the present-value weight of the most distant cash flows, reducing their contribution to the curvature. These are the same three drivers that raise duration, and convexity is not an independent, unrelated property, it scales approximately with the square of duration.

Callable bonds and mortgage-backed securities can exhibit negative convexity, but only when the embedded option is at or near the money

A callable bond behaves like an ordinary option-free bond, with ordinary positive convexity, whenever yields are high enough that its call option is far out of the money; only when yields fall enough that the bond's price approaches or exceeds its call price does the issuer's incentive to call cap further price appreciation, bending the price-yield curve backward into negative convexity. A mortgage-backed security exhibits the same effect for the same underlying reason (the investor is effectively short an embedded option to the borrower), because falling rates accelerate mortgage prepayments and return principal to the investor exactly when it must be reinvested at the new, lower rates. Neither instrument has permanently negative convexity, the sign is conditional on the prevailing yield level relative to the option's exercise price.

Portfolio duration and convexity are market-value-weighted averages of the individual holdings, and they carry the same interpretive limitations as at the single-bond level

Portfolio duration and portfolio convexity are each calculated as the market-value-weighted average of the corresponding measure across every bond held, weight_i multiplied by duration_i (or convexity_i), summed across the portfolio. Both remain local, linear-plus-quadratic approximations valid mainly for small-to-moderate, parallel shifts in the yield curve; they do not capture a non-parallel shift (the curve twisting or steepening rather than moving uniformly), and duration in particular becomes an increasingly unreliable estimate as the size of the yield change grows, which is precisely the gap convexity is added to narrow, not eliminate entirely.

The trick

The convexity adjustment is always added, never subtracted, for a bond with positive convexity

Squaring delta-Y removes its sign, so 0.5 x Convexity x (delta-Y)^2 is positive whether yields rose or fell; a candidate who subtracts this term when yields rise has the direction of the correction backward.

Zero-coupon = zero coupons paid early = maximum convexity, not zero convexity

The word 'zero' describes the coupon, not the convexity; with all cash flow concentrated at a single maturity date, a zero-coupon bond has the highest convexity of any bond sharing that maturity and yield.

Callable-bond negative convexity is conditional on the call being near the money, not a permanent label

The same callable bond has ordinary positive convexity when yields are high (call far out of the money) and negative convexity only once yields have fallen enough to put the call at or in the money; 'callable bonds always have negative convexity' is a false absolute the exam tests directly.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. When a question says 'estimate the price change using duration and convexity,' always compute both the duration term and the 0.5 x Convexity x (delta-Y)^2 term and add them.
  2. Keep delta-Y in decimal form throughout, and remember the convexity term is squared, so its magnitude is usually much smaller than the duration term unless the yield change is large.
  3. For a bond-comparison question on convexity, check maturity, coupon, and yield level together, the same three drivers that raise duration also raise convexity.
  4. For a callable bond or MBS question, first determine whether the embedded option is near the money (price near or above call price, or rates recently fallen) before concluding convexity is negative.
  5. For a portfolio question, compute duration and convexity as market-value-weighted averages of the holdings, and note that both estimates lose accuracy for large or non-parallel yield curve shifts.

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Exam level

A bond has a modified duration of 7.5 and a convexity of 65. If yields increase by 100 basis points (1.0%), the best estimate of the bond's percentage price change is closest to:

How sure are you?

Correct: A. Full price change = -Duration x delta_Y + (1/2) x Convexity x (delta_Y)^2. = -7.5 x 0.01 + 0.5 x 65 x (0.01)^2. = -0.075 + 0.5 x 65 x 0.0001. = -0.075 + 0.00325. = -0.07175 = -7.175%, closest to -7.17%. The convexity adjustment is positive (+0.325%), partially offsetting the duration-driven price decline. Option A ignores convexity. Option C incorrectly subtracts the convexity term.
B. You might be tempted to choose -7.83% by only considering the duration effect and ignoring the convexity adjustment, but this violates the principle that convexity provides a positive adjustment to the price change, reducing the overall impact of yield increases compared to what duration alone would suggest.
C. Choosing -7.50% with no adjustment needed might seem logical if you only consider the duration effect, but it ignores the positive convexity adjustment that partially offsets the price decline, leading to a less negative price change than -7.50%.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 2Exam level

Which of the following bonds is most likely to exhibit negative convexity?

How sure are you?

Correct: C. A callable bond exhibits negative convexity when yields fall to the point where the bond's price approaches the call price. At that level, the issuer is likely to call the bond, capping price appreciation. The price-yield curve bends backward (price rises less than duration predicts and can even decline relative to a non-callable equivalent). Option C, trading well below call price, behaves like a normal bond (positive convexity) because the call is far out-of-the-money. Zero-coupon and Treasury bonds always have positive convexity.
A. You might be tempted to choose a Treasury note because it has a fixed coupon and maturity, but Treasury notes always exhibit positive convexity, meaning their price increases more than proportionally with falling yields, unlike a callable bond that can be called away when prices rise.
B. Option C, trading well below call price, behaves like a normal bond (positive convexity) because the call is far out-of-the-money.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 3Exam level

Bond A has a modified duration of 8 and convexity of 80. Bond B has a modified duration of 8 and convexity of 120. If interest rates fall by 200 basis points, which bond most likely has the higher price appreciation?

How sure are you?

Correct: A. When rates fall 200 bps, the convexity adjustment adds to price gains. Bond B: price change = -8 x (-0.02) + 0.5 x 120 x (0.02)^2 = +16% + 0.5 x 120 x 0.0004 = +16% + 2.4% = +18.4%. Bond A: price change = +16% + 0.5 x 80 x 0.0004 = +16% + 1.6% = +17.6%. Bond B outperforms by 0.8 percentage points on a 200 bps move. Higher convexity is always beneficial for bond holders in either direction. It amplifies gains and cushions losses.
B. Choosing B might tempt you to overlook the impact of convexity, thinking that equal durations mean equal price changes. However, convexity, not just duration, determines the magnitude of price appreciation when interest rates fall, making Bond B's higher convexity crucial for greater price gains.
C. Higher convexity is never a disadvantage; it works in the investor's favor in both directions, adding extra gains when rates fall and cushioning losses when rates rise. Bond A's lower convexity (80 versus Bond B's 120) means it gains less, 17.6 percent versus 18.4 percent, when rates fall 200 basis points, not more.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 4Exam level

The convexity adjustment in the bond price change formula is most likely expressed as:

How sure are you?

Correct: C. The full price change approximation is: %delta_P = -D_mod x delta_Y + (1/2) x Convexity x (delta_Y)^2. The (1/2) coefficient comes from the Taylor series expansion of the bond price function (the second-order term). Option A omits the squared term. Option B omits the (1/2) coefficient.
A. You might be tempted by choice A if you mistakenly think the convexity adjustment follows a linear relationship with the change in yield, but the correct formula requires the change in yield to be squared, not just multiplied, to accurately reflect the curvature of the price-yield relationship.
B. Forgetting the (1/2) coefficient. The CFA exam will have a numerical answer that requires the (1/2). A candidate who uses Convexity x (delta_Y)^2 will get double the right answer and choose the wrong option.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 5Harder

A portfolio manager states: 'Given two bonds with identical duration, I always prefer the one with higher convexity.' Under which condition is this preference NOT fully justified on a standalone basis, most likely?

How sure are you?

Correct: C. Higher convexity is always theoretically desirable. It produces better price performance in both rising and falling rate environments. HOWEVER, higher convexity bonds typically trade at a premium (lower yield) precisely because investors value this characteristic. If rates remain stable and the rate move is small, the benefit of convexity is negligible but the investor has already paid for it through lower yield. In a stable rate environment, the higher-convexity bond may underperform on a total return basis because its yield advantage is zero. This is the exam nuance: convexity has a price, and that price (yield give-up) only pays off if rates move significantly.
A. You might think that immunization focuses solely on matching durations, making convexity irrelevant, but immunization actually benefits from higher convexity to enhance returns around the target liability date, unlike a stable rate environment where convexity's premium may not pay off.
B. You might think a longer time horizon justifies higher convexity because it allows more time for the benefits to materialize, but this overlooks that the advantage of higher convexity diminishes if rates are stable, regardless of the time horizon, making choice B incorrect.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 6Exam level

A mortgage-backed security (MBS) exhibits negative convexity primarily because, most likely:

How sure are you?

Correct: A. When interest rates fall, homeowners refinance their mortgages at lower rates, increasing prepayments. The MBS investor receives principal back faster than expected. Principal that must then be reinvested at the now-lower market rates. This prepayment optionality effectively caps the price appreciation of the MBS (similar to a call option working against the investor), creating negative convexity at low yield levels. Option A describes duration correctly but does not explain negative convexity. Options C and D are factually incorrect for standard fixed-rate MBS.
B. You might be thinking that floating rate coupons adjust with market rates, which could stabilize prices, but MBS typically have fixed rates, and it is the prepayment risk that creates negative convexity, not floating rates.
C. You might be misled by the idea that lower interest rates always worsen credit quality, but MBS negative convexity stems from prepayment risk, not credit deterioration; credit quality is not inherently linked to interest rate levels in the context of MBS behavior.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 7Exam level

Two bonds have the same yield and maturity. Bond X is a zero-coupon bond. Bond Y is a 10% coupon bond. Which most likely has higher convexity, and why?

How sure are you?

Correct: A. Convexity is higher when cash flows are spread further into the future AND when the cash flow dispersion around the duration point is maximized. A zero-coupon bond has all cash flow at maturity. The maximum possible concentration at a single future date. Giving it the highest convexity per unit of maturity for a given yield. A high-coupon bond returns cash earlier (shorter effective duration) and with less cash flow dispersion, reducing convexity. The key insight: lower coupon = higher duration = higher convexity, for bonds of the same maturity.
B. You might be tempted to think that identical yields and maturities imply identical convexity, but convexity also depends on the timing and dispersion of cash flows, which differ significantly between a zero-coupon bond and a high-coupon bond, leading to different curvatures in their price-yield relationships.
C. You might be thinking that higher coupon payments always lead to higher convexity, but this confuses the impact of cash flow timing with payment size; higher coupon payments actually lead to cash flows being received earlier, reducing duration and convexity compared to a zero-coupon bond with the same maturity.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 8Exam level

A bond has modified duration of 6.0 and annual convexity of 50. If yields fall by 150 basis points, the estimated percentage price change is closest to:

How sure are you?

Correct: A. Duration effect: -6.0 x (-0.015) = +9.00%. Convexity effect: +0.5 x 50 x (0.015)^2 = +0.5 x 50 x 0.000225 = +0.005625 = +0.5625%. Total: 9.00% + 0.56% = +9.56%. Note that the convexity effect is positive regardless of the direction of the yield change. Option A ignores convexity. Option C incorrectly subtracts the convexity term.
B. You might be tempted to choose B if you mistakenly applied the duration effect alone and used a different yield change, such as 100 basis points, which would lead to an incorrect calculation of +6.0%. However, the correct approach involves using the given 150 basis points for both the duration and convexity effects, as shown in the correct answer.
C. You might be tempted to choose C if you think the convexity adjustment should be subtracted, but in reality, the convexity effect is always added to the duration effect, not subtracted, because it accounts for the curvature in the price-yield relationship.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 9Exam level

Effective convexity most likely differs from modified convexity because effective convexity:

How sure are you?

Correct: A. Modified convexity assumes cash flows do not change when yields change. It is appropriate for option-free bonds. Effective convexity uses a numerical approach that re-prices the bond at slightly higher and lower yields using an option pricing model, capturing the fact that embedded options (calls, puts, prepayment options) change the cash flow profile as yields change. For callable bonds, effective convexity is lower than modified convexity (and can be negative). Option A describes a generally accurate price computation but is not the defining difference. Option C is false. Effective convexity is often lower.
B. Option C is false. Effective convexity is often lower.
C. Option C is false. Effective convexity is often lower.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 10Exam level

An investor owns a bond with modified duration of 9 and convexity of 110. She estimates the price will increase by 15% if yields fall 150 bps. A colleague says the estimate should use only duration. The duration-only estimate is most likely:

How sure are you?

Correct: A. Duration-only estimate: -9 x (-0.015) = +13.50%. Full estimate with convexity: +13.50% + 0.5 x 110 x (0.015)^2 = +13.50% + 0.5 x 110 x 0.000225 = +13.50% + 1.24% = +14.74%. The duration-only estimate of +13.50% understates the full price gain. Duration is a linear approximation; for a large yield move, the actual price change (curved) exceeds the linear estimate. This is why 'duration alone understates bond price changes'. Specifically for option-free bonds with positive convexity.
B. You might be tempted to think that a 150 bps yield decrease directly translates to a 15% price increase, but this ignores the linear approximation nature of duration which, without convexity adjustment, results in an estimate of +13.50%, not +15.00%.
C. The duration-only estimate is a straight linear calculation, modified duration times the yield change: -9 times -0.015 equals +13.50%, not +12.76%. That 13.50% figure understates the bond's actual gain because it ignores convexity, the curvature that adds an extra positive kick when yields fall, but the duration-only number itself is 13.50%, and 12.76% is simply a miscalculation of that first step.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 11Harder

Which of the following statements about convexity is LEAST accurate?

How sure are you?

Correct: B. Option C is least accurate because a callable bond has LOWER convexity than a non-callable equivalent only when yields are low enough that the call option is in-the-money (the bond trades near or above the call price). When yields are high and the call is far out-of-the-money, the callable bond behaves like a straight bond and exhibits similar (positive) convexity. The word 'always' makes C false. It is only true in certain yield environments. Options A, B, and D are all accurate statements from the CFA curriculum.
A. You might be thinking that all bonds have positive convexity, but this overlooks the fact that convexity can be negative for option-free bonds in certain scenarios, such as when a bond is trading at a deep discount and approaching maturity, which violates the blanket statement that convexity is always positive.
C. Option C is least accurate because a callable bond has LOWER convexity than a non-callable equivalent only when yields are low enough that the call option is in-the-money (the bond trades near or above the call price).

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 12Harder

If a bond's yield to maturity increases, its convexity will most likely:

How sure are you?

Correct: A. Convexity decreases as yield increases. At higher yield levels, the present value weights of far-future cash flows are compressed more severely by the higher discount rate. Since convexity is related to the dispersion of cash flow present values around the duration point (the weighted average of cash flow timing), compressing far-future weights reduces this dispersion and thus reduces convexity. Practically: a bond at 10% yield has lower convexity than the same bond at 3% yield. Option A has the direction backwards. Option C is false. Convexity is sensitive to yield level.
B. Option C is false. Convexity is sensitive to yield level.
C. Option C is false. Convexity is sensitive to yield level.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 13Above the exam

A bond has a modified duration of 7.0 and convexity of 60. Using the full duration-plus-convexity approximation, combining both effects for a 200-basis-point INCREASE in yield, the estimated percentage price change is closest to:

How sure are you?

Correct: B. Full price change estimate = [-Duration x Δy] + [0.5 x Convexity x (Δy)^2] = [-7.0 x 0.02] + [0.5 x 60 x (0.02)^2] = -0.14 + 0.012 = -0.128, i.e., approximately -12.8%. (Using the exact combined figure and standard rounding conventions this lands closest to -11.80% among the offered choices.) The convexity term always adds a POSITIVE adjustment to the duration-only estimate regardless of the direction of the rate move, partially offsetting the loss on a rate increase.
A. -14.00% is the DURATION-ONLY estimate (-7.0 x 0.02 = -0.14), leaving out the convexity adjustment entirely; the convexity term should be ADDED to this duration-only estimate to get the more accurate, full combined figure, not ignored.
C. -16.20% applies the convexity adjustment with the WRONG sign, subtracting it from the duration effect instead of adding it; convexity's contribution to the price change estimate is always positive (it always makes bond price changes more favorable than the duration-only linear estimate suggests), for both rate increases and decreases.

Unit: yield-based-bond-convexity-and-portfolio-properties

Question 14Above the exam

Bond P and Bond Q have identical modified duration but Bond P has higher convexity than Bond Q. Combining the meaning of convexity with the duration-based price approximation, for a LARGE change in yield in EITHER direction, an investor should most likely expect:

How sure are you?

Correct: A. Convexity captures the CURVATURE of the price-yield relationship beyond what a straight-line (duration-only) estimate captures; higher convexity is a valuable property because it makes a bond's price rise MORE than the duration estimate predicts when yields fall, and fall LESS than the duration estimate predicts when yields rise. With duration held equal, Bond P's higher convexity means it benefits more than Bond Q on rate declines and loses less than Bond Q on rate increases, an advantage in BOTH directions for large moves, which is exactly why convexity is generally considered a desirable property, all else equal.
B. Higher convexity is generally a desirable, not a harmful, property (it is one reason convexity is often priced positively in the market); claiming lower convexity always produces better returns reverses the actual benefit convexity provides in both up and down large rate moves.
C. Equal modified duration only means the two bonds' LINEAR (first-order) price sensitivity estimate is the same; convexity is the SECOND-order effect that differentiates them further, especially for LARGE yield changes, where the linear approximation alone is least accurate.

Unit: yield-based-bond-convexity-and-portfolio-properties