Practice: Yield-Based Bond Convexity and Portfolio Properties
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Fixed IncomeYield-Based Bond Convexity and Portfolio Properties
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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%.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.