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Duration & Price Sensitivity - Finance

  1. Except for zero-coupon bonds, a bond's Macaulay duration is always less than its time to maturity. The primary reason for this is that:
    Answer: interim coupon payments shorten the weighted-average time to receive the bond's cash flows. Duration questions test your understanding of how bond cash flows affect price sensitivity to interest rate changes. Think of Macaulay duration as measuring the weighted-average time it takes to receive a bond's cash flows. The key insight is that bonds make multiple payments over time, not just one payment at maturity. When you receive coupon payments before maturity, these interim cash flows effectively "pull forward" the average timing of when you get your money back. Since duration weights each cash flow by its present value and timing, earlier coupon payments reduce the overall weighted-average time compared to simply waiting until maturity for all cash flows. Zero-coupon bonds are the exception because they make no interim payments—you wait the full maturity period to receive anything, so duration equals maturity. Answer D correctly captures this: interim coupon payments do indeed shorten the weighted-average time to receive cash flows, making duration less than maturity. Answer A confuses duration with reinvestment risk, which affects total return but doesn't determine duration calculation. Answer B incorrectly suggests the discounting pattern causes the relationship—while principal is discounted more than early coupons, this doesn't explain why duration is shorter than maturity. Answer C makes an irrelevant comparison between premium and discount bonds that doesn't address the fundamental duration-maturity relationship. Study tip: Remember that duration measures the "balance point" of a bond's cash flows. Any payment before maturity shifts this balance point earlier, just like moving weight toward the fulcrum of a seesaw.
  2. An investor holds a 10-year, 6% annual coupon bond trading at its par value of $1,000. The bond's Macaulay duration is 7.80 years. If the bond's yield to maturity immediately falls by 1%, what is the approximate new price of the bond based on its duration?
    Answer: $1,073.58 First, since the bond trades at par, its yield to maturity (YTM) equals its coupon rate, so YTM = 6%. We need to convert Macaulay duration to modified duration: DMod=DMac/(1+YTM)=7.80/(1+0.06)=7.80/1.06≈7.3585 D_{Mod} = D_{Mac} / (1 + YTM) = 7.80 / (1 + 0.06) = 7.80 / 1.06 \approx 7.3585 . Next, calculate the estimated percentage price change for a 1% ( -0.01) fall in yield: %ΔP≈−DMod×Δy=−7.3585×(−0.01)=+0.073585 \% \Delta P \approx -D_{Mod} \times \Delta y = -7.3585 \times (-0.01) = +0.073585 . Finally, apply this percentage change to the initial price: New Price = $1,000×(1+0.073585)=$1,073.585 \$1,000 \times (1 + 0.073585) = \$1,073.585 . The closest answer is $1,073.58.
  3. A consol bond, which is a type of perpetuity, pays a fixed coupon forever. If such a bond has a yield to maturity of 5%, its Macaulay duration is closest to:
    Answer: 21.0 years. The formula for the Macaulay duration of a perpetuity is DMac=(1+y)/y D_{Mac} = (1 + y) / y , where y is the yield to maturity. Given y = 5% or 0.05: DMac=(1+0.05)/0.05=1.05/0.05=21 D_{Mac} = (1 + 0.05) / 0.05 = 1.05 / 0.05 = 21 years. The modified duration would be 1/y=1/0.05=20 1 / y = 1 / 0.05 = 20 years, a common incorrect choice.
  4. A 5-year, non-callable bond and a 15-year callable bond currently have the same modified duration and the same yield. If the yield curve experiences a large, parallel downward shift, which of the following outcomes is most likely?
    Answer: The price of the non-callable bond will increase by more than the price of the callable bond. While both bonds have the same duration, their convexity will differ. The non-callable bond has positive convexity, meaning its price increase will be slightly larger than predicted by duration. The callable bond will exhibit negative convexity as rates fall, because the increasing likelihood of it being called will cap its price appreciation. Therefore, the non-callable bond will experience a larger price increase than the callable bond.
  5. An analyst is comparing two bonds, both with a 10-year maturity and a 6% yield to maturity. Bond X has a 4% coupon, and Bond Z has an 8% coupon. Which statement about their price sensitivity to a 1% change in interest rates is most accurate?
    Answer: Bond X will have greater price sensitivity because its lower coupon rate results in a higher duration. For two bonds with the same maturity and yield to maturity, the bond with the lower coupon rate will have a higher duration. A higher duration implies greater price sensitivity to changes in interest rates. Therefore, Bond X, with its 4% coupon, will be more price-sensitive than Bond Z, which has an 8% coupon.
  6. A 20-year bond is callable in 5 years. The bond's duration-to-maturity is calculated to be 11.5, and its duration-to-call is 4.2. If an analyst expects interest rates to fall significantly, which duration measure is most relevant for assessing the bond's price sensitivity?
    Answer: Duration-to-call, as the issuer is more likely to call the bond in a lower interest rate environment. When interest rates fall, the issuer of a callable bond is more likely to exercise the call option to refinance its debt at the new, lower rates. Therefore, the bond's effective maturity is more likely to be the call date than the final maturity date. In this scenario, the duration-to-call becomes the most relevant measure for assessing the bond's price sensitivity.
  7. Assuming both bonds are option-free and have the same yield to maturity, which of the following bonds would most likely have the highest modified duration?
    Answer: A 20-year bond with a 6% coupon. Duration is positively related to a bond's maturity and inversely related to its coupon rate. To find the bond with the highest duration, we look for the combination of the longest maturity and the lowest coupon rate. Comparing the four options, the 20-year bond with a 6% coupon (Choice D) has the longest maturity and the lowest coupon rate among the 20-year options, guaranteeing it will have a higher duration than the other choices.
  8. A portfolio manager strongly believes that market interest rates will rise sharply over the next year. To best position the portfolio for this scenario, the manager should:
    Answer: decrease the portfolio's average duration by selling long-term bonds. If interest rates are expected to rise, bond prices are expected to fall. Bonds with higher duration are more sensitive to interest rate changes and will experience larger price declines. To mitigate potential losses, the manager should reduce the portfolio's interest rate sensitivity by decreasing its average duration. This is typically achieved by selling longer-maturity bonds and buying shorter-maturity bonds.
  9. For a conventional, option-free 10-year bond, as its yield to maturity decreases, what is the corresponding impact on its price and modified duration?
    Answer: Price increases; duration increases. There is a well-established inverse relationship between a bond's price and its yield to maturity; as yield decreases, price increases. There is also an inverse relationship between a bond's yield to maturity and its duration; as yield decreases, the present value of later cash flows (like the principal) increases relative to earlier cash flows, thus increasing the weighted-average time and thereby the duration.
  10. A portfolio's value is allocated as follows: 40% in a bond with a modified duration of 5.0 and 60% in a bond with a modified duration of 10.0. If yields on both bonds are expected to fall by 50 basis points, what is the estimated percentage increase in the portfolio's value?
    Answer: 4.00% First, calculate the portfolio's modified duration as the weighted average of the individual bond durations: Dp=(w1×D1)+(w2×D2) D_{p} = (w_1 \times D_1) + (w_2 \times D_2) Dp=(0.40×5.0)+(0.60×10.0)=2.0+6.0=8.0 D_{p} = (0.40 \times 5.0) + (0.60 \times 10.0) = 2.0 + 6.0 = 8.0 Next, use the duration formula to estimate the portfolio's percentage price change. Note that a fall in yields leads to a price increase. %ΔP≈−Dp×Δy \% \Delta P \approx -D_{p} \times \Delta y %ΔP≈−8.0×(−0.0050)=+0.040 \% \Delta P \approx -8.0 \times (-0.0050) = +0.040 , or a 4.00% increase.
  11. Which of the following statements most accurately describes the Macaulay duration of a 20-year, option-free, zero-coupon bond?
    Answer: Its Macaulay duration is exactly 20 years at issuance and declines proportionally with the passage of time. For a zero-coupon bond, there is only one cash flow: the principal repayment at maturity. Therefore, the weighted-average time to receive cash flows (the definition of Macaulay duration) is exactly equal to its time to maturity. At issuance, this is 20 years. As time passes, its remaining time to maturity decreases, and so does its Macaulay duration, declining linearly towards zero.
  12. An analyst states, "For any two option-free bonds, the one with the longer time to maturity will always have greater interest rate sensitivity." This statement is:
    Answer: incorrect, as a long-maturity bond with a sufficiently high coupon rate could have a lower duration than a shorter-maturity, low-coupon bond. The statement is incorrect. While duration generally increases with maturity, it is not an absolute rule. The coupon rate also has a significant impact. A bond with a very long maturity but also a very high coupon rate may have a lower duration (and thus lower interest rate sensitivity) than a bond with a shorter maturity but a much lower (or zero) coupon rate. The high coupon payments effectively shorten the weighted-average time to receive cash flows.
  13. A security that exhibits negative convexity is characterized by a price-yield relationship where:
    Answer: price appreciation for a yield decrease is smaller than the price depreciation for a yield increase of the same magnitude. When analyzing bond price behavior, convexity measures how the price-yield relationship curves. Most bonds exhibit positive convexity, but some securities like callable bonds can show negative convexity under certain conditions. Negative convexity creates an asymmetric price-yield relationship that works against bondholders. When yields fall, the security's price rises, but not by as much as you'd expect. Conversely, when yields rise by the same magnitude, the price falls by a larger amount. This happens because features like call provisions cap the upside potential - if rates drop significantly, the issuer will likely call the bond, limiting price appreciation. However, there's no corresponding floor when rates rise, so the full price decline occurs. Choice D correctly captures this asymmetric relationship where price gains from yield decreases are smaller than price losses from equivalent yield increases. Choice A describes positive convexity, which is the opposite of what we want. This is the normal, beneficial convexity that most straight bonds exhibit. Choice B describes low duration or interest rate insensitivity, which isn't the same as negative convexity - a security can still be sensitive to rate changes while exhibiting negative convexity. Choice C confuses duration with convexity and incorrectly suggests prices rise with yields, which would violate the fundamental inverse relationship between bond prices and interest rates. Remember that negative convexity is bad for investors because it creates "heads I lose a little, tails I lose a lot" price behavior - always look for the asymmetric relationship that favors larger losses.
  14. A bond has a modified duration of 9.0. If market interest rates are expected to increase by 100 basis points, the bond's price is expected to:
    Answer: decrease by approximately 9.0%, with the actual loss being slightly less than predicted due to positive convexity. Modified duration provides a linear estimate of a bond's price change. An increase in rates of 100 basis points (1%) would lead to an approximate price decrease of 9.0% (−9.0×0.01 -9.0 \times 0.01 ). However, duration is a first-order approximation. For conventional (option-free) bonds, the price-yield relationship is convex. Positive convexity means that for a given change in yield, the actual price will be higher than the linear estimate. Therefore, the actual price decline will be slightly smaller than the 9.0% predicted by duration.
  15. A portfolio consists of two zero-coupon bonds. 25% of the portfolio's value is in a 4-year bond, and the remaining 75% is in a 12-year bond. What is the Macaulay duration of the portfolio?
    Answer: 10.0 years When you encounter portfolio duration questions, remember that portfolio duration is simply the weighted average of individual bond durations, where the weights are based on market values. For zero-coupon bonds, there's a crucial simplification: the Macaulay duration equals the time to maturity. This is because zero-coupon bonds make no intermediate payments, so all cash flows occur at maturity. Here's the calculation: The 4-year bond has a duration of 4 years and represents 25% of the portfolio. The 12-year bond has a duration of 12 years and represents 75% of the portfolio. Using the weighted average formula: Portfolio Duration = (0.25 × 4) + (0.75 × 12) = 1 + 9 = 10 years Answer choice A (8.0 years) incorrectly uses equal weights: (0.5 × 4) + (0.5 × 12) = 8. This ignores the actual portfolio composition. Answer choice B (9.0 years) appears to use only the weighted contribution from the 12-year bond (0.75 × 12 = 9), forgetting the 4-year bond entirely. Answer choice C (11.0 years) might result from incorrectly weighting the 4-year bond at 75% and the 12-year bond at 25%, reversing the given allocations. The correct answer is D (10.0 years). Study tip: Always remember that for zero-coupon bonds, duration equals maturity. For portfolio duration problems, carefully identify the portfolio weights and apply the weighted average formula. Double-check that your weights match the problem statement and sum to 100%.
  16. A bond with a current market price of $980.00 has a modified duration of 8.2. If the bond's yield to maturity increases by 30 basis points, its new price will be closest to:
    Answer: $955.71 The formula for estimating the percentage price change of a bond using modified duration is: %ΔP≈−Dmod×Δy \% \Delta P \approx -D_{mod} \times \Delta y . First, calculate the estimated percentage price change: %ΔP≈−8.2×0.0030=−0.0246 \% \Delta P \approx -8.2 \times 0.0030 = -0.0246 or -2.46%. Next, calculate the dollar price change: ΔP=−0.0246×$980.00=−$24.108 \Delta P = -0.0246 \times \$980.00 = -\$24.108 . Finally, calculate the new estimated price: New Price = $980.00−$24.108=$955.892 \$980.00 - \$24.108 = \$955.892 . The closest answer is $955.71.
  17. A portfolio manager using duration to estimate price changes for a large interest rate shock finds that the actual price decline was larger than predicted. The portfolio most likely contains a significant concentration of:
    Answer: callable corporate bonds. The standard duration formula assumes positive convexity, where the linear estimate overstates price declines. If the actual price decline is larger than predicted, it implies negative convexity. Negative convexity is a characteristic of bonds with embedded call options, such as callable corporate bonds or mortgage-backed securities. When rates rise, the value of the call option decreases, exacerbating the bond's price decline.
  18. Holding all other factors constant, which of the following would most likely increase a conventional fixed-rate bond's modified duration?
    Answer: A credit rating upgrade from A to AA. A credit rating upgrade would decrease the bond's perceived risk, leading to a lower required yield to maturity (YTM). There is an inverse relationship between a bond's YTM and its duration. Therefore, a lower YTM resulting from the upgrade would increase the bond's modified duration. An increase in YTM or coupon rate, or a decrease in maturity, would all lead to a decrease in duration.
  19. Based on the table and assuming all three bonds have the same yield to maturity, which of the following correctly ranks the bonds from most sensitive to least sensitive to a small, parallel shift in the yield curve?
    Answer: M, L, K Price sensitivity is measured by duration. Holding yield constant: 1. Duration increases with maturity (coupon held constant). Comparing K (10-yr, 5%) and L (20-yr, 5%), Bond L has a longer maturity, so Duration(L) > Duration(K). 2. Duration decreases with the coupon rate (maturity held constant). Comparing L (20-yr, 5%) and M (20-yr, 3%), Bond M has a lower coupon, so Duration(M) > Duration(L). Combining these facts gives the ranking from highest duration (most sensitive) to lowest duration (least sensitive) as M > L > K.
  20. A bond with a modified duration of 6.5 is initially priced at $1,050. After a shift in the yield curve, the bond's price falls to $1,029. The change in the bond's yield to maturity was most likely a(n):
    Answer: increase of 32 basis points. First, calculate the actual percentage price change: %ΔP=($1,029−$1,050)/$1,050=−$21/$1,050=−0.02 \% \Delta P = (\$1,029 - \$1,050) / \$1,050 = -\$21 / \$1,050 = -0.02 or -2.0%. Next, use the duration formula %ΔP≈−Dmod×Δy \% \Delta P \approx -D_{mod} \times \Delta y to solve for Δy \Delta y : −0.02≈−6.5×Δy -0.02 \approx -6.5 \times \Delta y Δy≈−0.02/−6.5≈+0.003077 \Delta y \approx -0.02 / -6.5 \approx +0.003077 . Converting this to basis points gives 0.003077×10,000≈30.8 0.003077 \times 10,000 \approx 30.8 basis points. The closest answer is an increase of 32 basis points.

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