61 Curve-Based and Empirical Fixed-Income Risk Measures

• Effective duration accounts for the impact of embedded options on a callable bond's cash flows.
• Approximate percentage price change of a bond = (-)(effective duration)(ΔΥΤΜ).

Question 1 Example

• A callable bond trading at $1,000 has an effective duration of 5.
• The modified duration is 6.
• If the market yield increases by 1%, the bond's price will decrease by approximately $50.
• (-5)(1%) = -5%
• $1,000 × -5% = -$50

Key Rate Duration

• Key rate duration measures the price sensitivity of a bond or portfolio
• It measures sensitivity to a change in the interest rate at one specific maturity on the yield curve.

Convexity Effect

• The convexity effect (percentage price change due to convexity) formula: (1/2)convexity × (ΔΥΤΜ)2.

Question 3 Example

• If a bond has a convexity of 51.44 and rates rise by 150 basis points, the approximate percentage price change due to convexity is 0.58%.
• (½)(51.44)(0.015)2 = 0.0058.

Question 4 Example

• A straight bond has a duration of 1.89 and a convexity of 32.
• If interest rates decline by 1%, the estimated percentage price change of the bond is 2.05%.
• Total estimated price change = -1.89 × (-0.01) × 100 + (½)(32) × (-0.01)2 × 100 = 2.05%

Question 5 Example

• A 15-year bond has a 10% semiannual coupon and a modified duration of 7.6 years.
• If the market yield rises 75 basis points, the bond's approximate price change will decrease by 5.7%.
• ΔΡ/Ρ = -7.6(+0.0075)= -0.057, or -5.7%.

Question 6 Example

• A UK 12-year corporate bond with a 4.25% coupon, priced at £107.30, has a duration of 9.5 and a convexity of 107.2.
• If the bond's yield decreases by 125 basis points, the estimated price of the bond is closest to £120.95.
• Return impact ≈ –(9.5 × -0.0125) + (1/2) × (107.2) × (-0.0125)2 ≈ 0.1272 or 12.72%
• Estimated price of bond = (1 + 0.1272) × 107.30 = 120.95

Question 7 Example

• A bond has a maturity of 30 years, a modified duration of 16.9 years, and a yield to maturity of 6.5%.
• If the yield to maturity decreases by 0.75%, the bond's price will change by +12.675%.
• (-16.9)(-0.75%) = +12.675%

Question 8 Example

• An option-free U.S. corporate bond has a par value of $10 million, a convexity of 90, and a duration of 7.
• If interest rates increase 2% (200 basis points), the bond's percentage price change is closest to -12.2%.
• Percentage change in prices = Duration effect + Convexity effect = [(-7)(0.02) + (1/2)(90)(0.02)2] = -0.12 = -12.2%.

Question 9 Example

• A Treasury bond has an annual modified duration of 10.27 and an annual convexity of 143.
• If interest rates increase by 125 basis points, the estimated percentage price change in the bond is -11.72%.
• [-(10.27)(0.0125)] + [(1/2)(143)(0.0125)2] = −0.128375 + 0.011172 = -0.117203 = -11.72%.

Question 10 Example

• Vantana Inc.'s bond has a modified duration of 5.3 and an approximate convexity of 110.
• If yields increase by 1%, it will decrease by less than 5.3%.
• Price change = (-5.3 × 0.01) + (0.5 × 110 × 0.012)= -0.0475 = -4.75%.

Question 11 Example

• A bond has a duration of 4.5 and convexity of 87.2.
• If interest rates rise 100 basis points, the bond's percentage price change is closest to -4.06%.
• (-4.5)(0.01) + (1/2)(87.2)(0.01)² = -4.06%

Question 12 Example

• A bond's duration is 10 and its convexity is 200.
• If interest rates decrease by 100 basis points, the bond’s price will change by approximately +11.0%.
• percentage change in price = [(-10) (-0.01)] + [(1/2) (200) (-0.01)²] = 0.11 = 11%

Question 13 Example

• Sensitivity of a bond's price to a change in yield at a specific maturity is least appropriately estimated by using effective duration.
• Key rate duration, also known as partial duration, measures the sensitivity of a bond price to a change in yield at a specific maturity.
• Effective duration is used to measure the sensitivity of a bond price to a parallel shift in the yield curve.

Question 14 Example

• A bond has a duration of 8 and a convexity of 100.
• If yields decrease by 60 basis points, the approximate percentage price change of the bond is 4.98%.
• -8 × (-0.006) + (1/2)(100) × (-0.0062) = +0.0498 or 4.98%.

Question 15 Example

• A 9-year corporate bond is priced at 103.96, has a duration of 7.8, and a convexity of 69.8.
• If the bond's yield increases by 100 basis points, the impact on the bondholder's return is closest to -7.45%.
• -(7.8 × 0.0100) + (1/2) × (69.8) × (0.0100)2 ≈-0.0780 + 0.0035 ≈-0.0745 or -7.45%

Empirical Duration

• Empirical duration relies on using historical relationships between benchmark yield changes and bond price changes.
• Analytical duration approaches that are based on mathematical analysis include Macaulay, modified, and effective durations.

Question 17 Example

• If a bond's price is $101.76 when the term structure of interest rates is flat at 5$, using effective duration, the percentage price change of the bond assuming rates decrease by 0.5% is 1.74%.
• Effective duration = (105.56-98.46) / (2 × 101.76 × 0.01) = 3.49
• Percent price change = -3.49 × (-0.005) × 100 = 1.74%

Question 18 Example

• When yields increase, bond prices fall and flatten the price curve.
• A change in yield has a smaller effect on bond prices.
• When yields decrease, bond prices rise and steepen the price curve.
• A change in yield has a larger effect on bond prices.

Question 19 Example

• A non-callable bond has a modified duration of 7.26.
• If rates increase by 25 basis points, the approximate price change of the bond will be -1.820%.
• -(7.26)(0.25%) = -1.82%.

Question 20 Example

• Effective duration is more appropriate than modified duration as a measure of a bond's price sensitivity to yield changes when the bond contains embedded options.
• For option-free bonds, modified duration is similar to effective duration.
• Modified duration does not consider the effect of embedded options.
• Effective duration considers expected cash flow changes that may occur with embedded options.

Question 21 Example

• Key rate duration is most appropriate estimating price sensitivity when an analyst expects a non-parallel shift in the yield curve.
• Effective duration and Modified duration measure price sensitivity to a parallel shift in the yield curve.

Question 22 Example

• A bond with a modified duration of 7 and convexity of 100 will most likely increase by 7.5% if interest rates decrease by 1%.
• Percentage Price Change = –(7) (-0.01) + (1½)(100) (-0.01)2=7.5%.

Question 23 Example

• A bond priced at par ($1,000) with a modified duration of 8 and convexity of 100 will be closest to $1,041.25 if interest rates fall 50 basis points.
• (-)(8)(-0.005)+(100)(-0.005)2 = +0.04125, or up 4.125% $1,000 × 1.04125 = $1,041.25.

Question 24 Example

• If a bond has modified duration of 5.61 and convexity of 43.84, the estimated percentage price change in the bond, for a 75 basis point decrease in interest rates is 4.33%.
• [-(5.61)(-0.0075)] + [(½)(43.84)(-0.0075)²] = 0.042075 + 0.001233 = 0.043308 = 4.33%.

Question 25 Example

• If a bond with a modified duration of 1.93 has required yields increase by 50 basis points, the bond price would be expected to decrease by 0.965%.
• 0.5(1.93%) = 0.965%.
• Modified duration indicates the expected percent change in a bond's price given a 1% (100 bp) change in yield to maturity.

Question 26 Example

• The approximate percentage price change of a bond with a duration of 10.62 and a convexity of 182.92, for a 200 basis point increase in yield is -17.58%.
• -(10.62 × 0.02) + (1½)(182.92)(0.022) = -0.2124 + 0.0366 = -0.1758 or -17.58%.

Question 27 Example

• The appropriate measure of interest rate sensitivity for bonds with an embedded option is effective duration
• Effective duration is appropriate for bonds with embedded options because their future cash flows are affected by the level and path of interest rates.

Question 28 Example

• Bond J is callable with a maturity of 5 years and a par value of $10,000, Effective Duration is used.
• Bond M is option-free with a maturity of 5 years and a par value of $1,000, Modified Duration or Effective Duration is used.

Question 29 Example

• Estimates of empirical and analytical durations should be similar for a portfolio consisting solely of short-term U.S. government bonds.
• A portfolio consisting solely of short-term U.S. government bonds should closely resemble the performance of its government benchmark yield.