Question13 Consider the following four bonds:[table]
Bond | Time to Maturity (years) | Coupon rate | YTM 1 | 5 | 10% | 7% 2 | 3 | 10% | 7% 3 | 5 | 3% | 4% 4 | 5 | 3% | 7%
[/table] If the yield-to-maturity for all bonds changes by 1%, rank the bonds from the lowest percentage change in price to the largest percentage change in price based on the duration approximation. Select one alternative: A. 4,3,2,1 B. 1,2,3,4 C. 2,1,4,3 D. 2,3,4,1 ResetMaximum marks: 2 Flag question undefined

Question

BlackTom AI · Accepted Answer

Duration (modified) gives the approximate percentage price change for a small change in yield: ΔP/P ≈ −(Modified Duration) × Δy. Modified Duration = Macaulay Duration / (1 + y). Lower coupons, longer maturities, and lower yields increase duration.

Step-by-step calculations (par = 100, annual coupons, Δy = 1% = 0.01):

Bond 1: T = 5, coupon = 10, y = 7%
- Price = 10*(1 − (1.07)^(−5))/0.07 + 100/(1.07)^5 = 112.3006
- PV-weighted times: sum t·PV(CF_t) = 473.9875
- Macaulay D = 473.9875 / 112.3006 = 4.218
- Modified D = 4.218 / 1.07 = 3.944
- %ΔP ≈ −3.944 × 0.01 = −3.94%

Bond 2: T = 3, coupon = 10, y = 7%
- Price = 10*(1 − (1.07)^(−3))/0.07 + 100/(1.07)^3 = 107.873
- Sum t·PV = 296.1995
- Macaulay D = 296.1995 / 107.873 = 2.745
- Modified D = 2.745 / 1.07 = 2.565
- %ΔP ≈ −2.565 × 0.01 = −2.57%

Bond 3: T = 5, coupon = 3, y = 4%
- Price = 3*(1 − (1.04)^(−5))/0.04 + 100/(1.04)^5 = 95.5482
- Sum t·PV = 450.0199
- Macaulay D = 450.0199 / 95.5482 = 4.708
- Modified D = 4.708 / 1.04 = 4.527
- %ΔP ≈ −4.527 × 0.01 = −4.53%

Bond 4: T = 5, coupon = 3, y = 7%
- Price = 3*(1 − (1.07)^(−5))/0.07 + 100/(1.07)^5 = 83.5992
- Sum t·PV = 391.8916
- Macaulay D = 391.8916 / 83.5992 = 4.690
- Modified D = 4.690 / 1.07 = 4.383
- %ΔP ≈ −4.383 × 0.01 = −4.38%

Ranking by magnitude of percentage change (smallest to largest): Bond 2 (−2.57%), Bond 1 (−3.94%), Bond 4 (−4.38%), Bond 3 (−4.53%) → 2, 1, 4, 3.

Why other choices are incorrect:
- A (4,3,2,1) and B (1,2,3,4) order either put high-duration bonds first or misorder short vs long maturities/coupons.
- D (2,3,4,1) swaps bonds 3 and 4; bond 3 has lower yield and thus slightly higher duration than bond 4, so 3 should be last.

Final confirmation: the correct ranking (lowest to largest percent price change) is 2,1,4,3, which matches answer C.

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