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Answer: 3.1 and -205
Use the standard effective duration and convexity formulas with the up/down shocked prices. ### Given - Current price, \(P_0 = 584.2\) - Price when yield rises to 4.5% and PSA falls to 50%: \(P_{up} = 573.5\) - Price when yield falls to 3.5% and PSA rises to 150%: \(P_{down} = 591.9\) - Yield shock, \(\Delta y = 0.5\% = 0.005\) ### Effective duration \[ D_{eff} = \frac{P_{down} - P_{up}}{2P_0\Delta y} \] \[ D_{eff} = \frac{591.9 - 573.5}{2(584.2)(0.005)} = \frac{18.4}{5.842} \approx 3.15 \] So the approximate effective duration is **3.1**. ### Effective convexity \[ C_{eff} = \frac{P_{down} + P_{up} - 2P_0}{P_0(\Delta y)^2} \] \[ C_{eff} = \frac{591.9 + 573.5 - 2(584.2)}{584.2(0.005)^2} = \frac{-3.0}{584.2 \times 0.000025} \approx -205 \] So the approximate effective convexity is **-205**. ### Correct answer **B. 3.1 and -205**
Author: Manit Arora
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P2.T5.105. Effective duration and effective convexity of pass-through MBS
105.1. The current term structure is flat with a yield of 4.0%. Assuming 100%, the price of $500 million pass-through mortgage-backed security (MBS) is $584.2 million. Assuming a 4.5% yield and 50% PSA, the value of the MBS drops to $573.5 million; assuming a 3.5% yield and 150% PSA, the value of the MBS increases to $591.9 million. What are, respectively, the approximate effective duration and effective convexity of the MBS?
A
2.7 and -133
B
3.1 and -205
C
4.6 and -375
D
5.8 and -412
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