Answer-first summary for fast verification
Answer: 6.40.
## Explanation Effective duration measures the sensitivity of a bond's price to changes in interest rates. The formula for effective duration is: \[ \text{Effective Duration} = \frac{P_- - P_+}{2 \times P_0 \times \Delta y} \] Where: - \(P_-\) = Price when yields decrease - \(P_+\) = Price when yields increase - \(P_0\) = Current price - \(\Delta y\) = Change in yield (in decimal form) **Given:** - Current price \(P_0 = 102.31\) - Price when yields increase by 20 bps: \(P_+ = 101.12\) - Price when yields decrease by 20 bps: \(P_- = 103.74\) - \(\Delta y = 0.0020\) (20 bps = 0.20% = 0.0020) **Calculation:** \[ \text{Effective Duration} = \frac{103.74 - 101.12}{2 \times 102.31 \times 0.0020} \] \[ = \frac{2.62}{2 \times 102.31 \times 0.0020} \] \[ = \frac{2.62}{0.40924} \] \[ = 6.40 \] **Verification:** - \(2.62 / 0.40924 = 6.40\) Therefore, the effective duration is closest to **6.40**, which corresponds to option B.
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A bond is currently selling for 102.31. A valuation model estimates the price will fall to 101.12 if interest rates increase by 20 bps and rise to 103.74 if interest rates decrease by 20 bps. Using these estimates, the effective duration of the bond is closest to:
A
6.31.
B
6.40.
C
6.48.
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