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Answer: USD 41.17 million
## Explanation To solve this duration-matching problem, we need to: 1. **Calculate the current market value of the portfolio** - Face value: USD 88 million - Maturity: 5 years - Yield: 4% with continuous compounding - Market value: $P = 88 \times e^{-0.04 \times 5} = \text{USD } 72.05 \text{ million}$ 2. **Set up the duration equation** - Let W = weight of 1.5-year bonds - Weight of 5-year bonds = (1 - W) - Desired portfolio duration = 3 years - Duration equation: $1.5 \times W + 5 \times (1 - W) = 3$ 3. **Solve for W** - $1.5W + 5 - 5W = 3$ - $-3.5W = -2$ - $W = 0.5714$ - $(1 - W) = 0.4286$ 4. **Calculate the market value of 1.5-year bonds** - Value = $W \times \text{Total portfolio value}$ - Value = $0.5714 \times 72.05 = \text{USD } 41.17 \text{ million}$ **Why other options are incorrect:** - **A (USD 30.88 million)**: This is the value of the 5-year maturity bond portion - **B (USD 37.72 million)**: This is $(1 - W) \times 88$ million (using face value instead of market value) - **D (USD 50.28 million)**: This is $W \times 88$ million (using face value instead of market value)
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A portfolio manager holds USD 88 million face value of zero-coupon bonds maturing in 5 years and yielding 4%. The portfolio manager expects that interest rates will increase. To hedge the exposure, the portfolio manager wants to sell part of the 5-year bond position and use the proceeds from the sale to purchase zero-coupon bonds maturing in 1.5 years and yielding 3%. Assuming continuous compounding, what is the market value of the 1.5-year bonds that the portfolio manager should purchase to reduce the duration on the combined position to 3 years?
A
USD 30.88 million
B
USD 37.72 million
C
USD 41.17 million
D
USD 50.28 million
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