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Answer: B
The portfolio modified duration is calculated as the weighted average of individual bond durations: - Bond 1: $4,000,000 × 7.5 = $30,000,000 - Bond 2: $2,000,000 × 1.6 = $3,200,000 - Bond 3: $3,000,000 × 6 = $18,000,000 - Bond 4: $1,000,000 × 1.3 = $1,300,000 Total portfolio value: $10,000,000 Sum of (Value × Duration): $52,500,000 Portfolio modified duration = $52,500,000 / $10,000,000 = 5.25 For a 0.1% (10 basis points) increase in interest rates: ΔP ≈ -Modified Duration × P × Δy = -5.25 × $10,000,000 × 0.001 = -$52,500 This represents the approximate decrease in portfolio value due to the interest rate increase.
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| (A) Bond | (B) Value (USD) | (C) Modified Duration | (D) (B × C) | (E) (D/B) |
|---|---|---|---|---|
| 1 | 4,000,000 | 7.5 | 30,000,000 | |
| 2 | 2,000,000 | 1.6 | 3,200,000 | |
| 3 | 3,000,000 | 6 | 18,000,000 | |
| 4 | 1,000,000 | 1.3 | 1,300,000 | |
| SUM | 10,000,000 | 52,500,000 | 5.25 |
The portfolio modified duration is 5.25. This is obtained by multiplying the value of each bond by the modified duration(s), then taking the sum of these products, and dividing it by the value of the total bond portfolio.
The change in the value of the portfolio will be -10,000,000 × 5.25 × 0.1% = -52,500
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