Answer-first summary for fast verification
Answer: 5.9.
The correct answer is **A** because the modified duration of the portfolio is calculated as the weighted average of the modified durations of the individual bonds, using market values as weights. Here's the step-by-step breakdown: 1. **Calculate Modified Duration for Each Bond**: - Modified Duration (ModDur) = Macaulay Duration / (1 + Yield to Maturity) - ModDur of Bond 1 = 7.5 / (1 + 4%) = 7.211538 - ModDur of Bond 2 = 5.4 / (1 + 3%) = 5.242718 2. **Determine Weights Based on Market Values**: - Weight for Bond 1 = $200,000 / ($200,000 + $400,000) = 0.333333 - Weight for Bond 2 = $400,000 / ($200,000 + $400,000) = 0.666667 3. **Calculate Portfolio Modified Duration**: - Portfolio ModDur = (Weight for Bond 1 × ModDur of Bond 1) + (Weight for Bond 2 × ModDur of Bond 2) - Portfolio ModDur = (0.333333 × 7.211538) + (0.666667 × 5.242718) = 5.898992 ≈ 5.9 **Why Not B or C?** - **B** is incorrect because it uses par values instead of market values for weights, leading to a miscalculation. - **C** is incorrect because it mistakenly uses Macaulay duration instead of modified duration for the portfolio calculation.
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A bond portfolio consists of the following option-free annual-pay coupon bonds:
Bond 1 Bond 2
Par value $300,000 $450,000
Market value $200,000 $400,000
Yield to maturity 4% 3%
Macaulay duration 7.5 5.4
The modified duration of this portfolio is closest to:
A
5.9.
B
6.0.
C
6.1.