Explanation

To calculate the portfolio duration, we need to use a weighted average of the individual bond durations, where the weights are based on the market values of each bond.

Step 1: Calculate total market value of the portfolio

• Bond 1: $12 million
• Bond 2: $6 million
• Bond 3: $6 million
• Total market value = $12 + $6 + $6 = $24 million

Step 2: Calculate weight of each bond

• Bond 1 weight = $12 million / $24 million = 0.50
• Bond 2 weight = $6 million / $24 million = 0.25
• Bond 3 weight = $6 million / $24 million = 0.25

Step 3: Calculate weighted duration

• Bond 1 contribution = 0.50 × 3 = 1.50
• Bond 2 contribution = 0.25 × 7 = 1.75
• Bond 3 contribution = 0.25 × 6 = 1.50

Step 4: Calculate portfolio duration Portfolio duration = 1.50 + 1.75 + 1.50 = 4.75

Verification:

• (12/24 × 3) + (6/24 × 7) + (6/24 × 6) = (0.5 × 3) + (0.25 × 7) + (0.25 × 6) = 1.5 + 1.75 + 1.5 = 4.75

Key Concept: Portfolio duration is calculated as the market-value weighted average of the durations of the individual bonds in the portfolio. The par values are not used in this calculation - only market values matter for weighting purposes.