The table below gives the prices (per 100 of par value), coupon rates, maturities, and modified durations of three semiannual coupon bonds.

Coupon

Maturity

Market Price

D*

2.50%

5 years

102.248

4.687

2.75%

10 years

100.000

8.691

3%

30 years

95.232

19.393

What is the modified duration of a portfolio that is long $40 million face value of the 5-year bond, and $50 million face value of the 10-year bond?

(Round to 4 decimal places)

Question

The table below gives the prices (per 100 of par value), coupon rates, maturities, and modified durations of three semiannual coupon bonds.

Coupon

Maturity

Market Price

D*

2.50%

5 years

102.248

4.687

2.75%

10 years

100.000

8.691

3%

30 years

95.232

19.393

What is the modified duration of a portfolio that is long $40 million face value of the 5-year bond, and $50 million face value of the 10-year bond?

(Round to 4 decimal places)

BlackTom AI · Accepted Answer

Portfolio modified duration is the market-value-weighted average of the individual bonds’ modified durations. Modified duration measures price sensitivity per unit of yield and must be weighted by each bond’s market value (not face value).

Step 1 — compute market values (prices are per 100 of par):
- 5-year bond: face = $40,000,000, price = 102.248 per 100 → price factor = 102.248/100 = 1.02248.
  Market value = 40,000,000 × 1.02248 = $40,899,200.
- 10-year bond: face = $50,000,000, price = 100.000 per 100 → price factor = 1.00000.
  Market value = 50,000,000 × 1.00000 = $50,000,000.

Step 2 — total portfolio market value:
Total MV = 40,899,200 + 50,000,000 = $90,899,200.

Step 3 — compute weighted sum of modified durations:
- Modified duration of 5-year bond D1* = 4.687. Contribution = MV1 × D1* = 40,899,200 × 4.687.
  Compute: 40,899,200 × 4 = 163,596,800.
           40,899,200 × 0.687 = 28,097,750.4.
  Sum = 163,596,800 + 28,097,750.4 = 191,694,550.4.
- Modified duration of 10-year bond D2* = 8.691. Contribution = 50,000,000 × 8.691 = 434,550,000.0.
- Weighted sum = 191,694,550.4 + 434,550,000 = 626,244,550.4.

Step 4 — portfolio modified duration:
D_portfolio* = (weighted sum) / (total MV) = 626,244,550.4 / 90,899,200 ≈ 6.8894396.
Rounded to 4 decimal places = 6.8894.

Why this is correct:
- Using market-value weights is required because modified duration expresses price sensitivity per dollar invested; weighting by face value would misstate the portfolio sensitivity when prices differ from par.
- The semiannual nature and coupon rates are already reflected in the provided modified durations, so no further coupon adjustments are needed.

Common incorrect approaches:
- Weighting by face amounts (40M and 50M) instead of market values would give a different, incorrect duration.
- Averaging durations arithmetically without weighting by value would also be wrong.

Final answer (rounded to 4 decimals): 6.8894, which matches the provided answer.

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