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.

类似问题

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

Why have most international agreements for environmental protection been based on the honor system?

As was exhibited with the Montreal Protocol and the encouragement of reduced CFC production, trade can be an effective tool to compel compliance with international treaties.

The Stockholm Declaration and UNEP were the result of the Blank ______ first conference on the Human Environment in 1972.

In addition to the direct mission of the Convention on International Trade in Endangered Species (CITES), why was CITES so important?

President Trump hiring a corporate leader from the oil and gas industry to head the Department of Energy is an example of regulatory ; one intention behind this was to change environmental policy.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单