An investor allocated $10,000 to a one-year zero-coupon bond, $20,000 to a two-year zero-coupon bond, and $10,000 to a three-year coupon bond that pays coupons semi-annually. If the Macaulay duration of this bond portfolio is 1.77 years, what is the Macaulay duration of the coupon bond?

Round your answer to two decimal places. If your answer is "2.34567", enter it as 2.35.

Question

An investor allocated $10,000 to a one-year zero-coupon bond, $20,000 to a two-year zero-coupon bond, and $10,000 to a three-year coupon bond that pays coupons semi-annually. If the Macaulay duration of this bond portfolio is 1.77 years, what is the Macaulay duration of the coupon bond?

Round your answer to two decimal places. If your answer is "2.34567", enter it as 2.35.

BlackTom AI · Accepted Answer

We start by noting the portfolio consists of three parts with given investment amounts: $10,000 in a 1-year zero-coupon bond, $20,000 in a 2-year zero-coupon bond, and $10,000 in a 3-year coupon bond that pays semi-annual coupons.

For the Macaulay duration of a portfolio, we use the weighted average of each position's duration, weighted by the present value (or investment) in that position. The formula can be written as: D_portfolio = [PV1 * D1 + PV2 * D2 + PV3 * D3] / (PV1 + PV2 + PV3).

- The 1-year zero-coupon bond has D1 = 1 year, and PV1 = $10,000.
- The 2-year zero-coupon bond has D2 = 2 years, and PV2 = $20,000.
- The 3-year coupon bond has a Macaulay duration D3 (unknown), and PV3 = $10,000.

Let D_coupon denote the Macaulay duration of the coupon bond. The given portfolio duration is 1.77 years, and the total value is 10,000 + 20,000 + 10,000 = 40,000.

Plugging into the formula:
1.77 = [ (10,000 * 1) + (20,000 * 2) + (10,000 * D_coupon) ] / 40,000.

Multiply both sides by 40,000 to clear the denominator:
40,000 * 1.77 = 10,000 * 1 + 20,000 * 2 + 10,000 * D_coupon.

Compute the left side: 40,000 * 1.77 = 70,800. The right side terms: 10,000 * 1 = 10,000, and 20,000 * 2 = 40,000, so their sum is 50,000. Therefore we have:
70,800 = 50,000 + 10,000 * D_coupon.

Subtract 50,000 from both sides:
20,800 = 10,000 * D_coupon.

Divide by 10,000:
D_coupon = 2.08 years.

Thus, the Macaulay duration of the coupon bond is 2.08 years (rounded to two decimal places as required).

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