Consider a one-year loan with a face value of $100,000 and a coupon rate of 10%. The loan requires payment of accrued interest and 30% of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year. If the required yield is 18%, what is the duration of the loan in terms of years? (Please round your answer to three decimal places.)

Question

Consider a one-year loan with a face value of $100,000 and a coupon rate of 10%. The loan requires payment of accrued interest and 30% of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year. If the required yield is 18%, what is the duration of the loan in terms of years? (Please round your answer to three decimal places.)

BlackTom AI · Accepted Answer

We start by identifying the cash flows and the timing.

- The loan has a face value of $100,000 and a coupon rate of 10% annually. Since payments are described at 6 months and at 1 year, the coupon is effectively 5% of face value for each half-year period.
- At the end of 6 months, the borrower pays accrued interest for the first 6 months plus 30% of the principal. Accrued interest for 6 months = 0.10 × $100,000 × 0.5 = $5,000. Principal repayment at 6 months = 30% × $100,000 = $30,000. So the cash flow at t = 0.5 years is CF1 = $5,000 + $30,000 = $35,000.
- After the 6-month payment, the remaining principal is $100,000 − $30,000 = $70,000.
- At the end of 12 months, the remaining principal plus accrual for the next 6 months is due. Accrued interest for the next 6 months on the remaining $70,000 is 0.10 × $70,000 × 0.5 = $3,500. So the cash flow at t = 1 year is CF2 = $70,000 + $3,500 = $73,500.

Next, we compute the price of the loan using the required yield of 18% per year, with discounting at the appropriate times. Since one cash flow occurs at 0.5 years and another at 1 year, we discount by (1.18)^{t} with t in years:
- Present value of CF1: PV1 = $35,000 / (1.18)^{0.5} = $35,000 / sqrt(1.18).
- Present value of CF2: PV2 = $73,500 / (1.18)^{1} = $73,500 / 1.18.
- The price P of the loan is P = PV1 + PV2.

Now we compute the Macaulay duration, which is the weighted average time to receive the cash flows, with weights equal to the present value of each cash flow, divided by the price:
- Numerator for Macaulay duration: t1 × PV1 + t2 × PV2, where t1 = 0.5 years and t2 = 1 year.
  Numerically: 0.5 × PV1 + 1 × PV2.
- Denominator: P = PV1 + PV2.

Putting in the numbers:
- sqrt(1.18) ≈ 1.086287, so PV1 ≈ 35,000 / 1.086287 ≈ 32,238.
- PV2 ≈ 73,500 / 1.18 ≈ 62,288.
- Price P ≈ 32,238 + 62,288 ≈ 94,526.
- Numerator: 0.5 × 32,238 ≈ 16,119; plus 1 × 62,288 ≈ 62,288; total ≈ 78,407.
- Macaulay duration in years: Duration ≈ 78,407 / 94,526 ≈ 0.830 years.

Therefore, when rounded to three decimal places, the duration is approximately 0.829 (to three decimals).

FINS5530-Financial Institution Mgmt - T3 2025

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