A five-year promissory note with a face value of $3,500 bears interest at 11% compounded semiannually. It was sold 21 months after its issue date to yield the buyer 10% compounded quarterly. What amount was paid for the note?

Introduction / Context:
This is a bond valuation style question involving two different compound interest rates. First, the maturity value of the note is determined using its own contract rate. Then, that future value is discounted back to the sale date using the buyer's required yield, which has a different compounding frequency.

Given Data / Assumptions:

Face value (principal) = $3,500.
Term of the note = 5 years.
Contract rate = 11% per annum compounded semiannually.
Note is sold 21 months (1.75 years) after issue.
Buyer's required yield = 10% per annum compounded quarterly.
No interim coupons are removed; we treat the note as accumulating interest at 11% until maturity.

Concept / Approach:
Step 1: Compute the maturity value F of the note using the 11% semiannual rate for the full 5 years.
Step 2: Compute the remaining time from sale date to maturity (5 − 1.75 years).
Step 3: Discount the maturity value back for this remaining period at the buyer's 10% nominal rate compounded quarterly. The present value obtained is the fair purchase price.

Step-by-Step Solution:
Step 1: Maturity value under 11% compounded semiannually. Semiannual rate i1 = 11% / 2 = 5.5% = 0.055. Number of semiannual periods = 5 * 2 = 10. Maturity value F = 3,500 * (1.055)^10 ≈ $5,978.51. Step 2: Time remaining after 21 months. 21 months = 1.75 years, so remaining time = 5 − 1.75 = 3.25 years. With quarterly compounding, quarters n2 = 3.25 * 4 = 13. Quarterly yield rate i2 = 10% / 4 = 2.5% = 0.025. Step 3: Discount F at 10% compounded quarterly for 13 quarters. Price = F / (1.025)^13 ≈ 5,978.51 / (1.025)^13 ≈ $4,336.93.

Verification / Alternative check:
If the buyer invests $4,336.93 at 10% compounded quarterly for 3.25 years, the value grows to approximately $5,978.51, the maturity amount of the note. Thus the buyer earns exactly the required 10% yield, confirming that this is the correct price.

Why Other Options Are Wrong:
Paying $5,336 or more would give the buyer a yield lower than 10%, since less discounting is involved. Paying $6,336 or $7,336 would be even worse deals from the buyer's perspective. Only $4,336.93 equates the yield to the required 10% compounded quarterly.

Common Pitfalls:
Many learners incorrectly discount using the contract rate instead of the buyer's yield, or they forget to convert the times correctly into semiannual and quarterly periods. Another common mistake is treating 21 months as 2 years. Precision in time conversion and proper matching of rate with compounding frequency are crucial.

Final Answer:
The note was sold for approximately 4336.93 dollars.