A five year promissory note has a face value of $3,500 and bears interest at 11% per annum, compounded semiannually. The note is sold 21 months after its issue date. If the buyer wishes to earn 10% per annum, compounded quarterly, what amount should the buyer pay for the note?

Introduction / Context:
This problem involves evaluating the fair price of a promissory note that accrues interest at one rate but is sold at a time when the buyer wants a different yield. The note grows at 11% per annum compounded semiannually from issue to maturity, while the buyer requires a yield of 10% per annum compounded quarterly from the purchase date to maturity. We must determine the price that equates the buyer's required yield to the future payoff, which is a standard but advanced time value of money exercise.

Given Data / Assumptions:

• Face value (principal) of the note = $3,500.
• Maturity time from issue = 5 years.
• Interest on the note: 11% per annum, compounded semiannually.
• The note is sold 21 months (1.75 years) after issue.
• Buyer's required yield = 10% per annum, compounded quarterly.
• We assume no intermediate coupon payments; the accumulated amount is paid at maturity.

Concept / Approach:
First, find the maturity value of the promissory note after the full 5 years at 11% compounded semiannually. That is:
Maturity value FV = 3500 * (1 + 0.11 / 2)^(2 * 5)Next, from the buyer's perspective, this future value is discounted back from the sale date (which is 21 months after issue) to today at 10% compounded quarterly. The time from sale date to maturity is 5 years - 1.75 years = 3.25 years. With quarterly compounding, this is 3.25 * 4 = 13 quarters. The purchase price PV is then:
PV = FV / (1 + 0.10 / 4)^13

Step-by-Step Solution:
Step 1: Compute the semiannual rate of the note: i1 = 0.11 / 2 = 0.055 per half year.Step 2: Total number of semiannual periods over 5 years: n1 = 5 * 2 = 10.Step 3: Calculate the maturity value: FV = 3500 * (1.055)^10.Step 4: Numerically, (1.055)^10 is greater than 1.7, giving a future value above $5950. The exact calculation yields a specific FV.Step 5: Determine the time from sale date to maturity: 5 years total minus 21 months (1.75 years) equals 3.25 years.Step 6: Buyer's effective periodic rate: i2 = 0.10 / 4 = 0.025 per quarter, with n2 = 3.25 * 4 = 13 quarters.Step 7: The fair purchase price is PV = FV / (1.025)^13, which evaluates numerically to about $4336.93.

Verification / Alternative check:
To check the logic, imagine the buyer pays $4336.93 today. If this amount earns 10% per annum compounded quarterly for 13 quarters, its future value should equal the maturity value of the note. That is:
4336.93 * (1.025)^13 ≈ FVIf we have matched the maturity value from the seller's side and the buyer's required yield side, the pricing is internally consistent. This consistency check confirms that $4336.93 is a fair price for the given yield requirements.

Why Other Options Are Wrong:
$5336, $6336, and $7336 are all higher than the fair price and would result in a yield lower than 10% to the buyer, making the investment unattractive under the stated requirement. $3336.93 is too low and would offer a yield much higher than 10%, which contradicts the idea of a fair market yield at 10%. Only $4336.93 satisfies the time value of money calculations for the specified compounding conventions.

Common Pitfalls:
Errors often arise from confusing which rate applies over which period. Some candidates incorrectly discount using 11% instead of 10%, or they forget that the sale happens after 21 months, not at issue. Others mis-handle the compounding frequency, using 5 years instead of 10 half years, or 3.25 years instead of 13 quarters. A careful timeline and clear identification of periods and rates help to avoid these mistakes.

Final Answer:
The amount that the buyer should pay for the note, in order to earn 10% per annum compounded quarterly, is $4336.93.