Two payments of 10000 dollars each are to be made one year and four years from now. If money earns 9% interest compounded monthly, what single payment made two years from now would be equivalent to these two scheduled payments?

Introduction / Context:
This question is an application of the time value of money with multiple cash flows. It asks for a single equivalent payment at a different time that has the same present economic value as two separate payments. Because interest is compounded monthly at 9% per year, we must use an effective monthly rate and carefully shift each cash flow to the specified comparison time. Such problems are common in financial mathematics and engineering economics.

Given Data / Assumptions:

• Two payments of 10000 dollars each.
• First payment is due one year from now.
• Second payment is due four years from now.
• Interest rate r = 9% per year, compounded monthly.
• We want an equivalent single payment at time t = 2 years.
• We assume 12 months per year, so the monthly rate is r / 12.

Concept / Approach:
The key idea is equivalence at a chosen point in time using compound interest to move cash flows forward or backward. For the payment at year 1, we must move it forward by one year (till year 2). For the payment at year 4, we must discount it backward by two years (back from year 4 to year 2). We use the monthly interest rate i = 0.09 / 12 and the fact that one year equals 12 months, while four years equals 48 months. The equivalent payment at year 2 is the sum of the future value of the first payment and the present value at year 2 of the second payment.

Step-by-Step Solution:
Step 1: Compute the monthly interest rate: i = 0.09 / 12 = 0.0075. Step 2: The first payment of 10000 dollars occurs at the end of year 1, which is 12 months from now. To move it to year 2, we compound it for another 12 months. Step 3: Future value at year 2 of the first payment is FV1 = 10000 * (1 + i) ^ 12 = 10000 * (1.0075) ^ 12. Step 4: The second payment of 10000 dollars occurs at the end of year 4, or 48 months from now. To move it back to year 2, which is 24 months from now, we discount it over 24 months. Step 5: Value at year 2 of the second payment is PV2 = 10000 / (1 + i) ^ 24 = 10000 / (1.0075) ^ 24. Step 6: Numerically, (1.0075) ^ 12 is about 1.0938, so FV1 is about 10000 * 1.0938. Step 7: Similarly, (1.0075) ^ 24 is about 1.1964, so PV2 is about 10000 / 1.1964. Step 8: Calculating more precisely, the combined equivalent at year 2 is approximately 19296 dollars. Step 9: Therefore, the single payment at year 2 that is equivalent to the two scheduled payments is 19296 dollars.

Verification / Alternative check:
One way to verify is to convert everything to present value at time zero and compare. If X is the equivalent payment at year 2, its present value is X / (1.0075) ^ 24. The present value of the first payment is 10000 / (1.0075) ^ 12, and the present value of the second payment is 10000 / (1.0075) ^ 48. Setting these equal gives X / (1.0075) ^ 24 = 10000 / (1.0075) ^ 12 + 10000 / (1.0075) ^ 48. Multiplying through by (1.0075) ^ 24 yields X = 10000 * (1.0075) ^ 12 + 10000 / (1.0075) ^ 24, which is the same expression used previously for FV1 + PV2. This confirms internal consistency and supports the answer of approximately 19296 dollars.

Why Other Options Are Wrong:
The value 17296 is smaller than 19296 and would not be enough to match the combined effect of two 10000 dollar payments under the given interest rate. The option 13296 is much too small and fails to reflect the significant time value component of the later payment. The option 15296 is still below the correct equivalent and underestimates the required single payment. The option 20296 is larger than necessary and would create a present value greater than the sum of the present values of the scheduled payments. Only 19296 balances the cash flows correctly at 9% compounded monthly.

Common Pitfalls:
Students often forget to convert the annual rate to a monthly rate and instead use 9% as if it were a yearly rate for each period. Another frequent mistake is to treat all cash flows with simple interest instead of compound interest, which distorts the equivalence. Some candidates may move all amounts to the wrong comparison time or misuse forward and backward compounding. Careful attention to the direction of movement in time and consistent use of the periodic rate are essential in solving such multi cash flow problems accurately.

Final Answer:
The single payment two years from now that is equivalent to payments of 10000 dollars in one year and four years at 9% interest compounded monthly is approximately 19296 dollars, which is option C.