Two payments of $10,000 each are due one year and four years from now. If money earns 9% annual interest compounded monthly, what single payment made two years from now would be economically equivalent to these two scheduled payments?

Introduction / Context:
This problem involves the concept of equivalent cash flows under compound interest. Instead of two separate payments at different future times, we want to replace them with a single payment at an intermediate date that has the same economic value when interest is 9% compounded monthly. This type of calculation is common in loan restructuring and financial planning.

Given Data / Assumptions:

• Payment 1: $10,000 at t = 1 year
• Payment 2: $10,000 at t = 4 years
• Interest rate: 9% nominal per annum, compounded monthly
• We must find a single equivalent payment X at t = 2 years
• All comparisons are made using the same interest rate and compounding convention

Concept / Approach:
To find an equivalent payment at a different time, we move each original payment to the target time using compound interest. The value at t = 2 of each payment is:
Value at t = 2 of amount A at time t1: A * (1 + i)^n or A / (1 + i)^nwhere i is monthly rate and n is number of months moved forward or backward. Then we sum these equivalent values and set them equal to the single payment X at t = 2.

Step-by-Step Solution:
Step 1: Monthly interest rate.i = 9% / 12 = 0.09 / 12 = 0.0075 per monthStep 2: Move first payment from t = 1 year to t = 2 years (12 months forward).Value1 = 10000 * (1.0075)^(12)Step 3: Move second payment from t = 4 years to t = 2 years (24 months backward).Value2 = 10000 / (1.0075)^(24)Step 4: Equivalent single payment at t = 2 is:X = Value1 + Value2Using accurate calculation, X ≈ 19296.38, rounded to $19,296.

Verification / Alternative check:
If you discount the single payment X back to t = 0 and compare it with the present value of the original two payments, you will find they match. This confirms that the replacement stream is economically equivalent under 9% interest compounded monthly.

Why Other Options Are Wrong:
$19,396, $19,496, $19,596: These correspond to slightly different interest assumptions or compounding periods and do not result from the correct formula.$18,296: Too low; it undervalues the combined effect of the two $10,000 payments.

Common Pitfalls:
Students sometimes use annual compounding instead of monthly, or simply add the two payments without time adjustment. Another mistake is to move both payments in the wrong direction (both forward or both backward) or to use the wrong number of months. Careful attention to the time diagram and compounding frequency avoids these errors.

Final Answer:
The equivalent single payment two years from now is approximately $19,296.