PV Calculation for Compound Interest

To calculate the present value (PV) of an amount due in the future when compound interest is involved, we need to consider the interest rate, the time period, and the frequency of interest payments. In this case, we have an amount of $10,000 due in 2 years, with an interest rate of 5% per annum, and interest is paid on a half-yearly basis.

Step 1: Determine the interest rate per compounding period

Since interest is paid semi-annually, we need to adjust the annual interest rate to reflect the compounding periods. The formula to calculate the interest rate per compounding period is:

Interest rate per compounding period = Annual interest rate / Number of compounding periods per year

In this case, the annual interest rate is 5% and there are 2 compounding periods per year (half-yearly payments). Therefore:

Interest rate per compounding period = 5% / 2 = 2.5%

Step 2: Determine the number of compounding periods

To determine the number of compounding periods, we need to multiply the number of years by the number of compounding periods per year. In this case, we have 2 years and 2 compounding periods per year. Therefore:

Number of compounding periods = 2 years * 2 = 4 compounding periods

Step 3: Calculate the present value (PV)

The formula to calculate the present value of a future amount with compound interest is:

PV = FV / (1 + r)^n

Where PV is the present value, FV is the future value (the amount due in the future), r is the interest rate per compounding period, and n is the number of compounding periods.

In this case, the future value (FV) is $10,000, the interest rate per compounding period (r) is 2.5%, and the number of compounding periods (n) is 4. Therefore:

PV = 10,000 / (1 + 0.025)^4

PV = 10,000 / (1.025)^4

PV ≈ $9,511.62

Therefore, the present value of $10,000 due in 2 years at a 5% per annum compound interest rate with half-yearly interest payments is approximately $9,511.62.