Final answer:
To find out how long before one payment is equivalent to another, we need to calculate the time period necessary for them to have the same present value, given the compound interest rate. The logarithmic solution or a financial calculator can be used to solve the present value formula for the time period when considering semi-annual compounding.
Step-by-step explanation:
To determine how long before a scheduled payment of $3,238 a payment of $2,253 will be economically equivalent, given an interest rate of 5.25% compounded semi-annually, we need to apply the present value concept of finance.
We will find the time period where both payments have the same present value. This is a typical problem in time value of money calculations, which involves understanding compound interest.
Using the formula for the present value of future cash flows, which is PV = FV / (1 + r)^n, where PV is the present value, FV is the future value, r is the interest rate per period, and n is the number of periods, we can equate the present value of both payments and solve for n.
The semi-annual compounding means that interest is applied twice a year, so we need to adjust the interest rate and the number of periods accordingly.
Given the future value of $3,238 and a semi-annual interest rate of 5.25%/2 = 2.625%, n can be found using the equation $2,253 = $3,238 / (1 + 0.02625)^n.
This equation can be solved using logarithms or a financial calculator to find the precise number of half-year periods before the two payments are equivalent. Once n is found, the time in years is n divided by 2.
As noted, compound interest can make a significant difference over time, especially with larger sums and longer time periods.
This example is an application of that principle, considering the time value of money and the effect of compounding on making two different sums of money equivalent at different times.