How single cash formulas can be applied to determine the present sum and future sum in economic equivalence problems.
The correct answer and explanation is:
Single cash formulas are fundamental tools in economic equivalence problems used to determine the present sum (present value) and future sum (future value) of a single amount of money. These formulas rely on the concept of the time value of money, which states that a certain amount of money today is worth more than the same amount in the future due to its potential earning capacity.
The formula to calculate the future sum (future value) of a single cash amount is:
F=P(1+i)nF = P (1 + i)^n
Here, FF represents the future sum, PP is the present sum or principal, ii is the interest rate per period (expressed as a decimal), and nn is the number of periods. This formula shows how a present amount grows over time at a given interest rate.
Conversely, to determine the present sum (present value) from a known future sum, the formula used is:
P=F(1+i)nP = \frac{F}{(1 + i)^n}
This formula discounts the future amount back to the present by dividing the future value by the accumulation factor (1+i)n(1 + i)^n. This reflects the idea that money expected in the future is worth less today.
In economic equivalence problems, these formulas help compare cash flows occurring at different times by converting all values to a common point in time, either present or future. For example, if a project requires an investment today but will yield returns in the future, calculating the present value of those returns helps determine if the investment is worthwhile. Similarly, knowing the future value of current savings helps in financial planning.
Interest rates can be simple or compound, but in most economic equivalence problems, compound interest is assumed because it accounts for interest earned on previously accrued interest. The accuracy of these calculations depends on selecting the correct interest rate and compounding period.
In summary, single cash formulas serve as essential tools for decision-making in finance and economics by enabling comparisons of amounts at different times through present value and future value calculations. This approach allows for informed evaluation of investments, loans, and other financial transactions involving single cash flows.