Complete the following using present value. (Use the Table 12.3 provided.) Note: Do not round intermediate calculations. Round the "Rate used" to the nearest tenth percent. Round the "PV factor" to decimal places and final answer to the nearest dollar amount.
| Amount desired at end of period | | | Compounded | Period used | Rate used | | PV factor used | PV of amount desired at end of period |
|---|---|---|---|---|---|---|---|
| $ 6,000 | 8 years | 3 | Semiannually | | | | |
| | | | | | 1.5 | % | 0.7876 | $ |
Answer (2)
The present value of $6000 needed in 8 years at a 3% annual interest rate compounded semiannually is $4726. This was calculated by determining the number of compounding periods, the per-period interest rate, and using the present value factor. The calculation resulted in a present value of $4725.60, which rounds to $4726.
;
Calculate the number of periods: P er i o d s = 8 × 2 = 16 .
Verify the interest rate per period: 2 3% = 1.5% .
Calculate the present value: P V = $6000 × 0.7876 = $4725.60 .
Round the present value to the nearest dollar: $4726 .
Explanation
Understanding the Problem We are given the amount desired at the end of the period, the length of time, the interest rate, and the compounding frequency. We need to find the number of periods, the rate used per period, and the present value of the amount desired.
Calculating the Number of Periods First, we calculate the number of periods. Since the interest is compounded semiannually, the number of periods is the length of time in years multiplied by 2.
Determining the Period Used P er i o d s = 8 years × 2 = 16 periods Thus, the period used is 16.
Verifying the Interest Rate per Period The annual interest rate is 3%, so the semiannual interest rate is 2 3% = 1.5% . This matches the rate used in the table, which is 1.5%.
Calculating the Present Value Next, we calculate the present value (PV) by multiplying the amount desired at the end of the period by the present value factor.
Finding the Present Value P V = $6000 × 0.7876 = $4725.60 Rounding to the nearest dollar amount, we get $4726.
Final Answer Therefore, the present value of the amount desired at the end of the period is $4726.
Examples
Understanding present value is crucial in financial planning. For instance, if you want to have $6,000 in 8 years for a down payment on a car and your savings account earns 3% interest compounded semiannually, calculating the present value tells you how much money you need to deposit today. This concept is also fundamental in investment decisions, loan evaluations, and retirement planning, helping individuals make informed choices about their financial future.
