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 | Length of time | Rate | Compounded | On PV Table 12.3 | PV factor used | PV of amount desired at end of period |
|---|---|---|---|---|---|---|
| | | | | Period used | Rate used | |
| $ | 8 years | | 3 | Semiannually | | % | |
Answer (2)
Calculate the number of periods: 8 years × 2 = 16 periods .
Calculate the rate per period: 2 3% = 1.5% .
Find the PV factor from the table: PV factor = 0.78757 (approximately).
Calculate the present value: 0.78757 s , which rounds to 0.79 s .
Explanation
Understanding the Problem We are asked to complete a table using present value concepts. We need to determine the 'Period used', 'Rate used', 'PV factor used', and 'PV of amount desired at end of period'. We are given that the amount desired at the end of the period is 's', the length of time is 8 years, the rate is 3%, and it is compounded semiannually.
Calculating the Number of Periods First, we need to find the 'Period used'. Since the interest is compounded semiannually, this means it is compounded twice a year. Therefore, the number of periods is the number of years multiplied by the number of times it is compounded per year. So, the 'Period used' is 8 years × 2 = 16 periods .
Calculating the Rate per Period Next, we need to find the 'Rate used'. Since the annual rate is 3% and it is compounded semiannually, we divide the annual rate by the number of compounding periods per year. So, the 'Rate used' is 2 3% = 1.5% .
Finding the PV Factor Now, we need to find the 'PV factor used'. We would typically look this up in a present value table (Table 12.3 as mentioned). Using a period of 16 and a rate of 1.5%, the PV factor is approximately 0.78757. Rounding it to five decimal places, we get 0.78757.
Calculating the Present Value Finally, we need to find the 'PV of amount desired at end of period'. This is calculated by multiplying the amount desired at the end of the period (which is 's') by the PV factor. So, the PV is s × 0.78757 = 0.78757 s . Rounding to the nearest dollar amount, we have 0.79 s .
Examples
Understanding present value is crucial in financial planning. For instance, if you want to have $10,000 in a savings account after 5 years with a certain interest rate, calculating the present value tells you how much money you need to deposit today. This concept is also used in investment decisions, loan calculations, and retirement planning to determine the current worth of future cash flows.
To find the present value, we determined the number of compounding periods (16) and the rate per period (1.5%). The PV factor from Table 12.3 is approximately 0.78757, leading to a final calculation of the present value as s × 0.78757 .
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