Alice Wolf wants to know how much she will have to invest today in order to receive an annuity of $6,000 for four years if interest is earned at 10 percent annually. She will make all her withdrawals at the end of each year. How much should Alice invest? (Using the Calculator)
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45,000
675 peercent
To calculate how much Alice needs to invest today, we need to use the present value formula for an annuity.
The formula for calculating the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present value (amount Alice needs to invest today)
PMT = Payment amount (annuity amount Alice wants to receive each year, which is $6,000)
r = Interest rate per period (in this case, 10% annually, so we need to convert it to a decimal by dividing by 100, resulting in 0.10)
n = Number of periods (in this case, 4 years)
Let's plug in the values and calculate:
PV = 6000 * (1 - (1 + 0.10)^(-4)) / 0.10
Now, to solve this equation using a calculator, follow these steps:
1. Calculate the value inside the parentheses first: (1 + 0.10)^(-4)
- Press the "1" key.
- Press the "+" key.
- Press the "0" key.
- Press the "." key.
- Press the "1" key.
- Press the "0" key.
- Press the "y^x" or "x^y" (power) key.
- Press the "(" key. (This is important to ensure that the exponent applies to the entire expression inside the parentheses.)
- Press the "-" key. (This ensures that the exponent is negative.)
- Press the "4" key.
- Press the ")" key.
2. Calculate the second part of the equation: (1 - calculated value from step 1)
- Press the "1" key.
- Press the "-" key.
- Press the calculated value from step 1. (You can use the "ANS" button if supported by your calculator.)
3. Calculate the third part of the equation: (annuity amount / interest rate per period)
- Press the annuity amount, in this case, "6000".
- Press the "/"
- Press the interest rate per period, in this case, "0.10".
4. Calculate the final result: (second part of the equation from step 2) / (third part of the equation from step 3)
- Press the second part of the equation from step 2.
- Press the "/"
- Press the third part of the equation from step 3.
The final result will be the present value (amount Alice needs to invest today) that would provide her with an annuity of $6,000 for four years with an interest rate of 10% annually.