Questions: Unit 7 Lab Quiz Stephanie Sanders 11/05/24 9:41 PM This quiz: 40 point(s) Question 14 of 20 This question: 2 point(s) possible Submit quiz (Calculation shown below) How much money do you have to deposit today so that beginning 11 years from now you can withdraw 5,000 a year for the next 7 years (periods 11 through 17) plus an additional amount of 10,000 in the last year (period 17)? Assume an interest rate of 11 percent. The amount of money you have to deposit today is (Round to the nearest cent.) Time Remaining: 02:17:03 Next

Solution Steps

To solve this problem, we need to calculate the present value of a series of future cash flows. The cash flows consist of $5,000 per year for 7 years and an additional $10,000 in the last year. We will discount these future cash flows back to the present value using the given interest rate of 11%. The present value of each cash flow is calculated using the formula for present value of a future sum: \( PV = \frac{FV}{(1 + r)^n} \), where \( FV \) is the future value, \( r \) is the interest rate, and \( n \) is the number of periods. We sum the present values of all cash flows to find the total amount to deposit today.

The present value of the annual withdrawals of $5,000 for 7 years, starting from year 11, is calculated as follows:

\[ PV_{\text{annual}} = \sum_{i=0}^{6} \frac{5000}{(1 + 0.11)^{11 + i}} \]

Calculating this gives:

\[ PV_{\text{annual}} \approx 8297.8119 \]

The present value of the additional withdrawal of $10,000 in year 17 is calculated as:

\[ PV_{\text{additional}} = \frac{10000}{(1 + 0.11)^{17}} \]

Calculating this gives:

\[ PV_{\text{additional}} \approx 1696.3262 \]

The total present value required today to cover all future withdrawals is the sum of the present values calculated in the previous steps:

\[ PV_{\text{total}} = PV_{\text{annual}} + PV_{\text{additional}} \]

Calculating this gives:

\[ PV_{\text{total}} \approx 9994.14 \]

Final Answer