A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Determine which is the better buy by finding the present value of each business over the next 7 years (until the couple reach age 65).
A)the Video Palace
B)the Gift Shoppe

Question

A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by  (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by  (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Determine which is the better buy by finding the present value of each business over the next 7 years (until the couple reach age 65). ​
A)the Video Palace 
B)the Gift Shoppe

Quizplus · Accepted Answer

To compare the present values of the two businesses, we need to find the present value of the income stream for each business over the next 7 years, using the continuous compounding formula:

PV = FV / e^(rt)

where PV is the present value, FV is the future value, r is the annual interest rate (10% in this case), and t is the time period in years.

For the Gift Shoppe, the annual rate of flow is given by:

11eb0d3b_98a0_d285_9790_83a5e1a0b377_TB1243_11 = $10,000

So the total income over the next 7 years is:

I = $10,000 x 7 = $70,000

The present value of this income stream is:

PV = $70,000 / e^(0.1 x 7) = $36,966.46

For the Video Palace, the projected annual rate of flow is given by:

11eb0d3b_98a0_d286_9790_d124079520af_TB1243_11 = $20,000

So the total income over the next 7 years is:

I = $20,000 x 7 = $140,000

The present value of this income stream is:

PV = $140,000 / e^(0.1 x 7) = $73,932.93

Therefore, the Video Palace is the better buy, as it has a higher present value of income over the next 7 years.

A 58-Year-Old Couple Are Considering Opening a Business of Their