Question

A 63-year-old couple is considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Wine Boutique. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by G(t) = 38,700 (dollars per year). The Wine Boutique has a continuous income stream with a projected annual rate of flow at time t given by W(t) = 21,000e^0.08t (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Find the present value of each business over the next 2 years (until the couple reaches age 65) to see which is the better buy. (Round your answers to the nearest dollar.)

Gift Shoppe $

Wine Boutique $

Answer 1

To compare the present value of the Gift Shoppe and the Wine Boutique over the next 2 years, we calculate the present value of the income generated by each business using continuous compound interest. The present value of the Gift Shoppe is $31,983.47 and the present value of the Wine Boutique is $25,170.48.

Step-by-step explanation:

To compare the present value of the Gift Shoppe and the Wine Boutique over the next 2 years, we need to calculate the present value of the income generated by each business using the continuous compound interest formula. The present value (PV) of an income stream is given by the formula PV = CF / (1 + r)^t, where CF is the cash flow, r is the interest rate, and t is the time period.

Let's calculate the present value of the Gift Shoppe first:

1. CF = $38,700 per year, r = 10% (0.10), and t = 2 years.
2. Using the formula PV = CF / (1 + r)^t, we get PV = $38,700 / (1 + 0.10)^2.
3. Simplifying the equation, we have PV = $38,700 / (1.10^2) = $38,700 / 1.21 = $31,983.47.

Now let's calculate the present value of the Wine Boutique:

1. CF = $21,000e^0.08t per year, r = 10% (0.10), and t = 2 years.
2. Using the formula PV = CF / (1 + r)^t, we get PV = $21,000e^0.08(2) / (1 + 0.10)^2.
3. Simplifying the equation, we have PV = $21,000e^0.16 / (1.10)^2.
4. Evaluating the exponential term, we get PV = $21,000(1.172013) / 1.21 = $25,170.48.

Therefore, the present value of the Gift Shoppe is $31,983.47 and the present value of the Wine Boutique is $25,170.48.