Camille purchased a 10-yr franchise for a computer outlet store that is expected to generate income at the following rate measured in dollars/year.
R(t) = 400,000
If the prevailing interest rate is 10%/year compounded continuously, find the present value of the franchise. (Round your answer to the nearest whole number.)
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Well, Camille really lucked out with this computer outlet store! So, let's calculate the present value of the franchise using the continuous compounding formula.
The continuous compounding formula is given by the formula:
PV = R / (e^(rt))
Where:
PV = Present value
R = Income generated per year
r = Interest rate
t = Time period
In this case, the annual income generated (R) is $400,000, the interest rate (r) is 0.10 (since it's given as 10% per year), and the time period (t) is 10 years.
Now, let's plug in the values and calculate!
PV = 400,000 / (e^(0.10 * 10))
PV ≈ 400,000 / (2.71828^(1.0))
PV ≈ 400,000 / 2.71828
PV ≈ 147,023
So, the present value of the franchise is approximately $147,023. Don't spend it all in one place, Camille!
To find the present value of the franchise, we need to calculate the present value of the expected income over the 10-year period.
The formula for calculating the present value using continuous compounding is:
PV = R / e^(rt)
Where PV is the present value, R is the expected income per year, r is the interest rate, and t is the time period.
In this case, the expected income per year is $400,000, the interest rate is 10% (or 0.10), and the time period is 10 years.
Using these values in the formula:
PV = 400,000 / e^(0.10 * 10)
Calculating the exponent:
PV = 400,000 / e^1
Approximating the value of e^1 as 2.71828:
PV = 400,000 / 2.71828
PV ≈ 147,126
Therefore, the present value of the franchise is approximately $147,126.
To find the present value of the franchise, we need to calculate the present value of the expected income over the 10-year period, taking into account the prevailing interest rate.
The present value formula for continuously compounded interest is given by:
PV = R / e^(rt)
Where:
PV = Present Value
R = Future Value (expected income)
r = Interest rate (in decimal form)
t = Time period (in years)
e = Euler's number, approximately 2.71828
In this case, R(t) is constant at $400,000, the interest rate is 10% which is 0.10 in decimal form, and the time period is 10 years.
Let's substitute these values into the formula and calculate the present value:
PV = 400,000 / e^(0.10 * 10)
First, let's calculate e^(0.10 * 10):
e^(0.10 * 10) ≈ 2.71828^(1) = 2.71828
Now, let's calculate the present value:
PV = 400,000 / 2.71828 ≈ 147,019
Therefore, the present value of the franchise is approximately $147,019.