Suppose an investment is expected to generate income at the rate of
R(t) = 200,000
dollars/year for the next 6 years. Find the present value of this investment if the prevailing interest rate is 8%/year compounded continuously. (Round your answer to two decimal places.)
To find the present value of the investment, we need to calculate the integral of the income function over the 6-year period and discount it back to the present value using the prevailing interest rate.
The present value (P) can be calculated using the formula:
P = ∫[0,6] R(t) * e^(-rt) dt
Where:
- R(t) is the income function at time t,
- e is the mathematical constant approximately equal to 2.71828,
- r is the interest rate, and
- t represents time.
In this case, the income function R(t) is a constant value of $200,000/year for the next 6 years. Therefore, R(t) = $200,000.
The interest rate, r, is given as 8%/year, which can be written as 0.08.
Let's substitute these values into the formula and solve the integral:
P = ∫[0,6] $200,000 * e^(-0.08t) dt
To integrate this expression, we can use the power rule of integration for the exponential function:
∫ e^(-kt) dt = -(1/k) * e^(-kt) + C
Where C is the constant of integration.
Applying the power rule to our integral:
P = ∫[0,6] $200,000 * e^(-0.08t) dt
= [$200,000 / (-0.08)] * e^(-0.08t) | [0,6]
Evaluating the integral from 0 to 6:
P = [$200,000 / (-0.08)] * e^(-0.08 * 6) - [$200,000 / (-0.08)] * e^(-0.08 * 0)
Since e^0 = 1, the second term simplifies to zero:
P = [$200,000 / (-0.08)] * e^(-0.08 * 6) - [$200,000 / (-0.08)]
Now, let's evaluate the expression:
P = [$200,000 / (-0.08)] * e^(-0.08 * 6) - [$200,000 / (-0.08)]
≈ [$200,000 / (-0.08)] * 0.513 = -$2,565,000
Rounding the answer to two decimal places, the present value of the investment is approximately -$2,565,000. Note that the negative sign indicates that the investment has a negative present value, meaning it is not a profitable investment at the given interest rate.
To find the present value of the investment, we need to use the formula for continuous compound interest:
PV = C * e^(-rt)
Where:
PV is the present value of the investment
C is the cash flow (income generated) per year
e is the mathematical constant approximately equal to 2.71828
r is the interest rate
t is the time in years
In this case, the cash flow is $200,000 per year, the interest rate is 8% (0.08 in decimal form), and the time is 6 years.
Plugging these values into the formula, we have:
PV = 200,000 * e^(-0.08 * 6)
Calculating the exponent: -0.08 * 6 = -0.48
PV = 200,000 * e^(-0.48)
Using the value of e from the question (approximately 2.71828), we have:
PV = 200,000 * 2.71828^(-0.48)
Calculating 2.71828^(-0.48):
PV = 200,000 * 0.6191
PV ≈ 123,820
Therefore, the present value of this investment is approximately $123,820.