A person is considering an investment situation that requires the investment of $160,000 at time zero and $230,000 at year one to generate profits of $99,000 per year starting at year two and increasing 2,000 per year due to inflation through year 10 (a 9 year profit period) with projected salvage value of $160,000 at the end of year 10. Determine the compound interest rate of return for these end of period funds. Draw the cumulative cash position diagram for the time zero through end of year 10 at the project rate of return.
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To determine the compound interest rate of return for the end of period funds, we need to calculate the present value of the expected profits and the salvage value.
1. Present value of profits:
The profits generated starting from year two and increasing by $2,000 per year due to inflation can be treated as an annuity. The annuity payment is $99,000, and the number of periods is 9 years since the profit period spans from year 2 to year 10.
Using the formula for the present value of an ordinary annuity:
PV = P * [1 - (1 + r)^(-n)] / r
Where:
PV = Present value
P = Annual payment
r = Interest rate per period
n = Number of periods
Using this formula, we can calculate the present value of the profits:
PV of profits = $99,000 * [1 - (1 + r)^(-9)] / r
2. Present value of the salvage value:
The salvage value of $160,000 at the end of year 10 needs to be discounted to time zero as well. Since it is a single payment at the end of the project, we can calculate its present value using the formula:
PV = FV / (1 + r)^n
Where:
FV = Future value
PV = Present value
r = Interest rate per period
n = Number of periods
Using this formula, we can calculate the present value of the salvage value:
PV of salvage value = $160,000 / (1 + r)^10
3. Total present value of investment:
To calculate the total present value of the investment, we need to sum the present values of the profits and the salvage value:
Total PV = PV of profits + PV of salvage value
Now, we need to find the interest rate (r) that will make the total present value equal to the initial investment of $160,000 + $230,000:
Total PV = $160,000 + $230,000
Setting up the equation:
PV of profits + PV of salvage value = $390,000
$99,000 * [1 - (1 + r)^(-9)] / r + $160,000 / (1 + r)^10 = $390,000
This equation cannot be solved algebraically. We will need to use numerical methods or financial calculators/software to find the compound interest rate (r) that satisfies the above equation.
Once the compound interest rate (r) is determined, we can draw the cumulative cash position diagram from time zero through the end of year 10 at the project rate of return.
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