A company is considering the purchase of a forest that is estimated to yield an annual return of $50,000 for 10 years, after which the forest will have no value. The company wants to earn 8% on its investment and set up a sinking fund to replace the purchase price. If the money is placed in the fund at the end of each year and earns 6% compounded annually, find the price the company should pay for the forest.

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If I understand the investment instructions correctly, this is how it works:

The company would invest an amount P which should yield net 8% return each year for 10 years.

The revenue generated is constant each year at $50,000, and includes profit and capital repayment. The capital is repaid from a sinking fund that pays 6% p.a. and should generate the original capital at the end of 10 years. The annual payment into the sinking fund (out of the revenue) is $x.

In short,
P=initial investment
x=annual investment into sinking fund

For the 8% profit over 10 years, we have
P*(0.08*10) = 50000*10 - 10x ...(1)

and that the amount x over 10 years at 6% p.a. should yield exactly P:
P = x*1.06^10/(1.06-1.0) ...(2)

Substituting (2) into (1)
x*1.06^10/(1.06-1.0)*(0.08*10) = 50000*10 - 10x
Solve for x to get $14758.85
and from (2)
P = $440514.32

Substitute the amounts into the scenarios and check.

To find the price the company should pay for the forest, we need to calculate the present value of the annual returns for 10 years and the future value of the sinking fund.

Step 1: Calculate the present value of the annual returns:
The annual return is $50,000 per year for 10 years. To find the present value, we need to discount each of these cash flows back to the present using the required rate of return of 8%.

Using the formula for the present value of an ordinary annuity:
PV = C * [(1 - (1 + r)^(-n)) / r]

where PV is the present value, C is the cash flow per period, r is the discount rate, and n is the number of periods.

In this case, C = $50,000, r = 8%, and n = 10.

PV = $50,000 * [(1 - (1 + 0.08)^(-10)) / 0.08]
PV = $50,000 * (1 - 0.4665) / 0.08
PV = $50,000 * 0.5335 / 0.08
PV = $333,375

Step 2: Calculate the future value of the sinking fund:
The sinking fund is set up to replace the purchase price at the end of 10 years. To calculate the future value, we need to compound the annual payments made into the fund at a rate of 6% for 10 years.

Using the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r

where FV is the future value, P is the payment per period, r is the interest rate per period, and n is the number of periods.

In this case, P = $50,000, r = 6%, and n = 10.

FV = $50,000 * [(1 + 0.06)^10 - 1] / 0.06
FV = $50,000 * (1.790847 - 1) / 0.06
FV = $50,000 * 0.790847 / 0.06
FV = $658,872

Step 3: Calculate the purchase price:
The purchase price of the forest should be equal to the present value of the annual returns minus the future value of the sinking fund.

Purchase Price = PV - FV
Purchase Price = $333,375 - $658,872
Purchase Price = -$325,497

The calculation shows that the company should pay approximately -$325,497 for the forest. Note that this negative value means the company should receive this amount as a payment in order to earn an 8% return on its investment.