An investor wants to know the amount she should pay for an oil well expected to yield an annual return of 30000 for the next 30 years, after which the well will be dry. Find the amount she should pay to yield a 14% annual return if a sinking fund earns 10% annually. Round the answer to the nearst dollar.
43
75576
To find the amount the investor should pay for the oil well, we need to calculate the present value of the future cash flows. The present value represents the current worth of the future returns, taking into account the time value of money.
Here's how we can calculate the present value using a sinking fund:
1. First, let's calculate the amount the investor would receive each year from the oil well. Given that the expected return is $30,000 annually for 30 years, the future cash flows can be represented as an annuity.
2. Next, we need to calculate the present value of the annuity. We'll use the present value of an annuity formula, which is:
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 = $30,000, r = 14% (0.14), and n = 30.
3. Now, let's calculate the value of the sinking fund over the 30-year investment period. We'll assume the sinking fund earns a 10% annual return.
To find the value of the sinking fund, we can use the future value of a lump sum formula:
FV = PV * (1 + r)^n
where FV is the future value, PV is the present value of the sinking fund, r is the interest rate, and n is the number of years.
In this case, r is 10% (0.10) and n is 30.
4. Finally, we'll calculate the amount the investor should pay for the oil well by subtracting the present value of the sinking fund from the present value of the annuity.
Let's calculate the values step by step:
1. Calculate the present value of the annuity:
PV_annuity = $30,000 * [(1 - (1 + 0.14)^(-30)) / 0.14]
≈ $223,619.62
2. Calculate the value of the sinking fund:
FV_sinking_fund = $223,619.62 * (1 + 0.10)^30
≈ $2,861,756.88
3. Calculate the amount the investor should pay for the oil well:
Amount = PV_annuity - PV_sinking_fund
= $223,619.62 - $2,861,756.88
≈ -$2,638,137.26
The investor should pay approximately -$2,638,137 (rounded to the nearest dollar) for the oil well in order to achieve a 14% annual return, taking into account the 10% return on the sinking fund.