A design-build operate Engineering company in Texas that owns a sizeable (30) amount of land plans to lease the drilling rights (oil and gas only) to a mining and exploration company. The contract calls for the mining company to pay $15,000 per year for 20 years beginning 3 years from now (i.e. beginning at the end of year 3 and continuing through year 22) plus $ 8,000 six years from now and 12,000 sixteen years from now. Utilize engineering economy relations to determine the four equivalent values listed below at 15% per year (1) Total present worth Pr in year 0. (ii) Future worth F in year 22 (iii) Annual series over all 22 years. (iv) Annual series over the first 12 years.
The correct answer and explanation is:
Solution:
We will use standard engineering economy relations such as Present Worth (PW), Future Worth (FW), and Annual Worth (AW) formulas to calculate the required values based on the given cash flow details and an interest rate of 15% per year.
Given:
- Annual lease payment: $15,000 (years 3 to 22).
- Lump sums: $8,000 (at year 6) and $12,000 (at year 16).
- Interest rate (ii): 15% per year.
- Time period: 22 years.
Step-by-Step Calculation:
1. Total Present Worth (P0P_0) at Year 0:
Use present worth formulas for annual payments and individual lump sums.
Annual Lease Payments: P=A⋅P/A(i,n)⋅(P/F(i,t))P = A \cdot P/A(i, n) \cdot (P/F(i, t))
Where A=15,000A = 15,000, n=20n = 20 (number of payments), t=2t = 2 (shift to year 0). P=15,000⋅[1−(1+0.15)−200.15]⋅(1+0.15)−2P = 15,000 \cdot \left[\frac{1 – (1 + 0.15)^{-20}}{0.15}\right] \cdot (1 + 0.15)^{-2}
Lump Sums: P6=8,000⋅(1+0.15)−6P_{6} = 8,000 \cdot (1 + 0.15)^{-6} P16=12,000⋅(1+0.15)−16P_{16} = 12,000 \cdot (1 + 0.15)^{-16}
Sum up all present values to get P0P_0.
2. Future Worth (F22F_{22}) in Year 22:
The future worth is the value of all cash flows at year 22. F22=A⋅F/A(i,n)+F6+F16F_{22} = A \cdot F/A(i, n) + F_{6} + F_{16}
Where F6=8,000⋅(1+0.15)16F_{6} = 8,000 \cdot (1 + 0.15)^{16}, F16=12,000⋅(1+0.15)6F_{16} = 12,000 \cdot (1 + 0.15)^6.
3. Annual Series Over All 22 Years:
The equivalent annual series is: A=P⋅A/P(i,n)A = P \cdot A/P(i, n)
Where n=22n = 22.
4. Annual Series Over the First 12 Years:
For this, we consider the present worth of cash flows for the first 12 years and convert it to an equivalent annual series over n=12n = 12.
Explanation:
The goal is to evaluate cash flows from leasing drilling rights using time value of money principles. The engineering economy relations help transform cash flows occurring at different times into equivalent values at specific time points or uniform annual series.
- Present Worth (P0P_0) evaluates the total value of cash flows as if all occurred at year 0. It involves discounting the annual lease payments (starting from year 3) and lump sums to year 0. This calculation uses P/AP/A, P/FP/F, and shifts due to cash flow timing.
- Future Worth (F22F_{22}) accumulates all cash flows to year 22. Annual payments are compounded to year 22 using F/AF/A, while the lump sums are compounded appropriately. This gives the total value of cash flows at the contract’s end.
- Annual Series Over 22 Years (A22A_{22}) represents the uniform equivalent annual amount over the entire project timeline. By dividing P0P_0 into an equivalent annual series over 22 years, we ensure time-consistent decision-making.
- Annual Series Over 12 Years (A12A_{12}) focuses on the first 12 years. Cash flows within this period are isolated, converted to a present value, and transformed into a uniform annual equivalent over 12 years.
These computations provide stakeholders a clear understanding of project economics, enabling informed financial decisions about lease agreements. Let me know if you’d like detailed numerical results for each step!