The Interest Rate Is 4%. You Are Offered Either $1,000 Now

The interest rate is 4%. You are offered either $1,000 now or $1,081.60 in two years.

The assignment involves applying principles of advanced engineering economics, particularly focusing on time value of money and financial decision-making. The problems cover various topics such as present value, future value, loan amortization, investment appraisal, and project evaluation techniques like NPV and IRR. The core objective is to analyze different financial scenarios using given interest rates, cash flows, and project parameters to support sound economic decisions.

Paper For Above instruction

Financial decision-making in engineering economics hinges on the fundamental understanding of the time value of money, which asserts that a dollar today is worth more than the same dollar in the future due to its potential earning capacity. This principle underpins the majority of problems presented, including comparing investments, calculating future and present values, loan amortization, and project appraisal metrics like net present value (NPV) and internal rate of return (IRR).

The initial scenario involves a choice between receiving $1,000 now or $1,081.60 in two years at an interest rate of 4%. To analyze this, we compare the present value (PV) of the future sum discounted at 4% for two years. Using the formula PV = FV / (1 + r)^n, where FV is the future value, r is the interest rate, and n is the number of years, we find that PV = 1,081.60 / (1 + 0.04)^2 = 1,081.60 / 1.0816 ≈ 1,000. Hence, the present value of $1,081.60 in two years at 4% interest is exactly $1,000, indicating that both choices are equally valuable. Therefore, the choice depends on preference, but from a purely financial perspective, they are worth the same (Option C).

Next, when calculating how much $1,000 is worth in 6 years at 4%, we need to find its future value (FV). Using the FV formula, FV = PV (1 + r)^n, the calculation is FV = 1,000 (1 + 0.04)^6 ≈ 1,000 * 1.265319 ≈ $1,265.32. Closest match among options is $1,265, confirming the importance of compounding. Accurate future value calculations guide investors and engineers in assessing project profitability or investment growth over time.

Regarding loan amortization, a borrower takes a $5,000 loan at 5%. By paying $1,410.06 annually, the question is what the remaining balance will be at the end of year 2. Using amortization and present value calculations, this involves calculating the loan balance after two payments, considering interest accrued and payments made. The loan balance after two years would be approximately $2,621.88, aligning with the amortization process, where the borrower reduces principal over time with interest on remaining debt (Option B).

To determine the annual payment to fully amortize a $5,000 loan over 4 years at 5%, we use the loan amortization formula for an ordinary annuity. The calculation involves solving for A in the equation: A = PV * (r(1+r)^n) / ((1+r)^n - 1). Substituting the values yields an approximate annual payment of $1,410.06, validating Option B. This calculation is fundamental in project finance, ensuring consistent payments and full repayment by the end of the term.

The balance at the end of year 4, after making annual payments of $1,410.06 on a $5,000 loan at 5%, can be computed using amortization schedules. At that point, the remaining loan balance would be approximately $0, meaning the loan is fully paid off, assuming all payments are made as scheduled (Option A). Such calculations are crucial in managing debt and evaluating project feasibility.

Investing in a company's future sales involves discounted cash flow analysis. The company expects to generate revenues of $600,000 in years 3 through 6, with a required return of 6%. Using the present value of an annuity formula, PV = CF * [(1 - (1 + r)^-n) / r], the maximum investment aligns with the discounted sum of these cash flows. Calculations show the maximum feasible investment would be approximately $1.85 million (Option B). This approach helps investors assess whether anticipated cash flows justify the current investment.

Loan payments over multiple years at an interest rate involve calculating equal payments using the present value of an annuity formula. For a loan of $2,000 at 6% interest over 6 years, the annual payment is computed via PV/A formula, which yields approximately $376.40 annually. The calculation ensures that each payment covers interest and reduces the principal, ultimately paying off the loan within six years (Option C). Understanding such payment structures assists in planning and managing personal or project loans efficiently.

Choosing the best decision criterion in project evaluation depends on the context. While NPV considers the absolute value added and IRR indicates the rate of return, financial managers often prefer NPV for its direct measure of value. NPV reflects the dollar amount increase (or decrease) in value from undertaking a project, making it a superior metric for decision-making (Option A). This aligns with modern financial management principles emphasizing value creation.

When comparing two cash flow plans with the same interest rate, their present values are equal if the sum of their discounted cash flows is the same. For Plan 1, receiving $2,000 in one year and $1,081.60 in two years has the same PV as plan 2 with $1,000 in one year and $2,000 in two years, provided the interest rate remains constant. Since the sum of discounted cash flows equals, both plans have equivalent PV, making Option C the correct choice.

The IRR of a project equaling the discount rate where NPV is zero; if the IRR is 10%, and the project's NPV at a 10% discount rate is positive ($1,250), then the IRR must be higher, slightly above 10%. This is because IRR is the rate that zeroes out the NPV, and a positive NPV at 10% indicates the actual IRR exceeds 10%. Therefore, the statement is True.

Similarly, the profitability index (PI), calculated as the ratio of present value of future cash flows to initial investment, will be greater than 1 if the NPV is positive. With an NPV of $1,250 and an initial investment of $10,000, the PI is (NPV + initial investment) / initial investment = (1,250 + 10,000) / 10,000 = 1.125, which is greater than 1. Thus, the answer is Yes.

Calculating after-tax cash flow involves adjusting revenues and costs for taxes and accounting for non-cash items like depreciation. The formula is: (Revenues - Operating costs - Depreciation) (1 - tax rate) + Depreciation. Applying given figures: (2.5 million - 1.5 million - 1 million) (1 - 0.45) + 1 million results in an after-tax cash flow of approximately $750,000 (Option B). This measure indicates the actual cash generated after tax deductions, vital for assessing project viability.

If revenue increases while operating costs and depreciation remain constant, the after-tax cash flow would increase, given the positive correlation between revenue and cash flow. Since incremental revenues contribute directly to cash flow, the logical conclusion is Option A, increase.

Conversely, an increase in operating costs, all else equal, would decrease after-tax cash flow because higher costs reduce net income and cash available. Therefore, the appropriate answer is B, decrease. Similarly, an increase in the marginal tax rate reduces after-tax cash flow, as it increases tax obligations, meaning the correct choice is B, decrease.

Estimating IRR involves understanding the ratio of annual returns to initial investment. A project returning $250 annually on a $10,000 investment has an initial IRR estimate of 2.5%, but since the actual IRR accounts for the time value of money, the approximation suffices only as a first guess. The initial guess of 2.5% aligns with the simple ratio, confirming that the answer is Yes.

For a project returning $300 annually over 4 years with an initial investment of $1,000, estimating the IRR involves comparing the annuity payment to the investment. Using financial tables or approximation formulas, the IRR is between 6 and 7%, making Option C the best estimate. This method provides a reasonable initial estimate before more precise calculations.

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