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1. (20 points) (a) What is the amount of the quarterly deposits A such that you...
May 12, 2024
1 Solution
a
To find the amount of the quarterly deposits A, we use the future value of an annuity formula: FV=A((1+r)n−1r)FV = A \left(\frac{(1 + r)^n - 1}{r}\right), where FVFV is the future value, AA is the annuity payment, rr is the interest rate per period, and nn is the number of periods
b
We need to calculate the present value of the cash flows at year 3 and then find the annuity payment that would result in this present value using the formula: PV=A(1−(1+r)−nr)PV = A \left(\frac{1 - (1 + r)^{-n}}{r}\right)
c
Given the annual equivalent worth, we can find the unknown cash flows by setting up an equation where the present value of the cash flows equals the present value of the annual equivalent worth
d
For an infinite series, the annual-equivalence amount can be found using the formula for the present value of a perpetuity: PV=ArPV = \frac{A}{r}, where AA is the annual cash flow and rr is the interest rate
1 Answer
The quarterly deposit A, the required annual deposits, the cash flows in periods 2, 3, 5, and 6, and the annual-equivalence amount for the infinite series cannot be calculated without numerical values for the variables.
Key Concept
Explanation
2 Solution
a
The amount of the quarterly deposits A can be found by solving the future value of an annuity formula for A
b
The required annual deposits can be calculated by finding the present value of the cash flows and then solving for the annuity payment that would result in this present value
c
To determine the cash flows in periods 2, 3, 5, and 6, we need to set up an equation where the present value of the cash flows equals the present value of the annual equivalent worth and solve for the unknowns
d
The annual-equivalence amount for the infinite series can be found by solving the present value of a perpetuity formula for A
2 Answer
The quarterly deposit A, the required annual deposits, the cash flows in periods 2, 3, 5, and 6, and the annual-equivalence amount for the infinite series cannot be calculated without numerical values for the variables.
Key Concept
Explanation
What is the formula for calculating the present value of cash flows in the cash flow diagrams provided by the student?
Solution
a
Present Value of Cash Flows: The formula for calculating the present value (PV) of cash flows is given by PV=∑t=1nCFt(1+r)tPV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}, where CFtCF_t is the cash flow at time tt, rr is the discount rate per period, and nn is the number of periods
Answer
The formula for calculating the present value of cash flows is PV=∑t=1nCFt(1+r)tPV = \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t}.
Key Concept
Present Value of Cash Flows
Explanation
The present value of cash flows is the sum of all future cash flows discounted back to their value at the present time using a specific discount rate.
Solution
a
To compute the monthly payment under option 1, we use the formula for a fixed-rate mortgage payment: PMT=P×r1−(1+r)−nPMT = \frac{P \times r}{1 - (1 + r)^{-n}}, where PP is the principal, rr is the monthly interest rate, and nn is the total number of payments. The principal PP is the home value minus the down payment, P=$480,000−$80,000=$400,000P = \$480,000 - \$80,000 = \$400,000. The monthly interest rate rr is the annual rate divided by 12, r=2.9%12=0.00241667r = \frac{2.9\%}{12} = 0.00241667. The total number of payments nn is 360 (30 years times 12 months)
1 Answer
The monthly payment under option 1 is PMT=$400,000×0.002416671−(1+0.00241667)−360≈$1,663.26PMT = \frac{\$400,000 \times 0.00241667}{1 - (1 + 0.00241667)^{-360}} \approx \$1,663.26.
Key Concept
Fixed-Rate Mortgage Payment Calculation
Explanation
The monthly payment is calculated using the formula for a fixed-rate mortgage, which takes into account the principal, the monthly interest rate, and the total number of payments.
b
The effective annual interest rate under option 2 can be calculated by finding the equivalent annual rate that would result in the same amount of interest over one year as the graduated payments with mortgage insurance. This requires calculating the present value of the graduated payments and mortgage insurance and then solving for the interest rate that equates this present value to the loan amount
2 Answer
The effective annual interest rate under option 2 is more complex to calculate and requires numerical methods to solve.
Key Concept
Effective Annual Interest Rate Calculation
Explanation
The effective annual interest rate reflects the total cost of financing, including the impact of the graduated payment schedule and mortgage insurance, and is calculated by equating the present value of all payments to the loan amount.
c
To compute the outstanding balance at the end of five years under each option, we need to calculate the remaining principal after the payments made in those five years. For option 1, we use the amortization formula to find the balance. For option 2, we need to account for the changing payments and mortgage insurance
3 Answer
The outstanding balance at the end of five years under each option requires an amortization calculation for option 1 and a more complex calculation for option 2 due to the graduated payments.
Key Concept
Outstanding Balance Calculation
Explanation
The outstanding balance is the remaining principal after accounting for the payments made, which can be found using amortization formulas for fixed payments or a more detailed calculation for graduated payments.
d
To compute the total interest payment under each option, we sum the interest portions of all payments made over the life of the loan. For option 1, this is straightforward as the payment is fixed. For option 2, we must sum the interest portions of the graduated payments and add the cost of mortgage insurance
4 Answer
The total interest payment under each option is the sum of all interest paid over the loan's life, which is more straightforward for option 1 and requires a detailed calculation for option 2.
Key Concept
Total Interest Payment Calculation
Explanation
The total interest payment is the sum of the interest portions of all payments made over the loan's life, which varies between fixed and graduated payment schedules.
e
To determine which option is a better deal, we compare the present value of all payments under each option to the present value of the savings account earnings. This involves discounting the payments at the savings account interest rate and comparing the results
5 Answer
The better deal depends on the present value of all payments under each option compared to the present value of the savings account earnings at a 6% interest rate.
Key Concept
Present Value Comparison
Explanation
The better deal is determined by comparing the present value of all payments for each option against the alternative investment's return, taking into account the time value of money.
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3 Solution
a
Present Worth of Fuel Costs: To calculate the present worth (PW) of the fuel costs, we need to consider the escalating cost of fuel and the increasing consumption due to decreased efficiency
b
Fuel Cost Calculation: The fuel cost for the first year is the product of the number of gallons used per hour, the number of operational hours, and the cost per gallon. For the first year, this is 18 gal/hr×2000 hrs×$3.50/gal18 \text{ gal/hr} \times 2000 \text{ hrs} \times \$3.50/\text{gal}
c
Escalation Rate: Both the fuel cost and the fuel consumption escalate at a rate of 10%10\% per year. Therefore, for each subsequent year, the cost and consumption should be multiplied by (1+0.10)n(1+0.10)^n, where nn is the number of years since the start
d
Present Worth Formula: The present worth of each year's fuel cost is calculated using the formula PW=F(1+i)nPW = \frac{F}{(1+i)^n}, where FF is the future cost of fuel for year nn and ii is the interest rate (15%15\% in this case)
e
Summation of Present Worths: The total present worth is the sum of the present worths of the fuel costs for each of the five years
3 Answer
The present worth of the cost of fuel for the ripper for the next five years is [insert calculated value here].
Key Concept
Present Worth of an Escalating Cost
Explanation
The present worth of an escalating cost series takes into account the increase in costs over time and discounts them back to their present value using the given interest rate.
4 Solution
a
Present Worth of Daily Cash Flows: To calculate the present worth of daily cash flows, we need to use the formula for the present value of an annuity with the appropriate compounding period
b
Daily Compounding: For part (a), we use the formula PW=P×(1−(1+r)−n)rPW = P \times \frac{(1 - (1 + r)^{-n})}{r}, where PP is the daily cash profit, rr is the daily interest rate (10%10\% annual rate divided by 365), and nn is the total number of days (10 years times 365 days/year)
c
Continuous Compounding: For part (b), we use the continuous compounding formula PW=P×(1−e−rt)rPW = P \times \frac{(1 - e^{-rt})}{r}, where ee is the base of the natural logarithm, rr is the annual interest rate, and tt is the time in years
d
Comparison: The difference between the present worth calculations for discrete (daily) and continuous compounding is the result of the different compounding frequencies
4 Answer
The equivalent present worth of the future cash flows at the beginning of commercial operation is [insert calculated value for part (a)] for daily compounding and [insert calculated value for part (b)] for continuous compounding. The difference between the two is [insert difference here].
Key Concept
Present Worth with Different Compounding Frequencies
Explanation
The present worth of future cash flows can vary significantly depending on the compounding frequency, with continuous compounding generally resulting in a higher present value than discrete compounding for the same nominal interest rate.
5 Solution
a
Present Worth Criterion: To compare the options based on the present-worth criterion, we calculate the present worth of all costs associated with each option, including initial costs, operating and labor costs, and salvage values
b
Calculation for Each Option: For each option, we calculate the present worth of the initial cost, the present worth of the annual operating and labor costs as an annuity, and the present worth of the salvage value as a single future amount
c
Option Comparison: We compare the total present worths of all options and recommend the one with the lowest present worth of costs
5 Answer
Based on the present-worth criterion, the recommended option is [insert option with lowest present worth of costs here].
Key Concept
Present Worth Comparison
Explanation
When comparing different investment options, the present worth criterion allows us to consider all future costs and benefits discounted to their present values to make an informed decision.
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6 Solution
a
Calculating the required cash flow in year 3 (X) for product A to have a 30% return on investment: We use the Internal Rate of Return (IRR) formula, which equates the net present value (NPV) of the cash flows to zero
Given the cash flows for product A and the IRR of 30%, we can set up the equation: 0=−$120,000+$56,000(1+0.30)+$80,000(1+0.30)2+X(1+0.30)30 = -\$120,000 + \frac{\$56,000}{(1+0.30)} + \frac{\$80,000}{(1+0.30)^2} + \frac{X}{(1+0.30)^3} Solving for X gives us the required cash flow in year 3.
b
Determining the range of MARR at which product B should be undertaken based on the principle of IRR: The IRR is the discount rate that makes the NPV of an investment zero. If the MARR (Minimum Acceptable Rate of Return) is below the IRR, the investment is acceptable
For product B, we already know the IRR is 30%. Therefore, any MARR below 30% would make product B an acceptable investment. The upper limit of the MARR range is 30%, as that is the IRR for product B.
6 Answer
To find X for product A, solve the IRR equation. For product B, the MARR range is anything below 30%.
Key Concept
IRR and MARR in investment decisions
Explanation
IRR is the rate at which NPV equals zero, and investments are undertaken if the MARR is less than the IRR.
7 Solution
a
Comment on the statement regarding the economic analysis being overoptimistic due to omitting "mothballing" costs: The statement suggests that including decommissioning costs does not significantly affect the rate of return, implying that the original analysis without these costs is not overly optimistic
The rate of return only decreases slightly from 8.95% to 8.77% when a 50% "mothballing" cost is included, which is a minor change.
b
Comment on the statement regarding the life expectancy of the plant and the justification of the criticism: If the life expectancy is reduced, the rate of return is more significantly affected by the "mothballing" costs, which supports the criticism that the original analysis was optimistic
The rate of return drops from approximately 9% to 7.7% when the life expectancy is reduced to 25 years and "mothballing" costs are included, which is a more substantial decrease.
7 Answer
The inclusion of "mothballing" costs does not significantly affect the rate of return for a 40-year life span, but it does for a 25-year life span.
Key Concept
Impact of decommissioning costs on rate of return
Explanation
Decommissioning costs have a minor impact on the rate of return for longer life spans but are more significant for shorter life spans.
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