Engineering Economics

Essay 1:

You wish to purchase a home for $150,000 and you can put down 10% of this price as down payment. You can get a 20 year fixed rate mortgage loan for 6.0% with no points. You can optionally decide to pay 2 points to bring the mortgage rate down to 5.25%. Your closing fees (not including points) are expected to be $3,300. Your PMI (if applicable) is $150/month for the first 4 years (after which you will have 20% equity), your property taxes are $3,600/year and your casualty insurance is $2,400/year. Your lender will collect the PMI, property tax and hazard insurance as monthly escrow payments along with your principal and interest payments.

a. What is the initial TOTAL CASH required to buy this house AT CLOSING with the given mortgage fees and down payment provided that you choose the 5.25% loan option?

b. What is the initial monthly payment (including PMI and monthly escrow) for the 6% loan option?

c. If you intend to live in the home for 5 years or less, which loan option should you take (i.e. pay points or not)? Assume MARR to be 10% on your other investments. You must show your work to mathematically justify your answer in order to get full credit. (NOTE: you can analyze this on an annual savings basis by multiplying the monthly savings by 12 and perform a reverse look-up of your N value in the 10 percent interest tables that were provided with the exam).

To solve this problem, we need to calculate the initial total cash required to buy the house at closing with the given mortgage fees and down payment, as well as the initial monthly payment for the 6% loan option. Finally, we need to determine which loan option is better if you intend to live in the home for 5 years or less.

a. To calculate the initial total cash required with the 5.25% loan option, we need to consider the down payment, closing fees, and any additional costs.

1. Down payment: Since the house is priced at $150,000 and you are putting down 10%, the down payment will be 10% of $150,000 = $15,000.

2. Closing fees: The closing fees, excluding points, are given as $3,300.

3. Additional costs: The PMI, property taxes, and casualty insurance will be collected as monthly escrow payments with the principal and interest. However, we are only concerned with the initial cash required at closing, so we don't need to include these costs.

Thus, the initial total cash required with the 5.25% loan option will be the sum of the down payment and closing fees:
Initial total cash = Down payment + Closing fees

Substituting the values:
Initial total cash = $15,000 + $3,300

b. To calculate the initial monthly payment for the 6% loan option, we need to consider the principal and interest payment, PMI, property taxes, and casualty insurance.

1. Principal and interest payment: To calculate the monthly principal and interest payment, we can use the loan amount, interest rate, and loan term. The loan amount will be 90% of the house price, since you are putting down 10% as a down payment.

Loan amount = 90% of $150,000 = $135,000

Using the formula for the monthly payment on a fixed-rate mortgage, we can calculate the principal and interest payment using the loan amount, interest rate, and loan term:
Monthly payment = (Loan amount * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-n))

Where Monthly interest rate = Annual interest rate / 12 and n = Number of months in the loan term.

Substituting the values:
Monthly interest rate = 6% / 12 = 0.06 / 12 = 0.005
Number of months = 20 years * 12 months/year = 240

Monthly payment = ($135,000 * 0.005) / (1 - (1 + 0.005)^(-240))

2. PMI, property taxes, and casualty insurance: These will be collected as monthly escrow payments along with the principal and interest payment. However, we are only concerned with the initial monthly payment, so we don't need to include these costs.

Thus, the initial monthly payment for the 6% loan option will be the sum of the principal and interest payment, PMI, property taxes, and casualty insurance:
Initial monthly payment = Principal and interest payment + PMI + Property taxes + Casualty insurance

Substituting the values:
Initial monthly payment = Monthly payment + $150 + ($3,600 / 12) + ($2,400 / 12)

c. To determine which loan option is better if you intend to live in the home for 5 years or less, we need to compare the costs of the two loan options taking into account the time value of money. We will calculate the present value of the cash flows for both options.

1. Loan option with 6% interest rate: Calculate the present value of the monthly payments for 5 years. Use the monthly payment calculated in part b and discount it back to present value using a discount rate of 10% (MARR - Minimum Acceptable Rate of Return).

2. Loan option with 5.25% interest rate and 2 points: Calculate the present value of the monthly payments for 5 years. Use the monthly payment calculated in part b (assuming it for the 5.25% interest rate option) and discount it back to present value using a discount rate of 10%.

Then compare the present values of the two options to determine which one has a lower cost.

To calculate the present value of the monthly payments, we can use the present value formula for an ordinary annuity:

Present value = Monthly payment * [(1 - (1 + discount rate)^(-n))] / discount rate

Where discount rate = 10% / 12 and n = 5 years * 12 months/year.

Substituting the values and performing the calculations will give us the present values of the two options. The option with the lower present value will be the better choice if you intend to live in the home for 5 years or less.