Engineering Economics A3-g Suppose you want to borrow $200,000 to purchase a hom
ID: 2815162 • Letter: E
Question
Engineering Economics
A3-g Suppose you want to borrow $200,000 to purchase a home. You have found terms that include a 30-year note with a (nominal) rate of 7.0% compounded monthly. The loan includes payment of 2.5 points, which will be paid out of the loan at closing. (One point is equal to 1% of the loan value. A) Calculate the monthly mortgage payment. B) How much interest is paid in the 2nd month's payment? C) How much principal is paid in the 2nd month's payment? D) Assuming the points are the only relevant closing costs associated with the loan, what is the APR on this loan? The key concept in Case D is recognition that you will not have the full $200,000 loan value to apply against the purchase of the home (due to the points), but you will have to pay back $200,000 in principal payments plus the accrued interest.Explanation / Answer
The specs are : Loan Amount = $200,000 ; Term is 30 years or 360 months ; Interest Rate is 7% p.a. or 0.58% per month ; Closing Costs = 2.5% of loan amount = $5000.
Monthly Mortgage Payment = Loan Amount * [ r * (1+r)t ] / [ (1+r)t - 1 ] ; where r is the applicable monthly interest rate and t is tenure in months. Plugging in the values we get:
Monthly Mortgage Payment = 200000 * [ 0.58% * (1+0.58%)360 ] / [(1+0.58%)360 - 1] = $1330.60
In the first month, out of the payment of $1330,60, the interest will be = (200000 * 0.58%) = 1166.67 hence principal repayment will be only $ 163.94. So for the second month:
Interest Paid will be = (200000 - 163.94) * 0.58% = $ 1165.71
Principal repayment will be = (1330.60 - 1165.71) = $ 164.89
APR for the Loan: Given that the net loan amount is only $195000 but the repayment is for $200,000 loan, the APR is essentially the discount rate which will equate the monthly payment of $ 1330.60 paid over 360 months to the net loan amount of $195000 . This is like an IRR of the loan repayments. If we denote APR by x, then we have:
195000 = 1330.60/(1+x/12) + 1330.60/(1+x/12)2 + ...... 1330.60/(1+x/12)360
we can solve this equation usin excel or since this is like an annuity stream, we can use the present value of annuity formula to solve for x.
PV of annuity = Monthly Cash flow * [ 1 - (1+x/12)-t]/(x/12) ; we divide x by 12 to make it monthly interest rate. Plugging in the values:
1950000 = 1330.60 * [ 1- (1+x/12)-360] / (x/12) ; solving for x, we get x = 7.25%
hence the APR for this loan is 7.25% (which is higher than the nominal quoted rate).
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