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(Sample) Loan Project: Buying a House in Purcellville, Virginia Submitted by Suzanne Sands

Purpose: To analyze the financial implications of purchasing a house.

Task (1): House Description:

I decided to choose a house in my hometown. This house is typical of homes in the new developments around town. I have no interest in moving, though --- I am happy in my own house, which was built in the 1970s and does not look like this house at all!) Asking price: $439,000 Real estate taxes: $4,836 (found in the http://www.loudoun.gov property database) Purchase Price: $388,400 (I chose this value because it was the assessed value at the time this project was prepared, according to the county property database) Current market interest rate (30 yr fixed rate): 4.625% through AimLoan.com (found at http://www.bankrate.com, by doing a search for my zipcode) [NOTE: This sample project was completed quite some time ago, so interest rates, taxes, and purchase prices could be quite different now.]

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Task (2) 20% down payment: $77,680 (= 0.20 x $388,400) Amount financed (i.e., amount borrowed): $310,720 (= $388,400 – 77,680) Monthly mortgage payment: $1,597.53 (see below for details of calculation) Payment calculation:

• present value PV = $310,720 • annual interest rate r = 0.04625 • interest compounded m = 12 times per year • number of payments n = 30 x 12 = 360 The interest rate per month is i = r/m = 0.04625/12.

The monthly mortgage payment PMT is

��� = �� �

�� � ��� = 310,720

�.�����/��

� � � �.�����/������� ≈ 310,720

�.�� !����"

���.��� ���

≈ 310,720 0.005141396� = $1,597.53

Remark: Intermediate results must be computed with great precision, so that the final value has 6 digits of accuracy (an amount involving dollars and cents, with the dollar amount in the thousands).

Task (3): Monthly real estate taxes: $403.00 (= $4836/12)

Total monthly payment = Monthly mortgage payment + Monthly taxes = 1597.53 + 403.00 = $2000.53 The total monthly payment is used when determining the income level needed to afford the home. Task (4): Minimum income requirement

Total monthly payment ≤ 0.30 x Monthly Income (Total monthly payment)/0.30 ≤ Monthly Income Monthly Income ≥ (Total monthly payment)/0.30 = $2000.53/(0.30) = $6,668.43 Annual income = 12(Monthly Income) = 12($6668.43) = $80,021.16 In order to qualify for the loan, the monthly income must be at least $6,668.43, and the annual income must be at least $80,021.16. Task (5): Amortization Table --- See separate document (generated by using the amortization schedule calculator at http://www.bankrate.com , pasted into a spreadsheet, and then reformatted to paste into a Word document). Task (6) In this sample project, the mortgage was set up in December, 2008, with the first payment in January, 2009.

By consulting the amortization table, January, 2024 is the first month when the amount applied to the principal ($799.35) exceeds the amount of interest ($799.18) paid. This is the 181

st payment.

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Task (7) After 24 years, 24(12) = 288 payments have been made, so there are 360 - 288 = 72 payments remaining. To find the unpaid balance, calculate PV, the amount remaining to be paid off by 72 monthly payments of $1,597.53 at an interest rate of 4.625% compounded monthly.

�� = ��� � � � ���

� = 1,597.53

� � � �.�����/����()

�.�����/��

= 1,597.53 ���."�!�"*!�

�.�� !�����" = $100,274.55

The unpaid balance after 24 years is $100,275 (rounded to the nearest dollar) Notice that this amount agrees with the amount in the amortization table for the unpaid balance after 288 payments. (It is very close but not precisely the same as the amount in the amortization table, because in going month to month in the table, there are issues involving repeated rounding to the nearest cent, accumulating over time.)

Since $100,275/310,720 ≈ 0.323, the unpaid amount after 24 years is 32.3% of the amount financed.

Task (8) [Assuming the loan is held for the full 30 years] Total principal paid = Amount borrowed = $310,720

Total interest paid = 360($1597.53) – $310,720 = $264,390.80 Task (9) Conclusion For a 30-year loan of $310,720 at an interest rate of 4.625%, the monthly payment is $1,597.53, and the borrower would need a minimum annual income of about $80,000 to qualify for the loan.

It takes a bit over 15 years to reach the point where the monthly principal applied is larger than the monthly interest paid. After 24 years, nearly a third of the amount borrowed is still owed.

If the mortgage is held for the full 30 years, then $264,391 will be paid in interest, which is 85% of the amount borrowed.

Without understanding a loan analysis, a common misconception is that the unpaid balance is reduced by the same amount each year. However, interest is charged monthly on the amount owed, which is very large at the beginning and gradually diminishes. A quick scan of the amortization table confirms that in the start of the 30 year mortgage, almost all of the monthly payment is interest paid to the lender, and little is applied to reducing the principal. Indeed, for the loan in this project, it takes over half of the 30 year period before more of the monthly payment is applied to reducing the principal than in paying interest. And, when four-fifths of the 30 year period has elapsed, about a third of the amount borrowed is still owed. Extra credit:

Scenario: Smaller down payment Suppose the down payment is just 10% of the purchase price. (Assume the loan is still a 30-year fixed rate mortgage at the same interest rate.)

10% down payment: $38,840 (= 0.10 x $388,400) Amount financed: $349,560 (= $388,400 – $38,840) Monthly mortgage payment: $1,797.23 (see below for details of calculation) Payment calculation:

• present value PV = $349,560 • annual interest rate r = 0.04625 • interest compounded m = 12 times per year • number of payments n = 30 x 12 = 360 The interest rate per month is i = r/m = 0.04625/12.

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The monthly mortgage payment PMT is

��� = �� �

�� � ��� = 349,560

�.�����/��

� � � �.�����/������� ≈ 349,560

�.�� !����"

���.��� ���

≈ 349,560 0.005141396� = $1,797.23 Scenario: Shorter term Suppose the loan is a 15-year fixed rate mortgage. (Assume the loan has the same interest rate and that the down payment is 20% of the purchase price.)

20% down payment: $77,680 (= 0.20 x $388,400) Amount financed: $310,720 (= $388,400 – 77,680) Monthly mortgage payment: $2,396.89 (see below for details of calculation) Payment calculation: .

• present value PV = $310,720 • annual interest rate r = 0.04625 • interest compounded m = 12 times per year • number of payments n = 15 x 12 = 180 The interest rate per month is i = r/m = 0.04625/12.

The monthly mortgage payment PMT is

��� = �� �

�� � ��� = 310,720

�.�����/��

� � � �.�����/����+,� ≈ 310,720

�.�� !����"

���.��� ����!

≈ 310,720 0.007713972� = $2,396.89

Comparison of Mortgages Scenario Original

(20% down, 30 years)

Smaller Down Payment (10% down, 30 years)

Shorter Term (15 years, 20% down)

Purchase Price $388,400 $388,400 $388,400 Down Payment $77,680 $38,840 $77,680 Amount Financed (PV) $310,720 $349,560 $310,720 Annual Interest Rate 4.625% 4.625% 4.625% Number of Payments 360 360 180 Monthly Payment (PMT) $1,597.53 $1,797.23 $2,396.89 Total Interest Paid (If loan held for entire term)

$264,390.80 $297,442.80 $120,720.20

Conclusion: (Drawn from extra credit scenarios) By making a 20% down payment rather than a 10% down payment, the borrower saves about $33,000

( ≈ $297,443 - 264,391) in interest over the course of the loan. The 20% down payment also provides more of a cushion in case real estate values decline. With a 20% down payment, a borrower is less likely to be “underwater” later, owing more than the property is valued. By taking out a 15-year loan of $310,720 rather than a 30-year loan, the borrower saves over $143,600

( ≈ $264,391 - 120,720) in interest over the course of the loan. For the 15-year mortgage, $120,720 will be paid in interest, which is about 39% of the amount borrowed. However, since the monthly payment is $2,396.89, substantially higher than for the 30-year loan, the borrower will need a higher annual income. For this 15-year loan, the borrower would need a minimum annual income of about $112,000

[ ≈ 12(2397 + 403)/0.30] to qualify for the loan.