1. A basic 30-year ARM is made for $300,000 with an initial interest rate of 2%.
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Question
1. A basic 30-year ARM is made for $300,000 with an initial interest rate of 2%. Payments are monthly. The rate will reset every year. The index is the one-year Treasury and there is a 2% margin. There are no relevant caps. The Treasury rate at the beginning of year 2 is expected to be 3% and it is expected to be 4% at the beginning of year 3. You expect that the loan will be paid off at the end of year 3.
a. Calculate the payments and loan balances for years 1, 2, and 3.
b. If the lender charges 3 points, what is the yield on the loan?
Explanation / Answer
Answer (a)
Amount of Loan = $300,000
Initial Interest rate 2% per annum or monthly interest of 0.16667%
Period of Loan = 30 years or 30*12 = 360 months
Year 1
Monthly Payment = 300000 * [{(0.001667)*(1.0016667)^360 }/{(1.0016667)^360-1}]
= 300000 * [(0.001667 * 1.821231)/(1.821231 – 1)]
= 300000 * (0.003035445/0.821231)
= 300000 * 0.0036962
= $ 1,108.86 (rounded off)
Loan Balance after 12 months = 300000 * [{(1.0016667)^360 – (1.0016667)^12}/{(1.0016667)^360-1}]
= 300000 * [(1.821231 – 1.02018476)/(1.821231 – 1)]
= 300000 * [0.80104624/0.821231]
= 300000 * 0.975421 = $ 292,626.30 (rounded off)
Year 2
Loan Amount = $ 292,626.30
Interest rate = 3% + 2% = 5% per annum or 0.416667% monthly
Period = 360-12 = 348 months
Monthly Payment = $ 292,626.30 * [{(0.00416667)*(1.00416667)^348 }/{(1.00416667)^348-1}]
= $ 292,626.30 * [{(0.00416667)*(4.250291 }/{(4.250291-1}]
= $ 292,626.30 * (0.001770956/3.250291)
= $ 292,626.30 * 0.0054486
= $ 1,594.4059 or $ 1,594.41 (rounded off)
Loan Balance after 12 months = $ 292,626.30 * [{(1.00416667)^348 – (1.00416667)^12}/{(1.00416667)^348-1}]
= $ 292,626.30 * [(4.250291 – 1.0511619)/(4.250291 – 1)]
= $ 292,626.30 * [3.1991291/3.250291]
= $ 292,626.30 * 0.98425929 = $ 282,020.1532 or $ 282,020.15 (rounded off)
Year 3
Loan Amount = $ 282,020.15
Interest rate = 4% +2% = 6% per annum or 0.5% per month
Period of Loan = 348-12 = 336
Monthly Payment = $ 282,020.15 * [{(0.005)*(1.005)^336 }/{(1.005)^336-1}]
= $ 282,020.15 * [{(0.005*5.343142 }/{(5.343142-1}]
= $ 282,020.15 * (0.0026715712/4.343142)
= $ 282,020.15 * 0.00615124
= $ 1,734.7738 or $ 1,734.77 (rounded off)
Balance at the end of the year = $ 282,020.15 *[{(1.005)^336 – (1.005)^12}/{(1.005)^336-1}]
= $ 282,020.15 * {(5.343142 – 1.0616778)/(5.343142 – 1)}
= $ 282,020.15 * (4.2814642/4.343142)
= $ 282,020.15 * 0.9857988
= $ 278,015.1272 or $ 278,015.12 (rounded off)
Answer (b)
If the lender charges 3 points then the interest in the first year would 2%, second year 3% + 3% = 6% and third year = 4%+3% = 7%
Monthly interest year 1 = 0.0016667
Monthly interest in year 2 = 6%/12 = 0.005
Monthly interest in year 3 = 7%/12 = 0.0058333
Yield in year 1 = 1.0016667^12 – 1 = 0.020184 or 2.0184%
Yield in year 2 = 1.005^12 – 1 = 0.0616778 or 6.1678%
Yield in year 3 = 1.0058333^12 – 1 = 0.0722897 or 7.22897%
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