Variable Adjustable Loan Calculations

You took out an adjustable rate mortgage of $200,000 for 15 years to purchase a home in South Dakota. The interest rate is fixed at 4% for the first 10 years after which it will increase by 1% every year until the 15th year. The bank states that the interest rate will not exceed 10% over the course of the loan.

Here is what we do know:

  • We know that there are 180 months total in the life of the loan (15 years*12 months/yr= 180 months)
  • 120 of those months are fixed since 10 years*12 months/yr
  • In years 11-15, the interest rate goes up 1% each year (or 12 months)
  • Our interest rate will not exceed 10%; however, in this problem the rate stops at 9% so let's put together a table.
Year(s) Month Span Annual Interest Rate
1-10 1-120 4%
11 120-132 5%
12 132-144 6%
13 144-156 7%
14 156-168 8%
15 168-180 9%

1.) What is your monthly payment for the first 10 years when the interest rate is fixed?
(a) $1,588.12
(b) $1,098.53
(c) $800.16
(d) $1,479.38

Let's break this down:

  • We need to use 180 months instead of 120 because the life of the loan must be covered in this calculation. Therefore N = 180
  • Remember that 4% is for the year (APR) so we also need to get the effective interest rate per month which is:

EIR= (.04/12 months) * 100 = .33%, which is our interest rate per month.

  • $200,000 is our loan amount
  • Our formula should look like this: A= 200,000(A/P, i= .33%, N= 180)

Using excel or a calculator, the answer is $1,479.38, which is D

2.) What is your monthly payment for year 11?
(a) $1,988.32
(b) $1,625.17
(c) $2,000.58
(d) $1,515.89

This problem takes careful consideration so let's break it down:
*We know that 120 months just went by, and that there are 60 months remaining. We can't just use $200,000 again because we've paid off part of the principle, so we need to calculate what the new principle balance is by using the formula:

(1)
\begin{align} \textstyle{Balance~Remaining}= Payment(P|A,i=\textstyle{effective~per~period}, N= \textstyle{Payments~remaining}) \end{align}

Payment ( or A) = 1479.38 (We use the answer from above because that was our monthly payment for 120 months)
Payments Remaining = 60 months
Effective rate per period = .33 (We still use the same rate from question 1)

Put it together in the formula: Balance Remaining= 1479.38(P/A, i =.33%, N = 60)

We get a principle balance of $80,328.96

Steps to solve for monthly payment in year 11:

  • Our P= 80,328.96
  • Remember our interest rate for year 11 is 5% due to the 1% annual (12 month) increase. Calculating our new EIR we get: (.05/12 months)*100 = .417%, which is i
  • Now recall that we have 60 monthly payments remaining. This will be our N for year 11. It's the same as what we did with year 1 when there was 180 payments remaining.

Using 80,328.96(A/P, i= .417%, N= 60) we get $1,515. 89, which is answer D.

3.) What is the balance remaining at the end of year 11?
(a) $58,311.12
(b) $45,888.36
(c) $71,625.44
(d) $65,824.52

Steps to calculate balance remaining:

  • We already know that our monthly payments for year 11 were $1,515.89, so this will go into the balance remaining formula above.
  • Use the same effective interest rate of .417 because we aren't in year 12.
  • To find N we subtract both the 120 months for years 1-10, and the 12 months for year 11 from 180 total months of the loan. N= 48.
  • Solving with our formula 1,515.89(P/A, i= .417, N= 48) and the answer is D, $65,824.52.

Explanation to find monthly payments: All we do each time there is a rate increase at the beginning of the next year is first find the balance remaining or the principal amount for the year we are solving for. Once we do that, then use the number of payments left and the effective interest rate per month and solve for A.
Completing the table:

Months Year(s) Yearly Rate Monthly Payment Balance Remaining
1-120 1-10 4% $1479.38 $80238.17
120-132 11 5% $1515.89 $65824.52
132-144 12 6% $1545.89 $50815
144-156 13 7% $1569.02 $35044.23
156-168 14 8% $1584.96 $18220.25
168-180 15 9% $1593.39 $0
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License