Question

You are offered a Price level adjusted mortgage (PLAM) with the following terms:

- Amount: $600,000

- Interest Rate: 5.60%

- Term: 30 years with monthly payments

- Expected Inflation: 5.00% per year

Please show what is the adjusted loan balance at the end of Year 1, what is the monthly payment in year 2, and what is the adjusted loan balance at the end of year 2?

Please show your work, thanks.

Answer 1

Loan amount = 600,000 | Rate = 5.60% | Term = 30 years with monthly payments | Inflation = 5% per year

First we will find the monthly payment for the first 12 payments in the first year.

We will use Annuity formula for the calculation.

PV of Annuity = (PMT / R) * (1 - (1+R)^-T)

Using the formula and putting PV = 600,000 | Rate = 5.60% / 12 | Term = 30 * 12 = 360

=> 600,000 = (PMT / (5.60% / 12))*(1 - (1+5.60% / 12)^-360)

PMT = (600,000 * (5.60% / 12)) / (1 - (1+5.60% / 12)^-360)

Solving the above equation, Monthly payment for 1st Year = 3,444.47

Now we will find the remaining Balance at the end of 1st Year.

First we will find the Future Value of 12 monthly payments made in the first year.

FV of annuity = (PMT / R)*((1+R)^T - 1)

FV of 12 monthly payments = (3,444.47 / (5.60% / 12))*((1+5.60%/12)¹² - 1)

Note: I have not rounded intermediate calculations as these calculations are done on Excel. If you use value of 3444.47 in the formula, you may get a different answer than my answer. However, I will show a table for the payments to clear the doubts and provide all the results that I have got.

Solving the equation, FV of 12 monthly Payments = 42,411.26

Now we will calculate FV of the loan amount after 12 payments

FV of loan amount = Loan amount * (1 + 5.60% / 12)¹²

FV of loan amount at the end of 1st year = 600,000 * (1 + 5.60% / 12)¹²

Solving the above formula, FV of Loan amount at the end of 1st year = 634,475.96

Remaining Principal at the end of 1st year = FV of Loan amount at the end of 1st year - FV of 12 monthly Payments

Remaining Principal after 12 payments = 634,475.96 - 42,411.26

Remaining Principal after 12 payments = 592,064.70

With inflation rate at 5% per year, the Loan balance will be adjusted with a 5% increase.

Adjusted Balance at the end of Year 1 = 592,064.70 * (1+5%)

Adjusted Balance at the end of Year 1 = 621,667.93

Now we will calculate new monthly payment for 2nd year using the Adjusted Balance at the end of Year 1.

PV of Annuity = (PMT / R) * (1 - (1+R)^-T)

PV = 621,667.93 | R = 5.60% / 12 | T = 29 * 12 = 348

=> 621,667.93 = (PMT / (5.60% / 12))*(1 - (1+5.60% / 12)^-348)

=> PMT = (621,667.93 * (5.60% / 12)) / (1 - (1+5.60% / 12)^-348)

Solving the above formula, Monthly Payment for 2nd year = $ 3,616.70

Now to find the Remaining principal at the end of Year 2, we will calculate FV of 12 monthly payments in 2nd year and FV of Adjusted balance after Year 2.

FV of 12 monthly payments in Year 2 = (3,616.70 / (5.60% / 12))*((1+5.60% / 12)¹² - 1)

Solving the above expression, FV of 12 monthly payments in Year 2 = 44,531.83

FV of Adjusted balance at the end of Year 2 = 621,667.93 * (1+5.60% / 12)¹²

Solving the above expression, FV of Adjusted Balance at the end of Year 2 = 657,388.92

Remaining Principal at the end of Year 2 = FV of Adjusted Balance at the end of Year 2 - FV of 12 monthly payments in Year 2

Remaining Principal at the end of Year 2 = 657,388.92 - 44,531.83

Remaining Principal at the end of Year 2 = 612,857.10

With inflation rate of 5%, Ending balance for Year 2 will be adjusted with 5% increase.

Adjusted balance at the end of Year 2 = 612,857.10 * (1+5%)

Adjusted balance at the end of Year 2 = $ 643,499.95

Below are the results from calculations done on Excel using PMT and FV function: