Pey Soon has taken out a 20-year, $150,000 mortgage with monthly payments (made at the end of each month) at a stated mortgage rate of 6.8% per year compounded semi-annually. If she makes each payment on time, what will be the mortgage principal remaining after 10 years?

Please answer with the solution

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15

To calculate the mortgage principal remaining after 10 years, we need to use the formula for the present value of an ordinary annuity.

The present value of an ordinary annuity formula is:

PV = P × [1 - (1+r)^(-n)] / r

Where:
PV = Present Value (mortgage principal remaining)
P = Amount of each monthly payment
r = Monthly interest rate
n = Number of payments

Given:
Loan amount (mortgage principal) = $150,000
Loan term = 20 years or 240 months
Monthly interest rate = (6.8% / 2) / 100 = 0.034

To find out the monthly payment amount, we'll use the formula for the present value of an ordinary annuity:

150,000 = P × [1 - (1+0.034)^(-240)] / 0.034

Now, solve for P:

P = 150,000 × 0.034 / [1 - (1+0.034)^(-240)]

P ≈ $909.37 (approx.)

We rounded the result to two decimal places.

Now we know the monthly payment amount, P, is approximately $909.37.

To calculate the mortgage principal remaining after 10 years (120 months), we'll use the formula:

PV = P × [1 - (1+r)^(-n)] / r

PV = $909.37 × [1 - (1+0.034)^(-120)] / 0.034

PV ≈ $131,989.99 (approx.)

So, after 10 years, the mortgage principal remaining will be approximately $131,989.99.

To find the mortgage principal remaining after 10 years, we can use the formula for the future value of an ordinary annuity.

The formula for the future value of an ordinary annuity is:

FV = P * ((1 + r/n)^(n*t) - 1) / (r/n)

Where:
FV = Future Value of the annuity
P = Payment
r = Interest rate per year (as a decimal)
n = Number of compounding periods per year
t = Number of years

In this case, the mortgage payment is the annuity. The mortgage payment is $150,000, and it is made monthly for 20 years. Therefore, the number of compounding periods per year is 12 (monthly payments), and the number of years is 10.

First, let's calculate the interest rate per compounding period:
The annual interest rate is 6.8%, so the interest rate per period is 6.8% / 2 = 0.068 / 2 = 0.034.

Now, we can substitute the values into the formula to find the future value (mortgage principal remaining) after 10 years:

FV = $150,000 * ((1 + 0.034/12)^(12*10) - 1) / (0.034/12)

Calculating this expression, we get:

FV ≈ $104,691.51

Therefore, the mortgage principal remaining after 10 years will be approximately $104,691.51.