### Problem

You have a deposit saved for a house of 10,000 and decide that the maximum mortgage payment you can make is 800 per month at the end of each month. The annual mortgage interest rate is 6.2% and you want to finish paying the mortgage off after 25 years. What is the maximum house price you can afford?

### Solution

The monthly mortgage payments represent an annuity, and the present value of the annuity is the value of the home mortgage loan amount you can afford. As the mortgage loan payments are made at the end of each month, the problem is solved using the preset value of an annuity formula.

The mortgage loan amount (PV) is repaid using monthly installments (Pmt), so one period is one month, and the term of the loan is 25 years or 300 months (n). The interest rate has been quoted as an annual rate so the periodic (monthly) rate is 6.2%/12 per month (i).

The mortgage loan amount is calculated as follows:

Mortgage loan amount = Pmt x (1 - 1 / (1 + i)^{n}) / i Pmt = mortgage payment = 800 per month n = number of months = 25 x 12 = 300 months i = nominal rate = 6.2%/12 per month Mortgage loan amount = 800 x (1 - 1 / (1 + 6.2%/12)^{300}) / (6.2%/12) Mortgage loan amount = 121,843.10

Finally, to find the maximum house price, the mortgage loan amount needs to be added to the initial deposit.

Maximum house price = Mortgage loan amount + Initial deposit Maximum house price = 121,843.10 + 10,000.00 Maximum house price = 131,843.10

### Explanation

For the mortgage loan amount to be cleared at the end of 300 payments, the present value of the payments must be equal to the present value of the mortgage loan amount.

The solution to this problem simply uses the present value of an annuity formula to find the present value of the 300 payments of 800 per month, at a discount rate of 6.2%/12.

### Mortgage Loan Amount Examples

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