### Problem

A house is purchased for 130,000 using a 25 year mortgage with an interest rate of 6%. How much of the monthly payments will go towards the repayment of the principal in the first 2 years? Mortgage payments are made at the end of each month.

### Solution

This problem is solved in two steps.

- Calculate the monthly payments on the mortgage.
- Using the payments, calculate the balance on the mortgage after 24 months, and subtract this from the original principal (130,000).

#### Calculate the Monthly Payments

The value of the mortgage is the cost of the house 130,000 (PV). This mortgage is repaid using monthly installments, 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%/12 or 0.5% per month (i)

As the payments (Pmt) are made at the end of each period, and we know the present value, the problem is solved using the present value of an ordinary annuity formula as follows:

PV = 130,000 n = 300 i = 0.5% PV = Pmt x (1 - 1 / (1 + i)^{n}) / i

Rearranging the formula to solve for Pmt

Pmt = PV x i / (1 - 1 / (1 + i)^{n}) Pmt = 130,000 x 0.5% / (1 - 1 / (1 + 0.5%)^{300}) Pmt = 837.59

#### Calculate the Mortgage Balance

After 24 (n) months the balance on the mortgage is given by the future value of the initial principal 130,000 (PV), less the future value of the payments 837.59 (Pmt). The first part is solved using the future value of a lump sum formula, and the second part is solved using the future value of an annuity formula.

PV = 130,000 n = 24 i = 0.5% Pmt = 837.59 Balance = Future value of principal - future value of payments Balance = PV x (1 + i)^{n}- Pmt x ( (1 + i)^{n}- 1 ) / i Balance = 130,000 x (1 + 0.5%)^{24}- 837.59 x ( (1 + 0.5%)^{24}- 1 ) / 0.5% Balance = 125,229.17

From this we can calculate the reduction in the principal as 130,000 – 125,229.17 = 4,770.83, which must be the amount of principal repaid.

### Explanation

For the mortgage balance 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.

The solution to step one of this problem simply uses the present value of an annuity formula to find the 300 payments which, at a discount rate of 0.5%, will give a present value of 130,000.

For step two, the mortgage can be viewed as two separate cash flows, the first is the principal increasing as interest is added, and the second is the regular payments which can be treated as an annuity reducing the balance. Providing the payments are greater than the interest then the balance will decline.

To find the balance the future value of the principal is found using the future value of a lump sum formula, and the future value of the payments is found using the future value of an annuity formula. The net of the two figures will provide a value for the balance on the mortgage.

Finally, the reduction in the balance on the mortgage must equal the amount of principal paid by the payments.

### AN8 Annuity Examples

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