For a home buyer, a mortgage loan is a loan that you pay back over time (in theory). For a bank, a mortgage loan is an investment that earns a stream of interest income.
Let’s say there’s a beautiful house in historic Stony Point, Rockland County going for the bargain price of $120,000, that you can’t wait to buy while the market is down and interest rates are low. You borrow $100,000 at 5% interest compounded monthly, repayable over 20 years. From the bank’s perspective, the present value of their investment is $100,000, and it will produce a stream of revenue for 240 months.
To calculate the monthly payments you should expect, consider that you know the present value, the annual interest rate, and the total number of conversion periods. The formula linking these all these variables together with a regular stream of payments is:
PV = R x [{1 - (1 + r)-n} / r]
where PV is the present value, R is the regular payment amount, r is the interest rate per conversion period, and n is the total number of conversion periods.
First we have to convert the annual interest rate into the interest rate per conversion period. There are 12 conversion periods per year (one each month), since the interest is compounded monthly, so the monthly interest rate is r = 0.05/12 = 0.0042, or 0.4%.
Rearranging the formula to solve for R, the monthly payment amount, we get:
R = PV / [{1 - (1 + r)-n} / r]
Plugging in the numbers, we have:
R = $100,000 / [{1 - (1 + 0.0042)-240} / 0.0042]
and…
R = $662
Paying $662 per month for 240 months will eventually add up to $660 x 240 = $158,880. The borrower is able to purchase the house, but is ultimately paying $58,880 more for the house as a result of borrowing money. But this added cost is spread out over 20 years, which makes it more manageable. From the bank’s perspective, the loan brings in $158,880 – $100,000 = $58,880 in revenue over 20 years, a 58.88% return.
Maybe you should go for a bigger house. If you think you want to borrow $500,000, the only part of the formula that changes is the present value, PV. The other part, [{1 - (1 + r)-n} / r], known as the accumulation factor, won’t change unless the interest rate and number of conversion periods in your loan change. With r=0.0042 and n=240, the accumulation factor is equal to 151.
A $500,000 loan, also at 5% annual interest over 20 years, compounded monthly, would have regular payments of R = PV / accumulation_factor. Plugging in he numbers, we have R = $500,000 / 151 = $3,311. A $200,000 loan, with the same conditions, would have regular payments of $200,000 / 151 = $1,324.
Say you calculate that you can afford to make mortgage payments of $1,500 per month. What price house should you buy? At 5% annual interest over 20 years, compounded monthly, the accumulation factor is still 151. The formula is still R = PV / accumulation_factor. In this case, we want to solve for PV, so we rearrange the formula to be: PV = R x accumulation_factor. Plugging in the numbers, PV = $1,500 x 151 = $226,500.
What’s next? I feel like this one was just a rehash of interest
Yes, this is another variety of the interest math. I realized that I had failed to cover the formula for calculating present value based on interest and recurring payments in my last post (it only focused on calculating future value based on other variables). This post was meant to exhibit those mechanisms using mortgages as an example of a common use of such a formula.
what do you want to see next?
i want to see something about how debt is traded as a security
[...] an earlier post, I exposed the math of mortages in layman’s terms. Mortgages calculations are done using the formula for calculating Net [...]