The other day, JLP at All Things Financial put up a post demonstrating how to calculate the present value of an annuity. In it, he demonstrated how you can use the present value formula to calculate how much you’d need to invest today at a given rate of interest in order to generate a fixed income for a number of periods.

**On an amortized loan, payments are applied to both principal and interest (first towards interest and then principal).****The balance on an amortized loan goes down over time. In fact, the term “amortized” comes from the Latin word for death (i.e. the loan balance “dies off” over time).****The outstanding balance of an amortized loan at any point in time will be the Present Value of the remaining payments on the loan.**

Assume you take out a $200,000 mortgage with a 30-year term, monthly payments, and an annual interest rate of 6%. (1) What’s the monthly payment on the loan? (2) What will the outstanding balance on the loan be in a year’s time (i.e. after you’ve made 12 payments)? (3) How much of your total payments in the first year will go towards principal (i.e. reduction of the loan balance), and (4) how much towards interest?

**1) Calculating the monthly payment on the loan:**

= (1/0.005) – (1/0.005) x (1/1.005)^360 = 166.7916.

NOTE: in the notation abve, “r” is the interest rate *per month *(since it’s a monthly loan), “n” is the number of periods covered by the loan (360 in this case, since it’s a 30-year loan and it’s paid monthly) and the “^n” means “raised to the nth power”.

When calculating a loan payment, you divide the present value factor into the $200,000 initial balance on the loan. This give us a payment of 200,000/166.7916 = $1,190.10 per month.

Note: this is for principal and interest only. Your actual payment may also include property taxes, insurance, and PMI, but those are separate from the “basic” payment on the loan.

**2) Calculating the the outstanding balance on the loan be in one year’s time:**

PV factor = (1/0.005) – (1/0.005) x (1/1.005)^348

= 164.7438

*divided* the factor into the outstanding balance of $200,000 to get the monthly payment. Now, we’re going the opposite way – we know the payment, and we want to determine the outstanding balance. So, in this case we *multiply* the payment of $1,190.10 by the new present value factor to get an outstanding balance of 1,190.10 x 164.7438 = $197,543.98.

Note: after *two years *have gone by, the calculation would be based on “n” being 336 (28 years times 12 months per year). After two years in the mortgage, the present value factor would be 162.5688, and the outstanding balance would be $194,935.47.

**3) Calculating how much of the total payments in the first year goes towards principal:**

We began the first year with a balance of $200,000 on the loan and ended it with a balance of $197,543.98. The difference between these two numbers ($200,000 - $197,543.98) is the amount we paid towards principal during the first year. In other words, we paid off $2,456.02.

*per month, *so paying off only $2,456 in a year’s time seems like a rip-off, right? Keep in mind that you’ve borrowed $200,000, and the interest on the loan is 0.5% per month (i.e. 6% per year, divided by 12). So, the interest owed on the loan in the first month is $1,000. This means that in the first month of the loan, since you paid “only” $1,199.10, the first $1,000 goes towards paying the interest on the $200,000, leaving $199.10 to go towards paying down the loan. In the second month of the loan, the amount of the payment going towards principal is a tiny bit more than $199.10, since your outstanding balance is a bit less than $200,000 (remember, you paid down the loan by $199.10 in the first month). So, looked at it this way, the $2,456 paid off in the first year doesn’t seem too far off the mark. It’s a bit more than the $199.10 you paid off in the first month times 12.

**4) Calculating how much of the total payments in the first year goes towards interest:**

Just as a “seat of the pants” check to make sure that this is a reasonable figure, take the annual interest rate of 6% and multiply it times the initial balance of $200,000, and you’d get $12,000. The total interest paid over the first year’s time is actually a bit less than $12,000 because each payment decreases the outstanding balance a bit (remember – it’s an “amortized” loan, so the balance “dies off” over time). Therefore, since the interest owed each month is based on the outstanding balance, it goes down a bit each month, and the amount paid towards principal goes up.

So there you have it. Of course, most people don’t make loan calculations by hand – you’d use either a financial calculator or spreadsheet to do the calculations. But now (hopefully), you can see what goes on “behind the scenes) in a loan.

In a future post, I’ll show how to use the built-in functions of a spreadsheet like Microsoft Excel to do the same calculations in a much more efficient fashion.

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