Amortized Loan Overview
An Overview of How Amortized Loans Typically Work
Most people are familiar with some of the jargon associated with giving loans, and might understand what principal and interest are in a general sense. Principal is money given as a loan, interest is additional money assessed as the cost of getting a loan. People also understand that when you pay back a typical loan, the amount of each payment is usually the same and some portion of the payment is interest and another portion of the payment is principal. Fewer people, however, understand how those payment amounts are actually calculated in order to make sure the right amount of interest accrues and all the principal is repaid.
During the life of a loan, interest only accrues on the outstanding portion. Suppose I borrow $1000 and make a first payment of $100. After I make that payment, I will no longer be borrowing $1000 because I will have repaid a portion of it. Let’s say that $50 of my payment went to interest and $50 went to principal. If I repaid $50 of the $1000 borrowed, my remaining principal is only $950. Interest will accrue on that $950, so the next payment will still be $100, but a smaller portion of it will go toward interest.
For instance, here's an amortization schedule for a small loan, like you might see when financing a phone or laptop. The principal balance was $1000, and they had a 30% interest rate. Four quarterly payments were made, each at $297.87.
Period | Payment Amount | Interest | Principal | Loan Balance |
— | — | — | — | $1,000 |
1 | $297.87 | $73.97 | $223.90 | $776.10 |
2 | $297.87 | $56.77 | $241.10 | $535.00 |
3 | $297.87 | $40.02 | $257.85 | $277.15 |
4 | $297.87 | $20.73 | $277.14 | $0.01 |
Payments have to work out so that the borrower always pays the right amount of interest while also paying exactly enough principal so that there will no longer be an outstanding balance at the end of the loan term. In almost all cases, borrowers also want to make regular payments, giving the same sum of money at routine intervals.
The following mathematical formula is used to calculate the payment:
$$ P=\frac{r(PV)}{1-(1+r)^{-n}} $$
- P = Payment Amount, or what the borrower will pay each term.
- r = Interest rate for the loan period. This is the yearly rate divided by the number of periods in a year; for monthly payments, you'd divide the annual rate by 12. The rate is also expressed as a decimal. A 6% period rate would be written as 0.06, not 6.
- PV = Present value of the loan. Because this formula is calculated before any payments, this number is the starting principal of the loan.
- n = The total number of scheduled payments
If you'd like to plug it into a simple calculator, this downloadable spreadsheet file will let you see the formula in action. The file calculates the payment amount for a $10,000 loan with a 12% interest rate given for 36 months. That 12% rate is a annual, so we need to convert it to a period rate. Since payments will be made monthly and there are 12 months in a year, the rate for 1 month will be our 12% divided by 12 months in a year. So, the monthly rate is 1%. When we enter the rate into our formula, we will use the decimal equivalent, or 0.01. Here is the formula with our data entered:
$$ P=\frac{0.01(10000)}{1-(1+0.01)^{-36}} $$
Now, do some simple math. 0.01 x 10000 = 100, so the numerator of our fraction is 100. In the denominator, we will do the addition inside the parentheses first. 1 + 0.01 = 1.01. After this step, our formula looks like this:
$$ P=\frac{100}{1-(1.01)^{-36}} $$
Now, raise 1.01 to the -36 power. This will give 1/(1.01)^-36 or 1/1.43076878359. This fraction is equal to 0.69892494962. Now, we will do the subtraction of 1 – 0.69892494962. This gives 0.30107505037. Our equation is now:
$$ P=\frac{100}{0.30107505037} $$
If we do the division of 100/0.30107505037, we will find that the payment equals 332.14. Therefore, $332.14 is our payment amount.
Creating a Amortization Schedule
Once the payment is calculated, it is relatively simple to create the entire amortization schedule for the loan. Our loan amortization software makes this even simpler, and will calculate the schedule automatically as well as give you live updates.
Going back to our example, we now know that $332.14 should be paid in each period of the loan. However, we still need to calculate what portion of the payment is equal to interest. To do that, we will simply multiply the outstanding loan balance by the period interest rate.
In the first period of our loan, the outstanding loan balance is equal to the total loan amount of $10,000 because no principal has been repaid yet. That means that the interest portion of the payment equals our period rate, which we calculated as 0.01 multiplied by the total loan amount of $10,000. If we multiply 0.01 by 10,000, we get 100. The interest portion of our first payment is $100.
Now, we can calculate the principal portion of the payment by subtracting the interest portion from the total payment amount. This is written as $332.14 minus $100.00, or $232.14. Using this number, we can calculate the new loan balance by subtracting the principal portion from the current loan balance. In our case, that is $10,000.00 minus $232.14, or $9,767.86. We can continue this process through every period of the loan to get our full amortization schedule.
Full Amortization Schedule
Here's the full amortization schedule:
Period | Payment Amount | Interest | Principal | Loan Balance |
— | — | — | — | $10,000.00 |
1 | $332.14 | $100.00 | $232.14 | $9,767.86 |
2 | $332.14 | $97.68 | $234.46 | $9,533.39 |
3 | $332.14 | $95.33 | $236.81 | $9,296.58 |
4 | $332.14 | $92.97 | $239.18 | $9,057.41 |
5 | $332.14 | $90.57 | $241.57 | $8,815.84 |
6 | $332.14 | $88.16 | $243.98 | $8,571.85 |
7 | $332.14 | $85.72 | $246.42 | $8,325.43 |
8 | $332.14 | $83.25 | $248.89 | $8,076.54 |
9 | $332.14 | $80.77 | $251.38 | $7,825.16 |
10 | $332.14 | $78.25 | $253.89 | $7,571.27 |
11 | $332.14 | $75.71 | $256.43 | $7,314.84 |
12 | $332.14 | $73.15 | $258.99 | $7,055.84 |
13 | $332.14 | $70.56 | $261.58 | $6,794.26 |
14 | $332.14 | $67.94 | $264.20 | $6,530.06 |
15 | $332.14 | $65.30 | $266.84 | $6,263.22 |
16 | $332.14 | $62.63 | $269.51 | $5,993.71 |
17 | $332.14 | $59.94 | $272.21 | $5,721.50 |
18 | $332.14 | $57.21 | $274.93 | $5,446.57 |
19 | $332.14 | $54.47 | $277.68 | $5,168.89 |
20 | $332.14 | $51.69 | $280.45 | $4,888.44 |
21 | $332.14 | $48.88 | $283.26 | $4,605.18 |
22 | $332.14 | $46.05 | $286.09 | $4,319.09 |
23 | $332.14 | $43.19 | $288.95 | $4,030.14 |
24 | $332.14 | $40.30 | $291.84 | $3,738.30 |
25 | $332.14 | $37.38 | $294.76 | $3,443.54 |
26 | $332.14 | $34.44 | $297.71 | $3,145.83 |
27 | $332.14 | $31.46 | $300.68 | $2,845.14 |
28 | $332.14 | $28.45 | $303.69 | $2,541.45 |
29 | $332.14 | $25.41 | $306.73 | $2,234.72 |
30 | $332.14 | $22.35 | $309.80 | $1,924.93 |
31 | $332.14 | $19.25 | $312.89 | $1,612.03 |
32 | $332.14 | $16.12 | $316.02 | $1,296.01 |
33 | $332.14 | $12.96 | $319.18 | $976.83 |
34 | $332.14 | $9.77 | $322.37 | $654.45 |
35 | $332.14 | $6.54 | $325.60 | $328.85 |
36 | $332.14 | $3.29 | $328.85 | $0.00 |
Here's a spreadsheet version of this schedule so you can see it in action. That is how loans are typically calculated. This is a very simple scenario and doesn’t include calculating interest when the first payment period is longer or shorter than a regular period, what to do if payments aren’t made exactly on schedule or in the right amounts, etc.