Typical Loan Overview
An Overview of How Amortized Loans Typically Work
Most people are familiar with some of the jargon associated with giving loans. People 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. Few people understand how loan payment amounts are 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 of the loan. That means if I borrow $1000 and my first payment is $100. After I make that payment I will no longer be borrowing $1000 because I will have repaid a portion of the original $1000 that I borrowed. Let’s say that $50.00 of my payment went to interest and $50 went to principal. If I repaid $50 of the $1000 borrowed, now I am only borrowing $950. Interest is now accruing on the $950 amount that I am still borrowing. Payments have to work out so that the borrower always pays the right amount of interest, but also pays exactly enough principal so that at the end of the loan term there will no longer be an outstanding balance on the loan.
How is this accomplished? The following mathematical formula is used to calculate the payment:
$$ P=\frac{r(PV)}{1-(1+r)^{-n}} $$
P = Payment Amount
r = Interest rate for the loan period
PV = Present value of the loan (loan amount)
n = Number of loan payments
Click here to download a spreadsheet file and see this formula in action. Here is an explanation that will use the example from the spreadsheet. In the example, we will calculate the payment amount for a $10,000 loan with a 12% interest rate given for 36 months. The stated interest rate 12%. This is a yearly rate, so we need to convert it to a period rate. Since payments will be made monthly and there are 12 months in a year, then 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. So the formula now is:
$$ P=\frac{100}{1-(1.01)^{-36}} $$
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. So our equation is now:
$$ P=\frac{100}{0.30107505037} $$
If we do the division, we will find that the payment equals 100/0.30107505037 or 332.14. So, $332.14 is our payment amount.
Once the payment is calculated, it is relatively simple to create the entire amortization schedule for the loan. Especially when you’re using our loan amortization software, which will calculate the schedule automatically and give you live updates on the loan. We know now that $332.14 should be paid in each period of the loan. We need to calculate what portion of the payment is equal to interest. In order 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. 0.01 x 10,000 = 100 so 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 gives $332.14 – $100.00 or $232.14. Now 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 – $232.14 or $9,767.86. We can continue this process through every period of the loan to get our full amortization schedule:
Loan Balance | ||||
Payment Period | Payment Amount | Interest Portion | Principal Portion | $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 is a spreadsheet version of this schedule so you can see it in action. That is how loans are typically calculated. This is the simplest 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.
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