Primer on Loan Interest and Principal Payments

Aside from paying taxes, one of the most important things you should understand as an adult is how loans work. While it's always a good idea to live by the mantra to "never spend more than what you have" when it comes to finances, reality doesn't always play out that way. Even in an inflated housing market, living in the US often means needing a car, and that could mean an auto loan for a lot of people. I'm pretty sure I don't have to tell you guys what a student loan is either. Therefore, let's break down how loan payments work.

First, let's talk about what interest is and how it works. When you deposit money in a savings account in a bank, you're essentially loaning the bank your money. For that, the bank pays you interest on what you save with them every month. Interest is essentially a fee paid to the loaner for borrowing money. At the most basic level, interest is calculated as follows:

$LaTeX: P_t=P_0\left(1+r\right)^t$ $P_t=P_0\left(1+r\right)^t$

r represents the annual interest rate, in decimal form. Also known as APY (annual percentage yield).
t represents the number of years the money was loaned out or stored.
P₀ represents the initial amount loaned out.
P_t represents the final amount after t number of years have passed, assuming no money was withdrawn in that entire time span.

Interest rates for savings accounts vary from bank to bank, but generally hover close to whatever the Federal Reserve sets. Let's assume 5%, a number that was fairly common in 2024. Let's assume you deposit $10k when you open the new savings account and withdraw no money from this account for two years.

$LaTeX: P_2=10000\left(1+0.05\right)^2=11025$ $P_2=10000\left(1+0.05\right)^2=11025$

And so in two years' time, assuming that entire time you never withdrew any money this account since the original $10k deposit, you'll have earned $1025 in interest payments from the bank, making the bank balance at the end of two years $11025.00. This is known as Compounding Interest, where the first interest payment becomes part of the new principal for next year's interest calculation, and thus money grows exponentially.

Year 1: $LaTeX: P_1=10000\left(1+0.05\right)^1=10500$ $P_1=10000\left(1+0.05\right)^1=10500$
Year 2: $LaTeX: P_2=10500\left(1+0.05\right)^1=11025$ $P_2=10500\left(1+0.05\right)^1=11025$

Interest payments can be calculated yearly, monthly, daily, or even continuously. APY represents the annual percentage yield and are often the values advertised, but even then, most interest on loan payments or even your bank statements are calculated monthly. When these varying periods of payments are taken into account, the formula changes to the following:

$LaTeX: P_{nt}=P_0\left(1+\frac{r}{n}\right)^{nt}$ $P_{nt}=P_0\left(1+\frac{r}{n}\right)^{nt}$

n represents the number of payments in a year. For example, if a bank pays you interest monthly over your principal amount, then n = 12, as there are 12 months in a year. If interest is paid daily, then n = 365. (Continuously uses a slightly different formula, we'll skip that for now.) Sometimes, daily interest may use n = 360 (30 days per month model) or even n = 366 for leap years. However, n = 365 is the most common assumption used for calculating daily interest.

You can see that if we were to include n = 12 on the previous example,

$LaTeX: P_{24}=10000\left(1+\frac{0.05}{12}\right)^{24}=11049.41$ $P_{24}=10000\left(1+\frac{0.05}{12}\right)^{24}=11049.41$

And so, after 24 months (2 years since the original deposit), we have earned $1049.41 in interest. As you can see, getting paid interest monthly using the same APY nets you slightly more interest over the same amount of time. This is not a mistake, as this is the result of the interest becoming part of your new principal every month as opposed to every year, and so your money grows slightly faster as a result. Most banks advertise interest rates using APY and pay you interest monthly. To determine the interest payment in any one given month, you need to know the current balance and apply the interest rate to that amount alone.

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Next, let's go into the opposite scenario, when the bank loans you money so that you can buy a car, buy a house, or start a business. Unlike savings accounts, this is a fixed amount of money loaned to you, and this is a formal contract where you're expected to pay back the original loan amount plus interest within a certain amount of time via regular monthly payments.

Assuming these are fixed rate loans (meaning the interest rate does not change for the life of the loan), then what happens is that every month, part of your monthly payment goes to paying off the expected interest and the rest after interest payment goes towards paying off the principal balance. When the principal balance hits $0, your loan is paid off. Because of interest, you'll have paid way more than the original principal balance to buy whatever it is you just purchased, so hopefully it was worth it in the long run. Loan rates are often determined using APR (annual percentage rate, similar but slightly different from APY; still represented as r), and the amount of time you ask for the loan to be paid off will determine your monthly due (M, where n is fixed at 12). This is calculated as follows:

$M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{P_0\left(\frac{r}{n}\right)\left(1+\frac{r}{n}\right)^{nt}}{\left(1+\frac{r}{n}\right)^{nt}-1}$ $M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{P_0\left(\frac{r}{n}\right)\left(1+\frac{r}{n}\right)^{nt}}{\left(1+\frac{r}{n}\right)^{nt}-1}$

It looks scary, but this formula is actually not that complicated. What this formula is doing is that it's consistently taking into account the interest accrued as you're actively paying off the loan principal. By setting a fixed timeline that you want to pay off the entire debt, this formula determines the fixed payment you need to make every month (or whatever interval you're using) to pay off the debt. All variables retain the same definitions as previous formulas shown. The idea is that by paying this amount regularly, you'll pay off the interest and principal faster than the interest can accrue, thus eventually paying off the debt. This formula is sometimes known as Capital Recovery, as it's meant to help determine the regular cost of something so that revenue expectations can be set.

For example, say I take out a loan to buy a house (mortgage). I'm borrowing $500k. This will be a 30-year fixed rate mortgage with an interest rate set at 6%. I will pay off my mortgage monthly.

$M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{500000\left(\frac{0.06}{12}\right)}{1-\left(1+\frac{0.06}{12}\right)^{-12\cdot30}}=2997.75$ $M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{500000\left(\frac{0.06}{12}\right)}{1-\left(1+\frac{0.06}{12}\right)^{-12\cdot30}}=2997.75$

And so, this means that these loan terms will translate to needing to pay $2997.75 to the bank every month for 30 years straight to fully pay off this mortgage.

You'll notice that if you do the math by multiplying out this amount for every month over 30 years, the total you actually paid out is way, way higher than the original borrowed amount of $500k. (We paid $1,079,190.75 over the course of 30 years to pay off this loan! $500k for the original principal and the remaining $579,190.75 in interest fees.) This is primarily because we needed a very long time to pay off the loan, and so the interest accrued reflects that. If you want to avoid paying the bank a ton of interest, you'll want a shorter loan term, such as 15 years, but such a change means a much higher monthly payment to compensate. It causes more principal to be paid off sooner to avoid accruing interest any longer than desired. The idea is that a house is supposed to last far longer than the life of the loan and appreciates in value over time (meaning its market value goes up), so all of this interest is still worth paying.

As another example, let's quickly look at a car loan. Let's assume I borrow $40k to buy a car. Interest rate is set at 5%, and the loan is to be paid back in 5 years. Once again, I'll make monthly payments.

$M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{40000\left(\frac{0.05}{12}\right)}{1-\left(1+\frac{0.05}{12}\right)^{-12\cdot5}}=754.85$ $M=\frac{P_0\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}}=\frac{40000\left(\frac{0.05}{12}\right)}{1-\left(1+\frac{0.05}{12}\right)^{-12\cdot5}}=754.85$

Like before, we can multiply this amount out over 5 years and see that a grand total of $45,291.00 was paid out (rounding error for pennies can be ignored...the actual amount according to the calculator would be $45,290.96). Thus, given that the original loan was $40k, $5,291.00 was paid in interest overall.

Have you ever wondered why there are so many student loan debt horror stories of people making those debt payments for decades and still have debt left over? It's because they're not paying enough every month to pay off all interest AND reduce the principal. It can easily become a vicious cycle/debt trap if you don't pay enough monthly towards your debt. This usually isn't a problem for people with fixed rate loans, but adjustable rate loans exist, and those monthly payments can snap up quickly when interest rates spike for reasons outside of any one person's control.

In case you're wondering, unpaid credit card debt from the month prior accrues interest daily after the bill's original due date passes. If you can't pay off your entire month's bill, check your credit card's APR so you know what you're getting yourself into! Cash advances on credit cards accrue interest daily from the moment they're taken out, so be extra careful about using those!

Another side note: interest paid on certain types of loans (namely student, business, and mortgage) are tax deductible, meaning interest on these types of loans will subtract from your normal income for the year and thus you end up paying less in taxes for that year. In other words, this is how the government incentivizes people to take out loans, and hopefully boost the economy in the long run. (This policy is not without significant flaws and biases though.)

-------------- You can stop here and proceed with the first problem of Homework 9 before reading on --------------

Which leads us to our next topic: amortization.

Amortization is basically breaking down each loan payment such that you can see exactly how much goes into paying off interest and paying off principal each month. First, use the formula given in the previous section to determine the monthly amount owed. We'll use the exact same home loan example as before.

Next, let's start from the beginning of our payment cycle. The current interest owed on any given month is determined as follows:

$LaTeX: I_i=P_{i-1}\left(\frac{r}{n}\right)$ $I_i=P_{i-1}\left(\frac{r}{n}\right)$

Where I represents the current amount of interest due and P represents the current principal remaining. P_i-1 represents the previous month's principal, used to determine the current interest payment. P_i represents the new principal after the monthly payment is applied. Lower case i represents the current month of payment since taking out the loan. Lower case r and n retain the same definitions as before.

We started out with $500k. On the first month of payment (i = 1), you get the following:

$LaTeX: I_1=P_0\left(\frac{r}{n}\right)=500000\left(\frac{0.06}{12}\right)=2500$ $I_1=P_0\left(\frac{r}{n}\right)=500000\left(\frac{0.06}{12}\right)=2500$

And so, for our first monthly payment, we owe $2500.00 in interest. That's as big as interest will ever get for one month. Our monthly payment was previously determined to be $2997.75. Simple subtraction determines that, for the first month, we pay $497.75 towards the principal.

$LaTeX: P_i=P_{i-1}-\left(M-I_i\right)$ $P_i=P_{i-1}-\left(M-I_i\right)$

So, for this example,

$LaTeX: P_1=P_0-\left(M-I_1\right)=500000-\left(2997.75-2500.00\right)=499502.25$ $P_1=P_0-\left(M-I_1\right)=500000-\left(2997.75-2500.00\right)=499502.25$

And so, after the first month, $499502.25 of the principal remains to be paid off. Doesn't seem like much of a dent, but because the principal decreased, that means we pay less in interest the following month. Because our monthly payments are consistent, more money goes into paying off principal in every subsequent month. Eventually, it'll all be paid off after 30 years as planned. Let's check out what the second month looks like.

$LaTeX: I_2=P_1\left(\frac{r}{n}\right)=499502.25\left(\frac{0.06}{12}\right)=2497.51$ $I_2=P_1\left(\frac{r}{n}\right)=499502.25\left(\frac{0.06}{12}\right)=2497.51$

$LaTeX: P_2=P_1-\left(M-I_2\right)=499502.25-\left(2997.75-2497.51\right)=499002.01$ $P_2=P_1-\left(M-I_2\right)=499502.25-\left(2997.75-2497.51\right)=499002.01$

We paid off another $500.24 towards the principal during the second month's payment. Now we rinse and repeat for 360 months straight (30 years), and our loan will be fully paid off. Laying this all out in a table (abbreviated for simplicity) we can trace the timeline of how our loan is fully paid off. This process applies to any kind of loan payment. Simply adjust r and n accordingly depending on your loan payment terms like before.

$500k, 30-year loan @ 6% fixed interest = $2997.75 paid monthly:

Amortization Table
Payment # (Month)	Principal Balance ($)	Interest Due ($)	Principal Paid ($)	Principal Remaining ($)
1	500000.00	2500.00	497.75	499502.25
2	499502.25	2497.51	500.24	499002.01
3	499002.01	2495.01	502.74	498499.26
...	...	...	...	...
358	8904.07	44.52	2953.23	5950.84
359	5950.84	29.75	2968.00	2982.84
360	2982.84	14.91	2982.84	0

If you did your math correctly, you'll see that, assuming you made consistent payments every month over the life of the loan (not including additional principal payments, junk fees, or even late fees) the remaining principal will be exactly 0 on the final month.