[slavik81]

You don't need to know any of that to understand a loan. That's somebody's math homework, not an actual financial transaction.

When you get a loan, they tell you the interest as annual percentage rate (APR). That annualized rate plugs right into the simple interest formula they teach you in high school math class. You just calculate each year one at a time, plugging in the amount you owe at the start of the year and you'll get how much is owed at the end of the year.

If you make payments or if the rate changes during the year, you can use the same formula to calculate what you owe at the time of change/payment, and continue on from there. Using the effective annualized rate for less than a year will give you an answer that's slightly off, but it will be very close.

You'd have a very difficult time using differential equations for a real loan. That question has the buyer making an infinite number of infinitesimally small car payments, because a monthly installment would make it a bad differential equations homework problem. The sensible way to solve most real-world problems is just to calculate the result iteratively using basic multiplication and addition.

[ransom1538]

I suppose? The above mentioned method is really inaccurate. Student interest is calculated daily.

Here is a top search result for calculating student loan interest. They suggest not to do a yearly approach like you suggest. It even mentions doing a daily compounded function. I added a [LEAP] where the author left the reader hanging.

--- To understand how compound interest works, let’s look at an example. Consider a Direct loan with a $10,000 balance and a 4.45% interest rate.

First, you figure your daily interest rate by dividing 4.45% by 365 to get 0.012%. On $10,000, that works out $1.20. That $1.20 is added to your loan balance, bringing it to $10,001.20. That’s your new balance, and when interest is compounded the following day, you’ll pay interest on that total amount.

[LEAP]

By the end of the year, you’re looking at paying $455.02 in interest, rather than the $445 you’d pay if your interest was compounded just once a year instead of daily.

https://studentloanhero.com/featured/how-student-loan-intere...

[slavik81]

That's not using the APR percentage, which would be 4.55% for that loan. Had they used that, they would have gotten the correct value.

> By the end of the year, you’re looking at paying $455.02 in interest, rather than the $445 you’d pay if your interest was compounded just once a year instead of daily.

They used the wrong percentage and got $445/year. That's off by $10/year. Still, it's close enough to use for yearly financial planning and for judging if the loan is worthwhile. It should be sufficient to prevent any surprises like the article author's.