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Discussion
Say you want to make a big purchase, like a new phone or car, but you don't have all the cash on hand to buy it upfront.
A loan can bridge the gap between "I wish I had enough money to buy that" and "I can get it today." When you take out a loan (from a bank, financial institution, or sometimes another person), you're essentially borrowing money with the promise to pay it back over time.
Why might a bank want to give out a loan? What do they get out of it?
Banks lend because they make money from the extra charge called interest.
By the time you've paid off your entire loan, you've paid a little bit extra to the bank in interest. This extra "fee" is essentially the lender's reward for loaning you the cash now instead of spending it themselves. Plus, they take on the risk that you might not pay it back.
That extra set of interest payments (on top of the original balance owed) is why banks agree to give out loans. Otherwise, it wouldn't make sense for them to risk putting money in someone else's hands!
A loan is borrowed money that you agree to repay over time plus an extra fee called interest. Payments are made at a certain interval, called the period, until the full balance is paid off at the end of the term.
Here's an example of a very simple loan structure:
Alice lends Bob $1000, which Bob is to repay over five years, so Bob will have to pay $200 per year. Bob will also pay additional interest of $50 per year. The following table shows the payments Bob makes each year, and the amount he still owes Alice, which we call the outstanding balance.
Notice that Alice has made a profit of $250 over the course of the loan.
But how do we decide how much interest is paid?
The interest will be bigger for a bigger loan. The amount you pay should be related to the outstanding balance. As you pay off the loan, the amount of the lender's money you have is decreasing.
One logical solution would be for the interest to be calculated as a percentage of the outstanding balance. That would mean the interest payments get smaller with each period.
Discussion
In reality, it makes more sense for the payments to be the same each period.
Why might borrowers and lenders both prefer a fixed payment amount per period?
For borrowers, having a fixed payment amount each period provides predictability and stability. You always know exactly how much you need to pay, which makes it easier to budget and plan your finances. This consistency can help reduce stress and uncertainty, since you do not have to worry about your payment suddenly increasing if the bank's interest rates change.
For lenders, a fixed payment schedule creates a reliable and steady stream of income. This regularity makes it easier for banks or other lenders to manage their own cash flow and financial planning.
This structure benefits both parties by making the loan process more transparent and manageable for everyone.
In reality, what we do is calculate a fixed payment amount such that the loan is fully repaid, including interest, at the end of the term.
This graph shows the outstanding loan balance as a function of time. Try adjusting the sliders to see how they affect repayments. What do you notice?
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You may have noticed that higher interest rate loans are more costly, and as a result take longer to pay off. Additionally, increasing the payments per year and the size of said payments both help pay off loans faster.
When you take out a loan in real life, you typically want a low interest rate and to pay it off as soon as you can, to avoid paying anything more than you have to.
You don't need to know the formula to calculate this payment amount. On the IB, you will always use the TVM Solver to find that fixed payment amount (PMT).
Discussion
Below is a table showing the monthly payment on a $1,000 loan, assuming the stated annual interest rates are compounded monthly across three common loan terms.
What trends do you notice?
Longer terms equates to smaller monthly payments. Stretching a loan over more time will typically drive the monthly cost down, but keep in mind the total interest paid is much higher.
Higher rates hurt more on long loans. At 7%, a 1-year loan only rises from $85.15 to $86.53 (a relatively small jump). But on a 30-year loan, 7% interest pushes payments up from $4.77 to $6.65—a 40% increase—because interest compounds over so many more periods.
Your monthly payment is a function of the interest rate, loan length, and amount borrowed. These tables give you the foundation to understand how each factor drives your budget and total cost.
Checkpoint
Imagine you take out a loan for $3,000 at a 5.5% interest rate compounded monthly for 30 years.
What would your monthly payment be?
Select the correct option
Positive & Negative Cash Flows (TVM)
Whenever you use the Finance App (TVM Solver) on your calculator, it's critical that you enter and interpret the signs correctly:
When you receive money from a bank or savings account, that value is positive, because you're gaining money.
When you send money to a bank, that value is negative, because you're losing money.
In the context of loans:
PV = Positive (you're borrowing money now)
PMT is Negative (because money is leaving your pocket each time you make a payment)
FV is Negative or 0 (you've repaid some or all of the debt)
Using TVM Solver (Calculator) - Loans
In IB, loans are paid off at the end of a number of periods (N) and have an annual interest rate (I%), an initial balance (PV), a fixed payment (PMT), and an outstanding balance (FV). Payments per year and compounds per year typically occur at the same frequency (P/Y, C/Y).
You can use the TVM solver with loans to find any of those variables if you know all the others.
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