The Basics
Whenever I borrow anything, the lender and I must agree on how much rent I need to pay, and how often I need to pay it. Borrowing money is no different. The money I borrow at the beginning of a loan is called the principal, and the rent I pay for borrowing it is called interest. Unlike rents on other items, the interest I pay when I borrow money is not a fixed amount. Rather it's (usually) a fixed interest rate--a percentage that tells me how much of the amount borrowed I need to pay in rent. For example, I might borrow $500 and agree to pay 2% of the borrowed amount every month. Then, after the first month, I would owe 2% × $500 = $10 in interest. (Recall that 2% = 0.02.)Note: The 2% interest rate in my example is what I call the periodic interest rate--the percent charged every period when interest is collected/assessed. By convention, we usually speak in terms of annual interest rates--the percent due each year. Therefore, the first step of any interest calculation is usually to convert an annual interest rate into the more useful periodic interest rate. In my example, the annual interest is 24%, the period is 1 month, and therefore the periodic interest rate I need for my calculations is 24% ÷ 12 = 2%.
The One Law Of Interest Payments
Every period, the borrower is charged an amount of interest equal to the periodic interest rate (r) multiplied by the borrowed amount (b).Initially, the borrowed amount b is simply the principal (the amount of money initially borrowed). Surprisingly, this is the only law you ever need to calculate interest payments.
Let's return to the example where I borrow $500 and I'm charged 2% interest each month. We already determined (using The One Law Of Interest Payments) that I owe 2% × $500 = $10 in interest for the first month. But suppose I pay only $7. Since I owed $10 but paid only $7, I've effectively borrowed another $3. Now I've borrowed a total of $503, so The One Law Of Interest Payments tells me that next month I'll owe 2% × $503 = $10.06 in interest. The extra 6 cents is really interest on the interest I didn't pay. Now, $7 is an unusual payment, so let's look at some more common payment plans.
Payment Plan #1: Simple Interest
In this plan, I pay the full amount of interest every period, and at the end of the loan I also pay back the principal in full. Again, suppose I've borrowed $500, and I'm charged 2% interest each month, which works out to be $10. In this plan, I pay exactly $10. The next month, since I've still borrowed $500, I again owe $10. By paying the full interest every month, the amount I've borrowed doesn't change, which keeps the math simple. At the end of the loan, I pay back the $500. If the loan lasts for 1 year, then I've paid 12 × $10 = $120 in interest. In general, I'll pay a total of t × r × p in interest, where t is the number of periods in the loan, r is the periodic interest rate, and p is the principal.Payment Plan #2: Compound Interest
In this plan, I'm charged interest each period (as always) but I don't pay any of it. At the end of the loan, I pay back the principal in full, along with all the accumulated interest. So, this time when I'm charged $10 interest on my $500 loan, I pay $0. That means I've effectively borrowed an additional $10, so I've now borrowed a total of $510. Next month, The One Law Of Interest Payments tells me I'll owe 2% × $510 = $10.20 in interest. Again, I pay $0, so I've now borrowed $510 + $10.20 = $520.20, and next month I'll be charged 2% × $520.20 = $10.404, and again I'll pay $0, and so on. But how much will I have to pay back at the end of the loan?Originally, I borrowed $500.
After 1 period, I've borrowed $500 + 0.02 × $500 = $500 × 1.02.
After 2 periods, I've borrowed ($500 × 1.02) × 1.02 = $500 × (1.02)2.
After 3 periods, I've borrowed $500 × (1.02)3.
Each period, the amount I owe multiplies by 1.02 (and is therefore growing exponentially). At the end of 1 year, I'll owe $500 × (1.02)12 = $634.12, where $134.12 of that is interest ($14.12 more than I paid using the simple interest payment plan.)
In general, after t periods, I'll owe a total of p(1 + r)t.
With compound interest, my debt can accumulate quite quickly. If I borrow that $500 for 10 years, then the simple interest payment plan would cost me $1,200 in interest, while the compound interest payment plan would cost me $4882.58 in interest. That's why people talk of "the power of compound interest."
Payment Plan #3: Overpaying
Suppose I need $100,000 to buy a house. A loan to buy a house is called a mortgage, but it's essentially like any other loan. My bank would be crazy to lend me that money using a simple or compound interest payment plan. Let's see why.Suppose the bank offers to loan me the money for 30 years at 6% interest, and interest is assessed every month. That makes the periodic interest rate r = 6% ÷ 12 = 0.5% (or 0.005).
I would owe 0.5% × $100,000 = $500 in interest for the first month. If I pay exactly $500 each month, I end up with a simple interest plan, in which I still owe $100,000 at the end of the loan. If I pay less than $500 each month, I'll owe even more at the end of the loan (the principal plus accumulated interest). In the most extreme case, I pay $0 each month, resulting in a compound interest plan in which I owe over $600,000 at the end of the loan. The bank is never going to trust me to be able to make a single large payment of $100,000 or $600,000 at the end of the loan.
To reduce the amount I'll owe at the end of the loan, my monthly payment will need to be more than $500. That way, every month I'll pay off all of the interest and some of the principal.
Suppose I pay $550 each month. Again, The One Law Of Interest Payments guides us. After one month, I owe $500 in interest, but I pay $550, which covers all of the interest and also pays off $50 of the loan (the principal). That means I've effectively borrowed $50 less, so I've now borrowed "only" $99,950. Since I've borrowed less, I'll now be charged less interest next month: 0.5% × $99,950 = $499.75. Again, I pay $550, which now pays off all of the interest plus $50.25 of the principal. The interest calculations for the first 3 months of this loan are summarized in the following table.
Month
|
Borrowed
|
Interest
|
Principal Paid Off
|
1
|
$100,000
|
0.5% × $100,000
= $500
|
$550 - $500
= $50
|
2
|
$100,000 - $50
= $99,950
|
0.5% × $99,950
= $499.75
|
$550 - $499.75
= $50.25
|
3
|
$99,950 - $50.25
= $99,899.75
|
0.5% × $99,899.75
= $499.49875
|
$550 - $499.49875
= $50.50125
|
Notice that each month, I've borrowed less, so I owe less interest, so I pay off more of the principal, so I've borrowed even less, and so on. OK, so how much will I owe at the end of 30 years? To answer that, we'll derive the general formula for interest payments.
The General Formula
So far, we've seen three payment plans, and in each one, my regular payment x is a constant. These plans are summarized in the following table.
Payment Plan
|
Regular Payment
|
Simple Interest
|
x = pr
|
Compound Interest
|
x = 0
|
Overpaying
|
x > pr
|
Clearly, the more I pay during the loan (larger x), the less I'll owe at the end of the loan. In this section, we'll derive a general formula for how much I owe at the end of the loan, as a function of x (and p, r, and t).
Initially, I borrow p. After one period, The One Law Of Interest Payments tells me I'll owe a total of p(1 + r), which includes both the amount I borrowed (p) and one period of interest (pr). I make a payment of x, which means now I've only borrowed p(1 + r) - x. Notice that we multiplied the amount borrowed by (1 + r) and then subtracted x. This same process will repeat every period, so after my next payment, I will multiply p(1 + r) - x itself by (1 + r) and then subtract x, resulting in the following pattern:
Originally, I borrowed p.
After 1 period, I've borrowed p(1 + r) - x.
After 2 periods, I've borrowed p(1 + r)2 - x(1 + r) - x.
After 3 periods, I've borrowed p(1 + r)3 - x(1 + r)2 - x(1 + r) - x.
After t periods, I've borrowed p(1 + r)t - x(1 + r)t-1 - x(1 + r)t-2 ... - x(1 + r) - x
Recognizing that (1 + r)t-1 + (1 + r)t-2 ... + (1 + r) + 1 is a geometric series whose sum is
[(1 + r)t- 1] ÷ r, our formula simplifies to
p(1 + r)t - x[(1 + r)t - 1] ÷ r
which tells me how much I owe after borrowing p at a periodic interest rate of r, and making payments of x for t periods. This is The General Formula.
Using The General Formula
If this formula is really so general, we should be able to use it to calculate both simple and compound interest, and indeed we can.
In a compound interest payment plan, x = 0 (since I don't pay interest until the end of the loan). Plugging x = 0 into the general formula, the ugly righthand term immediately disappears, and we are left with p(1 + r)t, which we recognize as the formula for compound interest.
In a simple interest payment plan, x = pr (since I pay off the interest each period). When we plug x = pr into the general formula, most of the formula cancels out, and we're left with just p. And this makes perfect sense, because no matter how long I borrow for, I've always paid off all interest, and I therefore only owe the principal p at the end of the loan.
In my mortgage example, p = $100,000, r = 0.005, t = 360 (the number of months in 30 years), and x = $550. Plugging these values into the formula, I get
($100,000)(1.005)360 - ($550)[(1.005)360 - 1] ÷ 0.005 = $49,774.25
I still owe almost $50,000 at the end of the loan. Apparently, I should have made higher monthly payments. But how high?
Payment Plan #3 (Revised): Overpaying (to pay off the principal exactly)
I want to determine the regular payment x that will completely pay off the full loan (principal and interest) in exactly 30 years. In this plan, at the end of the loan, I will owe $0, meaning that the general formula had better give me $0.p(1 + r)t - x[(1 + r)t - 1] ÷ r = 0
By solving this equation for x, we can determine the correct monthly payment:
x = pr(1 + r)t ÷ [(1 + r)t - 1]
This is the mortgage payment formula you'll find elsewhere (in some equivalent form).
In my mortgage example, p = $100,000, r = 0.005, and t = 360. Plugging these values into the formula, I get
x = ($100,000)(0.005)(1.005)360 ÷ [(1.005)360 - 1] = $599.55
Therefore, if the bank offers me a 30-year loan of $100,000 with a fixed (annual) interest rate of 6%, my monthly mortgage payment will be $599.55. After 360 payments of $599.55, I will have paid off the loan in full, having paid a total of $215,838 for the privilege of borrowing $100,000 for 30 years. But, of course, now we know there's nothing arbitrary about that $215,838. It's the logical consequence of applying The One Law Of Interest Payments.
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