Don’t forget about the leap years!
The 60-day 6% or 90-day 4% rules
The 60-day 6% and 90-day 4% rules are two shortcut estimations of interest.
These rules are great because both 60 and 90 divide 360 evenly and form nice fractions when you divide them by 360. For instance, take a look:
360 60
6 1
360 90
4 and 1
= =
When you couple the fractions formed by dividing 60 or 90 by 360 with inter- est rates of 6% or 4% or 12%, you can do quick calculations of the interest (as long as the combination of the fraction and the percentage comes out to be a nice number).
For example, if you have a 60-day loan at 6%, you can multiply 1⁄6×6% = 1%.
A 90-day loan at 8% gives you 1⁄4×8% = 2%. The rest of the computations involving multiplying fractions and interest rates work the same, and, hope- fully, you can do them in your head.
As an example, use the 90-day 4% method to estimate the interest on a loan of $25,000.
This one is so easy. First, multiplying 4% by 1⁄4gives you 1%. Then multiplying
$25,000 by 0.01 gives you $250. Not too bad.
Looking into the future with present value
When you’re talking money, the expression present valuedescribes itself fairly well, but it’s also a bit confusing. Just remember this: What you’re really looking forward to is having a certain amount of money in the future when you talk about present value. In other words, if you have some goal for a par- ticular amount of money in the future, you want to know how much money to deposit now(in the present) so that the addition of simple interest will create the sum of money that you need.
For example, if you want $10,000 to buy a new piece of machinery 3 years from now, do you invest $8,000, $9,000, or $9,500 today? When these amounts of money earn interest, will the added interest bring you up to $10,000? Well, you can’t really answer this question yet, because I haven’t given you any interest rate. So this type of question will be a two-pronged one. One plan of attack will be to determine how much money to deposit if you know the inter- est rate, and the other approach will be to find an interest rate if you know the amount of money available for deposit.
In general, you can solve for the principal needed or the rate needed (or even the amount of time needed) after you know what your money goal is. The for- mulas used in the computations are all derived from the basic A= P(1 + rt) formula for the total amount of money resulting from simple interest. (You can find more on this basic formula in the earlier section, “Stepping it up a notch: Computing it all with one formula.”) Here are all the formulas that you’ll use when dealing with present value:
A P rt P
rt A r Pt
A P t
Pr A P
1 1
= + =
+
= -
= -
^ h
where Ais the total amount after adding simple interest, Pis the principal, ris the interest rate as a decimal, and tis the amount of time in years.
Now it’s time to try your hand at these formulas with a few examples.
Say that 3 years from now, you want to buy a new storage shed — and you need to have $27,000 set aside at that time. You intend to deposit as much money as is necessary right now so that when it earns simple interest at 53⁄4% it will be worth $27,000 in 3 years. How much do you have to deposit?
You know the total amount needed, A, the interest rate, r, and the amount of time, t. So, use the formula to solve for the principal needed:
.
$ ,
.
$ ,
$ ,
P rt
A
1 1 0 0575 3
27 000
1 1725 27 000
23 028
= . + =
+ ^ h=
As you can see, you need to set aside about $23,000 right now.
Here’s the same problem with a twist: Three years from now, you want to buy a new storage shed — and you need to have $27,000 set aside by that time.
You have $20,000 at hand, which means that you need to find someone who will give you a high enough simple interest rate so that the $20,000 will grow to $27,000 (by adding the interest) in 3 years’ time. What interest rate do you need?
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Chapter 9: Computing Simple and Compound Interest
This time you know the total amount, A, the principal, P, and the time, t.
Use the formula to solve for the rate needed, like this:
$ ,
$ , $ ,
$ ,
$ , .
r Pt A P
20 000 3 27 000 20 000
60 000 7 000
0 1167
= - = - .
^ h =
You need an 11.67% interest rate. Lots of luck finding that rate.
Getting to Know Compound Interest
The biggest difference between simple interest and compound interest is that simple interest is computed on the original principal only (see the earlier sec- tion, “Simply Delightful: Working with Simple Interest,” for details). In other words, no matter how long the transaction lasts, the interest rate multiplies only the initial principal amount.
Compound interest,on the other hand, multiplies the interest rate by the origi- nal principal plus any interest that’s accumulated during the time period of the transaction. As you can imagine, compound interest is pretty powerful.
After all, the principal keeps growing, making the interest amount increase as time passes. The bigger interest amount is added to the principal to make the interest amount even bigger.
Compound interest is the type of interest that financial institutions use. In this case, your money grows exponentially, so exponents (those cute little superscripted powers) are a part of the formula for determining how much money you have as a result of compound interest. Of course, for compound interest to have its full effect, you can’t remove the interest earned; com- pound interest is based on the premise that you leave the money alone and let it grow.
Figuring the amount of compound interest you’ve earned
Computing the total amount of money that results from applying compound interest takes a jazzy formula. Besides involving multiplication, addition, and division, this formula also requires you to work with exponents (and who but mathematicians like to do that?). The hardest parts of working with the formula are entering the values correctly into the formula and performing the operations in the right order.
The formula for computing compound interest is A P 1 nr
nt
= c + m
where Ais the total amount of money accumulated (principal plus interest), Pis the principal (the amount invested), ris the rate of interest (written in decimal form), nis the number of times each year that the compounding occurs, and tis the number of years.
Some scientific and graphing calculators allow you to type the numbers into the formula pretty much the same way you see them. But if you don’t have a calculator with all the bells and whistles, you still can do this problem correctly — as long as you perform the steps of the equation in the correct order. You need to apply the order of operations.(If you don’t know how the order of operations works, refer to Chapter 5, where I discuss it in detail.) For the compound interest formula, you need to perform the operations in the following order:
1. Determine the value of the exponent by multiplying n×t (the number of times compounded each year times the number of years).
2. Inside the parentheses, divide the interest rate, r, by the number of compoundings each year, n.
3. Add 1 to the answer in Step 2.
4. Raise the result from Step 3 to the power that you got in Step 1.
5. Multiply your answer from Step 4 by the principal, P.
Put these steps to use in the following examples.
How much money has accumulated in an account that’s earning interest at the rate of 4% compounded monthly if $10,000 was deposited 7 years ago?
Fill in the formula letting P= $10,000, r= 0.04, n= 12, and t= 7:
$ , .
A 10 000 1 12 0 04 12 7( )
= c + m
Now go through these steps (the order of operations) to find your answer:
1. Multiply 12 ×7 = 84.
This is the value of the exponent.
2. Divide 0.04 ÷ 12 = 0.0033333. . . .
Round this answer to five decimal places to get 0.00333.
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Chapter 9: Computing Simple and Compound Interest
3. Add 1 to the answer in Step 2.
After adding, you get 1.00333.
4. Raise 1.00333 to the 84th power.
So 1.0033384≈1.32214.
5. Multiply 1.32214 ×$10,000 = $13,221.40.
Now you have the total amount accumulated in 10 years.
Curious to know how this compares to investing the same amount of money for the same amount of time at the same rate using simple interest? Well, use the formula for the total amount, A= P(1 + rt), to find out. $10,000(1 + 0.04 ×7) =
$12,800. By using compound interest, you earn about $400 more than you would have with simple interest.
Noting the difference between effective and nominal rates
When you walk into a bank, you’re faced with an easel or other display that’s covered with decimal numbers. You see the latest on rates for car loans and home equity loans and the update on the effective interest rates for various