The power of compound interest
4. Divide the amount of money, A, by the denominator
Say, for example, that you want to have $400,000 in 5 years to purchase a new piece of equipment. How much do you need to deposit today at 63⁄4% interest, compounded monthly, so there’s $400,000 in your account when you need it?
To solve, you just need to plug your numbers into the formula for present value, like so:
. ,
.
, $ , .
P
1 12
0 0675 400 000
1 005625 400 000
285 690 83
( )
12 5 60
= +
= =
c m ^ h
After doing the math, you can see that you need to deposit about $286,000 right now. But look how much it grows in just 5 years!
Determining How Variable Changes Affect Money Accumulation
Many different banks are available these days, and there are just as many options, offers, and arrangements that can be made with your money at the different institutions. For instance, do you choose the free checking or the unlimited account transfers? Decisions, decisions. I can’t help you in choos- ing one bank over another, but I can show you just how much the interest rate, the number of compoundings, or the time invested actually affect the end result. The rest is for you to sort through and decide.
As you may know from reading this chapter, the compound interest formula has four different variables,or things that can change. The variables are
P, the principal or amount of money invested r, the interest rate, entered as a decimal
n, the number of times each year that the interest is compounded t, the time in number of years
If you adjust any of the four variables, you change the output or end result.
In general, increasing any of the variable numbers increases the output. But how much is the increase, and which variable has the greatest impact on an increase? I show you in the following sections.
Comparing rate increases to increased compounding
Say that you have $10,000 and need to determine where to invest that money to earn the greatest amount of interest. If you invest your money at 4% inter- est, compounded annually, you’d have $10,000(1 + 0.04) = $10,400 at the end of one year. (Refer to the earlier section, “Getting to Know Compound Interest,”
for the details on how to do this computation.) Knowing that, your next ques- tion is this: Will you do better to invest your $10,000 with an institution that increases the interest rate by one-quarter of a percent, or should you stick with the same interest rate and go someplace that compounds interest quarterly?
To answer this question, you need to determine the total amount of money in an account if $10,000 is deposited for 1 year at 41⁄4% compounded annually.
Then you simply compare that total amount with the result of $10,000 being
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Chapter 9: Computing Simple and Compound Interest
deposited in an account that earns interest at 4% compounded quarterly.
Here’s what the computations look like:
One year at 41⁄4%, compounded annually: A $10 000 1, . $ , 1
0 0425 10 425
( ) 1 1
= c + m =
One year at 4%, compounded quarterly: A $10 000 1, . $ , . 4
0 04 10 406 04
( ) 4 1
= c + m =
Clearly, the increase in the interest rate has the greater impact on the total amount after a year. A 1⁄4% increase may not be realistic, however. In banking circles, a quarter of a percent is big money.
Imagine that the total amount in an account where a deposit of $10,000 is earning 4% interest compounded quarterly is $10,406.04 at the end of 1 year.
What are the effects of increasing the interest rate by one-hundredth of a per- cent at a time and applying annual compounding?
The following list shows the end results of interest rates of 4.01%, 4.02%, and so on, with annual compounding:
A= $10,000(1 + 0.0401)1(1)= $10,401 A= $10,000(1 + 0.0402)1(1)= $10,402 A= $10,000(1 + 0.0403)1(1)= $10,403 A= $10,000(1 + 0.0404)1(1)= $10,404 A= $10,000(1 + 0.0405)1(1)= $10,405 A= $10,000(1 + 0.0406)1(1)= $10,406
It only takes a six-hundredths of a percent increase in the interest rate for the end result to equal the effect of compounding quarterly instead of annually.
Comparing rate increases to increases in time
Maybe you’re stuck with one type of interest compounding. For instance, maybe you only deal with institutions that compound quarterly. Now you want to compare the effect of increasing the interest rate with the effect of increasing the amount of time you leave the money in the account. If you have your money invested at 4% compounded quarterly, would you be better off increasing the rate of interest to 41⁄4% or leaving it in the original account for an extra quarter of a year?
Your math for solving this problem looks like this:
One year at 41⁄4% compounded quarterly:
$ , . $ , .
A 10 000 1 4
0 0425 10 431 82
( ) 4 1
= c + m =
One and a quarter years at 4% compounded quarterly:
$ , . $ , .
A 10 000 1 4
0 04 10 510 10
( . ) 4 1 25
= c + m =
As you can see, the increase in interest rate resulted in a smaller total than the increase in time.
Using the information from the previous example, determine what rate of interest, compounded quarterly, earns the same amount of interest on
$10,000 invested for one and a quarter years.
Here’s how to get started:
$ , . $ ,
x 10 000 1 x
4
0 04 10 000 1
4 4%, comp. quarterly,1
4
1 year %, comp. quarterly,1 year
( . ) ( )
4 1 25 4 1
=
+ = +
c m c m
Next, divide each side by $10,000, and then simplify the exponents and the terms that are inside the parentheses, like so:
. x
1 01 1 4
5 4
= +
^ h c m
Now finish by computing the power on the left, taking the fourth root of each side, and then solving for the value of x:
. . . . .
x x x x x 1 05101005 1
4 1 05101005 1
4 1 012515586 1
4 0 012515586
4 0 050062344
4
4
4 4
= +
= +
= +
=
= c
c m
m
So, as you can see, you need an interest rate of about 5% to equal the effect of leaving the money in for 3 extra months.
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Chapter 9: Computing Simple and Compound Interest
The problems involving comparing interest rates and compounding and time show you that many variations are available when investing your money or borrowing money. Find some interest charts and look to see how much your money earns at each rate and amount of time. Sometimes, though, it’s more a matter of convenience and service (rather than a few extra dollars here and there) that draws you to a particular institution — and that’s okay.