Compound Interest Formula

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COMPOUND INTEREST FORMULA

Jump right to the exercises!

If you put $1000 in a bank that offers 5% simple annual interest,
then your interest is added in only after one year has passed.
If you withdraw your savings one day before the year is over,
you'll have only your original $1000 (drat).
However, if you wait the extra day (making one full year),
then you'll have $1000 + 5%($1000) = $1050 .

Wouldn't it be better to add in 112th of your annual interest after the first month,
giving a slightly greater amount to earn interest for the next month?
Or, why not add in 1365th of your annual interest after the first day,
giving a slightly greater amount to earn interest for the next day?

When you add in interest at regular intervals, this is called compound interest.
With compound interest, you are EARNING INTEREST ON YOUR INTEREST,
not only on your original principal!
The Compound Interest Formula is a non-recursive formula that is convenient for working with compound interest situations,
and is the subject of this section.

Before deriving the compound interest formula, let's go back to that $1000 with 5% interest, and see how much benefit is gained from:

compounding monthly (adding in interest each month);
there are 12 months in one year, so the monthly interest rate is 112 (0.05)=0.05 12
compounding weekly (adding in interest each week);
there are 52 weeks in one year, so the weekly interest rate is 152 (0.05)=0.05 52
compounding daily (adding in interest each day);
there are 365 days in one year, so the daily interest rate is 1365 (0.05)=0.05 365

EFFECTS OF COMPOUNDING
$1000 initial deposit, 5% interest rate
(all units are DOLLARS, rounded to the nearest cent)

TIME	Simple Annual Interest	Compounding Monthly	Compounding Weekly	Compounding daily
1 day	1000	1000	1000	1000+ 0.05365( 1000)=1000.14
2 days	1000	1000	1000	1000.14+ 0.05365( 1000.14)=1000.28
1 week	1000	1000	1000+ 0.0552( 1000)=1000.96	1000.96
2 weeks	1000	1000	1000.96+ 0.0552( 1000.96)=1001.92	1001.92
1 month	1000	1000+ 0.0512( 1000)=1004.17	1004.17	1004.18
2 months	1000	1004.17+ 0.0512( 1004.17)=1008.35	1008.36	1008.37
6 months	1000	1025.26	1025.30	1025.31
1 year	1000 + 0.05(1000) = 1050.00	1051.16	1051.25	1051.27
2 years	1050 + 0.05(1050) = 1102.50	1104.94	1105.12	1105.16
10 years	1628.89	1647.01	1648.33	1648.66

The added savings from earning interest on interest is perhaps not quite as much as you'd hope.
For example, in one year you'd earn an additional $1051.27 - $1050 = $1.27 over simple annual interest, by adding in interest daily.
In ten years, you'd earn an additional $1648.66 - $1628.89 = $19.77 over simple annual interest, by adding in interest daily.
As the length of time and the amount of money invested increase, though, the savings do go up—and every little bit helps!

DERIVATION OF THE COMPOUND INTEREST FORMULA

The compound interest formula results from using variables to represent a general investing situation,
writing down several computations, and seeing a pattern emerge.

In a nutshell, you're going to invest P dollars at annual interest rate r ,
add in interest n times per year, and see how much you have after t years.

Here are the details:

You are investing P dollars.
You will start the clock at the moment you make this initial investment;
i.e., let t=0 correspond to the initial investment time.
Let r denote the (simple) annual interest rate for this investment.
Express r as a decimal. For example, 5% corresponds to r=0.05 .
Assume that interest is added in at regular intervals, n

n=12

n=52

n=365

compounding periods

Let A represent the total amount of money (principal plus interest) after t years.

The table below shows the accumulations after various numbers of compounding periods:

after this time...	you'll have this much money...	NOTE:
1 compounding period	P+r n(P) =P(1+ rn) =P(1+ rn )1	factor out P
2 compounding periods	P(1+ rn) +rn ⋅P(1+ rn)=P(1+ rn) (1+r n)=P (1+r n) 2	factor out P(1+ rn)
3 compounding periods	P(1+ rn )2+ rn⋅ P(1+r n) 2=P( 1+r n)2 (1+ rn)= P(1+r n) 3	factor out P(1+ rn )2
...	...	...
n compounding periods = 1 year	P(1+r n) n	notice the emerging pattern
2n compounding periods = 2 years	P(1+r n) 2n
tn compounding periods = t years	A=P(1 +rn )tn =P(1+ rn )nt	the compound interest formula!

Thus, we have:

THE COMPOUND INTEREST FORMULA
(see above for descriptions of variables)

A=P(1 +rn )nt

USING THE COMPOUND INTEREST FORMULA

QUESTION:   Suppose $2500 is invested at 3% annual interest, compounded daily, for seven years.
How much money will you have?

SOLUTION:   I recommend doing a crude computation first to get a "ballpark" figure.
This catches a lot of calculator mistakes.
(3%)($2500) = $75;    7($75) = $525;   you'll have more than $2500 + $525 = $3025.

P=2500 ,   r=0.03 ,   n=365 ,   t=7

A=2500(1 +0.03365 )(365⋅7) =3084.17    (compare with $3025; believable!)
Remember to put the exponent computation inside parentheses when you key this into your calculator!

You'll have $3,084.17 .

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.