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# Compound Interest

See how to find the final value/time of the investment in compound interest
(yearly, monthly/daily, continuously).
4 examples and their solutions.

## Compound Interest: Yearly

### Formula

A = A0(1 + r)t

A: final value
A0: initial value
r: rate of change
t: time
A compounded interest means
add the principle and the interest,
calculate the next interest,
and repeat this process.
The amount of money shows exponential growth.

### Example

\$1,000 investment is at a rate of 6% per year, compounded yearly. Find the estimated value of the investment 5 years later.
(Assume 1.065 = 1.338.)
Solution

### Example

\$1,000 investment is at a rate of 6% per year, compounded yearly. After how many years will the investment worth more than \$1,800?
(Assume log 1.8 = 0.255, log 1.06 = 0.025.)
Solution

## Compound Interest: Monthly, Daily

### Formula

A = A0 (1 + rn)tn

n: 12 (monthly), 365 (daily)
If the investment is compounded monthly or daily,
and if the unit of the rate r is '/year',
then use this formula.
The rate for each period is decreased.
rr/n
But the number of the period is increased.
ttn
Then the final value A increases.

### Example

\$1,000 investment is at a rate of 6% per year, compounded monthly. Find the estimated value of the investment 5 years later.
(Assume 1.00560 = 1.349.)
Solution

## Compound Interest: Continuously

### Formula

A = A0ert

A: final value
A0: initial value
e: constant number (= 2.71828...)
r: rate of change
t: time
Constant e

### Example

\$1,000 investment is at a rate of 6% per year, compounded continuously. Find the estimated value of the investment 5 years later.
(Assume e0.3 = 1.350.)
Solution