Compound Interest: Final Value

How to find the final value of the compound interest investment (yearly, monthly, continuously): 3 formulas, 3 examples, and their solutions.

Formula: Yearly Compounded Interest

Formula

A Compounded interest means
add the principle and the interest,
calculate the next interest,
and repeating this process.

The amount of money shows exponential growth.

To find the final value
of a compound interest investment,
use the exponential growth formula:
A = A0(1 + r)t.

Use this formula when
the unit of the rate r [per year]
and the unit of the compound period [yearly]
are the same.

Example 1

Example

Solution

The initial value of the investment is $1,000.
So A0 = $1000.

The investment is at a rate of 6% per year.
So r = 0.06/year.

Write the unit [per year].

The final value is the value 5 [years] later.
And the unit of the rate is [per year].

So write 5 years in [years]:
t = 5 years.

A0 = 1000
r = 0.06/year
t = 5 years
The investment is compounded yearly.

The unit of the rate r [per year]
and the unit of the compound period [yearly]
are the same.

Then the final value A is
A = 1000(1 + 0.06)5.

(1 + 0.06) = 1.06

It says
assume 1.065 = 1.338.

So 1000⋅1.065 = 1000⋅1.338.

1000⋅1.338 = 1338

The initial value A0 is in $.

So the final value A is $1,338.

So $1,338 is the answer.

Formula: Monthly, Daily Compounded Interest

Formula

If the units of the rate r and t (year)
and the unit of the compound period (monthly or daily)
are different,
use this formula:
A = A0(1 + r/n)t⋅n.

Change the units of r and t
to the unit of the compound period:
r (per year) → r/n (per month, day)
t (years) → t⋅n (months, days).

n: [12 months/year], [365 days/year], ...

Example 2

Example

Solution

The initial value of the investment is $1,000.
So A0 = $1000.

The investment is at a rate of 6% per year.
So r = 0.06/year.

The final value is the value 5 [years] later.
And the unit of the rate is [per year].

So write 5 years in [years]:
t = 5 years.

A0 = 1000
r = 0.06/year
t = 5 years
The investment is compounded monthly.

The units of r and t [year]
and the unit of the compound period [monthly]
are different.

Then the final value A is
A = 1000(1 + 0.06/12)5⋅12.

r: 0.06 per year = 0.06/12 per month
t: 5 years = 5⋅12 months

0.06/12 = 0.01/2

5⋅12 = 60

0.01/2 = 0.005

(1 + 0.005) = 1.005

It says
assume 1.00560 = 1.349.

So 1000⋅1.00560 = 1000⋅1.349.

1000⋅1.349 = 1349

The initial value A0 is in $.

So the final value A is $1,349.

So $1,349 is the answer.

Compare this answer
to the answer of the previous example
(compounded yearly: $1,338).

As the compound period gets shorter
(yearly → monthly),
the total investment gets bigger
($1,388 → $1,349)

Formula: Continuously Compounded Interest

Formula

As the compound period gets shorter,
the total investment gets bigger.

So, to maximize the investment,
the compound period should be minimized (r/n → 0)
and the number of period should be maximized (t⋅n → ∞).

This is the case of
continuously compounded interest.

To find the final value
of a continuous compound interest investment,
use the continuous exponential growth formula:
A = A0ert.

Example 3

Example

Solution

The initial value of the investment is $1,000.
So A0 = $1000.

The investment is at a rate of 6% per year.
So r = 0.06/year.

The final value is the value 5 [years] later.
And the unit of the rate is [per year].

So write 5 years in [years]:
t = 5 years.

A0 = 1000
r = 0.06/year
t = 5 years
The investment is compounded continuously.

Then the final value A is
A = 1000⋅e0.06⋅5.

0.06⋅5 = 0.3

It says
assume e0.3 = 1.350.

So 1000⋅e0.3 = 1000⋅1.350.

1000⋅1.350 = 1350

The initial value A0 is in $.

So the final value A is $1,350.

So $1,350 is the answer.

Compare this answer
to the answers of the previous examples
(compounded yearly: $1,338, compounded monthly: $1,349).

As you can see,
the total investment is maximized
by continous compounding.