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.
FormulaCompounded Yearly
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
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.
FormulaCompounded Monthly, Daily
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
$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)
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)
FormulaCompounded Continuously
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
$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.
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.