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Continuously Compounding Interest

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This lesson applies a formula for finding the value of an account when interest compounds continuously.

Tutorial

- Compound Interest
- Continuously Compounding Interest
- Solving for Account Balance
- Solving for Growth Time

**Compound Interest**

Bank and investment accounts appreciate in value according to compounding interest. For many, the account has an annual percentage rate (APR) that is compounded periodically throughout the year. This means that a portion of the APR is applied each time the interest is compounded. For example, if interest is compounded quarterly, one-fourth of the interest is applied each quarter, for a total of four times per year.

We can represent compounding interest using the following formula:

where,

- A is the account balance
- P is the principal (initial starting balance)
- r is the annual percentage rate (expressed as a decimal)
- n is the number of times per year interest is compounded
- t is time in years

**Continuously Compounding Interest**

When interest is compounding continuously, we can think of the variable n (the number of times per year interest is compounded) as being infinitely large. How do we make sense of our formula if n is an infinitely large number? Let's isolate the part of our formula that relies on the variable n, and consider when n approaches infinity:

For these purposes, it isn't so important that we understand this notation, but this denotes the limit of the expression as n approaches infinity. The limit represents a value that the expression approaches, but will not exceed.

As it turns out, as n gets infinitely larger, the expression simplifies to e^{rt}. Recall that e is a mathematical constant, approximately equal to 2.718282. Having this in mind, we can adjust our interest formula for continuously compounding interest:

where

- A is the account balance
- P is the principal (initial starting balance)
- e is the mathematical constant, approximately equal to 2.718282
- r is the annual percentage rate (expressed as a decimal)
- t is time in years

**Solving for Account Balance**

Now that we have a formula to use when interest is compounded continuously, let's use it to solve some problems with account balances. Consider this scenario:

An account has an initial balance of $1150.00, which has an APR of 2% that is compounded continuously. What is the balance of the account after 3 years, assuming no additional deposits or withdrawals are made?

Let's identify values we can plug in for the variables in our formula:

- P = 1150
- r = 0.02
- t = 3

If your calculator has the e button, use it to get the most accurate answer. If not, use the approximation 2.718282 and use as many decimal digits as possible during your calculations, and round to the nearest cent at the very end only. If you round too often during the calculations, you may get the same dollar amount, but the cents will likely be off.

**Solving for Growth Time**

We can also use the formula for continuously compounding interest to solve for the time it takes for the account to reach a certain value. Consider the following scenario:

A savings account has a balance of $4,500. The interest rate of the account is 3.5% annually, which is compounded continuously. How long will take for the account to reach a value of $7,000, assuming no additional deposits or withdrawals are made?

Again, let's match up given information with variables in our formula

- P = 4500
- r = 0.035
- A = 7000

Here, we want to solve for t, so this remains our unknown; and we want to solve for t when the account has a balance of $7,000, so our A value is 7000. So the equation for this situation is:

How can we solve for the variable t? One thing to note right away is that the variable t is an exponent here. The exponent 0.035t is applied *only* to the mathematical constant e, not the principal balance, P. So our first step should always be to divide both side of the equation by the principal.

Now we are ready to undo the exponent in order to isolate t. This requires applying a logarithm to undo the exponent. The important thing here is to consider the base of the log. Since the base of the exponential is e, we should use the natural log to solve for t.