Today we're going to talk about continuously compounded interest. So we're going to start by deriving the formula for continuously compounded interest by looking at the compound interest formula. And then we'll do some examples using that formula.
So let's start by reviewing compound interest. Suppose I have $1,000 that I invest into a bank account that earns 3% annual interest rate. We can look at the value of the account using a formula for compound interest.
Compound interest is when banks apply the interest more than one time per year, so maybe monthly or daily. And that formula for compound interest looks like this. A is going to be the value of the account after a certain number of time, a certain amount of time.
P is the principal, or the initial value, of the account. r is going to be our annual interest rate. n this is the number of times compounded in the year-- so how many times do they apply the interest in a year.
And finally t is our time expressed in years. So we can use this formula to answer questions about savings accounts that have interest that's compounded-- interest that's compounded.
So let's look at our formula for continuously compounding interest. And we're going to start with the formula we already know, compound interest. So some banks or financial institutions will compound interest continuously, meaning an infinite number of times per year. So our value for n is going to be approaching infinity.
And as n approaches infinity, this part of our formula is going to be approaching a mathematical constant, e, raised to the power of r, where r is still our annual interest rate. And e is a mathematical constant that is equal to approximately 2.718281.
So some calculators will have a button on it with this e symbol, or the letter-- lowercase letter e on it. And so you can use that button. If you don't have that button, then you should use this approximation.
So going back to our formula, the rest of our variables or terms will come down. And so now we have a formula for continuously compounding interest. And in this formula, A is still our value of the accountant. P is still our initial deposit, or the initial amount in our account. r is our annual interest rate, and t is our time in years.
So let's do an example using our formula for continuously compounded interest. Let's say I invest $1,000 into a bank account where the interest rate is 3% annual, and it's compounded continuously. I want to find the value of the account after 15 years.
So looking at my formula, I know that I want to find the end value of the account, so I'm looking for-- I'm trying to find A, which means I need values for P, r and t. We already know e is our mathematical constant.
So using my formula, I'm going to have A is equal to P. My initial amount is $1,000. e is going to stay e-- that's our mathematical constant. r, my interest rate, 3%, I'm going to write as 0.03. And t, my time is 15-- 15 years.
So to simplify this to determine what A is, I'm going to start by multiplying the two values in my exponent. And that's going to be 0.45. So this becomes e to the 0.45, bring down my other terms. e to the 0.45-- again, you can just evaluate that using your calculator, either with using the button for e or by using the approximation for e.
So this evaluates to approximately 1.568. And then finally, I can just multiply 1,000 times 1.568. And I find that A, the value of my account after 15 years, will be approximately $1,568.
So for my last example, let's say I invest $10,000 into an account that earns 1.5% annual interest rate, and that's compounded continuously. We're going to assume that we make no other withdrawals or deposits from or into the account.
And we want to find how long it's going to take for my initial investment of $10,000 to grow to $50,000. So we're solving for our variable t, the number of years. So I'm going to substitute 50,000 in for my value for A. I'm going to substitute 10,000 for P. e is my mathematical constant.
r, my interest rate, my value for r is going to be 0.015. I took my interest rate, 1.5%, and divided by 100. And t is what I'm trying to solve for, so I'm going to leave that t.
So I'm going to start by simplifying. I'm going to divide both sides by $10,000. Here this will cancel. And on this side, I have 5, which be equal to e to the 0.015t.
Now I can use the logarithm to solve this. And because-- this is an exponential form-- because the base is e, I know a logarithm with a base e is the natural log. So I'm going to cancel out the exponent operation by taking the natural log of both sides.
So now I have the natural log of 5 is equal to the natural log of e to the 0.015t. I know there's a property of logarithms which says that I can turn this exponent into a factor being multiplied in front of the logarithm. So now my equation becomes 0.015t, and we're going to multiply that by the natural log of e.
So I want to continue to isolate my t variable, so now I'm going to divide both sides of my equation by this value. We know that this is just a constant value. And here this will cancel.
The natural log of 5 divided by the natural log of e is going to give me approximately 1.609. And on the other side, I still have 0.015t. Finally, I'm going to divide both sides by 0.015. And I find that t is approximately equal to 107.3 years.
So let's go very key points from today. Some bank accounts gain interest continuously. With continuously compounding interest, n, the number of times interest is compounded per year is an infinitely large number. The continuously compounded interest formula uses the mathematical constant e, which is equal to approximately 2.718281.
A the account balance, P is the principal, or initial value, r is the annual interest rate, and t is time in years. So I hope that these key points and example is helped you understand a little bit more about continuously compounded interest. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.