Calculating future value of annuity

0:00- [Instructor] We're told that Maya deposits $250 into an account at the end of each month. The account earns 5.4 annual interest compounded monthly. She makes deposits for four years. What is the future value of the annuity to the nearest cent? So pause this video, have a go at that.

0:20Alright, so the future value of the annuity, that's just what all of these deposits combined are going to be worth in four years. So we have a formula for that, which we prove in other videos. It's a little bit out of the scope of this one, but we know that the future value is going to be equal to your monthly payment.

0:37And this is for a regular annuity, like what we're dealing with right over here, times one plus our rate that we are compounding at. And we're going to want to do this monthly. So we're going to think about how many monthly periods there are to the n. n is how many monthly periods, how many months we have in four years, minus one divided by r.

1:02Now, what is the payment? Well, the payment is $250. That right over there is the payment. What is r? Well, r is going to be 5.4% divided by 12 to get monthly. So 0.054 over 12. We'll figure out what that is in a second. And then what is n? Well, four years is equal to 48 months.

1:28I'm just multiplying by 12. So n is equal to 48. So, our future value is going to be equal to $250 times 1 plus... actually, let me get the calculator out. So if we get 0.054 divided by 12, that gets us 0.0045. So this is going to be 1.0045 to the 48th power, 48th power minus one, all of that over 0.0045.

2:09So let's get our calculator out to figure out what this is. So, here we go. Let me say it's 250 times open parentheses 1.0045 to the 48th power and then minus one, making sure that I'm getting the order of operations right. Yes, and I could just take that numerator and then divide by the denominator and it still works out the same way.

2:38So divided by 0.0045, get a little bit of a mini drum roll. I get rounded to the nearest penny, 13,361.18. Alright, so let us write that down. What is the value of the annuity to the nearest cent? It is $13,361 and if I round to the nearest cent, $0.18 right over there.

3:09Now we aren't done yet. There's more parts of this problem. The next part is asking us, let me make sure I can... The next part is suppose Maya deposits the money at the beginning of each month instead with the same deposit amount, interest rate and time period.

3:28What is the future value of this annuity to the nearest cent? And this will oftentimes be called an annuity due because you are paying at the beginning of the month as opposed to an ordinary annuity where you pay at the end of each month. Well, you could look up the formula for an annuity due, but it really is just this formula over here times 1.1 plus r.

3:52Now why is that? Well, every payment now has an extra month. So that extra month you compound by 1 plus r. So you just take whatever value we got here and we multiply by 1.0045. That is 1 plus r. So instead of memorizing two formulas, you could just think about this formula and say, okay, if I'm paying at the beginning of the month, I'm just going to compound everything by one more month.

4:19And so that's going to be... So if I just go back over here and I just multiply this times 1.0045, so I'm just compounding everything by an extra month. I will get $13,421.30. So I get $13,421.30. And there's one last part to this question. Content creators wanted to keep me busy with this one.

4:53Oh, that overwrote my answer to the last one a little bit, but let me see, I can scroll that one down a little bit. Alright. Compared to the graph of the ordinary annuity, what the graph of the annuity do lies and this dropdown would go above or below the graph of the ordinary annuity.

5:12Well, if the ordinary annuity has some graph that looks like this, in the annuity due where everything is shifted a month ahead. You always, at any given point, you're just making that one plus 1.0045% more for any given month. So the annuity due is going to be right above it.

5:31It's going to look something like that. And it probably won't be that pronounced because this isn't that much of a difference, but this is, if you were to zoom in, you would see that it is shifted up, lies above the graph of the ordinary annuity.