Calculating and graphing exponential functions for investing
About this video
Avery saves $100 a month from 25 to 55, while Blake saves $210 a month from 35 to 55, both earning 6% a year compounded monthly. Using the future value of an annuity formula with r = 0.005 per month, Avery ends with about $100,451.50 and Blake about $97,028.59. Starting 10 years earlier lets Avery finish over $3,400 ahead despite saving less each month.
You'll be able to
- convert an annual interest rate and number of years into a monthly rate and number of months.
- use FV = P x ((1 + r)^n - 1) / r to find the future value of monthly deposits.
- compare two savings plans and explain why starting earlier can beat saving more later.
Key words
- Compounded monthly
- Interest is added every month, at the annual rate divided by 12.
- Future value
- How much money an account will hold at a later date.
- Annuity
- A series of equal payments made at regular intervals, like monthly deposits.
- Compounding period
- Each interval when interest is added; here, one month.
- Retirement account
- An account people pay into during their working life to save for retirement.
Transcript
- [Instructor] We're told that two employees each contribute to a retirement account through their company. Avery deposits $100 at the end of each month from age 25 to age 55. Blake deposits $210 at the end of each month from age 35 to age 55. Each account earns 6% annual interest compounded monthly.
At age 55, who will have more money, and about how much more? So pause this video and have a go at this before we do this together. Okay, let's do this together, and before I even break out the math, let's just think about it a little conceptually. Blake is starting 10 years later, but putting more money per month into the retirement account.
And that's why it's not obvious who's going to have more, well, we're going to do the math. They're both getting the same interest compounded monthly. So let's do the math now. So the key formula we're going to use, and we're not going to prove it in this video, the math is a little bit more math, you have to learn a little bit about geometric series, which we do in other places on Khan Academy.
But the future value of these payments is going to be equal to the amount of each payment times one plus the rate at which you are compounding to the number of compounding periods, minus one, over that rate once again. And I know this might look complicated, and once again, this can be proven, but for this video we will just use it, but let's use this formula now for each of them.
Let's use it first for Avery. So for Avery, the future value. How much is Avery putting in every month? $100. So that's the payment, it equals $100. And then the rate, so you might want to put 6% in here, but remember we're compounding monthly. So if you have 6% annual interest compounded monthly, that means you get 1/12 of this 6% every month.
And so another way to think about that, that is 6% divided by 12 would be 0.5% per month, per month, or another way to think about it is 0.005 per month as a decimal. So this is going to be 1 plus 0005, or 0.005, so that's 1.005 to the... Now, what's the number of periods that we're compounding?
You might want to say 20, but remember, we're compounding monthly. So it's 20 years, we have 20 years. So n is equal to 20 years times 12 months per year, which is going to be 20 times... Oh sorry, this is 30 years, sorry, I said... 25 to 55 is 30 years, not 20 years, times 12 months.
So that's going to be 360 months. 360 months, that's the number of periods we're compounding, minus one, divided by that rate, 0.005. So that's what we're going to get for Avery, and then for... Actually, let me get my calculator out and just calculate that. So if I take 1.005 to the 360th power, I get that minus one is equal to that, and then divided by 0.005 gets me to, it looks like, let's see, did I...
Oh, I have to be careful. And then I have to multiply times this 100 right over here, times our payment. So times 100 gets me to $100,451.50, let me write that down. So this is going to be equal to $100,451.50. And now we can do the same thing for Blake. So Blake, let me write it over here.
Blake, future value is going to be equal. Blake is paying $210 per month. The rate is going to be the same here, so 1.005. Here we have 20 years, not 30 years. So 20 years times 12 months per year is 240 compounding periods, minus one, all of that over our rate, 0.005.
Let's get our calculator back out, make sure we can see what we're trying to do, all right? So I am going to do 1.005 to the 240th power this time. Subtract 1 from that, and then divide that by 0.005. Gives me that, and then I have to multiply that times 210. And I get $97,028.59.
So this is, and actually I'm rounding in both of them, so let me make it approximately. Approximately $97,028.59. So who will have more money at age 55? It looks like Avery is doing better. So even though they've put less per month, they started earlier, and those early months are compounding.
You're getting that half a percent of interest every month for 10 more years before Blake even starts, even though Blake is putting money in at a higher rate. And about how much more? Well, let's see, $100,451. This looks like it's a little over $3,000 more by age 50.
It's actually a little more than $3,400 more by age 55.
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