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Math behind saving for college

Calculation of what to save to pay for a child's college tuition. Created by Sal Khan.

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Video transcript

Let's say you've just had a child, and you expect him or her to go to college in 18 years, and you at least want to have the option of being able to pay for his or her college education. You want to know how much money can I start putting aside every month so that I can pay for their college education 18 years in the future. I've done some research just so we can start with some assumptions, and all of what I do in this video is all based on assumptions. I encourage you to do the same computation, maybe with different assumptions, based on what you believe is going to happen. I've done some research and the current full cost of a year of a private education is approximately $45,000. A public education- the average cost is roughly half that. It's approximately 22,500. For the sake of this video, I'll just focus on the private one, but we know if the back of our mind public is half of that. So whatever payment we come up with for private, the public we could just do half the payment and that would be sufficient. So how do we think about what tuition is going to be in 18 years? I think we all have a sense that it's going to be a higher number. Well, I did a little bit of research there, and it looks like the last few years the full cost of a college education- the annual cost- has grown by about 3% a year. I guess we can extrapolate that for. That's a reasonable assumption. If you think it's gonna grow faster than 3%, you can put in that assumption. If you think it's going to grow slower, you can put that assumption in. But I'm just going to assume 3%. If we just wanted to grow this for one year by 3%, we would multiply by 1.03. If we wanted to grow it for another year by 3%, we'd multiply by another 1.03. If we want to do this for 18 years, we'd have to multiply by 18 1.03's. Or, there's a shorthand for that. We remember from our middle school mathematics. That's the same thing as 1.03 to the 18th power. If we do that- Let's see 45,000 times 1.03 to the 18th power gets us to 76,000. I'll just round up to the nearest dollar: $76,609. Seventy- I'll say approximately equal to- $76,609. I have a short memory. Okay, yeah, that's what it was. So this would be one year. One year- the full- The one year if, based on our assumptions of room and board and supplies and tuition at a private university. You might be tempted. Oh, let's just multiply this by four to get the cost of a full four year education, but we have to remember that this inflation in cost might continue. So year two is going to be, we can assume, 3% more. So what's that going to be? So, times 1.03. So this is the previous answer grown by 3%. Gets us 78,907. Let me write these numbers down. So I'm gonna have year one is going to be 76,609. Year two is going to be 78,90- I'll round, actually- 908. Round to the nearest dollar- 78,908. So this is 78,908, and then year three is going to be times 1.03. It's going to be 81,275- 81,275. Then year four- at this price, you better hope that your child finishes in four years. (laughs) Let's see. Then we're gonna grow by another 3% times 1.03. The senior year is gonna cost 83,713- 83,713. These are kind of eye-popping numbers, but frankly, someone who lived 18 years ago or whose child was born 18 years ago, these were probably going to be eye-popping numbers. But regardless, if we take the sum of this, we'll get the sense of the four year college education. These aren't precise numbers. These are based on our assumptions, and it's unlikely that the tuition will go up by exactly 3% every year. Once again, there's a bunch of different universities all with different levels of tuition. Let's figure out what this sum is. We have 76,609 plus 78,908 plus 81,275 plus 83,713. It gets us to $320,505. So this is $320,505 for the full cost of four years at a private university. If we assume these ratios hold and the inflation and the public is the same as the private, then the public will be roughly half of this. Now that we know the lump sum that we need in 18 years, how much do we have to put aside each month? One way to think about it is, well, let's see. I'm gonna put a payment this first month of my child's life. I'm using a new color. I'm going to put one payment this first month of my child's life, and then the second month and then the third month. I'm going to do this for 18 years. 18 years times 12 months a year. That's 216 months. I'm gonna have a total of 216. I'm gonna have a total of 216 payments, and they're going to have to add up to $320,505. Another way to think about this is 216 times P- 216 times P- is equal to 320,505. Or to solve for P, you could just divide by 216. 320,505 divided by 216 is going to get you- Divided by 216 is going to get you almost $1,500. So 1,484 approximately. So P is approximately 1,484. This wouldn't be a bad approximation, but you're saying, well, look. Hey, I'm not just gonna take these payments. I'm not just going to take these payments and stuff them into a mattress. I'm gonna put them into some type of an account. Maybe I'll buy some long-term bonds that mature in time for them to be liquid when my child goes to college. Maybe I'll invest this in the stock market. At least I'm going to put it into a savings account. Whatever it might be; a CD maybe. I'm gonna get some return on this. So this isn't really a fair calculation. If you said that, you'd be absolutely right because on this first payment you're going to get 216 months of interest on that. This one you're gonna get 215 months. This one you're gonna get 214 months of interest. So the math gets a little bit fancier when you start trying to calculate that. If you're really curious about it, I encourage you to go to the Khan Academy video on sums of finite geometric series, but in those videos I prove the formula for how to calculate this payment. This is also the way that things like mortgage payments are calculated. The formula that we derive in those videos is that your payment if you factor in that you're gonna get some return on it, is going to be the amount that you're saving for. That, in this case, is $320,505 times your monthly return that you think you're going to get. Remember, if you're gonna get, say- If you were gonna- Let's say your monthly return is say .3% per month, that'll turn into an annual return of a little bit more than 3.6% if you compound it. If you just multiply it by 12, you get 3.6, but you're compounding every month so it'll be a little bit more than 3.6% per month, or 3.6% per year. Let's just assume that just to make the numbers easy. Let's just assume that our monthly growth rate is .3%. So we're gonna grow by- we're gonna essentially- 1.003- this is how much we're gonna grow every- Or, we're gonna be 1.003. If we start the month at $1,000, at the end of the month we're gonna have $1,003. This, once again, translates into a little bit more than 3.6% return. If you think you're gonna get a higher return, you'd put a higher number here. If you think you're gonna get a lower return, you'd put a lower number here. But the key is that this is monthly. We're thinking about monthly growth. That's where that number comes from. Then you take that, and you subtract 1. So this whole expression is essentially able to just give you that .003, or that .3%. Then all of that over- all of that over- our 1.003, to the number of periods that we're compounding. That's 216 periods- 216 months- minus 1. We definitely need our calculator for this. This is going to give us 320,505 times- This is just going to be .003. I just lose the one. I'm subtracting one there. So that's our numerator. Then divided by- We definitely need the calculator for this. 1.003 to the 216th power minus 1. I think we deserve a drum roll right about now. We get a payment of about $1,057, approximately $1,057. Notice it's lower than this number because here we're assuming that each of these payments are growing. This one grows for 18 years. This one grows for 17 years and 11 months. Because of that, you can afford to put a little bit less aside based on these assumptions. Now everything I did was for private. If you did public, it would be half this number.