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# Math behind saving for college

## 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 and so it lets anyone as well how much money can I start putting aside every month so that I can pay for their college education 18 years in the future and so 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 so 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 twenty two thousand five hundred for the sake of this video I'll just focus on the private one but we'll you know in the back of our mind public half of that so whatever payment we come up with a 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 it looks like the last few years the full cost of a college education the annual cost has grown by about 3% a year so I guess we can extrapolate that forward that's a reasonable assumption if you think it's going to grow faster than 3% you can put it in that assumption if you think it's going to grow slower you can put that assumption age but I'm just going to assume 3% so if we just wanted to grow this for one year by 3% we would multiply by 1.03 and 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 eighteen 1.03 s or there's a shorthand for that we remember from our middle school mathematics that's the same thing as 1.8 1.0 3 to the 18th power and if we if we do that let's see 45,000 times one point zero three to the 18th power gets us to seventy-six thousand I'll just round to the nearest dollar seventy six thousand six hundred nine dollars seventy I'll say approximately equal to seventy six thousand six hundred and nine dollars I have a short memory okay yeah that's 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 in tuition at a private at a private university now you might be tempted to let's just multiply this by 4 to get the cost of a full four-year education but we have to remember that well this inflation and cost might continue so year 2 is going to be we can assume 3 percent more so what's that going to be so times 1.03 so this is the previous answer grown by 3% guess there's 78,000 907 let me write these numbers down so I'm going to have year one is going to be seventy-six thousand six zero nine year two is going to be 78 nine oh I'll round actually 908 to round to the nearest aisle or 78 908 so this is this is 78 908 and then year three is going to be times one point O 3 is going to be 81 275 81 275 and then year for this price you better hope that your child finishes in four years let's see and then we're going to grow by another three percent times 1.03 so the for the senior year is going to cost eighty three thousand seven hundred and thirteen eighty three thousand seven hundred thirteen and these are kind of eye-popping numbers but frankly someone who lived 18 years ago or whose child was born just 18 years ago these were probably going to be i popping numbers but regardless if we take the sum of this we'll get the sense of the four year four year college education and these aren't precise numbers these are based on our assumptions it's unlikely that the tuition will go by exactly three percent every and once again there's a bunch of different universities all the different levels of tuition let's figure out what this sum is so we have seventy six six oh nine plus 78 908 plus 81 275 + 83 713 and gets us to three hundred twenty thousand five hundred and five dollars so this is three hundred and twenty thousand five hundred and five dollars for the full cost of four years at a private university and if we assume these ratios hold and the inflation the public is the same as the private then the public will be roughly half of this so now that we know the lump sum that we need in 18 years how much do we have to put aside each month well one way to think about is well let's see I'm going to put a payment this first month of my child's life um using a new color so 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 so I'm going to have a total of 216 I'm going to have a total of 216 payments and they're going to have to add up to 320 thousand five hundred and five dollars and so another way to think about this is 216 times P 216 times P is equal to three hundred twenty thousand 505 or to solve for P you could just divide by 216 three hundred twenty thousand 505 / 216 is going to get you / - sixteen is going to get you a little almost$1,500 so 1484 approximately so P is approximately one thousand four hundred and eighty four and this wouldn't be a bad approximation but you're saying well okay I'm not just going to take these payments I'm not just going to take these payments and stuffing them into a mattress I'm going to 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 or at least I'm going to put it into a savings account whatever it might be a CD maybe so I'm going to get some return on so this isn't really a fair calculation and if you said that you'd be absolutely right because on this first payment you're going to you're going to get two hundred and sixteen months of your going at two hundred sixteen months of interest on that this one you can get two hundred fifteen months this one you're going to get two hundred and fourteen months of interest and so the math is a little bit fancier when you start trying to calculate that and if you're really curious about I encourage you to go to the Khan Academy video on sums of finite geometric series of the series but in those videos I proved the formula for how to calculate this payment and this is actually this is also the way that things like mortgage payments are calculated so what we get it when the the formula that we derive in those videos is that your payment your payment if you factor in what that you're going to get some return on it is going to be it's going to be the amount that you're saving for so that in this case is three hundred and three hundred twenty thousand five hundred and five dollars times times your monthly return that you think you're going to get so remember if you if you're going to get say if you were going to if you see a monthly return to say point three percent per month that'll turn into an annual return of a little bit more than three point six percent if you compounded if you just multiply it by 12 you get by three point six but you're compounding every month so it'll be a little bit more than three point six percent per month and so three point six percent per year and so let's just assume that just to make the numbers easy so let's just assume let's just assume that our monthly growth rate is point three percent so we're going to grow by we're going to essentially one point zero zero three this is how much we're going to grow every or we're going to be one point zero three if we start a month at a thousand dollars at the end of the month we're going to have a thousand and three dollars and this once again translates into a little bit more than three point six percent return if you think are going to get a higher return you'd put a higher number here if you think you're lower in return you'd put a lower number here but the key is that this is monthly we're thinking about monthly monthly growth that's where that number comes from and then you take that you subtract one and so this whole expression is essentially it will just give you that point zero zero three or that point three percent and then all of that over all of that over our one point zero zero three two the number of periods that were compounding that's two hundred and sixteen periods two hundred sixteen months minus one and so we definitely need our calculator for this this is going to give us three hundred and twenty thousand 505 times this is just going to be point zero zero three and I just lose the one I'm subtracting one there so that's our numerator and then divided by and we definitely do calculate it for this one point zero zero three to the 216 power minus one and I think we deserve a drumroll right about now we get a payment of about one thousand and fifty seven dollars approximately one thousand and fifty seven dollars I notice it's lower than this number because here we're assuming that each of these payments are growing this one grows for eighteen years this one grows for seventeen years and eleven months and 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
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