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so a special case of the cash flow diagrams that we were looking at is one where you have a number of cash flows that happen at uniform amounts for a specified period of time so the same amount happens over a period of time so the way that that may look is something like this you have a number of periods over time and let's actually look at at the case where you know over this these number of periods you actually are you know let's say that you are having to deposit into a bank account so this would actually be from the perspective of the bank account right it's these deposits into the bank account okay and there's a reason why I'm starting here at at one right rather than at zero okay and then after having deposited this same amount each time okay so let's actually I tell you what I'll go ahead and put a number to it let's say that this is a hundred bucks each time so you're saving up for something right so you say every month I'm gonna put a hundred bucks into my bank account okay and then after all of this time I am going to make a withdrawal from my bank account okay this is a little bit different cuz we're not even bringing the wallet into this at all now what we're doing is we're looking at the bank account as being the storage device right so anyway but this is let's let's say we're doing this and we have to figure out how much could we withdraw at the end if this is what we had been saving and you know again I'll say what if those are months and let's say what what interest rate do you want for this period of time okay three percent let's say we have three percent annual interest okay and you're going to compound it monthly okay and what you want to know is how much can you take out at the end if this is how you have been saving and based on what we just did what would you have to do to calculate this okay you do one at a time you'd basically say that you would have a hundred okay this would be your what you could have is your future value would have a hundred times one plus okay now what we need to do is look at that three percent but we're doing this these are months down here I said right okay so how do I deal with that okay so we have this three percent point O three over twelve that gives me my monthly interest rate and then what do I do okay he says the power of four and that's the reason he's saying it is that there are four time frames four periods between that cash flow of a hundred bucks and now what else will I have so that's that's power 4 so plus a hundred times one plus three percent over twelve raised to the third plus a hundred times one plus 3% over twelve raised to the second okay plus a hundred times one plus three percent over twelve raised to the first power and then finally a hundred times one plus three percent over twelve raised to what zero okay so we've basically those five terms allow us to take care of each of those cash flows and we can figure out how much will we have there at the end okay to make sure that we're you know doing this correctly the what we need to do is make sure you know that we have consistent units that were applying so in this case we have a monthly interest rate that we create by dividing our annual interest rate by twelve the question there is is our exponent than in months since we have a monthly interest rate that we have now created by dividing by 12 yes right because this is we're actually the whole thing lasts less than one year right it's just five months of savings right so right so you know just be very careful look at it carefully to make sure your matching consistently that the time units that you have in your exponent are the same as the basis of your interest rate okay well I want to show you something that you're probably gonna like okay well before I show you anything else let me actually show you that there's another reason why you might like your calculator okay most of you have probably seen a for a summation construct like this I'm wearing your math stuff that you have done we can do this okay what would our formula look like if we're gonna do a sum first of all the amount is a hundred bucks right so that's easy okay every one of them has one plus 0.03 over twelve right okay so that doesn't change for any of the terms what is the thing that does change okay we could take this to a power of a variable okay X and then as long as we pick the right limits for our sum here what do our limits have to be okay it has to start at zero and go to four okay and that'll tell us how much we will have at the end we didn't earn a bunch of interest right but we you know we earned a little bit there 500 and 251 okay now not everyone has that calculator or the the TI equivalent if you don't have that that's a lot of typing right to get all of that put in there and there's a good chance you could mess something up as you're typing it all in so there is actually a an interesting thing that we can do for these uniform series to try to take what we did there and turn it into something that's a little bit easier to use than having to manually go through and type in each of these terms which isn't as bad if you have a summation function on your calculator but it would be nice for us to go ahead and solidify that into an actual formula so let's do that let's do it kind of like we did whenever we first looked at compounding interest let's take a couple of examples so first of all let's say that this is our basic cash flow diagram I'll draw it over here so a basic cash flow diagram is that we're gonna start with nothing at zero but then we will start in in year one okay and we'll start depositing these amounts okay and I tell you what we'll start with three deposits and then at the end of that we will make a withdrawal the question is if these are each a okay and we have an interest rate of I per period and these are our periods one two and three question is how much can we withdraw just kind of like a future value all right well let's actually start you know figuring that out so we we say the deposit that happened at the end of year three okay what is that one gonna be okay so you know period three deposit what is that one worth at the end a right did it have any time to accumulate interest the one I'm referring to is this one right here did it have time any time to accumulate interest okay what about the period to deposit okay so that one did have time to earn interest and how much will it be worth there at the end raised to the first power right okay what about in in one okay a times 1 plus I to the second okay and then did we have anything in 0 no okay so let's actually expand these out so this is actually a this is a times 1 plus I and this is a times 1 plus 2i plus I squared okay and if I add all of these up that gives me my amount at the end correct okay so when we add these up a is just this factor out here so we really can just sort of add up this is a times 1 right so we can really just add up since they a is in all of them we can add up the stuff that's in the parentheses do you agree with that so I should be able to do this a times what 3 plus 3i plus I squared interesting interesting let's change it up just a little bit so that we start looking at that we go I might be seeing a pattern there but I'm not sure let's go to four okay let's change this problem let's say we'll go one more period yeah no one saw anything all right so we we extend this by one more period okay and what do we have now okay what we actually have is that we can slide all this down we extended this by one more period so we're going out to four now and we're gonna take all of these all right I tell you what I'm gonna do I am NOT gonna try to slide these at all I'm going to erase them okay and what we're going to do is we're going to write in new numbers over here we are going to put this is 4 this is 3 this is 2 and now what do we need to do okay we probably need to instead of coming up with this result here at the end what do we need to do we need to add a 1 and what happens for one okay if we expand that one plus I to the third what do we find one plus 3i plus 3i squared plus I cubed okay so now let's let's add these up okay and what we end up with there is we have a times four plus what plus 6i plus four I squared plus I cubed okay and remember was our last one when we when we ended here what do we end up with a times three plus 3i right plus I squared do you start seeing the beginnings of something of a pattern okay here's the problem we don't have the whole pattern right some of you are looking at those and saying I see a pattern that looks like binomial theorem but it's not quite right to actually be the the binomial theorem right as we saw in their previous lecture so what do we do about it okay and this is by the way this is our future value here's what we can do about it if it doesn't look exactly like we want it let's manipulate it in some ways that we know aren't wrong okay we're not sure they're right but we know they're not wrong okay so we're gonna manipulate this a little bit and the first thing we will do is we are going to add and subtract excuse me we're gonna multiply and divide this the first thing we're gonna do multiply and divide by I okay so what I'll do basically is say this is going to be a times 4i plus 6i squared plus 4i cubed plus I to the fourth over I that legal okay all I did is I multiplied and divided by I okay next what I'm going to do is add and subtract one so here what I'll do is I'll say we have a okay how do I do this okay right so basically what we say here is that if we add and subtract right in the same place then you know we get away with this right so one way of looking at it actually is put a down here is it okay for me to write this as 1 plus 4i plus 6i squared plus 4i cubed plus I to the fourth minus 1 over hi because we still have that that still lingers on ok well now what have I made here okay if you if you recognize that that becomes just 1 plus i raised to what power the fourth power okay so this says that F is equal to a times 1 plus I raised to the fourth power minus 1 over I okay if I had gone through that whole process up here remember this is what we had before what would this look like could I multiply and divide by I okay so that would basically be I over here over I that gives me what that would be a times okay this would be 3i plus 3i squared plus I cubed over I and then what would I do if you add and subtract 1 you would end up with a times and I'll just cut to the chase it'll be a times 1 plus I to the third power minus 1 over I so do we see a pattern okay we do and that pattern continues to hold if you keep on adding more periods that you do this so basically what this tells us is that we have F is going to be equal to a times 1 plus I to the N minus 1 over I okay this is if you have constant amounts that's right that a is a constant value each time that you put it in there okay so this is probably a good idea when we write this up here to go ahead and also put our cash flow diagram all right make sure that we understand air all the details about the cash flow diagram the first thing to note there are no cash flows for this equation that happen in the you know like right now right the the assumption is that the first cash flow happens at the end of that first segment at the beginning of the number one right so that would be where it begins it continues on at that same level of a each time so in the next one you'd also have a value of a okay in that second one and then you know you can expand that out for however long you want out to N and you do have a cash flow of a in that last one as well and that's where you can pull out your value of F okay so this is a formula that's extremely handy for us to to have questions yet at this point okay say what now if you were to include one at zero what should you do yeah you could shift over everything one right you could actually turn your zero into one all right if you wanted to have one right there at the very beginning you might have to shift your viewpoint on what your time frame is okay a bigger question might be what happens so this is this is a good example of if you are saving up for something over time right that was the example I did right up here you're you're saving up a hundred bucks a month over these five months and you know you want to know how much have you accumulated over that period of time often what we want to know is how big alone or how big a payment am I going to have for instance if I get a loan of a certain amount which is a present value right so the question there is how can I change this formula so that I end up with a present value okay what do you think do we have a formula that relates future value to present value okay so since we have that what we can do is actually just do a substitution and say here then that for this formula we have P times 1 plus I to the N is equal to a times 1 plus I to the n minus 1 over I and what that ends up meaning is that P is equal to a times 1 plus I to the n minus 1 over I times 1 plus I to the end thank you okay so what does this cash flow diagram look like for that formula does it look kind of similar and here's kind of what it looks like right you have a time line what happens here is like you might need money right now right so you receive money but then what happens you have to pay it back in little bits over time okay and again there could be a gap in these number of times that you have to pay back right out to the end time zero one two each of these is a and what you get out is this present value that's what you get right at the beginning okay so this is kind of like a savings account this is kind of like a loan and this right here is almost exactly what your mortgage will be like whenever you get one okay it'll most likely be compounded monthly but that is basically at its most fundamental level how a mortgage works it's usually how car payments work as well okay well should we do a problem all right let's do this problem you just came to college okay your parents didn't buy you a car okay mine didn't either okay so you're at college and you want a car you want a $5,000 car you're not asking for much okay your parents aren't gonna buy it for you you don't have the $5,000 you're thinking about taking out a loan okay have you all looked into how much loans cost okay what's what's a typical interest rate for for purchasing a car what's that too much okay I've seen 6% if no one has any you know quarrels with that I'll say let's say that you shop around you find a 6% interest rate okay how long do you want to be paying on your car okay there's a lot of times there's like three year loans sometimes there's four year loans for car loans okay what do you want to do let's say it let's actually say it's four years we'll do it four year loan okay this is an annual interest rate okay we'll do a four year loan compounded what do you want to say monthly by fortnightly ok 4 year loan compounded monthly okay so and what you want to know is how much is your payment gonna be okay so what's interesting here is we're solving the problem here not what we're trying to find P right we know P what's P $5,000 well we're trying to find here yeah we're trying to find a that's that payment amount of a so what one things that might be helpful to us here is to actually take this formula and rearrange it slightly what if you solved this formula for a yeah it's not tricky you just flip the fraction right so you just say this is going to be equal to P times I times 1 plus I to the N over 1 plus I to the N minus 1 okay so what do we put in here a will be equal to what $5,000 times okay 6% good question here is though how are we compounding so we really need a monthly interest rate right then what okay one plus 0.06 divided by twelve okay and we're talking about four years which is 48 months right so you can either do that as four times twelve up here to kind of remember that you're going to monthly instead of annually or you can just put in forty-eight if you feel comfortable just putting that in directly all right what about in the denominator 1 plus 0.06 over 12 raised again to the 4 times 12 minus 1 and what is your payment going to be okay we will have 5,000 times there's a fraction here I'll actually put in another fraction of point zero six over 12 times 1 plus 0.06 over 12 okay that gets going to get raised to the 4 times 12 okay then in the denominator we put 1 plus a fraction again 0.06 over 12 and that will be raised to the 4 times 12 then what minus 1 so if you can swing 117 dollars and 43 cents or so a month then you can at least make your car payment of course that doesn't buy you gas that doesn't buy your repairs all right doesn't buy you tires insurance right but you can at least afford to make the purchase of your car all right well let's solve a different problem now let's actually say that your again you're gonna say you know what I don't necessarily need a car right now you know I've got friends that have cars you know if I need to go to Walmart there's a Walmart just right over across the road I can walk over there all right there's not really a big reason I need a car and so I'm gonna forgo having a car for four years and instead of buying the car now I'm gonna plan on buying a $5,000 car when I get out of college because that's a point in time where I feel like I'm really gonna need it okay so again $5,000 car will say what kind of interest rate do you think you can get if you just save your money okay $5,000 okay we'll talk about someone says 2% interest rate I think you might be able do a little bit better than that let's go let's call it 3% okay 3% interest rate okay so now this was up here this was a loan this is now a savings idea okay we're gonna do four years of savings okay how much do you have to save each month to buy the car just straight up by your $5,000 car okay how does this change yeah so we're looking at a future value kind of a problem here right because you're trying to figure out how much will you have at the end after having saved a certain amount each period okay and so again though we're trying to find a so we're trying to do is take this formula that I have up here and rearrange it yeah stuffs not really working real well today so we'll just go ahead and rewrite it what is f it's a times 1 plus I to the N minus 1 over I okay we rearrange this where we solve for a and it gives us F times what I over 1 plus I to the N minus 1 okay how much do we have to save each month to be able to buy the car okay again it's $5,000 up here we will take 3% of course we want that as a monthly because that's the way a lot of a lot of savings accounts compound so here we've got 1 plus 3% over 12 raised to the what again 4 times 12 right 48 months worth of savings while you're in school minus 1 all right so 5000 times 0.03 over 12 then we have 1 plus 0.03 over 12 raised to the 4 times 12 minus 1 okay so if you can save 98 dollars and 17 cents a month over 4 years then you can afford to buy this $5,000 car whenever you get out of school okay so my question here is this so this is kind of a bigger question which side of this coin would you like to be on in life okay so my point with this is is this right in either case you're getting a $5,000 car you had to go through four years of your life where you didn't have a car right the other option you have is you could buy something that was way cheaper to get you buy for those first four years all right my first few cars I bought were in the neighborhood of about $1,000 each so what's that well kind of actually you know I I like working on cars so my first car was a Volkswagen Karmann Ghia and I I restored it you know so that it was it was a lot of fun and it taught me a lot as well but you know he didn't have to cost me a whole lot of money right to have that vehicle and so that gave me transportation in the early part of my life and then you know as time goes on if you stock away some money in every one of the times you get a paycheck and you decide that is how I have a car payment instead of having a car payment that it's a loan I sort of have a car payment that is a storage of resources to the point where I need them right and it costs you a lot less and once you get into that cycle in your life it continues to cost you a lot less and you are not living any lower quality of life right you can still have the same sort of value of vehicle and you know by doing it from the savings end rather than from the loan end it helps you out okay so that's that's kind of my I just wanted to illustrate that just a little bit because what's the I mean that's a pretty big difference if you think of percentages right and these are not ridiculous numbers right there there these are all relatively reasonable numbers that I have up here for this problem right sure all right any other questions so that is the idea of uniform amounts that you apply each period and how that accumulates over time present or future value for that type of a cash flow all right awesome appreciate it