 Well welcome to episode 33. This episode is on annuities and payments for installment purchases. This particular episode I think you'll find interesting not only for business majors who will be talking about these formulas and using them in business courses but also just for the general student who wants to know more about how how they can save money, how they determine payments on loans and this sort of thing. Let's look at the at the list of objectives for this episode. First of all we want to find out what is an annuity and then we'll look at a formula for calculating the amount of annuity. Then we'll look at what's called the present value of an annuity. You'll hear these terms in business courses and we'll look at a formula for calculating the present value and then finally we'll look at how monthly payments are computed on a loan. Okay well now to begin with when I say an annuity what I mean is it's a sum of money that has been collected from regular payments and interest that are put into an account. For example you might hear you might hear people talk about a retirement annuity. I have a retirement annuity and every month I put in a certain amount of money into that and so by way of my contributions and by the compounding of interest hopefully it'll be a sizable amount by the time that I'm ready to retire. Let's look at the next graphic and we'll see this sort of analyzed. An annuity is a sum of money that is the result of regular payments and then I mentioned this example of retirement annuities. The regular payment that you make into an annuity is called the rent on that account so I'll be using the letter R to represent that. Okay so first of all I'd like to figure a far come up with a formula that will show us what will be the value of an annuity at the end of a certain amount of time. If we know the interest rate, if we know how many payments are going to be made into the account, how many periodic payments and what those payments would be. Now we have a formula on the next graphic that we want to derive but let's go ahead and look at this. A formula for the amount of an annuity. Now a sub f that refers to the final amount a sub f is equal to the rent times 1 plus i to the n minus 1 over i. i is the periodic interest rate, r is the periodic payment that you make into the account and n is the number of payments that you make into the account. Okay to derive that I'm going to go over here to the whiteboard and to begin with let's let me just remind you of a couple formulas that we've seen in the past that we want to use. First of all from the last episode I want to use the formula for the sum of payments for the sum of a geometric series and that was s sub n is equal to a times 1 minus r to the n over 1 minus r now we've just seen that formula recently this is for the sum of the terms in a geometric series. Another formula that I want to use is this one and that says that if you deposit some money say the principal into a bank account and you want to figure out what is the amount that that will grow to then the amount is the initial principal times 1 plus i to the n power. I think we ought to talk about this just a little bit because we'll be using this one a lot and we saw this formula in an episode gosh maybe 10 or 12 episodes back so it's been a while since we've seen this. So let's just talk about this formula why don't I just leave it up here so we can refer to it. Suppose first of all that we deposit this amount of principal into a bank account that pays i percent, pays a rate of i percent per period and it's going to be compounded for n periods so we have n periods of compounding then the amount in the account after no periods have lapsed is just the amount that we put in the account because you have to wait until the end of the first period before any interest is added to that. The amount at the end of one period would be the principal plus the interest that's added on to it and the interest would be i times p. You take the interest rate times the principal and that would be the interest and you know this is equal to p times 1 plus i. Okay now if I wait two periods then at the end of the second period I will have the amount of money that I started that period with. This is how much money I entered the second period with plus I'll have the interest on that money which is i times p plus 1 plus i. In other words this is how much money I started with in the second period and then this is the interest rate times that amount of money that's the interest and you notice that if I factor out the common factor of 1 plus 1 plus i let's put that in front I guess 1 plus i then I'll be left with p plus ip and I can factor a p out of that if so if I factor p out of that expression then I have 1 plus i times 1 plus i again and that's going to give me p times 1 plus i to the second power. Well I think we can see a pattern if if I wait zero periods the amount of money in the account is just the principal that I've deposited if I wait one period it's p times 1 plus i if I wait two periods it's p times 1 plus i squared and as a general rule after n periods I'll put a little subscript in on that then it'll be p times 1 plus i to the n power and in fact in this case that formula even applies because this is p times 1 plus i to the zero power r1 okay now with this with this formula in mind let's look at how we could come up with a formula for the for the sum total that is the annuity given a certain rent okay so let's assume that r represents our periodic payments or the rent into the account. Suppose i is the interest rate per period so this is the rate per period now by the way that period doesn't have to be the end of a year it could be a monthly payment or a weekly payment but this would have to be the interest rate per that per period and I'm going to be assuming that this is being compounded according to this period that is to the period of the payment so if we're making monthly payments then it's being compounded monthly if we're making if we're making a semi-annual payments then it's being compounded semi-annually and let's suppose n represents the number of periods so this is the number of periods of compounding and what I want to find is a sub p well excuse me a sub f which is the amount of the annuity so I'm calling that a sub f for the final amount in that case okay well let's see um suppose I were to make um suppose I were to make just to keep it simple suppose I were to make only four there were only four periods so I would make four payments in the account so and that total would give me would give me a sub f I'll put a sub f over here to give me a little room for that okay now um let's see in the first period um I would have in the first period I would my principle would be r and it would be left in the account for n minus one periods after that by the way I'm assuming that I'm making this payment at the end of the period so there will be n minus one periods left over that would be three periods left over for it to compound in now the payment or the second payment I make if I make it at the end of the period then it's only going to be in there for two more periods before um I've come to my final amount so one plus i squared and then the payment I make in the third period is only going to be compounded for one period so that'll be r times one plus i to the first power and then the amount that I make in the third period uh doesn't have a chance to do any compounding so it's just r so I have the rent here here here in here but the one that I make earliest is the one that grows the most because it has more compounding attached to that so if I figure the sum or the value of all of those rental payments that is my final amount now you know if I turn this around to me this looks like a geometric series and it looks to me like the first term a is r and it looks like the common ratio is the little r is one plus i okay now let's go to our summation formula for a finite geometric series you remember I could write that as a times one minus r to the n over one minus r that's the formula that we derived in the last episode so in this case for a I'll put r and for little r I'll put one plus i and in this case that's going to be to the fourth power because I'm adding up four terms over one minus one plus i now if I reduce this that's going to be r times one minus one plus i to the fourth over let's see what is this going to reduce to b class one minus one plus i how much is that i it's going to be a negative i actually you see one minus one minus i it's going to be a negative i and now just take care of that negative why don't we multiply top and bottom by negative one and when I multiply by negative one here I'll just reverse the order so this says r times the quantity one plus i to the fourth minus one all over i now this is the formula when you make four payments how do you think I would write that formula if I were going to make n payments instead of four how do you think it would change well I tell you what happens is I just change that exponent so if I'm making n payments into an annuity then the formula is r times one plus i to the nth power minus one all over i okay let's look at the next graphic and I think we'll see that formula okay now here we have a formula for the amount of an annuity and this is the formula that we've just arrived on the board a sub f is r times one plus i to the n minus one all over i where each of those terms is explained again let's go to the next example and we'll see how we can apply this it says what will be the value of an annuity that consists of quarterly payments of a hundred dollars for 20 years at an annual interest rate of six percent compounded quarterly okay well first of all class let me ask you this in 20 years how many quarters will there be how many quarterly payments will be made 80 the 80 payments sure because there are four quarters in every year so we'll have 80 payments so 80 payments of one hundred dollars so that means we're putting in actually eight thousand dollars of our money now we would expect that due to compounded interest that the value of the annuity is going to be considerably more than eight thousand dollars but at least we have a kind of a lower bound on how much will be in this account so now let's try figuring the exact value of this annuity and I'll use the formula that we have just presented here so we have a sub f is r times one plus i to the n power minus one all over i now the rent is a hundred dollars that's the amount of the payment the interest rate is not six percent because this has to be the interest rate per quarter so i'll have to divide that by four uh so what is six percent divided by four one and a half one and a half percent yeah so that'll be zero point zero one five that i'll put in here for one plus i and i'll raise this to n now n is the number of payments the number of periods and we said that would be 80 so it's not 20 years but it's 80 quarters minus one and then we divide by i and i again is zero point zero one five well i can let's say i better put square brackets around that portion now i can reduce this a little bit by calling this 100 times the quantity 1.015 to the 80th power minus one all divided by zero point zero one five okay so i'd like to carry out that calculation on my calculator and let me just lay this right in the middle here if you can zoom in on my calculator i'm going to be i'm going to be computing that amount okay so first of all we said that was going to be 100 times the quantity um one point zero one five uh raised to the 80th power minus one close parentheses now i'm going to divide that amount by point zero one five and we get a total of $15,271 and uh you see that is more than the 8,000 that we put in initially in fact our money is just about doubled if you look at the actual face value of the money we put in um we we now have a little over $15,000 this is at the end of 20 years now you may say well jeez Dennis that's not really that much but you know we're only putting in a hundred dollars per quarter so we're not really putting enough money in there to make that grow too rapidly let's go to another problem another example about the Jeffersons what does it take okay in this problem it says the Jeffersons plan to retire in 40 years and have decided they need a retirement annuity of $600,000 at the end of that time what payment should they make into this annuity if these accounts typically pay 4.8 percent interest and let's assume monthly compounding okay well let's see if we come to the to the green screen here we're going to be using the same formula and that is once again that a sub f the final amount of the annuity is r times uh one plus i to the nth power minus one all over i uh by the way i should mention to you that when you have when we cover this material on the on the final exam because this is the only place that you'll that you'll see this that i'll give you all of these formulas all right we're going to have two other formulas that are come up they're going to come up in this episode and i'll give you these formulas since these are rather specific to business majors i won't expect you to memorize them okay now you see this time the question is how much should the jeffersons be depositing in their account every month so we're looking for the rent and because i'm now looking for the rent it means i have to know what is the final amount and we said that should be six hundred thousand dollars so i'll put six hundred thousand over here on the left uh equals r times now let's see we were assuming that the interest rate was 4.8 percent that was the annual interest rate and so monthly compounding let's see 4.8 percent if i divide that by 12 that's going to be 0.04 percent uh per month so here i'll put 1.004 that's the one plus i and i'll raise this to let's see now with this was for 40 years how many months are in 40 years uh let's see i think that'll be 480 compounding periods or months and i'll divide it by one or rather subtract off a one and then divide it by 0.004 you know let me write that 480 a little bit clear i think it's probably a little harder to read on the camera so this is 1.004 raised to the 480th power uh so what i'll need to do here is to solve for r well it looks like r is going to be 600,000 times 0.004 divided by the quantity in this bracket so 600,000 times 0.004 divided by 1.004 raised to the 480th power minus one and we want to figure out how much that is now you know if i want to get this down to the exact penny when i go to round off to the nearest penny i'm going to round up because if i round down they theoretically won't have they won't have 600,000 they'll come up just a few cents short so i'll be rounding up and also i'm assuming that there's the same interest rate for 40 years so that's one of the assumptions that we've made here now if you can zoom in on my calculator uh i'm having to lay this right over the numbers but that's what i'm going to that's what i'm going to compute and this will be 600,000 times 0.004 and then i have to divide by uh i have to divide by a difference let me just show you i have to divide by this difference i'm going to put this in i'm going to put that in parentheses um so parentheses 1.00404 uh raised to the 480th power because there are 480 compounding periods in in 40 years minus one closed parentheses and i get 414 and 15 cents but i'm going to round that up to 16 cents because 15 cents means they're going to come up a little bit short so we'll say 16 so i mean we're getting a little technical here but around 414 dollars so we'll say this ends up being 414 dollars and 16 cents is what they should put in monthly if they want this to become 60 thousand 600 thousand dollars at the interest rate now if we go back to that graphic let's look at one other that the the last part of the question let's go back to what does it take um what if the account pays only 3% and what if it pays only 6% so i tell you what on my screen here i'm going to copy this answer and uh we said that at 4.8% the rent would be 414 16 okay that's what we've just computed here now if i change the percent to 3% what will i have to change in this expression right here it'll still be 600 thousand dollars but what will i what would i put here on the bottom what will be i now this is supposed to be the monthly interest rate well if you take 3% and divide by 12 that's going to be 1 fourth of 1% which is 0.0025 that's what i need to put for my monthly interest rate so i'll put that right here 0.0025 and while i'm at it i'll put that up here 1.0025 now what do you think is going to happen to the rent on the account do you think it's going to get larger or smaller larger it's going to get larger because they're not compounding interest uh as rapidly so they're going to have to make bigger payments now let's go back and calculate that one so if we zoom in then once again this is going to be 600 000 uh times 0.0025 uh divided by the quantity 1.0025 raised to the 480th power minus one closed parentheses and now we get 647 dollars and 91 cents 647 91 okay if we if we look then at the at the compare these results then it looks like their monthly payment has gone up by more than 50% they've gone from a little over 400 to almost uh 650 dollars so it's gone up by a little more than 50% when the interest rate drops to about 60% of what it was initially okay now the last question that i asked in that graphic is what happens if the interest rate had been 6% and i tell you what i don't think i'll work that one out because i think you see the procedure what i'll do is have to change uh this to be one half of one percent per month that'll be 1.005 and i'll divide by 0.005 and of course this will end up being the smallest amount of all did any of you compute that have any of you computed that at your seats some of the people in here were working on the calculator so that maybe they'd gotten that answer okay so i'll leave that one open but it'll certainly be less than 414 dollars in fact i wouldn't be surprised if this isn't uh perhaps even under 300 dollars what that would be okay um well let's see let's go to the next graphic okay this one says how long must she suffer now we're talking about heather here heather has a hankering for a hybrid now you know i want to say a hybrid i'm referring to these uh dual powered automobiles that run on gas and uh and uh well they run on gasoline and they run on electricity so uh heather refuses to pay interest to anyone uh don't ask that's just the way she is so heather figures she can save 350 dollars each month in a savings account that pays four and a half percent interest per year compounded daily how long will it take her to save 30 000 dollars well you see uh this time we're given the total amount of the annuity that she wants to collect 30 000 but we also know the rent on the account is 350 what we don't know is the number of payments that have to be made so we'll be solving for little in okay well if we come back to the green screen here once again here's our formula you know i i think you're going to be in this section you'll be using this formula enough that you will just know the formula even though the fact that i said i would give it to you on the exam um so once again we have this formula and we're looking for in this time the number of payments so she wants 30 000 dollars actually i don't know how much a hybrid costs so i just pick that number in this problem and uh she can save 350 dollars a month which sounds like a nice tidy sum in fact you might say well gee why doesn't she just go ahead and buy it now but why isn't she buying it now interest she didn't want to pay interest yeah don't ask that's just the way she is you see okay so she's going to put this money into an account and uh the account paid let's see 4.5 percent um let's see that was 4.5 percent but it's compounded daily and uh so you know she's making her payments monthly so what i'm thinking we should do is to say why don't we just assume this is compounded monthly because the compounded interest rates are very close to one another as we saw a few episodes ago so even though the money isn't being compounded monthly let's take this to be the interest rate compounded monthly and if i take a 12th of that let's see 45 over 12 if i divide top and bottom by 3 that's going to be 1.5 over 4 percent and then if i divide by 4 that's going to be 0.0375 percent so i've divided by 12 and this is the interest rate i get per month so this will be 1.00 uh oops i have one too many zeros in there that should be 0.3 uh so 1.00375 raised to the power of n we don't know n minus 1 divided by 0.00375 well now wait a minute how am i going to solve for n in a problem like this what would what would you do to solve for n we'll have to get it uh that term alone and then use logarithms yeah we're going to use logarithms on this problem so this is going to be 30 000 multiplied by 0.00375 while we're at it let's go ahead and divide by the 350 and that's going to equal 1.00375 raised to the nth power minus 1 so if i add the 1 on the other side 30 000 times 0.00375 over 350 plus 1 and that's 1.00375 to the end okay so here i have to use logarithms so this is a good opportunity for us to review what a logarithm can do um on my calculator i have a log base 10 and a natural log button i can use either one of those i don't see that anyone's more appropriate in this problem so let's say we take a common log on both sides so i'll take the log of the quantity 30 000 uh times 0.00375 over 350 my uh plus 1 and i'll take the log of this expression now you remember when you take the log of an exponential you can bring the exponent out in front times the log of 1.00375 so to solve for n i think i just have enough room here to write this i'm going to divide by that logarithm over here so i'm going to divide by the log of 1.00375 and then i can remove that from the other side okay so we're going to have to go to our calculator again and compute this and i see everybody in the class is already working on it so let me try to catch up with them if you'll zoom in on my calculator um i'm going to take uh the log of the quantity 30 000 times 0.00375 and then i'm going to divide that by uh 350 that was the the amount that was the rent on the account and then i'm going to add one and close parentheses so that's the logarithm or that's the numerator which is that logarithm and then i have to divide by the logarithm that was on the bottom and that's the log of 1.00375 and this should give me n and i get 74 and well about 74 and a half now you remember this is measured in months this is the number of months it's going to take her to save that much money so either she's going to save this in 74 months or 75 months 74 she'll come out a little short so we'll say this ends up being 75 months now let me move my calculator off the screen here and you'll see that i'm going to round up because if i round down she won't quite have the 30 000 dollars so if we go to the green screen yeah so i'm putting in in equal 75 and what i just computed was this was this ratio over here and this is a good application for using using logarithms it's going to take her about 75 months how many years is that 75 months about six and a quarter years yeah it's six years and three months six years and three months so it's going to take heather quite a while to save up this amount of money at 350 dollars per month okay now the other question if we go back to that graphic of how long must heather suffer uh what if what if she can make payments of 450 dollars a month let's see when i work this out before class i think the number ends up being around 59 or 60 months so she can save about a year if she raises her payments by a hundred dollars but in the interest of time i'm not going to work that out but if you come back to the green screen i think it would be very easy to adjust this you notice all what i have to do is put in 450 where i have 350 because would only be changing the rent on the account so if you just put a 450 right there you can calculate that and i think you get about five years in that case okay uh now let's look at a different sort of problem this is referred to as the present value of an annuity so let's go to the next graphic um the present value of an annuity this says a lump sum investment that provides regular withdrawals such as a say a trust fund or something uh of a fixed amount and on a regular basis is called the present value of an annuity this is the present value of the regular payments that will be drawn onto the account on the account okay so for example if you put a lump sum of money in a savings account and you decide you want to take a regular amount out of that um for a certain number of for a certain length of time uh then the amount that you put in is called the present value of the annuity whereas the value of all those things compounded at the other end would be the would be the final annuity okay so what we need to do is find a formula that will compute the present value of an annuity in fact well we have that on the first on the next graphic here let me show you the formula and then we're going to derive it up here at the board so a formula for the present value of an annuity i'm calling that a sub p for present value of the annuity the present value is equal to the rent which would be the the regular withdrawals you make on the account times one minus one plus i to the negative in exponent don't overlook that negative in there all over i so let's see where that comes from now um what we're assuming then is that we put some money into a bank account and then we're going to take in payments out of that in regular payments uh and the amount of our dollars so we'll be withdrawing the same amount every time and it's going to be compounded at an interest rate of i uh the interest rate is i per unit per per period and then we have in payments each one for that period okay so how would i figure out what's the present value in that account well let's see uh let's just keep things simple suppose i were going to just withdraw four uh payments so i'd be withdrawing r and r and r and r now where does that money come from well of course it's somehow in this account but i have to have a certain amount of money to cover that payment i'll call that a sub one and then i'll have to have a certain amount of money extra to cover this payment a sub two and then a certain amount of money to cover the next payment and some money to cover the fourth payment now let's see um if if this fourth payment represents the last payment that i'm going to withdraw then this is probably the smallest one of these four amounts because this has the most time to compound whereas this one only gets to compound for one period if i put the money in the account and then i wait one period and then i start withdrawing the money immediately then this guy only compounds one time this compounds for two periods for three periods and for four periods okay now i'm going to use the formula that we derived earlier a equals p times one plus i to the n that's my fundamental compounding formula and i figure this is the amount i've deposited in the account to cover the first withdrawal which is one period later so a sub one times one plus i to the first power is going to have to give me an amount of r dollars at the end of that period so that i can withdraw that so if i solve for a one a one is equal to r over one plus i okay so i know how much that's going to have to be r divided by one plus i what about better put a box around that what about the uh the the second amount now this is going to be sitting the account for two periods before i withdraw it for the second rent payment so i'll have to deposit a sub two it's going to be the in the account for two periods and at the end of two periods it's going to have a value of r dollars so that i can withdraw that that money so a sub two is r divided by one plus i squared let's put that one right below it a sub two is r times one plus i squared well i think you can see the pattern here a sub three is going to be r over one plus i cubed and a sub four is going to be r over one plus i to the fourth power so if i add those four quantities together that's how much money i need in the account to cover those four payments so let's just write that down up here the present value that i need to deposit is going to have to be r over one plus i plus r over one plus i squared plus r over one plus i cubed plus r over one plus i to the fourth power that's how we computed those uh those quantities now of course if there if i were going to make a hundred payments out of this account i'd write down a hundred terms like that but i just pick forward to keep it brief so you could kind of see what's going on now it looks to me like this is a geometric series and for the geometric series it looks like the first term a is one over one plus i and it looks like the multiplier what i multiply by every time is one over one plus i and i'm going to write that as one plus i to the negative one power it'll take up a little bit less space and therefore the sum of after these of these four terms is what i'm calling a sub p and my formula for this is a times one minus r to the n over one minus r so for a i'll substitute r over one plus i and i'll multiply by one minus now let's see r raised to the nth power that's going to be one plus i to the negative n one plus i to the negative n power divided by one minus r and for r i'm going to write it as this fraction one over one one plus i okay now to simplify this i think what i'll do is multiply on top and bottom by one plus i one plus i one plus i and you see what that's going to do is it's going to cancel out this denominator and it's going to cancel out this denominator so i think this will give me r times because this is now cancelled one plus no one minus uh one plus i to the negative n power over now when i multiply here i'll get one plus i minus one one plus one plus i minus one which reduces to just be little i so i'll just erase that and put i right there now this gives me the present value of the account oh by the way this should have been for a power of four because i was adding up four terms but in general if there had been n terms there then i should be putting an n in that expression i guess i was thinking of the ultimate formula that i'm going to derive so that's going to be an n and if i were adding up those four terms i'd just put a negative four right there now let's go back to that graphic and i think you'll see that that's the formula that we've just derived yes a sub p is r times one minus the quantity one plus i to the negative n power all over i okay well let's go to an application of this formula and try solving a present value problem so if we can go to the the next example here we are it says what sum of money should be deposited if we plan to make monthly withdrawals of a thousand dollars each month gee i wish i could for the next 10 years assume the account pays 7 interest compounded monthly and these are monthly withdrawals okay now the formula that we want to use the formula that we've just derived says that a sub p is r times one minus the quantity one plus i to the negative n power all over i okay so what do we know here what is the rent on this account well we're wanting to make a thousand dollar withdrawal so that'll be the rent one thousand dollars times one minus the one plus i now the interest rate this is a 7.2 percent per year if i divide that by 12 if you divide that by 12 what would you get let's do it right below here 7.2 divided by 12 you know 12 goes into 72 six times so that'll be 0.6 so this is 0.6 tenths of a percent per month so i'll be adding on six tenths of a percent that'll be 1.006 and this is raised to the negative n power now what is in well we're making monthly withdrawals for 10 years how many months will that be 120 i'll put a negative 120 power on that and then i'll divide by 0.006 so if i compute that amount uh that is how much i need to put in the account now to allow me to do this so let's go to the calculator and we'll enter 1000 times the quantity one minus 1.006 raised to the 120 and oh i should have put my negative in there first you know there are some calculators where you have to enter the negative uh ahead of time and this is one of them so i'll insert here a negative okay so negative 120 and then close parentheses and then i have to divide that by 0.006 and this gives me an amount of 85,366 dollars and 57 cents in other words roughly 85,400 dollars should be plenty but well let's write down that exact value there 85,366 dollars and 57 cents now how much money are we actually withdrawing from the account 120 payments of a thousand dollars each how much is that 120 payments or withdrawals of a thousand dollars 120 thousand 120 thousand dollars so let's just compare these two we are withdrawing 120 thousand but we put in only 85,000 and that's due to all the compounding that occurs along the way so we don't have to put in that full amount to be able to withdraw a thousand dollars a month okay we have another example our last example and you know what you are a winner yes you are a winner congratulations you've just won a million dollars in the lottery now the money will be paid in equal installments of 50,000 dollars every for every year for 20 years or you can accept a smaller lump sum payment right now if you prefer now assume that you can expect to get 8 percent interest to compound it annually actually that's pretty high to get expect to get 88 percent interest certainly right now for the next 20 years if they offered you 600,000 dollars now would you accept it well of course there are lots of reasons for accepting a lump sum now or lots of reasons for spreading this out other than financial reasons you may need the money right now or you may not expect to live for 20 years and you may want to be able to spend all that money on for yourself and maybe not for not by your heirs so for whatever reason you may decide to take a lump sum or you may decide to take the equal installments but you notice that 20 years 50,000 dollars a year that is a million dollars that you would receive but you don't get it all at once so i think what we should do is figure out what is the present value of these 50,000 dollar payments every year for 20 years and find out if the present value is more or less than 600 and then we can make our decision as to whether we'd take all the money now or not okay so let's come to the green screen and try solving this present value problem okay once again our formula is the present value is the rent times one minus one plus i raised to the negative in power all over i so um it's this fundamental formula now that we'll we'll be using once again now what would you say is the rent in this case on this account 50,000 dollars 50,000 dollars yeah that's going to be the payment we're going to be making or we're going to be receiving so the rent is 50,000 dollars um let's see now what is the interest rate now we were assuming that we could get eight percent per year compounded annually and these are annual payments so i will be eight percent so this will be one minus one point zero eight raised to the negative uh now let's see annual payments for 20 years that'll be a negative 20 power divided by 0.08 okay now if we calculate that let's see what we get has anybody calculated it yet out of your seats i guess not let's let's just try that calculating it right here 50,000 dollars times the quantity one minus 1.08 raised to the negative eight power close parentheses oh excuse me that should be negative 10 power let's back up here and change that the negative 10 power close parenthesis no negative 20 negative 20 power we're getting there close parentheses uh divided by 0.08 and we get 490,000 dollars and 907 so let's round it off and say it's about 491 dollars so it's approximately 491,000 dollars and you know this is it this is it compounding at 8 percent annually that's that's relatively high you might be able to do that but i don't think you could do that necessarily continuously through um 20 years unless you invest in something with a fixed return and i don't i don't think it i think that would be hard to find right now so probably in the long run you wouldn't you wouldn't actually have even this present value if you were going to invest it so uh from a strictly dollar point of view should you accept the 600,000 dollars now let's go back and look at the problem one more time you see at the very end it says if they offered you 600,000 dollars now would you accept it now if you're strictly interested in getting the most money should you accept the 600,000 or the 50,000 dollar payments every year 50,000 well let's see what did we figure the present value to be let's come back to the green screen the present value of those payments is less than half a million dollars or would you take 600,000 dollars now and then you could invest that if you wanted at the same rate of 8 percent so it looks like 600,000 dollars is the more attractive offer this is less than half a million dollars is the is the actual present value of those 50,000 dollar payments and they would give you 600,000 dollars in cash so this has a higher value what you do with this money is up to you you could put this in the bank and you could draw interest on it and every year you could take out more than 50,000 dollars a year Matt for what about not counting inflation or deflation is 20 years from now would a 50,000 dollar payment be worth as much well you see I think that's actually taken into account in the compounded interest yeah so the fact that you're getting compounded interest on that money means that that that money's you're going to get more payments out of it so you may only be getting 50,000 dollars per year but you're getting 20 years of payments out of that due to the compounded interest so hopefully the compounding will counteract inflation you know another problem is if you take the 600,000 dollars now you'd probably have to pay income tax on all of that whereas if you take these smaller payments of 50,000 you wouldn't have you don't have to pay interest or income tax on 50,000 dollars every year for 20 years so that's something to take into account too that we haven't mentioned here so perhaps with income tax included maybe this is the better deal because the taxes may eat up so much of that 600,000 okay well let's go to the next graphic and I think there's a our third formula and last formula for this episode okay now here we have a formula for installment buying suppose for example you take out a loan and you have monthly payments to repay the loan for example if you're buying a car or a house here's a formula that will calculate your periodic payments and it says that the rent or the payment is i times a sub p the present value which is the amount that you borrowed divided by 1 minus 1 plus i to the negative n now that formula looks a little complicated but actually it's just a it's just a variation of our previous formula if you come to the green screen i can show you why here's our present value formula a sub p equals r etc etc now i'm going to solve for r in this formula and if i solve for r i'm going to have i times a sub p the present value divided by 1 minus the quantity 1 plus i to the negative n power now you see if you purchase a car and let's say that car sells for twenty thousand dollars that's the present value of the loan that if you borrow twenty thousand dollars that's the present value that the bank or what are the lender is giving up at that moment now you have to make payments to pay back that amount so that is like that that would be equivalent to saying if you took this money a sub p put in a bank account and you made these regular withdrawals and you and you had just enough money to make all of the payments on the car that would be in payments then that would be the rent on that present value account so what i've done is i've solved for r and just made up this variation of the formula so for example if you if you bought it bought a twenty thousand dollar car and if the interest rate per month were let's say zero point zero zero seven percent then if i divide by one minus one point zero seven and let's say we're going to pay this off in sixty months i'd put a negative sixty there if i calculate that this would tell you how much your monthly payment would be although there'd probably be some other charges included with this like loan charges and so forth but but this would be basically the the amount should have to repay per month let's go to the next graphic and we'll see an application of this formula this is heather's dilemma you remember heather has a hankering to buy to buy a hybrid now what we found out was that if she saves three hundred and fifty dollars per month it'll take her about seventy five months to save up thirty thousand dollars to buy this car now let's consider this alternative uh what if she were to make a fifteen hundred dollar down payment and finance the balance at nine percent for the next seventy five months what would her payments be not including the loan charges well let's come to our formula and find out now let's say it's a thirty thousand dollar car but she's paying fifteen hundred dollars up front so that says that a sub p is only going to be twenty eight thousand five hundred so we have i times a sub p over one minus the quantity one plus i to the negative n okay now let's see the amount she's financing is twenty eight thousand five hundred dollars and the interest rate was nine percent uh per year so per month i'd have to divide that by twelve and that would be zero point zero zero seven five and then i'll divide that by one minus one point zero zero seven five and this was for seventy five months that'll be a minus seventy five the reason i chose seventy five months to repay the loan is uh that's how long it was going to take her to save up this money now let's see how much her payments would be so i'm going to go to the calculator and this will be point zero zero seven five times twenty eight thousand five hundred divided by the quantity one minus one point zero zero seven five raised to the negative seventy five power and i'll close parentheses equals and i have four hundred ninety eight dollars let's say it's five hundred dollars to round it off it's about five hundred dollars okay so what if we found out here if she were saving her money into an account pay cash down the line should be paying three hundred and fifty dollars into the account for seventy five months on the other hand if she makes a small down payment then her payments would be around uh remember her down payment was fifteen hundred dollars then uh she would have to uh should have to make payments of around five hundred dollars a month to pay off the car in seventy five months so you notice this is increased dramatically and the difference is if you're making this car payment of five hundred dollars you're having to pay all the compound interest that lies ahead on the other hand if you're saving three hundred and fifty dollars you are collecting the compound interest rather than spending the compound interest so there's a drastic difference here in what you would have to save now to buy the car later and what should pay now to have the car now of course the difference is uh with the five hundred dollar payment you you're able to drive the car right now the three hundred and fifty dollar payment you're having to wait over six years before you can purchase the car with cash okay so there's a that's the distinction between the payment now um the savings payment now and the loan payment right now okay well let's see we've had three formulas in this episode let me just remind you how they go the first one was to find the value of an annuity then we found the present value of an annuity and then we had the installment loan payment on on a loan now those three formulas once again will be given to you on a on the on the final exam and so you should only need to know how to use it be sure to bring your calculator with you you can use a calculator on that last exam and um I will see you next time you know I thought we'd take a chance to visit with a couple students in the class uh this is Steven over here on the far side from me on on your left and then this is Matt on my right you know we've been filming two episodes a day uh Steven's been in every episode I think and Matt comes for the second episode because he has another class that he attends uh Steven what is your major or do you have a major right now I'm actually undeclared right now you're you're undeclared Matt what about you undeclared undeclared now you know both of these fellows have actually had College Algebra from me already Matt had College Algebra it was in my College Algebra class about what a year ago or so a little over a year little over a year and Steven was in my College Algebra class last semester so if you see that these guys seem to give a lot of good answers it's because they've actually seen this material before but I invited them to sit in on this class um what other courses are you taking this semester Steven what are you taking um I'm taking a computer science course in C plus and I'm also taking a statistics course oh really yeah and an English course in technical writing in technical writing yeah and Matt what about you what else are you taking calculus information systems 1120 intro and also taking modern philosophy oh modern philosophy oh very good now uh let's see Matt here says he's taking calculus one you know to take the regular engineering calculus you need to have college algebra and then trigonometry so you had trigonometry after you had the college algebra course and uh Steven I think may take calculus in the future but uh you know this raises a good point for those of you at home we plan on taking the regular calculus sequence math 1210 1220 and uh 2210 you need to take college algebra and trigonometry both before you go into those courses if you're a business major if you're a biology major uh and you're planning on taking what they call math 1100 intro to calculus you don't need to have trigonometry just math 1050 college algebra and then you can go right into those courses next semester or that course uh we'll see you next time for episode 34