this is level one of the cfa program the topic on quantitative methods and the reading on time value of money which simply means that money accumulates value over time and i'm guessing that you learned this way back in kindergarten that if you saved an amount today and you invested in a financial asset that had a positive rate of return then it's sometime in the future you would have a larger amount and that's really the concept of time value of money this is known as compounding into the future and you'll see uh that point as i take you through some of these learning outcome statements now notice that the action words ask us to interpret explain and demonstrate so those are important losses in fact i think the first two are probably super important because we need to figure out the components of a given interest rate we need to interpret those different types of interest rate and then we need to use a timeline but the challenge will be for those of you who have not used a financial calculator in several years to re-familiarize yourself with the calculator for those of you who've never used a financial calculator it's your challenge to kind of figure it out and that's what i will do for both uh sets of candidates here we'll work through some problems so that we can and look at those middle losses calculate calculate calculate and we'll do things like future value and present value we'll go ahead and solve for the interest rate and we'll view some problems that have different frequencies of compounding and that'll take us through those losses and let me just remind you that any loss is fair game for a question on the exam so let's start with a basic question if i gave you a choice between uh receiving a thousand dollars today and receiving a thousand dollars in five years this is really a no-brainer right you'd say hey jim let me have that one thousand dollars today and i might ask you i might say hey tell me why you want that one thousand dollars today and some students answer that question they'll say well i want it so that i can spend it and some students say well i want it today so i can spend it because prices might rise in the future and so those are two pretty okay answers but still in the absence of inflation you'd still rather have that one thousand dollars today because you can invest it and as long as the interest rate is positive after five years you're going to have boy dare i say it way more than one thousand dollars in five years and so this is really uh the basis of this concept called time value of money because the dollar day today is that is received today is worth more in fact some would argue that it's worth way more than it's than an identical amount received sometime in the future now look at that last embedded circle point that we've written down there and this is an interesting kind of an addition to our understanding of time value of money allows you the opportunity to postpone consumption and earn interest and this is going to be super important when we when we go through let me go back here quickly when we go through that second loss and we divide an interest rate into its components but let's start with a super simple example let's consider an investor with 100 today and this investor hopes to purchase an asset one year from today now if that savings rate is ten percent the one hundred dollar savings that initial savings will grow to 110 right that's an obvious answer everybody i'm guessing that's watching this recording will know that 100 grows to 110. but what happens if those inputs are not quite so obvious in other words suppose i changed that one hundred dollars today to fourteen thousand eight hundred and sixty two dollars and the savings rate was four point three four six seven percent how much would that grow to after uh after one year well my brain can't figure that out maybe you guys are smarter than i am and you can you can figure it out but let's go ahead and and do some math here at the risk of offending all of your historical mathematical knowledge bear with me as i go through what i call the old long way math and by the way that's that's jim's term that's not uh that's not a cfa term it's not really a finance term and it's probably not even a math term but you'll see what i mean here in just a second so how would we do this we would take the 100 multiply it by 10 and we would get 10 worth of interest on our initial investment and then of course we would add the 100 initial investment to get the 110 but notice how inefficient that old long way math model is we had to multiply and we had to add two mathematical functions wouldn't it be awesome if we could combine those two mathematical functions into one mathematical function and that's of course what i'm calling the new shortway math and to do this we're going to use something that i call the time value factor and i'll show you this equation in a later slide but for now just envision this time value factor as the number one plus the decimal form of the interest rate raised to the number of periods in the problem so all we're going to do in this new short way math is to take our original 100 multiply it by 1.10 raise that to the one power and that gets us 110 and so clearly the new shortwave math is much more efficient than the old long way math so let's make life even more interesting let's suppose that this investor is not going to make a purchase one year from today but is going to make a purchase two years from today now you might be tempted to say wait a minute jim this is super obvious if we earn ten dollars worth of interest the first year then we're going to earn ten dollars of interest in the second year so you might be tempted to say that that future purchase amount is going to be 120 right ten and year one and ten in year two however however this ignores the value of compounding because what's going to happen is of course we're gonna earn ten dollars in interest in that first year but then we're gonna earn interest on top of that ten dollars during the second year as well as another ten dollars so go ahead and work through the old long way math with me here if you want to so we're going to say 10 times 100 that gives me that gives me the 10 dollars right uh add the 100 to it that gets me 110. so now i multiply that 110 times the 10 and i get 11. ah there's that extra one dollars of compounding on top of interest on top of principal or initial savings then multiply i'm sorry add that to the 110 and you get 121. so look you might have been tempted to say 120 is the answer but that's not correct because it ignores the value of compounding and once again this is the nature of time value of money so that future purchase amount is going to be 121. so let's do the new shortway math down at the bottom all i'm going to do is the same thing i did at the top i'm going to take one plus the decimal form of the interest rate and i'm going to square it because it's two periods and that gets me my 121. all right so what i want to do now is show you the financial calculator steps all right let me show you how to use a financial calculator to solve for the 110 and the 121. now notice i have a financial calculator app opened in this slide deck and i'm guessing that you guys noticed that this is not the ba 2 plus it is a different kind of a calculator but i just want to let you know that texas instruments does not have an app that can be used on a mac so i had to use this one here if texas instruments were to have allowed me to use the app here i surely would have done it but the inputs are identical so we should be okay let me tell you a couple of things about this financial calculator when you turn it on you probably have two decimals so i want to show you how to change that quickly so we're going to do a second format notice it says decimals equals two and i usually have three so i'm just gonna do watch this three and then i'll come here and hit enter so that changes my decimals to three you can do four or five uh i think it allows you to do seven or eight i don't think it allows you to to do nine or any number above but for those of you who feel like you need nine decimal places uh uh probably you don't now i also want to make sure that you do this hit your second p slash y button that needs to be at one payment per year the default is 12 so just go ahead and hit uh so whatever you have there do a one and then hit enter and then that should be fine so that's how you want to set your payments per year so that we compound annually and we'll make some slight adjustments to that as we go through all right so you ready for this so let's do let's get our 110 here as our first result so what we're going to do is we're going to hit the number first and then we're going to hit one of these five time value of money buttons that are in the third row so what we're going to do is we're going to input four of them and then solve for the fifth it doesn't matter the order of the inputs but of course it matters which one you're solving for so you want to follow along with me so let's do 100 is our initial amount we're going to call that present value we're going to say 10 percent so leave this in percent form that's the i slash y which stands for interest per year we did this over one period so we're going to make one n now we don't know anything about a payment but do you remember that oh what was that the fourth or so maybe the fifth uh los there was a phrase in there to solve for ordinary annuity and an annuity do so that's going to be our pmt i always tell students i say hey look for the a n button on your calculator which there is no a n button so the pmt button is the annuity button so we're going to do we don't know anything about that so there's no payment so we'll do that and then you're tempted to just hit future value now those of you who have the hp calculator you're going to do the same thing here and all you have to do is hit your future value but the texas instruments people you see the cpt button so you need to compute future value and there is our 110 notice it's a negative 110 and that's because you told the calculator that somebody gave you a hundred dollars we had we entered it as a plus which means at the end of year one you owe that person 110 dollars and so the calculator doesn't care whether you are the borrower or the lender time value of money applies equally to both sides of the contract i will say that if if we had entered that 100 as a minus input then this future value would have been a positive 110. now let me show you something that's really cool with the calculator if we want to do this second one back here the 121 i can go i can just say 2 is n and then i can recompute future value and there's there's my 121. all right let's move on to a series of time value of money questions and i'll show you how to solve these super simple problems with the calculator all right you ready so what did we do in the previous problem we went from 100 today to 110 in the future and then 121 in the future well not all time value of money problems are let's solve for that future amount we call that future value sometimes we want to solve for present value sometimes we want to solve for the interest rate sometimes we want to solve for the payment of the pmt so let's go ahead and take a look at these five problems here once again these are really simple problems problems that you should be able to get using the old math that i described to you earlier but let's go ahead and do this with our financial calculator all right notice i have the question stem in black and i have the input solution in red so let's go ahead and do these as quickly as we can and feel free to stop and practice this on your own bob saves 500 today compute the value of the savings in five years if the relevant interest rate is seven percent all right so this is we're going to say 500 is being saved today so that's a present value and notice how cool the texas instruments operates it tells you what you just inputted there now remember you don't have to clear anything so we'll just go to the next step so we'll say 5 is n over that time period and we'll say 7 is the interest rate and we don't know anything about payment yet we'll do that here in just a second and then we're going to solve for future values so we're going to do compute future value there's our 701 and then i'll go ahead and go clear this so betty needs a thousand in 11 years what do we have to save today ah now we're given we're given that future value and then we need to figure out uh what the goal is how much must be saved today all right so all we're going to do and notice i kind of broke up the order here just to prove to you that there's no uh no reason for you to continue or to believe that you need to put these in some exact order the inputs don't matter the order of the inputs don't matter i should say that the actual input matters but the order doesn't so let's do uh 11 is n and nine is i zero is payment we'll do something with a payment in a little bit thousand is future value and so all we're going to do is hit compute present value there's our 387. getting the hang of this now with your calculators all right bill has 49 today and needs 92 in six years ah this is super important here so let me go down and clear this remember when we did our very first calculation we entered 100 and our answer was a minus 110. so when we're solving for the interest rate we need to be super sensitive about positivity and negativity so watch what i'm going to do here i'm going to say what does bill start with 49 and i'll make it negative so hit the plus minus button don't hit the minus sign there hit the plus minus so what is that that's present value and then 92 is future value 6 is n and 0 is payment and we're going to compute i so there's our 11.07 now think about it if you if you leave the 49 positive and the 92 positive you'll get the calculator will say no solution the calculator will say wait a minute you told us that somebody gave you 49 and then sometime later somebody gave you 92 there is no interest rate that links those two positive cash flows so this is what i tell students all the time i want you guys to take out your pencil and i want you to write this down when solving for the interest rate comma make present value negative now you could make present value or future value negative it doesn't matter but there's going to be a time when we do some capital budgeting problems where we're going to have to make present value negative so you might as well just get in the habit of making present value negative all right let's see bonnie promises to pay 200 per year for eight years uh beginning one year from today all right so let's go ahead and let's take a look at this 200 per year every year for eight years that's our definition of an annuity a series of consecutive equal payments that begin one year from today that's an ordinary annuity and that's going to be the pmt button on the calculator you might think to yourself hey i wish i could cross off the pmt and put a and n on there and that's that's a that's an okay thought process but we're going to solve this the same way that we've always done so we're going to say since nothing happens today we'll say 0 is present value we'll say 8 is n we'll say 11 is i and we'll say 200 is now payment and we want to know how much this is going to be worth at some time in the future 23.71 let's go ahead and solve one final problem bruce has 9000 today promises to pay his sons every year for seven years compute the annuity awesome now we're solving for that pmt we're solving for the annuity so we're just going to do 9000 as present value 7 is n 3 is the interest rate 0 is future value solve for payment and there's our 14 44 55. so hopefully you get the sense of you're entering four variables and you're solving for the fifth one and the fifth one could be future value that could be present value it could be the interest rate make sure you're sensitive to positivity and negativity there it could be payment and what i didn't go through was a problem to solve for n but we could have done that as well all right those simple examples kind of gave us a good sense of the concept of time value of money and notice that in four of those problems we were given an interest rate and in one of those problems we had to solve for it and so let's go ahead and specifically address this loss although i've kind of hinted at it throughout the first couple of minutes of this slide deck notice we're asked to interpret the interest rate as all right a required rate of return so i want you to think of this required return as you and i enter a contract and it could be as simple as that very first example where we had a hundred dollars today and 110 dollars a year from now and so you and i agree to make that exchange it doesn't matter which direction they flow initially or at the end and so what's happening here is that the lender is saying something like let's suppose i'm lending you that hundred dollars the lender is saying something like hey i'm requiring 10 percent in other words i'm going to lend you that 100 and i'm going to demand i'm going to require you pay me that 10 which would give me 110 a year from now but from your perspective what you're thinking is okay i'm borrowing this 100 from jim what am i going to do with it well i need to do something more with it than 10 because i need to invest it at a minimum of 10 percent in order to satisfy my contract with jim so what does this read the minimum rate of return an investor must receive in order to accept an investment so we can view that from my perspective too that ten percent is hey you better do something with it i don't care what you do with it but you better at least get ten percent because i want my 110 dollars uh a year from now now it's also used and interpreted as this term a discount rate this is a generic term that is used in time value of money and other applications that we'll get to in a few future readings in which we need an interest rate to discount future cash flows back to the present and now the third interpretation is something like okay if i lend you that hundred dollars today for one year uh what am i thinking i'm thinking well i sure hope a really super profitable opportunity doesn't come by over the next year because i'm going to lament the fact that i lent you that 100 and all i asked for was 10 percent suppose somebody comes to me next month or in six months and says hey jim do you have a hundred dollars i have this great opportunity we can double our money you know let's forget about the legality of it or or the ethics of it and so i'm going to think to myself oh my gosh i wish i didn't lend that 100 out because i might have something else in the future so look at this opportunity cost you know this from um from your economics days the value of the best foregone alternative uh and the interesting thing about this foregone alternative is that you may or may not know what that is today but we'll talk about that at some time in the future all right so be able to answer an ello a question on the exam that addresses this loss required rate of return discount rate and opportunity cost now which brings us to the second loss which i think is a really super cool one if i were creating exams i would i would craft a handful of questions relating to explain an interest rate as the sum of a real risk-free rate and premiums uh that investors require as compensation for bearing are you ready for this distinct types of risk all right so look inside the gray box here real risk-free rate plus plus plus plus so we have a bunch of things in there now there's irony in life and there's coincidence in life i just gave this lecture to my students uh yesterday it's an undergraduate financial institutions class and so we talked about this exact uh los so what i want to do uh over the next couple of minutes is i want to go ahead and give you that quick lecture and i want to refer back to uh our original interest rate like 10 so when i gave you that 10 interest rate on that very first slide we didn't ask the question where did it come from right i just made it up jim just made it up pulled it right out of the air but what we want to do is we want to divide that interest rate which was ten percent in my example could have been eight percent could have been 12 could have been 2 we want to break that up into its component parts so notice that we have five of them here so this is this is essentially five really good exam questions so let's start with this concept of a real risk-free interest rate and i want to address the risk-free component of it first so let's suppose that i come to you and i say i want two loans i want a 50 000 loan and i'll pay you back in a month but i also want another 50 000 loan and i'll pay you back in one year so we have two loans identical amounts both the same borrower but two time periods all right so i want to talk about this risk-free component first so you're going to say to me all right jim i have a hundred thousand dollars way over here and i'm happy to lend it to you but what do i get out of it and before i start answering that question i reach into my pocket and i say how about if i give you this as collateral and this as collateral is the pink panther diamond now those of you who are old enough to have watched the old peter sellers pink panther movies will know that they are super funny those of you who are young enough to watch the steve martin pink panther movies will know that those are extremely funny so that's your homework assignment when you have a free moment go watch uh go watch those movies but anyway those of you who watch those movies know that the pink panther diamond is priceless so if i use that as collateral in the signing of this contract right fifty thousand dollar loan here fifty thousand dollar loan here you're going to accept that as collateral because it turns it into a risk free proposition now of course when we sign this contract i'm going to say look when i when i repay you all of your capital and your interest i want i want my pink panther diamond back all right so i'm going to prevent you from taking that pink panther diamond and selling it over there or i'm going to prevent you from re-hypothecating that pink panther diamond so that you can't use it collateral for your own for your own uh borrowings so i want you to think about that there's no chance that you're not going to get your 100 000 plus interest back because you have this as collateral now i want to address the real component of this and to do the real component of the real risk-free interest rate i also have to consider the conversation of inflation so real a real interest rate whether it's risk-free or not a real interest rate means a compensation for the change in your status in life over time all right so you want to be better off think of real as a measure of the improvement in the quality of your life so let's think about this let's suppose that a loaf of bread costs a dollar today and you expect a loaf of bread to cost a dollar ten a year from now well clearly clearly the change in the value of a loaf of bread let's call that the change in the price of a loaf of bread let's call that the rate of inflation of a loaf of bread is 10 well if you lend me my capital and just charge me 10 then you're no better off a year from now than you were today you can still buy just a loaf of bread so the real risk-free rate of interest tells us how much better off you want to be in the absence of inflation and then of course we have to add an inflation component so the real risk-free rate of interest is the starting point plus you want protection against that rate of inflation so you might charge me two percent real rate and then you might charge me 10 because inflation is 10 so if we add real plus inflation you might charge me 12 so that gives you a real return plus it protects you against inflation hopefully that makes perfect sense but then what you're going to do is add a couple of other premiums or premium depending on which word you prefer to your nominal interest rate that you're going to charge me you're going to charge me a default risk premium so you're going to look at me and you're going to say all right jim what do your assets look like what does your mortgage payment and car payment look like how much money do you make what kind of income do you have outside of your college teaching job so you're going to do a credit analysis on me and so you're gonna add a default risk premium maybe it's three percent right so where are we you're gonna charge me you're gonna charge me two percent for the real rate you're gonna charge me 10 for inflation you're gonna charge me two percent for default so now we're up to 15 but notice we have some more plus signs we're going to add a liquidity premium so here's where we get a differential between the one month loan and the one year loan all right and this has a lot to do with that opportunity cost conversation that we just had a moment or two ago so i want you to think about this you're gonna be without fifty thousand dollars for one month on that first loan you're gonna be without that fifty thousand dollars for one year on that second loan so you are giving up liquidity you are giving up access to that 50 000 for a month and 50 000 for a year so you're gonna charge me a liquidity premium for giving up that liquidity the access to that capital but here's the thing you might charge me one percent for the one month loan and you might charge me four percent for the one year loan so now we're up to 16 or 19 ah do you see how cool this is we're arriving at different interest rates for different time periods and then we're going to add a final maturity premium which might sound a little bit like the liquidity premium but it has its own subtleties you're going to charge me less for the one month loan on this liquidity premium than you will on the one year loan because you ready for this let's suppose that over the next uh week or two that you want your money back and so you go to the secondary market and you try to sell your loan well you're going to be able to sell your one-month loan for a much higher price than you will receive for the 12-month loan that's the maturity risk premium the fact that you're at risk and the risk is largely due to changes in interest rates so notice in the end you might charge me but i'm going to make up some numbers here you might charge me 20 for the one month loan and 24 for the one year loan based on these five factors so let's go ahead and look at these embedded circle points the real risk-free rate is the single period interest rate for a completely risk-free security if no inflation was expected all right that makes perfect sense my pink panther diamond makes that a risk-free security now of course i'd probably be a nut to who hand over the pink panther priceless diamond to you as collateral for this i'd probably go to a bank or something and and i'd probably want to keep the pink panther diamond in a safe deposit box or some place safe but anyway the key point is no inflation so then we add an expected inflation premium to that right and then we add a default risk premium let me go ahead and read that the possibility that jim will fail to make a promised payment at the contract to time and in the contracted amount so there is default risk on this because i may not pay you now of course there is default risk even though you're holding this collateral because when i do pay you you have to you have to turn that collateral back to me we're going to add a liquidity liquidity premium notice in bold we have if the investment needs to be converted to cash quickly that's what i was saying to you about selling it and then the maturity premium accounts for additional risks for the longer time to maturity all right moving on to calculate and interpret the effective annual interest rate uh given a stated annual interest rate and the frequency of compounding so let's go ahead and introduce this concept of hey sometimes the banker is going to pay you interest that's compounded every year but the banker doesn't have to pay you interest compounded annually the banker could pay you every six months or every four months or every month or every week or could pay you every third monday or every third month i mean the banker could offer interest over any time period that he or she or she chooses but what we need to do is we need to convert something like that into what we call an effective annual interest rate so there are two equations there that you probably should memorize and of course as we always do here in these slide decks we try to give you a good example so you ready for this suppose we're given a stated interest rate of 10 percent so the nominal interest rate the stated interest rate is almost always given to you as an annual number and then in parentheses or as we have here comma compounded quarterly here is what we get for the uh ear so we're going to take that here are you ready for this that time value factor that i explained to you before right it was 1 plus the decimal form of the interest rate so in that first slide we had 1.10 but now we're chopping that 10 percent into four different time periods three months then three months then three months then three months so what we need to do is we need to divide that decimal form of the interest rate by four so it's one plus .10 divided by four and then we're going to re-raise it to the four time periods because we're compounding quarterly subtract out the one and there we get the 10.38 effective annual rate so here's our calculate the effective annual rate so there we do that quarterly and notice what we give it to you here let's suppose we do it monthly we just divide by 12 raise it to the 12 and of course we're going to get a higher ear 10.47 versus 10.38 versus the 10 stated or the nominal annual interest rate so look at the look at the two arrow points at the bottom and notice that we have bolded this the more frequent the frequency of compounding the higher is going to be the effective annual rate now here's this slide that i was telling you about earlier this is the this is the financial mathematic formula that i was describing to you just a few moments ago so here we go ready future value is equal to present value times 1 plus r 1 plus the interest rate and remember and see you guys know this at this point that r is the decimal form of the interest rate and we're going to raise it to the n power so this is the process of compounding notice that equation there we're taking a present value we're multiplying it by 1 plus a positive interest rate so we're just making it bigger as we go out into the future and so right this is what you want to do this is what i want to do i want to save an amount today right let's suppose it's 100 000 and i want to retire in oh let's say 40 years and i want to earn 10 so i want to retire with you know a billion dollars and this is the value of compounding we're taking an initial savings amount and we're making it bigger in the future now of course let's go ahead and stand on our heads you ready let's stand on our heads let's take that top equation and instead of solving for future value let's go ahead and solve for present value so you do some quick algebra i think that's called cross multiplication and you arrive at present value is equal to future value divided by that time value factor and so those are two super critical formulas that provide the fundamental support for the financial calculator right here's here's my can you guys see my calculator so we're using those five time value of money buttons and the calculator is programmed to use those handful of equations here let me put this one up i love this calculator here this is my favorite calculator of all times i carry both of them my family thinks that i'm a little bit of an oddball but i don't go anywhere without my financial calculators in case someone pops up and say hey jim solve this problem for me now of course the los asks us to worry about different compounding periods so we need to make the adjustment and so the adjustment is here let me go back here to present value so look at that and if we're compounding over multiple periods during the year we need to worry about the n right and we need to worry about the r and so here's what we do so all we do is take the r divided by the m right and then the m times the end and so this is going to be able to identify for us different compounding periods all right you ready for this suppose that you need ten thousand dollars in your savings account at the end of year three right so that's a future value but the account offers a return of 10 per year so if we stopped right there if we stopped right there we would just enter those thing those inputs just like we did before 10 000 would be future value n would be 3 10 would be i zero would be payment and we would solve for present value right but now notice that we have this extra thing in there compounded quarterly all right so how are we going to do this all right so let's go ahead and look at a timeline and this helps us with our our final loss let's go ahead and look at a timeline so notice we have 0 and then 1 2 3 but instead of a one two three we have four quarters eight quarters and twelve quarters so this is what we need we need ten thousand at the end of year three how much would we need to save today in order to solve that problem and so there's our good old formula 7435 and notice the adjustments that we make in the denominator because what we're doing is we're saying boy we're not getting 10 every quarter right we're only getting 10 divided by four every quarter so what's that two and a half percent every quarter and it's not three years but the number of periods is now twelve quarters so we divide by four and we multiply by three to get that seventy-four thirty-five now of course you can go ahead and use your financial calculator to make that same computation now notice what we've done here in the previous slides i went ahead and get out my financial calculator and showed you that but here i'm just showing you the steps so that this is probably a little bit more efficient so get out your financial calculator and go ahead and work through the inputs and notice we have for the ba two plus and we also have for the hp 12c and here let me let me do this for you what we're doing is we're going to go back to this equation here where am i this equation here present value is just discounted future value and we're making a slight adjustment here if we want to do it oh man i don't want to say this the long way math but let's go ahead and do it with the financial calculator so we need to make a slight adjustment so notice the slight adjustment is that the n is not three and that the i slash y is not ten the n is now twelve and the i slash y is now two and a half and so you'll get that same here let me go back here you'll get that 7 74 35 whether you use the time value factor formula or the slight adjustment for the for the financial calculator now let me give you a piece of advice here lots of students want to say something like wait a minute jim i can hit my p slash y button my payments per year and i can i can change that to four and yes you can you can solve that by changing that uh setting on your calculator but my concern is that if you do it then you may forget to change it back to p slash y equals 1 for the next problem now there are exceptions to my rule so i think it's and we think that it's much more efficient to go ahead and make the slight adjustment here make n12 and make i two and a half and you get the 74 35. now i know that some of you who are way smarter than i am and you can just look at this equation right here and you can say now i'm going to use this formula so go ahead and use that one remember the cfa institute doesn't really care how you obtain it the answer as long as it's your answer and not your neighbor's answer of course so do it this way or do it that way how about we do another quick one 50 000 12 year loan interest rate of one percent per month all right so we're going to go out 12 times 12. so there we go 144 periods right those are months we're taking out this loan 50 000 today and we're being charged one percent per month so what is that future value so there we go we're going to go ahead and raise the 1.01 right to the 144. so there we go 209 530 dollars whoops i went ahead too quickly so those of you are super quick with your brains you can use it do it that way or you can use the inputs here and so note we're making the slight adjustment again n is now 144 and i is one so you get the 209 530 whether you use either calculator or whether you use this equation back here i gave you this definition of annuity just a few moments ago right an annuity is a series of consecutive equal cash flows and of course it's finite meaning that it occurs uh over a fixed period like 10 years or 100 years all with the same value your calculator assumes that when you hit the pmt button that it is an ordinary annuity so let me give you that definition right off the bat here ordinary annuity is a series of consecutive equal payments that begin one year from today or one period from today we all are probably familiar with our mortgage payment what do we know we say pay to the order of mortgage banker thousand dollars a month we do that every month for 10 years or 20 years or 30 years annuity due on the other hand is a series of consecutive equal payments but the first payment is due d-u-e it's due today insurance premiums of course you know i make my uh car pre premiums uh quarterly and so my my bill is due my turn my year is due in november so on november 1st i make a quarterly payment and then i do it again you know over the next time periods and so it's kind of like uh it's kind of like prepaid insurance for that uh for that time period so annuity due is relevant for insurance premiums now of course we need to make some slight adjustments to the present value and future value of the ordinary annuity given our time value of money time value factor equations that we've become familiar with through this slide deck and so here we are and you guys can memorize this if you want or you can just use your financial calculator which i'm guessing most of you will now the present value and future value of the annuity do is a little bit more complex so notice what we have to do here we need to multiply here let me go back here we need to take those two equations and multiply them by 1 plus the interest rate and the reason that we do that is i'm going to do this here with my hands so here's here's an annuity an ordinary annuity right so one two three four so let's suppose that's four years right so in year one two three four can you guys see that or is that backwards how about if i had to go like this does that help how about if i go like that i don't know what helps but anyway you should get my point of the timeline right but in annuity due the payments are due like this right or should i go like that or should i go like that i don't know so i go like this so what we need to do is compound that out one more time or or what you can do is you can here let me skip ahead let me skip ahead or you can change your calculator mode so let's do this problem here all right you ready sorry i was skimming through there quickly i hope i didn't give anybody a migraine so here we go dave buys an annuity regular series of payments 200 per year 15 years he will receive equal payments at the beginning of every year ah so what premium should maxwell be willing to pay for this annuity assuming 13 and a half percent all right so there's the key and this is crucial i do this to students all the time on my exams i don't ever say hey it's an ordinary annuity or hey it's an annuity do i couch the question stem in the following manner i say suppose we have a series of regular payments that begin one year from today ah that's the ordinary annuity or that begin today ah that's the signal for the annuity do so be prepared to try to solve that problem uh on your level one exam so here we go method one using the formula we get 14 30. using the calculator ah here we go let's go ahead and switch the calculator from end mode which is the default and which our calculators have been set as such set it to beginning mode and so those of you who have the ba2 plus you need to use your second function so do your second begin and then second set so look at your second set is up under the enter button until the begin displays and then you need to do second quit so you need to do your second second second second so this one here takes a few moments for you to get it uh you guys that have this wonderful calculator here just look up uh under the is it the seven and the eight look up under the seven the eight and yours is just a simple uh blue function to set to the begin mode so if you do that then your inputs are exactly like they were in the past and you go ahead and you get that 14 30. so 1430 whether you use the time value of money factor formula adjusted for the annuity and then adjusted for the annuity due or you take the simple approach and set your calculator to begin mode now let's take a deep breath are you ready for this let me warn you when you change this to begin mode and you solve this problem on the exam immediately set it back to the end mode so that you don't solve the next series of problems in the annuity do mode which will inevitably give you the incorrect answer which you inevitably know in these multiple choice questions will be one of your answer choices now we've done timelines here i've done these i've shown you a couple here is a physical illustration and so my advice is to go ahead and construct a timeline no matter what time value money problem that you're being asked to solve because it helps you visualize the cash flows what you can then do as you can see regular amounts you can see them as annuities but if they have unequal amounts you can see them as a series of one-time lump sum future values and so uh let's go ahead and take a look at a visualization here notice the loss just asks us to demonstrate this so let's suppose that we have a hundred dollars in period one 150 and period two 250 and period three three hundred and period four 250 in period five assuming that the interest rate is five percent present value now remember this loss doesn't ask us to calculate this loss just asks us to demonstrate now of course at some future reading we're going to have to calculate a series of consecutive unequal cash flows their present value but not today so there's the there's the timeline right zero is time period today then there's the one two three four five and so in order to compute those present values all we're gonna do is just discount them back using the regular old time value factor and we get uh well we get those intermediate answers and then we get 889 i know i've spent through those quickly but go ahead and feel free to stop the slide deck if you want to go ahead and do the math i will say this that in a future reading we will show you how to use your are you ready for this your cash flow button in order to solve for present value but for now i think we've done enough for this reading and this satisfies the los that final one of demonstrate the use of the timeline so at the end of every slide deck i like to give candidates kind of my thoughts about which ones that i think are more important and remember my opinion is just my opinion the institute can ask questions based on any los so my advice to my student candidates is to just know it all and and and beyond with your life that's really what i say but here clearly that second one is super important explain the components of an interest rate i love that second one because that's going to give you a really super foundation as we move through these readings in level one because we're going to continually talk about liquidity and maturity and opportunity cost so i think that's a super important one and then of course make sure you are familiar with the steps leading to calculator outputs [Music]