Industry sign banking missouri presentation later
today we want to start a new topic we're going to talk about asset values and interest rates and so anyway as we start this let's just talk about interest rates what those are I have heard a million people possibly less than a million people I've heard many people to talk about interest rates and here's what they like to say and you should not write this down because I'm going to tell you something's wrong here's what they say interest rate price of money know the interest rate is the price of credit what's the difference well first of all what do we mean by the price of something we mean what you have to give up in order to get it right and so if we were talking about the price of money the price of money is what you have to give up more to get it and you can give up your labor services you can give up goods and services products and so if we talk about the price of money the price of money is expressed in terms of the goods and services we give up more what what we give up in order to get it and really done sound ride that didn't sound like the interest rate credit is alone and so the interest rate or interest it's what we pay to get a loan and it could be borrowed money but you know what if you borrow something else some other merchandise not money then you still have to pay for the use of that something and technically that would be interest also back in the old days before there was money and there was commodity monies and things like this are just commodities people would borrow commodities and pay interest for that loan and nowadays we might use a term rent or something like that you might go to well anyway don't go too far down that path but people paid for the use of various commodities so anyway a loan refers to credit and the interest rate is the price of credit and we are eventually in this unit going to get around and talk about what determines that price and guess what and there'll be a lot of details that go into it but prices in markets get determined by two things what are they supply and demand and so we're going to talk about supply and demand for credit and then that will determine the price of credit the interest rate and so anyway we'll eventually get around to that there's different ways of thinking about why I shouldn't say that the price of credits what it is but it plays various roles in our economy one is this is it is an incentive to people to postpone spending okay for example you go to work you earn a thousand bucks you got a thousand bucks what are you going to do with it couldn't just go out and spend it well if somebody comes along and says hey if you'll only that money I'll give you interest and what they're really saying about that is don't spend that money let me spin let me use it and then I'll pay you back later and you can spend right and so when we get interest the way we get that interest is we don't spend everything we have and we postpone our spending until later and so interest gives us an incentive to do that and also let's just say you earn a thousand dollars and you spend $1,000 and you want to spend more you want to spend $2,000 you have to go out and borrow and then you have to pay interest and so when you go out and pay interest you're saying oh man that hurts maybe I ought not to spend maybe I ought to wait and again we're back to this idea of postponing spending okay and so anyway this is an important role that is played by interest in our economy let's kind of shift gears we're going to talk about interest a lot in this this unit of material let's kind of shift gears and start talking about something else that will have interest blended into the discussion future value okay and this starts off kind of simple kind of slow suppose that you have P dollars today I'll put a little dollar sign up there P and that could just be any number it could be ten dollars a million dollars whatever you want but let's say you had P dollars today and then you put that in the bank account or you invest in the bond or something like this and you just let it sit there and now those dollars sent there and then you come back one year hence when you're from today and you say hey how much do I have not account well F dollars and by the way you'll see this all in a few minutes anyway but I'm using P and F these are just you know symbols like I say x and y but I'm using a P because we're talking about the present amount of money okay today and then F a future amount of money and so I'm using letters here rather than the number because we're just using generalized notation whatever we talk about if this were ten dollars the discussions the same as if it's a million the specific numbers work out differently so anyway we're just using these these symbols to represent a certain number of dollars in the present a certain number of dollars in the future so how much is this an answer as well I would get back my P dollars of course you know in the future I would have the original amount plus I'd have interest right if I put this money in a bank account and then I go back a year from now I would get back my P dollars and I would also get back whatever the interest rate is I'll use this interest rate percent and we'll put it in decimals so I give it's five percent point zero five but anyway so if I go back a year from now how much would be in my account in the future of my original P dollars plus interest multiplied by that original P dollars and so if we let that be 5% and let's say P dollars are $100 today then I go back at the end of a year I'd have my original $100 plus 5% point zero five times the hundred dollars equals a hundred plus five equals a hundred and five dollars everybody with me on that that's kind of simple right and by the way I could factor a little bit here and then it would look like this my F dollars how much I would have in the future equals P times 1 plus I not taking that dollar sign out of there just because we don't need that forever ok pretty straightforward right well what if I go back in two years we've got a simple case there I went back one year from now what if I go back in two years I put down the P dollars today and then I leave it there and then I get F dollars and I wait two years well then here's how much is there it's my original P dollars and then for the first year I got interest on that P dollars and then in the second year I got interest on that P dollars again is that it anybody can you hear me out there perhaps there's a glass wall here is that it interest on the interest so here's what happened here's what would happen I put my dollar my money there to work and it stays there and it works for one year and here's when I would have 105 dollars right but I don't collect it I just let it stay there and so in the second year I should get interest not only on my original hundred dollars but here's another five bucks and so here's the interest on the original $100 for year two here's year one but also in year two I'll put year two here I get interest on the first year's interest right and this I times P that was the five dollars and so now it's just interest on interest how much is five percent of five dollars pardon me fifty cents 25 cents right so that's not a lot is it that extra 25 cents and so what would we have we would have $100 plus five percent of a hundreds five dollars five percent of 100 is five dollars and five percent of $5.25 and so then what would it have one hundred and ten dollars and twenty five cents except we don't have to keep doing it this way what I could do is do a little bit of factoring just as I did before and here's what I get P times one plus and then erase a little bit here pause IP IP huh - I okay and then here's a P and in I squared well that's a nice expression now this is back where you remember in fourth grade in your sin why do I have to do that learn to factor things because this can be simplified to P times I mean the the P isn't being simplified but this thing right here is 1 plus I to the second power squared I don't know if you did that but that's what I said fourth grade why do I have to be able to factor I'll never use that again no just for the rest of my life well let's go back over here after one year I'm putting one up there but that was a wound up there P times 1 plus I to the first power after 2 years P times 1 plus I to the second power well guess what I can just write a more general formula and here's what it looks like f equals P my original amount of money F is how much I get back in the future by the way I'm going to put a little footnote down or a subscript down here I'll say T T is in some point in the future some time period okay equals my original amount of money times 1 plus I this is the interest rate in decimals raised to the T power however many years I have to wait now here T is a subscript with the F and it's just telling us T time how many years do I have to wait to get my money back and then there's T though it has a mathematical value we are raising this one plus I to a power and so if I put that money there and I wait 30 years then this would be a hundred dollars times 1.05 there's my interest and then if I wait 30 years 30 is my exponent okay however long you wait that's how much would be there in that account now I'm not going to do this calculation I started to do it but I want to do something else not only get sidetracked here so this is our general formula the first two years one year two years I kind of expanded that to kind of show you what a person would do we could have come back here and said oh I wait three years and then add another IP and then another interest on this interest in another years we could do that but this is just the general statement right here so if you invest some money in the present this is today how much you invest one is just one is the loneliest number one is just the number one the I is the interest rate that we're earning or the rate of return we're earning in decimals right and so then and T is how long that money is working for us okay and so there's how much would show up in that account later on now we did this so I want to come back and talk about it just a little bit more moment ago this is 25 cents you remember this is interest on interest what do we call that compound interest interest on interest it's compound interest what do we call this interest on the principal amount simple interest and so here's what we had we add $10 this is with the two-year investment we had $10 in simple interest we had 25 cents in compound interest and you know you could just I mean a person could say gosh 25 cents who cares almost who cares but here's the deal compound interest matters not so much in two years and not so much in three but over the long term it matters a lot I just mentioned 30 years let's talk about 30 years for just a second let's say that we just said well compound interest is so small let's ignore it and just leave it out of the story and let's say all we got with simple interest just interest on our original hundred dollars how much would be there in 30 years well we'd have our original hundred dollars right and then we'd have five dollars an interest a year simple interest on our principal our original amount times 30 years so that would be a hundred and fifty dollars one hundred plus 150 250 bucks and so if we lead the compound interest off makes it a lot simpler doesn't it then if we get into all those exponents and things like that if we leave the compound interest off and just go with the simple interest we got that number and you know if we done that here and left the 25 cents off rather than send 110 dollars and 25 cents we said 110 and you might go you know that's close enough for government work it's not exact but it's pretty close so how close is it if it's a 30-year deal though is it 250 how far off is at 250 and so let me get the old calculator out and I'll say $100 is my present value of my investment five percents the interest rate and I'm going to let it go for 30 years I'm going to do that again couldn't be 14 million dollars could it that's not what it showed up okay I've got a microphone on and I said edit that out four hundred and thirty two dollars and nineteen cents that's what the compound interest in here with the two-year deal it was worth 25 cents and you could almost ignore it wouldn't buy a soda but the compound interest what's minus 250 the compound interest in this particular case is what a hundred and eighty two dollars and nineteen cents compound interest is quite a bit then if I draw you a graph and I like to draw the graphs first of all yeah here's our dollars and we'll start off with a hundred bucks here's our half and we'll start off with that amount and then we'll this is T and so what we see is this if we have simple interest then what happens is every year we get another five bucks and if we have compound interest what happens it starts going up like that okay and this is simple plus compound interest this is f plus what would be I T P maybe I should do in a different order t IP the number of years multiplied by what the simple interest and then this formula is just what I wrote before equal to F times 1 plus I to the T and the difference between these two the vertical distance at any single point in time we want to choose after one year two year three years five years whatever the difference between those is the compound part and so yeah it's very small like it in one year there's no difference but after that what we have at two years this with 25 cents difference and yeah you can't ignore that but it just gets bigger and bigger and you know what the longer it stays the bigger it gets and it gets huge over time and I mean huge and I will just leave that as an experiment you can perform at home if you like to expand out make it go a hundred years and see what happens I could do that 100 years I think it was 150 yeah and so then what you have how much would it be in 100 years there's only simple interest it'd be a hundred times five that's 500 interest plus your original hundred six hundred bucks so this would go to six hundred well with compound interest thirteen thousand so there's twelve thousand five hundred dollars worth of compound interest the longer you let that grow the bigger it gets now so I guess what I'm doing I'm doing a couple things one is I'm telling you about this compound interest and sort of showing you how it's growing exponentially the other thing I'm telling you this is if you hope to live until retirement you hope to be not poor and one way of not being poor when you retire is to save money and the sooner you save the more that compounding works for you and if what you do is wait until you're 40 and let's say you want to retire when you're 65 you wait till you're 40 retired sixty-five you say for 20 years and that compound interest is working for you yeah but if you start saving when you're 25 and you retire 65 then you've got 40 years with the compounding and boy I mean this thing gets steep and if you could wait you know what was the story about the the Europeans who came and bought Manhattan Island from the Indians living there and they paid like an $12 worth of money beads if you took that $12 and invested it for about six hundred years that wouldn't be that long would it be how many years would be from like 1600 till today be four or five hundred years you invest twelve dollars for four or five hundred years and guess what it's a it runs into billions and so the longer you put that money away the more the compounding works for you and it's not just working for you a little it's exponential Oh where do you find interest at 5% a lot of places five percents not an outrageous interest rate your savings account is at 0.5% well that's not where you found it then okay but bonds corporate bonds you can buy a mutual fund that's investing you can buy today and interest rates change over time but today the interest rate on 30-year Treasury bonds is how much four point two percent 4.25 percent so that's not quite but I mean that is it's safe assuming the government doesn't go bankrupt that is safe but you could go out to corporations and so forth and buy a corporate bond and get about five percent and I don't mean to say just any c
rporation but you buy ideally a mutual fund that is investing in corporate bonds and it's investing in corporate bonds from you know large companies that have been in business for a hundred years or more and are very likely to stay in business pay their bills you know that sell all kinds of products a General Electric and IBM and AT&T and things like that and so you can get above 5% none your savings account not in a passbook savings account probably has no maturity date on and so forth interest rates are all unknown right now but anyway there's a lot of places to get then I had bonds issued by banks years ago and they were paying 12% and you know that was pretty good and there was only one problem is that bank went out of business and I got my money back I didn't really want it back I want it back someday but I wanted to earn 12% for 30 years and then they went broke in about three so there you go anyway let's go back I want to use this and talk about something else a rule of thumb called the rule of 72 and it basically it's consistent with all this stuff but it's a rule of thumb and when we say rule of thumb we do not mean this is exact outfit and so forth we're talking about something that is you know it's a kind of a rule that you can remember and maybe do some calculations on the back of the envelope sometime when you don't have a calculator with you and what the rule of 72 is about is this question how long for my investment to double in value we've kind of switch gears a little bit different letters here but you'll be able to figure this out fine annual rate of return interest if you like and then tea time for investment to double double in value so in this particular case and this is not as exact as what we're doing before but in this particular case we're going to let this rate of return we're going to express that in whole numbers okay if we don't want to do that when we'd have a rule of 0.72 we're not going to do that anyway rule of 72 and so now if we had 5% we just put a 5 up here rather than point zero five okay so anyway I'm going to take an easier number to work with here's what the rule of 72 says 72 equals if let's say we get a 6% rate of return I put a six in my formula times tea time to double the money divide each side of that formula by six Kancil you know about the canceling and so then that's 12 right 12 equal t what's 12 mean it means this if you invested some money and it's earning 6% a year 12 years from now then you'd have twice as much money this is when we don't take anything out and that's what we were talking about before we're not taking anything out each year we're just leaving it there for 12 years and then there'd be twice as much money if it were $100 then there'd be 106 dollars after the first year and the second year there'd be a hundred and twelve dollars and 36 cents there'd be twelve dollars and simple interest after two years and then that 36 cents to be interest on interest it'd be six percent of six dollars in the first year and so forth and it would grow and it would compound and then over the years it would be double suppose you get 8% what's our answer 72 equals eight times T how much is is T now nine equal T now I express this in terms of an investment it's not just an investment our economy grows on average over the long run the economy US economy in real terms grows about 3% a year and so if the economy grows 3% a year and we come back 72 equals 3 times T then what we would find out is that in about 24 years the size of the economy will double in value okay in recent years we've had oh I don't know let's say 2% inflation if prices rise 2% a year then in 36 years prices will be twice as high as they are today the average price not each price if we go back over a generation prices have gone up at about a 3% rate and so prices rise at a three percent rate then every twenty four years they would double in value a house that's $100,000 a day be about $200,000 in 24 years a car this 20,000 be 40,000 okay suppose that you weigh 150 pounds and suppose your weight goes up two percent how much is that that's three pounds a year right and so you weigh 150 and then next year you weigh 150 three and you say you know that's not much and so now your weights going up two percent a year I'm sorry yeah two percent a year in 36 years okay maybe you're 20 today 36 years you're 56 years old and you weighed 300 pounds well this has been 92 when you're 92 you weigh 600 pounds you don't want to gain 2 percent of years what I'm saying find that optimum and lock it in or just don't grow at a compound rate just grow by one pound a year or two pounds or three pounds but don't grow at 2% that's what we get you is the compound weight gain right if you're growing at 2% a year hmm and if you can just get that simple growth just an extra three pounds a year you move up this curve I don't know if this is making any sense to you because it doesn't make any sense but the numbers are there and the numbers make sense now where is this rule of 72 come from well some people just worked with the numbers I have a colleague that wrote a paper about this some people worked with the numbers and they just came up with it and how accurate is it how I could do let's do something like this I've got a calculator I'll do the calculation suppose that we started off with a hundred dollars today and we use that formula we had before the F equals P times 1 plus I to the T what a rule of 72 is telling us is this if this is point zero six and this is 12 right there's our rule of 72s is double your money let me do that calculation I'll say we start off with a hundred dollars six percent a year twelve years now what would our formula tell us that we have in that account two hundred and one dollars and 22 cents so is that pretty close you know that's pretty close that's pretty close from just being able to bought a piece of paper and just do this in a few seconds to say I got an answer it's not exact I mean that's not going to get you through an accounting class or a math class but that's what the rule of 72 is going to tell us and so very often it's a it's kind of a useful thing to have what this colleague found out when he did this study is sometimes you be it depends sometimes you're a little bit better off of the rule of 71 and sometimes you're a little bit better off of the rule of 73 you know that when were working with rules of thumb it's not always exact the problem was 71 and 73 and almost any other number is those are difficult numbers to manipulate in your mind there's a lot of things that go into 72 though you can put two into it three four six eight nine twelve a lot of numbers will divide into 72 and so it's pretty close you can just kind of use the rule something you here's when it's further off when it when it the error is bigger is when these two things the our entity when they are further apart okay when they're like but if they are close like eight and nine you get a pretty good rule of 72 but if they're further apart let's just say that you're getting one percent a year so you start off with a hundred dollars okay then oh no what I was going to do is differently let's say you got seventy two percent a year there we go and you start off with $100 you get 72 percent a year at the end of the year you got 170 a few dollars well rule of 72 would say yeah one year at doubles well that's pretty far off now we're 28 bucks off so the rule of 72 is not so good when these are far apart when they're closer together it's pretty good so anyway there's our story on future value and there's our V our story on rule of 72 a couple things I would add to it and one of them is this that we have is the term compound interest interest on interest what I've assumed here is that we compound once a year that you put the hundred dollars in it stays there for a year and now it's 105 dollars I was working to 5% and then you leave it for another year and you're getting 5% on the previous year what if you get compounding every day what if they were paying interest on a daily basis what if they took that 5% and divided by 365 and said you know I'm going to pay you that much interest a year so then we would have daily compounding we could have monthly compounding quarterly compounding and so what I'm saying to you is that we started off with this example up here where we just have well not that that's a rule of 72 but that formula we were just saying Express this on an annual basis and then that was the end of our story so and then what we would say is it's a annual compounding if we want to have more frequent compounding then we would come in and express both of these things on a different basis and so if we were compounding let's say daily and we're talking about 3 years then we would express our interest rate annual rate 5 percent divided by 365 and that would go in as the interest rate and then if it were 3 years it would be 3 times 365 that many compounding periods at many not years but compound periods if we were compounding on a quarterly basis then what we'd do is we'd say oh well we'd have five percent divided by four so there'd be four compounding periods a year and then if it were three years then there would be what four quarters per year times three years this would be a twelve twelve power and so anyway we can make adjustments to this formula what we are going to do is use annual compounding but I just want you to know that there is no sort of problem with this it's just a matter of how you express these time periods and your interest rate but once you get the time periods an interest rate correct then you can just work with sort of any compounding period you want okay anyway one final thing what would the final thing be when I talk about interest we I've always used like five percent we also want to be able to work with some fractional interest rates like if it's five and a quarter percent then what we would use in our formula is point zero five two five right so again we're always going to be using decimals and in this formula not in a rule of 72 and the other thing I would mention to you is this is and this is just terminology of people who do business in financial markets they would say this is 25 this is the fractional part of a percent 25 basis points that's the point two five well really this is 500 basis points right so the five point two five percent another way of expressing it would be five hundred 25 basis points that's a terminology a hundred basis points per one percent then that is the terminology used in financial markets to describe interest and I'm saying that now because there are other occasions when this can come up questions about this okay well here's the good news the next topic is just exactly the same as this only the opposite is that the good news and here's what I mean by that we write the same formula down we've got some future money and the value of that future investment or they value that investment in the future is what we start off with today times 1 plus I to the T power this would be and just kind of think about this suppose you went to the bank and you said hey mr. banker or hey miss banker I want to give you $100 now and I'll come back in three years or five years or eight years whatever the number is I'll come back how much we'll be in my accountant and that's what this formula would be for sometimes we have investments where that's not the question sometimes we have investments where this number is locked in the future amount for example suppose you buy a bond if you buy a bond today you hand over some money today the present amount of money but the amount you get back in the future that's locked in a bonds a contract it's got the specific amounts of money that you get back on specific dates and in that particular case when this is locked in the future amount of money oh I'm buying a bond it matures it's worth a thousand dollars thirty years from today when that is locked in then the question you start asking is gosh how much would I be willing to pay right now how much would I be willing to put down right now to get that thousand dollars thirty years from today so sometimes that's the question we ask so here's what we do we say hey we've got a formula the thing I want to solve for now is how much am I willing to pay to buy that bond or that future promise of money the future promise of money is locked in there's a contract and so here's what we do we divide both sides by this thing in parentheses right that's mathematically illegal to divide both sides of a formula by the same thing over here it cancels and so let me just rewrite this and I'm going to reverse sides but anyway here's what it says P equals F T over 1 plus I to the T I'm going to annotate this so we've got all the details here future value of investment T years hence two years from now we've talked about these other things this is our rate of return in decimals here's the years hence and then what we would say about this is this is the present value of f dollars to be received t years from today t years hence and just because I don't want to wait until the day before the test yes there's the answer yes you need to know this formula so you can do the the present value calculations you'll be able to use a financial calculator yeah but guess what there are some calculations that are a little bit sophisticated your financial calculators not fixed up to do that so I mean if you know everything about your financial calculator you could probably get to an answer but you can't just like push four buttons to say there's my answer and so it will be bad if you knew that formula and did a couple examples I will bring you some homework to do and it won't be where you turn it in for credit but some homework just for practice so that you be able to do this so anyway I've got an investment it's going to give me some money in the future suppose that somebody said this to you hey I'll give you a thousand dollars five years from now a thousand dollars five years ago would you give $1,000 for that promise if you had a thousand dollars today would you give it to somebody and say just give me a thousand dollars back five years from now no if you've got a thousand bucks why give it up to get back a thousand bucks later you might have some use for that thousand bucks in the meantime and also you could take that thousand bucks and earn some interest on it and so know don't give it to somebody and just say give it back to me later and so the rule that we're going by here is a present dollar is more valuable than a future dollar that's kind of like saying a burden two hands worth two in the bush well here's what we do if somebody promises you future dollars we decrease their value when we start talking about what's that when we translate what's their value presently this process right here you see how we decrease their value here is a promises of some future money the way we decrease that to see its present value as we divide it by something bigger than one this process is called discounting the present value discounting what discounting future dollars to present value okay discounting present or future dollars to their present value okay and you can see that it's discount we've got a future number and we're converting it to a present number but we're dividing that future number by one if it were only one that'd be the end of it right if our interest rate were zero if we just said you know I don't want any interest I couldn't earn any interest in the story then this would be one plus zero that's hey that's one and so there'd be no discounting really a future dollar in a present dollar equally valuable but as long as there is any interest there then we're going to divide that future dollar by a number bigger than one and so then when we convert it over in today's value it'll be smaller than that future dollar the future number of dollars let's do a calculation let's say something like this F equals I don't know 500 bucks let's say and let's say the interest rate is equal to we'll use 5% because we've used it before and then let's say that the T is I don't know three years is three and we'll put this into our formula present value equals five hundred dollars divided by 1.05 to the th
rd power so here we've converted our denominator and as I said before we're going to divide that future amount of money by a number bigger than one and here it's about 16% not quite 16 percent bigger than one so we're kind of discounting those future dollars by about 16 percent but anyway when we do this let me do the calculation here so what five hundred divided by one point one five seven six two five four hundred thirty one dollars and ninety two cents and what I'm saying is this these have equal value to you if you've got the scales there in your head or in your calculator or whatever in your financial calculator and you're saying gosh what's equal what's if somebody promises made five hundred dollars three years and now how much would you have to put on the scales over here to make me feel equally good today an answer is four hundred thirty one dollars ninety two cents and then the scales would be balanced and you say gosh I don't care if I have five hundred dollars three years from now or four 3192 today they're all the same and if they're not then this interest rates not five percent if that just if you are discounting those at five percent a year then those are of equal value and you just flip the coin and say I don't care which one I get any questions about this this thing right here this five percent I told you this term is called discounting to present value this interest rate right here is called the discount rate and later in the semester we will talk about the Federal Reserve and we'll have a different discount rate you need to be able to figure out what I'm talking about by the context how it's being used but the discount rate is the interest rate that we use in these calculations to produce future dollars to their present value and that is what we'll pick up with next time but we will work more with this present value formula so long see you next time
