Understanding Compounding and Financial Calculations

Now I would like to discuss a little bit the compounding periods. In particular, the fact that in finance, it's a general convention, it's a common rule that R is always quoted on annual basis. So R is annual rate of return. Now why is it important? Because very often even though R is quoted on annual basis, in reality it's compounded or discounted depending on whether it's future or present value on a more frequent basis. For example, if you put money in the bank, you would be quoted the annual interest rate. In fact, it actually is going to be annual effective rate, something we're going to talk about in a moment, but it's always quoted on the annual basis. So let's say R equal to, I don't know, 12% annually. Okay. But what you also realize is that in reality, instead of getting 12% every year, you're getting 1% every month. Okay. So that would be a typical example of annual rate compounded monthly. So in this case, your R is equal to 12%. Okay. but your m is equal to 12. Okay, so what it means, it means that m, which is frequency of compounding within a year, represents the frequency with which your interest is accrued during the year. So when m is equal to 1, you're talking about annual compounding. So m equal to 1, that's going to be just your regular annual. When m equal to 2, that would be semi-annual. Okay, and so on. Another example could be m equal to 3, that would be quarter. And finally, well, two more, m equal to 6. No, actually six we don't have. Six, that's uncommon. So M equal to 12, that would be your monthly. Okay. You could also have daily compounding. So M is equal to something like 365. Okay. And finally, you could have so-called continuous compounding continuous compounding is is a completely different concept the formula changes completely okay but continuous compounding is some kind of imaginary situation so it does happen sometimes but it's hard to imagine so the idea is that you get interest like every fraction of a second okay so let's not worry about it let's just focus only on discrete cases so if that happens In finance, regardless of your M, R will always be quoted on annual basis. It will be your job to figure out what happens with your future value or present value if that annual R is compounded more frequently than on the annual basis. So that brings us to this concept of more frequent compounding periods and effective annual rate. So for example, in our case here, if you have... So let me first do one... Let's just see something here. Okay. So let me do, I'm just trying to make it consistent with what you have there on these slides. So let me start with an example. So let's say you have bank A and then you have bank B. Okay, so let's say bank A offers you 12% compounded annually. Actually, you know, let me do, sorry about that. Let's say bank A offers you 12.5%, okay? compounded annually okay and bank b offers you just 12 but compounded monthly okay So let me explain what it means. Let me start with bank A. What it means is that your future, let's say you put money for one year only. You know, when you extend it to two, three, four, it doesn't really matter. The result is going to be still the same in terms of which bank you prefer. But if you simply look at these two offers, what really jumps at you is that one bank gives you higher overall rate of return. Looks like 12.5 is better than 12.5. than 12 but the catch is it offers you 12.5 compounded annually so you're getting 12.5 percent only once a year Bank B on the other hand offers you 12 percent annual rate but compounded monthly which means that instead of getting 12 percent every year you're going to get 1 percent every month okay the question is what is better well if I I told you 12% compounded annually I hope you could immediately tell me that bank bay is definitely better because everything else equal more frequent compounding is better and the logic is very simple once I get my first 1% the next 1% will be compounded on a big your base so i have more opportunity to generate that interest on interest that so-called snowball effect so everything else equal 12 compounded monthly is definitely better than 12 compounded annually so i hope that's clear so you know but we don't look at that we look at something a little bit uh you know more complicated so you have more frequent compounding versus less frequent but with higher rate so the question is which option is better so there are two ways of doing it one is just to simply calculate the future value in which case in each case so it's going to be some initial amount present value whatever you put in thousand dollars hundred dollars as it turns out it doesn't really matter and then you have one plus 0.25 0.125 so that's what you're gonna get a year from now if you put money in bank a if you put money in bank b the formula changes so you have present value times one plus but instead of having your r you have r divided by m So again, if you look at this formula here, you have r divided by m, where r is your stated annual rate, and m is frequency of compounding. So it will be 0.12 divided by 12, which results in 1% monthly. That's basically what we thought it's going to be. But now, here is to the power of 1, because it's only one year. Here it's one year multiplied by 12 months. So you're talking about 12 periods. So the question is what is bigger 1 plus 0.25 to the power of 1 or 1 plus 0.12 divided by 12 to the power of 12. Okay and let me just do the calculation. So bear with me. So I'm just gonna use so 1.25 I'm sorry. So yeah, let me just write that down first. So that's going to be PV times 1.125. And here, that's where we need the calculation. So it would be PV times and all I need to do now is to have 1.01 to the power of 12. And that would be 1. 12 6 8 so as you can see the difference is not huge but bank b gives you a better option okay because again i don't care about initial amount of money because that would cancel out when you compare. What matters is this factor here. Okay, the overall accumulation. So as you can see after one year I will have 12.5% more money if I invest in Bank A. but I will have 12.68% more money if I invest in bank B, even though the actual rate is lower. So that's basically something that you really need to take into account. So if you have more frequent compounding or more frequent discounting, if you're using present value instead, you need to be careful. You need to understand that the actual rates will always be quoted on annual basis. You need to make modification. Now, a better way to do that is to always calculate annual equivalent. Okay, so when you cross shop, you know, across different alternatives, if you have bank A, bank B, bank C, you have multiple banks or multiple investment opportunities with different frequencies. The best option for you is to always convert everything to the annual basis compounded annually. So for bank A, so-called effective annual rate is already 12.5%. So we don't need to do any modification because it's already compounded annually. But for bank B, the effective annual rate... would be an annual rate compounded annually that would make you indifferent between getting 12% compounded monthly or getting that X rate compounded annually. And as you can see already it will be 12.68%. Okay, so it's going to be equal to 12.68%. All right, so that's actually how it's done. So instead of finding actual future values or present values, what you do instead, you immediately realize. that the rate that you're getting quoted on annual basis if it has more than annual compounding more frequent compounding than annual you need to immediately convert it into effective rate okay turns out that there is a very very nice formula for it you could just figure it out right away so first of all i'm going to skip this example here because it just does exactly what i did except that it has 50 initial value so okay so what i'm going to do i'm going to skip right to here okay so as you can see it's also 12.68 it's exactly the same stuff i just I just did a little different example. But the point is, you don't have to do any of these calculations. You could jump to effective annual rate formula, which is equal to 1 plus r over m to the power of m minus 1. So in our case, right, if I look at that, you know, bank B, okay, my r is 12, my m is 12. Okay, all I have to do is just to plug numbers right here. So I have 1 plus 0.12, excuse me, divided by 12 to the power of 12 and then minus 1. And, you know, if you do calculations, you will get your 0.1268 or... 12.68 percent okay so that's that's basically uh how you do it so that's called effective annual rate or ear formula okay now for many standard financial products like deposits, bonds, mortgages, you know different types of car loans, the effective annual rate is already given to you by the financial institution. By law the banks and other financial institutions are required to disclose their interest rates on effective annual basis. The reason for it is because the governments do not want financial institutions to confuse people so for example again if you are not in finance if you have no idea you go to Bank a you go to Bank B you might be confused you might be actually lured towards Bank a because it has higher annual rate but Bank B is actually better it turns out so governments understand that consumers don't have the knowledge so they require every financial institution to convert all of their rates into so-called effective annual basis. That's why you see something like APR or APY. APY is your annual percentage yield on deposits that is already converted into effective annual basis. APR is interest on loan, on mortgage for example, or on car loan that is also converted to the annual basis, annual compounding basis. Because all of those products require you to make more frequent payments. For example, on mortgage, I'm not paying annually, I'm paying monthly. Similarly, if I put money into deposit, I'm not getting my interest annually, I'm getting it monthly. But the interest is still on an annual basis. So how do we take into account that difference in compounding? Well, again, financial institutions are required to do it for you and disclose it. so that you don't get confused. Now, the question then is why would we need this formula right here? Well, because remember, in finance, we don't deal with only standard products. Sometimes we deal with certain cash flows that are generated by different projects that have nothing to do with mortgage or deposit account or a bond. That would be a project that actually generates our cash flows. So some projects will generate cash flows on annual basis, some of them will generate on monthly, some of them will generate on daily basis, so different frequencies. So it will be your job as a finance manager to make sure that you always take into account the frequency and incorporate your effective annual rate analysis. Okay, all right, so the last thing we're gonna do in this lecture is to keep we will keep talking about multiple cash flows but now we are going to introduce so-called simplifications. So these days we don't need those simplifications because you know the spreadsheet software like excel is very powerful so I mean that simplification doesn't really matter but in the past when computers were not well developed The finance people ask mathematicians to help them to figure out how to calculate those present or future values when you have multiple cash flows. OK, how can we simplify them? And as a result, those simplifications become so common in our finance lives that now a lot of actual standard products in finance are a result of those simplifications. So. what are those simplifications to the first one is perpetuity so let me just start talking about it right away so what is the perpetuity excuse me so perpetuity is a constant stream of cash flows that lasts forever okay so instead of having c1 c2 c3 and c4 excuse me you have C C C C that lasts forever So you might wonder what does it mean? Well, first of all, perpetuity does not have future value. And that's obviously why, because there is no maturity, it lasts forever. So there is no point in talking about future value. However, perpetuity does have present value. And you might wonder how is it possible that, excuse me, you have multiple cash flows that have infinite time horizon. okay, that go all the way to infinity. But somehow you sum them up and you get the finite number. The reason for it is because each of the subsequent cash flow is getting smaller and smaller and smaller. So in the limit when your time t approaches infinity, each additional cash flow that happened here, for example if you have c divided by one plus r, you know to the power of 1000. This one is going to be so small that we're going to approximate it to zero. So basically what's going to happen because each of the additional cash flow is getting smaller and smaller and smaller and smaller, mathematically we can prove that this entire sequence pretty much collapses to C over R. Okay, so the present value of each of those cash flows when you sum them up is going to be simply equal to c divided by r. Okay, so now that we understand the general idea, let me derive that result. So the only reason I'm deriving it is because I want to make sure that you understand the logic here. So from math, we know that 1 plus a plus a squared plus a to the power of 3 and so on all the way to infinity. would be equal to 1 divided by 1 minus a if a is less than 1. Okay so what does it mean? So think about it this way why do we need a to be less than 1? Because whenever you take a number that is less than 1 and you raise it to the power the higher the power the smaller the number is going to be. Okay, so for example, if you have 0.5, you get 0.5 squared. It's going to be a smaller number. 0.5 to the power of 3 is going to be even smaller. If a is greater than 1, then every subsequent term will get bigger and bigger and bigger, and that would be infinity. Okay, so it's important to understand that when a is less than 1, then 1 plus a plus a squared plus a to the power of 3 all the way to infinity is going to be equal to 1. divided by 1 minus a. It's a known mathematical fact. Now let's look at our present value. Okay so it's equal to c divided by 1 plus r plus c over 1 plus r squared plus c over 1 plus r to the power of 3 and so on. So what I'm going to do now I'm going to take c over 1 plus r out, okay? Factorize it. And now I have 1 plus 1 over 1 plus r plus 1 over 1 plus r squared and so on, okay? Now look what's interesting here. I can claim that 1 over 1 plus r is my a and 1 over 1 plus r squared is my a squared. Can I legitimately say that? Yes, because a is less than 1. 1 over 1 plus r is less than 1 unless r is equal to 0. So if r is equal to 0, then yes, it blows up, it becomes infinity. But as long as r is less than 1, excuse me, less, excuse me, greater than 0, which is normally the case, 1 over 1 plus r is going to be less than 1. Okay, so now I can utilize the formula. So it's going to be C over 1 plus R. And then I'm going to have 1 divided by 1 minus A. But my A is 1 over 1 plus R. Okay, the rest of it is just very simple. So it's C over 1 plus R. Then 1 plus R goes on top. And I'm having 1 plus r minus 1. So cancels out. 1 and 1 cancel out. So I have c divided by r. Okay. So yes, it took us a couple of minutes, but at least, you know, we are being thorough here. So you see where the number is coming from. I don't like giving formulas that absolutely are out of nowhere. So I prefer to derive as much as I can unless it requires some... more mathematical knowledge than you guys might have. So in this case it's a very simple derivation. So anyway, turns out that perpetuity is a very useful and simple case. So even though you have multiple cash flows that last forever, the pricing or present value collapses to a very simple formula C over R. Turns out that perpetuities exist in reality. There are some financial instruments that are perpetuity. The most prominent example is British Consul. So British Consul is a bond that has no maturity. Another example of perpetuity could be a preferred stock. as you will see later preferred stock just promise you a fixed dividend that lasts forever okay there is no maturity because it's a stock and the dividend does not vary because it's a preferred stock so it's guaranteed and it's fixed so that's another example of of perpetuity so let's look at british council for example you know that it promises you to pay let's say 15 pounds the reason is pound is because british bond forever and let's say the interest rate or discount rate is equal to 10 percent the question is what is the present value and it turns out that the present value is very simple simply to calculate it's just 15 pound divided by 0.1 and again that gives you 150 dollars as you will see later it means that the price of a bond is going to be equal to 150 dollars excuse me pounds in this case so again i'm trying to stay away from pricing right now i'm just trying to find present value but sort of going a little bit ahead keep in mind that in the you will learn in the future that the price the market price of a financial instrument is just the present value of all the cash flows that instrument will generate that's why it's so important to understand these concepts here. Okay, so that was perpetuity. Some perpetuities are growing, which means that the cash flows are not the same, but they are growing at a constant rate. So that is also a very interesting example. So you see changes over time, it's getting bigger and bigger and bigger, but Each subsequent cash flow is connected to the previous one through the growth rate. And it turns out that I can apply exactly the same formula, exactly the same derivation strategy as I did before. And I can show you, which I'm not going to do this time, that the present value in this case is simply equal to C over R minus G. How can you verify that this formula makes sense? Well. Try to set g equal to zero. So if there is no growth, it just becomes c over r, which is exactly what we got for regular perpetuity. So yes, looks like that formula makes sense. Okay, so what could be an example of a growing perpetuity? Regular common stock. So unlike preferred stock or British Consul that has a fixed dividend, regular common stock might have dividends that are growing over time because the company is growing the company is generating more and more earnings the growth rate might not be very high but it's stable so in finance we like to approximate dividends paid by a stock as a growing perpetuity we try to forecast approximate growth rate and apply that growth rate to dividends that the stock will generate okay So for instance, let's say you have a stock that has a first coming dividend equal to 1.3. That's your first next expected dividend, but then it is expected to grow at 5% forever. If discount rate is equal to 10%, what is the present value? Well, you should immediately recognize that it's a growing perpetuity. So it's going to be your initial cash flow, which is 1.3 divided by R minus G. So that gives you $26 right here. Okay, so again, you will see that it's equal to the price. So the price would equal to $26. Okay. All right, we will again, we will get to pricing, but that's just an example of growing perpetuity. Okay. The next one is even more common and that is called annuity. So annuity is extremely common in finance. Everything that you can hear about like mortgage, car loan, student loan, a bond, you know at least the coupon payment portion of a bond. Those would be all examples of annuities. We just like to have those fixed payments. and those fixed payments that actually end at particular point in time. So unlike perpetuity that has fixed income stream that does never end, annuity has maturity. Annuity has finite time, it has finite t. So therefore annuities are very very popular. Again mortgages, a lot of fixed loans, a lot of deposits that have fixed interest, they all represent annuities. So, again, you just need to discount each of your cash flows up to period T. You can do it manually or you could put it in spreadsheet or you could use a formula. And as you can see, the formula looks scary, but again, it actually is derived, all right, using the perpetuity formula. Okay, so let me just show you quickly where that is coming from. Okay, so it shouldn't be that bad. So the trick is to realize, so if I have... time 0 and then time 1, 2 and so on all the way to let me see if we use time t or n okay we use t all the way to period t so I have c1 c2 and then ct the trick to figure out the the present value of annuity is to realize that if I look at from here all the way to infinity that would be your typical perpetuity so that would be c over r but then if i start from period t plus one okay so uh by the way it's not c1 i'm sorry it's actually the same c because it's uh it's a constant stream starting from period t plus one i again have c and so on so if i take my cash flow stream starting from period t plus one from here that is technically also a perpetuity and that was also b equal to c over r the present value would also be equal to c over r the only difference is all i need to do now is to figure out this part right here and this part would be equal to big perpetuity here which is C over R minus smaller perpetuity which is equal to C over R but now if I use C over R only it would be as of period t. I want it as of period zero so what it's going to be is going to be C over R times 1 over 1 plus r to the power of t. That's it. And that would be your present value of annuity. So again, present value of annuity is just the difference in two present values of two perpetuities. One is longer and one is shorter. Okay, and that's exactly what you have right here. Okay, now you don't have to remember that derivation. It's just for you, for some of you who are more inclined to look in things a little bit deeper. But if you want to, you could just use the formula right away. So here's an example. So let's say you have a car payment that you desire, so you can afford $400 monthly car payment. How much car can you afford? Meaning what is the value of a car that you can afford if interest rates are 7% on a 36-month loan? So this is the present value of annuity. The question is, how do we know? And that's going to be the most challenging part of this material. Not so much putting numbers in the formulas, but more about, you know, realizing which formula to use. Okay, so you need to be able to read the wording and recognize what formula to utilize under certain circumstances. So you need to understand the situation. So I'm going to claim that it's present value of annuity. But the question is why? So let me explain. So you are the customer of a bank. You want to borrow money in order to buy a car. So as a customer of a bank you promise that you are going to pay $400 monthly. So now you're putting yourself in the shoes of a bank. So as a banker I'm going to get $400 every month for 36 months. In exchange, I'm willing to give X amount of dollars to my customer today to buy that car. The question is, what is the maximum amount of money I'm willing to give up today in exchange for cash flow stream that is equal to $400 monthly payment for 36 months? Given that, I want to earn at least 7% rate of return. So what I'm doing, I'm discounting. each of those future $400 monthly payments. using 7% discount rate. That is a typical present value of annuity application. So in other words, I want to find out what is the present value of all of those $400 monthly payments to me as a banker today. And when I calculate present value, I incorporate my opportunity cost. So my opportunity cost is what can I make in the market? and in the market i can make seven percent annually okay so all is left now is just to use the uh the present value of annuity formula the only thing that you need to realize here you gotta be very careful is that this is your c 400 this is your r 0.07 but that's on annual basis remember i told you that sometimes you can have cash flows that have more frequent discounting or more frequent compounding that is the case here so instead of getting seven percent annually you're going to get seven divided by 12 but you're going to do it on a monthly basis okay so you modify your r you modify your t so your t is not going to be you know three years it's going to be 36 months and you end up getting you know roughly 13 000 so what does it mean now it means that the bank is willing to give up $12,954 today in exchange for $400 car payments over the next 36 months. Given that that's how much bank is willing to give up, that's exactly the amount of loan that you will receive as a customer and that's the amount of money you will pay for your car. So the answer how much car can you afford is around $13,000. Okay, so it is present value of annuity. Sometimes annuities could be delayed. For example, here you have a four-year annuity that makes first payment two years from now. So not today. So you are here. Okay, but your annuity payment happens two years from now. So what's the present value? Well, one option is to take each of cash flows and discount them manually to period zero. That way you will never make a mistake. Alternatively, you could apply the annuity formula to the whole cash flow stream. Right? So you will just use... oops... you will use this formula right here. But when you do that, you have to be careful because if you apply the present value of annuity formula straight you will receive this number right here but that would be as of period one because by definition the annuity formula assumes that your first cash flow happens exactly one period from now but in your case it happens two periods from now so if you calculate present value of annuity blindly without thinking what the formula really mean you will actually overestimate the present value. So what you need to do is either discount each of cash flows separately all the way to period zero or you do apply present value of annuity formula but then you take that number and you discount it again to period zero. So that's exactly what's going on right here. So your present value in period zero would be your present value of annuity divided by 1 plus r in which case r is equal to 9 okay we are not going to utilize that a lot so don't worry about it too much it's just some something that you need to be aware of it's called delayed annuity okay and finally let me give you formula for future value of annuity so unlike perpetuity that does not have future value annuity does have future value and here is the formula for it so the question is under what circumstances would we apply excuse me would you apply a future value of annuity well one example could be something like this you invest let's say thousand dollars every month in college fund at let's say 10%. annual R. How much money will you have, let's say, in, you know, 20 years? Okay, so you want your child to go to college and you start saving early In fact in this case before your child even born because most kids go to college around 17 18 years old So, you know you'll be an extremely conservative you're trying to be proactive and you just figured okay I will have a child very soon so let me just start investing right now so that I have 20 years until I actually send my kid to college so you put this decide $1,000 every month. The question is, what would be the value that you're going to generate? That would be your typical example of future value of annuity. Unlike previous example with car loan, where you needed to know what is the value now, because you wanted to figure out how much bank is willing to give you today. In this case, the question is very different. You're asking how much money will you have in the future. future if you invest on a regular basis in that annuity so you would use this formula you will have fv equal to so your c is one thousand dollars okay then you have one plus r your r is 10 percent but remember you're trying to invest every month so you need to make modifications you will do point one divided by twelve And then your t is going to be 30 years times 12. Okay, so it's going to be 360 months instead. Then again, here, you're going to have 0.1 divided by 12. And then minus 1 divided by 0.1 divided by 12. So as you can see, it gets a little bit complicated in terms of actual calculations. So if you use calculator, you got to be very careful. I suggest that you do. do it step by step a much better solution is to use excel because in excel you have a present value future value formulas already built in all you need to do is just to specify parameters the excel allows you to show the frequency the time period it makes all the adjustment by itself it's very very convenient so i'll show you in a moment how to use excel so i'm not going to do calculation here you know i don't want to spend time on that I will come back to this example when we do Excel and we will figure out how much money you will have in that college fund in 20 years. Okay, so the last thing we're going to do in terms of those so-called simplifications is growing annuity. So growing annuity, just like growing perpetuity, is when each additional cash flow is connected to another to the previous cash flow. flow through the fixed growth rate so again cash flows themselves are not fixed but the growth rates are so each cash flows is growing at a constant rate so if that is true then you have you know this formula right here but again you don't need to memorize anything you don't need to derive it in this case but again you could see that if g is equal to zero then your formula would be just your regular present value of annuity It's not going to be growing annuity anymore. And you could verify that it's going to be exactly the same as we had before. So what is the example of growing annuity? The most typical example would be some kind of retirement payments that you receive that are adjusted for inflation. Another example could be a bond that is also adjustment for inflation. Something that is called TIPS, Treasury Adjusted Price Security. So. So, for instance, you know, at least in the United States, you could get a government bond that will pay you coupon payment that is adjusted for inflation. So every time there is inflation, let's say 3-4%, you're getting 3-4% more and more and more with each additional coupon payment. So you're being protected from inflation. Or alternatively, as an example here, you have some kind of retirement plan that offers you $20,000 initially. for 40 years, but then... annual payments increase three percent every year perhaps to adjust for inflation so that would be a typical example of growing annuity okay so you plug in numbers you know and you get for example present value of growing annuity equal to 265 000 so that's by the way is another example of how to use these formulas you know um The retirement plan is a perfect example of present value of annuity formulas. Very often people have money accumulated before they retire, but they don't know how to use it. They are afraid that if they simply keep that money in bank account, it will not be growing fast enough. On the other hand, they're afraid to put it in stock market because it might crash. So what they often choose when they retire They just trust someone else to worry about this money. Instead, they exchange that money for some kind of fixed income stream. So they buy the annuities. And if you go to the website of any decent bank, most banks, especially larger banks, as well as some other financial companies, offer such service. You can buy annuity from financial institution. So what you're doing, you are giving up money right now. In exchange, you're buying yourself a predetermined fixed income, so you don't need to worry about anything. Now it's going to be bank job to figure out how to deal with that money. It's not your problem anymore. So that would be a typical example of present value of growing annuity. Let's say you have, you know, a plan to get $20,000 every year initially, but then it grows with 3% annually for 40 years. Well... you go to the bank and you buy that annuity and it turns out that you can buy it for 265 000 so if you give up 265 000 today the bank promises you to keep paying you 20 initially and then 20 times 103 and so on and so forth okay so that's again a very prominent prominent example of present value of annuity in this case growing annuity all right finally The last thing I want to discuss is that aside from standard financial instruments like bonds, stocks, something that we're going to talk about a little bit later in the course, the concept of present value analysis is widely used to value the firm itself. So basically, conceptually, you need to understand that conceptually, the value of the company is simply the present value of its future cash flows. We don't care about buildings, we don't care about assets, we don't care about facilities unless they generate future cash flows. Nobody pays money to buy a company only because it looks beautiful. People buy companies, people buy stocks on the basis of the ability of that firm to generate future cash flows. Therefore, again, conceptually, the value of the firm is determined by the present value of all of its future cash flows. Now, theoretically, it sounds very simple, but in practice is extremely difficult to do. Because number one, nobody knows for sure what those cash flows are going to be. A lot of it is just forecasting. And number two, we don't know the risk. So the size. the timing and probably the trickiest part the risk of those future cash flows are all variables you really need to be able to figure that out and that's what makes finance fascinating and also very tricky because a lot of valuation involves doing just that figuring out size timing and risk of future cash flows when you see stock prices going up and down because some analyst comes out and says you know my target value of stock is $800 suddenly stock price goes up another analyst influential analyst comes out and says my target price is $600 and stock price goes down it's because markets believe in the expertise of those prominent financial analysts but what they really do they simply have a very good ability probably very good tools and good knowledge of Estimating the size and the timing and the risk of the future cash flows that that company will generate. Once you know that, you simply apply present value analysis to figure out the value of the firm and as a result, the value of its stocks. Okay, so we will come back to that concept later on, but just so that you know fundamentally, it just turns out that the value of the company is just the present value of its shares. All right, okay last but not the least, sorry, I want to show you very quickly how to use Excel to calculate present and future values. So first of all let me tell you it turns out that in Excel present value or future value functions are kind of all-inclusive. So whether you want to calculate let's say future value of single amount or future value of annuity you're going to use exactly the same formula okay so how would you do that so for instance if i want to know future value of one thousand dollars uh in five years let's say at five percent annual rate Okay, so let me explain. If you look at the timeline, you have $1,000 today, so you basically invest. So I'm going to put it as minus $1,000, and you want to know five years from now how much you're going to have. This is not annuity. That's just a single amount. So if you use formula, you would have future value is equal to $1,000 times 1 plus... r and in our case it's five percent to the power of five okay now how would you do it in excel you would use that fv function and you will put the following inputs you would say equal to fv in a moment i'll show you that in excel but then in parentheses you will start putting the inputs the rate is your r so you will put 0.05 NPR is number of periods, that's your 5. Now things become a little bit tricky. PMT is your annuity payment. Do you have annuity payment here? No, because it's not an annuity, it's just regular single amount. So what you're going to do, you will either put 0 or you just put another comma, you just skip it. So there is no annuity payment here. Instead, You put PV, which is your present value, and I suggest that you put it with negative sign, minus 1000. Okay, so that's your PV, just to be sure. The reason why you put it with negative sign is because Excel likes to see it that way. Excel will understand that you're investing now. You have cash outflow, and then the future value would be your cash inflow. If you don't put it with negative sign, the future value will give you the negative result. It would still be correct in absolute terms, but it will come out with negative sign for some reason. So to avoid it, just put negative PV right away. Okay, so that would be your future value of single amount. Now what if instead I want to know future value of annuity that contributes let's say $500 every year for five years. Okay, so now the the problem I'm right I'm running out of space here, but it would be something like this 0, 1, 2, 3, 4, and then 5. And you have 500, 500, 500, 500, and 500. So as you can see, you know, you... have several contributions so it's not single amount anymore it's actual annuity so what you're going to do you could either use formula or again you would use excel you would say equal to fv and then your parameters your r is still five percent your number of periods is still five but now you have pmt so your pmt your payment your annuity payment is going to be minus 500 but you're not going to have pv so as you can see pmt and pv are mutually exclusive you cannot have both so in this case you do have annuity so you you put it as annuity payment but you don't put pv and then you close the parentheses by the way you don't need to worry about type as you can see there is type here you don't need to worry about it keep it as a default so by default it's equal to zero it it only needed for annuity and zero assumed that that annuity payments occurs at the end of each period that's our assumption sometimes annuity happens in the beginning of the period therefore you would need to put type equal to one but again for our purposes just leave it as it is by default it's equal to zero so as you can see these formulas you know give you a pass opportunities to calculate pretty much anything you want so that's how you do future value if you want to do present value instead you use PV function self-explanatory exactly the same but now instead of having PV you have FV as a parameter sometimes you need to know number of periods okay so let's say you know present value in our future value but you want to know number of periods that it takes for example you know let's say it's an annuity you know how much you want to put aside let's say thousand dollars a month You know your goal, that you want to generate $2 million. The question is, how long it's going to take, given certain rate of return. That would be your number of periods. So you're solving for that T. So for that, you would use NPER function. Sometimes you want to know annuity payment itself. Again, let's say you want to accumulate $2 million. You know that you want to do it over 30 year period. The question is what should be your monthly contribution. So that would be your PMT function. Sometimes you need to know rate. Again, you need to have $2 million. You want to put aside $1,000 every month. Okay, the question is, what kind of rate will allow you to do that? In other words, how aggressive your investment portfolio should be. If you need 10 or 12% rate to do that, you need to put it all in stock market. So sometimes you want to know that rate, so your sole info are. Anyway, I'm not going to show you all of these functions because they are all self-explanatory. Once you understand one of them, the rest of them is very simple. So I already wrote to you. how exactly to do it let me just try to quickly pull out Excel if I could let me just see where it is on the iPad I don't use Excel for iPad that often so hopefully it will work out so let's see not now there you go okay so I would say equal to Let's see Hold on a second oh there you go man i'm not i don't use ipad that often for those purposes so equal to uh so let's say we want to find uh uh the future value of that college fund remember we said that you want to put decide what you want to invest $1,000 a month for 20 years I think at at 10% annual rate how much money will you generate in your college fund okay so let's do it as future value for annuity so you have FV okay then you open parentheses okay actually I'm surprised There you go. So that way you know I could have done it just straight but that way you could see parameters again. So in our case the rate was 10% annual I think but remember we wanted to contribute on a monthly basis. So what we can do and that's again what's beautiful about Excel it allows you to actually put expressions as opposed to actual rates. So what I could do, I could put 0.1, which is my rate, divided by 12. So Excel will figure it out. Excel will understand that it's 10% annual rate, but it's done on a monthly basis. Okay. Then in terms of number of periods, again, I have 20 years. Okay. But I'm going to contribute. Sorry. over 12 months so it's going to be 20 times 12. now this is annuity right because i'm contributing thousand dollars a month so it's going to be pmt so i will have a minus wow okay there you go so i'm gonna have minus one thousand okay pv i'm going to skip and type i don't need to worry so now i'm ready i can just click enter so if i click return looks like i will generate 759 000 over 20 year period that's not bad at all so if you put 1 000 a month for your child and if you invest in relatively aggressively at 10 annual rate which is not unreasonable if you put everything in stocks for example or most of it in stocks And if you do it over 20 year period, you will be able to generate quite substantial amount of money. Okay, so that was just an example of how to use Excel for these formulas. I strongly encourage you to try NPR, PMT, PV, rate function. But again, we will come back to all of these functions later on when we do some applications. That concludes... the discussion of discounted cash flows the next would be the application for capital budgeting and identification of investment opportunities

Now why is it important? Because very often even though R is quoted on annual basis, in reality it's compounded or discounted depending on whether it's future or present value on a more frequent basis. For example, if you put money in the bank, you would be quoted the annual interest rate.

In fact, it actually is going to be annual effective rate, something we're going to talk about in a moment, but it's always quoted on the annual basis. So let's say R equal to, I don't know, 12% annually. Okay. But what you also realize is that in reality, instead of getting 12% every year, you're getting 1% every month.

Okay. So that would be a typical example of annual rate compounded monthly. So in this case, your R is equal to 12%.

Okay. but your m is equal to 12. Okay, so what it means, it means that m, which is frequency of compounding within a year, represents the frequency with which your interest is accrued during the year. So when m is equal to 1, you're talking about annual compounding. So m equal to 1, that's going to be just your regular annual. When m equal to 2, that would be semi-annual.

Okay, and so on. Another example could be m equal to 3, that would be quarter. And finally, well, two more, m equal to 6. No, actually six we don't have.

Six, that's uncommon. So M equal to 12, that would be your monthly. Okay. You could also have daily compounding.

So M is equal to something like 365. Okay. And finally, you could have so-called continuous compounding continuous compounding is is a completely different concept the formula changes completely okay but continuous compounding is some kind of imaginary situation so it does happen sometimes but it's hard to imagine so the idea is that you get interest like every fraction of a second okay so let's not worry about it let's just focus only on discrete cases so if that happens In finance, regardless of your M, R will always be quoted on annual basis. It will be your job to figure out what happens with your future value or present value if that annual R is compounded more frequently than on the annual basis. So that brings us to this concept of more frequent compounding periods and effective annual rate. So for example, in our case here, if you have...

So let me first do one... Let's just see something here. Okay. So let me do, I'm just trying to make it consistent with what you have there on these slides. So let me start with an example.

So let's say you have bank A and then you have bank B. Okay, so let's say bank A offers you 12% compounded annually. Actually, you know, let me do, sorry about that.

Let's say bank A offers you 12.5%, okay? compounded annually okay and bank b offers you just 12 but compounded monthly okay So let me explain what it means. Let me start with bank A. What it means is that your future, let's say you put money for one year only.

You know, when you extend it to two, three, four, it doesn't really matter. The result is going to be still the same in terms of which bank you prefer. But if you simply look at these two offers, what really jumps at you is that one bank gives you higher overall rate of return.

Looks like 12.5 is better than 12.5. than 12 but the catch is it offers you 12.5 compounded annually so you're getting 12.5 percent only once a year Bank B on the other hand offers you 12 percent annual rate but compounded monthly which means that instead of getting 12 percent every year you're going to get 1 percent every month okay the question is what is better well if I I told you 12% compounded annually I hope you could immediately tell me that bank bay is definitely better because everything else equal more frequent compounding is better and the logic is very simple once I get my first 1% the next 1% will be compounded on a big your base so i have more opportunity to generate that interest on interest that so-called snowball effect so everything else equal 12 compounded monthly is definitely better than 12 compounded annually so i hope that's clear so you know but we don't look at that we look at something a little bit uh you know more complicated so you have more frequent compounding versus less frequent but with higher rate so the question is which option is better so there are two ways of doing it one is just to simply calculate the future value in which case in each case so it's going to be some initial amount present value whatever you put in thousand dollars hundred dollars as it turns out it doesn't really matter and then you have one plus 0.25 0.125 so that's what you're gonna get a year from now if you put money in bank a if you put money in bank b the formula changes so you have present value times one plus but instead of having your r you have r divided by m So again, if you look at this formula here, you have r divided by m, where r is your stated annual rate, and m is frequency of compounding. So it will be 0.12 divided by 12, which results in 1% monthly.

That's basically what we thought it's going to be. But now, here is to the power of 1, because it's only one year. Here it's one year multiplied by 12 months.

So you're talking about 12 periods. So the question is what is bigger 1 plus 0.25 to the power of 1 or 1 plus 0.12 divided by 12 to the power of 12. Okay and let me just do the calculation. So bear with me.

So I'm just gonna use so 1.25 I'm sorry. So yeah, let me just write that down first. So that's going to be PV times 1.125.

And here, that's where we need the calculation. So it would be PV times and all I need to do now is to have 1.01 to the power of 12. And that would be 1. 12 6 8 so as you can see the difference is not huge but bank b gives you a better option okay because again i don't care about initial amount of money because that would cancel out when you compare. What matters is this factor here.

Okay, the overall accumulation. So as you can see after one year I will have 12.5% more money if I invest in Bank A. but I will have 12.68% more money if I invest in bank B, even though the actual rate is lower.

So that's basically something that you really need to take into account. So if you have more frequent compounding or more frequent discounting, if you're using present value instead, you need to be careful. You need to understand that the actual rates will always be quoted on annual basis. You need to make modification. Now, a better way to do that is to always calculate annual equivalent.

Okay, so when you cross shop, you know, across different alternatives, if you have bank A, bank B, bank C, you have multiple banks or multiple investment opportunities with different frequencies. The best option for you is to always convert everything to the annual basis compounded annually. So for bank A, so-called effective annual rate is already 12.5%. So we don't need to do any modification because it's already compounded annually.

But for bank B, the effective annual rate... would be an annual rate compounded annually that would make you indifferent between getting 12% compounded monthly or getting that X rate compounded annually. And as you can see already it will be 12.68%. Okay, so it's going to be equal to 12.68%.

All right, so that's actually how it's done. So instead of finding actual future values or present values, what you do instead, you immediately realize. that the rate that you're getting quoted on annual basis if it has more than annual compounding more frequent compounding than annual you need to immediately convert it into effective rate okay turns out that there is a very very nice formula for it you could just figure it out right away so first of all i'm going to skip this example here because it just does exactly what i did except that it has 50 initial value so okay so what i'm going to do i'm going to skip right to here okay so as you can see it's also 12.68 it's exactly the same stuff i just I just did a little different example. But the point is, you don't have to do any of these calculations.

You could jump to effective annual rate formula, which is equal to 1 plus r over m to the power of m minus 1. So in our case, right, if I look at that, you know, bank B, okay, my r is 12, my m is 12. Okay, all I have to do is just to plug numbers right here. So I have 1 plus 0.12, excuse me, divided by 12 to the power of 12 and then minus 1. And, you know, if you do calculations, you will get your 0.1268 or... 12.68 percent okay so that's that's basically uh how you do it so that's called effective annual rate or ear formula okay now for many standard financial products like deposits, bonds, mortgages, you know different types of car loans, the effective annual rate is already given to you by the financial institution.

By law the banks and other financial institutions are required to disclose their interest rates on effective annual basis. The reason for it is because the governments do not want financial institutions to confuse people so for example again if you are not in finance if you have no idea you go to Bank a you go to Bank B you might be confused you might be actually lured towards Bank a because it has higher annual rate but Bank B is actually better it turns out so governments understand that consumers don't have the knowledge so they require every financial institution to convert all of their rates into so-called effective annual basis. That's why you see something like APR or APY. APY is your annual percentage yield on deposits that is already converted into effective annual basis.

APR is interest on loan, on mortgage for example, or on car loan that is also converted to the annual basis, annual compounding basis. Because all of those products require you to make more frequent payments. For example, on mortgage, I'm not paying annually, I'm paying monthly. Similarly, if I put money into deposit, I'm not getting my interest annually, I'm getting it monthly.

But the interest is still on an annual basis. So how do we take into account that difference in compounding? Well, again, financial institutions are required to do it for you and disclose it. so that you don't get confused. Now, the question then is why would we need this formula right here?

Well, because remember, in finance, we don't deal with only standard products. Sometimes we deal with certain cash flows that are generated by different projects that have nothing to do with mortgage or deposit account or a bond. That would be a project that actually generates our cash flows.

So some projects will generate cash flows on annual basis, some of them will generate on monthly, some of them will generate on daily basis, so different frequencies. So it will be your job as a finance manager to make sure that you always take into account the frequency and incorporate your effective annual rate analysis. Okay, all right, so the last thing we're gonna do in this lecture is to keep we will keep talking about multiple cash flows but now we are going to introduce so-called simplifications. So these days we don't need those simplifications because you know the spreadsheet software like excel is very powerful so I mean that simplification doesn't really matter but in the past when computers were not well developed The finance people ask mathematicians to help them to figure out how to calculate those present or future values when you have multiple cash flows. OK, how can we simplify them?

And as a result, those simplifications become so common in our finance lives that now a lot of actual standard products in finance are a result of those simplifications. So. what are those simplifications to the first one is perpetuity so let me just start talking about it right away so what is the perpetuity excuse me so perpetuity is a constant stream of cash flows that lasts forever okay so instead of having c1 c2 c3 and c4 excuse me you have C C C C that lasts forever So you might wonder what does it mean? Well, first of all, perpetuity does not have future value. And that's obviously why, because there is no maturity, it lasts forever.

So there is no point in talking about future value. However, perpetuity does have present value. And you might wonder how is it possible that, excuse me, you have multiple cash flows that have infinite time horizon. okay, that go all the way to infinity. But somehow you sum them up and you get the finite number.

The reason for it is because each of the subsequent cash flow is getting smaller and smaller and smaller. So in the limit when your time t approaches infinity, each additional cash flow that happened here, for example if you have c divided by one plus r, you know to the power of 1000. This one is going to be so small that we're going to approximate it to zero. So basically what's going to happen because each of the additional cash flow is getting smaller and smaller and smaller and smaller, mathematically we can prove that this entire sequence pretty much collapses to C over R.

Okay, so the present value of each of those cash flows when you sum them up is going to be simply equal to c divided by r. Okay, so now that we understand the general idea, let me derive that result. So the only reason I'm deriving it is because I want to make sure that you understand the logic here.

So from math, we know that 1 plus a plus a squared plus a to the power of 3 and so on all the way to infinity. would be equal to 1 divided by 1 minus a if a is less than 1. Okay so what does it mean? So think about it this way why do we need a to be less than 1? Because whenever you take a number that is less than 1 and you raise it to the power the higher the power the smaller the number is going to be. Okay, so for example, if you have 0.5, you get 0.5 squared.

It's going to be a smaller number. 0.5 to the power of 3 is going to be even smaller. If a is greater than 1, then every subsequent term will get bigger and bigger and bigger, and that would be infinity.

Okay, so it's important to understand that when a is less than 1, then 1 plus a plus a squared plus a to the power of 3 all the way to infinity is going to be equal to 1. divided by 1 minus a. It's a known mathematical fact. Now let's look at our present value. Okay so it's equal to c divided by 1 plus r plus c over 1 plus r squared plus c over 1 plus r to the power of 3 and so on.

So what I'm going to do now I'm going to take c over 1 plus r out, okay? Factorize it. And now I have 1 plus 1 over 1 plus r plus 1 over 1 plus r squared and so on, okay?

Now look what's interesting here. I can claim that 1 over 1 plus r is my a and 1 over 1 plus r squared is my a squared. Can I legitimately say that?

Yes, because a is less than 1. 1 over 1 plus r is less than 1 unless r is equal to 0. So if r is equal to 0, then yes, it blows up, it becomes infinity. But as long as r is less than 1, excuse me, less, excuse me, greater than 0, which is normally the case, 1 over 1 plus r is going to be less than 1. Okay, so now I can utilize the formula. So it's going to be C over 1 plus R. And then I'm going to have 1 divided by 1 minus A.

But my A is 1 over 1 plus R. Okay, the rest of it is just very simple. So it's C over 1 plus R.

Then 1 plus R goes on top. And I'm having 1 plus r minus 1. So cancels out. 1 and 1 cancel out.

So I have c divided by r. Okay. So yes, it took us a couple of minutes, but at least, you know, we are being thorough here. So you see where the number is coming from. I don't like giving formulas that absolutely are out of nowhere.

So I prefer to derive as much as I can unless it requires some... more mathematical knowledge than you guys might have. So in this case it's a very simple derivation.

So anyway, turns out that perpetuity is a very useful and simple case. So even though you have multiple cash flows that last forever, the pricing or present value collapses to a very simple formula C over R. Turns out that perpetuities exist in reality. There are some financial instruments that are perpetuity. The most prominent example is British Consul.

So British Consul is a bond that has no maturity. Another example of perpetuity could be a preferred stock. as you will see later preferred stock just promise you a fixed dividend that lasts forever okay there is no maturity because it's a stock and the dividend does not vary because it's a preferred stock so it's guaranteed and it's fixed so that's another example of of perpetuity so let's look at british council for example you know that it promises you to pay let's say 15 pounds the reason is pound is because british bond forever and let's say the interest rate or discount rate is equal to 10 percent the question is what is the present value and it turns out that the present value is very simple simply to calculate it's just 15 pound divided by 0.1 and again that gives you 150 dollars as you will see later it means that the price of a bond is going to be equal to 150 dollars excuse me pounds in this case so again i'm trying to stay away from pricing right now i'm just trying to find present value but sort of going a little bit ahead keep in mind that in the you will learn in the future that the price the market price of a financial instrument is just the present value of all the cash flows that instrument will generate that's why it's so important to understand these concepts here. Okay, so that was perpetuity. Some perpetuities are growing, which means that the cash flows are not the same, but they are growing at a constant rate.

So that is also a very interesting example. So you see changes over time, it's getting bigger and bigger and bigger, but Each subsequent cash flow is connected to the previous one through the growth rate. And it turns out that I can apply exactly the same formula, exactly the same derivation strategy as I did before. And I can show you, which I'm not going to do this time, that the present value in this case is simply equal to C over R minus G. How can you verify that this formula makes sense?

Well. Try to set g equal to zero. So if there is no growth, it just becomes c over r, which is exactly what we got for regular perpetuity.

So yes, looks like that formula makes sense. Okay, so what could be an example of a growing perpetuity? Regular common stock.

So unlike preferred stock or British Consul that has a fixed dividend, regular common stock might have dividends that are growing over time because the company is growing the company is generating more and more earnings the growth rate might not be very high but it's stable so in finance we like to approximate dividends paid by a stock as a growing perpetuity we try to forecast approximate growth rate and apply that growth rate to dividends that the stock will generate okay So for instance, let's say you have a stock that has a first coming dividend equal to 1.3. That's your first next expected dividend, but then it is expected to grow at 5% forever. If discount rate is equal to 10%, what is the present value? Well, you should immediately recognize that it's a growing perpetuity.

So it's going to be your initial cash flow, which is 1.3 divided by R minus G. So that gives you $26 right here. Okay, so again, you will see that it's equal to the price. So the price would equal to $26.

Okay. All right, we will again, we will get to pricing, but that's just an example of growing perpetuity. Okay.

The next one is even more common and that is called annuity. So annuity is extremely common in finance. Everything that you can hear about like mortgage, car loan, student loan, a bond, you know at least the coupon payment portion of a bond. Those would be all examples of annuities.

We just like to have those fixed payments. and those fixed payments that actually end at particular point in time. So unlike perpetuity that has fixed income stream that does never end, annuity has maturity.

Annuity has finite time, it has finite t. So therefore annuities are very very popular. Again mortgages, a lot of fixed loans, a lot of deposits that have fixed interest, they all represent annuities. So, again, you just need to discount each of your cash flows up to period T. You can do it manually or you could put it in spreadsheet or you could use a formula.

And as you can see, the formula looks scary, but again, it actually is derived, all right, using the perpetuity formula. Okay, so let me just show you quickly where that is coming from. Okay, so it shouldn't be that bad. So the trick is to realize, so if I have... time 0 and then time 1, 2 and so on all the way to let me see if we use time t or n okay we use t all the way to period t so I have c1 c2 and then ct the trick to figure out the the present value of annuity is to realize that if I look at from here all the way to infinity that would be your typical perpetuity so that would be c over r but then if i start from period t plus one okay so uh by the way it's not c1 i'm sorry it's actually the same c because it's uh it's a constant stream starting from period t plus one i again have c and so on so if i take my cash flow stream starting from period t plus one from here that is technically also a perpetuity and that was also b equal to c over r the present value would also be equal to c over r the only difference is all i need to do now is to figure out this part right here and this part would be equal to big perpetuity here which is C over R minus smaller perpetuity which is equal to C over R but now if I use C over R only it would be as of period t.

I want it as of period zero so what it's going to be is going to be C over R times 1 over 1 plus r to the power of t. That's it. And that would be your present value of annuity.

So again, present value of annuity is just the difference in two present values of two perpetuities. One is longer and one is shorter. Okay, and that's exactly what you have right here.

Okay, now you don't have to remember that derivation. It's just for you, for some of you who are more inclined to look in things a little bit deeper. But if you want to, you could just use the formula right away. So here's an example.

So let's say you have a car payment that you desire, so you can afford $400 monthly car payment. How much car can you afford? Meaning what is the value of a car that you can afford if interest rates are 7% on a 36-month loan?

So this is the present value of annuity. The question is, how do we know? And that's going to be the most challenging part of this material. Not so much putting numbers in the formulas, but more about, you know, realizing which formula to use. Okay, so you need to be able to read the wording and recognize what formula to utilize under certain circumstances.

So you need to understand the situation. So I'm going to claim that it's present value of annuity. But the question is why?

So let me explain. So you are the customer of a bank. You want to borrow money in order to buy a car.

So as a customer of a bank you promise that you are going to pay $400 monthly. So now you're putting yourself in the shoes of a bank. So as a banker I'm going to get $400 every month for 36 months.

In exchange, I'm willing to give X amount of dollars to my customer today to buy that car. The question is, what is the maximum amount of money I'm willing to give up today in exchange for cash flow stream that is equal to $400 monthly payment for 36 months? Given that, I want to earn at least 7% rate of return.

So what I'm doing, I'm discounting. each of those future $400 monthly payments. using 7% discount rate.

That is a typical present value of annuity application. So in other words, I want to find out what is the present value of all of those $400 monthly payments to me as a banker today. And when I calculate present value, I incorporate my opportunity cost. So my opportunity cost is what can I make in the market?

and in the market i can make seven percent annually okay so all is left now is just to use the uh the present value of annuity formula the only thing that you need to realize here you gotta be very careful is that this is your c 400 this is your r 0.07 but that's on annual basis remember i told you that sometimes you can have cash flows that have more frequent discounting or more frequent compounding that is the case here so instead of getting seven percent annually you're going to get seven divided by 12 but you're going to do it on a monthly basis okay so you modify your r you modify your t so your t is not going to be you know three years it's going to be 36 months and you end up getting you know roughly 13 000 so what does it mean now it means that the bank is willing to give up $12,954 today in exchange for $400 car payments over the next 36 months. Given that that's how much bank is willing to give up, that's exactly the amount of loan that you will receive as a customer and that's the amount of money you will pay for your car. So the answer how much car can you afford is around $13,000.

Okay, so it is present value of annuity. Sometimes annuities could be delayed. For example, here you have a four-year annuity that makes first payment two years from now. So not today. So you are here.

Okay, but your annuity payment happens two years from now. So what's the present value? Well, one option is to take each of cash flows and discount them manually to period zero.

That way you will never make a mistake. Alternatively, you could apply the annuity formula to the whole cash flow stream. Right?

So you will just use... oops... you will use this formula right here. But when you do that, you have to be careful because if you apply the present value of annuity formula straight you will receive this number right here but that would be as of period one because by definition the annuity formula assumes that your first cash flow happens exactly one period from now but in your case it happens two periods from now so if you calculate present value of annuity blindly without thinking what the formula really mean you will actually overestimate the present value. So what you need to do is either discount each of cash flows separately all the way to period zero or you do apply present value of annuity formula but then you take that number and you discount it again to period zero.

So that's exactly what's going on right here. So your present value in period zero would be your present value of annuity divided by 1 plus r in which case r is equal to 9 okay we are not going to utilize that a lot so don't worry about it too much it's just some something that you need to be aware of it's called delayed annuity okay and finally let me give you formula for future value of annuity so unlike perpetuity that does not have future value annuity does have future value and here is the formula for it so the question is under what circumstances would we apply excuse me would you apply a future value of annuity well one example could be something like this you invest let's say thousand dollars every month in college fund at let's say 10%. annual R.

How much money will you have, let's say, in, you know, 20 years? Okay, so you want your child to go to college and you start saving early In fact in this case before your child even born because most kids go to college around 17 18 years old So, you know you'll be an extremely conservative you're trying to be proactive and you just figured okay I will have a child very soon so let me just start investing right now so that I have 20 years until I actually send my kid to college so you put this decide $1,000 every month. The question is, what would be the value that you're going to generate? That would be your typical example of future value of annuity. Unlike previous example with car loan, where you needed to know what is the value now, because you wanted to figure out how much bank is willing to give you today.

In this case, the question is very different. You're asking how much money will you have in the future. future if you invest on a regular basis in that annuity so you would use this formula you will have fv equal to so your c is one thousand dollars okay then you have one plus r your r is 10 percent but remember you're trying to invest every month so you need to make modifications you will do point one divided by twelve And then your t is going to be 30 years times 12. Okay, so it's going to be 360 months instead.

Then again, here, you're going to have 0.1 divided by 12. And then minus 1 divided by 0.1 divided by 12. So as you can see, it gets a little bit complicated in terms of actual calculations. So if you use calculator, you got to be very careful. I suggest that you do.

do it step by step a much better solution is to use excel because in excel you have a present value future value formulas already built in all you need to do is just to specify parameters the excel allows you to show the frequency the time period it makes all the adjustment by itself it's very very convenient so i'll show you in a moment how to use excel so i'm not going to do calculation here you know i don't want to spend time on that I will come back to this example when we do Excel and we will figure out how much money you will have in that college fund in 20 years. Okay, so the last thing we're going to do in terms of those so-called simplifications is growing annuity. So growing annuity, just like growing perpetuity, is when each additional cash flow is connected to another to the previous cash flow. flow through the fixed growth rate so again cash flows themselves are not fixed but the growth rates are so each cash flows is growing at a constant rate so if that is true then you have you know this formula right here but again you don't need to memorize anything you don't need to derive it in this case but again you could see that if g is equal to zero then your formula would be just your regular present value of annuity It's not going to be growing annuity anymore. And you could verify that it's going to be exactly the same as we had before.

So what is the example of growing annuity? The most typical example would be some kind of retirement payments that you receive that are adjusted for inflation. Another example could be a bond that is also adjustment for inflation. Something that is called TIPS, Treasury Adjusted Price Security.

So. So, for instance, you know, at least in the United States, you could get a government bond that will pay you coupon payment that is adjusted for inflation. So every time there is inflation, let's say 3-4%, you're getting 3-4% more and more and more with each additional coupon payment.

So you're being protected from inflation. Or alternatively, as an example here, you have some kind of retirement plan that offers you $20,000 initially. for 40 years, but then... annual payments increase three percent every year perhaps to adjust for inflation so that would be a typical example of growing annuity okay so you plug in numbers you know and you get for example present value of growing annuity equal to 265 000 so that's by the way is another example of how to use these formulas you know um The retirement plan is a perfect example of present value of annuity formulas. Very often people have money accumulated before they retire, but they don't know how to use it.

They are afraid that if they simply keep that money in bank account, it will not be growing fast enough. On the other hand, they're afraid to put it in stock market because it might crash. So what they often choose when they retire They just trust someone else to worry about this money.

Instead, they exchange that money for some kind of fixed income stream. So they buy the annuities. And if you go to the website of any decent bank, most banks, especially larger banks, as well as some other financial companies, offer such service.

You can buy annuity from financial institution. So what you're doing, you are giving up money right now. In exchange, you're buying yourself a predetermined fixed income, so you don't need to worry about anything. Now it's going to be bank job to figure out how to deal with that money.

It's not your problem anymore. So that would be a typical example of present value of growing annuity. Let's say you have, you know, a plan to get $20,000 every year initially, but then it grows with 3% annually for 40 years. Well...

you go to the bank and you buy that annuity and it turns out that you can buy it for 265 000 so if you give up 265 000 today the bank promises you to keep paying you 20 initially and then 20 times 103 and so on and so forth okay so that's again a very prominent prominent example of present value of annuity in this case growing annuity all right finally The last thing I want to discuss is that aside from standard financial instruments like bonds, stocks, something that we're going to talk about a little bit later in the course, the concept of present value analysis is widely used to value the firm itself. So basically, conceptually, you need to understand that conceptually, the value of the company is simply the present value of its future cash flows. We don't care about buildings, we don't care about assets, we don't care about facilities unless they generate future cash flows.

Nobody pays money to buy a company only because it looks beautiful. People buy companies, people buy stocks on the basis of the ability of that firm to generate future cash flows. Therefore, again, conceptually, the value of the firm is determined by the present value of all of its future cash flows.

Now, theoretically, it sounds very simple, but in practice is extremely difficult to do. Because number one, nobody knows for sure what those cash flows are going to be. A lot of it is just forecasting.

And number two, we don't know the risk. So the size. the timing and probably the trickiest part the risk of those future cash flows are all variables you really need to be able to figure that out and that's what makes finance fascinating and also very tricky because a lot of valuation involves doing just that figuring out size timing and risk of future cash flows when you see stock prices going up and down because some analyst comes out and says you know my target value of stock is $800 suddenly stock price goes up another analyst influential analyst comes out and says my target price is $600 and stock price goes down it's because markets believe in the expertise of those prominent financial analysts but what they really do they simply have a very good ability probably very good tools and good knowledge of Estimating the size and the timing and the risk of the future cash flows that that company will generate. Once you know that, you simply apply present value analysis to figure out the value of the firm and as a result, the value of its stocks. Okay, so we will come back to that concept later on, but just so that you know fundamentally, it just turns out that the value of the company is just the present value of its shares.

All right, okay last but not the least, sorry, I want to show you very quickly how to use Excel to calculate present and future values. So first of all let me tell you it turns out that in Excel present value or future value functions are kind of all-inclusive. So whether you want to calculate let's say future value of single amount or future value of annuity you're going to use exactly the same formula okay so how would you do that so for instance if i want to know future value of one thousand dollars uh in five years let's say at five percent annual rate Okay, so let me explain. If you look at the timeline, you have $1,000 today, so you basically invest.

So I'm going to put it as minus $1,000, and you want to know five years from now how much you're going to have. This is not annuity. That's just a single amount.

So if you use formula, you would have future value is equal to $1,000 times 1 plus... r and in our case it's five percent to the power of five okay now how would you do it in excel you would use that fv function and you will put the following inputs you would say equal to fv in a moment i'll show you that in excel but then in parentheses you will start putting the inputs the rate is your r so you will put 0.05 NPR is number of periods, that's your 5. Now things become a little bit tricky. PMT is your annuity payment. Do you have annuity payment here? No, because it's not an annuity, it's just regular single amount.

So what you're going to do, you will either put 0 or you just put another comma, you just skip it. So there is no annuity payment here. Instead, You put PV, which is your present value, and I suggest that you put it with negative sign, minus 1000. Okay, so that's your PV, just to be sure.

The reason why you put it with negative sign is because Excel likes to see it that way. Excel will understand that you're investing now. You have cash outflow, and then the future value would be your cash inflow.

If you don't put it with negative sign, the future value will give you the negative result. It would still be correct in absolute terms, but it will come out with negative sign for some reason. So to avoid it, just put negative PV right away. Okay, so that would be your future value of single amount.

Now what if instead I want to know future value of annuity that contributes let's say $500 every year for five years. Okay, so now the the problem I'm right I'm running out of space here, but it would be something like this 0, 1, 2, 3, 4, and then 5. And you have 500, 500, 500, 500, and 500. So as you can see, you know, you... have several contributions so it's not single amount anymore it's actual annuity so what you're going to do you could either use formula or again you would use excel you would say equal to fv and then your parameters your r is still five percent your number of periods is still five but now you have pmt so your pmt your payment your annuity payment is going to be minus 500 but you're not going to have pv so as you can see pmt and pv are mutually exclusive you cannot have both so in this case you do have annuity so you you put it as annuity payment but you don't put pv and then you close the parentheses by the way you don't need to worry about type as you can see there is type here you don't need to worry about it keep it as a default so by default it's equal to zero it it only needed for annuity and zero assumed that that annuity payments occurs at the end of each period that's our assumption sometimes annuity happens in the beginning of the period therefore you would need to put type equal to one but again for our purposes just leave it as it is by default it's equal to zero so as you can see these formulas you know give you a pass opportunities to calculate pretty much anything you want so that's how you do future value if you want to do present value instead you use PV function self-explanatory exactly the same but now instead of having PV you have FV as a parameter sometimes you need to know number of periods okay so let's say you know present value in our future value but you want to know number of periods that it takes for example you know let's say it's an annuity you know how much you want to put aside let's say thousand dollars a month You know your goal, that you want to generate $2 million.

The question is, how long it's going to take, given certain rate of return. That would be your number of periods. So you're solving for that T. So for that, you would use NPER function.

Sometimes you want to know annuity payment itself. Again, let's say you want to accumulate $2 million. You know that you want to do it over 30 year period. The question is what should be your monthly contribution. So that would be your PMT function.

Sometimes you need to know rate. Again, you need to have $2 million. You want to put aside $1,000 every month.

Okay, the question is, what kind of rate will allow you to do that? In other words, how aggressive your investment portfolio should be. If you need 10 or 12% rate to do that, you need to put it all in stock market. So sometimes you want to know that rate, so your sole info are. Anyway, I'm not going to show you all of these functions because they are all self-explanatory.

Once you understand one of them, the rest of them is very simple. So I already wrote to you. how exactly to do it let me just try to quickly pull out Excel if I could let me just see where it is on the iPad I don't use Excel for iPad that often so hopefully it will work out so let's see not now there you go okay so I would say equal to Let's see Hold on a second oh there you go man i'm not i don't use ipad that often for those purposes so equal to uh so let's say we want to find uh uh the future value of that college fund remember we said that you want to put decide what you want to invest $1,000 a month for 20 years I think at at 10% annual rate how much money will you generate in your college fund okay so let's do it as future value for annuity so you have FV okay then you open parentheses okay actually I'm surprised There you go.

So that way you know I could have done it just straight but that way you could see parameters again. So in our case the rate was 10% annual I think but remember we wanted to contribute on a monthly basis. So what we can do and that's again what's beautiful about Excel it allows you to actually put expressions as opposed to actual rates.

So what I could do, I could put 0.1, which is my rate, divided by 12. So Excel will figure it out. Excel will understand that it's 10% annual rate, but it's done on a monthly basis. Okay. Then in terms of number of periods, again, I have 20 years.

Okay. But I'm going to contribute. Sorry. over 12 months so it's going to be 20 times 12. now this is annuity right because i'm contributing thousand dollars a month so it's going to be pmt so i will have a minus wow okay there you go so i'm gonna have minus one thousand okay pv i'm going to skip and type i don't need to worry so now i'm ready i can just click enter so if i click return looks like i will generate 759 000 over 20 year period that's not bad at all so if you put 1 000 a month for your child and if you invest in relatively aggressively at 10 annual rate which is not unreasonable if you put everything in stocks for example or most of it in stocks And if you do it over 20 year period, you will be able to generate quite substantial amount of money.

Okay, so that was just an example of how to use Excel for these formulas. I strongly encourage you to try NPR, PMT, PV, rate function. But again, we will come back to all of these functions later on when we do some applications. That concludes...

the discussion of discounted cash flows the next would be the application for capital budgeting and identification of investment opportunities

Transcript for:Understanding Compounding and Financial Calculations

Transcript for:
Understanding Compounding and Financial Calculations