- How do we decide whether to take a financial decision? How do we compare investments when they might pay off in different time periods? And how do we make investment decisions when people might disagree? Somebody might prefer a short-term investment and somebody else might prefer something long term. And how do we deal with investments, which are quite complicated and pay lots of different cash flows in many different time periods. So these are the questions that I'll be exploring today in my third lecture in my Gresham College series on the principles of finance. Today's lecture is called how to make financial decisions. Thank you very much to everybody for attending. So maybe a good place to start is to think about, well, what is an investment to begin with? Now, when most people think about investments, they probably have in mind buying stocks and shares. So that was what I considered in my lecture last October, how the stock market works. Now with shares, what you're doing is you're spending some money now to buy the shares and you're getting something in the future. What do they shares give? They give dividends and capital gains. But if an investment is something that costs you money now and gives you some benefits in the future, then it extends to more than just buying shares. And that's really important because if the only thing I was going to look at here is how to buy shares, that's not a common decision that we make every day instead what I want to stress is that we make financial decisions all the time. And therefore the power of knowing how to make financial decisions is actually much more important and much more common and much more relevant than we might think. So what I'd look to look at is to extend the discussion from financial investments to real investments. So what is a real investment? It's anything that costs us money and gives us a benefit in the future and that benefit is something with intrinsic value in and of itself. What does it mean to have intrinsic value? Well, if you are, for example, a company and you're buying, building a new factory, that factory has value. That factory will be churning out cars or toys or clothes and those cars and toys and clothes are worth something in and of themselves. It's not just that you're getting money. Like you are with a stock paying dividends. Similarly, as a person, if you're going to choose to renovate your house or to renovate your kitchen, that new kitchen also has intrinsic value, right? Because that's something that you can enjoy. Now, interestingly, we can think of wheel investments as not only tangible investments, where that benefit you get is something that you can see and touch, but also some investments have intangible value. And so that's something that you can't see, but it still has a lot of value. For example, just attending this lecture, hopefully there's intrinsic value to this. It might help you make financial decisions in the future, or maybe it's just interesting. Why? We might take the sessions because we enjoy it. And that's something which really matters. I know economists only like to think about things, which we can convert into monetary terms, but welfare and happiness and utility, these are all really important reasons for why we take decisions. And what about a company? What's an example of something within tangible value. Let's say you're going to choose to give more parental leave to your employees. That's something that makes the employees more motivated and more happy and more likely to stay. It's something you can't see, but clearly it has future value to those employees and therefore to you. So the defining feature of any investment is something that costs us money today, but gives you some benefits in the future. Some of those benefits might be direct cash benefits like dividends of a share, or some of those other benefits are not in cash, but they might have monetary value. For example, if you're factually your belt, but making some clothes, you might be able to sell those clothes, or you might be able to sell to a house for more now that it has a better kitchen. Now you might think, well, how do we make financial decisions? Isn't that really simple? Don't we just compare all the future benefits that we're getting, add them all up and then compare them to the cost of the investment. And you can't do that because they're not apples to apples for two reasons. The first reason is that the cost of the investment today is something certain, but the future benefit is risky. For example, if we're building any factory, how do we know that people will even want to buy our clothes in the future? How do we know what the state of the economy will be? Maybe there'll be a recession and nobody's going to want to buy them. Now that is something we're not going to look at today. We're going to look at the next lecture on how to measure and manage risk. The fact that some of these cash flows are certain, but the benefits in the future are going to be uncertain and risky. But what we are going to look at today is the fact that these cash flows occur at different time. When we're making an investment, we incur that cash cost today, but the benefits come in the future and we're going to explain that cash in the future is worth less than cash today because of a fundamental concept called the time value of money. And that's what I'm going to be focusing on in today's lecture, how we can compare different investments in different cash flows when those cash occur at different time periods. You might think the idea of the time value of money. Well, that's quite subjective. Yes, we've got some sense that we prefer cash today to cash in the future, but doesn't that preference depend on who we are. For example, let's take a little child or a little child's trust fund. Now this little child probably is willing to wait many, many years, perhaps decades for an investment to pay off. So maybe he's going to be quite patient. He doesn't mind in investing in a project, let's say an electric car factory, which might take many decades to pay off. But then in contrast, let's take a pension fund, which exists to make some returns for pensioners. Now, pensioners might not be around in several decades time. So they might want a different project which pays back cash flows today or next year. Now that has the potential to create a lot of conflict. Why? Because most companies are owned by more than one shareholder. And those shareholders might disagree as to what the optimal investment decision is. Let's say you're a car company and you think, do I make traditional vehicles or do I invest in self-driving cars or electric cars? You might have to consult all your shareholders and ask them, do you want to quick buck today ar you're willing to wait many years, maybe decades for the future return. And that's going to be extremely cumbersome process. And what happens if they disagreed or they fight it out or do they have a debate or do they have a vote that will make corporate decision-making really difficult. But what we're going to show is actually there are ways in which both the little child's trust fund and the pension fund will agree on what investment decision to take. Even though the horizons are different, the power of what I'm going to show you is that we can get agreement on whether to take an investment. So how do we get to this agreement? Well, as I promised, I want to talk about the time value of money. So what is this? Why does money have time value? If you go back to my first lecture in the series, how to save and invest money has value because if you have money now rather than in the future, you can do something with that money. You can save it. And why is saving useful? Because it gives you interest. That is the return that you get for putting your money away. So let's go to just basic examples with some simple numbers. If you have one pound today and the interest rate is 10%, then it's going to be worth 1.10 in the future. So if you have the money now, rather than one year from now, then it's going to be worth something because you can put it away at the interest rate of 10%. Now, I know this is an elementary first example, but why I'm stressing this is that both the child's trust funds and the pension fund will agree on the time value of money. Money today is worth 10% more than money next year. Why? Because everybody can invest at the 10% interest rate. If banks were offering you a 10% interest rate then that is the time value of money. It has nothing to do with preferences or time horizons. It's something that everybody will agree on. Now let's more generalize this let's say the interest rate is not 10%, but it's an amount or percent. So maybe it's 1%, maybe 8%, whatever it is, then the time value of money is given by the r. And so one pound a day is going to grow to one plus R in the future. And let's generalize this even more. Let's say, we're not going to invest for one year. Maybe we're not invest for T years where T could be 5, 10 50, anything. What we looked at back in lecture one is how do we find the time value of money? You multiply it by one plus the interest rate, and you turn it to the power of T. You compound it T times. So just to sum up, what was we had in lecture one, and why is this relevant today? What we talked about back then was the future value of money. Money today is worth much more in the future. Why? Because we can invest it. We can compound it by investing it at that same rate every year for those T years, the future value of one pound is given by one times one plus r all raised to the power of T. So that's what we looked at previously. But what we're going to look at now today is what is the present value. If we have money in the future, what is that money going to be worth back to today? So what we looked at previously was converting money from now into the future. Now, what we're looking at today is the reverse. If we have some money, which we know we're going to get in the future, why? Because we're going to sell clothes from the factory. How much is that future money worth today? And that is exactly going to be this, but just in reverse. So we previously established that one pound a day, it's worth 1.10 next year. Then in contrast, if my clothes factory is going to be generating one pound 10 for me next year, how much is that worth in today's terms. We just do that calculation and do it in the opposite direction. The one pound 10 in the future is going to be worth one pound today, because one pound today, we could invest for that 10% and it would grow to be one pound 10 in the future. So these two things are worth exactly the same. We should begin different between one pound today and one pound 10 in the future. And again, so if the same straight forward, but let me just belabor the point. People will typically say more money is always better, but it's not true. What matters is when you're getting the money. Here, this is more one pound 10 is greater than one, but it's not better because it occurs in the future, it has less time value. And again, one other thing I want to stress to highlight, why this is non-obvious. Everybody agrees on what the present value of this one pound tenets. Why? Because they all divide it by this 10% interest rates, even though the child's trust fund and even though the pension fund, they might have different time horizons because some are more patient than others. Everybody agrees that cash flows in the future are worth less than cash flows today by this amount, 10%. Why? Because that is what you could invest money in if you had it today rather than in the future. So the new terminology I'm going to introduce is the discount rate. What do I mean by this? Well, we've talked about the interest rates. And the interest rate is what we're applying to grow money from now to the future. When we think about the future value of today's money, what the discount rate, the first two is the present value of future money, what we do in reverse to convert something in the future back to today. And so let me just go through some more terminology, just so that we're all on the same terms. What I did on the last slide is when I bring something from the present to the future, we are compounding the cashflow. What we're doing when we're moving things in reverse, as we're going to be discounting the cash flows. So anything in the future is going to be discounted. It is worth less because it is not benefiting from that time value of money. And again, what we can do is generalize this. So let's move from just one year and move from just 10% to any interest rate and any number of years, right? Because it might be, we're investing in the electric car factory, which might not pay off for 30 years. So when the interest rate is r percent, one pound, this year is worth one plus r next year in contrast, if we're getting now one pounds next year, how much is it worth today? What we now do is we divide by one plus r, okay? So to move from today to next year's money, we multiply by one plus r. When we moved from next year, back to today, we divide, okay. So this is how we're converting back. And so what we'll do is we just divide by one plus the interest rate. And then if we want them to convert, it's something which occurs many years from now, two years from now. Well, what did we do? We just do exactly the same as previously. We're going to be dividing. But remember we're using that power T as we did many, many slides ago, right? So whenever we have a cashflow and we're going through many, many years, all we do is we raise this right here, the compounding rate or discount rate to the power T, which shows you the number of years that we're investing for, or we're bringing it back to. And so what I've shown you here is the general formula for a present value. So the present value of any cashflow that you're receiving at some point in time in the future is simply achieved by getting that cashflow, dividing it by one plus the interest rates and raising it to the power, which is given by the number of years that we are investing. And the key concept here, which is one of the most fundamental concepts in finance is the whole idea of an opportunity cost. Why is it bad to have cash in the future other than today? Because if you're only getting cash in the future, you fall go the opportunity of doing something else with that cash. And what is the other thing you can do is you can invest it elsewhere. And because both the old man and the child's trust fund could invest it in the exact same opportunities, everybody has the same opportunity cost. It does not depend on preferences. And therefore everybody will be able to agree on the time value of money, because they all agree on what the opportunity cost is. And that is given by the interest rate, which everybody agrees on. Now, I realize I've shown you a lot of numbers and RS and equations point that we have an example. To put this into practice and show you, how do we use it to make an actual financial decision? I'm not going to do that right now. I'm going to do it in one more slide from now. But because I need to introduce a couple of things before. One thing I want to do is show you why this matters. So what this graph looks up is the present value of one pound and how all this depends on the interest rate, the discount, rate and how it depends on how many years we're investing for. Now, if you start here, if you're only investing for maybe one or two years, it really doesn't matter what the interest rate is. There isn't that much of a difference. However, if you're investing for 25 years, then whether the interest rate is 5% or 10% or 15%, that really, really matters. Why? Because if I'm having to wait 25 years before my electric car factory actually pays some dividends, then I'm going to be having to full go the interest rate for many, many years. So I really care about whether I'm giving up 5% per year or 15% per year, because what I'm giving up by building that electric car factory, I'm going to have to give it up for many many years. And then let's lean out, link this to the real world. Let's think of Tesla. So why does Tesla stock price go up and down so much and change so much? Well, one thing which causes it to change is the interest rate. What changes in the interest rate from 5% to 10 or 15, that will cause a huge fluctuation in the value of Tesla. Why? Because its cash flows are so far in the future that the opportunity cost of investing in Tesla rather than investing in the bank is really, really high. Final thing that I want to go through before I go through the numerical example is the fact that investments often make many cash flows. All I've shown you so far is what happens when you have one cashflow. But if you have a car factory, it's going to make you some cash next year and two years from now, three years from now, maybe it will make your cash flows 100 years from now, if people are still using electric cars. So how do things change when investments pay cash at many different points in time. The nice thing is it doesn't get that more and more complicated. All we do is we take each individual cashflow, take the present value of it, bring it back today, and then once you bought it back to you today, all you do is you sum up those cash flows. So let's say, take your factory. A factory pays you one pound one year from now, what is that one pound worth today? Remember we discounted, we divide it by one plus the interest rate. And so that now becomes a present value as of today. Let's say we also have the factory and it also makes us one pounds two years from now, how much is that worth today? We discount it. We bring it back by one plus the interest rate squared, because the opportunity cost is now twice. And that also brings it back to a cash flow today. And maybe we're also going to get another cash flow in two years from now. We can bring that back similarly, all the way back to today by using the T year discount rates. And the beauty is that because all of these are cash flows in the same year that all cash flows today because we've already discounted them. We can now add them up. But we can't add these up. They're not apples to apples. They occur at different periods, but once we've converted all the cash flows into today's cash flows, they are apples to apples. We can sum them up and we can say, well, the present value of a sequence of cash flows is simply the sum of the present value of each individual cashflow. Here if you're getting one pound each year, then you just convert each of those one pounds into the present values and then sum. So maybe now is the time that we're going to go through our much needed example to put this into practice. All I've done here is summarized everything I've done up to the point or we're doing when we take an investment, is we can take the cashflow and every individual year in the future, convert it back to today. We can sum up all those cash flows and we can compare the sum of all those cash flows to the cost of the investment. How much did it cost to build the factory? And if the costs are less than the benefits, then we want to go ahead and build that factory. So let's now go to this example. So the example I'm going to give you is let's say we're considering building a gym. Now, what we think is that the gym will cost us 1000 pounds to build. And let's say the gym is going to last for three years. And we're predicting, we're going to get revenue. We're going to get profit of 200 pounds in the first year. It's not going to make much money because we're still trying to get customers. We're going to be batch one year to 350 and we're going to do do even better in year three and we're going to get 600 and then we're just going to shut down the gym. Now, how do we figure out whether to take the gym as an investment? Now, one thing you might do is you might think, well, let's just add up all of these different cash flows. So minus 1,00, plus 200, plus 350, plus 600, all of these that adds up to 1,150, we get more out than what we put in. We get 150 more than what we put in. So you might think, well, we should want to take the gym, but we should know that that's not the right thing to do because these things occur at different time periods. So instead what we do is we take each individual cashflow we discount it by the interest rate. And interest rate here is given by 8%. So what do we? Do when we take the investment, it costs us 1000 today, it's giving us 200 pounds next year, and that is discounted by 1.08. It gives us 350 two years from now. We discount it by 1.08 squared. And finally the cashflow in the third year is 600. We discount that by 1.08 cubed. And after we do this and put it into a calculator what we get is minus 38. So what this is is negative. What this means is that the present value of all the future benefits of the gym are greater than the present cost. If the benefits are less than the costs, the nobody should build the gym. And you might think, well, why should it be nobody? What happens if you have a child trust fund with no other need for the money, you're willing to wait? Yeah. Maybe the return is not that great, but you're still getting more than what you put in. But if you're the child's trust fund, the other thing you should do is you should just put the money in the bank, because if you put it in the bank, you get the 8% return and you'd end up with 1,260 in three years, rather than this 1150. Let's just take a step back from all of the equations and four of the numbers, and just look at the power of the methodology that we've derived. But many decisions we make in life are consumption decisions. But what wine do we choose? What do we choose to eat? Where do we choose to go on holiday. And for consumption decision where we're going to get some immediate benefits. That depends on taste. So there's no possibility of having a general rule that applies to everybody on what wine to choose. It depends on weather where a liker of say sweet wines or full body dread and suddenly Whetstone holiday, but different people have different preferences. What book to read, what concert to go to nobody's ever going to have a possibility of coming up with a general rule or knowing which one to choose. But the power of looking at an investment decision where what we're getting is benefits, where you can at least roughly put those benefits in monetary terms is that we can come up with a rule, which is the net present value that I've shown where, what we do is convert this into some financial benefits. And then we bring them back to today at an interest rate that everybody agrees on. Then there is one objective way to make a decision. And that objective, why is something which is independent of preferences. That's one point I wanted to draw out. The second point I want draw out was, well, if indeed there was one objective way of evaluating investment decisions. Why is that that investment decisions are so difficult to make. And why do people still disagree? I've just argued. I've made this really big thing about why the old man and the child's trust fund should agree. So was that so much disagreement? Why some people say actually this car company should invest in self-driving cars and others say, well, that's just too risky. They should just build traditional vehicles. But the reason why people disagree is because they will disagree about the future cashflow. Like people don't really know whether the gym is going to be a success, maybe it's not going to get 600 in year three. Maybe it's only going to get 400. Maybe we're just over optimistic about how well it's going to do. So that is where people disagree. They don't know what the cash flows are going to be in the future. But all I'm telling you is that given a set of cash flows, there is an objective way that everybody can agree on in order to bring those cash flows back to today. And you might think, well, why does that really help us? Because it's really difficult to forecast cash flows to begin with, but there are certain types of investments where actually the cash flows are quite simple to forecast. For example, investing in certain bonds or maybe some housing and so forth. And even if there is some uncertainty, what we can do is we can play through different scenarios and look at how does this investment decision change under an upside case with the gym is really successful and a downside case where the gym is not so successful and see how that changes the investment decision. So again, I'm giving you a framework so that once we were given some cash flows, we can figure out whether the investment decision is good. I fully acknowledged that we don't know what the cash flows are going to be, but we can just repeat this with different cash flows and see what's going to turn out. So let's try to do this. So let me actually change something. And what I'm going to change here is not so much the cash flows, but I'm going to change the interest rate. So in the previous slide, the interest rate was 8%. Let's say the government just cuts interest rates to 5%. So I not the government, the central bank cuts interest rates to 5%. What changes? Well, if the cash rates have stayed the same, but now we're discounting by 5%, not by 8%. And so what this means is because we're dividing by a smaller number. Now, the net present value of the project, which is how much the benefits are exceeding the cost by, this now becomes a positive number. It becomes 26. So now everybody should want to build the gym. Why is this? Let's go back to the concept of opportunity cost. So opportunity cost is if you did not build the gym, you could put your money in the bank, but now putting your money in the bank is less attractive because the bank is only paying you 5%. And therefore the gym with the returns that's giving is a more attractive option. And this is why now this becomes a positive net positive value decision when previously it was negative. And again, let me take a step back from the numbers, but this is why central banks can influence investment decisions by changing the interest rate. But when the Bank of England chooses to increase the interest that set to be a way of cooling the economy and reducing economic activity. And why is this? Because it means it's less attractive now for companies to make some investments, to go out and buy stuff and build stuff, because it's much more attractive to save that money in the bank and that's one way of the economy and reducing some inflationary repression. So now the gym example you might think, okay, great, I came up with an example, but that example was not that realistic because the gym only lasted for three years. In reality, maybe a gym will last for 50 years. And if the gym lasts for 50 years, then you're going to have to calculate all of these different cash flows for 50 years time. You can see, I had lots of fun writing this particular slide. And that's not something that is going to be that easy to do. So how can we come up with perhaps a shortcut to save us, having to have cash flows for 50 years? Now, one thing we can look up is what happens if we have not cash C this for 50 years, but in fact, cash flows for infinity years? What do I start with infinity, not 50? Because this is when we actually have the simplest possible shortcuts. What if we had an investment which paid the same cashflow every year forever? So this is something which is called a perpetuity. It's paying a perpetual cash flow until the end of time. So here, the shortcut is really simple. If you have an investment which pays a cashflow of C every year until the end of time, then you would find the present value by taking that cashflow and simply dividing it by the interest rate. Well, that makes sense because the lower the interest rate is the higher, the present value. Again, a lower interest rate means a low opportunity cost, investments become more of attractive when the opportunity cost is lower. And you might think, well, that's a great shortcut, but it's completely useless because no investment last forever, but actually some do, right? So there are companies where these companies have been existing for many hundreds of years and might keep existing for hundreds of years going forward. It's not clear that the products they're making, let's say clothes will ever stop being in glistens. And even if they will only exist for maybe 500 years, maybe that's sort of close to good enough and we can still use this approximation. And there are some investments literally that last forever. So in the 18 hundreds, after the Napoleonic wars, the British government wanted to consolidate all of the debt that they had accumulated. And so what they did was they introduced a bond, which they called a consol and this bond would never be paid back, but it would pay you interest every year until the end of time. And then the consol don't just exist in the UK. Here's an example of an American consul. If you look at this, it says 4% consols. What does that mean? It means they are going to be paying an interest rate of 4% and it will pay paying the interest rate of 4% on the amount of $50, which is the face value of that bond. So let's again, use an example. Let's try to apply that shortcut formula that I've just made this big thing about introducing. So let's say we are now in the US the common interest rate is 3%. What that means is we could put our money in the bank and earn 3%, but we own one of these beautiful looking bonds, which says, we're going to pay you 4% on a face value of $50. How much should that bond cost? Remember what the formula is. This is something which is going to give us how much interest every year. Is it a pious 4% on the face value of 40.$% of 50 is $2. That bond entitles us to $2 every year until the end of time, I'm pretty boring investment, but a really safe investment. We're going to get $2 now and at every point in the future. Remember the formula was the present value of a perpetuity is going to be the cashflow that we're getting divided by the interest rate of 3%. And this is going to be worth $66.7. Now, one thing that I've tried to stress throughout my principles of finance course is the importance of sanity checking and think, well, do these things make sense? Step back from the equations and think about, is that something which passes common sense? And the answer is yes. Why? The face value of the bond is 50. The bond is only worth $50 on paper. Why? That's what it says. It says it's a bond, which is worth 50. However, why is it that you're willing to pay $66 for something with a paper value of only $50? It's because the interest rate that is offering of 4% is greater than the 3% that banks are offering. So the bond is paying more than the opportunity cost. If you had your money, if you took your money and you put it in the bank, you'd earn 3% given this bond is going to be paying you 4% you should be willing to pay than the face value of $50 for it. And that's why you're willing to pay as much as $66.70. And so this is the concept of a premium bond, which you might remember in lecture one, once you have a bond where the interest rate is greater than the opportunity cost, it should trade at a premium to its face value. Why? Because it's given you a higher return than you could get elsewhere. Now then let's look at what happens when the interest rate changes. Let's now change the interest rate to 4%. Now what will happen? We do the same calculation. Now the discount rate is 4%. And so the value of the bond is 50. Exactly the same as the face value. Does that make sense? Yes. Well, the bond is offering you 4%, bank offering you 4%. The bond is neither a good deal, nor a bad deal it's often only just the same as you could get elsewhere. And therefore you would pay exactly what it's worth. You pay exactly that $50. And let's move this one more time in the same direction. Let's now increase the interest rate to 5%, we do the same calculation. We find that when we discount by 5%, the present value, the value of that bond now falls to $40. Does that make sense? Yes. What the bonds face value is $50, but why are you only willing to pay $40 for a bond with a paper value 50? It's because the bond is only giving you a 4% interest rate whereas you could get 5% with the bank because that is the opportunity cost, that is the outside option. So what I've highlighted is how the value of anything, a bonds, a gym, any investment goes in the opposite direction to the interest rates. With the gym, remember that was initially something we didn't want to invest in, but when the interest rate fell, we did want to invest in the gym. And same for the sponsors here, the lower the interest rates, the higher the value of the bond, the bond is worth more to us. And you might think, well, does that really makes sense. Let's do the sanity check that Alex has always been emphasizing. Well, bonds pay interest. Shouldn't a bond be worth more, the more interest it pays. But remember, the interest rate I'm talking about here is the outside interest rates. That is what we could get elsewhere, if we were to put the money in the bank instead. So that is why we have the negative relationship. This is the opportunity cost. This is what else you could do with your money, if you didn't buy the bond, if you didn't invest in the factory, if you didn't renovate your kitchen. The better the outside option, the lower the desirability of building the factory or renovating your kitchen and so on. The worst, the outside option, the more attractive the investment decision is. Well, this is just like, if we had the World Cup final on at the moment, nobody would be in this lecture if maybe the strictly conducting final was on or anything else, which is really attractive, that would reduce the attractiveness of making the decision to come here. But if there isn't so much else on, then an it decision to come here, it's going to be more attractive whenever we make any decision, which costs us something, be it money, be it time, this is something with an outside opportunity costs. So what we need to look at is what is the cost of the outside option and here it's given by the interest rate. So there's a couple more shortcuts that I want to go through before closing and opening it to questions. You might think, well, the bond that I showed you is really boring. It's just paying you the amount of 4% every year, forever. In reality, many investments that you might realistically want to take will grow over time. So let's say we're going to bail that close factory again. That closed factory might generate maybe 100 pounds in the first year, but each future year, that will grow maybe you're going to make the clothes more better maybe you're going to be building a brand and so on and so you're going to be having some growth here. So how do you things change when, what you have is a growing perpetuity where the cashflow is not this boiling amount, see every year, but it grows every year by a growth rate given by g. How does the present value change rather than having C divided by r we're now having C divided by r minus G. So in the denominator, we're reducing the denominator? Does that make sense? The answer was yes. The higher the growth rate, the faster the cash flows from an investment grow a lot higher growth rate means a lower denominator, which means a higher value. So something is more valuable when it's growing. And so going back to companies like Tesla, Tesla doesn't produce much cash right now. However, because the growth prospects are so high, this is why the company could be trading at much, much, many, many times it's common cash flows. Why? Because the expected growth rate is so high. That's what justifies the extra astronomical valuations. Again, let's put this into practice with an example. Let's go back to our gym. And let's say that as before, the gym is costing us 1,000 today, and it's going to give us 200 pounds one year from now. And let's say, then that 200 pounds is going to grow at a rate of 2% every year forever. So how do we calculate whether the gym is worth it or not? The interest rate is 8% as previously. We're going to use the exact same formula as before. We said to calculate the value of a growing perpetuity, which is growing until the future, take the first year's cashflow divided by r minus g. The interest rates minus the growth rates, 8% minus 2%. This gives you the value of all the future cash flows. And remember, we will not the net present value. We want to net off the initial costs. We pay 1,000 for something worth this. And so if all, what we have is something worth 2,333, the value is much, much higher than the cost. And therefore everybody would want to go ahead and build this gym. The final thing I'm going to go through in terms of a final shortcut is, well, what happens if we don't have something which is going to last forever? Well, okay. Yeah, we get it consul as they last forever. And maybe we've got a business model, which is going to be not clearly terminating at some point, we can assume that it might last forever. But there's some things which last for short period of time. And so what we're going to look at is that type of cashflow. It's called an annuity. An annuity is something which pays cash for certain number of periods, and then just stops. And one big example of that is a mortgage. So what do we do when we take out a mortgage, I'm going to borrow it from a bank. I'm going to pay back to the bank, the same amount every month, that's not changing and then hopefully after 25 years, I have fully paid back the mortgage. So the bank is getting money from me every month, but I only for 25 years, they're not getting it in perpetuity. I'm only slaved to them for 25 years. So the question is, is if you do have this investment, which is paying for only a finite length of time, how much is it worth? Is here, the stream of cash flows is C every year for an amount of time, which is T periods. It's not coming on forever. And what we have is this formula here as being the shortcut. So it is like, it starts off being a perpetuity, but it's worth less than a perpetuity because of this minus term here. And that kind of makes sense, right? Because an annuity stops at a certain point in time is going to be worth less. And that's reflected by the fact that we're multiplying by this number, which is less than one. And obviously the fortress stops the lower T, when T is lower, this is higher. And so when T is lower, this whole thing here is going to be higher. We're subtracting a bigger number and therefore the whole present value is also going down. So this gives me my final example. And I'm getting, can I use the example of a mortgage? I'm taking out a mortgage for 100,000 pounds at a fixed interest rate of 3%. The mortgage will be repaid each year. I know that in reality it's months, but let me keep it simple and have it years, over 25 years. How much will I need to pay back every year? So what am I doing when I'm paying back a mortgage I'm making an amount every year for 25 years, it has to be that the present value of all of those future 25 payments has to equal 100,000 pounds today. And so how do I calculate the present value of all of those future 25 payments? I use the annuity formula. It has to be that whatever my annual payment is every year, that when I take that over 25 years has to give me 100,000 pounds. Why? I'm boring a hundred thousand pounds from the bank. What I have to pay back the bank in the future every year for 25 years at an interest rate of 3%, must be worth the 100,000 pounds that I'm borrowing from it to begin with. And when I put this into calculator, what I get is I get the cashflow is 5,743. How much I need to pay back to the bank every year, is this for me to have fully repaid the loan? The question is, well, does that make sense? And the answer is yes, because if I type that and I multiply it by 25 years, then I'm going to be paying back overall 143,570 to the bank. And that makes sense because I need to pay the bank some interest. So it makes sense that I'm bothering 100,000 now, and I'm paying back an amount much higher than a hundred thousands because I need to pay interest. Now, before today's lecture, I'd asked you this question, you might think if I'm having to pay back 100,000 pounds over 25 years, how much will I pay back every year? 100,000 divided by 25, 4,000 pounds per year, but that's clearly wrong because that doesn't take the interest rate into account that shows you how to do this. And this gives you the answer here, which indeed makes sense. Before I close and answer the Q&A, let me just go through some final practical tips of how to do these things in practice, right? I've shown you the formulas and I've shown you how to apply them but as I alluded to earlier, so often some of this skill of this is to think about what numbers to put into the formulas to begin with rather than actually how to implement the formula. And one really important thing is to realize that you need to take all incremental cash flows into account. So what does it mean by incremental? Any new cashflow, which comes as a result of having to put on at the investment, this needs to be taken into account. So example of Gresham College is going to be trying to introduce a new subject. And let's say, we're going to do a new series of lectures on this new topic. If this means that we need to hire out the Museum of London to have that new lecture series, then those cash flows need to be incorporated. However, it is that this requires some marketing and that marketing we already have, the website, we've already built it, then we might not think that that is something which is incremental. So we only to include things that are going to be added as a result of having to launch that new investment. On the flip side, what we need to do is ignore sunk costs. So what do you mean by a sunk cost? That is a cost that happened in the past. It's the bygone. It's not something that we can change. For example, let's say I have spent as a company,$10 million developing this new drug. And now I'm going to think about, do I actually commercialize the drug? And if I was to commercialize the drug, I would get revenues of 5 million, but it would cost me 6 million to do that. Now you might think, well, it's obvious I shouldn't do it. I'm going to pay six and get five, but you might think, well, at the back of my mind, I've already spent 10 million developing the drug. If I don't commercialize that that 10 million would have been wasted. But that 10 million what's already wasted what you can't change the fact that that 10 million has already been spent. The only thing that you care about now is any future cost, which comes from taking that investment decision. And the only relevant thing is the six it's going to cost me to develop the drug, verse the five I'm going to get by doing so. And because the benefits are less than the costs, I'm not going to want to go ahead. So when you have a witness at a trial, the witness has to swear to tell the truth, the whole truth and nothing but the truth. And that principle applies to the cash flows. Like we want to have the whole truth, any incremental new cashflow, which comes about that needs to be included, but nothing but the truth, anything which is irrelevant because it was already spent in the past and is not affected by the new investment decision, we should make sure that we're not going to be including. Well, thank you very much for everybody's attention. We're going to have time for some questions. Let's invite Claire on stage.(audience clapping)- So the first question is, how do you decide on investing in connection with inflation and you kind of dressed a little bit here. For example, if you buy the PV bond at $50 and its value when it matures to 67, has the overall value of the bond decreased because of inflation.- Yes. Thanks very much. And that's actually something I didn't come to just an interest of times. I'm glad you've asked it. So all I've done here is I've always considered nominal cash flows. And so what are those? Those are cash flows which incorporate inflation. So why is that? When I look at the discount rate, that is a nominal financial discount rate, which does take inflation into account. So all of the cash flows and the numerator should also incorporate inflation. So when I think about how fast the cash flows of the gym are going to grow every year, that should include any inflation as well. So what I'm looking at is the cash flows in financial terms, how much money does this give you, including any effect of inflation? And that is apples to apples with the fact that the discount rate, the interest rate, that also is something which incorporates inflation as well. So I have inflation in the numerator and inflation the denominator.- [Man] I want to ask about now it seems the age of cheap credit is coming to an end. Inflation is skyrocketing, because of huge government spending it seems, how would that affect house prices and mortgage price is more like a mortgage rate and and government debt. So, I mean, particularly in house prices, there's mortgage rates go up, house prices also go up. What would you say about house buying?- Actually I think when mortgage rates and interest rates go up, house prices will go down rather than going up. So why is that? Because the opportunity cost is instead of buying a house, what it could do is put your money in the bank and that's going to be a more attractive option when interest rates go up. Alternatively, you might say, well, in order to buy the house, you need to finance that house purchase with a mortgage. That mortgage is going to be much less affordable with a higher interest rate. And therefore that the house is going to be a less attractive proposition. So typically when interest rates go up, then house prices will typically go down. Now, if the market is efficient, then the market should have ready have factored that in. So the current house prices right now should already take into account the fact that we think that the age of cheap credit is going to be over. So what should then it determine whether or not you would want to buy the house is not whether interest rates are going up because everybody agrees that they are, but whether your view is more optimistic or more pessimistic than the current view. So let's say everybody thinks interest rates are going to go up to, let's say, just to pick a number 4% by 2025, you think you're interested is going to go up, but maybe they're only going to go up by to 3%. You would still buy the house, even though interest rates are rising. Why? Because you're not pessimistic as everybody else. So what matters is your view on interest rate, not compared to where they are now, but compared to where the market thinks it's going.- Thank you all very much for attending and for again, for a fantastic lecture, please do come back for Professor Edmans next lecture, which is on how to measure and manage risk. And that will be held on Thursday, the 24th of February

00 PM back here in the museum and online. Thank you very much for coming this evening and thank you to our online audience as well.(audience clapping)