ECON 251: Financial Theory
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ECON 251 - Lecture 8 - How a Long-Lived Institution Figures an Annual Budget; Yield
Chapter 1. Yale’s Budget Set [00:00:00]
Professor John Geanakoplos: So we’re spending a couple classes these days learning basic facts and vocabulary about finance, and along the way we’re trying to apply the simple lessons that Irving Fisher taught about turning a financial problem into a general equilibrium problem and making use in particular of the budget set.
That very simple budget set we wrote down at the very beginning turns out to be quite useful and people often can get quite confused. So the last issue we ended with, I’m going to take up again. Suppose that you’ve got a very long-lived institution like Yale. How should Yale think of how much to spend every year? What is Yale’s budget set?
Almost every big institution like Yale creates a fiction of an annual budget and they talk about the deficit and having to bring the deficit under control and making cuts to close the deficit gap, but really there is no such thing as a one-year budget set. I mean, why one year? Why not one month? Why not one day? Nobody expects Yale to balance its budget every day. Some expenditure comes in one day. They have to hire an electrician to fix something unexpected. They’re going to spend more money than they take in student tuition that day. So the fact that they’re supposed to budget the balance every year is just a fiction.
Irving Fisher taught us that Yale really has–if you can borrow and lend and you don’t have to worry about risk there’s one infinite-lived budget set. It’s an infinite horizon budget set where you just take the present value of all the expenditures, that’s the left hand side, and the present value of all the revenue, that’s the right hand side, and make sure that the left hand side is smaller than the right hand side over the whole course of Yale’s life.
So that simple principle has a tremendous implication which was overlooked, to the chagrin of the last Yale president. So as I said, the issue was in 1997, I believe, it could have been ‘96 something like that, 1997, Benno Schmidt, who was then the Yale president, released a white paper, he called it, documenting the fact that Yale had deferred maintenance in the buildings, he called it, and a study that he commissioned, a very good study that he commissioned, argued that the deferred maintenance–Yale could be brought up to snuff and then go on afterwards as a normal running institution provided it spent 100 million dollars a year for ten years, and that included fixing every college–they’re going to do more than one a year over a 10 year period.
So Yale’s total budget, as I told you, was about 1 billion dollars at the time, and then here all of a sudden was this 100 million dollar a year expense for 10 years. That’s 10 percent of the Yale budget. And a lot of the costs you can’t change. You have to have the lights on. You have to heat the buildings.
You can’t really reduce those costs. So Benno Schmidt came to the conclusion that he’d have to reduce the costs he could change by 15 percent in order to balance the budget, to cut about 100 million dollars a year out of the budget. So he announced one day that he was going to fire 15 percent of the faculty by attrition. If they were junior faculty he wouldn’t promote them, and if they were senior faculty, wait until they retired and not replace them. So this, needless to say, caused a tremendous commotion among the faculty, and as I told you a committee was formed and I had to present the report.
Chapter 2. Analysis of Yale’s Expenditures and Endowment [00:03:37]
So actually the report went pretty carefully through all the calculations made in the white paper, but the heart of the report was simply to apply the lesson of Irving Fisher. So what is the lesson of Irving Fisher?
Let’s suppose that there’s no inflation so that when they say 100 million dollars a year they mean 100 million real dollars a year. So Irving Fisher would say Yale’s going to live forever. Let’s suppose that Yale wants the same quality of education every year forever, so it should have the same real spending every year forever after it compensates for inflation. So at the moment we’re assuming there’s no inflation. So what does that mean?
That means that you just look at the right hand side and you say, what’s Yale’s revenue? Well, whatever it was before we were told by this report of the president that as long as you did the deferred maintenance Yale would be back in balance. So what’s the loss of revenue on the right hand side? It’s 10 years of 100 million dollars a year as you can see.
Now, you need an interest rate. What should the interest rate be? Well, should it be the nominal interest rate or the real interest rate? Well, we’re supposing now there’s no inflation so it should be the real interest rate because all the hundreds have no inflation in them. So what real interest rate shall we use?
Well, the white paper used 5 percent because they thought that was the number that Yale could earn after inflation pretty reliably every year. They think that’s the real rate of return Yale gets, and so they discounted it 5 percent. Now the real rate of interest typically in the economy is 3 percent, but let’s suppose that we calculated this at 5 percent. The present value of 100 for 10 years is 772.
Now, how could you do that in your head? Well, we know that 5 percent is going to double every 14 years, 5 into 72 is about 14. If it doubles every 14 years 10 years is going to be less than half the value of the bond. So if you got 100 forever at 5 percent interest that would be 2 billion, and we know for only 10 years it’s less than half the value, so considerably less than half the value, so it’s not one billion it’s something less than that. It’s 772.
So I just did that in Excel and I calculated 772, but in your head you know that if it had gone on for 14 years then the present value, our formula, our famous formula is that you would take the coupon 100 divided by the interest rate .05 times (1 - 1 over (1 + R) to the 10th). And so we know that if this were 14 instead of 10 you’d get a half here. So this is 20 times 100 which is 2 billion times 1 half would be 1 billion, but since it’s only 10 years and not 14 years it’s less than a billion, so 772.
So in your head you could have probably figured that out approximately. So you could be sitting there in the audience hearing Benno Schmidt talk and be computing in your head that we’re talking about something under 1 billion, like 3 quarters of 1 billion. So now, how much does that mean reduction in every year?
Well, if Yale’s going to spend the same amount every year that means it should be spending 5 percent of 772 less every year. That’s 38.6 million dollars less every year. So that’s a drastically smaller number than 100 million dollars a year. It’s crazy just to think that because you’ve got these expenditures for 10 years and then no expenditures after that, that you should cut the budget by a 100 million and then let it go up after 10 years. So you’d only need to cut it by 38 million.
Now, by the way, 5 percent is a pretty arbitrary number. Suppose you put 3 percent here? Well, 3 percent would give you a much higher present value, but then when you multiply it by 3 percent at the end for the annual reduction it will give you a much smaller number. Anyway, this number, which I computed in Excel, but again you can do it sort of in your head, is 853 million, but you multiply that by 3 percent and you get 26 million a year. So now the reduction is starting to sound like it’s not such a frightening thing.
So let’s stick with the 5 percent which is what the white paper, take all the assumptions of the white paper and take it literally, and then notice that they never said anything about inflation. So actually, this calculation of present value loss–there’s inflation, and say the inflation at that time is around 4 percent. In fact all the Yale contracts that are still in place assume a 4 percent inflation even though inflation’s less than that now.
But anyway, so that 100 million a year of dollars is actually less in present value terms because what should you discount by? If you look at the present value of 100 million dollars over 10 years and you take into account its dollars you should be discounting by the real interest rate times the inflation, so by 9 percent, a tiny bit over 9 percent. So if you discount that by 9 percent you get 641 million as the present value loss to Yale.
Now, given that there’s inflation how much should you be spending every year? You should be spending in real dollars, reducing your expenditure how much in real dollars? Well, by 5 percent of the 641 million. So if you have 641, that’s today’s present value. There hasn’t been any inflation yet. So that’s the real loss in dollars. So if you ask, what’s the real expenditure reduction every year, it’s 5 percent of 641 and that’s 32 million.
So 32 million is a far smaller number than 100 million and requires a far smaller drop in expenses, so our committee recommended that we cut the faculty by 6 percent instead of by 15 percent, and 6 percent–there are a lot of people leaving every year. You can do 6 percent pretty quickly. So the upshot of this is that it is a simple application of present value, a very elementary calculation. It came as somewhat of a revelation to our administrators, I’m afraid, and the day after the report the provost of the university resigned.
Two weeks later the dean of the university resigned and two months later the president of the university resigned. And Rick Levin, the current president, took over and he cut the faculty by 6 percent, but by no more than that. And then, of course, the finances of Yale got much better and he’s since added back that 6 percent plus a little bit more than that.
So just to tell you, though, something good about the Yale administration, the provost who resigned that next day happens to be a friend of mine. I’ve had dinner with him every month for the last 12 years since this happened and he’s never once criticized me or shown the slightest discomfort about the report that basically ended his administrative career. He cares so much about Yale and was so determined to do the right thing. He just thought he made a miscalculation and stepped aside.
So I believe that he wanted to do the right thing for Yale and just made the wrong calculation, not that he had some political agenda or something to cut such a huge part of the faculty. So it’s the most honest and most Yale-loving administrator that you could imagine. And I didn’t know what the reaction would be of someone like him after giving the report. I was quite terrified, actually, that they’d be–they were still the president, the provost and the dean, that they would be quite angry at our committee, but they responded with tremendous integrity. Yes?
Student: Why did you multiply the present value by the interest rate to find the <>?
Professor John Geanakoplos: You tell me. Why did I do that? So someone else tell me. So that’s a good question. Why is that? Yes?
Student: It’s like finding the coupon of the perpetuity with a 5 percent interest rate, because it’s like you’re rearranging C over r equals the present value, so it’d make it present value times the interest rate equals C.
Professor John Geanakoplos: Right. So the question was why, after I figured out the present value loss like 772 million, why did I multiply that by 5 percent to figure out how much Yale should reduce its spending every year.
And the answer that was given down here is that I’m assuming that Yale is going to go on forever. So Yale can reduce its expenditures every year forever and by doing that make up for the same present value loss, so forever means perpetually, so it’s a perpetuity. So how much do you have to reduce–what is the coupon reduction, the expenditure reduction every year at 5 percent interest that just makes a present value decline of 772 million?
Well, it’s 5 percent of the principal. If you have 772 million in the bank and every year 5 percent of that you throw away, you’ve thrown away the whole value, at 5 percent interest, of the 772 million. So by reducing your expenditures by 5 percent every year you defray the 772 million dollar loss.
So the critical thing, the critical mistake that the administration made is they had a short run problem with a bunch of short run costs, but Yale’s going to live forever and Yale should share the cost and the loss over all future generations, not just make the current faculty, and the current students, and the current city bear all the costs of this one shot loss, one shot problem that Yale faced. So they weren’t thinking in Fisher’s terms of taking the present value over the whole course of the lifetime of the institution. They were thinking, well, we’ve got to spend 100 million dollars this year we better cut costs by 100 million dollars.
But that’s obviously crazy. Like suppose you had to spend an extra 100,000 dollars in one day. Does that mean you should lay off your faculty for one day so you can find the money to pay that? And of course not, so you have to spread the loss over the lifetime of the university. Are there any other questions about this principle? Yes?
Student: If you wanted to return to say spending 100 million dollars a year again after 10 years would you spend 77 million a year for those just 10 years if you wanted to incur the whole cost over a fixed period of time?
Professor John Geanakoplos: If I wanted to incur the whole cost over 10 years?
Professor John Geanakoplos: Then I would have to reduce my expenditures by 100 million dollars a year. If at the end of 10 years I wanted to be back even then I would have to spend less. I would have to cut my expenditures by exactly the money I was pouring into the buildings. So if I wanted to reduce my expenditures by an equal amount every year it would have to be 100 million dollars for 10 years, right?
If the maintenance costs go over a 10-year period and I want the expenditures to go over a 10-year period, right? So if I want the expenditures to be reduced evenly over a 10-year period I’d have to do 100 million dollars a year. If I wanted to eat away the costs all in 1 year–I guess I didn’t understand you’re question. You’re saying if I wanted to reduce expenditures entirely in 1 year and then return next year to my usual pattern of expenditure then I’d have to cut the 1 billion dollar budget by 772 million. So that would involve basically firing the whole faculty and saying take a year off you’re on furlough, sort of what they’re doing in California. Yes?
Student: Can you also spread the present value cost evenly over 10 years and not like 100 million dollars every year, but 100 million in present value terms.
Professor John Geanakoplos: I could do that if I wanted. I could rearrange the 100 million any way I wanted to. So you would cut less than 100 million dollars this year out of the budget and a little bit more every year after that?
Student: Assuming you wanted to get these costs over with in a 10-year timeframe.
Professor John Geanakoplos: Well, one way to do the costs over with in a 10-year timeframe is reduce costs, reduce paying the faculty by 100 million dollars every year for 10 years, right? That would obviously do it, because that’s the money I need to get.
You could now rearrange that by reducing costs a little less at the beginning and a little more at the end of the 10 years, but I would say that doesn’t sound–why do that? That’s kind of what bad politicians do. They say, “It’s not our fault. We’ll just make those guys in year 10 get totally crushed,” and pretty soon they are in year 10 and then they’ve got 200 million they have to cut costs. Yes?
Student: So say in 25 years Yale wants to do another round of rebuilding all of the buildings they’ll still be paying for the buildings that they built 25 years prior?
Professor John Geanakoplos: Right. So there’s a good question.
So actually some people, the administration, that was their best response, they said, “Well even though the white paper said this was a one shot thing, and after we do this 10 year plan Yale is back in good shape, and of course every year Yale has allowed maintenance expenses. That’s part of the budget is maintenance. So after we get the buildings back in tip top shape we’re going to keep them in tip top shape by doing these normal expenditures every year so we should never have another period where you have to do something drastic like that.”
That’s what the white paper said, but after the report they responded just like you said. They said, “Well, we didn’t really mean that. Maybe in 25 years we’re going to have to do another remodeling effort.”
And so well, if you need to that then it wasn’t just a one-time deferred maintenance. It means that you’ve drastically underestimated the cost of keeping up the buildings over Yale’s whole future, so then you would have these reductions in expenditures for the first 10 years and then in year 35 you’d have to have more reductions, and then in year 70 you’d have more reductions like that.
So you have to take the present value of all those things and then figure out how much to reduce expenditures on an even basis and so it would be much more than 32 million. It would be 60 or something million or 50 million. So you’re exactly right, but that’s not what they said in the white paper. So I took them literally what they meant.
So what happened after that? Yale has done much more building expenditures than that, but that’s because Yale’s endowment went up to 23 billion. So from 3 billion which it was at the time it’s now 23 billion so Yale’s launched an incredible program of building construction. Basically they’ve done two things. They’ve built a huge number of construction jobs and they’ve hired a lot of administrators and stuff.
So the faculty is still not that much bigger. It went down 6 percent. It’s now back a little bigger than it was before. So the plan to expand the college and expand the faculty hasn’t happened yet. So Yale faces another choice now. You know the endowment went to 23 billion and then this past year they managed to lose down to 17 billion, 30 percent got lost. So we’re down to 17 billion now. So again we have the same question. We just lost 6 billion dollars. How much should we reduce expenditures every year? So what’s your answer to that?
Student: About 30 percent?
Professor John Geanakoplos: Not 30 percent. So how much would you reduce expenditures? Why wouldn’t it be 30 percent? Because Yale spends a lot of money it doesn’t get from the endowment, right? It gets money from tuition, for example. So what should Yale do? What do you think Yale’s going to have to take out of the budget now?
Student: <> calculation.
Professor John Geanakoplos: Yeah, and let’s say it’s still 5 percent real interest then what would you do?
Student: Present value’s 6 billion dollars and we assume we’re at 5 percent interest.
Professor John Geanakoplos: Yeah, so what’s that?
Student: 6 over .05?
Professor John Geanakoplos: 6 times .05, so what’s that? Hard to do these things in your head, right, but what is it?
Student: 300 million.
Professor John Geanakoplos: 300 million, so Yale’s got to somehow cut 300 million out of its budget so it’s not going to do it in one year, but over the course of the next few years it’s going to have to cut 300 million. Now the budget is well over 2 billion so that’s 15 percent of the budget, though, Yale’s going to have to cut. So this is a serious thing. How do you cut 15 percent of the Yale budget?
Student: Firing faculty.
Professor John Geanakoplos: Firing faculty, well, I hope they don’t do that. I think they learned their lesson so I doubt if they’ll do that. Things are already changing. They’re charging for long distance telephone calls and all kinds of stuff like that.
That doesn’t get you quite 300 million, but there’s going to be a bunch of stuff like that. So anyway, we have another budget problem, by the way. So these kinds of budget problems are happening all over the country. I gave a talk at Albany University and they’re going to abolish their graduate economics program, SUNY Albany. These are serious problems losing that much money.
But in any case 6 billion translates to, right, the difference here is 6 billion and you multiply that by .05 just as we said and that equals 300 million a year. So Yale won’t do it right away, but over the course of a few years Yale’s going to have to reduce its budget by 300 million, so they’re going to obviously choose to do a lot less building and presumably some of the new people that got hired they’re going to not keep. So any other questions? Yep?
Student: Doesn’t that presume that the endowment stays that way?
Professor John Geanakoplos: Yes.
Student: Could you also, once the stock market and the real estate goes back up, can you have some assumption that they don’t need to cut as much?
Professor John Geanakoplos: Right, very good question. So in the front he’s saying this presumes that we know for sure that the endowment lost 6 billion. It’ll never recover it. Maybe that’s a temporary drop in the stock market and it’ll go back up, and basically the principle he’s applying is he’s saying you can’t make these drastic reductions in annual expenditures, firing people and then two years later realizing you’ve got a lot of money trying to hire them back because you won’t be able to hire them back.
So clearly Yale has to have a more complicated rule about how it gradually adjusts its spending when there’s a change in the endowment. And so we’re going to talk about that later because it involves uncertainty and how to think about uncertainty.
But you’re absolutely right. So Levin did not announce a 300 million dollar reduction immediately, but he announced a big reduction immediately, and you’re going to expect next year if the stock market doesn’t drastically improve for there to be another reduction. And, by the way, this number could go down as well as it goes up. So we’re going to come back to Yale’s investments and what they’re like.
A lot of Yale’s investments are called private equity investments that are very hard to value. So for all we know this 17 is a lot worse than that, but we’ll be finding out in the next year or two. It’s not like a hedge fund where you have to value all your assets by what the market will be willing to pay. A lot of these assets there is no market so they just sort of make up what the number is.
Anyway, we’re going to come back and discuss this. It’s a very interesting question. So one last thing about this present value calculation, one last obvious thing, it’s hard to keep in your mind the difference between real and nominal. So let’s just do a very simple thing. The mortgage, mortgages are traditionally nominal fixed payments. So for example, a 100,000 dollar 30 year mortgage at 2.3 percent is about 4,600 dollars per year.
How did I do that so quickly in my head? Well, because I know if the interest rate is 2.3 percent and you’re going to pay it forever you’d pay 2,300 a year. We know at 30 years at 2.3 percent, 2.3 percent doubles almost in 30 years. That’s 69. That’s getting pretty close to 72, so maybe it takes 31 years or something to double.
So after 30 years the remainder is worth half the mortgage. So you’ve lost half of the value by only getting it for 30 years. So instead of paying 2,300 you have to pay 4,600. So the coupon over .023 times (1 - 1 over 1.023 to the thirtieth), that’s 1 - 1 half about, so if this coupon equals 100,000 the payment is going to be–since this is 1 half the payment isn’t going to be 2,300. It has to be twice that, 4,600. So it’s 4,600 per year.
So if there’s no inflation that means you’re making the same real payment every year. Now what happens if there’s inflation? What if inflation goes up? Now what’s going to happen to what you have to pay? How would you figure that out?
Well, if inflation is another 2.3 percent or something, then the nominal interest rate 1 + i is going to equal the real interest rate times the rate of inflation. So let’s say this is 1.023 and this is also 1.023, so that’s 1.046, a little bit more than 4 6, almost a little bit more than 4 6. So you know that the interest rate the mortgage companies are now going to charge is going to be 1.046. So the 4.6 is going to be the mortgage interest rate and so you can figure out by the same calculation what the coupon’s going to be. So what’s the coupon going to be?
Well, instead of doubling every 30 years at 4.6 percent it’s going to double approximately every 15 years. So this is going to be doubling twice. This is 1 quarter, so this’ll be 3 quarters here. And so if you multiply everything, 2,300 a year times 4 thirds–am I doing the right calculation here? I’m telling you it’s so easy to compute in your head and meanwhile.
Oh, I forgot to change this to 4 6, so the interest rate is 4 6. So this to the other side is 4,600 times 4 thirds and that’s 6,000 about. So the annual payment is going to go up to 6,000 instead of 4,600. It was 4,600. The interest rate went up because there was inflation, so of course they’re going to ask you for more money every year, because if you pay the same amount every year and this is the real payment–if you make the same amount and this is time in terms of inflation corrected dollars you’re paying less and less every year.
So clearly if you started with no inflation and a number like this, so no inflation and now you’ve got inflation but the same real interest rate, and the present value of your expenditures, the real present value, right–the mortgage company’s going to want, the lender’s going to want to get the same amount back in real terms as it got before because the inflation hasn’t changed the real world. So Irving Fisher would say the inflation is just a veil.
Everybody’s going to want the same real interest rate and so the mortgage is going to have to return the same real thing it did before. The present value in real terms, or the real payments is still going to be 100,000. So if the real payments go down over time and have the same present value they had before it’s got to be that they’re higher at the beginning and in real terms lower at the end. So sure enough 6,000 is a much higher number at the beginning than 4,600. So of course when inflation went up and everybody knows it’s up the mortgage companies are going to ask for higher annual payments so it’d be 6,000 a year instead of 4,600 a year.
But now if you inflation correct that, the 6,000 every year is going to be less, in terms of real goods, less and less every year, but the present discounted value of this thing has to be the same as where you started. So the effect is the young borrowers are going to be spending a lot more in real goods when they’re young and a lot less when they’re old.
So inflation has an unfortunate impact on mortgages quoted in nominal dollars that it makes the repayments happen earlier. So the young who have less money are having to pay a huge amount, and when they get old the inflation’s so high that that same 6,000 dollars is practically nothing. So when they’re 50 and 60 they’re paying practically–they’re peanuts to them, but when they were young it was really a hardship. So there’s a big problem with nominal mortgages, which is that in inflationary times it kills the housing market.
Fortunately we’re not in inflationary times. Any questions about that? All right, so that’s the basic lesson of taking the present value. And again, you’ve always got to sort out the nominal from the real, and look though the veil, and don’t get all mixed up by the fact that there’s inflation. It’s the real thing that you want to concentrate on as much as you can. So that’s it for the obvious lesson of present value.
Chapter 3. Yield to Maturity and Internal Rate of Return [00:31:51]
Now I want to introduce another word which is very famous in finance. It’s called the yield or yield to maturity. And I’m going to do it, unlike the way I’ve presented in the notes, I’m going to do it in terms of a hedge fund, so if you can see this? So yield, the next topic, or yield to maturity is a way of trying to compute one number that summarizes how good a bond is, or how good, how well a hedge fund has done. So I think the more interesting case, and the less obvious one, is to start with a hedge fund.
How do you measure how well a hedge fund’s doing or how well it’s done in the past? Yeah, how well it’s done in the past. We’re going to spend a lot of the course talking about this in various ways, but the first way to do it involves yield to maturity.
So let’s see why the problem is a little complicated. So I imagine that there are three investors in this hedge fund. So every year some of the investors are going to decide what to do, and they’re going to decide whether to withdraw money. Here’s investor one. Maybe he’s going to withdraw money. The hedge fund’s just beginning. He’s going to put in 100 dollars. The other two guys haven’t done anything. So now the hedge fund before this guy put in his money had nothing. It’s just beginning. He’s put in his 100 dollars, so the hedge fund’s got 100 dollars. So that’s it.
So I’m imagining these all happen, they usually happen quarterly or annually or something. They don’t happen every day. There’s a fixed moment at which everyone deposits their money. So let’s say they happen annually. The guy puts in 100 dollars at the beginning of the year. For the rest of the year nobody can do anything. They can’t take money out. They can’t put money in. So the hedge fund, let’s say, manages to put the 100 dollars to work and finds a 7 percent return. So it’s now got, the hedge fund all together has got 107 dollars. So let’s just go to the hedge fund all together.
The hedge fund had 100 after these guys, because only one guy put in money and now the hedge fund’s got 107 dollars. Well, that 107 dollars is all the first guy’s money, because nobody else has put anything in. He still owns it. So, so far the hedge fund got a 7 percent return.
Well, now the next year, we’re now at the beginning of year 2, our first investor thinks to himself, “Well, they did fine, 7 percent, not great, but I’m okay. I’m not going to do anything, won’t take any money out or put any in. A rich second investor puts in 1,000 and another guy puts in 200. So now what’s happened to the NAV of the fund? Well, the first guy, at the moment they put in the money, the first guy still owns 107 of the dollars. The second guy’s now got 1,000 in the fund and the third guy’s got 200 in the fund and the hedge fund now has 1,200 plus the 107.
That’s 1,307. That’s how much money is in the fund. So that’s at the beginning of year 2 if you’re still following this. If you’re not following it interrupt me. Sorry. So what happens in the beginning of the next year, year 3, well let’s say our guy–so the hedge fund makes money and this time it made 3 percent, a crappier, sorry, a less good return, only 3 percent. God, it’s on film. I’m glad that’s going to live for posterity. Only a 3 percent return, and so the hedge fund which ended the year at 1,307 now by the next of this next year it’s made 3 percent on that so it’s up to 1,346, so 39 dollars, 3 percent on 1,300 so 1,346.
Now of that who’s got the money? Well, our original guy he’s now made–everyone made 3 percent so his 107 turned into 110. The second guy’s got 1,030 in the fund, and the third guy’s got 206 in the fund and the total fund is 1,346, Now let’s suppose that our guy, this is the beginning of year 3, our first guy says, “3 percent, that’s a terrible return. I’m taking my money out. I’ve had it. It’s 110 dollars. That’s what I have. I’m taking it out,” and no one else does anything.
So at the end of the year now he’s down to zero and everybody else is where they were and the hedge fund thing has gone down a little bit. Well now next year the thing does even worse. It makes a 0 percent return. So everybody’s money is just the same except that the second guy decides this is really getting lousy and he takes half his money out.
So this is taking half his money out. What was his money? He was 1,112 and half of that is 556. So he takes half of it out leaving half behind, and the column on the right reduces what the hedge fund’s total cash is. But now the hedge fund has a great year and it makes 50 percent.
Having made 50 percent–sorry, so this guy takes half his money out. At the beginning of the year the fund does badly then the fund–I skipped the 8 percent. There was an 8 percent return, sorry. Oh, what an idiot. Anyway, so the next year the fund returned not 0 percent it returned 8 percent. So the first guy after the 3 percent return this guy took his money out. The other guys left it in and then the fund had an 8 percent return. So it’s a little bit better, but this guy decides to take half his money out. Then the fund has a 0 percent return and after that this guy decides to take his money out, half his money out, but then finally the last year the fund gives a 50 percent return, which is fantastic, so everybody does well.
And now let’s say they all decide to take their money out. So now there’s nothing left in the fund, and they withdrew the total 934. So what I’ve done here, just to summarize it, is every year people are putting in money or taking out money at the beginning of the year. You can never take out more than you have or you can put money in. The fund earns returns over the whole year and then people, again, decide to take money out or put money in and then the fund earns a different return the next year, and eventually the fund returns all the money or people withdraw the money. So the question is how has the fund done?
How would you summarize in one number how the fund has done over its 1, 2, 3, 4, 5 years of earning returns? That’s the question. So this is a standard–this is obviously what happens every day with hedge funds. So how do hedge funds report how they’ve done historically? So do you have any suggestions? What would you do to summarize how the hedge fund has done? If you had to pick one number what would it be? How good an investor is the fund? Yeah?
Student: Just multiply the return. Take a geometric average.
Professor John Geanakoplos: So one thing you could do is you could say–he said literally multiply all of these returns. What does that mean? That means if you put 1 dollar in the fund at the beginning you’d earn 7 percent. If you left it there and never took it out you get another 3 percent. Then you’d get 8 percent on top of that, then 0 percent, then 50 percent. Of course this is a multiplicative thing, so he says you’d get 1.07 times 1.03 times 1.08 times 1 times 1.5.
Multiplying all that would give you the number of dollars you’d have at the end of 5 years given that you put 1 dollar in at the beginning and never took it out. And if you wanted to annualize that he said take the geometric average, the fifth root of that and that’s the constant rate of return that would have given you the same amount of money at the end that you would get by having left 1 dollar from the beginning in the fund all the way to the end. Is that clear to everybody?
That seems like a logical thing to do. That’s what money hedge funds do do, in fact. That’s the number they tell you. Now, why might not that be a great number? Did everyone follow what his suggestion was? So by the way, his number’s going to come out to be–well, I don’t know. Anyway it’s going to be some number. We could do that. In fact it wouldn’t be that hard to do.
Let’s just do it. Sum, oh this wasn’t a very–ah! How about equals? Oh, dear, all right, circular reference. Equals sum. Oh, why did I get zero here? Oh, because I’m trying to multiply these. I’m adding instead, so equals. I’ll just have to do it one by one. That times the next one, times the next one. There’s obviously a much faster way of doing this. Times the next one, times that one and zero, that’s what 1 dollar would have gotten you if you had put it in and kept it until the end, and now he’s saying take this to the fifth power. So I’ll take up this .2 enter.
So 12.2 percent, now why isn’t that the right number? Why might you think there should be another number? What’s another number? What’s the matter with that–way back there?
Student: It’s grossly inflated by that last year’s 50 percent return.
Professor John Geanakoplos: Well, is it grossly inflated, but why is it grossly inflated by that?
Student: There are going to be lots of years where it doesn’t do that.
Professor John Geanakoplos: That’s true, but you’ve taken that into account. So the years we didn’t do that well like the 0 percent return that brought the average down, so why is that a problem exactly? Yeah, it’s averaging the good years with the bad years, so. Yeah?
Student: As far as a measure of past performance it doesn’t take into account that after three mediocre years a lot of money was removed from the fund.
Professor John Geanakoplos: That’s the crucial thing. So I didn’t do extreme enough numbers. The crucial thing to take into account is that suppose you have a fund that starts off, many funds like my fund started off with very little money, but we did it at the right time because we knew that was a good time. You’re going to see when the leverage cycle, we talk about it. It was at the bottom of the leverage cycle just like this past year. We’re up 30 percent this year. So at the bottom of the leverage cycle you’re going to have a great year.
Of course we hardly had any money because the fund was just starting, and so we made 50 percent the first year. But then everybody said, “Oh, these guys must be geniuses,” and they poured a huge amount of money into our fund, and let’s say the next year we did 10 percent. Actually we had another great year the second year, but let’s say that we did 10 percent the second year.
So the young man in the front is saying you made 50 percent on pennies and then you made 10 percent on a gigantic amount of money. It isn’t right to take the average of 50 percent and 10 percent because almost all the money that you managed you made 10 percent on not 50 percent on. That’s his point. So how would you deal with his point? How could you figure out a way of computing the right return to compensate for the fact that some years you have a lot more money at stake than you have other years? Yes?
Student: Take a weighted average?
Professor John Geanakoplos: Take a weighted average. Well, how would you take the weighted average? Sounds a little complicated. That’s what I’m going to do, but it’s not immediately obvious how to do it. So I wouldn’t have expected you to be able to answer that. You’re right on the right track. So does anyone have anything else to say? Yep?
Student: I guess you multiply each <>, and then you add it all up and divide it by the <>.
Professor John Geanakoplos: All right, so I’m going to now give you the answer which is a little bit like that. She’s saying do some dollar weighted thing, and so you somehow weight the numbers by how much money there was invested for that year. So because there was a lot more money invested in year 3 than there was in year 1 that number should somehow have a bigger weight.
And it’s not exactly clear how the weights are going to get in there, but it’s obvious that that’s something like that you ought to do. So here’s the mathematical solution that the internal rate of return or yield gives. So it says every year let’s look at what happened in the fund.
We shouldn’t care about which investor put in which dollar. We care about how the fund managed dollars. The names of the investors don’t make any difference. It’s how did the fund manage its money. So in the first year the fund got 100 dollars. So as for producing money it produced a negative 100. Money went into the fund. That was the one guy invested. We don’t care who it was. The total that went into the fund was 100.
The second year 1,200 went into the fund. Those are the second two guys. The third year 110 came out. That was the first guy. The fourth year 556 came out. That was the second guy. This is beginning of the fifth year. The third guy took out 155 dollars, and then the last year the second and third guys took out everything that was left. So from the point of view of money creation and use this is every year what went into the fund and what came out of the fund, the net inflow out.
So there are a bunch of numbers. So the question is, so the one number summary is what rate of return, it’s called the internal rate of return which if you discounted all these numbers what would Fisher say if you discounted all these numbers? If you had to use just one interest rate, Fisher would say, what’s the present value of these cash flows?
Well, if the interest rate was 0 you just add up all the numbers and you’re going to get a big positive. If the interest rate is infinity then that means that after the first year you’re discounting everything to 0 and you’re going to get negative 100. So you could say, what interest rate could you discount all the cash flows at to produce 0?
That would be like saying at that interest rate, all the fund has done is rearranged its money. It’s taken money in and put money out, but at that same constant interest rate the present value is 0. So it’s allowed you to trade money across periods at this internal rate of return, the interest rate which makes the present value zero.
So 10 percent turns out to be the right number. So if you discounted things by 10 percent, you see what the formula is, you take the inflow and you discount it by 1.10 to the first power and this you discount by 1.10 to the second power, the inflow, the net outflow, I guess, which is 110 you discount by 1.10 to the third power etcetera.
You keep discounting by that and you get all the discounted net flows. You add them all up and you get practically nothing. That’s adding them up and this is taking the square, and I used Solver to figure out what the right discount rate was to make all this zero. So that’s the simple way of averaging, dollar weighting everything. You don’t care about who the people are. You don’t care about whether one guy’s putting money in, another guy’s taking it out at the same time. You just care about the net, and if you’ve got a bunch of net numbers, that’s the net outflow, and you’re trying to say, what’s the single rate of return?
The idea is to say at what interest rate? If there were a bank paying an interest rate and all those things discounted gave you zero that would be like saying the fund is functioning just like a bank. No matter when people put the money in or take it out they’re always getting this rate of return which is 10 percent. So they’re getting a constant 10 percent rate of return no matter when the money goes in or out and that kind of gives you a measure of how well the fund’s doing.
So that’s the internal rate of return, or the yield to maturity. It says take the net cash flows every year. Find the number which when you discount at that number you get present value equal to zero. So it’s 10 percent which is a different number from 12 percent. Now before we got the geometric average of 12 percent.
Now very typically this is the case that this internal rate of return is lower than the dollar return from putting the money in at the beginning, assuming the fund doesn’t just collapse and go to zero at the end. So why is that? For funds that have survived typically that number, 12 percent, is higher than 10 percent. Why would that be? What does that tell you about the world? Someone who hasn’t–well, go ahead.
Student: That it’s easier to make money with a smaller amount of money?
Professor John Geanakoplos: It tells you that, and how is it that the hedge fund–it could tell you that. It tells you that the hedge funds are doing better when they have a smaller amount of money than when they have a larger amount of money, exactly. You’ve concluded that it’s easier to make money when you have a small amount. What’s another possible explanation?
Student: People normally invest a lot after a big year <>.
Professor John Geanakoplos: Right. I think that’s the reason. The reason is that people pour money into hedge funds just after they’ve done incredibly well, and they keep pouring money in, then eventually there’s a blow up. And so when the blow up comes the hedge funds have zillions of dollars and they’re losing a lot of money. Then everybody pulls their money out and that’s when the cycle is going up and then all of a sudden the hedge funds have these huge returns again, but they hardly have any money. So, any other questions about this?
Chapter 4. Assessing Performance of Coupon Bond [00:51:52]
All right, so internal rate of return is a way–we are going to see that it has many shortcomings, but it’s a way at getting at the idea, as several of you have said, that you can’t just take the geometric average, which is what every hedge fund like ours always produces. That’s the number we tell everybody because it’s a better number than the other one. So the geometric average of the returns are, if you’re an investor who puts 1 dollar in at the beginning and leaves it there forever what’s your geometric average of all your returns.
That’s not a good reflection of how the hedge fund’s done necessarily because some years the fund had a lot of money to work with. And so the average dollar didn’t do that well, and now there’s a question about how should you measure the average dollar, and I’ve given the internal rate of return.
There are actually other formulas you could give. This is the most famous one. Now let’s see how this internal rate of return is used all the time on Wall Street. By the way, if you’re not following what I’m saying you should please interrupt me. So what if you took a bond, a simple coupon bond? What is a simple coupon bond? The yield to maturity of a simple coupon bond, I’m now on this lecture called yield, what is a simple coupon bond?
A simple coupon bond pays the same coupon every year and then pays the principal and the coupon at its maturity. So the yield to maturity is going to be the price, which is like a negative payment–suppose you knew the price of this bond. If you knew the price of the bond, and so the bond is promising all these payments, how good a deal is that?
Well, how good a deal it is they would say is you simply take this first negative payment and all these positive payments and find the unique interest rate which when you discount it will give you present value of 0. And so if this is a coupon bond, say paying 7 percent, say the face is 100, it pays 7 dollars forever, and 107 at the end, and the price is 105, what do you think the yield to maturity is going to be?
Can you say anything qualitative about it? Suppose it’s a 7 percent coupon bond, face of 100, 10-year bond, no one thinks that it’ll default, but its price is 105. What is the yield to maturity in that case do you suppose? Just a vague guess, I just want a qualitative number.
Professor John Geanakoplos: 6.7, that’s qualitatively wrong. Well, I mean it’s not qualitatively wrong. No, it’s qualitatively right. It could have been better. So he’s right. So what if the price were 100 what would the yield to maturity be?
Student: 7 percent.
Professor John Geanakoplos: 7 percent, that’s obvious, right? If it’s a 7 percent coupon bond on face of 100 and its price is 100, price = to the face, then obviously the thing that discounts you back to 100 is going to be 7 percent interest. So if the price were 100 the yield to maturity would just be 7 percent.
But I told you the price was 105, which is a lot more expensive. So it’s a bad deal. So it’s not going to be as good as 107. So it’s not going to be 7 percent and he said 6.7. So I think it would be a little worse than that, but that’s qualitatively just what I asked for, something worse than 7 percent. So who said 6.7 percent? You did.
Student: I just took the price of the bond over the coupon payment and that’s 6.7, but it’s not the only <>. That’s just…
Professor John Geanakoplos: So that’s another number. So I’m going to come back to your question. It’s a good question. So do you all see that if you measured the yield to maturity on this bond–so the bond, remember, pays 7, 7, 7, 107 and its price is 105. So the yield to maturity is going to be that number such that 105 = 7 over (1 + the yield) + 7 over (1 + the yield) squared + 7 over (1 + the yield) cubed + … + 107 over (1 + yield) to the tenth, that’s 105.
So what we observed is that Y has to be less than 7 percent because if Y were exactly equal to 7 percent this would give us 100. But this bond is more expensive, so you’re paying more to get the same payments you would–than the face. You’re paying more than the face to get the same payment. If the price were equal to the face it would be a 7 percent yield. So since you’re paying more you’re getting a worse deal, so it’s pretty obvious that if you want to discount this number to more than 100, namely 105, Y is going to have to be less than 7 percent. So that’s the first thing we said.
Now what did he do? He gave a number and he said 7 over 105 which he said was about 6.7 percent. So that number that he gave is called the current yield. It’s another number people give, and he figured that was 6.7 percent. So let’s believe him that’s 6.7 percent. How does that number compare to the yield to maturity, to Y? Well, we could compute this out on Excel since I’m doing so brilliantly at it now. So we could go 7, 7, 7, 7, 7, 7, 7, 7, 7, and 107. Those are our payments and then we could try some yield to maturity, internal rate of return and let’s guess 1.067.
And now we’d say the cash flow is going to be this, is going to be the left. Oh dear, I have to number things. So again, there’s probably some clever way of doing this. So this’ll be–I forgot to write the year. Let’s just add this. Equals up + 1 enter. So I’m just going to copy this. All right, so now I’ve numbered all the years. Equals up + 1 enter and now control copy. So I’ve numbered all the years here and here are the payments. And now at this yield to maturity I’m going to go equal the thing on the left divided by the yield to maturity that we’re guessing raised to this power. Now I can just copy that and I’ve got all these cash flows. Oh, what did I do that time?
Professor John Geanakoplos: What?
Professor John Geanakoplos: Control copy, sorry.
Professor John Geanakoplos: Control copy, right? So I just want to copy this?
Professor John Geanakoplos: Oh, I see. Yeah, yeah, yeah, right, right, right, so here I’ve got the–you are right. So there’s a trick here which I forgot which is the discount rate, the internal rate of return E I’ve got to put a dollar sign, dollar E, dollar 1 so that way it remembers the spot. So now when I copy it it’s going to remember that.
Professor John Geanakoplos: What’s B1?
Professor John Geanakoplos: Oh, B1. Thank you, so there are two mistakes, B1. Now, control copy. So I’m supposed to be showing you how easy it is to do this, but–all right, so anyway that’s it. So if you see for each number here I’ve got the payments for every year and I’ve discounted them by taking this internal rate of return, and now I just have to sum all this. Equals sum parenthesis. Oh shit.
Professor John Geanakoplos: I’m trying to sum it, you’re right. So here equals, I want to sum all these. So what did I do wrong?
Student: C1 colon C10.
Professor John Geanakoplos: C1 colon C10, that’s what I thought I did, but obviously I didn’t. Good, and now we can do this and square it. So that’s the thing I want to minimize, and so now I’m going to Tools, Solver, Min, C12 by concentrating on this number, and solve.
Professor John Geanakoplos: All right, what did I do?
Professor John Geanakoplos: Here’s the sum, here’s the…
Student: The original price.
Professor John Geanakoplos: Oh, the original price. Ah ha! Thank you very much. So we need the original price of 105. Very good, so I see that preparing this would have helped. So here’s the sum. So we just summed all these things, and here we’re going to–so we’re summing those, right? So I summed all these and then I’ve got the original price 105 enter. Now we’ll add equals this plus this.
Professor John Geanakoplos: You’re right, this minus that. And now we want to square this, equals up squared enter and now we’ll do Solver. Hopefully there isn’t another mistake. So Tools, Solver, so I want that times E1. Oh, that’s very bad. Anyway, you can solve it using Solver and it’s extremely simple to do, and a child could learn how to do it. So we get Solver and we solve all this, and you should be able to use Excel with no problem at all.
So the question is what are the relationships between the current yield and the yield to maturity, and suppose there’s some actual interest rate. So there’s an actual market interest rate of, say, 6 percent or something. So those are the things that we want to sort out now in the next five minutes. So let me go back to the notes and we’ll re-ask all the questions here. You can see that’s Solver. I’ve broken Solver.
Suppose that there were an actual interest rate in the economy and our bond, the bond that was 7, 7, 7, 107, if the actual interest rate was, say, 6 percent the price of the bond would be more than 100. So some price which will be the present value will be greater than 100, obviously, because it’s paying–if the interest rate’s only 6 percent this is giving you more than 100.
Now, the current yield is 7 over the present value, exactly analogous to what he said. So if someone calls you on the phone, you haven’t gotten these calls yet, but when you get older you’ll get–they’re screened now, so it’s harder to get these, but it used to be a few years ago you’d get called on the phone quite often and somebody would say he’s running a bond fund, and the bond fund is really doing great, and he wants to tell you that the bonds they have in the fund last year paid a current yield of 7 over the present value which is much bigger than the interest rate of 6 percent, and therefore you should buy his bonds.
Now, what can you say about that? Suppose somebody tells you there’s a market price for the bonds, which we know is the present value at the interest rate of 6 percent, and he tells you that, “Look, look at the market value I got last year, the yield I got; the current yield I got on these bonds. The present value was some number bigger than 100, but I take 7 over the present value and I get something bigger than 6 percent. At your bank you’re only getting 6 percent, so therefore you should invest in my fund.”
Is that a good argument to invest in his fund? Why not? Now, this bond is called a premium bond because the price is bigger than the face, and a discount bond means the price is less than the face, and a par bond, the price equals the face. So just because a bond pays a coupon of 7–it may be when the bond was issued everybody thought the interest rate was going to be 7 percent forever, so that’s why they picked a coupon of 7 so that the price when they first issued it would be equal to its face value.
But maybe the next day, so it’s still a 10-year bond, practically no time has changed, but unexpectedly the interest rates fell to 6 percent. If you take the same coupon bond paying 7 all the time at 6 percent interest its price is obviously going to go up in the market because everybody is going to discount the 7s, not at 7 percent, but at 6 percent and get a number that’s bigger than 100. So you’re going to have to pay more for the bond because the present value’s higher.
However, people who now will try and market the bond they’re going to tell you, “Well, look at the market price,” whatever the market price is. So the current yield is the market price. They’ll say, “Look at the market price. This is what we bought the bonds for. I’m a fund. I went out and bought these bonds. Look at the price I paid and I got 7 dollars for these bonds this year in income and 7 over the market price of the bond is bigger than 6 percent, so I was doing a great job. You should invest in my fund.”
So, that can’t be a correct thing to say or a persuasive thing to say, because the market price reflected the fact that the interest rates were 6 percent. Everybody was properly computing the present value, and let’s say the market price was equal to the present value, the present value would indeed be greater than 100 and in fact the current yield would be more than the interest rate of 6 percent. So why is that?
So, theorem, if market price equals present value at a going interest rate then the current yield on a premium bond is always greater than the interest rate. So why is that? So in this case, if I hadn’t screwed up the Excel we would have calculated the present value.
So 7, 7, 7, 107 there’s only a 6 percent interest. So everybody taught by Irving Fisher computes the new present value which of course is bigger than 100 and that’s the market price. Some unscrupulous salesman starts a fund, buys the bond for whatever this present value, the market price is, then goes out to a bunch of clients, potential clients, investors, and says, “Look, my very first year in business I spent a little more than 100 dollars and I got 7 dollars as a coupon and 7 over this little more than 100 is giving me at current yield that’s more than 6 percent. I beat the interest rate. You should invest in me, and by the way, I’ll charge you a little fee to do that because I’m doing so great.”
Now, that’s always going to be the case so it has to be that he really hasn’t accomplished anything at all. So why is it easy to see that whenever I computed the present value it was going to have to be more than–this current yield would always be more than 6 percent. How do I know that? Yep?
Student: Because of the face value that’s going to be at the end at maturity isn’t going to reflect that increased interest or relatively higher interest. It’s going to be the original face value.
Professor John Geanakoplos: Exactly. So let’s say, to keep this simpler, let’s suppose these were 10s everywhere, and the interest rate went down to 5 percent. Now, what’s going to be the present value of this bond? It’s not going to be 200, right? If the bond paid 10, 10, 210, suppose I have something paying 10-10-10 and 110, and the coupon is 5 percent. The present value is going to be less than 200. The coupon is always double the interest rate, so it looks like that’s what you’d get if you had 200 dollars, but at the end, as he’s saying, you’re only going to get 110 and not 210. So this present value of this thing has to be less than 200.
So if you double the coupon–if you halve the interest rate, the interest rate was originally 10 percent, if you cut the interest rate in half to 5 percent it looks like your annual coupon is double the interest rate, but at the end you don’t get principal that’s double the original principal. All you did was double–relative to the interest rate of 5 percent, the coupon’s twice as big as you normally expect, but the face isn’t.
So therefore, this has to be worth less than 200. If it were 200 at 5 percent it would give you 10, 10, 10, 10, 10, 210, but this gives you 10, 10, 10, 110, so obviously the present value is less than 200, but therefore 10 over something less than 200 is going to be more than 5 percent.
So that’s his intuitive proof, which is the essence of the thing, that if ever you have a coupon bond that’s a premium bond then the current yield is always above the interest rate. So you can always advertise it as having a spectacular current yield when in fact it’s just priced perfectly fairly. So I’m going to just continue this story of what’s the right way to measure things and how you can get confused by measuring the wrong way next class.
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