ECON 252: Financial Markets (2008)
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Financial Markets (2008)
ECON 252 (2008) - Lecture 4 - Portfolio Diversification and Supporting Financial Institutions (CAPM Model)
Chapter 1. Introduction [00:00:00]
Professor Robert Shiller: Today’s lecture is about portfolio diversification and about supporting financial institutions, notably mutual funds. It’s actually kind of a crusade of mine — I believe that the world needs more portfolio diversification. That might sound to you a little bit odd, but I think it’s absolutely true that the same kind of cause that Emmett Thompson goes through, which is to help the poor people of the world, can be advanced through portfolio diversification — I seriously mean that. There are a lot of human hardships that can be solved by diversifying portfolios. What I’m going to talk about today applies not just to comfortable wealthy people, but it applies to everyone. It’s really about risk. When there’s a bad outcome for anyone, that’s the outcome of some random draw. When people get into real trouble in their lives, it’s because of a sequence of bad events that push them into unfortunate positions and, very often, financial risk management is part of the thing that prevents that from happening.
The first — let me go — I want to start this lecture with some mathematics. It’s a continuation of the second lecture, where I talked about the principle of dispersal of risk. I want now to carry that forward into something a little bit more focused on the portfolio problem. I’m going to start this lecture with a discussion of how one constructs a portfolio and what are the mathematics of it. That will lead us into the capital asset pricing model, which is the cornerstone of a lot of thinking in finance. I’m going to go through this rather quickly because there are other courses at Yale that will cover this more thoroughly, notably, John Geanakoplos’s Econ 251. I think we can get the basic points here.
Chapter 2. Evaluation of Efficient Portfolio Frontiers [00:02:37]
Let’s start with the basic idea. I want to just say it in the simplest possible terms. What is it that — First of all, a portfolio, let’s define that. A portfolio is the collection of assets that you have — financial assets, tangible assets — it’s your wealth. The first and fundamental principle is: you care only about the total portfolio. You don’t want to be someone like the fisherman who boasts about one big fish that he caught because it’s not — we’re talking about livelihoods. It’s all the fish that you caught, so there’s nothing to be proud of if you had one big success. That’s the first very basic principle. Do you agree with me on that? So, when we say portfolio management, we mean managing everything that gives you economic benefit.
Now, underlying our theory is the idea that we measure the outcome of your investment in your portfolio by the mean of the return on the portfolio and the variance of the return on the portfolio. The return, of course, in any given time period is the percentage increase in the portfolio; or, it could be a negative number, it could be a decrease. The principle is that you want the expected value of the return to be as high as possible given its variance and you want the variance of the return on the portfolio to be as low as possible given the return, because high expected return is a good thing. You could say, I think my portfolio has an expected return of 12% — that would be better than if it had an expected return of 10%. But, on the other hand, you don’t want high variance because that’s risk; so, both of those matter. In fact, different people might make different choices about how much risk they’re willing to bear to get a higher expected return. But ultimately, everyone agrees I — that’s the premise here, that for the — if you’re comparing two portfolios with the same variance, then you want the one with the higher expected return. If you’re comparing two portfolios with the same expected return, then you want the one with the lower variance. All right is that clear and — okay.
So let’s talk about — why don’t I just give it in a very intuitive term. Suppose we had a lot of different stocks that we could put into a portfolio, and suppose they’re all independent of each other — that means there’s no correlation. We talked about that in Lecture 2. There’s no correlation between them and that means that the variance — and I want to talk about equally-weighted portfolio. So, we’re going to have n independent assets; they could be stocks. Each one has a standard deviation of return, call that σ. Let’s suppose that all of them are the same — they all have the same standard deviation. We’re going to call r the expected return of these assets. Then, we have something called the square root rule, which says that the standard deviation of the portfolio equals the standard deviation of one of the assets, divided by the square root of n. Can you read this in the back? Am I making that big enough? Just barely, okay.
This is a special case, though, because I’ve assumed that the assets are independent of each other, which isn’t usually the case. It’s like an insurance where people imagine they’re insuring people’s lives and they think that their deaths are all independent. I’m transferring this to the portfolio management problem and you can see it’s the same idea. I’ve made a very special case that this is the case of an equally-weighted portfolio. It’s a very important point, if you see the very simple math that I’m showing up here. The return on the portfolio is r, but the standard deviation of the portfolio is σ/√(n). So, the optimal thing to do if you live in a world like this is to get n as large possible and you can reduce the standard deviation of the portfolio very much and there’s no cost in terms of expected return. In this simple world, you’d want to make n 100 or 1,000 or whatever you could. Suppose you could find 10,000 independent assets, then you could drive the uncertainty about the portfolio practically to 0. Because the square root of 10,000 is 100, whatever the standard deviation of the portfolio is, you would divide it by 100 and it would become really small. If you can find assets that all have — that are all independent of each other, you can reduce the variance of the portfolio very far. That’s the basic principle of portfolio diversification. That’s what portfolio managers are supposed to be doing all the time.
Now, I want to be more general than this and talk about the real case. In the real world we don’t have the problem that assets are independent. The different stocks tend to move up and down together. We don’t have the ideal world that I just described, but to some extent we do, so we want to think about diversifying in this world. Now, I want to talk about forming a portfolio where the assets are not independent of each other, but are correlated with each other. What I’m going to do now — let’s start out with the case where — now it’s going to get a little bit more complicated if we drop the independence assumption. I’m going to drop more than the independence assumption, I’m going to assume that the assets don’t have the same expected return and they don’t have the same expected variance. I’m going to — let’s do the two-asset case. There’s n = 2, but not independent or not necessarily independent. Asset 1 has expected return r1. This is different — I was assuming a minute ago that they’re all the same — it has standard — this is the expectation of the return of Asset 1 and r2 is the expectation of the return — I’m sorry, σ1 is the standard deviation of the return on Asset 1. We have the same for Asset 2; it has an expected return of r2, it has a standard deviation of return of σ2. Those are the inputs into our analysis. One more thing, I said they’re not independent, so we have to talk about the covariance between the returns. So, we’re going to have the covariance between r1 and r2, which you can also call σ12 and those are the inputs to our analysis.
What we want to do now is compute the mean and variance of the portfolio — or the mean and standard deviation, since standard deviation is the square root of the variance — for different combinations of the portfolios. I’m going to generalize from our simple story even more by saying that, let’s not assume that we have equally-weighted. We’re going to put x1 dollars — let’s say we have $1 to invest, we can scale it up and down, it doesn’t matter. Let’s say it’s $1 and we’re going to put x1 in asset 1 and that leaves behind 1-x1 in asset 2, because we have $1 total. We’re not going to restrict x1 to be a positive number because, as you know or you should know, you can hold negative quantities of assets, that’s called shorting them. You can call your broker and say, I’d like to short stock number one and what the broker will do is borrow the shares on your behalf and sell them and then you own negative shares. So, we’re not going to — x1 can be anything and x — this is x2 = 1-x1, so x1 + x2 = 1.
Now, we just want to compute what is the mean and variance of the portfolio and that’s simple arithmetic, based on what we talked about before. I’m going to erase this. The portfolio mean variance will depend on x1 in the way that if you put — if you made x1 = 1, it would be asset 1 and if you made x1 = 0, then it would be the same as asset 2 returns. But, in between, if some other number, it’ll be some blend of the — mean and variance of — the portfolio will be some blend of the mean and variance of the two assets. The portfolio expected return is going to be given by the summation i = 1 to n, of xi*ri,. In this case, since n = 2 that’s x1r1 + x2r2, or that’s x1r1 + (1 - x1)r2; that’s the expected return on the portfolio. The variance of the portfolio σ² — this is the portfolio variance — is σ² = x1² σ1² + x2² σ2² + 2x1 x2 σ12; that’s just the formula for the variance of the portfolio as a function of — Now, since they have to sum to 1, I can write this as x1² σ1² + (1 - x1)² σ2² + 2x1 (1 - x1) σ12 and so that together traces out — I can choose any value of x1 I want, it can be number from minus infinity to plus infinity. That shows me then for any value of x1, I can compute what r is and what σ² is and I can then describe the opportunities I have from investing that depend on these.
Now, one thing to do is to solve the equation for r and x1 and I can then recast the variance in terms of r ; that gives us the variance of the portfolio as a function of the expected return of the portfolio. Let me just solve this for — let’s solve x1 for r. I’ve got — this should be x — did I make a mistake there — so it says that r1 - r2 = x1 r1 - r2, so x1 = (r - r2)/(r1 - r2) and I can substitute this into this equation and I get the portfolio variance as a function of the portfolio expected return r. That’s all the basic math that we need. If I do that, then I get what’s called the frontier for the portfolio. I have an example on the screen here, but it shows other things. Let me just — rather than — maybe I’m showing too many things at once. Let me just draw it. I’ll leave that up for now but we’re moving to that. What we’re doing here is the — with two assets, if I plot the expected annual return r on this axis and I plot the variance of the portfolio on this axis, what we have — I’m sorry, the standard deviation of the portfolio return. It tends to look — it looks something like this — it’s a hyperbola; there’s a minimum variance portfolio where this σ is as small as possible and there are many other possible portfolios that lie along this curve. The curve includes points on it, which would represent the initial assets.
For example, we might have — this is asset 1 — and we might have something here — this could be asset 2. Depending on where the assets expected returns are and the assets’ standard deviations, we can see that we might be able to do better than — have a lower variance than either asset. The equally-weighted case that I gave a minute ago was one where the two assets had — were at the same — had the same expected return and the same variance; but this is quite a bit more general. So that’s the expected return and efficient portfolio frontier problem.
I wanted to show an example with real data that I computed and that’s what’s up on the screen. The pink line takes two assets, one is stocks and the other is bonds, actually government bonds. I computed the efficient portfolio frontier for various — it’s the efficient portfolio frontier using the formula I just gave you. The pink line here is the efficient portfolio frontier when we have only stocks and bonds to invest. You can see the different points — I’ve calculated this using data from 1983 until 2006 — and I computed all of the inputs to those equations that we just saw. I computed the average return on stocks over that time period and I computed the average return on bonds over that time period. These are long-term government bonds and I — now these are — since they’re long-term, they have some uncertainty and variability to them. I computed the σ1, σ2, r1, and r2 for those and I plugged it into that formula, which we just showed. That’s the curve that I got out. It shows the standard deviation of the return on the portfolio as a function of the expected return on the portfolio.
I can achieve any combination — I can achieve any point on that by choosing an allocation of my portfolio. This point right here is, on the pink line, is a portfolio 100% bonds. Over this time period, that portfolio had an expected return of something like a little over 9% and it had a standard deviation of a little over 9%. This is a portfolio, which is 100% stocks, and that portfolio had a much higher average return or expected return — 13% — but it also had a much higher standard deviation of return — it was about 16%. So, you can see that those are the two raw portfolios. That could be investor only in bonds or an investor only in stocks, but I also show on here what some other returns are that are available. The minimum variance portfolio is down here. That’s got the lowest possible standard deviation of expected return and that’s 25% stocks and 75% bonds with this sample period. I can try other portfolios; this one right here — I’m pointing to a point on the pink line — that point right there, 50% stocks, 50% bonds.
You can see–You can also go up here, you can go beyond 100% stocks, you can have 150% stocks in your portfolio. That means you’d have a leveraged portfolio, you would be borrowing. If you had $1 to invest you can borrow $.50 and invest in a $1.50 worth of stocks. That would put you out here; you would have very much more return, but you’d have more risk. Borrowing to buy stocks is going to be risky. You could also pick a point down here, which is more than 100% bonds — how would you do that? Well, you could short in the stock market, you could short $.50 worth of stocks and buy $1.50 worth of bonds and that would put you down here. Any one of those things is possible it’s just the simple math that I just showed you.
Do you have any idea what you would like to do, assuming this? Well if you’re an investor, you don’t like variance. So, you probably don’t want to pick any point down here, because you’re not getting anything by picking a point down there because you could have a better point by just moving it up here. You’d have a higher expected return with no more variance. It’s getting kind of complicated, isn’t it? We started out with just a simple idea: that you don’t want to put all your eggs in one basket and if you had a lot of independent stocks you would want to just weight them equally. But now, you see there are a lot of possibilities and the outcome of your portfolio choice can be anything along this line. I’m not going to tell you what you want to do except to say, you would never pick a point below the minimum variance portfolio, right? Because, if you did, then you would always be dominated. You could always find a portfolio that had a higher expected return for the same standard deviation. But beyond that, if you were confined to just stocks and bonds, it would be a matter of taste where along this frontier you would be. You would call it an efficient portfolio frontier. It would be anywhere from here to here, depending on how much you’re afraid of risk and how much you want expected return.
Chapter 3. The Significance of Portfolio Diversification [00:26:59]
Now, we can also move to three assets and, in fact, to any number of assets. The same formula extends to more assets. In fact, I have it — suppose we have three assets and we want to compute the efficient portfolio frontier, the mean and variance of the portfolio. What I have up there on the diagram are calculations I made for the efficient portfolio frontier with three assets. So, now we have n = 3 and in the chart are stocks, bonds, and oil. Oil is a very important asset, so we want to compute what that — so now we have lots of inputs.
Let’s put the inputs — r1, r2, and r3 are the expected returns on the three assets. Then, we have the standard deviations of the returns of the three assets and we have the covariance between the returns on the three assets. There are three of them — σ12, σ13, and σ23. That’s what we have to know to compute the efficient portfolio frontier with three assets.
To make this picture, I did that. I computed the returns on the stocks, bonds, and oil for every year from 1983 and I computed the average returns, which I take as the expected returns, I took the standard deviations, and I took the covariance. These are all formulas, I just plugged it into formulas that we did in the second lecture. What is the portfolio expected return? The portfolio expected return — we have to choose three things now: x1, x2, and x3. x1 is the amount that I put into the first asset, x2 is the amount that I put into the second asset, and x3 is the amount I put into the third asset. I’m going to constrain them to sum to one. The return on the portfolio is x1 r1 + x2 r2 + x3 r3. The variance of the portfolio, σ², is x1² σ12 + x2² σ2² + x3² σ3² — then we have to the count of all of the covariance terms — + 2x1x2 σ12 + 2x1x3 σ13 + 2x2x3 σ23. Is that clear enough? It seems like a logical extension of that formula to three assets and you can easily see how to extend it to four or more assets, it’s just the logical extension of that.
What I did in this diagram is I computed the efficient portfolio frontier — now it’s the blue line with three assets. Now, once you have more than three — more than two assets — it might be possible to get points inside the frontier. But I’m talking here — this is the actually the frontier — the best possible portfolio consisting of three assets. You can see that it dominates the pink line. When you add another asset, you do better when you have three assets, you do better than if you just had two because there’s more diversification possible with three assets than with two. Oil, bonds, and stocks are all independent — somewhat independent — they’re not perfectly independent, but they’re somewhat independent and, to the extent that they are, it lowers the variance. You should see that the blue line is better than the pink line because, for any expected return, the blue line is to the left of the pink line, right? So, for example, at an annual expected return of 12% if I have a portfolio of stocks, bonds, and oil I can get a standard deviation of something like 8% on my portfolio. But if I would confine myself just to stocks and bonds, then I would get a much higher standard deviation. Are you following this?
The general principle of portfolio management is: you want to include as many assets as you can. You want to get it — if you keep adding assets, you can do better and better on your portfolio standard deviation. You can see some of the points I’ve made along the blue line here. This is — let’s see if I have it. This is a portfolio, which has all oil and stocks and it has no bonds. This portfolio, the minimum variance portfolio, is 9% oil, 27% stocks, and 64% bonds and most of the — many choices you can make. The first — you see, the idea here is that in order to manage portfolios what we want to do is calculate these statistics, which are the expected returns on the various assets, the standard deviations of the various assets, and you’ve got to know their covariances because that affects the variance of the portfolio. The more they covary — they move together — the less they cancel out. So, the higher the covariance is, generally, the higher — you can see from here — the higher the σ² of the portfolio. Is that clear?
There’s one more thing that we can do. I have three assets shown here. I have stocks, bonds, and oil but I want also to add one more final asset, we’ll call it the riskless asset, which is the asset–long-term bonds are somewhat uncertain and variable because they’re long-term. If we have an annual return that we’re looking at, we can find a completely riskless asset with an annual return — it would be a government bond that matures in one year. Now, assuming that we trust the government — I think the U.S. Government has never defaulted on its debt — we’d take that as a riskless return. It probably has some risk, but the way we approximate things in finance, we take the government as riskless. With the government expected return, we want to make that expected return as a fourth asset — we could call it r4 but I’ll call it rf — it’s a special asset. So rf is the riskless asset. So, for it, σf = 0. It’s like a fourth asset but we’re using a special feature of this asset: that it has no risk. Moreover, the correlation — the covariance between any of these σ1f = 0, etc. It just doesn’t have any risk to it, it has no variability.
If we want to add that asset to the portfolio, what it does is it produces an efficient portfolio frontier that is now a straight line; I show that on the diagram. The best possible portfolio that you can get would be points along this straight line. That is the final aspect of the efficient portfolio calculations. Again, I’m not able to give this as much of a discussion as I would like because I don’t want to spend too much time on this. In the review sections I’m hopeful that your T.A.’s can elaborate on this more.
There’s a very important principle that finally comes out here, it is that you always want to reduce the variance of your portfolio as much as you can. That means that you want to pick, ultimately, a point on this — this line is tangent to the efficient portfolio frontier with all the other assets in it. Tangent means that it has the same slope, it just touches the efficient portfolio frontier for risky assets at one point, and the slope of the efficient portfolio frontier, including the riskless asset, is a straight line that goes through the tangency point, here. That is the — I think that’s the end of my mathematics.
What I have shown here is how you calculate your portfolio management. The way you would go about it, if you’re a portfolio manager, is you have to come up with estimates of the inputs to these formulas — that means the expected returns, the standard deviations, and the covariances. You take all the risky assets and you analyze them first to get their — you have to do a statistical analysis to get their expected returns, their variances, and their covariances. Once you’ve got them together, then you can compute the efficient portfolio frontier without the riskless asset. Then finally, the final step is to find what is the tangency line that goes through the riskless rate. It doesn’t show it on this chart, it goes through 5% at a standard deviation of 0, then it touches the risky asset efficient portfolio frontier at one point, then from there it goes up above in terms of higher expected returns for the same variance. That’s the theory of efficient portfolio calculation.
Chapter 4. The Tangency Portfolio and the Mutual Fund Theory [00:38:43]
There’s something that — a fundamental principle — and this is leading us now to the institutional topic of this course — is that there’s only one tangency portfolio and that portfolio is called the tangency portfolio, where a line drawn from the risk — from the x-axis at the riskless rate is tangent to the efficient portfolio frontier. The tangency portfolio is the portfolio that one should hold. The tangency portfolio gives rise to what’s called the mutual fund theorem in finance, which says that all investors need is a single mutual fund. Now, I haven’t defined mutual fund yet. A mutual fund is an investment vehicle that allows investors to hold a portfolio.
The theory of mutual funds is: nobody is supposed to be holding anything other than — the ideal theory of mutual funds is holding something other than this tangency portfolio. So, why don’t we set up a company that creates a portfolio like that and investors can buy into that portfolio. What this — if my analysis is right — namely, if I’ve got all of the right estimates of the expected returns on stocks, bonds, and oil, and the standard deviations and covariances — and assuming the interest rate is 5%, which is what I’ve assumed here, that this line, if you go to 0 it hits that 5%. It hits the vertical access at 5%. Then everyone should be holding the tangency portfolio. What is the tangency portfolio in this case? It’s 12% oil, 36% stocks, and 52% bonds. That’s what I got using this sample period. Some people might disagree with that, they might not take my estimates. They might say my sample period was off, but that’s what the theory — using my data for the sample period that I computed — the expected returns and co-variances says one should do.
The theory says everybody should be investing in these proportions and this theory then — it doesn’t leave any latitude for individual choice except that you can choose which mixture of the mutual fund and the riskless asset you want. Somebody who is very risk averse could say, I want to hold only the riskless assets because I just don’t like any risk at all. That person — I should have maybe included that in the diagram — that person could get 5% return with no risk. Somebody else might say, well I want to just hold this point, I want to hold the tangency portfolio. That’s attractive to me because I could then get a bigger expected return, I could get almost 12% return per year, and I’d sacrifice — I’d have some standard deviation of like 8%; but, if that’s what I want, if I have different tastes about risk, then — and that’s what I want — then that’s the optimal thing to do.
Other people might say, well you know, I’m really an adventurer — I don’t care too much about risk — I want the much higher return. Such a person might pick a point up here and that would be a portfolio with — a leveraged portfolio. That would be a portfolio where you borrowed at the riskless rate and you put more than 100% of your money into the tangency portfolio. What you could do is, say, borrow $.50 on your $1 and put $1.50 into a portfolio, which consisted of 9% oil, 27% stocks, and 64% bonds. Everyone would do that, no one would ever hold some other portfolio because you can see that this line is the lowest — it’s far — you want to get to the left as far as possible. You want to, for any given expected return, you want to minimize the standard deviation, so it’s the left-most line and that means that everyone will be holding the same portfolio.
I don’t find that my analysis is profound in the final answer, I just took some estimates using my data and, again, we could — if someone wanted to argue with us they could argue with my estimates of the expected returns of the standard deviations and the covariances, but not with this theory. This theory is very rigorous. If you agree with my estimates, then you should do this as an investor, you should hold only some mixture of this tangency portfolio, which is 9% oil, 27% stocks, and 64% bonds. You see what we’ve got here? I started out with the equally-weighted — I was talking about stocks — about n stocks that all have the same variance and are all independent of each other. But, I’ve dropped that assumption and now I’m going on to assuming that they’re taking account of their dependence on each other, taking account of their different expected returns, and taking account of their different covariances and variances; so that’s what we’ve got. This is a famous framework. This diagram is, I think, the most famous diagram in all of financial theory and it’s actually the first theoretical diagram. I did it myself using my data, but it would always look more or less like this — slightly different positions if people use different estimates.
I actually showed this diagram — I went to Norway with my colleague — I actually have a couple more pictures here. That’s my colleague, Ronit Walny, and I, we’re posing in front of the Parliament Building in Oslo. We went to Norway to discuss the — with the Norwegian Government — their portfolio. This is a slide that I showed them. I showed them the slide that I just showed you, showing the optimal portfolio, and then I looked at the Norwegian Government’s position. The Norwegian Government has pension fund assets in the amount of, as of 2006, just under two trillion Norwegian Kroner; but they also own North Sea Oil. If you know that, it’s kind of divided between the UK and Norway. Norway has a much smaller population than the UK and they have a lot of oil up there in the North Sea. As of then, I calculated the value of their oil in the North Sea and that’s what I got — it’s worth about 3.5 billion Norwegian Kroner. Do you see the difference? In fact, the assets that the Norwegian Government owns is about two-thirds oil and one-third government pension fund assets. This government pension fund, I guess, in dollars, is about $200 billion. It’s a huge amount of money that they’re managing, but I was trying to convince them that they should do something to manage their oil risks because they’re way over-invested in oil.
Where are they on the efficient portfolio frontier? They have 64% oil in their portfolio. Where does that put them? Well it’s not — it’s really off the diagram. The furthest point that I recorded was 28% oil — that puts them there — so if they — if you wanted to, where would it be? It would be somewhere over there off the charts. What the Norwegian Government is doing wrong is — it’s a little bit controversial, my pointing this out to them. I ended up in the newspaper the next day for having argued that they are way off on their investment opportunities because oil is such a volatile thing. They’ve got so much of their assets tied up in oil. I got a good hearing. I went to the Ministry of Finance and we went to the Norges Bank, which is their central bank, and I think the answer I got from them was, yes you’re right. I never got it quite like that — something like that.
There was a conditional agreement: yes, Norway should manage this oil risk but it’s politically difficult and that’s the problem — is that we’re not able to do our optimal management. Maybe there are a number of structural problems that prevent them from doing that and they think, maybe, that — I think Norway may be moving that way, so we’ll see in the future. I went to the Bank of Mexico; I tried to convince the — I met the president of the Bank of Mexico and tried to tell them that Mexico is too reliant on oil, too much oil — they have to get rid of their risk. I’m going to the Russian Stabilization Fund in — I think I’ve got an arrangement to meet with them in Moscow in March — that would be during the semester. I’ll tell you what reaction I hope I get from the Russians. Oil is very important to the Russian economy and are they managing the risk well? I bet not. I’m going to do a diagram like this for Russia. The countries that really matter are the ones that — the Arabian — are the Persian Gulf countries; I was just talking to people at the World Economic Forum about that. Some of those countries are really reliant on oil, so they really have to do — they really should do a calculation of efficient portfolio frontiers.
Well, one of the lessons of this course is we have a wonderful theory, but we don’t manage it well, mostly. And I don’t mean to be criticizing foreign countries. The same criticism applies to the United States as well. We’re in a different position. Where are we on this frontier regarding oil in the United States? Well, we don’t have much oil relative to the size of our portfolio. I don’t know what percent; our oil reserves in the U.S. are pretty small, so we’re kind of lying somewhere inside — maybe on this pink line. So, people in the U.S. don’t have the optimal portfolio either. I set up a theoretical framework here and I wanted to give you — I mentioned oil because it seems to make it, to me, so clear what we’re talking about here. It’s talking about not getting tied up in risks and so — I was talking to someone at the World Economic Forum from a Persian Gulf country and I said, aren’t you worried about reliance on oil? He said, of course we’re worried on reliance, so much of our GDP and of our government revenues is oil-related. We’ve seen the price of oil, lately, move all over the map. It went up to $100 recently and it was just as late as late 1990s that it was under $20 and people just don’t know where it’s going to go. I think that these countries are somewhat trying to manage this oil risk but they can’t yet get onto the frontier. That’s a sign to me that we’re not there yet and we have a lot more to do in finance.
Chapter 5. The Capital Asset Pricing Model [00:51:46]
There’s one more equation that I wanted to write down and I’m going to not — I’m not going to spend a lot of time explaining this because it’s going to take awhile. This is the equation that relates the expected return on an asset. It’s so called capital asset pricing model. In the capital asset pricing model, in finance — this is the most famous model in finance. It’s abbreviated as the CAPM and I’m not going to do it justice here, I’m sorry, but there are so many ins and outs of this. You should really take ECON 251 to learn this more. It was — the model divided by James Tobin here at Yale, was the original — who got the original precursors, but it was more invented by William Sharpe, John Lintner, Harry Markowitz. Everyone of these, except Lintner I think, won the Nobel Prize. I think he died too young — that’s one of the misfortunes of living. Did I say that right? One of the misfortunes of scholarship, you have to live long enough to get your accolades.
The asset pricing model — and this is critical — assumes everyone is rational and holds the tangency portfolio. That is a wild assumption, but it’s fun to make, because I know pretty well — I know very well — that people aren’t doing this. They have lots of maybe good — maybe it’s not because they’re irrational, it’s that they’re political or they’re constrained by tradition or laws or regulations, all sorts of things; but, they’re not holding the tangency portfolio. It’s a beautiful theory to assume, to see what would happen if they did. That would mean that everybody is holding that same portfolio of risky assets and nobody is different, they’re only different in how — what proportions they hold the risky — the tangency portfolio. It implies that the tangency portfolio has to equal the actual market portfolio and that means then — it’s a very simple implication of the theory.
In my diagram, I said that the tangency portfolio — I estimated that the tangency portfolio is 9% oil, 27% stocks, and 64% bonds. If we’re all doing that, then that has to be what the outstanding is. If we’re all holding the same portfolio, that has to be the total, so that would mean that 9% of all wealth is oil, 9% of all — is oil — 27% of all wealth is stocks, and 64% is bonds. If you accept my estimates and you accept the capital asset pricing model, that would have to be true. Again, I don’t want to make too much of my estimates because different people would estimate these things in different ways, but that’s the theory. The theory says that the tangency portfolio equals the optimal portfolio and that gives us the famous equation — I’m not going to derive this, but there is the most famous equation in finance — says — can you read that? That ri, the expected return on the ith asset, is equal to the riskless rate plus something called the beta for the ith asset times the expected return on the market minus the riskless rate. Again, I’m not going to spend much time on this, but the β of the ith asset is the regression coefficient when you regress the return on the ith asset on the return of the market portfolio. rm is the expected return on the market portfolio, which is the portfolio of all assets.
The market portfolio is, if you took all the stocks and bonds, and oil, and real estate, anything that’s available to invest in, in the whole world, put them all together in one portfolio. It’s the world portfolio, it’s everything and we compute the expected return on that portfolio, that’s rm. Also, we need to know how much individual stocks are correlated with rm; we measure that by the regression coefficient. The β of a stock is how much it reacts to movements in the market portfolio. If β = 1, it means that if the market portfolio goes up 10% in value then this asset also goes up 10% in value. If β is two, it means that if the market goes up 10% in value, the stock tends to go up 20% in value and so on. These are the basic theoretical structures that you incidentally need for the problem set, the first problem set.
The first problem set — I guess you’ve — you can turn in your first problem set here before you leave because it was due today. The second problem set is about this model. I realize I’ve given you some difficult mathematics, but — it’s not that difficult actually — but I kind of went through it quickly. So, we are setting up our — you have already gotten email and you’ve talked about setting up your review sessions — so I have a few more minutes here.
Chapter 6. Implications of the Equity Premium and Conclusion [00:59:09]
What I want to do, I want to talk about Jeremy Siegel’s book and the equity premium puzzle. Underlying this analysis, we have estimates of the expected returns on assets, notably, the expected returns on stocks and bonds. Jeremy Siegel, in his book, which is assigned for this course, is really emphasizing this capital asset pricing model, emphasizing the kind of efficient portfolio frontier calculations that I’ve done. What Siegel emphasized — the book is really about this — he talks about what is the expected return of stocks and what is the expected return of bonds and so on. We’re going to call asset one stocks in the U.S. and we’re going to call asset two bonds. He estimates, for his purposes, and he shows you calculations of the efficient portfolio frontier.
I want to just talk a little bit about his estimate. He has data for a very long time period, 1802 to 2006, for the U.S. Over that very long time interval, the expected return that we got for stocks was 6.8% a year in real terms, this is real inflation corrected; whereas, for bonds, it was, over this whole time period, 2.8% a year, real. Then he also computes σ1 and σ2 and σ12 but I’m not going to talk about that right now. Then he computes the efficient portfolio frontier. Now, he’s using a much longer sample than I did, so he’s not going to get this tangency portfolio that I did.
The thing that is very interesting that he finds is the difference, 4%, between the historical real return on stocks and the historical real return on bonds. This is called the equity premium. Actually, I should take that back, this is really rf — he shows all three. rf that I’m reporting is the riskless rate. There’s also — if you look on Table One in Chapter One, it shows r1 stocks, r2 long-term bonds, and these are short-term that I’m recording here. The equity premium is the — this short-term 2.8% is the riskless return, historically, for a period of almost 200 years. This is the return on stocks for a period of almost 200 years. Stocks have paid 4% more a year over this incredibly long time period than short-term bonds. Also, they’ve paid much more than long-term — well it’s not so different. I don’t have the data in front of me, so let’s emphasize the difference between r1 and rf. It surprises many people that stocks have paid such an enormous premium over the return on short-term debt.
The theme of Siegel’s book is, can we believe this? I mean, do you really — you might wonder, aren’t people missing something if the excess return is so high of stocks over short-term bonds? Why would anyone not just hold a lot of — a really large number of stocks? That’s the theme of his book and his book is at — his conclusion is that he largely believes that this is true, that the returns that we have seen in the U.S. in the stock market have exceeded those of other assets by quite a substantial margin. That means — his calculations are very different than the ones I have because he has a much longer sample period. I was using only from 1983 to the present. For him, the optimal portfolio should be very heavily into stocks. Now, this is a controversial view, but that is the view that he advances in the book and I think it’s a very interesting analysis.
So, what it means then is that the optimal portfolio should not be the one I show there, but should be one that’s very heavily into stocks. This is — I’m showing here U.S. data, but Siegel also argues in the latest edition that the equity premium is also high for advanced countries over the whole world. If you look at his book, again in the first chapter, he gives a list of countries — it’s based on an analysis that some others — that Professors Dimson, Marsh, and Stanton used in their 2002 book. They show the expected return on stocks versus bonds or short-term debt for a whole range of countries. In every one of these countries, since 1901, there has been a very high equity premium. I’ll let you read Siegel and his discussion of this possibility, but I think that — what I want to get now is not necessarily any agreement on whether you believe that there’s such a high excess return for stocks. but I just want you to understand the basic framework. The first problem set asked you to manipulate the model that I just presented — the model of how you form portfolios and the model of the capital asset pricing model.
I have one final question, just about the mutual fund industry. The mutual fund industry is supposedly, and according to theory, doing this for you. The ideal thing would be that the mutual fund does these calculations and it puts it all together for you. At least in some approximate sense that’s what they are doing. I asked you in the final question to just get on a couple of websites, The Federal Reserve and the ICI, which is a mutual industry website, and write a little bit, a couple paragraphs about what this industry is really achieving or how it’s trending. Next Wednesday–well, there are three lectures this week, Wednesday will be about the insurance industry and I’ll see you in two days then.
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