by Filip Palda
The goal of all sciences is to simplify. When you start giving importance to individual events separately from others, you vastly complicate the problem of solving for the outcome of the system. In this case the outcome is equilibrium leading to a stock price and return. Equilibrium would come about and a stock price settled upon when people had adjusted their purchases of stock, and stock prices had changed such that everyone was just indifferent between giving up a certain dollar of income for the utility gain (not the expected money gain) to be expected from investing in the stock.
Yet this approach is bad, or at least awkward science because it is really not a very practical characterization of the problem. The main item of disappointment is that expected return disappears from the calculation, to be replaced by the expected utility of income in the different states of the world corresponding to different possible levels of stock return. Return still figures in the problem, but in a highly fragmented manner. We are no longer able to speak simply of expected return, which is the sum of all possible returns weighted by their probabilities. Now we have to consider each possible return to a stock individually. Discovering what the equilibrium stock return should be is a nightmare.
One way to get out of the mess of figuring out asset prices using a model of maximizing the expected utility of investing in stocks is to make assumptions about either preferences or the probabilities of the different possible states of the world. Nobellist James Tobin (1958) took this line and discovered that in some cases you do not need to worry about the utility of income in thousands of states, and the attached probabilities, to solve the consumer’s choice on how to spread income among states. When preferences contain only a linear and a squared term (a case of diminishing returns) or the probabilities of different stock returns follow a normal distribution (an equation that contains a linear and squared terms as parameters), a simple formulation of a person’s investment choices becomes possible. Under Tobin’s assumptions we can reformulate the person’s decision problem as being one of trading off risk and expected return. Risk, or more precisely the variance of your investment portfolio creates spread in the returns you expect. People are willing to assume more risk only if compensated by a higher level of expected return. One can thus think of a tradeoff people are willing to make between risk and expected return. They invest in risky assets to the point at which their willingness to trade off risk and return is equal to the rate at which they able to trade them off. It is difficult to exaggerate how brilliant is the simplification of the investment problem that flows from these assumptions. Instead of worrying about the investor’s optimization problem in potentially millions of possible states of the world, one need only worry about how the investor can trade off risk and return in the stock market.
The risk-return frontier
OF COURSE NO stock market sells you risk and return directly, just as no supermarket sells you nutrition and flavor directly. In stock markets investors combine stocks into portfolios that produce different levels of risk and return, just as grocery shoppers combine purchased ingredients into a final culinary product. Is there some analogue in the uncertain world of the arbitrage condition that in a certain world gave such clarity to the equilibrium return of stocks? In fact there is, and as you might guess, this arbitrage condition is the only factor determining expected stock returns in a partial equilibrium sense. Recall that as yet we have no way of predicting how the final payout to stocks come about. That is a datum arising from full equilibrium in product markets that have nothing to do with finance models. This separation of equilibria, as it were, is the reason why finance models focus on returns rather than stock prices. Expected returns are creatures uniquely of arbitrage once we take as given the final payout.
So what is the arbitrage condition in question? In a certain world the arbitrage condition for equilibrium is that all stock prices should adjust so that all stocks have zero return. In an uncertain world people are willing to accept risk to get greater return. Now a whole range of positive returns is possible, accompanied in a consistent manner by varying levels of risk. If there is a stock or portfolio of stocks that offer more return at a given level of risk than others at that level of risk, people will rush to buy stocks in that portfolio until their prices rise to lower the expected return back to that in the herd of stocks in other portfolios grazing at the level of risk in question. Or the prices of the stocks in these portfolios could fall. All that matters is that no stock or portfolio offering a higher level of return than others should have an equivalent or lower level of risk. That is the arbitrage condition in question.
The risk-return arbitrage condition is believed to be a powerful force driving stocks to their equilibrium levels of expected return because if the condition is violated a savvy investor can make a great deal of money. All you need do is short-sell, that is, “borrow” the portfolio with similar risk but inferior returns, sell it, and buy up the portfolio with similar risk but higher return. You then sell this portfolio at a higher price to investors wishing to efficiently manage risks in their portfolios and pay back the person who “lent” you the inferior portfolio.
The arbitrage condition is more complicated to describe than in the case of certainty but is no more complicated in principle. The complicated part is in seeing how investors go about building the so-called “efficient frontier” of tradeoffs between portfolio risks and returns, and equilibrium comes about.
After Tobin’s signal contributions to the field, economists had no answers. The task fell to finance jocks, the first and foremost being Harry Markowitz. He showed how an individual investor could build an efficient frontier for substituting risk and return. His student William Sharpe then showed how this led to stock market equilibrium. Let us see how these two thinkers founded the modern field of financial economics.
The Markowitz algorithm
THE MATHEMATICAL METHOD for combining available stocks to give a risk-return frontier was discovered by future Nobellist Harry Markowitz in the 1950s. His insight was that, given some group of stocks, you could combine them in different proportions (with different “weights”) to produce a level of variance of the return and given expected return. His “algorithm” for finding the efficient frontier was to postulate a level of risk and ask what weights on some or all the stocks created the greatest expected return for this level of risk. You gave the problem to a computer and let it solve the highest possible return for every risk level and that gave you the efficient frontier. In following such an algorithm the investor could choose the portfolio that best satisfied her desire for tradeoff between risk and return while assuring herself that she was squeezing the most return out of her chosen portfolio given the risks involved.
Despite its simplicity, Markowitz’s efficient frontier is often misunderstood to be a tool which tells people how to invest. This is as mistaken an idea of its meaning as it is to think that a consumer’s budget constraint tells him what to buy in the supermarket. The confusion is understandable. It arises because the efficient frontier is derived in relation to preferences; the wish to minimize risk and maximize return. Ordinary consumer budget constraints have nothing to do with preferences, but only with the prices and incomes given to people. Markowitz’s efficient frontier is a strange creature in the menagerie of economic ideas because it is a hybrid of peoples’ risk-return preferences and the possibilities of combining stocks to attain different levels of risk and return. The frontier seems to filter out the worst risks, but then it is still up to the investor to decide situation himself thereupon by choosing the appropriate portfolio.
Markowitz’s formula is also misunderstood to be by itself a description of how stock markets work. It is in fact Markowitz’s view of how an individual investor should eliminate inefficient portfolios from his or her set of feasible choices. As such it is a purely personal prescription for how to conceive of what the most efficient portfolios might be. It thus fell one step short of helping to understand how stock markets determined ass
et returns.
We saw in the insurance example that the zero-profit assumption led to an asset price, in this case insurance being the asset, which was proportional to the odds-ratio for the two states in question. But what was the analogue of the zero-profit condition in the multi-state portfolio model? More generally, what was the “equilibrium condition” that would achieve consistency between the formula investors should follow and the actual price of assets in the stock market? Markowitz opened the door to the answer but did not step through.
The answer was provided by Markowitz’s student William Sharpe in 1964, and independently by John Lintner in 1965, and Jan Mossin a year later. They were the most prominent expositors of Markowitz, but not the only ones. Other academics worked toward a solution, and pretty much came to the same conclusions, though in different contexts and by expressing themselves in different terms. The purveyors of great ideas sometimes arrive, whips cracking, like jockeys at a photo finish. In this race, Sharpe nosed out the others. He won the Nobel Prize in 1990 along with his teacher Markowitz.
Stock market equilibrium
AS MARKOWITZ WROTE in his Nobel Prize lecture, “My work on portfolio theory considers how an optimizing investor would behave, whereas the work by Sharpe and Lintner on the Capital Asset Pricing Model (CAPM for short) is concerned with economic equilibrium assuming all investors optimize in the particular manner I proposed” (1991, 469). The wording of this sentence is important.
The equilibrium Sharpe and Lintner proposed works like this: suppose you have a group of people looking at anticipated stock returns, that is, how much you can expect to gain by investing. If one portfolio with an expected rise in price is dominated by a portfolio with the same rise in price (the return) but lesser variability, then the stocks in this first portfolio need to fall in price so that the rate of anticipated price rise increases. The prices of all stocks must adjust so that they can justify their presence in portfolios lying on Markowitz’s efficient risk, expected-return frontier. Sharpe showed how millions of investors following the Markowitz algorithm of weeding out inefficient portfolios would drive stock prices and hence returns to conform to a market-wide efficient risk-return frontier. Sharpe also showed that in the presence of a risk free asset there is only one efficient combination of stocks. The risk-return frontier narrows to a straight line joining the expected returns of this efficient “market portfolio” with the returns on the risk-free asset. All that is left to individuals is to choose the combination of the risk-free asset and the unique efficient portfolio, called the market portfolio that most closely suit their aversion to risk and desire for return. The one efficient portfolio is what determines the unique returns to stocks. Because there is only one relevant portfolio all individuals should hold, variations in individual preferences do not figure in returns. This makes good sense. The efficient frontier is derived from everyone’s quest to exploit arbitrage opportunities. Since everyone has the same quest to very simplistically make money from arbitrage, variations in complex individual desires do not figure in the determination of stock returns.
Once again we see a similarity to insurance where in equilibrium the parameters of demand were absent. In the stock market these parameters are only felt indirectly through the efficient portfolio which is based on the very simple assumption that less risk for the same return is good but does not involve preferences in any more explicit manner. In the insurance market what made demand irrelevant to insurance price was the zero-profit condition. In the stock market, demand has no influence on stock returns because nature imitates art. The artist in question is Harry Markowitz. If only one person were to follow his algorithm for holding a mean-variance efficient portfolio he would become rich, and attract attention from other investors who would imitate him until there was no longer any opportunity to “beat the market”. Stock prices and hence returns would have shifted to the mean-variance efficient frontier by the efforts of investor’s following the Markowitz method.
All of which makes me wonder sometimes whether Markowitz might not have done better by keeping his mouth shut. He had the perfect recipe for beating the market but by revealing it he allowed everyone to catch up and wipe out the possibility of extraordinary profits. Instead, people following the Markowitz algorithm are picking up crumbs. It also makes me wonder whether someone just as bright, but less attracted to the spotlight than Markowitz might have already come up with a better way of managing risk and returns and is quietly amassing a fortune. If I were her I too would keep mum.
Diversification, efficiency
THERE ARE TWO aspects of the Capital Asset Pricing Model I have not elaborated upon because a full discussion would lead us deeper into the field of finance that we need go to understand its place in the economics of chance. But the two aspects do deserve brief mention, one because of its utter weirdness, the other because of its connection to the notion of an “efficient market”.
The first aspect of the CAPM that many find difficult to grasp is that the return to a stock depends on the portfolio in which it is held. To see this note that a single stock suffers from two sorts of risk. One risk is that inherent to the company, the so-called idiosyncratic risk. Another risk is shared, perhaps unequally, with other stocks and arises from economic upheavals that touch all enterprises. Suppose you combine one stock with another that to some extent countervails the risk in the first, just by pure luck. Sometimes when the first share is up, the second share by chance is down. This share is like the new member of an insurance pool whose idiosyncratic risk cancels out the idiosyncratic risk of another member. It is possible to reduce to negligible proportions the idiosyncratic risk when you pool roughly fifty randomly chosen stocks. But these are details. The point then is that the rate of return of a stock should not depend on its individual risk.
People have trouble swallowing the notion that idiosyncratic risk should be absent from expected stock returns. If ACME Corporation has an emotionally labile CEO this should not have any influence on its expected rate of return. If the market thought the idiosyncrasies of the CEO were a negative feature, then a clever investor could buy up shares of the ACME and pool them into a portfolio where other companies have CEOs with equally up-and-down character traits. The effects of these character traits would cancel each other out on average in a portfolio. The investor then beats the market because he has neutralized the risk by including the stock in a diversified portfolio. If other investors think this way they will bid up the price of ACME and subsequently drive down the expected return of the stock. The stock market neutralizes idiosyncratic risk just as well as does the insurance market. As William Sharpe explained in parsimonious terms, “Through diversification, some of the risk inherent in an asset can be avoided so that its total risk is obviously not the relevant influence on its price” (1964, 426).
Allied to the concept of idiosyncratic risk is the notion of system-wide or “undiversifiable risk”. After diversifying away idiosyncratic risk you are still left with some risk all stocks share, but unequally and in a tied manner. One way of characterizing the common risk is to think of how the expected return on a stock varies with the expected return on the entire portfolio. In finance it is called the “beta”. If it varies by more than the portfolio then it is adding risk “pollution”. This pollution has to be compensated for by a lower price, hence higher return, at which you are willing to include the stock in your portfolio.
The point about both idiosyncratic risk and commonly felt risks is that the returns to a stock in equilibrium should depend on the portfolio in which that stock is situated. It is one of the weirder aspects of finance and arises because of statistical laws and arbitrage relations.
The second aspect of the CAPM worth noting is its role in testing whether stock markets are “efficient”. The CAPM provokes ire by claiming that stock markets are efficient. It should not. The notion of efficiency in the CAPM is circumscribed to relative stock prices and returns. The CAPM does not pretend to be able to predic
t the price of stocks. It can only tell you if a stock’s expected return is too low relative to other similar stocks.
In 2013 when arbitrage specialists won the Nobel Memorial Prize in economics the media reported they had done so for predicting stock prices. Not quite so. Their work was focused on whether the prices of some stocks were out of line with others according to the arbitrage criterion embedded in the Capital Asset Pricing Model. Prediction is not the goal of these models. Sniffing out opportunities for arbitrage is the goal. Investors in the stock market may be savvy when it comes to ensuring the efficiency of relative pricing but they may be dim-witted when it comes to divining the absolute value of stocks. A market crash can happen in the context of a market that is a blazon of efficiency in the CAPM sense. A crash can take place because almost no one was able to predict what would happen to markets in an absolute sense.
Here we return to the anticipated payoffs to a stock. Profits determine crashes and booms. The CAPM implicitly includes anticipations of such events into its formulation. But what explicitly determines the expected profits of a firm? Going deeper, what determines the expected demand and price for their product? The CAPM is not equipped to answer these questions. It can only frame the question. For a sense of how people form expectations of the prices of the goods and services upon which the value of firms ultimately must be based, we must turn to the field of rational expectations economics.
Rational Expectations
WE JUST SAW that asset returns depend on an undiversifiable risk factor encapsulated in the market beta of a stock. Asset prices are a trickier quantity. Yes, once you have determined asset return and once you know the anticipated payouts of stocks, then calculating the anticipated stock price is a trivial exercise. But where does the anticipated payout come from? How do you predict it? The CAPM is mute on this topic. The CAPM is not a model that is sufficient to determine stock price. It is a model based on an arbitrage principle that is highly specialized into predicting asset returns.