Investment Management Part III: A Quantitative Approach to Equity Asset Prices

This is the third in a series of Quantdary articles that will seek to demystify the investment management process to the general audience, and to arm investors with enough knowledge so as not to be led into investing in securities that are ill-suited for them. Much of the insights are derived from conversations with professors, portfolio managers and employees of financial institutions around Montreal. They are hopeful that the ongoing cultural  shift in the business and investment management landscape will promote the welfare of investors.

There was a time when securities trading was dictated not by financial theories or any measurement of risk, but rather by heuristics developed through the years that sometimes offer proverbial and conflicting advice. The leading investment books, from Graham and Todd’s Security Analysis to Williams’s The Theory of Investment Value, were devoid of any general theory towards investment, which was due to a lack of realistic economic models at that time.

There’s an old joke that tells of a group of economists who were stranded on an island. A crate of canned food is washed ashore one day, and the starving economists must think of a way to open the cans given the lack of tools at their disposal. One of them offered, as a way to solve the problem, “Assume we have a can opener.” This joke isn’t that far off from characterizing the microeconomic model in the early 20th century, where equilibrium models were derived from the clumsy assumption that agents have perfect foresight.

As Morgenstern mused at one point in his career, such an approach is a logical impossibility (as narrated by Justin Fox in “The Myth of the Rational Market”). Morgenstern offered the example of Sherlock Holmes and Dr Moriarty to illustrate the point. Suppose Holmes is running away from Moriarty, and if Moriarty catches Holmes, Holmes is killed. In his attempt to escape, Homes boarded a train to Dover that offered an intermediate stop. However, Holmes saw Moriarty board another train which also offers the same intermediate stop, and must now decide if he should get off at the intermediate stop or simply go to Dover. So Holmes reasons that if Moriarty is to get off at Dover, he should then get off at the intermediate stop. Moriarty reasons if Holmes thinks that he is going to Dover, then he should go to the intermediate stop, since that is what the logical Holmes is bound to do. But if Holmes foresees this to be the case, then Holmes should really get off at Dover. But then Moriarty would foresee that Holmes foresees his reasoning, then he would go to Dover, which in turn means that Holmes should get off at the intermediate step. And so on. As such, there is no way we can have good results without being able to deal with the issues of modelling uncertainty.

This motivation led to the development of the expected utility framework of Morgenstern and von Neumann, which served as the building block in decision making and modern portfolio theory. The notion is quite simple: let the agents introduce the probabilities across different states that could conceivably occur. Then agents choose their actions that weighs the utility of each state with its associated probability. The action that maximises the utility under such uncertainties are then chosen as the best course of action. What is neat about such an approach is that it need not mean that the agents are always right, but rather that they are acting rationally given the information that they have. This was a powerful insight at the time, and it allowed the field of Finance to usher in a new era of scientific approach towards investing, beginning with the work of Harry Markowitz.

The notion of expected utility provided the perfect stage for Harry Markowitz to expand upon a theory of investing that would incorporate uncertainty. As a graduate student at the University of Chicago, Markowitz was interested in developing a general theory of investment. His insight, derived from statistics, was quite powerful and simple. Shopping for securities is a bit like shopping for clothing. What matters is not only how attractive the garment is, but how it complements one’s wardrobe. Similarly, Markowitz argued that what is important when it comes to a security is not only its variance, but how its addition affects the movement of the entire portfolio of securities. After all, a mean-variance investor is constantly making the trade-off between risk and return–would a higher rate of return from adding any one security to the portfolio be worth the “swing” (captured by the variance of the portfolio) that might cause the investor to miss a targeted return?

Markowitz’s message was uncomplicated and conforms to the intuition of diversification that we all understand. If you add a couple of securities that are not perfectly correlated together, you get a better return to volatility profile to your portfolio. As an example, if you have Walmart and Apple in your portfolio, when the economy is doing poorly, Apple might be doing poorly too, but then Walmart’s good performance offsets Apple’s poor performance, with the result being that across different states of the economy you get a more “stable” return. The question that remained then, was how does one get an efficient portfolio with the best return and the smallest unit of volatility possible (the square root of the variance)?

William Sharpe offered an answer, assuming the Capital Asset Pricing Model was correct. According to Sharpe, if we all share the same beliefs and are mean-variance investors, then we would all hold the same portfolio with the same weight. Thus the market portfolio, where we add up everyone’s holdings, is then the efficient portfolio.

With these assumptions, Sharpe has all the necessary ingredient to set up an equation that optimizes the utility of an investor. What amounts to an efficient allocation of each stock at the margin is the excess return (that is the return of the stock in excess of the risk-free rate) over its co-variance with the market portfolio. In terms of our shopping example, this is tantamount to evaluating the attractiveness of a garment by taking into account how it complements your wardrobe. So if we, the mean-variance investors, take care to make sure that the excess returns of the stocks over their covariance with the market is equated across all stocks, we thus have an equation that is equivalent to running a regression of a stock’s return on the market portfolio’s return. The risk and thus compensation that stock should offer, is thus how much it causes the market portfolio to swing when it is added to the market portfolio. As such, there exists a market risk-premium, and according to the CAPM a stock’s price merely reflect the extent to which it is exposed to this market risk premium. The “level” of exposure is the regression coefficient on the market risk premium, which is widely referred to as the Market Beta.

Not only did Sharpe offer an equation to price stocks, he also offered a way to allocate asset provided that we are willing to accept more assumptions. In particular, if we are willing to assume that all investors have perfect access to credit, all we need is to allocate our assets in two portfolios, a risk-free portfolio and a risky portfolio. A more risk-averse investor could tilt away from the market portfolio and invest more in a risk-free portfolio, while a less risk-averse investor could borrow from the risk-free portfolio in order to increase the exposure to the market portfolio. Moreover, if everybody behaves the same way, what the CAPM give us is thus a General Equilibrium model, where the aggregate demand and supply of stocks intersect in a market-clearing price, so the market is not characterised by a shortage or glut of securities to trade. It is also with this insight that some investment managers offered the 60-40 rule, allocate 60% of one’s investment in stocks (the risky portfolio), and 40% in bonds (the “less risky” portfolio). It is noteworthy that an empirical work studying the investment decisions of Swedish households find that they seem to heed the CAPM investment advice, as their asset allocations seem not to veer too far away.

The bold prediction of the CAPM naturally subjected itself to intense scrutiny and testing. Throughout the years, scholars have put the CAPM under internal and external validation. In an internal validation, the assumptions are examined and tweaked around to look at what the resultant implications would be. In an external validation, the predictions of the CAPM are compared to empirical observations. The consensus is that while the CAPM’s predictions are right qualitatively, it does not seem to be the only explanatory factor for a stock’s return. The jarring observation is that there seems to be additional and systematic factors that could explain the returns of stocks alongside the market risk premium, which manifests in a statistically significant intercept term in a CAPM regression.

This statistically significant term, called “alpha,” is widely touted by active managers (that is, managers who claim that they are good at finding bargain stocks through their research and skills) as the value added that they deliver to clients. As their arguments go, if you invest in a fund that purely tracks the market-portfolio, the excess return of your portfolio – excess returns means the return of your portfolio minus the risk-free rate. After all, there is always an opportunity cost for investing in a risky portfolio, by putting your money is risky securities you give up the opportunity to use them to invest in risk-free securities) – is simply your portfolio’s exposure to the market risk-premium. Had you invested with these active managers, you are compensated with this extra “alpha” that is not related to the market-risk premium. Hence, with high alpha comes highly “justified” management fees.

As research in empirical asset pricing advanced, we began to acquire some insights on how this performance summary of “alpha” can be gamed. Researchers doing empirical work have documented that small capitalization stocks and value stocks offered a superior return profile compared to large capitalization stocks and growth stocks. A strategy that buys small and value stocks and sells large and growth stocks are thus able to generate a positive alpha when regressed on the market-risk premium, simply because in our regression analysis we omitted the small cap and value premiums. Hence, just because a strategy offers us a positive alpha need not mean that the strategy is generating value, it might be loading on other risk-sources that are not captured in a uni-variate CAPM regression. In fact, this understanding has led to a more general version of asset pricing model, which is essentially a multi-variable version of the CAPM.

Finding a general asset pricing model has thus turned into the holy grail in the field of empirical asset pricing. Some of the factors offered to explain a stock’s returns appear ad-hoc, and were mainly a result of data-exploratory exercises. On the other hand, some researchers began by consulting the literature on the internal validation of CAPM, and worked by loosening some of the assumptions first offered by William Sharpe. For instance, by assuming that not all investors have perfect access to credit to alter their portfolio allocation, researchers have constructed a “Betting Against Beta” (BAB) factor that was able to explain away the alpha generated by Warren Buffett, a highly respectable active manager whose strategy have consistently generated a positive alpha above the market risk premium. As the researchers offered, there are two factors, the BAB and QMJ (Quality Minus Junk) that are capable of explaining the alpha generated by Warren Buffett.

What these empirical researcher have given us, alongside Markowitz and Sharpe’s insight is the simple truth that asset prices are determined by aggregate demand and supply conditions, which are in turn caused by investors with different risk-profiles and endowments. There are risk-premiums that are attractive, but they are not for everyone. There are risk-strategies that pay off consistently, but might suffer high losses that only patient and wealthy investors could stomach. The lesson is that there is no easy and fast way of summarizing a strategy by just looking at the alpha–if anything we should try and control for as many documented risk-source as possible to be sure that the ostensibly high alpha isn’t a result of disproportionate exposure to some risk-sources.

The process of investment management is starting to lean towards a model where advisers are hired for advice, and not to peddle investment products. This serves well to counter a moral-hazard issue, so that investors are not pushed into buying products that are ill-suited for them. It would serves both investment advisors and households well to at least have an intuitive understanding of the quantitative approach to equity asset prices, and to be able to discuss the results of the advisory analyst in a healthy, yet skeptical way.


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