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06/08/2010

Crisis Mode: Modern Portfolio Theory Under Pressure


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During financial crises correlations among security returns approach one.

Therefore, diversification fails just when you need it.

Therefore, MPT (modern portfolio theory) is of no use.

The first of the statements above is correct; the second is partially true; the last statement is false. In order to understand the assertions of the last sentence, we need to review what MPT is and is not, what it offers and what it does not promise.

IDIOSYNCRASIES OF RISK

The above statements are most easily explained if we do not go back to my general formulation of MPT (Markowitz 1952; Markowitz 1959), but use William Sharpe’s (1963) simplified model of portfolio theory instead. While my work considers how to diversify given any pattern of correlations among risks, Sharpe assumes a “one-factor model.” Specifically, Sharpe’s simplified model assumes that each security is correlated with every other security because all are correlated with the market.

The return on a security is equal to:

A constant, called the security’s alpha,
plus
another constant, the security’s beta,
times
an underlying factor sometimes called the return on the market,
plus
an independent, random idiosyncratic term.

A security’s alpha (α) and beta (β) are constants, whereas the underlying factor and the idiosyncratic term are random. The last two terms—the common underlying factor and the idiosyncratic term—are the two sources of period-to-period (day-to-day, month-to-month, or year-to-year) fluctuations. Sharpe’s one-factor model assumes that the idiosyncratic random term of one security is uncorrelated with that of any other security and uncorrelated with the underlying factor. Thus, in this simplified model, one security is correlated with another only because both are correlated with the market.

The return on the portfolio as a whole also equals the portfolio’s alpha plus its beta times the underlying factor plus the portfolio’s idiosyncratic term. It follows from Sharpe’s assumptions about security returns that the portfolio’s idiosyncratic term is uncorrelated with the underlying factor, as are the idiosyncratic terms of the individual securities.

But the idiosyncratic term of the portfolio is not uncorrelated with the idiosyncratic terms of the individual securities, since it is composed of these terms. The alpha of the portfolio is a weighted average of the alphas of the individual stocks, weighted by portfolio holdings. The beta of the portfolio is a weighted average of the betas of the individual stocks, again with weights equal to portfolio holdings. Finally, the idiosyncratic term of the portfolio during any interval of time (day, month, year) is the weighted average of the idiosyncratic terms of the individual securities. However, the variance of the idiosyncratic term is not the weighted sum of the securities’ idiosyncratic variances. Since the securities’ idiosyncratic terms are uncorrelated, they tend to diversify: some will almost certainly do well when others do poorly.

With a sufficiently large and well-diversified portfolio the variance of the portfolio’s idiosyncratic term is negligible. Note, though, that this is not true of the variance due to what is called the “systematic risk,” that is, the risk due to the beta of the portfolio times the possible up or down movement of the market. Systematic risk, due to beta, does not diversify away; unsystematic risk does.

A simple example will illustrate this crucial point. Suppose that all securities in a portfolio have the same alpha; then the portfolio will have this alpha. Also suppose that each security has the same beta; then the portfolio will have the same beta as each of its securities. But if the variances of the idiosyncratic terms of the securities are the same, the variance of the idiosyncratic term for the portfolio will not be the same: it will be smaller than that of each of its securities. The idiosyncratic risks diversify away. The systematic risk (due to beta times the market) does not diversify away.

MPT BEFORE, DURING, AND AFTER A CRISIS

In Sharpe’s model, a crisis is, by definition, a period of time containing a large downward movement of the underlying factor—perceived as a large movement in the market. This period of time may be a day when the market drops dramatically—for example, October 19, 1987; or an extended period—such as October 1929 through 1933. Even assuming that the alphas, betas, and variances of the idiosyncratic terms of the individual securities stay constant throughout the crisis, the fact still remains that almost all securities and asset classes will have moved in the same direction.

This does not mean that individual securities are no longer subject to idiosyncratic risks. It means, rather, that the systematic risk swamps the unsystematic risk during this period.

Nor does it mean that all securities or asset classes fall equally during the period. A high beta security will probably have a very large move, unless it is lucky with its idiosyncratic term; a low beta security will probably have a low downward move unless it is unlucky with its idiosyncratic term. (Few, if any, securities have negative betas.)

The year 2008 is a good example. The S&P 500 fell approximately 38.5%; the higher-beta emerging-markets asset class fell much farther. Corporate bonds fell in value, but much less than equities, and government bonds went up. Small stocks went down, but not as much as might have been expected, given their historical betas. On average, they had a lucky idiosyncratic term during that interval of time. Generally, asset classes moved roughly in proportion to their historical betas.

Some investors (usually institutional investors) do their formal MPT, or mean-variance analysis, once a year, basing it on asset-class capital-market estimates proposed to and reviewed by an investment committee. These estimates, and the risk–return frontier that flows from them, may be reviewed quarterly, or perhaps more frequently, but usually are not thoroughly reconsidered more often than annually. Other investors (again, generally institutional investors) may make estimates (usually based on some forecasting model) monthly, weekly, or even daily.

Those who review estimates quarterly or annually usually make estimates and do portfolio analyses for asset classes. They then parcel out these asset-class investment requirements to money managers, investment companies, or exchange-traded funds.

The investors may estimate expected returns, variances, and correlations of the Markowitz general portfolio selection model, or they may estimate the alphas, betas, and idiosyncratic variabilities of Sharpe’s model. They may do so frequently or less frequently. Their estimates may change slightly or significantly from one time to the next. None of this alters the fact that the estimates used in a portfolio analysis should always be forward-looking estimates for the forthcoming period. (If the estimates are changed frequently, or change a good deal each time, then turnover constraints on the choice of portfolio are desirable. In any case, these changes of estimates should always anticipate the next “spin of the wheel.”)

In terms of Sharpe’s simplified model, whether or not the investor shifts beliefs a lot or a little, the investor will admit a range of possibilities both with respect to the underlying factor and the idiosyncratic terms. In other words, the investor (or investment advisor, or consultant) will attach a positive variance to the underlying factor and to each idiosyncratic risk. (Only a fool would believe that he or she could predict any of this with certainty.)

This forward-looking uncertainty encompasses various possible scenarios. In particular, the forward-looking estimate of the variance of the underlying factor implicitly or explicitly takes into account various possible movements of the market. If, in fact, a scenario occurs in which the underlying factor moves little, then the idiosyncratic terms will dominate the observed movements during the period. There will be little systematic movement in one direction and a substantial amount of relatively idiosyncratic movement in contrary directions.

Conversely, if the systematic risk factor goes up or down considerably, the systematic risk will dominate the unsystematic risk. The undiversifiable risk will swamp the diversifiable risk. If the underlying factor rises up a great deal nobody will complain; we’ll all consider ourselves geniuses. If it goes down considerably there will certainly be complaints from everyone who failed to account for the undiversifiable risk.

Those with high beta portfolios—those whose portfolios are heavily weighted with high beta securities—will probably have large upward or downward movements, depending on the direction of the underlying factor. Those whose portfolios are heavily weighted in low beta securities will probably have lesser moves. If the factor moves up, cautious investors with their low beta portfolios will look with envy at their neighbors. If the factor moves down, they will pride themselves on their prudence.

Unfortunately for investors, and fortunately for portfolio analysts, the estimates and efficient frontiers (risk–return trade-off curves) must be computed before the fact. The sophisticated institutional or individual investor knows that there is uncertainty, that there is a risk–return trade-off, that higher return entails higher risk. If you want greater certainty you must give up return in the long run. MPT never promised high return with low risk. You pays your money and you takes your choice.

THE MANY-FACTOR MODEL

Sharpe’s one-factor model assumes that the idiosyncratic terms of each security are uncorrelated with each other. But experience has shown that the idiosyncratic terms of various pairs of securities are correlated among themselves, reflecting the existence of additional factors.

Soon after the publication of Sharpe’s one-factor model, King (1966) presented convincing evidence for the existence of industry factors. Rosenberg (1974) presented a model, which became widely used, that included “fundamental factors” as well as industry factors. A simpler and now popular model is the Fama and French (1995) three-factor model, which adds the large-cap–versus–small-cap factors and the value-versus-growth factors to the general market factor.

On occasion, other factors, usually negligible, push their way to the front. For example, at times of higher oil prices the firms in some industries suffer, while others benefit. Sometimes no great premium is placed on liquidity, and illiquid assets are deemed attractive if they have moderately high expected returns. In other cases, liquidity is priceless.

If the one-factor model were true, one could control a portfolio’s risk by controlling the portfolio’s beta and dividing funds among a large number of any securities. When industry and other nonmarket factors are taken into account, it is clear that having many securities does not guarantee satisfactory diversification if those securities are concentrated in a few industries or are heavily exposed to some fundamental factor.

The fall of LTCM (Long-Term Capital Management) provides a memorable example of this. LTCM systematically went short liquid assets, such as 30-year Treasuries, and long a closely related illiquid security, in this case the 29-year-until-maturity Treasuries. In addition, LTCM leveraged heavily. When Russia defaulted, the market placed a great premium on liquidity. LTCM’s leveraged bet on illiquid over liquid securities brought the company down when this particular attribute of securities pushed itself to the front as a risk factor.

DIVERSIFICATION VERSUS CONCENTRATION

If diversification is of little use during crises, then concentrated portfolios should do perfectly well. Consider concentrated portfolios during 2008—their performance clearly depended on the areas in which they were concentrated.

A portfolio concentrated in government bonds would have done quite well. The problem with putting all one’s wealth in government bonds, of course, is that they usually yield less than corporate bonds, and over the long run their performance is much lower than that of equities. On the other hand, if a portfolio was concentrated in AIG, Citigroup, General Motors, or the financial or auto industries generally, the result would have been tragic.

THE LAW OF AVERAGE COVARIANCE

The terminology used above with regard to systematic and unsystematic risks arose from Sharpe’s simplified one-factor model. The original Markowitz model allowed an arbitrary matrix of correlations and therefore offered no simple separation between systematic and unsystematic risk. It did, however, imply a quite general result, which I now call the “law of average covariance” (Markowitz 1959, chapter 5).

The practical implication of the law of average covariance is that, in the face of correlated risks, even a widely diversified portfolio has substantial risks. For example, if all securities had the same standard deviation, sigma-S (σs), and all pairs had the same correlation, rho (ρ), then the standard deviation σρ of a highly diversified portfolio would be the square root of rho, √ρ, times the standard deviation of one security (σs).

Thus, with a correlation of ρ=0.25, therefore √ρ=0.5, the standard deviation of the portfolio would be σs ∕ 2. The volatility of the highly diversified portfolio would be 50% as great as if you had put all your money in one security!

It’s apparent that, while diversification does reduce volatility, its efficacy is limited in the face of correlated risks. To lower risk further, you must add cash or bonds.

SPIN THE WHEEL

We look at the past when we make our estimates and decisions for the future. Yet the future is always uncertain, particularly in times of market volatility. When we make our best estimates for the next spin of the wheel, we must choose an appropriate point on the efficient frontier, the risk–return trade-off curve. Depending on our risk aversion, we may select a more cautious portfolio, loaded with lower beta securities or asset classes, or we may choose a point higher on the frontier, with higher yield but with higher beta securities or asset classes. But unless our portfolios are comprised entirely of short-term government bonds, we’ll be dealing with a level of risk for which MPT prescribes following an old and true adage: “Don’t put all your eggs in one basket.”

REFERENCES

Fama, Eugene F., and Kenneth R. French. 1995. “Size and Book-to-Market Factors in Earnings and Returns.” Journal of Finance, vol. 50, no. 1. 131–155.

King, Benjamin F. 1966. “Market and Industry Factors in Stock Price Behavior.” Journal of Business, vol. 39, no. 1, part 2. 139–190.

Markowitz, Harry M. March 1952. “Portfolio Selection.” Journal of Finance, vol. 7, no. 1. 77–91.

———. 1959. Portfolio Selection: Efficient Diversification of Investments. New York, NY. Wiley. (Yale University Press 1970; Basil Blackwell 1991.)

Rosenberg, Barr. 1974. “Extra-Market Components of Covariance in Security Returns.” Journal of Financial and Quantitative Analysis, vol. 9, no. 2. 263–273.

Sharpe, William F. January 1963. “A Simplified Model for Portfolio Analysis.” Management Science, vol. 9, no. 2. 277–293.

Harry M. Markowitz received a PhD in economics from the University of Chicago in 1955. In 1989 he received the von Neumann Theory Prize from the Institute of Management Sciences and the Operations Research Society of America, and in 1990 he shared the Nobel Prize in Economics with William Sharpe and Merton Miller. Currently Dr. Markowitz consults and lectures (HarryHMM@aol.com) and is an adjunct professor at the Rady School of Business, University of California at San Diego.

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