Modern Portfolio Theory, Financial Engineering, and Their Roles in Financial Crises Harry M. Markowitz Professor of Finance University of California San Diego

Both mean–variance analysis (also known as modern portfolio theory) and financial engineering, which develops products and procedures based on the Black–Scholes– Merton equation, use mathematical techniques to advise investors. Each has been of great benefit, but each has also been associated with great disasters—Black Monday (19 October 1987), LTCM (Long-Term Capital Management), and the most recent financial crisis.

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n this presentation, I will first discuss modern portfolio theory and then move to financial engineering. These discussions will provide the background so that I can examine three recent financial disasters (Black Monday, Long-Term Capital Management, and the global financial crisis) and the roles that modern portfolio theory (MPT) and financial engineering (FE), however inadvertently, played in each. I will next raise the issue of why elegant mathematics sometimes leads to disastrous policies and end with observations on “best practice MPT.”

Modern Portfolio Theory At the heart of the MPT process, shown in Figure 1, is the portfolio optimizer. Like most computer programs, it has outputs and inputs. The portfolio optimization calculation produces two outputs. The first output is the familiar risk–return trade-off, known as the efficient frontier, which shows the expected return of the portfolio on the y-axis and the risk (typically represented by the standard deviation of the portfolio) on the x-axis. The investor chooses an efficient combination of risk and return from those available. The second output of the optimizer is the portfolio that produces this risk–return combination. This pr esentation comes fr om a sp eech given to the CF A Society of Orange County (Los Angeles) on 17 September 2009.

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The optimization process has three inputs. The first input is expected returns for each security in some universe of securities. These securities may be individual stocks and bonds or asset classes. It is sometimes assumed that historical average returns must be used as the requisite expected return estimates. This is a misconception. In principle, forward-looking estimates, for the “next spin of the wheel,” should be used. The second input to the optimizer is a covariance matrix, which consists of a standard deviation or variance for each security in the universe and a covariance (frequently provided by a factor model) or correlation for each pair of securities. Pairwise correlation makes the key insight of MPT possible. Unless securities are perfectly positively correlated, any portfolio formed from those securities will be less risky than the weighted average risk of the securities composing the portfolio. Pairwise correlation matrices are typically derived from historical data. The final input to the optimizer is constraints on the choice of portfolio. For example, constraints often include upper bounds on the proportions invested in various securities, industries, or asset classes and/or constraints on the turnover of the portfolio. The choice of constraints may depend on how frequently one plans to reoptimize, the liquidity of the securities or asset classes involved, and possible risk factors that the investor fears may not be adequately represented in the covariance matrix.

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Figure 1.

MPT Investment Process

Return

Expected Return Estimates

Risk

Variance/Covariance Estimates

Portfolio Optimization

Constraints on Portfolio Choice (e.g., turnover constraints)

Further details about the mean–variance (MPT) investment process, including proposed models of covariance, can be found in Markowitz (1952, 1959), Markowitz and Todd (2000), Sharpe (1963), Rosenberg (1974), Fama and French (1995), and Markowitz and Perold (1981a, 1981b).

Financial Engineering Options are investment vehicles whose value is derived from other securities. In addition, they display asymmetrical return payoffs. Financial engineering involves the creation of investment vehicles with these option-like characteristics. Therefore, to understand all FE-based products, one needs to understand the Black–Scholes–Merton (BSM) option-pricing model. On the one hand, some of the inputs to BSM, such as stock price, exercise price, interest rate, and expiration date, are directly observable. On the other hand, the standard deviation of the underlying investment is not. It must be estimated, typically from historical data. BSM also assumes that the standard deviation is constant, or at least predictable, over the life of the option. Furthermore, BSM assumes that stock prices do not experience extreme jumps, such as might occur as the result of a takeover offer. Finally, BSM assumes that there are no transaction costs.

Recent Financial Disasters The last 25 years have generally been a period of global economic prosperity. And the world’s financial markets have fully reflected that fact. But three financial disasters occurred during this time that

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Risk−Return Efficient Frontier

Choice of Portfolio

marred this overall success and that can be traced back, at least partially, to MPT and FE. B l a c k M o n d a y . On 19 October 1987, stock markets crashed globally. That day has come to be known as Black Monday. The cause of Black Monday, interestingly, can be found by looking at how FE uses option-pricing theory. Consider a plain-vanilla American call option, which represents the right, but not the obligation, to buy something at a fixed strike price on or before a specified expiration date. A synthetic call attempts to replicate the payoff profile of a call as a function of the price of its underlying security. BSM’s analysis implies that, in the frictionless world they assume, a call is redundant: That is, the same outcome can be produced by shifting between cash and the stock as the stock price moves up and down in relationship to the option’s strike price. If an investor purchases an actual call option, the investor cannot lose more than the price paid for the option. But if the underlying security goes up in value, the call purchaser will make an amount equal to the gain in the stock over the strike price of the call, less the price of the call. To replicate this outcome, the synthetic call must increase its position in the security when the latter’s price rises and decrease its position when the security price falls. In particular, the replication procedure must be completely out of the security quickly enough that—no matter how low the price of the security falls—the replication procedure cannot lose more than the price of the option being replicated. (Because this may be a nonexistent, theoretical option being replicated, “price” means the theoretical price of that option.)

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Modern Portfolio Theory, Financial Engineering, and Their Roles in Financial Crises

Portfolio insurance is an attempt to capture the upside of markets without having to endure the downside, such as by creating a synthetic call on an index. So, in the late 1980s, to obtain the risk–return exposure of the S&P 500 Index, an investor could have bought all the stocks of the index, or bought an investment company’s shares, or bought futures contracts against the index.1 Of these three alternatives, the buying and selling of S&P 500 futures was by far the most practical because S&P 500 futures trade throughout the trading day and have a very liquid market. The reason for executing a synthetic call against the S&P 500, rather than buying an actual call, is that synthetic calls can be executed in greater volume than actual calls can. For example, according to Brady (1988), “Various sources indicate that $60 to $90 billion of equity assets were under portfolio insurance administration at the time of [Black Monday]” (p. 29). To replicate the action of a call option against the S&P 500, portfolio insurers have to increase their position in the S&P 500 futures contract as the S&P 500 Index rises and decrease it as it falls so as to be completely out of their futures positions quickly enough that the maximum loss incurred from a large market decline does not exceed the BSMdetermined price of the replicated option. As it happened, on the Wednesday through Friday of the week prior to Black Monday, the market fell substantially. As a result, portfolio insurance formulas directed its participants to sell index futures contracts. This selling pressure on the futures contract caused a disequilibrium in the relationship between the futures price and the spot market price. Normally, index arbitrageurs step into this disequilibrium by buying futures and selling the spot market to restore the equilibrium relationship. But on Black Monday, the steep market decline placed so much pressure on portfolio insurers to sell their futures positions that it overwhelmed index arbitrageurs’ ability to take the other side. The problem was that one of the key assumptions of the BSM formula is that markets are continuous in time and infinitely liquid. In fact, they are not. Black Monday was a “black swan,” in the sense of Taleb (2007)—that is, a many-standard-deviation adverse move. Although not calling it that, the black swan phenomenon was analyzed by Markowitz and Usmen (1996a, 1996b). Markowitz and Usmen considered the extent to which a Bayesian would shift beliefs among various hypotheses concerning the probability distribution of daily

returns based on the log of the S&P 500 today divided by the S&P 500 yesterday. We found that posterior belief, given the data, should be shifted massively against the assumption that daily S&P 500 returns are normally distributed in favor of the Student’s t-distribution with between four and five degrees of freedom. The latter distributions have fat tails and, therefore, are subject to black swans. In contrast, if one adds together the 250 or so trading days’ worth of Student’s t daily returns to obtain annual returns, the result is close to normal. (Recall we are dealing with logs. Adding logs is equivalent to compounding returns.) Thus, although black swans on a daily basis are not all that rare, black swans on an annual basis have never occurred. In particular, for the year as a whole, 1987 was a nonevent year. The 38.5 percent fall in the S&P 500 in 2008 was between a two and two-and-a-half standard deviation move (based on the Ibbotson large-cap series since 1926). If annual returns were normally distributed, a downward move in excess of two standard deviations would happen roughly two-and-one-half percent of the time: 1 year in 40. Thus, the investor’s mix of equities and fixed income should take into account the occurrence of such two to three standard deviation moves from time to time. Long-Term Capital Management. LTCM was a large, highly leveraged hedge fund—notable for having two Nobel laureates (Robert Merton and Myron Scholes) on its board—that was bailed out under the supervision of the U.S. Federal Reserve in 1998 because of fear that its collapse would create a crisis of unprecedented proportions. LTCM’s approach was to go short the liquid side of many markets and long the related illiquid side.2 For example, it went long the 29-year U.S. Treasuries and short the 30-year Treasuries. When Russia defaulted in 1998, there was a great demand for liquidity and a corresponding flight from illiquid assets. This flight to quality would not have been a problem if LTCM had not been highly leveraged. As it was, its account was marked to market and was soon wiped out. LTCM used MPT rather than option-pricing theory in judging the safety of its portfolios. Thus, although Merton and Scholes were personally involved with LTCM, unlike portfolio insurance, the LTCM blowup is not an example of a questionable application of the Black–Scholes–Merton option-pricing analysis. 2

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Now, one can also buy an exchange-traded fund (ETF), but ETFs were not available at the time of Black Monday.

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This description of LTCM procedures is based on a lecture given by David Modest at IBM Research in Yorktown Heights, New York, a few months after the LTCM meltdown.

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One moral of the story is that when extreme leverage is used, any error in assumptions is likely to be fatal. Had LTCM been fully invested in these positions but without leverage, the Russian default event would have been inconsequential. If LTCM could have waited out the year, the 30-year bond would have become a 29-years-to-go bond and would have lost its liquidity premium. But LTCM’s leverage, and consequent mark-to-market requirement, precluded such from happening. LTCM’s small return per position required high leverage to obtain a reasonable return on capital. Thus, a corollary to this moral is that if a particular application of any method of investment becomes so competitive that great leverage is required to make a decent return, then it is time to look for another line of business. But it is probably too much to expect of any human to voluntarily give up a legal and highly profitable line of business because of what is perceived as a small chance of great harm to clients. The Latest Financial Crisis. The current financial crisis provides another example that highlights the importance of understanding the nature of correlation risk. In this crisis, CMOs (collateralized mortgage obligations) pooled together mortgages, some of which were themselves highly leveraged (e.g., a no-money-down mortgage is infinitely leveraged). Then the CMOs added their own leverage to the mixture. Some of the tranches of the CMOs went into CDOs (collateralized debt obligations), which added more leverage. Some of the tranches of the CDOs or CMOs were insured by CDS (credit default swaps), which were quite inadequately backed, especially considering that these were, in fact, insurance policies against correlated risks rather than uncorrelated risks. This situation was obscure as well as leveraged. Even the information technology departments of large players in this space could not figure out the indirect exposures of the instruments they held. These structured products of financial engineering were a major cause of the recent financial crisis. Another cause was the insistence by the U.S. Congress and the relevant regulatory agencies that Fannie Mae, Freddie Mac, and the banking system generally finance mortgages for low-cost housing. In effect, Congress and the regulatory agencies mandated subprime loans. These subprime loans were the grist for the mill processed by the financially engineered structured products just discussed. For typical mean–variance investors who held a reasonable mix of stocks and bonds, because they were not short or leveraged, their investments were not marked to market. Suppose, for

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example, that a couple started 2008 with a 60/40 mix of the S&P 500 and bonds (half corporate, half government)—namely, $600,000 of an S&P 500 ETF or index fund and $400,000 of bonds. By the end of the year, the $600,000 in stocks would have fallen to a little less than $400,000, whereas the $400,000 (equally split between corporate and government bonds) would have remained about the same. The couple clearly would not be happy watching their 401(k) shrink from a million dollars to less than $800,000. But they would be alive to invest another day, as compared with many investors in highly leveraged structured products who were wiped out.

Why Does Great Math Often Lead to Disastrous Results? Continuous-time financial models are elegant. The frequent objective of financially engineered structured products and procedures is to redistribute risk from those who wish to trade some return for less risk to those who wish to have the opposite side of the bargain. Why are such elegant models and noble objectives often associated with disastrous consequences? My own observations on this question are clearly from the point of view of a portfolio theorist, but this is a question that the financial engineering community itself should try to answer, especially so that it can avoid such consequences in the future. My answer is as follows. All financial models are an attempt to describe an infinitely complex reality. As a result, to achieve any success at all, portfolio theorists must make certain simplifying assumptions. But the problem with financial models is not that most of these assumptions are incorrect. Quite the opposite. These simplifying assumptions are generally true most of the time. The problem is that they are not always true. It is precisely at the point where the assumptions break down that financial models, pushed to their limits, lead to disastrous consequences, which is why I believe we should adopt what I call “best practice MPT” to protect ourselves from the exposures that represent the points at which models are most apt to break down. By this point, some of these exposures should be apparent. The first such sensitivities would be illiquidity and market impact. At times of crises, liquid securities can become illiquid very quickly. Illiquid securities may only be traded with a huge market impact, if they can be traded at all. And of course, leverage compounds these problems. The LTCM experience shows that MPT cannot protect an investor from the dangers of extreme leverage.

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Modern Portfolio Theory, Financial Engineering, and Their Roles in Financial Crises

For investors who wish to invest in illiquid securities, I offer a few commonsense suggestions. First, limit the amount of an illiquid security that you are willing to buy or sell during any one week. Also, limit the total amount of illiquid securities accumulated relative to total portfolio size. And of course, when dealing with illiquid securities, avoid leverage. An investor who is fully invested but not leveraged can wait out occasional market panics. As obvious as the preceding points are, one of the more subtle aspects of a best practice MPT concerns the insurance of correlated risk. A portfolio of credit default swaps is, in fact, a portfolio of insurance policies guaranteeing correlated risks. Chapter 5 of Markowitz (1959) presents what I now call the “law of the average covariance”: If you spread your money equally among many correlated securities, as the number of securities increases, the variance of the portfolio approaches the average covariance. For example, suppose all securities have the same variance, VS , and thus the same standard deviation, S, and that all pairs of securities have the same correlation coefficient, . In this case, every covariance will equal VS . As diversification increases, the variance of the portfolio approaches

VP → ρ VS . Also, because standard deviation is the square root of variance, then

σP → ρσS . For example, if  = 0.25 for all pairs of securities, then

σP → 0.5σS . This is shocking! The volatility (standard deviation) of the portfolio will never fall below 50 percent of that of a single security, even with unlimited diversification. If the correlation coefficient is a mere 0.1, then P will still be more than 30 percent of S. Diversification reduces volatility, but its efficacy is limited in the face of correlated risks. For the individual investor, this means that he or she must not expect wide diversification in an all-

equity portfolio to provide stable returns. Bonds or cash must be included in a portfolio to reduce volatility to an acceptable level. This analysis also has implications for issuers of CDS, which, as previously mentioned, is a product that insures against correlated risks. Specifically, the reserves behind a portfolio of such correlated risks need to be substantially greater than those behind a portfolio of nearly uncorrelated risks (such as life insurance policies). It may be that if the required reserves for a portfolio of default swaps are computed correctly, the level of reserves required will make this line of business unprofitable!

Conclusion Excessive leverage is bad; too many illiquid assets are dangerous; and writing insurance against correlated risks without reinsuring, or without quite large reserves, is an accident waiting to happen. More generally, evaluating risks one at a time rather than considering them as a portfolio is an all-toocommon error. Worse still is to be exposed to a combination of these bad practices, such as taking positions in correlated assets that may all fall off a cliff at the same time and whose illiquidity prevents one from raising cash to pay off creditors. As a result, I recommend the following: • Do not take a position in a derivative security unless you have a clear understanding of how it works and its relationship to any quantity to which it is derivative. • Recall that panics and black swans happen as often as water heaters leak. If you have nontrivial leverage, you will be wiped out. Last, but far from least, • Think about the portfolio as a whole. In particular, remember the law of the average covariance. This article qualifies for 0.5 CE credits.

R EFERENCES Brady, Nicholas F. 1988. “Report of the Presidential Task Force on Market Mechanisms.” U.S. Government Printing Office.

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

Fama, E.G., and K.R. French. 1995. “Size and Book-to-Market Factors in Earnings and Returns.” Journal of Finance , vol. 50, no. 1 (March):131–155.

———. 1959. Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons. (1991, 2nd ed., Cambridge, MA: B. Blackwell).

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CFA Institute Conference Proceedings Quarterly Markowitz, H.M., and A.F. Perold. 1981a. “Portfolio Analysis with Factors and Scenarios.” Journal of Finance , vol. 36, no. 4 (September):871–877. ———. 1981b. “Sparsity and Piecewise Linearity in Large Portfolio Optimization Problems.” In Sparse Matrices and Their Uses. Edited by I.S. Duff. New York: Academic Press:89–108. Markowitz, H.M., and P. Todd. 2000. Mean-Variance Analysis in Portfolio Choice and Capital Markets. New Hope, PA: Frank Fabozzi and Associates. Markowitz, Harry M., and Nilufer Usmen. 1996a. “The Likelihood of Various Stock Market Return Distributions, Part 1:

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Principles of Inference.” Journal of Risk and Uncertainty , vol. 13, no. 3 (November):207–219. ———. 1996b. “The Likelihood of Various Stock Market Return Distributions, Part 2: Empirical Results.” Journal of Risk and Uncertainty, vol. 13, no. 3 (November):221–247. Rosenberg, B. 1974. “Extra-Market Components of Covariance in Security Returns.” Journal of Financial an d Quantitative Analysis, vol. 9, no. 2:263–273. Sharpe, W.F. 1963. “A Simplified Model for Portfolio Analysis.” Management Science, vol. 9, no. 2:277–293. Taleb, N.N. 2007. The Black Swan: The Impact Improbable. New York: Random House.

of t he Highly

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