Revisiting Modern Portfolio Theory and Portfolio Construction

FEATURE | Revisiting Modern Portfolio Theory and Portfolio Construction

By Bryce James

ears ago, I was speaking at an invest- Figure 1: Standard Deviation Probabilities from the Mean Return ment industry conference when a Probability of Return at 1, 2, and 3 Standard Deviations Yman in the front row yelled out, “Is this just another rubber chicken pre- 5% 1σ sentation on asset allocation?” I responded 4% with, “Fasten your seatbelt!” and I’ll say it 3% again here, because this is not your generic 2% 2σ 2σ article on post-modern portfolio theory. 1% I’m going to challenge you to question what Probability Density 3σ 3σ 0% you think you know about asset allocation 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% and ask you to keep an open mind to what –9% –8% –7% –6% –5% –4% –3% –2% –1% 10% –10% is possible. DJIA Daily Return 99.7% Probability of Return 95.4% Probability of Return 68.3% Probability of Return Birth of a New Technology Source: Internal calculation using Excel. Data from Thomson-Reuters Computer technology was born in the 1950s with the first commercial com- return, and linear correlation (ρ) for which academics named the efficient market puter—the UNIVAC 1—delivered in 1951, correlation. hypothesis (EMH). The person credited followed by the first hard disk in 1955, 3. Optimize the portfolio using a covari- with this integration of the EMH was Fortran computer language in 1957, and ance matrix to seek the optimal effi- Eugene Fama at the University of Chicago. the modem in 1958. This era also gave cient frontier, i.e., asset mix. birth to modern portfolio theory (MPT). Standardized risk metrics in finance took You know about Harry Markowitz and In a classic MPT model, which is based hold in the 1950s when top economists, what he accomplished. All asset allocation upon probability theory, many assumptions such as Maurice Kendall (London School of work henceforth is a manifestation of his are made. The core assumption was Economics), Paul Samuelson (MIT), Harry concepts, founded on the principles of founded on the work of Louis Bachelier Markowitz (University of Chicago), and diversification. Implementing MPT is to (Mandelbrot and Hudson 2004, 9), who in William Sharpe (University of Washington apply a process called mean-variance 1900 introduced a probability-based model and University of California, Los Angeles), optimization (MVO). following the random walk theory, which is observed that over the long term, changes named after the unpredictable path of a in security prices plotted on a histogram An MVO model is a three-step process for drunken man (Fox 2009, 40–41; Sornette chart (frequency distribution) resembled portfolio construction: 2002, 38). Bachelier postulated that the the shape of the symmetrical bell curve market is like a 50-50 coin toss and that (Gaussian distribution) (see figure 1). 1. Collect a large amount of historical most market moves are measurable. This Combining all these pieces gave birth to price data on each security in your led him to conclude that markets are effi- the “science of investing.” fund universe (typically 30 years of cient. He calculated that 68.3 percent of daily prices and dividends). the changes up and down are small, falling As MPT took hold in the early 1960s, folks 2. Calculate the risk and return of each within one standard deviation (1σ) from such as Jack Treynor and William Sharpe security and the correlation between the mean return; that 95.4 percent of the introduced the capital asset pricing model pairs of securities. In traditional MPT, changes fall within 2σ; and that 99.7 per- (CAPM), improving upon Markowitz’s this is performed using standard devia- cent fall within 3σ. Markowitz (1952, 1959) capital allocation line. Risk finally could tion (σ) for risk, mean return (μ) for fit snugly into this random walk theory, be measured, as ratios, in various ways

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through the metrics created by Sharpe, Data bonds instead of non-callable bonds. In Treynor, and Sortino. Today, these theo- In 2005, a law firm representing a group of addition, they were using bond proxies ries, known for their risk metrics, stand as investors hired me as an expert witness. The when no long bond was issued (which is the foundation of investing. But as tech- clients, all retired, claimed their broker was most of the time) in calculating the long- nology has significantly changed over the imprudent in managing their assets, creating term bond index. The net effect was an over- past half century, is it possible finance has losses exceeding 40 percent. The brokerage allocation to equities and an undervaluing been asleep? firm’s asset allocation software, based on an of bond returns. The net result, according to MVO model, recommended the usual 60/40 Ryan, created a massive performance dif- MPT assumes all information is priced into mix. However, the firm chose to use 18 years ference of 45.3 percent (from 1941–1991). the security and that yesterday’s price has of historical data rather than the 28 years Furthermore, investors would have experi- no influence on today’s price; each price recommended by the software vendor. As enced a standard deviation of 9.04 percent change is independent from the previous a result, the Monte Carlo simulations esti- using the Ibbotson Long Treasury portfolio move. The market, and each security, is mated a 16-percent return on equities into compared to 5.96 percent standard devia- fairly priced and moves are completely ran- the future instead of the 9-percent estimate tion with the actual Treasury composite; dom, with no regard for human emotion. (as of December 31, 1999) using the software a difference of 45.6 percent more risk Therefore, MPT assumes: (1) returns are vendor’s default 28-year time series. Yet this (Ryan 1992). normally distributed, (2) risk (σ) is accu- was not the main issue. The broker himself, rately captured using a normal distribution, using his own personal asset allocation soft- In a response to Ryan’s claims, Ibbotson (3) historical prices (mean-variance) can be ware, decided to load only three years’ worth colleagues Laurence B. Siegel and Scott L. used to predict future prices, (4) linear cor- of history (ending December 31, 1999) into Lummer (1993) wrote: relation accurately represents the relation- the asset allocation software. Recall that this ship between pairs of securities, (5) data was one of the best-performing periods for Today, however, only a very naïve investor acquired from data providers is accurate, the stock market, if not the best, in history. In would use our 20-year constant maturity and (6) markets are efficient, thus ignoring doing so, the broker’s MVO model forecasted series as a benchmark for evaluating a the behavioral science of fear and greed. a 27-percent return, which he presented to diversified bond portfolio. Mr. Ryan I’m here to tell you that all these assump- his retirement-age clients. As you know, the makes a valid point in suggesting an asset tions are false. Stock markets do not follow dotcom bubble soon burst and it took the allocator could be fooled by the Ibbotson random walks (see, e.g., Lo and MacKinlay S&P a decade to break even and 14 years data into underweighting in bonds. This 1988; Grossman and Stiglitz 1980; for the NASDAQ. The case was an easy win is a danger only if the asset allocator liter- Colander et al. 2009). for our side. Selection of time series is a form ally believes the disappointing historical of momentum. return on long-term bonds will be It’s so hard to let go of judgments and repeated. Again, this is a naïve view. The beliefs. It’s human nature to hang on to The time series is such a critical piece of expected return on a default-free bond is what’s familiar. In our business, it’s best to data, yet so overlooked. I argue that all its yield. This, not the historical return, stay safe by doing what everyone else does. models are momentum models, because should be used for asset allocation. We also stay safe by keeping things simple. any time you select a time series, you are If it’s not simple, clients’ eyes roll back in making an assumption and making a bet. Asset allocation models should be expec- their heads and you’ve lost the sale. This is To buy and hold is making a bet that past tational; they require expected returns as why even the newest financial technology— performance is indicative of future results, inputs. For bonds, the expected return is robo-advisors—rely upon the MPT/MVO contrary to our marketing disclaimers. observable in the market as a yield and math from 1952; it’s fast and easy. The first does not have to be extracted from his- person to expose the naked truth about the Back in the days of the UNIVAC 1, the tory. To use a historical average as the mathematics was Benoit Mandelbrot term GIGO (garbage in, garbage out) was expected return for bonds, as Mr. Ryan (1964). He criticized the use of normal dis- born. Financial companies have been built has done, is conceptually improper and tributions and warned that markets are around delivering quality data. I’m here to leads to poorly constructed portfolios. influenced by human behavior. He should tell you it’s still problematic. You’ve seen the know—he worked with Markowitz, and he Ibbotson chart for most of your career—the No apologies were given by Siegel, was Fama’s PhD advisor. one showing the growth of $1,000 over Lummer, or Ibbotson; they just moved time for each major asset class. What if I blame to the user. But note that Ibbotson Asset Allocation Engine— told you that one of the single biggest indi- no longer shows long bonds on its charts. Looking under the Hood cators was grossly wrong for decades? It The basic parts to the asset allocation turns out the creator of the Lehman Risk engine include data (including time series), Brothers Bond Index (now Barclays), Ron What is risk? If we agree that risk is a mea- risk, return, correlation, and optimization. Ryan, called out Ibbotson for using callable sure of volatility (thereby agreeing to ignore

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all other risks), then it’s prudent to compare Table 1: Odds of Exceeding Each Standard Deviation risk models. We have two classes of risk models, one based on normal distributions Risk MPT Estimate Actual Standard Deviation Error and another on a class of non-normal dis- 1σ 6 days 8 days 18% less tributions. Normal distributions include 2σ 44 days 35 days 27% more the risk metrics: standard deviation (entire 3σ 3 years 0.54 years 5.4x variance), semi-variance (downside vari- 4σ 123 years 1.9 years 65.2x ance), and Value-at-Risk (VaR), or tail risk 5σ 14,000 years 3.6 years 3,828x (see figure 2). Normal distributions are measured following a Gaussian bell curve. 6σ 4 million years 10 years ~4,000x 7σ 3.052 billion years 15 years ~201 million x Table 1 shows the odds of exceeding each 8σ 6.279 trillion years 30 years ~208 billion x standard deviation. For example, in the 9σ 34.611 quadrillion years 30 years ~1143 trillion x EMH, the odds of a 4σ event is every 123 10σ 513 sextillion years 60.5 years ~8.5 sextillion x

Figure 2: Three Measures of Risk (Normal Distributions)

Standard Deviation Semi-Variance Value at Risk (VaR) Volatility around mean return Measures probability of loss 1% chance of losing x% or more on a given day 1952 1952 1993 5% 5% 5%

4% 4% 4% x% 3% 3% 3%

2% 2% 2% or more

Probability Density 1% Probability Density 1% Probability Density 1%

0% 0% 0% –1% .5% 2% 0% 1% 2% 3% 4% 0% 1% 2% 3% 4% 5% 6% –4% –3% –2% –1% –6% –5% –4% –3% –2% –1%

Actual Distribution VaR Normal Distribution Source: Internal calculation using Excel. Data from Thomson-Reuters

Figure 3: DJIA Daily Returns—Outliers

Normal, Actual, and "Heavy Tail" Distribution (left tail) 1985–2016

10 9 8 7 6 5 4 Instances 3 2 1 0 –8.0% –7.5% –7.0% –6.5% –6.0% –5.5% –5.0% –4.5% –4.0% –3.5% –3.0% –2.5%

Daily Return

Actual Count Normal Distribution Count

Source: Internal calculation using Excel. Data from Thomson-Reuters

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years, when in actuality it’s every 1.9 years. appears, and kurtosis, which estimates the The men who created GARCH, Clive In EMH, the odds of a 2008 event is one in extent of fat tails in the distribution of data. Granger and Robert Engle, won the Nobel three lifetimes of the universe, not every The red line in figure 3 depicts the proba- Memorial Prize in Economics in 2002. 60.5 years, which is what has occurred his- bility distribution of a normal distribution. torically. That is why we have so many out- Any gold bar above the red line represents The magic of GARCH is its ability to capi- liers, because of bad mathematics. an outlier. Table 2 lists specific daily losses, talize on the newer information, like the date, and the probability of occurrence EWMA, but discount the data as it moves VaR moves beyond the probability of losing using the bell curve. At what point do we too far from its traditional mean-return. money (semi-variance) and aims to mea- raise the white flag and surrender the nor- The disadvantage of GARCH is the whip- sure catastrophic risk, known as tail risk. mal distribution? sawing effect during trading ranges in the VaR picks a point on the distribution, say market. Our research indicates the positives the last 1 percent of the bell-curve area, and Non-normal distributions (NNDs) allow far exceed the negatives with GARCH. estimates the probability of losing more more than two moments to be seen. A wide than the average loss of the tail surface area. selection of NNDs can be fitted such as Figure 4 shows SPY, the exchange-traded For example, SPY has a 1-percent chance of Student’s-t, log-normal, skewed-t, paired-t, fund (ETF) replicating the S&P 500. The losing −2.09 percent or more today. VaR stable, and combinations of distributions. green line is the price growth of SPY; the grew so popular that it became the stan- Any of these NNDs can significantly black line is the daily price change (up and dard risk measure for all banks and insur- improve your results. For perfectionists, down); and the red line is a rolling three- ance companies as part of the Basel II scientists have nice mathematical tests to month VaR. Note how frequently the black Accord. But how does one measure tail risk compare and contrast forms of distributions. line exceeds the red line to the up and when one can’t see the tail? So far within downside. Focusing on the downside, the my research, we have yet to find an equity In addition, you can find which method has number of exceedances over the past two security that fits within the paradigm of a a history of producing the least number of years (ending November 4, 2016) at the normal distribution. VaR definitely did not outliers (aka exceedances, extreme events, 1-percent VaR level totals 17. The blue line help Long-Term Capital Management black swans); this can be measured at both is a combination of an NND and GARCH; (LTCM) in 1998, nor the banks and insur- tails, in absolute numbers and as a percent- we call this expected shortfall with GARCH ance companies during the 1987, 2000, and age. Now you have more-accurate risk tools features, or ES for short. Using ES, for the 2008 crashes (see figure ).3 replacing the inputs of the old ratios of past two years, we experienced three Sharpe, Sortino, and Treynor. Beyond VaR, exceedances at the 1-percent level versus 17 VaR’s core problem is that it uses an better tail metrics, such as Conditional VaR with a rolling VaR. assumption of a normal distribution. (CVaR), can now be better applied. Normal distribution can measure only two Nobel Prize-winning physicist Philip moments: mean return and variance (how Utilizing an NND significantly improves Anderson states: far a distribution is spread out from its your estimation of loss and lowers outliers mean). You can measure for normality with CVaR. However, you still are using a Much of the real world is controlled as using the Jarque-Bera normality test, which single long-term average number to predict much by the “tails” of distributions as by is a goodness-of-fit test of whether sample the future risk (and return). Fortunately means or averages: by the exceptional, data have the skewness and kurtosis match- there are models that allow for more-recent not the mean; by the catastrophe, not the ing a normal distribution. But actual return information to gain more importance, such steady drip; by the very rich, not the distributions have third and fourth as exponentially weighted moving average “middle class.” We need to free ourselves moments, called skewness, which is an (EWMA) and GARCH (generalized auto- from “average” thinking (Ramalingam expression of how lopsided a distribution regressive conditional heteroskedasticity). 2014, 219).

Table 2: Seven Worst Days (in 30 Years) and the Probability of Loss (Gaussian Bell Curve)

Return Date Description Probability (Years) −22.61% 10/19/1987 ‘87−88 Bear Market, Black Monday 377,928,949,357,521 + 25 more zeros −8.04% 10/26/1987 ‘87−88 Bear Market, Program Trading 22,546,897,547 −7.87% 10/15/2008 Financial Crisis Bear Market 7,348,618,460 −7.70% 12/1/2008 Financial Crisis Bear Market 2,449,539,487 −7.33% 10/9/2008 Financial Crisis Bear Market 241,966,705 −7.18% 10/27/1997 Asian Currency Crash 97,981,579 −7.13% 9/17/2001 9/11/2001 Attack 72,811,999

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I’ve never met the person who can accu- Figure 4: SPY—State Street Global Advisors (SSgA) S&P 500 Index rately predict stock prices. So the next Increasing Volatility question is: Can you predict volatility? 200 Low or Decreasing Volatility Volatility is most notably tracked with an 200 150 instrument called the VIX. The VIX is the 150 100 ticker symbol for the Chicago Board 100 50 Options Exchange (CBOE) Volatility Index, 50 0 which shows the market’s expectation of 0 0.2 30-day volatility. Amazingly, ES with 0.2 0.1 GARCH features is highly correlated (as an 0.1 inverse) to the VIX, which is based upon 0.0 0.0 the Black-Scholes pricing model. GARCH –0.1 –0.1 is using historical distributions of daily –0.2 –0.2 prices. In comparing the VIX (S&P 500) to –0.3 –0.3 1995 2000 2005 2010 2015 SPY (ETF proxy for S&P 500) we find a 2006 2008 2010 2012 2014 2016 very high correlation (see figure 5). Price Return Exp. Shortfall at 1% Value-at-Risk at 1% Price Return Exp. Shortfall at 1% Value-at-Risk at 1% Source: Internal calculation using Excel. Data from Thomson-Reuters The advantage of ES with GARCH features is the ability to create a VIX-like indicator Figure 5: Dynamic Risk (S&P, SPY) for every security or portfolio and also per- form a fairly accurate forecast of risk. 90 25% Figure 5 illustrates the similarities between 80 the VIX and ES using the ETF: SPY. 70 20% 60 15% Essentially, ES with GARCH can be a pre- 50 dictor of price. In other words, you can pre- 40 VIX Level 10% dict price by forecasting risk, what I call 30 ES (%) 1-day “directional risk,” and by analyzing the level 20 5% of risk, what I call “risk regime.” Figure 6 10 shows directional risk in relation to price. 0 0% Magenta lines (on the upper chart) are decreasing prices, and on the lower chart 1/10/1992 1/10/1993 1/10/1994 1/10/1995 1/10/1996 1/10/1997 1/10/1998 1/10/1999 1/10/2000 1/10/2001 1/10/2002 1/10/2003 1/10/2004 1/10/2005 1/10/2006 1/10/2007 1/10/2008 1/10/2009 1/10/2010 1/10/2011 1/10/2012 1/10/2013

increasing risk; visa-versa for the purple lines. VIX SPY (ES 1%)

Source: Internal calculation using Excel. Data from Thomson-Reuters Note the high correlation between price and volatility. Note also that fractal behavior and the power law come into play Figure 6: SPY—State Street Global Advisors (SSgA) S&P 500 Index during the 2008 crash, just as Mandelbrot and Hudson (2004) wrote in The (Mis) Increasing Volatility 200 Behavior of Markets. Low or Decreasing Volatility 200 150 150 100 Return 100 50 Recall that Siegel and Lummer (1993) said 50 0 that “only a fool” would rely on historical 0 0.2 returns. Yet they taught it to all of us for 0.2 0.1 decades. The truth is, the entire academic 0.1 and professional organizational network 0.0 0.0 taught us MPT and its derivative method- –0.1 –0.1 ologies, such as CAPM, arbitrage-pricing –0.2 –0.2 theory, and the three-factor models. –0.3 –0.3 1995 2000 2005 2010 2015 2006 2008 2010 2012 2014 2016 Price Return Exp. ShortfallAs weat 1% reflect Value-at-Riskupon mean- varianceat 1% for Price Return Exp. Shortfall at 1% Value-at-Risk at 1% estimating returns, we must accept that it Source: Internal calculation using Excel. Data from Thomson-Reuters ignores market cycles, the momentum

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Figure 7: Correlation between Two Securities

(A) Highly Correlated Securities (B) Non-Correlated Securities 6% 5%

2% 0% 0, 00, 0 FFIDX 0% FEMKX

–2%

–4% –5% –6% –4% –2% 0% 2% 4% 6% –4% –3% –2% –1% 0% 1% 2% 3% 4% FEQTX FAGIX History Simulated History Simulated Source: Internal calculation using Excel. Data from Thomson-Reuters

effect, economic and geopolitical events, Correlation: The Golden Goose up with a single number to represent the and social and technological trends. We Correlation is the foundation of asset allo- relationship between two securities. This is must accept that Brownian motion might cation because diversification is the under- like monitoring a couple who’ve been mar- work well in some sciences, such as botany, lying principal for reducing risk and ried 30 years and averaging their emotions, but not when emotions or dynamic enhancing returns. The importance of then defining their relationship as one changes in market forces come into play. reducing nonsystemic risk is the corner- emotion. Life doesn’t work that way. We all The world is more fractal than linear, stone of Markowitz’s work, but it’s really know correlations move toward 1 on big up which is why we replaced the Farmer’s driven home by Brinson, Hood, and and big down days. Clearly one number Almanac with Doppler radar for forecast- Beebower (1986) and Brinson, Singer, and cannot represent all the dots all the time, ing the weather. Forecasting returns using Beebower (1991). Xiong et al. (2010) com- which range from highly correlated to Monte Carlo simulations with MPT/MVO pletely re-analyzed the situation and made a highly negatively correlated (see figure 7). is the Farmer’s Almanac of finance. stronger case for where returns are sourced. Copulas are another form of correlation Forecasting returns is as problematic as the Xiong et al. (2010) reveals that the market model.2 By switching to a rank-correlation weather. Can you really count on analysts movement component accounts for about methodology, such as Kendall’s tau rank to give you accurate forecasts? Are they not 80 percent of the total return variations of a correlation coefficient, we can explore biased by investment banking fees or group portfolio. Of the returns in excess of market more-flexible correlation models, includ- think? So do we use historical mean- return (i.e., attributable to manager perfor- ing copulas with NNDs.3 Copulas, in the variance, or an analyst’s estimates, or third- mance), 90 percent of variability of returns wrong hands, can be bad. They didn’t party research, or create our own, and then across time is explained by asset allocation work well for the banks and insurance apply a tool like the Black-Litterman model policy, 40 percent of variation between companies who blamed copulas for their (BLM)?1 In a way, BLM is not so different funds is explained by differences in asset large losses during the financial crash. A from GARCH models. BLM uses long-term allocation policy, and 100 percent of return copula is only as good as its underlying performance as a baseline for estimating amount is explained by asset allocation pol- distribution. If you run a copula on a returns, then brings in views, such as esti- icy. Furthermore, with market movements normal distribution you ignore the tails, mates (with probabilities), to tilt the return removed, asset allocation and active just as VaR, semi-variance, and standard forecast. Using robust research and testing management are equally important in deviation do. Copulas also need NNDs methodologies, one could create a superior determining portfolio return differences to see the tails. return function using one or more views. within a peer group, with each accounting Each view can be whatever you want it to for around 20 percent. In figure 8B, note the high correlation in be, such as a technical strategy, a trading the tails on big up and down days; linear algorithm, or a collection of fundamental Correlation is still a key driver to returns correlation (figure 8A) ignores this or economic data. There is no way for me and risk. However, like distributions, not all dynamic nature. Traditional models such to compare all possible return functions correlations are the same and linear cor- as MPT depend upon linear correlation here. What I can say is that mean-variance relation can be the most limiting. As intelli- matrixes to define intermarket relation- is way down the list of optimal return func- gent beings, we somehow find it okay to ships. We know correlation, price, and risk tions. After all, past performance is not take a bunch of historical plots and draw a are dynamic, not static. Thus the adage that indicative of future results. least-squared line through them and come the only thing to go up in a down market is

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correlation. Likewise, Gaussian copulas do Figure 8: Comparing Dependency Models not account for tail correlation. (A) Linear Correlation (ND) (B) Copula (NND)

One such model that does account for tail 0.9266 correlation is Student’s t-copula. This now means we can model increased correlations during market extremes.

What concerns me about correlation is the long-term trend. Correlations have contin- ued to increase since 2000. Maybe the markets are becoming more efficient, or maybe there is a shortage of traders—who knows? At this pace, everything we know about Source: Internal calculation using Excel. Data from Thomson-Reuters asset allocation and diversification may go away. Clearly risk models are not alike; nor are been a meaningful contributor to his per- momentum models. Momentum can be formance (Bello 2008; Hou et al. 2015). Optimization achieved with technical indicators, factor Fama has long since reversed his views of Figure 9 shows actual returns for a models (such as Fama or Black-Litterman), an efficient market. moderate-risk target date fund offered by or econometric models. To say that momen- Morningstar to establish a baseline risk tum doesn’t work is simply not true. An Evolution Waiting for You level. We concluded that the average Eugene Fama was the father of the efficient As we moved through time, some fixes to expected shortfall of this fund over a markets hypothesis; he built an empire at MPT were Band-Aids and others were 10-year period (August 2004−August 2013) Dimensional Fund Advisors with his evolutionary processes. From MPT came was −1.5-percent ES at the 1-percent level. three-factor model. Then in 1997, he read MVO, then CAPM, followed by arbitrage In other words, this fund should lose a research abstract by Mark Carhart, who pricing theory with its factor models, then, −1.5 percent or more, 1 percent of the time. added a fourth factor—momentum—to the Black-Litterman multi-factor model The actual results are illustrated as orange the three-factor model by using 11 months and the French-Fama factor models. It’s an lines in figure 9A. By simply converting to of previous momentum. Fama quickly evolutionary process, but we remain stuck ES with GARCH, the exceedances were added this strategy to his offering and it’s in the old paradigm because too much reduced and performance improved.

Table 3: Target-Risk Methodology The new target-risk methodology (ES with GARCH) still deviates based on intramonth Target Risk > New (ES) Old (σ) S&P 500 risk changes, but overall these deviations Annual Growth 7.41% 6.16% 7.67% are shorter and smaller in magnitude. Annual Volatility 9.33% 10.15% 14.28% Intermonth hedging can also add value Maximum Drawdown −25.24% −36.10% −50.94% (see table 3).

Figure 9: Realized Expected Shortfall versus Targeted Risk

(A) Target-Risk (Moderate)—Realized ES (B) New Target-Risk (Moderate)—Realized ES 0 0 –0.02 −0.02 −0.04 −0.04 −0.06 −0.06 −0.08 −0.08 −0.1 −0.1 Morningstar Moderate Risk Smart Moderate Risk −0.12 (Expected Shortfall) −0.12 (Expected Shortfall) Moderate 'Target' Risk Moderate 'Target' Risk −0.14 −0.14 8/12/2004 8/12/2005 8/12/2006 8/12/2007 8/12/2008 8/12/2009 8/12/2010 8/12/2011 8/12/2012 8/12/2013 8/12/2004 8/12/2005 8/12/2006 8/12/2007 8/12/2008 8/12/2009 8/12/2010 8/12/2011 8/12/2012 8/12/2013

Source: Internal calculation using Excel. Data from Thomson-Reuters

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would need to change and some firms correct and relevant.” How long must we References Bello, Zakri. 2008. A Statistical Comparison of the CAPM could lose significant market share. It’s keep convincing ourselves? to the Fama-French Three Factor Model and the all about the Benjamins. With dynamic Carhart’s Model. Global Journal of Finance and Banking 2, no. 2: 1–23. models, style boxes would be rendered In the beginning, one man pointed out the Black, Fischer. 1986. Noise. Journal of Finance 41, no. 3 meaningless, as would investment-policy flaws in MPT and EMH—Benoit (July): 529–543. Brinson, Gary P., L. Randolph Hood, and Gilbert L. Beebower. questionnaires the way they are written Mandelbrot, a brilliant mathematician 1986. Determinants of Portfolio Performance. Financial today. Case law would need to change as whose work advanced myriad fields of Analysts Journal 42, no. 4 (July/August): 39–44. Brinson, Gary P., Brian D. Singer, and Gilbert L. Beebower. would the onslaught of robo-advisors and study, from physics to finance. In 2004, 1991. Determinants of Portfolio Performance II: An consulting firms. Not until more people Mandelbrot wrote a book about finance Update. Financial Analysts Journal 47, no. 3 (May/ June): 40–48. get sued for outdated methodologies will called The (Mis)Behavior of Markets. In it, Carhart, Mark. 1997. On Persistence in Mutual Fund things start to change. he stated, “If there is one message I’d like to Performance. Journal of Finance 52, no. 1 (March): 57–82. pass on … it is this: Finance must abandon Colander, David, Hans Föllmer, Armin Haas, Michael Imagine you could combine the best of its bad habits and adopt a scientific Goldberg, Katarina Juselius, Alan Kirman, Thomas Lux, and Brigitte Sloth. 2009. The Financial Crisis and the these tools. Imagine that you had access to method.” He also wrote, “Extreme Value Systemic Failure of Academic Economics. Kiel Institute clean data, that you can choose any data Theory, borrowed from the insurance for the World Economy Working Paper 1489 (February). Fama, Eugene. 1991. Efficient Capital Markets: II. Journal distribution you desire based on valid test- industry, is on the right track; it assumes of Finance 46, no. 5 (December): 1,575–1,617. ing, that you could apply GARCH features prices vary wildly, with fat-tails that scale.” Fox, Justin. 2009. The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street. to make the risk, return, and correlation New York: HarperCollins Publishers. numbers behave more like a Doppler radar, It’s time to take a hard look in the mirror Grossman, S., and J. Stiglitz. 1980. On the Impossibility of Informationally Efficient Markets. American Economic that you could modify your return function and ask ourselves if we use the traditional Review 70, no. 3 (June): 393–408. using Black-Litterman (for the return func- tools because they get us the results we Hou, Kewei, Chen Xue, and Lu Zhang. 2015. Digesting Anomalies: An Investment Approach. Review of tion) and you could incorporate copulas want and need or because they’re easy and Financial Studies 28, no. 3 (March): 650–705. using NNDs to dynamically calculate cor- safe and everyone else uses them. We tell Lo, Andrew, and A. Craig MacKinlay. 1988. Stock Markets Do Not Follow Random Walks. Review of Financial relation. The concept of creating models ourselves that our planet is at the center of Studies 1, no. 1: 41–66. that scale is the foundation of extreme the universe because we fear persecution, Mandelbrot, Benoit. 1964. Random Walks, Fire Damage Amount and Other Paretian Risk Phenomena. value theory (EVT). When I combine EVT but we do so at the risk of being foolish. Operations Research 12, no. 4 (July/August): 582–585. with GARCH features and newer correla- Watching from the outside is like visualiz- Mandelbrot, Benoit, and Richard L. Hudson. 2004. The (Mis)Behavior of Markets: A Fractal View of Financial tion methodologies, I create a dynamic risk ing Einstein’s definition of insanity. I’m Turbulence. New York: Basic Books. theory implemented with a dynamic opti- guessing Galileo and Mandelbrot might Markowitz, Harry. 1952. Portfolio Selection. Journal of Finance 7, no. 1 (March): 77–91. mization model. have had a lot in common. ———. 1959. Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. Validating Galileo Galilei Bryce James is president of Smart Ramalingam, Ben. 2014. The Devil Is in the Dynamics. Chapter 11 in Aid on the Edge of Chaos: Rethinking MPT, unlike the sun, is not the center of Portfolios, LLC and president of Portfolio International Cooperation in a Complex World. New the universe. In 1992 the Vatican, after Mason, LLC. He earned a BS in accounting, York: Oxford University Press. Ryan, Ron. 1992. Bond Data Problems Eyed. Pensions & 350 years, finally renounced condemning finance,and marketing from Central Investments (December 7). Galileo for stating the Earth revolved Washington University. He is a former Sharpe, William F. 1964. Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal around the sun. Likewise, markets do not CIMA® certificant. Contact him at of Finance 19, no. 3 (September): 425–442. follow the efficient market hypothesis [email protected]. Siegel, Laurence B., and Scott L. Lummer. 1993. Criticisms of Ibbotson Bond Data Answered. Pensions (i.e., the random walk theory). Nor is MPT, & Investments (January 11). MVO, or any risk, return, or correlation Endnotes Sornette, Didier. 2002. Why Stock Markets Crash: Critical Events in Complex Financial Systems. Princeton, NJ: 1. The Black-Litterman model is an asset allocation model built on normal distributions legiti- Princeton University Press. model that was developed by Fischer Black and Xiong, James X., Roger G. Ibbotson, Thomas M. Idzorek, mate in portfolio construction. Robert Litterman of Goldman Sachs. It is essentially and Peng Chen. 2010. The Equal Importance of Asset a combination of the two main theories of MPT: Allocation and Active Management. Financial Analysts CAPM and MVO. The main benefit of the Black- Journal 66, no. 2 (March/April): 22–30. Fama (1991) rejected the random walk the- Litterman model is that it allows the portfolio manager ory and promoted the idea that expected to use it as a tool for producing a set of expected returns within the MVO framework. This can allow the returns vary with time. Through decades of manager to avoid certain problems or issues inherent research and managing assets, Fama has in the MVO framework, such as the concentration of portfolio assets in only a handful of the assets under evolved; it’s time we all do. optimization. 2 . In probability theory and statistics, a copula is a mul- tivariate probability distribution for which the marginal Perhaps Fischer Black (1986) stated it probability distribution of each variable is uniform. best: “In the end, a theory is accepted not Copulas are used to describe the dependence between random variables. because it is confirmed by the conventional 4. In statistics, the Kendall rank correlation coefficient, empirical tests, but because researchers commonly referred to as Kendall's tau coefficient (after the Greek letter τ), is a statistic used to measure the persuade one another that the theory is ordinal association between two measured quantities.

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