The New Efficient Frontier Asset Allocation for the 21st Century
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Morningstar Direct 1
SM
The New Efficient Frontier
Asset Allocation
for the 21st Century
by he did not stand idly by in the area of portfolio cure what Savage [2009] calls the “flaw of
modeling, but continued to make improvements averages.” In general, the flaw of averages is a
Paul D. Kaplan, Ph.D., CFA in his own models and to influence the set of systematic errors that occur when
Vice President, models of others. Few of these improvements, people use single numbers (usually averages)
Quantitative Research however, were picked up broadly in practice. to describe uncertain future quantities.
For example, if you plan to rob a bank of $10
Going Supersonic million and have one chance in 100 of
Sam Savage, Ph.D. Because Markowitz’s first effort was so simple getting away with it, your average take is
Stanford University and powerful, it attracted a great number $100,000. If you described your activity
of followers. The greater the following became, beforehand as “making $100,000,” you would
Author, The Flaw of Averages the fewer questioners debated its merits. be correct on average. But this is a terrible
Markowitz’s original work is synonymous with characterization of a bank heist. Yet as Savage
When the Wright Brothers pioneered powered Modern Portfolio Theory and has been [2009] discusses, this very “flaw of averages”
flight in 1903, their genius lay in conquering taught in business schools for generations and, is made all the time in business practice, and
the three axes of control: pitch, yaw, and not surprisingly, is still widely used today. helps explain why everything is behind
roll. Over the years, technologies advanced, schedule, beyond budget, and below projection,
planes crashed, and aviation evolved to Then came the crash of 2008, and at last and was an accessory to the economic
compensate. By 1952, the Wright’s original people are starting to ask questions. The catastrophe that culminated in 2008.
airplane was barely recognizable in a world of confluence of the recent economic trauma and
jets and even supersonic aircraft, which the technological advances of the past Harry Markowitz’s 1952 mean-variance model
were nonetheless still governed by the same few decades make today the perfect time to attempted to cure the flaw of averages by
three principles of control. describe the supersonic models that distinguishing between different investments
can be built around Markowitz’s fundamental with the same average (expected) return,
In 1952, another pioneer, Harry Markowitz, principles of risk, reward, and correlation. but with different risks, measured as variance
invented portfolio optimization. His genius was In a recent paper, we assert that Markowitz’s or its square root, standard deviation. This
also based on three principles: risk, reward, original work remains the perfect framework was a breakthrough at the time that ultimately
and the correlation of assets in a portfolio. Over for applying the latest in economic thought garnered a Nobel Prize for its inventor.
the years, technologies advanced and markets and technology. We dub our updated model However, the use of standard deviation and
crashed, but the portfolio-optimization models “Markowitz 2.0.” covariance introduces a higher-order
used by many investors did not evolve to version of the flaw of averages, in that these
compensate. This is surprising in light of the Markowitz 2.0 concepts are themselves a version of averages.
fact that Markowitz himself was a pioneer The Flaw of Averages C ON T I N UE D ON N E X T PAGE
of technological advancement in the field of The 1952 mean-variance model of Harry
computational computer science. Furthermore, Markowitz was the first systematic attempt toThe New Efficient Frontier Asset Allocation for the 21st Century 2
Adding Afterburners to Traditional The Scenario Approach right, while others have distributions that are
Portfolio Optimization One of the limitations of the traditional skinnier or fatter in the tails than others.
By taking advantage of the very latest in mean- variance optimization framework is that
economic thought and computer technology, we it assumes that the distribution of returns of the Over the years, various alternatives have been
can, in effect, add afterburners, or more thrust, assets in the optimization can be adequately put forth to replace mean-variance optimization
to the original framework of the Markowitz described simply by mean and variance with an optimization framework that takes
portfolio-optimization model. The result is a alone. The most common depiction of this into account the non-normal features of return
dramatically more powerful model that is more assumption is to draw the distribution of each distributions. Some researchers have proposed
aligned with 21st century investor concerns, asset class as a symmetrical bell-shaped using distributions curves that exhibit skewness
markets, and financial instruments such as curve. However, as illustrated in Exhibit 1, the and kurtosis (that is, have fat tails) while
options. return distributions of different asset others have proposed using large numbers of
classes don’t always follow a symmetrical scenarios based on historical data or Monte
Traditional portfolio optimization, commonly bell-shaped curve. Some assets have Carlo simulation.
referred to as mean-variance optimization, or distributions that are skewed to the left or C ON T I N UE D ON N E X T PAGE
MVO, suffers from several limitations that can
easily be addressed with today’s technology.
Exhibit 1: Basic Series: Summary Statistics of Annual Total Returns: 1926–2008
Our discussion here will focus on five practical
Series Geometric Arithmetic Standard Distribution %
enhancements: Mean % Mean % Deviation %
Large Company 9.6 11.7 20.6 9
1 First, we use a scenario-based approach to allow 6
Stocks
for “fat-tailed” distributions. Fat-tailed return 3
distributions are not possible within the context of –90 0% 90 0
traditional mean-variance optimization, where Annual Total Return
return distributions are assumed to be adequately
Small Company 11.7 16.4 33.0 12
described by mean and variance. 9
Stocks* 6
3
2 Second, we replace the single-period expected
–90 0% 90
return with the long-term forward-looking
0
geometric mean (GM), as this takes into account
accumulation of wealth. Long-Term 5.9 6.2 8.4 30
Corporate Bonds 20
10
3 Third, we substitute Conditional Value at Risk
–90 0% 90
(CVaR), which only looks at tail risk, for standard
0
deviation, which looks at average variation.
Long-Term 5.7 6.1 9.4 30
4 Fourth, the original Markowitz model used a Government Bonds 20
10
covariance matrix to model the distribution
–90 0% 90
of returns on asset classes; we replace this with a 0
scenario-based model that can be generated
with Monte Carlo simulation and can incorporate Intermediate-term 5.4 5.6 5.7 40
30
any number of distributions. Government Bonds 20
10
–90 0% 90
5 Finally, we exploit new statistical technologies 0
pioneered by Sam Savage in the field of Probability
Management. Savage invented a new technology Treasury Bills 3.7 3.8 3.1 60
called the Distribution String, or DISTTM, which 40
encapsulates thousands of trials as a single data 20
element or cell, thus eliminating the main –90 0% 90 0
disadvantage of the scenario-based approach—the
need to store and process large amounts of data. Inflation 3.0 3.1 4.2 60
40
20
–90 0% 90 0
*The 1933 Small Company Stocks total return was 142.9% Source: Morningstar (2009)The New Efficient Frontier Asset Allocation for the 21st Century 3
The scenario-based approach has two main represent return distributions. For example, segment shows that the 5th percentile return
advantages over a distribution curve Exhibit 2 shows the distribution curve of annual under our model is –25.8 percent, meaning
approach: (1) it is highly flexible; for example, returns of Large Company Stocks under there is 5% probability of a return of less than
nonlinear instruments such as options our approach. Comparing Exhibit 2 with the –25.8 percent. However, under the lognormal
can be modeled in a straightforward manner, Large Company Stock histogram in Exhibit 1, model, the probability of the return being
and (2) it is mathematically manageable; for we can see that the smooth distribution less than –25.8 percent is only 1.6 percent.
example, portfolio returns under the scenarios curve retains the properties of the historical This illustrates how a mean-variance model can
are simply weighted averages of asset-class distribution while showing the distribution woefully underestimate the probability of
returns within the scenarios. In this way, in a more esthetically pleasing and precise tail events.
the distribution of a portfolio can be derived form. Furthermore, our model makes it possible
from the distributions of the asset classes to bring all of the power of continuous As Kaplan et al. [2009] discuss, tail events
without working complicated equations that mathematics previously enjoyed only by models have occurred often throughout the history of
might lack analytical solutions; only based on continuous distributions to the capital markets all over the world. Hence,
straightforward portfolio arithmetic is needed. scenario approach. it is important for asset-allocation models to
assign nontrivial probabilities to them.
In standard scenario analysis, there is no In Exhibit 2, the green line curve is what
precise graphical representation of return distri- we get when we use mean-variance Geometric Mean versus Single Period
butions. Histograms serve as approximations analysis and assume that returns follow a Expected Return
such as those shown in Exhibit 1. We augment lognormal distribution. The blue line is In MVO, reward is measured by expected
the scenario approach by employing a what we get when we use our smoothed return, which is a forecast of arithmetic mean.
smoothing technique so that smooth curves scenario-based approach. The area under However, over long periods of time, investors
the blue solid line to the left of the vertical C ON T I N UE D ON N E X T PAGE
Exhibit 2: Smooth Distribution Curve for Annual Returns on Large Company Stocks 1926–2008
Scenario-Based Distribution Lognormal Distribution
–90 –80 –70 –60 –50 –40 –30 –20 –10 0% 10 20 30 40 50 60 70 80 90
Annual Total ReturnThe New Efficient Frontier Asset Allocation for the 21st Century 4
are not concerned with simple averages of as the measure of risk in strategic asset investment. CVaR is the expected or average
return; rather, they are concerned with allocation. While any one of these could be loss of capital should VaR be breached.
the accumulation of wealth. We use forecasted used, our preference is to use Conditional Value Therefore CVaR is always greater than VaR.
long-term geometric mean (GM) as the at Risk (CVaR). For example, the 5% CVaR for the distribution
measure of reward because investors who plan shown in Exhibit 2 is 35.8%, or $3,580, on a
on repeatedly reinvesting in the same CVaR is related to Value at Risk (VaR). VaR $10,000 investment.
strategy over an indefinite period would seek describes the left tail in terms of how
the highest rate of growth for the portfolios much capital can be lost over a given period of Scenarios versus Correlation
as measured by geometric mean. time. For example, a 5% VaR answers a In mean-variance analysis, the covariation
question of the form: Having invested $10,000, of the returns of each pair of asset classes
Conditional Value at Risk versus there is a 5% chance of losing $X or more is represented by a single number, the
Standard Deviation in 12 months. (The “or more” implications of correlation coefficient. This is mathematically
As for risk, much has been written about VaR are sometimes overlooked by investors, equivalent to assuming that a simple linear
how investors are not concerned merely with with serious implications.) Applying this idea to regression model is an adequate description of
the degree of dispersion of returns (as returns, the 5% VaR is the negative of how the returns on the two asset classes
measured by standard deviation), but rather the 5th percentile of the return distribution. For are related. In fact, the R-square statistic of a
with how much wealth they could lose. example, the 5th percentile of the distribution simple linear regression model for two
A number of “downside” risk measures have shown in Exhibit 2 is –25.8% so its 5% series of returns is equal to the square of the
been proposed to replace standard deviation VaR is 25.8%. This means there is a 5% chance correlation coefficient.
of losing $2,850 or more on a $10,000 C ON T I N UE D ON N E X T PAGE
Exhibit 3: Geometric Mean—Conditional Value at Risk Efficient Frontier
10%
9
Geometric Mean
8
7
6
5
0% 5 10 15 20 25 30 35 40
Conditional Value at RiskThe New Efficient Frontier Asset Allocation for the 21st Century 5
However, for many pairs of asset classes, a encapsulate thousands of trials as a single References
linear model misses the most important data element. The use of DISTs greatly saves 1 Kaplan, Paul D., Thomas Idzorek, Michele Gambera,
features of the relationship. For example, on storage and speeds up processing time, Katsunari Yamaguchi, James Xiong, and
during normal times, non-U.S. equities so that a Monte Carlo simulation consisting of David M. Blanchett, “The History and Economics of
Stock Market Crashes.” In Siegel, Laurence B., ed.,
are considered to be good diversifiers for U.S. thousands of trials can be performed on a
Insights into the Global Financial Crisis,
equity investors. But during global crises, personal computer in an instant. While not all CFA Institute, 2009.
all major equity markets move down together. asset-management organizations are prepared 2 Kaplan, Paul D. and Sam Savage, “Markowitz at
to create the DISTs needed to drive the Mach 1,” Morningstar research paper,
Furthermore, suppose that the returns on two GM-CVaR optimization we described in Kaplan December 2009.
asset classes indexes were highly correlated, and Savage [2009], some outside vendors, such 3 Markowitz, Harry M., “Portfolio Selection,”
but instead of including direct exposures as Morningstar Ibbotson, can fulfill this role. Journal of Finance, 7 (1), 77-91, 1952.
to both in the model, one was replaced with an 4 Ibbotson Stocks, Bonds, Bills, and Inflation
option on itself. Instead of having a linear Another facet of Probability Management is 2009 Yearbook, Morningstar, Inc., 2009.
relationship, we now have a nonlinear interactive simulation technology, which 5 Poundstone, William, Fortune’s Formula, Hill
relationship that cannot be captured by a can run thousands of scenarios through a model and Wang, 2005.
correlation coefficient. before the sound of your finger leaving the 6 Savage, Sam, The Flaw of Averages, John Wiley
key reaches your ear. These supersonic & Sons, 2009.
Fortunately, these sorts of nonlinear relation- models allow much deeper intuition into the 7 Sheikh, Abdullah Z. and Hongtao Qiao,
ships between returns on different investments sensitivities of portfolios, and encourage the “Non-normality of Market Returns,” J.P. Morgan
Asset Management research paper, 2009.
can be handled in a scenario-based model. user to interactively explore different portfolios,
For example, in scenarios that represent normal distributional assumptions, and potential
times, returns on different equity markets black swans. A sample of such an interactive
could be modeled as moving somewhat apart model will be available for download from
from each other while scenarios that represent www.ProbabilityManagement.org in 2010.
global crises could model the markets as
moving downward together. Finale: The New Efficient Frontier
Putting it all together, we form an efficient
Ultrasonic Statistical Technology frontier of forecasted geometric mean
Because it may take thousands of scenarios to and Conditional Value at Risk as shown in
adequately model return distributions, until Exhibit 3, (Page 4) incorporating our
recently, a disadvantage of the scenario-based scenario approach to covariance and new
approach has been that it requires large statistical technology. We believe that
amounts of data to be stored and processed. this efficient frontier is more relevant to
Even with the advances in computer hardware, investors than the traditional expected return
the conventional approach of representing versus standard deviation frontier of MVO
scenarios with large tables of explicit numbers because it shows the trade-off between reward
remained problematic. and risk that is meaningful to investors;
namely, long-term potential growth versus
The phenomenal speed of computers has given short-term potential loss. K
rise to the field of Probability Management,
an extension of data management to probability Learn More
distributions rather than numbers. The key To arrange a demonstration:
component of Probability Management is the Call: +1 866 910-0840
Distribution String, or DIST , which can
TM
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