Part I discusses the book's section on asset allocation. Part II discusses the book's other insights, my critique, and the author's responses.
Historical statistics should not be blindly fed into an optimizer.
Paul D. Kaplan, Ph.D.
The quote above comes from page 273 of Kaplan's Frontiers of Modern Asset Allocation (384 pages, Wiley Finance 2012). This quote captures the spirit of the book, which combines an exhaustive knowledge of quantitative portfolio management with an acute awareness of its limitations. I recommend this book for institutional money managers and quant geeks, especially those who would like to address the limitations of Modern Portfolio Theory. Kaplan is more than qualified for the task: He is the quantitative research director at Morningstar Europe, he serves on the editorial board of the Financial Analysts Journal, and he has a doctorate in economics from Northwestern University. Fortunately for the reader, Kaplan's writing is clear and unpretentious.
The book covers a wide range of topics that span two decade's of Kaplan's research. The 27 chapters are organized in four sections:
- Equities (index construction, small-stock betas, etc.)
- Fixed income, Real Estate, and Alternatives
- Crashes and Fat Tails
- Doing Asset Allocation
Markowitz and Optimization
I found Kaplan at his best when discussing asset allocation, the fourth section of the book. He begins with a brief overview of Modern Portfolio Theory, which was based on the 1952 paper by Harry Markowitz. This pioneering work on asset allocation, uncertainty, and diversification led to Modern Portfolio Theory (MPT), and to the efficient frontier, which shows the trade-off between risk and return as a series of optimal portfolios. MPT also led to portfolio optimization, also called Mean Variance Optimization (MVO), and to the "fish hook" charts that adorn many client presentations.
The Limitations of Mean Variance Optimization
Anyone who has worked with portfolio optimization knows that the results are extremely sensitive to the inputs. This leads to "estimation errors," a wonderful euphemism for what could be an investment catastrophe. Consequently, Kaplan emphasizes the importance of the assumptions for expected returns, standard deviation, and the correlation of asset classes. He repeatedly notes that optimizers are extremely sensitive to these assumptions.
Kaplan also cautions against cutting our asset class slices too thin: If the asset classes are too similar, high correlations cause the correlation matrix to become "ill-conditioned," and small moves along the efficient frontier result in large changes in asset allocation. Kaplan's work suggests that optimization has limits when used for thin slices of the same wedge, as when stocks are divided into categories such as micro-cap, small-cap, mid-cap, large-cap, and mega-cap.
In chapter 26, Kaplan addresses four key problems with the original Markowitz model, and he offers solutions to each. Kaplan and co-author Sam Savage of Stanford University dub this "Markowitz 2.0," and Harry Markowitz himself approves this nomenclature in an interview (364). The following four sections describe how Kaplan addresses the limitations of the original Markowitz model.
- From Normal Distributions to Fat Tails: The most familiar objection to MPT is that the world is full of so-called Black Swan events, and these happen far more often than quant models suggest. Kaplan acknowledges that returns are not normally distributed in a bell curve, so he suggests that we replace a log normal distribution with a log stable distribution. The mathematics are beyond me, though it seems clear that a log stable distribution allows for fat tails (skewed distributions that reveal additional downside risk). In one of the interviews, British author and asset manager George Cooper explained the theories of the late economist Hyman Minsky on the causes of the business cycle in the economy and booms and busts in the capital markets. Minsky believed that these cycles are caused by feedback loops. He noted that self-reinforcing phenomena cause "sudden jumps" Cooper relates these jumps to the fat tails that Benoît Mandlebrot had found in price change data and modeled using non-normal distributions Mandlebrot concurred with Cooper in this exchange (243-4).
- From Expected Returns to Geometric Returns: Mean-Variance optimization models emphasize expected returns for a single period: But most investors are interested in building wealth over time, so Kaplan suggests a shift towards geometric returns, which measures the long-term rate of portfolio growth. Geometric returns are always lower than expected returns, and geometric returns are much lower when the asset class is volatile. (The distinction is critical, so Kaplan offers a 5-page explanation in the introduction.)
- From Value at Risk to Conditional Value at Risk: Mean-Variance Optimization measures risk in terms of volatility, and it describes the potential for losses in terms of Value at Risk (VAR). If the VaR is $4,000, then the portfolio has a 5% chance of losing $4,000 or more during a given period of time. Conditional VaR takes this one step further by incorporating "or more" into the answer. Thus, CVaR answers a new question: "Given a 5% chance that things go wrong, what is the average loss of capital I should expect?" CVaR is always higher than VaR, and I believe it is a better way to prepare investors for the worst.
- From Co-Variance to a Scenario Approach: MVO uses a matrix of values to show the correlation among asset classes. For Markowitz 2.0, Kaplan replaces these with scenario analysis using Monte Carlo simulations. These allow for much more robust analysis in general, but is particularly useful when a model includes distributions being made during retirement. Portfolio losses are especially tough during the first few years of retirement, and a scenario approach can help model this risk. The result is often given in terms of probabilities such as: "With this portfolio, you have a 57% chance of outliving your money."
Putting It All Together
The four changes above help define the Markowitz 2.0 process. Kaplan illustrated this in chapter 26, which I have replicated below.
Source: Adapted from chart 26.3 on page 332.
This chart shows the geometric returns on the vertical axis. These are the long-term returns that will compound over time (rather than the arithmetic returns, which are higher, and potentially misleading when cited out of context). The chart shows the CVaR on the horizontal axis, and this illustrates how much capital the investor can lose with a given portfolio. In this illustration, a portfolio with a forecast long-term return of 8% would risk losing an average of 15% of its capital in a year in the worst 5% of possible annual returns.
Overall, the Markowitz 2.0 framework offers a more robust illustration of the trade-off between risk and return.
Additional disclosure: See here.