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PAPERS
The Myth of Diversification
D. Chua, M. Kritzman and S. Page
The Journal of Portfolio Management, Fall 2009
Perhaps the most universally accepted precept of prudent investing is to diversify, yet this precept grossly oversimplifies the challenge of portfolio construction. Correlations, as typically measured over the full sample of returns, often belie an asset’s diversification properties in market environments when diversification is most needed. Moreover, upside diversification is undesirable. The authors first describe the mathematics of conditional correlations assuming returns are normally distributed. Then they present empirical results across a wide variety of assets, which reveal that, unlike the theoretical conditional correlations, empirical correlations are significantly asymmetric. Finally, the authors show that a portfolio construction technique called full-scale optimization produces portfolios in which the component assets exhibit relatively lower correlations on the downside and higher correlations on the upside than mean-variance optimization portfolios.
Risk Budgets
G. Chow and M. Kritzman
The Journal of Portfolio Management, Winter 2001
A risk budget conveys the notion that a portfolio's risk is apportioned to various categories, but such disaggregation is impossible for many descriptions of risk. For example, a portfolio's value at risk depends in part on its standard deviation, which is not additive, and on interaction effects that cannot be disentangled. The authors explore these issues in the hope of arriving at a clear and uniform understanding of risk budgets and their role in the investment process. They begin with a review of value at risk, which is the typical unit of measurement that is addressed in a risk budget. Next they review several definitions of risk budgets that have been proposed by members of the academic and investment communities. This review reveals that in some instances risk budgets are applied inappropriately. They discuss the appropriate role of a risk budget and describe what properly should be called risk attribution.
Risk, Regimes and Overconfidence
M. Kritzman, K. Lowry and A-S VanRoyen
Revere Street Working Papers, 272-2
Investors typically think of risk as the uncertainty of wealth at the end of their investment horizon. By focusing on the dispersion of ending wealth, investors ignore the effect of interim losses, no matter how severe. Investors also measure risk as though returns come from a single regime, which may understate the likelihood and severity of interim losses. The authors argue that the perception of risk as fully represented by the distribution of terminal wealth, together with the assumption of a single regime, leads to overconfidence. They apply first-passage probabilities to compare the risk of loss during an investment period with the risk of loss at the end of a horizon. Application of a methodology to measure risk based on quiet or turbulent regimes shows the extent to which the traditional measurement of risk understates exposure to loss. The authors present a forecasting procedure to assess the relative likelihood of quiet and turbulent regimes, and show how to use this information to structure portfolios that are regime-sensitive.
Optimal Portfolios in Good Times and Bad
G. Chow, E. Jacquier, M. Kritzman and K. Lowry
Financial Analysts Journal, May/June 1999
Experience with emerging market investments and hedge funds highlights the fact that risk parameters are unstable. To address this problem, the authors introduce a procedure for identifying multivariate outliers and use them to estimate a new covariance matrix. They suggest that a covariance matrix estimated from outliers better characterizes a portfolio's riskiness during market turbulence than a full sample covariance matrix. They also introduce a procedure for blending an inside sample covariance matrix with one from an outlier sample. This procedure enables investors to express views about the likelihood of each risk regime and to differentiate their aversion to them. This framework collapses to the Markowitz mean-variance model if: 1) the probabilities of the inside and outlying covariance matrices are set equal to their empirical frequencies, 2) investors are equally averse to both risk regimes, and 3) the inside and outlying covariances are estimated around the full sample mean.
Risk Containment for Investors with Multivariate Utility Functions
M. Kritzman and D. Rich
The Journal of Derivatives, Spring 1998
There are many financial situations in which investors care about joint occurrences, such as when: 1) a manager is evaluated against both an absolute target and a relative target; 2) an investor seeks protection from currency losses only when they coincide with unfavorable returns from the underlying portfolio; and 3) an investor wishes to structure an incentive fee that is conditioned on the simultaneous attainment of two objectives. Conventional approaches to risk containment assume implicitly that investor utility depends on a single random variable, and that risk is defined as the variability of this random variable. Investor behavior suggests, however, that investors care about multiple dimensions of risk. This article develops a risk containment model in which investor utility is explicitly contingent on more than one random variable. The framework offers option-based hedging strategies that protect investors from the joint occurrence of negative outcomes. The model is also applied to incentive fees that are conditioned on more than one measure of performance. Finally, the authors combine these notions in order to engineer a hybrid collar that sacrifices concurrent favorable outcomes to finance protection from concurrent negative outcomes.
The Mismeasurement of Risk
M. Kritzman and D. Rich
Financial Analysts Journal, May/June 2002
Investors typically measure risk as the probability of a given loss or the amount that can be lost with a given probability at the end of their investment horizon. This view of risk only considers the result at the end of the investment horizon, whether the horizon lasts for one day, one week, one year, or many years. It ignores what might happen along the way. The authors argue that investors perceive risk differently. They care about exposure to loss throughout their investment horizon and not just at its conclusion. They introduce two new risk measures: within horizon probability of loss and continuous value at risk. These new risk measures reveal that exposure to loss is substantially greater than investors normally assume.
Re-engineering Investment Management
M. Kritzman and L. Thomas
The Journal of Portfolio Management, September 2004
The conventional approach to building portfolios of investments is inefficient. The usual hierarchy of investment decisions imposes constraints on active management which are, under most circumstances, unnecessary and produce mean-variance inefficient portfolios. We propose a new approach to portfolio design, which eliminates these harmful constraints. It is based on the concept of alpha portability, and also relies on an analogous idea, beta portability. We present our position both conceptually and mathematically, and we illustrate the benefit of our approach with numerical examples. Our analysis reveals that alpha independence and separation are necessary but insufficient conditions for mean-variance efficiency. These features must be combined with leverage to effect portability and achieve the best expected outcome.
Optimal Hedge Fund Allocations: Do Higher Moments Matter?
J. Cremers, M. Kritzman, S. Page
Revere Street Working Papers, 272-13
Hedge funds have return peculiarities not commonly associated with traditional investment vehicles. Specifically, hedge funds seem more inclined to produce return distributions with significantly non-normal skewness and kurtosis. There is also growing acceptance of the notion that investor preferences are better represented by bilinear utility functions or S-shaped value functions than by neo-classical utility functions such as power utility. Many investors have therefore concluded that mean-variance optimization is not appropriate for forming portfolios that include hedge funds. We apply both mean-variance and full-scale optimization to form portfolios of hedge funds, given a wide range of assumptions about investor preferences. We find that higher moments of hedge funds do not meaningfully compromise the efficacy of mean-variance optimization if investors have power utility. We also find, however, that mean-variance optimization is not particularly effective for identifying optimal hedge fund allocations if preferences are bilinear or S-shaped. Finally, we show that investors with bilinear utility dislike kurtosis and that, contrary to conventional wisdom, investors with S-shaped preferences are attracted to kurtosis as well as negative skewness. Mean-variance optimization is insensitive to these preferences.
Mean-Variance Analysis versus Full-Scale Optimization
T. Adler and M. Kritzman
Mean-variance versus full-scale optimisation: In and out of sample, Journal of Asset Management, Vol. 7, 5, 302–311, 2007
For three decades, mean-variance analysis has served as the standard procedure for constructing portfolios. Recently, investors have experimented with a new optimization procedure, called full-scale optimization, to address certain limitations of mean-variance optimization. Specifically, mean-variance optimization assumes that returns are normally distributed or that investor preferences are well approximated by mean and variance. Full-scale optimization relies on sophisticated search algorithms to identify the optimal portfolio given any set of return distributions and based on any description of investor preferences. Full-scale optimization yields the truly optimal portfolio in sample, whereas the mean-variance solution is an approximation to the insample truth. Both approaches to portfolio formation, however, suffer from estimation error. Mean-variance analysis requires investors to estimate the means and variances of all assets and the covariances of all asset pairs. To the extent the out-of-sample experience of these parameters departs from the in-sample parameter values, the mean-variance approximation will be even less accurate. Full-scale optimization requires investors to estimate the entire multivariate return distribution. To the extent it varies from the insample distribution, full-scale optimization will also yield sub-optimal results out of sample. We employ a bootstrapping procedure to compare the estimation error of fullscale optimization to the combined approximation and estimation error of mean-variance analysis. We find that, to a significant degree, the in-sample superiority of full-scale optimization prevails out-of-sample.
Are Optimizers Error Maximizers: Hype Versus Reality?
M. Kritzman
The Journal of Portfolio Management, Summer 2006
Small input errors to mean-variance optimizers often lead to large portfolio misallocations when the assets are close substitutes for one another, which is why optimizers are sometimes referred to as error maximizers. When the assets are close substitutes, however, the return distribution of the presumed optimal portfolio is similar to the distribution of the truly optimal portfolio. Contrary to conventional wisdom, therefore, mean-variance optimizers are usually robust to small input errors when sensitivity is measured properly.