Chapter 1: What Is an Asset Class?

• The composition of an asset class should be stable.
• The components of an asset class should be directly investable.
• The components of an asset class should be similar to each other.
• An asset class should be dissimilar from other asset classes in the portfolio as well as combinations of the other asset classes.
• The addition of an asset class to a portfolio should raise its expected utility.
• An asset class should not require selection skill to identify managers within the asset class.
• An asset class should have capacity to absorb a meaningful fraction of a portfolio in a cost‐effective manner.

Chapter 2: Fundamentals of Asset Allocation

• A portfolio's expected return is the weighted average of the expected returns of the asset classes within it.
• Expected return is measured as the arithmetic average, not the geometric average.
• A portfolio's risk is measured as the variance of returns or its square root, standard deviation.
• Portfolio risk must account for how asset classes co‐vary with one another.
• Portfolio risk is less than the weighted average of the variances or standard deviations of the asset classes within it.
• Diversification cannot eliminate portfolio variance entirely. It can only reduce it to the average covariance of the asset classes within it.
• The efficient frontier comprises portfolios that offer the highest expected return for a given level of risk.
• The optimal portfolio balances an investor's goal to increase wealth with the investor's aversion to risk.
• Mean‐variance analysis is an optimization process that identifies efficient portfolios. It is remarkably robust. It delivers the correct result if returns are approximately elliptically distributed, which holds for return distributions that are not skewed, have stable correlations, and comprise asset classes with relatively uniform kurtosis, or if investor preferences are well described by mean and variance.

Chapter 3: The Importance of Asset Allocation

• It is commonly assumed that asset allocation explains more than 90 percent of investment performance.
• This belief is based on flawed analysis by Brinson, Hood, and Beebower.
• The analysis is flawed because it implicitly assumes that the default portfolio is not invested.
• Also, this study, as well as many others, analyzes actual investment choices rather than investment opportunity. By analyzing actual investment choices, these analyses confound the natural importance of an investment activity with an investor's choice to emphasize that activity.
• Bootstrap simulation of the potential range of outcomes associated with asset allocation and security selection reveals that security selection has as much or more potential to affect investment performance as asset allocation does.

Chapter 4: Time Diversification

• It is widely assumed that investing over long horizons is less risky than investing over short horizons, because the likelihood of loss is lower over long horizons.
• Paul A. Samuelson showed that time does not diversify risk because, though the probability of loss decreases with time, the magnitude of potential losses increases with time.
• It is also true that the probability of loss within an investment horizon never decreases with time.
• Finally, the cost of a protective put option increases with time to expiration. Therefore, because it costs more to insure against losses over longer periods than shorter periods, it follows that risk does not diminish with time.

Chapter 5: Error Maximization

• Some investors believe that optimization is hypersensitive to estimation error because, by construction, optimization overweights asset classes for which expected return is overestimated and risk is underestimated, and it underweights asset classes for which the opposite is true.
• We argue that optimization is not hypersensitive to estimation error for reasonably constrained portfolios.
• If asset classes are close substitutes for each other, it is true that their weights are likely to change substantially given small input errors, but because they are close substitutes, the correct and incorrect portfolios will have similar expected returns and risk.
• If asset classes are dissimilar from each other, small input errors will not cause significant changes to the correct allocations; thus, again the correct and incorrect portfolios will have similar expected returns and risk.

Chapter 6: Factors

• Some investors believe that factors offer greater potential for diversification than asset classes because they appear less correlated than asset classes.
• Factors appear less correlated only because the portfolio of assets designed to mimic them includes short positions.
• Given the same constraints and the same investable universe, it is mathematically impossible to regroup assets into factors and produce a better efficient frontier.
• Some investors also believe that consolidating a large group of securities into a few factors reduces noise more effectively than consolidating them into a few asset classes.
• Consolidation reduces noise around means but no more so by using factors than by using asset classes.
• Consolidation does not reduce noise around covariances.

Chapter 7: 1/N

• It has been argued that equally weighted portfolios perform better out of sample than optimized portfolios.
• The evidence for this result is misleading because it relies on extrapolation of historical means from short samples to estimate expected return. In some samples, the historical means for riskier assets are lower than the historical means for less risky assets, implying, contrary to reason, that investors are occasionally risk seeking.
• Optimization with plausible estimates of expected return reliably performs better than equal weighting.
• Also, equal weighting limits the investor to a single portfolio, regardless of the investor's risk tolerance, whereas optimization offers a wide array of investment choices.

Chapter 8: Necessary Conditions for Mean‐Variance Analysis

• It is a widely held view that the validity of mean‐variance analysis requires investors to have quadratic utility and that returns are normally distributed. This view is incorrect.
• Mean‐variance analysis is precisely equivalent to expected utility maximization if returns are elliptically distributed, of which the normal distribution is a more restrictive special case, or (not “and”) if investors have quadratic utility.
• For practical purposes, mean‐variance analysis is an excellent approximation to expected utility maximization if returns are approximately elliptically distributed or investor preferences can be well described by mean and variance.
• For intuition of an elliptical distribution, consider a scatter plot of the returns of two asset classes. If the returns are evenly distributed along the boundaries of concentric ellipses that are centered on the average of the return pairs, the distribution is elliptical. This is usually true if the distribution is symmetric, kurtosis is relatively uniform across asset classes, and the correlation of returns is reasonably stable across subsamples.
• For a given elliptical distribution, the relative likelihood of any multivariate return can be determined using only mean and variance.
• Levy and Markowitz have shown using Taylor series approximations that power utility functions, which are always upward sloping, can be well approximated across a wide range of returns using just mean and variance.
• In rare circumstances, in which returns are not elliptical and investors have preferences that cannot be approximated by mean and variance, it may be preferable to employ full‐scale optimization to identify the optimal portfolio.
• Full‐scale optimization is a numerical process that evaluates a large number of portfolios to identify the optimal portfolio, given a particular utility function and return sample. For example, full‐scale optimization can accommodate a kinked utility function to reflect an investor's strong aversion to losses that exceed a particular threshold.

Chapter 9: Constraints

• Investors constrain allocation to certain asset classes because they do not want to perform poorly when other investors perform well.
• Constraints are inefficient because, of necessity, they are arbitrary.
• Investors can derive more efficient portfolios by expanding the optimization objective function to include aversion to tracking error as well as aversion to absolute risk.
• Mean‐variance‐tracking error optimization produces an efficient surface in dimensions of expected return, standard deviation, and tracking error.
• This approach usually delivers a more efficient portfolio in three dimensions than constrained mean‐variance analysis.

Chapter 10: Currency Risk

• Investors improve portfolio efficiency by optimally hedging a portfolio's currency exposure.
• Linear hedging strategies use forward or futures contracts to offset currency exposure. They hedge both upside returns and downside returns. They are called linear hedging strategies because the portfolio's returns are a linear function of the hedged currencies' returns.
• Investors can reduce risk more effectively by allowing currency‐specific hedging, cross‐hedging, and overhedging.
• Nonlinear hedging strategies use put options to protect a portfolio from downside returns arising from currency exposure while allowing it to benefit from upside currency returns. They are called nonlinear hedging strategies because the portfolio's returns are a nonlinear function of the hedged currencies' returns.
• Nonlinear hedging strategies are more expensive than linear hedging strategies because they preserve the upside potential of currencies.
• A basket option is an option on a portfolio of currencies and therefore provides protection against a collective decline in currencies.
• A portfolio of options offers protection against a decline in any of a portfolio's currencies.
• A basket option is less expensive than a portfolio of options because it offers less protection.

Chapter 11: Illiquidity

• Investors rely on liquidity to implement tactical asset allocation decisions, to rebalance a portfolio, and to meet demands for cash, among other uses.
• In order to account for the impact of liquidity, investors should attach a shadow asset to liquid asset classes in a portfolio that enable investors to use liquidity to increase a portfolio's expected utility, and they should attach a shadow liability to illiquid asset classes in a portfolio that prevent an investor from preserving a portfolio's expected utility.
• These shadow allocations allow investors to address illiquidity within a single, unified framework of expected return and risk.

Chapter 12: Risk in the Real World

• Investors typically evaluate exposure to loss based on a portfolio's full‐sample distribution of returns at the end of their investment horizon.
• However, investors care about what happens throughout their investment horizon and not just at its conclusion.
• They also recognize that losses are more common when markets are turbulent than when they are calm.
• First passage time probabilities enable investors to estimate probability of loss and value at risk throughout their investment horizon.
• The Mahalanobis distance allows investors to distinguish between calm and turbulent markets.
• Investors can assess risk more realistically by applying first passage time probabilities to the returns that prevailed during turbulent subsamples.

Chapter 13: Estimation Error

• When investors estimate asset class covariances from historical returns, they face three types of estimation error: small‐sample error, independent‐sample error, and interval error.
• Small‐sample error arises because the investor's investment horizon is typically shorter than the historical sample from which covariances are estimated.
• Independent‐sample error arises because the investor's investment horizon is independent of history.
• Interval error arises because investors estimate covariances from higher‐frequency returns than the return frequency they care about. If returns have nonzero autocorrelations, standard deviation does not scale with the square root of time. If returns have nonzero autocorrelations or nonzero lagged cross‐correlations, correlation is not invariant to the return interval used to measure it.
• Common approaches for controlling estimation error, such as Bayesian shrinkage and resampling, make portfolios less sensitive to estimation error.
• A new approach, called stability‐adjusted optimization, assumes that some covariances are reliably more stable than other covariances. It delivers portfolios that rely more on relatively stable covariances and less on relatively unstable covariances.

Chapter 14: Leverage versus Concentration

• Theory shows that it is more efficient to raise a portfolio's expected return by employing leverage rather than concentrating the portfolio in higher‐expected return asset classes.
• The assumptions that support this theoretical result do not always hold in practice.
• If we collectively allow for asymmetric preferences, nonelliptical returns, and realistic borrowing costs, it may be more efficient to raise expected return by concentrating a portfolio in higher‐expected‐return asset classes than by using leverage.
• However, if we also assume that an investor has even a modest amount of skill in predicting asset class returns, then leverage is better than concentration even in the presence of asymmetric preferences, nonelliptical distributions, and realistic borrowing costs.

Chapter 15: Rebalancing

• Investors typically rebalance a portfolio whose weights have drifted away from its optimal targets based on the passage of time or distance from the optimal targets.
• Investors should approach rebalancing more rigorously by recognizing that the decision to rebalance or not affects the choices the investor will face in the future.
• Dynamic programming can be used to determine an optimal rebalancing schedule that explicitly balances the cost of transacting with the cost of holding a suboptimal portfolio.
• Unfortunately, dynamic programming can only be applied to portfolios with a few asset classes because it suffers from the curse of dimensionality.
• For portfolios with more than just a few asset classes, investors should use a quadratic heuristic developed by Harry Markowitz and Erik van Dijk, which easily accommodates several hundred assets.

Chapter 16: Regime Shifts

• Rather than characterizing returns as coming from a single, stable regime, it might be more realistic to assume they are generated by disparate regimes such as a calm regime and a turbulent regime.
• Investors may wish to build portfolios that are more resilient to turbulent regimes by employing stability‐adjusted optimization, which relies more on relatively stable covariances than unstable covariances, or by blending the covariances from calm and turbulent subsamples in a way that places greater emphasis on covariances that prevailed during turbulent regimes.
• These approaches produce static portfolios, which still display unstable risk profiles.
• Investors may instead prefer to manage a portfolio's asset mix dynamically, by switching to defensive asset classes during turbulent periods and to aggressive asset classes during calm periods.
• It has been shown that hidden Markov models are effective at distinguishing between calm and turbulent regimes by accounting for the level, volatility, and persistence of the regime characteristics.