Investors constrain their 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 the dimensions of expected return, standard deviation, and tracking error.

       This approach usually delivers a portfolio that is more efficient in three dimensions than the portfolio that is produced by constrained mean-variance analysis.

       Some investors prefer to construct portfolios from asset classes because asset classes are readily observable and directly investable.

       Other investors prefer to allocate to factors because they believe asset classes are defined arbitrarily and do not capture the fundamental determinants of performance as directly as factors do. Also, some factors carry risk premiums that are not directly available from asset classes.

       Investors can have it both ways by continuing to invest in asset classes but augmenting the Markowitz objective function to include a term that penalizes deviation from a desired factor profile.

       Investors rely on liquidity to implement tactical asset allocation decisions, to rebalance a portfolio, and to meet demands for cash, among other uses.

       To account for the impact of liquidity, investors should attach a shadow asset to the liquid asset classes in a portfolio that enable them 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 them 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.

       Investors may improve portfolio efficiency by optimally hedging a portfolio's currency exposure.

       Linear hedging strategies use forward or futures contracts to offset cur- rency 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. These strategies retain exposure to currencies that diversify the portfolio and reduce exposure to currencies that do not.

       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.

       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, the 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 to 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.

       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 con- centration even in the presence of asymmetric preferences, nonelliptical distributions, and realistic borrowing costs.

       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.