averages are poor forecasts because they fail to capture this time variation.
But extrapolating from a short sample of recent history is ineffective because it introduces noise and assumes a level of persistence that does not reliably occur.
As an alternative, investors often regress these asset class variables on economic variables to derive forecasts, but this approach may not help because additional variables contribute noise along with information.
The investor's challenge is to maximize the information content and minimize the noise, thereby generating the most reliable predictions.
The prediction from a linear regression equation can be equivalently expressed as a weighted average of the past values of the dependent variable in which the weights are the relevance of the past observations of the independent variables.
Within this context, relevance has a precise mathematical meaning. It is the sum of statistical similarity and informativeness.
Statistical similarity equals the negative of the Mahalanobis distance of the past values for the independent variables from their current values, and informativeness equals the Mahalanobis distance of the past values of the independent variables from their average values. In other words, prior periods that are more like the current period but different from the historical average are more relevant than those that are not.
Investors may be able to improve the reliability of their forecasts by filtering historical observations for their relevance and using this mathematical equivalence to produce new forecasts.
Chapter 14: The Stock–Bond Correlation
Investors rely on the stock–bond correlation to construct optimal portfolios and to assess risk.
Investors should care less about how stocks and bonds co-move from month to month as they do about their co-movement over the duration of their investment horizon.
The most common approach to estimating the longer-term correlation of stocks and bonds is to extrapolate the correlation of monthly returns over a prior period. This approach is decidedly unreliable, because the autocorrelations and lagged cross-correlations of stock and bond returns are nonzero.
As an alternative, investors may consider estimating the stock–bond correlation from longer-horizon returns, but this approach is unreliable because the stock–bond correlation varies over time.
To address these problems, this chapter introduces the notion of a single-period correlation that measures the extent to which stock and bond returns move synchronously or drift apart over the course of the investment horizon.
In addition, this chapter introduces several fundamental variables to predict the longer-horizon stock–bond correlation, some of which are expressed as paths rather than as single-period average values.
This chapter also describes how to filter historical observations for their historical relevance, as discussed more fully in Chapter 13.
Together, these innovations significantly improve the reliability of the forecast of the stock–bond correlation.
Chapter 15: Constraints
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.
Chapter 16: Asset Allocation Versus Factor Investing
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.
Chapter 17: Illiquidity
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.
Chapter 18: Currency 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.
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