How to manage portfolio risk and return

Portfolio management aims to take positions in financial instruments that achieve the desired risk-return trade-off regarding a benchmark. In each period, a manager selects positions that optimize diversification to reduce risks while achieving a target return. Across periods, the positions will be rebalanced to account for changes in weights resulting from price movements to achieve or maintain a target risk profile.

Diversification permits us to reduce risks for a given expected return by exploiting how price movements interact with each other as one asset's gains can make up for another asset's losses. Harry Markowitz invented Modern Portfolio Theory (MPT) in 1952  and provided the mathematical tools to optimize diversification by choosing appropriate portfolio weights. Markowitz showed how portfolio risk, measured as the standard deviation of portfolio returns, depends on the covariance among the returns of all assets and their relative weights. This relationship implies the existence of an efficient frontier of portfolios that maximize portfolio returns given a maximal level of portfolio risk.

However, mean-variance frontiers are highly sensitive to the estimates of the input required for their calculation, such as expected returns, volatilities, and correlations. In practice, mean-variance portfolios that constrain these input to reduce sampling errors have performed much better. These constrained special cases include equal-weighted, minimum-variance, and risk-parity portfolios.

The Capital Asset Pricing Model (CAPM) is an asset valuation model that builds on the MPT risk-return relationship. It introduces the concept of a risk premium that an investor can expect in market equilibrium for holding a risky asset; the premium compensates for the time value of money and the exposure to overall market risk that cannot be eliminated through diversification (as opposed to the idiosyncratic risk of specific assets). The economic rationale for non-diversifiable risk is, for example, macro drivers of the business risks affecting equity returns or bond defaults. Hence, an asset's expected return, E[ri], is the sum of the risk-free interest rate, rf, and a risk premium proportional to the asset's exposure to the expected excess return of the market portfolio, rmover the risk-free rate:

In theory, the market portfolio contains all investable assets and will be held by all rational investors in equilibrium. In practice, a broad value-weighted index approximates the market, for example, the S&P 500 for US equity investments. βmeasures the exposure to the excess returns of the market portfolio. If the CAPM is valid, the intercept component, αi, should be zero. In reality, the CAPM assumptions are often not met, and alpha captures the returns left unexplained by exposure to the broad market.

Over time, research uncovered non-traditional sources of risk premiums, such as the momentum or the equity value effects that explained some of the original alpha. Economic rationales, such as behavioral biases of under or overreaction by investors to new information justify risk premiums for exposure to these alternative risk factors. They evolved into investment styles designed to capture these alternative betas that also became tradable in the form of specialized index funds. After isolating contributions from these alternative risk premiums, true alpha becomes limited to idiosyncratic asset returns and the manager's ability to time risk exposures.

The EMH has been refined over the past several decades to rectify many of the original shortcomings of the CAPM, including imperfect information and the costs associated with transactions, financing, and agency. Many behavioral biases have the same effect, and some frictions are modeled as behavioral biases.

ML plays an important role in deriving new alpha factors using supervised and unsupervised learning techniques based on the market, fundamental, and alternative data sources discussed in the previous chapters. The inputs to a machine learning model consist of both raw data and features engineered to capture informative signals. ML models are also used to combine individual predictive signals and deliver higher-aggregate predictive power.

Modern portfolio theory and practice have evolved significantly over the last several decades. We will introduce:

  • Mean-variance optimization, and its shortcomings
  • Alternatives such as minimum-risk and 1/n allocation
  • Risk parity approaches
  • Risk factor approaches
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