Classic Econometric Models. Part 3: Portfolio and Risk

Exploring the foundations of portfolio optimization

In this third part of the series on econometric models, I will explore Portfolio and Risk models. The previous parts discussed modeling returns, volatility, and probability. This section focuses on classic models for portfolio construction and risk management, which, like the earlier topics, form the foundation for developing algorithmic trading strategies.

Photo by Dave Hoefler on Unsplash

Portfolio Models

CAPM (Capital Asset Pricing Model)

One of the most widely recognized portfolio models is the Capital Asset Pricing Model (CAPM; Sharpe, 1964). This model helps measure the performance of a portfolio at a given level of risk by examining the relationship between expected return and systematic risk through the beta coefficient. The beta reflects the sensitivity of the portfolio to market movements, providing a framework for understanding risk-adjusted returns. CAPM is useful for estimating the cost of equity and evaluating whether a security is fairly valued based on its risk-return profile.

Extensions of CAPM include the Zero-Beta CAPM (Black & Fischer, 1972), which addresses portfolios with no systematic market risk to aid diversified investors, and the Intertemporal CAPM (ICAPM; Merton, 1973), which spans multiple periods to account for dynamic changes in risk preferences and investment opportunities See Fama & French (2004), Campbell et al. (1997) or Paleologo, 2021 for more in-depth introduction into related topics.

Arbitrage Pricing Theory (APT)

The Arbitrage Pricing Theory (Ross, 1976) extends CAPM by incorporating multiple risk factors. Unlike CAPM, APT does not assume a single market factor but instead allows for several macroeconomic or firm-specific factors that influence asset prices. This flexibility makes APT a powerful tool for explaining returns in multi-dimensional risk environments. APT assumes no arbitrage opportunities, relying on linear factor models to capture relationships between returns and risk drivers.

Multifactor Models

Factor models are an extension of CAPM, introducing additional risk factors that better explain asset returns. These models decompose returns into contributions from various systematic factors, such as market risk, size, value, and momentum.

Key multifactor models extend CAPM by adding systematic factors such as size, value, profitability, investment, and momentum. Examples include the Fama-French models, which add size and value factors (Three-Factor; Fama & French 1993) and extend further with profitability and investment factors (Five-Factor; Fama & French 2015), the Carhart (1997) model with momentum, and Campbell’s models, which incorporate time-varying risks and economic indicators (Campbell & Shiller, 1988).

Black-Litterman Model

The Black-Litterman (1992) model combines CAPM principles with Bayesian statistics to improve portfolio optimization. It allows investors to use their subjective views on expected returns into the optimization process, balancing these views with market equilibrium assumptions. The model generates stable, intuitive portfolios by addressing shortcomings of traditional mean-variance optimization.

Mean-Variance Optimization (Markowitz Portfolio Theory)

Markowitz’s (1952) Mean-Variance Optimization remains a base of modern portfolio theory. The model aims to minimize risk for a given level of return or maximize return for a given level of risk. By constructing efficient portfolios that lie on the efficient frontier, investors can make informed trade-offs between risk and return.

Factor Analysis and PCA

Factor analysis, including Principal Component Analysis (PCA; Jolliffe, 1986), is often employed to identify the underlying factors driving asset returns. PCA simplifies complex datasets by reducing them to their principal components, helping determine the number of relevant factors in a multifactor model.

Modern Portfolio Approaches

Modern portfolio management approaches include Post-Modern Portfolio Theory (PMPT), which prioritizes minimizing downside risk, Risk Parity, which balances risk across assets, and Efficient Frontier Analysis, which identifies portfolios that optimize the trade-off between risk and return (see Sortino & van der Meer, 1991; Asness et al. 2012).

Risk Models

Risk models provide frameworks for quantifying and managing uncertainty in financial portfolios. Many models overlap with those in portfolio management, emphasizing the role of volatility (calculated differently, for example using GARCH models) as a primary indicators of risk. Below are key categories and models used in risk assessment.

GARCH (Generalized Autoregressive Conditional Heteroskedasticity)

GARCH models are integral to risk modeling due to their ability to capture time-varying volatility. By modeling the conditional variance of returns, GARCH provides insights into periods of heightened risk, which are critical for forecasting and stress testing. I have covered GARCH and other volatility models in the previous section of the series (see Teräsvirta, 2009).

Value at Risk (VaR)

Value at Risk estimates the potential loss in portfolio value over a specified period with a given confidence level. Common tolerance levels are set at 5% or 1%, as guided by the Basel Accords (which currently recommends the more conservative 1% threshold). VaR methodologies encompass approaches such as Variance-Covariance VaR, which assumes a normal distribution for returns; Monte Carlo Simulation VaR, which explores potential outcomes through simulation; and Historical Simulation VaR, which leverages past market data to estimate risks (see Holton, 2003; Jorion 2007).

Conditional Value at Risk (CVaR or Expected Shortfall)

CVaR, also known as Expected Shortfall, measures the expected loss that exceeds the VaR threshold, offering a deeper understanding of tail risk by focusing on the severity of losses in extreme scenarios. Unlike VaR, which only provides a cutoff point, CVaR calculates the average loss beyond this threshold, making it particularly effective for capturing risks in volatile or non-normal distributions. Its more comprehensive nature makes CVaR a preferred metric in regulatory stress testing and portfolio optimization, as it aligns closely with the risk aversion of investors and regulatory standards. Additionally, CVaR can be extended to multi-asset portfolios, allowing for the evaluation of aggregate risks across diverse asset classes (see Artzner 1999; Acerbi 2001).

Risk Metrics Framework

This standardized approach to portfolio risk measurement includes robust tools for assessing volatility, correlation, and exposure, ensuring a consistent and comprehensive framework for evaluating portfolio risk. The RiskMetrics methodology laid the groundwork for many modern risk practices and also introduced advanced quantitative techniques, such as stress testing frameworks, and standardized risk factor modeling, that continue to influence the development of risk assessment models, making it a fundamental approach in the field of financial risk management (see J.P. Morgan & Co., 1996).

Other Risk Measures

Additional models address specific dimensions of risk, including CreditMetrics, which evaluates default risk; Portfolio Beta Analysis, measuring systematic risk relative to the market; previously mentioned Sortino Ratio, emphasizing downside risk over total volatility; Liquidity at Risk (LaR), assessing potential liquidity shortfalls during stress; and Drawdown Analysis, which examines peak-to-trough losses to highlight portfolio vulnerabilities (see Crouhy et al., 2000 for more on the topic).

Summary

While this selection of models doesn’t cover all that is currently available, it provides a solid foundation for understanding the most widely used tools in this area. If you think I’ve missed something important or have suggestions, feel free to share—I may consider creating a more in-depth follow up article. Thanks for reading, and see you next time!

References

Acerbi, C., Nordio, C., & Sirtori, C. (2001). Expected shortfall as a tool for financial risk management. arXiv. arXiv:cond-mat/0102304

Artzner, P., Delbaen, F, Eber, JM., and Heath, D. (1999). Coherent Risk Measures. Mathematical Finance, Vol. 9, No. 3, pp. 203–228.

Asness, C. S., Frazzini, A., & Pedersen, L. H. (2012). Leverage Aversion and Risk Parity. Financial Analysts Journal, 68(1), pp. 47–59.

Black, F. (1972). Capital Market Equilibrium with Restricted Borrowing. The Journal of Business, Vol. 45, No. 3, pp. 444–455.

Black, F., and Litterman, R. (1992). Global Portfolio Optimization. Financial Analysts Journal, Vol. 48, No. 5, pp. 28–43.

Box, G. E. P., & Jenkins, G. M. (1970). Time series analysis: forecasting and control. Holden-Day.

Campbell, J. Y., Lo, A. W., & MacKinlay, A. C. (1997). The Econometrics of Financial Markets. Princeton University Press.

Campbell, J. Y., and Shiller, R. J. (1988). The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors. The Review of Financial Studies, Vol. 1, No. 3, pp. 195–228.

Carhart, M. (1997). On Persistence in Mutual Fund Performance. The Journal of Finance, Vol. 52, No. 1, pp. 57–82.

Crouhy, M., Galai, D., & Mark, R. (2000). Risk Management: The New Approach. McGraw-Hill.

Fama, E. F., and French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, Vol. 33, No. 1, pp. 3–56.

Fama, E., F., and K. R. French. 2004. The Capital Asset Pricing Model: Theory and Evidence. Journal of Economic Perspectives, 18 (3): 25–46.

Fama, E. F., and French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, Vol. 116, No. 1, pp. 1–22.

Holton, G. A. (2003). Value-at-Risk: Theory and Practice. Academic Press.

J.P. Morgan & Co. (1996). RiskMetrics – Technical Document. J.P. Morgan & Co.

Jolliffe, I. T. (1986). Principal Component Analysis. Springer Series in Statistics. Springer-Verlag.

Jorion, P. (2007). Value at Risk: The New Benchmark for Managing Financial Risk. McGraw-Hill.

Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, Vol. 7, No. 1, pp. 77–91.

Merton, R. C. (1973). An Intertemporal Capital Asset Pricing Model. Econometrica, Vol. 41, No. 5, pp. 867–887.

Paleologo, G. A. (2021). Advanced Portfolio Management: A Quantitative Approach for Producing Superior Returns and Controlling Risk. Wiley.

Ross, S. A. (1976). The Arbitrage Theory of Capital Asset Pricing. Journal of Economic Theory, Vol. 13, No. 3, pp. 341–360.

Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, Vol. 19, No. 3, pp. 425–442.

Sortino, F. A., and van der Meer, Richard (1991). Downside Risk. The Journal of Portfolio Management, Vol. 17, No. 4, pp. 27–31.

Teräsvirta, T. (2009). An Introduction to Univariate GARCH Models. In T. Mikosch, J.-P. Kreiß, R. A. Davis, & T. G. Andersen (Eds.), Handbook of Financial Time Series (pp. 17–42). Springer.

Tsay, R. S. (2010). Analysis of Financial Time Series. John Wiley & Sons.