Introduction: Portfolio Choice with Mean-Variance Preferences
Portfolio theory, pioneered by Harry Markowitz in the 1950s, revolutionized the field of finance by introducing a systematic approach to portfolio construction. The fundamental principle behind portfolio theory is that investors can achieve an optimal balance between risk and return by diversifying their investments. By considering mean-variance preferences, investors can identify the most efficient combination of assets that maximizes expected returns for a given level of risk or minimizes risk for a given level of expected return.
Portfolios with Two Assets
A simple case of portfolio theory involves combining two risky assets. By analyzing the correlation between the assets and their individual risk-return profiles, investors can achieve diversification benefits. When the correlation between the two assets is less than perfect, the combined portfolio's risk is lower than the weighted average of the individual asset risks.
Portfolio Frontier with Multiple Risky Assets
As the number of risky assets in a portfolio increases, the portfolio frontier, or efficient frontier, represents the set of optimal portfolios that offer the highest expected return for a given level of risk. The frontier is convex, indicating that there are diminishing marginal benefits of diversification as more assets are added to the portfolio.
Portfolio Choice with a Safe Asset
When a risk-free asset, such as a government bond, is included in the portfolio, the efficient frontier becomes a straight line originating from the risk-free rate on the vertical axis. This line is called the capital market line (CML), and it represents the maximum Sharpe ratio achievable at any given level of risk.
Analytics of the Portfolio Frontier
The portfolio frontier is derived from the optimization problem of maximizing the Sharpe ratio, subject to the constraints of the investors' risk tolerance and the available investment opportunities. The analytics of the portfolio frontier involve solving a quadratic programming problem, which can be computationally intensive for large portfolios.
Properties of the Tangency Portfolio
The tangency portfolio is the portfolio on the efficient frontier that has the highest Sharpe ratio, i.e., the point where the CML is tangent to the efficient frontier. This portfolio offers the best risk-return trade-off and is considered the optimal portfolio for all investors.
Optimality of the Tangency Portfolio
The tangency portfolio is optimal because it maximizes the Sharpe ratio and offers the best risk-return trade-off. In other words, no other portfolio can provide a higher expected return per unit of risk.
Non-Mean-Variance Objective
Some investors may have objectives other than mean-variance optimization, such as utility maximization, minimizing the probability of loss, or considering non-financial criteria like environmental, social, and governance (ESG) factors. Portfolio theory can be adapted to accommodate these objectives, but the optimal portfolio may differ from the tangency portfolio.
Capital Asset Pricing Model (CAPM)
The CAPM, developed by Sharpe, Lintner, and Mossin, is an extension of portfolio theory that describes the relationship between an asset's expected return and its beta, a measure of systematic risk. The model asserts that an asset's expected return is equal to the risk-free rate plus a risk premium proportional to its beta.
Risk and Return under CAPM
Under CAPM, the total risk of an asset is divided into systematic risk (market risk) and unsystematic risk (firm-specific risk). Investors are rewarded for taking systematic risk, as reflected by an asset's beta, but not for taking unsystematic risk, which can be eliminated through diversification.
CAPM Pros and Cons
The CAPM has several advantages, including its simplicity, wide applicability, and usefulness as a benchmark for performance evaluation. However, it also has limitations, such as its reliance on several assumptions, including investors holding well-diversified portfolios, frictionless markets, and homogeneous investor expectations. Empirical evidence has revealed some deviations from the model's predictions, but the CAPM remains a fundamental tool in finance.
CAPM vs. APT
The Arbitrage Pricing Theory (APT), proposed by Stephen Ross, is an alternative to the CAPM that considers multiple sources of systematic risk. APT assumes that asset returns are driven by a set of factors, and investors can construct well-diversified portfolios to eliminate unsystematic risk.
For Portfolio Management
Both CAPM and APT are useful for portfolio management, but they differ in their assumptions and methods. While CAPM focuses on a single systematic risk factor, APT considers multiple factors. APT may provide a more comprehensive view of risk, but it can be more complex to implement.
For Capital Budgeting
In capital budgeting, both models are used to estimate the cost of equity. While CAPM is more popular due to its simplicity and intuitive appeal, APT may offer a more nuanced understanding of risk factors affecting a project's cash flows.
Compared to Statistical Averages of Expected Returns
Both CAPM and APT are based on theoretical foundations, while statistical averages of expected returns rely on historical data. The latter approach may not account for changes in market conditions or risk factors and can be less effective in predicting future returns.
Security Market Line
The Security Market Line (SML) is a graphical representation of the CAPM, showing the relationship between an asset's expected return and its beta. The SML slopes upward, indicating that higher beta assets have higher expected returns.
Why is the Security Market Line Flat in Empirical Data?
The SML may appear flat in empirical data due to various market anomalies, such as the low-beta anomaly, which suggests that low-beta stocks outperform their high-beta counterparts, contrary to the CAPM predictions.
Empirical Failures of CAPM
Some empirical evidence challenges the CAPM, such as the existence of size and value effects, which suggest that small-cap and value stocks outperform the market on a risk-adjusted basis.
CAPM Violations Tend to be Short Lived
Although there are empirical deviations from the CAPM, these violations tend to be short-lived, as market participants often exploit and correct these anomalies over time.
CAPM is the Leading Model for Capital Budgeting
Despite its limitations, the CAPM remains the leading model for capital budgeting, as it provides a simple and intuitive framework for estimating the cost of equity and evaluating investment projects. However, for investment portfolio analysis one likely would be better served by APT or the Black-Literman model due to empirically observed shortcomings of the CAPM.
CAPM is a Useful Benchmark Model
Even with its imperfections, the CAPM serves as a useful benchmark model in finance, helping investors and practitioners estimate expected returns, assess risk, and make informed investment decisions.
Merton's I-CAPM vs. APT (Mathematically Equivalent)
Merton's Intertemporal CAPM (I-CAPM) extends the traditional CAPM by allowing for time-varying investment opportunities. Although it is mathematically equivalent to APT, the models differ in their economic interpretation and assumptions.
Intuition of Merton's I-CAPM
Merton's I-CAPM incorporates investors' reactions to changing investment opportunities over time. It accounts for the impact of time-varying risk premia, which can be driven by factors such as economic conditions, technological advancements, or shifts in investor preferences.
Conclusion
Portfolio theory and its extensions, such as the CAPM and APT, have profoundly impacted the field of finance by providing a systematic approach to portfolio construction, risk management, and performance evaluation. Despite some empirical deviations and limitations, these models remain central to understanding the relationship between risk and return, guiding investors in their pursuit of optimal portfolios. As the financial landscape continues to evolve, researchers and practitioners will undoubtedly refine and expand upon these foundational theories to better address the complexities of modern markets and the diverse needs of investors. Ultimately, portfolio theory and its related models serve as invaluable tools for navigating the dynamic and uncertain world of investing.