Markowitz model

What Is Markowitz Model?

The Markowitz model, also known as Mean-Variance Optimization, is a mathematical framework for assembling an investment portfolio of assets to maximize expected return for a given level of risk, or equivalently, minimize risk for a given expected return. It is a foundational concept within portfolio theory, a broader financial category that examines how investors can construct portfolios to achieve specific financial goals while managing risk. The Markowitz model emphasizes that the risk and return of an individual asset should not be evaluated in isolation but rather by how it contributes to the overall risk-return tradeoff of the entire portfolio. This approach revolutionized the understanding of diversification by quantifying how combining assets can reduce overall portfolio volatility, a key aspect of portfolio optimization.

History and Origin

The Markowitz model was introduced by economist Harry Markowitz in his seminal 1952 paper, "Portfolio Selection," published in The Journal of Finance. His work laid the theoretical groundwork for modern investment management by shifting focus from analyzing individual securities to considering the entire portfolio. Markowitz's groundbreaking ideas were recognized with the Nobel Memorial Prize in Economic Sciences in 1990, jointly awarded with Merton Miller and William Sharpe, "for their pioneering work in the theory of financial economics"⁴. He developed a method of portfolio analysis aimed at finding diversified portfolios that offered the maximum return for a given risk level, or minimum risk for a given return level. This fundamental concept became the cornerstone of modern financial engineering, applying mathematical tools to practical investment problems³.

Key Takeaways

• The Markowitz model focuses on optimizing portfolio risk and return simultaneously.
• It quantifies the benefits of diversification by considering the correlation between assets.
• The output of the model is the efficient frontier, representing optimal portfolios.
• It operates on specific assumptions about investor rationality and market efficiency.
• The model forms the bedrock of Modern Portfolio Theory.

Formula and Calculation

The Markowitz model utilizes statistical measures to determine the expected return and risk (as measured by standard deviation or variance) of a portfolio, taking into account the correlation between asset returns.

For a portfolio of (n) assets, the expected return of the portfolio ((E_p)) is the weighted average of the expected returns of individual assets:

$E_p = \sum_{i=1}^{n} w_i E_i$

Where:

• (E_p) = Expected return of the portfolio
• (w_i) = Weight (proportion) of asset (i) in the portfolio
• (E_i) = Expected return of asset (i)

The portfolio variance ((\sigma_p^2)), which quantifies risk, is more complex as it considers the covariance between asset returns:

$\sigma_p^2 = \sum_{i=1}^{n} w_i^2 \sigma_i^2 + \sum_{i=1}^{n} \sum_{j=1, j \neq i}^{n} w_i w_j \sigma_{ij}$

Where:

• (\sigma_p^2) = Variance of the portfolio's return
• (w_i) = Weight of asset (i) in the portfolio
• (\sigma_i^2) = Variance of asset (i)'s return
• (\sigma_{ij}) = Covariance between asset (i)'s return and asset (j)'s return ((\sigma_{ij} = \rho_{ij} \sigma_i \sigma_j), where (\rho_{ij}) is the correlation coefficient between asset (i) and asset (j))

Interpreting the Markowitz Model

Interpreting the Markowitz model revolves around the concept of the efficient frontier. This curve represents the set of all optimal portfolios that offer the highest expected return for a defined level of risk, or the lowest risk for a given expected return. Investors seeking to apply the model would use mean-variance analysis to plot various portfolio combinations on a risk-return graph. Portfolios lying below the efficient frontier are considered suboptimal because they offer either lower returns for the same risk or higher risk for the same return. An investor's optimal portfolio on the efficient frontier depends on their individual risk tolerance; a more risk-averse investor would choose a point further to the left (lower risk, lower return), while a more aggressive investor might choose a point further to the right (higher risk, higher return).

Hypothetical Example

Consider an investor, Sarah, who wants to build an investment portfolio using two assets: Stock A and Stock B.

• Stock A: Expected Return = 10%, Standard Deviation = 15%
• Stock B: Expected Return = 15%, Standard Deviation = 25%
• Correlation between Stock A and Stock B = 0.20

Sarah wants to determine the optimal [asset allocation](https://diversification.com/term/asset%20allocation\) to achieve a specific balance of risk and return.

If Sarah allocates 50% to Stock A and 50% to Stock B ((w_A = 0.50, w_B = 0.50)):

Expected Portfolio Return (E_p = (0.50 \times 0.10) + (0.50 \times 0.15) = 0.05 + 0.075 = 0.125) or 12.5%.

To calculate portfolio variance, first calculate covariance:
(\sigma_{AB} = \rho_{AB} \sigma_A \sigma_B = 0.20 \times 0.15 \times 0.25 = 0.0075).

Portfolio Variance (\sigma_p^2 = (0.50^2 \times 0.15^2) + (0.50^2 \times 0.25^2) + (2 \times 0.50 \times 0.50 \times 0.0075))
(\sigma_p^2 = (0.25 \times 0.0225) + (0.25 \times 0.0625) + (0.50 \times 0.0075))
(\sigma_p^2 = 0.005625 + 0.015625 + 0.00375 = 0.025).

Portfolio Standard Deviation (\sigma_p = \sqrt{0.025} \approx 0.1581) or 15.81%.

By running these calculations for various weight combinations (e.g., 60/40, 40/60, etc.), Sarah can identify different portfolios with their associated expected returns and risks. Plotting these points would reveal the curve of the efficient frontier, allowing her to select the portfolio that best matches her risk tolerance.

Practical Applications

The Markowitz model's principles are widely used in various areas of finance and investment management. Fund managers and institutional investors employ portfolio optimization techniques derived from the Markowitz model to construct balanced portfolios for their clients. The concept of the efficient frontier guides the strategic asset allocation process, helping investors understand the optimal balance between risk and reward. It has also contributed to the development of other key financial models, such as the Capital Market Line, which incorporates a risk-free rate for further optimization, and the Sharpe ratio, a measure of risk-adjusted return. Many commonly used investment vehicles, such as target-date funds and diversified exchange-traded funds (ETFs), implicitly or explicitly adhere to the tenets of Modern Portfolio Theory, which the Markowitz model underpins. Investors seeking broad diversification, often through low-cost index funds, are also aligning with the principles of MPT².

Limitations and Criticisms

Despite its foundational importance, the Markowitz model has several limitations and has faced criticisms, primarily concerning its underlying assumptions. One significant critique is its reliance on historical data for expected return and standard deviation (risk), assuming that past performance is indicative of future results, which may not hold true. The model also assumes that asset returns follow a normal distribution, which often isn't the case in real financial markets, especially during periods of extreme volatility.

Furthermore, the Markowitz model relies on investors being perfectly rational and risk-averse, always preferring higher returns for the same risk or lower risk for the same return. It measures risk solely by variance or standard deviation, which penalizes both upside and downside volatility equally, when in reality, investors are typically more concerned with downside risk. Issues arise when dealing with high-dimensional portfolios, as accurately estimating the correlation and covariance between a large number of assets becomes computationally intensive and prone to error¹. Additionally, factors like transaction costs, taxes, and liquidity constraints are generally not explicitly accounted for in the basic model. The model also struggles with assets that do not exhibit a clear linear relationship in their risk and return, or those with non-normal return distributions, which can impact the accuracy of portfolio risk measurements, including concepts like beta.

Markowitz Model vs. Modern Portfolio Theory

The Markowitz model is often used interchangeably with Modern Portfolio Theory (MPT), but it's more accurate to view the Markowitz model as the cornerstone or the foundational component of MPT. MPT is a broader mathematical framework that extends and builds upon the concepts introduced by Markowitz. While the Markowitz model specifically focuses on mean-variance optimization to construct efficient portfolios and derive the efficient frontier, MPT encompasses this core idea and integrates other elements such as the Capital Asset Pricing Model (CAPM) and the concept of the Capital Market Line, which further define the relationship between risk and return in a diversified portfolio in the presence of a risk-free asset. In essence, the Markowitz model provides the mathematical engine for portfolio construction, and MPT is the comprehensive theoretical structure that houses it and its subsequent advancements.

FAQs

How does the Markowitz model help in diversification?

The Markowitz model quantifies the benefits of diversification by considering not just the individual risks and returns of assets, but also how their returns move in relation to each other (their correlation). By combining assets that are not perfectly positively correlated, the model shows how the overall portfolio risk can be reduced without necessarily sacrificing expected return.

Is the Markowitz model still relevant today?

Yes, the Markowitz model remains highly relevant as a foundational concept in finance and portfolio theory. While practical applications often use more sophisticated variations and account for its limitations, the core principles of mean-variance optimization and the concept of the efficient frontier continue to guide investment professionals in structuring portfolios and managing risk.

What is the efficient frontier in simple terms?

The efficient frontier is a curve that represents the best possible combinations of assets that an investor can hold. Any portfolio on this curve offers the highest possible expected return for a given level of risk, or the lowest possible risk for a given expected return. Investors aim to construct portfolios that lie on this frontier, tailoring their choice to their individual risk tolerance.