Modello di markowitz

What Is Modello di Markowitz?

The Modello di Markowitz, also known as the Markowitz Model or Mean-Variance Model, is a foundational concept within Portfolio Theory. It is a mathematical framework for portfolio optimization that assists investors in constructing an investment portfolio to maximize expected returns for a given level of risk, or conversely, minimize risk for a target expected return. Introduced by Harry Markowitz in 1952, the model emphasizes the importance of diversification by analyzing the relationships between the securities within a portfolio, rather than viewing each asset in isolation. Its core insight is that a portfolio's overall risk is not merely the sum of the individual risks of its components, but also depends on how their returns move together, or their covariance.

History and Origin

Prior to Harry Markowitz's seminal work, investment analysis often focused solely on selecting individual securities with the highest anticipated returns. Risk, if considered, was typically approached from a qualitative standpoint. Markowitz revolutionized this perspective with his paper "Portfolio Selection," published in The Journal of Finance in 1952. He recognized that investors should evaluate the entire portfolio of investments as a whole, rather than focusing on individual securities³⁴, ³⁵.

Markowitz's imaginative leap was to quantify risk using the variance of a portfolio's returns and to rigorously demonstrate how a diversified portfolio could lower overall risk without sacrificing returns³³. His theory mathematically formalized the long-held adage, "Don't put all your eggs in one basket." This groundbreaking contribution earned him, along with Merton H. Miller and William F. Sharpe, the Nobel Memorial Prize in Economic Sciences in 1990, for their pioneering work in the theory of financial economics.³¹, ³²

Key Takeaways

• The Modello di Markowitz, or Markowitz Model, provides a mathematical framework for constructing optimal investment portfolios.
• It focuses on the relationship between expected return (mean) and risk (variance or standard deviation) of a portfolio.³⁰
• A central tenet is that effective diversification can reduce a portfolio's overall risk by combining assets whose returns do not move perfectly in sync.²⁸, ²⁹
• The model helps investors identify the Efficient Frontier, representing portfolios that offer the highest possible expected return for a given level of risk.²⁶, ²⁷
• It assumes investors are rational and risk-averse, seeking to maximize returns for a chosen level of risk or minimize risk for a desired return.²⁵

Formula and Calculation

The Modello di Markowitz quantifies portfolio risk and return using the expected return of individual assets, their individual variances, and the covariances between all pairs of assets.

For a portfolio consisting of (n) assets, the expected return of the portfolio ((E[R_p])) is the weighted average of the expected returns of the individual assets:

Where:

• (E[R_p]) = Expected return of the portfolio
• (w_i) = Weight (proportion) of asset (i) in the portfolio
• (E[R_i]) = Expected return of asset (i)

The variance of the portfolio ((\sigma_p^2)), which represents the portfolio's risk, is calculated using the following formula:

Where:

• (\sigma_p^2) = Variance of the portfolio's return
• (w_i), (w_j) = Weights of asset (i) and asset (j) in the portfolio
• (\text{Cov}(R_i, R_j)) = Covariance between the returns of asset (i) and asset (j)

If (i = j), then (\text{Cov}(R_i, R_i)) is simply the variance of asset (i), often denoted as (\sigma_i^2). The covariance term is crucial, as it accounts for how the returns of different assets move together. A positive covariance indicates that assets tend to move in the same direction, while a negative covariance suggests they move in opposite directions. The correlation coefficient is a standardized measure derived from covariance.

Interpreting the Modello di Markowitz

The Modello di Markowitz provides a powerful framework for investors to understand and manage the risk-return tradeoff in their portfolios. The central output of the Markowitz Model is the Efficient Frontier – a curve representing the set of all optimal portfolios. Any portfolio on this frontier offers the maximum possible expected return for its given level of risk, or the minimum possible risk for its given expected return. Portfolios below the Efficient Frontier are considered sub-optimal because they offer less return for the same risk, or more risk for the same return.

²⁴Investors, depending on their individual risk aversion, can then select a portfolio from this frontier that aligns with their specific financial goals and comfort level with risk. A highly risk-averse investor might choose a portfolio on the lower left of the frontier, aiming for lower risk and a corresponding lower return. Conversely, an investor with a higher tolerance for risk might opt for a portfolio further up the curve, seeking higher returns despite greater volatility. The concept of the Capital Allocation Line further extends this, allowing investors to combine a risk-free asset with a portfolio on the efficient frontier to achieve a wide range of risk-return combinations.

Hypothetical Example

Consider an investor, Sofia, who wants to build a portfolio using two assets: Stock X and Stock Y.

Assumptions:

• Stock X: Expected Return = 8%, Standard Deviation = 15%
• Stock Y: Expected Return = 12%, Standard Deviation = 25%
• Correlation Coefficient between X and Y = 0.30 (positive, but not perfectly correlated)

Sofia wants to achieve an expected return of 10%. Using the Modello di Markowitz principles, she can determine the weights for Stock X ((w_X)) and Stock Y ((w_Y)) that minimize the portfolio's risk for this target return.

1. Set up the expected return equation:
   (0.10 = w_X (0.08) + w_Y (0.12))
   Since (w_X + w_Y = 1), we have (w_Y = 1 - w_X).
   Substituting: (0.10 = w_X (0.08) + (1 - w_X) (0.12))
   (0.10 = 0.08w_X + 0.12 - 0.12w_X)
   (0.10 - 0.12 = (0.08 - 0.12)w_X)
   (-0.02 = -0.04w_X)
   (w_X = 0.50)
   Therefore, (w_Y = 1 - 0.50 = 0.50).
   So, 50% in Stock X and 50% in Stock Y.

2. Calculate the portfolio standard deviation (risk) for these weights:
   First, convert standard deviations to variances:
   Variance of X ((\sigma_X^{2)) = (0.15}2 = 0.0225)
   Variance of Y ((\sigma_Y^{2)) = (0.25}2 = 0.0625)

   Now, calculate covariance:
   (\text{Cov}(R_X, R_Y) = \text{Correlation}(R_X, R_Y) \times \sigma_X \times \sigma_Y)
   (\text{Cov}(R_X, R_Y) = 0.30 \times 0.15 \times 0.25 = 0.01125)

   Portfolio Variance ((\sigma_p^2)):
   (\sigma_p^2 = w_X^2 \sigma_X^2 + w_Y^2 \sigma_Y^2 + 2 w_X w_Y \text{Cov}(R_X, R_Y))
   (\sigma_p^2 = (0.50)^2 (0.0225) + (0.50)^2 (0.0625) + 2 (0.50)(0.50)(0.01125))
   (\sigma_p^2 = 0.25 (0.0225) + 0.25 (0.0625) + 0.50 (0.01125))
   (\sigma_p^2 = 0.005625 + 0.015625 + 0.005625)
   (\sigma_p^2 = 0.026875)

   Portfolio Standard Deviation ((\sigma_p)) = (\sqrt{0.026875} \approx 0.1639) or 16.39%.

By applying the Modello di Markowitz, Sofia found that a 50/50 asset allocation offers her desired 10% expected return with a portfolio standard deviation of approximately 16.39%. This demonstrates how the model helps quantify the relationship between return on investment and risk within a diversified portfolio.

Practical Applications

The Modello di Markowitz, as the bedrock of Modern Portfolio Theory, is widely applied in various areas of finance and investment management.

²³* Investment Management: Portfolio managers and financial advisors routinely use the principles of the Markowitz Model to construct and manage client portfolios. They aim to build diversified portfolios that align with an investor's desired risk-return profile. This involves selecting various asset classes, such as stocks, bonds, and other instruments, and determining their optimal weights to achieve the most efficient portfolio for a given risk tolerance.
*²¹, ²² Institutional Investing: Large institutional investors, including pension funds, endowments, and mutual funds, leverage the Markowitz Model for strategic asset allocation. Its framework helps them optimize the balance of risk and return across vast and complex portfolios.

• Regulatory Guidance: Financial regulators, such as the U.S. Securities and Exchange Commission (SEC), emphasize the importance of diversification as a key principle for investors. While not mandating specific models, their guidance on investor protection and prudent investing aligns with the core tenets of the Markowitz Model, encouraging investors to spread their investments to manage risk.
  *¹⁹, ²⁰ Financial Software and Tools: The quantitative nature of the Markowitz Model lends itself well to implementation in financial software and analytical tools. These programs enable investors and professionals to quickly calculate portfolio expected returns, variances, and correlations, and to visualize the efficient frontier, facilitating informed decision-making.

¹⁷, ¹⁸## Limitations and Criticisms

Despite its revolutionary impact and widespread adoption, the Modello di Markowitz is not without its limitations and criticisms.

• Reliance on Historical Data: A primary critique is the model's heavy reliance on historical data for estimating expected returns, variances, and covariances. The assumption is that past performance is indicative of future results, which is not always true, especially in dynamic and unpredictable markets.
  *¹⁵, ¹⁶ Assumption of Normal Distribution: The model assumes that asset returns follow a normal distribution, meaning returns are symmetrically distributed around the mean. In reality, market returns often exhibit "fat tails," implying a higher probability of extreme events (both positive and negative) than a normal distribution would suggest. This can lead to an underestimation of true portfolio risk.
  *¹³, ¹⁴ Computational Complexity: For a large number of assets, the calculation of all pairwise covariances can become computationally intensive, making practical application challenging without specialized software.
  *¹² Static Nature: The Markowitz Model is typically a single-period model, meaning it provides an optimal portfolio for a specific point in time without accounting for changes in market conditions or investor preferences over time. Real-world investment portfolio management requires continuous monitoring and rebalancing.
  *¹¹ Ignoring Transaction Costs and Taxes: The basic Modello di Markowitz does not typically factor in real-world constraints such as transaction costs, taxes, or liquidity issues, which can significantly impact net returns.
  *¹⁰ Rational Investor Assumption: The model assumes investors are perfectly rational and risk-aversion maximizers of expected utility. Behavioral finance has demonstrated that investors often exhibit biases and irrational behaviors that deviate from these assumptions. F⁸, ⁹or example, diversification, a core tenet of the Markowitz Model, can sometimes feel like a "regret-maximizing strategy" during bull markets, leading investors to question its value when a concentrated portfolio performs exceptionally well.

⁷## Modello di Markowitz vs. Capital Asset Pricing Model (CAPM)

The Modello di Markowitz and the Capital Asset Pricing Model (CAPM) are both fundamental concepts in Capital Market Theory, but they serve different purposes. The Markowitz Model is a portfolio optimization tool that helps investors construct an efficient portfolio by considering the risk and return of all assets and their covariances to find the Efficient Frontier. It is prescriptive, telling an investor how to build an optimal portfolio based on their risk tolerance.

⁶In contrast, the CAPM is an asset pricing model that builds upon the Modello di Markowitz, particularly the concept of the Capital Allocation Line. The CAPM simplifies the complexity of Markowitz by focusing on the relationship between an asset's expected return and its systematic risk, measured by beta. It provides a theoretical framework for calculating the appropriate required return on investment for any given asset, assuming efficient markets and rational investors. While Markowitz focuses on the entire portfolio's risk through variance, CAPM isolates systematic risk (non-diversifiable risk) as the only risk that investors should be compensated for.

⁵Essentially, the Modello di Markowitz helps construct the optimal diversified portfolio, while the CAPM helps determine the theoretical fair price (and thus expected return) of an individual asset or portfolio based on its market risk.

FAQs

What problem does the Modello di Markowitz solve?

The Modello di Markowitz addresses the problem of how to construct an investment portfolio that offers the best possible expected return for a given level of risk, or the lowest possible risk for a desired expected return. It provides a quantitative method to achieve effective diversification.

⁴### Is the Markowitz Model still relevant today?
Yes, the Markowitz Model remains highly relevant. It forms the theoretical bedrock of modern asset allocation and portfolio optimization strategies. While more advanced models have emerged, the core principles of understanding portfolio risk through the interaction of assets (covariance) and the concept of the Efficient Frontier are still fundamental in finance.

³### What are the main assumptions of the Markowitz Model?
Key assumptions include that investors are rational and risk-averse, aiming to maximize utility (return for a given risk), and that asset returns are normally distributed. It also assumes investors have access to all necessary information to make informed decisions and that there are no transaction costs or taxes.

²### How does the Markowitz Model define risk?
In the Markowitz Model, risk is primarily defined and measured by the standard deviation (or variance) of a portfolio's returns. Higher standard deviation implies greater volatility and thus higher risk. The model's insight is that portfolio standard deviation is reduced through diversification, especially when assets have low or negative correlation coefficients.¹