INTRODUCTION TO MODERN PORTFOLIO THEORY

4.1 Framework for Constructing Portfolios - Modern Portfolio Theory

Modern Portfolio Theory (MPT) serves as a mathematical framework for constructing and selecting investment portfolios. It is based on the fundamental "conventional wisdom" of not putting all eggs in one basket, but it goes beyond simple intuition by quantifying the benefits of diversification.

Introduced by Harry Markowitz in his 1952 article "Portfolio Selection," MPT changed the landscape of finance by introducing the statistical notions of covariance and correlation between investment assets. Markowitz mathematically demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk.

Key Takeaway: MPT allows investors to select portfolios based on expected performance and their specific risk appetite, focusing on how assets move in relation to one another rather than evaluating them in isolation.

4.2 Assumptions of the Theory

To function as a mathematical model, Modern Portfolio Theory relies on several core assumptions regarding investor behavior and market conditions:

• Risk-Return Optimization: Investors seek to maximize return for a given level of risk. If two assets offer the same return, the investor will always choose the one with lower risk.
• Probability Distributions: Investment alternatives are viewed as probability distributions of expected returns over a specific holding period.
• One-Period Utility: Investors aim to maximize their expected utility over a single period and assign utility scores to various portfolio choices.
• Decision Criteria: Investment decisions are based solely on expected return and risk (variability of returns).
• Risk Estimation: The risk of a portfolio is estimated based on the variability of the expected returns of its constituent assets.

4.3 Definition of Risk Averse, Risk Seeking, and Risk Neutral Investor

Risk in investment is defined as the possibility that actual earnings will differ from expected outcomes. Investors demand a risk premium to compensate for bearing this uncertainty.

The Utility Function

Financial experts use a utility function to assign a score (U) to portfolios based on their risk-return profile: U = E(r) - ½ Aσ²

• E(r): Expected return.
• A: Index of the investor’s risk aversion.
• σ²: Variance of the investment returns.

Investor Classifications

1. Risk Averse: These investors prefer risk-free assets or investments with a positive risk premium. They assign higher utility scores to portfolios with higher returns and lower risk.
2. Risk Seeking: These investors expect a higher return for taking on higher risk (Sample Question 5).
3. Risk Neutral: These investors are indifferent to risk levels, making decisions based purely on expected returns.

Comparison with Sharpe Ratio: While the Sharpe Ratio assumes all investors have homogenous risk perceptions (higher ratio is always better), the Utility concept acknowledges heterogeneity—different investors will choose different portfolios based on their unique individual risk aversion.

4.4 Expected Rate of Return for Individual Security

The Expected Rate of Return (also known as Ex-Ante Return) for a single security is the sum of all possible returns multiplied by their respective probabilities. This differs from Ex-Post Returns, which are calculated based on historical data.

Formula: RA = Σ (Probability × Return in that state)

Example: If a stock has a 30% chance of a 15% return (Boom), a 50% chance of a 10% return (Normal), and a 20% chance of a 2% return (Recession), the expected return is: (0.3 × 15%) + (0.5 × 10%) + (0.2 × 2%) = 9.9%.

4.5 Variance of Return for Individual Security (Ex-Ante Risk)

Risk is defined as the variability of an asset's return. The most common measure for this is Standard Deviation, which is the square root of the Variance.

Calculation Process:

1. Calculate the Expected Return [E(Ri)].
2. Find the difference between each possible return and the expected return [Ri - E(Ri)].
3. Square those differences and multiply by their probability [(Ri - E(Ri))² * Pi].
4. Sum these values to find the Ex-Ante Variance (σ²).
5. The square root of variance is the Ex-Ante Risk (Standard Deviation).

4.6 Expected Rate of Return for a Portfolio (Ex-Ante Return)

The expected return of a portfolio is simply the weighted average of the expected returns of the individual investments within that portfolio.

Formula: E(Rport) = Σ (Wi * Ri)

• Wi: The percentage of the portfolio invested in asset i.
• Ri: The expected rate of return for asset i.

Key Term: Weights are determined by the proportion of value each asset holds in the total portfolio.

4.7 Variance of Return for a Portfolio

Unlike expected return, portfolio variance is not a simple weighted average of individual variances. It must account for the co-movement between assets.

Variables Affecting Portfolio Risk:

1. Weights of individual investments.
2. Individual risks (standard deviation) of investments.
3. Co-movement (Covariance or Correlation) between each pair of assets.

Correlation Coefficient (r): Varies from -1 to +1.

• +1 (Perfect Positive Correlation): Assets move together linearly; no diversification benefit exists.
• -1 (Perfect Negative Correlation): Assets move in opposite directions; can theoretically yield a zero-risk portfolio.

4.8 Graphical Presentation of Portfolio Risk/Return

Plotted on a graph where the Y-axis is return and the X-axis is risk (standard deviation):

• Perfectly Correlated Assets (+1.0): Represented by a straight line connecting the assets; risk is exactly the weighted average.
• Less than Perfectly Correlated (< 1.0): The portfolio risk is less than the weighted average, causing the plot to bulge to the left. This bulge represents the essence of diversification.

Key Takeaway: As long as the correlation is less than +1, benefits of diversification occur. The lower the correlation, the higher the benefits.

4.9 Efficient Frontier

When multiple securities are combined in infinite weight combinations, they form an umbrella-shaped curve.

• Definition: The Efficient Frontier represents the set of portfolios that provide the maximum rate of return for a given level of risk, or the minimum risk for a given level of return.
• Portfolio Efficiency: A portfolio is "efficient" if no other portfolio offers a higher return for the same risk, or lower risk for the same return.

4.10 Portfolio Optimization Process

To find the optimum portfolio (the combination meeting specific objectives for a given set of constraints), a manager must estimate:

1. Expected returns for every asset in the universe.
2. Standard deviations for each asset.
3. Correlation coefficients for every pair of assets.

Constraint Consideration: Managers must also input any specific investor constraints (e.g., liquidity needs, tax status) to reach the final Optimal Portfolio.

4.11 Estimation Issues

The primary challenge of MPT is the accuracy of statistical inputs. The number of required correlation estimates grows exponentially as more securities are added.

Formula for Number of Correlations: Number = (n² - n) / 2

• For a portfolio of 50 securities, a manager must estimate 1,225 correlations.

Estimation Risk: Potential errors arising from these numerous estimations are referred to as estimation risk, which can significantly impact the reliability of the portfolio allocation output.

Chapter 4: Important Terms and Key Takeaways

Final Exam Tip: Remember that portfolio risk decreases if the correlation coefficient between stocks in the portfolio decreases, even if all other factors remain constant.