Markowitz Portfolio Model

Risk and Return for 2 and 3 Asset Portfolios

Core Concept: The Markowitz Portfolio Model, developed by Harry Markowitz, is a foundational concept in modern portfolio theory (MPT). It provides a framework for constructing efficient portfolios by considering the expected returns, risks (standard deviations), and correlations between assets. The model aims to identify the portfolio that maximizes expected return for a given level of risk or minimizes risk for a desired level of expected return. Key Inputs:

• Expected Returns: The anticipated returns for each asset in the portfolio.
• Standard Deviations (Risks): The measure of volatility or uncertainty associated with each asset's returns.
• Correlation Coefficients: The measure of how the returns of different assets move in relation to each other.

1. Two-Asset Portfolio

• Portfolio Return:

  • Formula: Rp = w1 * R1 + w2 * R2
    • Where:
      • Rp = Portfolio return
      • w1 = Weight of asset 1 in the portfolio
      • w2 = Weight of asset 2 in the portfolio
      • R1 = Expected return of asset 1
      • R2 = Expected return of asset 2

• Portfolio Risk (Standard Deviation):

  • Formula: σp = √[(w1^2 * σ1^2) + (w2^2 * σ2^2) + (2 * w1 * w2 * ρ1,2 * σ1 * σ2)]
    • Where:
      • σp = Portfolio standard deviation
      • w1 = Weight of asset 1 in the portfolio
      • w2 = Weight of asset 2 in the portfolio
      • σ1 = Standard deviation of asset 1
      • σ2 = Standard deviation of asset 2
      • ρ1,2 = Correlation coefficient between asset 1 and asset 2

• Example:

  • Asset 1: Expected Return (R1) = 10%, Standard Deviation (σ1) = 15%

  • Asset 2: Expected Return (R2) = 15%, Standard Deviation (σ2) = 20%

  • Correlation Coefficient (ρ1,2) = 0.3

  • Scenario 1: Equal Weights (w1 = 50%, w2 = 50%)

    • Portfolio Return: Rp = (0.50 * 0.10) + (0.50 * 0.15) = 0.05 + 0.075 = 0.125 = 12.5%
    • Portfolio Risk:
      • σp = √[(0.50^2 * 0.15^2) + (0.50^2 * 0.20^2) + (2 * 0.50 * 0.50 * 0.3 * 0.15 * 0.20)]
      • σp = √[(0.25 * 0.0225) + (0.25 * 0.04) + (0.5 * 0.3 * 0.03)]
      • σp = √[0.005625 + 0.01 + 0.0045] = √0.020125 ≈ 0.1419 = 14.19%

  • Scenario 2: Unequal Weights (w1 = 70%, w2 = 30%)

    • Portfolio Return: Rp = (0.70 * 0.10) + (0.30 * 0.15) = 0.07 + 0.045 = 0.115 = 11.5%
    • Portfolio Risk:
      • σp = √[(0.70^2 * 0.15^2) + (0.30^2 * 0.20^2) + (2 * 0.70 * 0.30 * 0.3 * 0.15 * 0.20)]
      • σp = √[(0.49 * 0.0225) + (0.09 * 0.04) + (0.42 * 0.3 * 0.03)]
      • σp = √[0.011025 + 0.0036 + 0.00378] = √0.018405 ≈ 0.1357 = 13.57%

  • Observations: By changing the weights, we can adjust the portfolio's risk and return. Lowering the weight of the riskier asset (Asset 2) reduces both the expected return and the portfolio risk.

2. Three-Asset Portfolio

• Portfolio Return:

  • Formula: Rp = w1 * R1 + w2 * R2 + w3 * R3
    • Where:
      • Rp = Portfolio return
      • w1, w2, w3 = Weights of assets 1, 2, and 3 in the portfolio
      • R1, R2, R3 = Expected returns of assets 1, 2, and 3

• Portfolio Risk (Standard Deviation):

  • Formula: The formula for a three-asset portfolio becomes more complex:
    • σp = √[ (w1^2 * σ1^2) + (w2^2 * σ2^2) + (w3^2 * σ3^2) + (2 * w1 * w2 * ρ1,2 * σ1 * σ2) + (2 * w1 * w3 * ρ1,3 * σ1 * σ3) + (2 * w2 * w3 * ρ2,3 * σ2 * σ3) ]
    • Where:
      • σp = Portfolio standard deviation
      • w1, w2, w3 = Weights of assets 1, 2, and 3
      • σ1, σ2, σ3 = Standard deviations of assets 1, 2, and 3
      • ρ1,2, ρ1,3, ρ2,3 = Correlation coefficients between assets 1 & 2, 1 & 3, and 2 & 3

• Example:

  • Asset 1: Expected Return (R1) = 8%, Standard Deviation (σ1) = 12%

  • Asset 2: Expected Return (R2) = 12%, Standard Deviation (σ2) = 18%

  • Asset 3: Expected Return (R3) = 15%, Standard Deviation (σ3) = 22%

  • Correlation Coefficients:

    • ρ1,2 = 0.4
    • ρ1,3 = 0.2
    • ρ2,3 = 0.6

  • Scenario: Equal Weights (w1 = 33.33%, w2 = 33.33%, w3 = 33.33%)

    • Portfolio Return: Rp = (0.3333 * 0.08) + (0.3333 * 0.12) + (0.3333 * 0.15) ≈ 0.0267 + 0.04 + 0.05 ≈ 0.1167 = 11.67%
    • Portfolio Risk:
      • σp = √[ (0.3333^2 * 0.12^2) + (0.3333^2 * 0.18^2) + (0.3333^2 * 0.22^2) + (2 * 0.3333 * 0.3333 * 0.4 * 0.12 * 0.18) + (2 * 0.3333 * 0.3333 * 0.2 * 0.12 * 0.22) + (2 * 0.3333 * 0.3333 * 0.6 * 0.18 * 0.22) ]
      • After Calculating approximately, σp will be around 15.4%.

  • Observations: The three-asset portfolio provides more diversification opportunities. By carefully selecting assets with different risk-return profiles and correlation coefficients, it's possible to achieve a better risk-return trade-off compared to a two-asset portfolio.

3. Key Takeaways and Implications of the Markowitz Model

• Diversification is Key: The Markowitz model demonstrates that diversification can reduce portfolio risk without sacrificing return.
• Correlation Matters: The correlation between assets is a critical factor in determining portfolio risk. Lower correlations lead to greater diversification benefits.
• Efficient Frontier: The model helps investors identify the efficient frontier, which represents the set of portfolios that offer the best possible risk-return combinations.
• Investor Preferences: The optimal portfolio for an investor depends on their individual risk tolerance and investment goals.
• Limitations: The Markowitz model relies on several assumptions, including:
  • Investors are rational and risk-averse.
  • Asset returns are normally distributed.
  • Transaction costs and taxes are ignored.
  • Accurate estimates of expected returns, standard deviations, and correlation coefficients are available. (In reality, these are forecasts and subject to error)
• Use in Practice: Despite its limitations, the Markowitz model provides a valuable framework for portfolio construction and asset allocation. It is widely used by institutional investors and financial advisors.

Conclusion:

The Markowitz Portfolio Model provides a rigorous approach to portfolio analysis by considering the risk and return characteristics of individual assets and their correlations. By constructing efficient portfolios based on these principles, investors can improve their risk-adjusted returns and achieve their investment goals more effectively.