Expected Returns and Variance: Calculating Risk and Return in Investment Portfolios

25.4.1 Expected Returns and Variance

In the realm of finance and investment, understanding the expected returns and variance of assets and portfolios is crucial for making informed decisions. This section delves into the mathematical foundations and practical applications of these concepts, equipping you with the tools to assess and manage investment risk effectively.

Understanding Expected Return

The expected return of an asset is a fundamental concept in finance, representing the average return an investor anticipates earning from an investment over a specific period. It is calculated by considering all possible outcomes, their probabilities, and the returns associated with each outcome.

Formula for Expected Return of an Asset

The expected return \( E(R_i) \) of an asset \( i \) is calculated using the following formula:

• \( E(R_i) \): Expected return of asset \( i \).
• \( p_j \): Probability of state \( j \).
• \( R_{ij} \): Return of asset \( i \) in state \( j \).

This formula highlights the weighted average of all possible returns, where the weights are the probabilities of each state occurring.

Calculating Expected Return of a Portfolio

A portfolio is a collection of assets, and its expected return is the weighted average of the expected returns of the individual assets. The formula for the expected return of a portfolio \( E(R_p) \) is:

• \( E(R_p) \): Expected return of the portfolio.
• \( w_i \): Weight of asset \( i \) in the portfolio.

This calculation requires knowing the proportion of the total investment allocated to each asset and their respective expected returns.

Variance and Standard Deviation as Risk Measures

Variance and standard deviation are statistical measures used to quantify the risk associated with an asset’s returns. Variance measures the dispersion of returns around the expected return, while standard deviation is the square root of variance, providing a more intuitive measure of risk.

Variance of an Asset

The variance \( \sigma_i^2 \) of an asset \( i \) is calculated as follows:

This formula calculates the average of the squared deviations from the expected return, weighted by the probabilities of each state.

Standard Deviation

The standard deviation \( \sigma_i \) is simply the square root of the variance:

Standard deviation provides a more understandable measure of risk, as it is expressed in the same units as the returns.

Calculating Portfolio Variance

The variance of a portfolio is not just the weighted average of the variances of the individual assets. It also depends on the correlation between the assets. For a two-asset portfolio, the variance \( \sigma_p^2 \) is calculated as:

• \( \rho_{1,2} \): Correlation coefficient between assets 1 and 2.

This formula accounts for the covariance between the assets, which is influenced by their correlation.

The Impact of Diversification

Diversification is a strategy used to reduce risk by investing in a variety of assets. The key to effective diversification is selecting assets with low or negative correlations, as this can significantly reduce portfolio variance without sacrificing expected returns.

Diversification Effect

• Negative or Low Correlation: When assets are negatively or lowly correlated, the portfolio variance decreases, reducing overall risk.
• Risk Reduction: Diversification allows investors to achieve a desired return with lower risk compared to investing in a single asset.

Sample Calculation

Let’s consider a practical example to illustrate these concepts.

Example: Calculating Portfolio Expected Return and Variance

• Asset A: \( E(R_A) = 10% \), \( \sigma_A = 15% \).
• Asset B: \( E(R_B) = 8% \), \( \sigma_B = 10% \).
• Correlation (\( \rho_{A,B} \)): 0.5.
• Portfolio Weights: \( w_A = 0.6 \), \( w_B = 0.4 \).

Calculate Portfolio Expected Return:

Calculate Portfolio Variance:

Breaking it down:

• \( (0.6)^2 \times (15%)^2 = 0.36 \times 0.0225 = 0.0081 \)
• \( (0.4)^2 \times (10%)^2 = 0.16 \times 0.01 = 0.0016 \)
• \( 2 \times 0.6 \times 0.4 \times 15% \times 10% \times 0.5 = 0.072 \times 0.075 = 0.0054 \)

Adding these components gives:

The portfolio standard deviation \( \sigma_p \) is:

Key Concepts and Limitations

Key Concepts

• Risk and Correlation: The portfolio’s risk is influenced by the individual risks of the assets and their correlations.
• Diversification Benefits: Diversification is most effective when assets are negatively correlated, reducing overall risk.

Limitations

• Historical Data: Calculations often rely on historical data, which may not accurately predict future performance.
• Real-World Constraints: Practical constraints, such as transaction costs and market liquidity, can limit diversification opportunities.

Conclusion

Understanding expected returns and variance is essential for constructing and managing investment portfolios. These quantitative tools enable investors to evaluate the risk-return profile of their investments and make informed allocation decisions. By leveraging diversification and considering asset correlations, investors can optimize their portfolios to achieve their financial goals.

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