Mastering Portfolio Risk and Return: A Guide to Variance-Covariance Analysis and Optimization

Introduction

Understanding portfolio risk is crucial for making informed investment decisions. Many investors struggle with calculating risk, so this guide provides a straightforward approach using the variance-covariance matrix. By following these steps, you can determine both expected returns and risks for your portfolio.

Theory: Portfolio Risk and Return

Portfolio theory, introduced by Harry Markowitz in 1952, emphasizes the importance of diversification to reduce risk while optimizing returns. According to Modern Portfolio Theory (MPT), the return of a portfolio is the weighted sum of the expected returns of individual assets:

where:

• is the expected return of the portfolio,

• is the weight of each asset in the portfolio,

• is the expected return of individual assets.

Portfolio risk, measured by variance or standard deviation, is given by:

where:

• is the portfolio variance,

• is the variance of individual assets,

• is the covariance between assets.

Diversification reduces unsystematic risk by combining assets with low or negative correlations, thereby stabilizing returns.

Step 1: Data Collection

Collect 60 months of historical prices for each security. You can download this data from Yahoo! Finance or other financial data providers.

Step 2: Calculate Expected Portfolio Return

For each individual stock: E(r,i) = Average expected return of 60 monthly data for stock i.

For the entire portfolio: E(r,p) = Expected return of the portfolio.

Example with Three Securities:

E(r,p) = 1.09% monthly

Step 3: Calculate Portfolio Risk

Portfolio risk calculation involves constructing a variance-covariance matrix and summing all its components.

Variance-Covariance Matrix Construction:

1. Single Stock: 1 component (1² = 1)

2. 

3. Two Stocks: 4 components (2² = 4)

4. 

5. Three Stocks: 9 components (3² = 9)

6. 

7. Four Stocks: 16 components (4² = 16)

8. 

9. Five Stocks: 25 components (5² = 25)

10. 

11. Six Stocks: 36 components (6² = 36)

12. 

... and so on. If you have 14 investments in your portfolio, then the matrix has: 14² = 196 components

Optimization

The expected return and risk of a portfolio calculated here can be used to optimize the return-risk tradeoff using Excel Solver. Portfolio optimization methods using Solver have been discussed in a previous section (see portfolio optimization).

Conclusion

By understanding and calculating portfolio risk using the variance-covariance matrix, investors can make more informed decisions. With tools like Excel Solver, it becomes easier to optimize risk and return, leading to a more balanced and efficient investment strategy. Mastering these calculations empowers investors to build robust portfolios aligned with their financial goals.