The Role of Linear Algebra in Finance in Simple Terms

Suppose you have a simple portfolio consisting of two assets, Asset A and Asset B, with the following information:

Expected Returns:

• Asset A (Expected return): 8%
• Asset B (Expected return): 12%

Covariance Matrix:

• Variance (*) of Asset A returns: 0.036 
• Variance of Asset B returns: 0.062
• Covariance between Asset A and Asset B returns: 0.025

(*) Variance = volatility squared

 

Portfolio Weights:

• Weight of Asset A in the portfolio: 60%
• Weight of Asset B in the portfolio: 40%

Now, let's calculate the portfolio's risk:

 

1. Construct Vectors and Matrix:

• Weight vector (w) is [0.6, 0.4].
• Covariance matrix (Cov) is as follows:

[0.036 0.025]

[0.025 0.062]

 

The weight vector (w) represents how you allocate your investments across different assets in the portfolio. Each element (w_i) in the vector corresponds to the weight or proportion of your investment in a specific asset.

The covariance (*) matrix (Cov) captures the relationships between the returns of different assets. It contains information about how these returns move together or diverge from each other. 

 

2. Calculate w^T*Cov *w: 

• Multiply the weight vector (w^T) by the covariance matrix (Cov), and then multiply the result by (w) again.

[0.6 0.4] * [0.036 0.025] * [0.6]

                  [0.025 0.062].   [0.4]

 

Transposing the weight vector (w) into w^T effectively changes it from a column vector to a row vector. This is necessary for matrix multiplication and ensures that the dimensions match correctly.

Multiplying the weight vector (w) by its transpose (w^T) essentially results in the square of the portfolio's volatility or standard deviation. When you perform the matrix multiplication (w^T*Cov *w), you are effectively calculating a weighted sum of the covariances between the assets in the portfolio. This weighted sum quantifies how the assets' returns collectively contribute to the overall risk of the portfolio.

 

3. Calculate the Square Root:

• Take the square root of the result obtained in step 2. This gives you the portfolio's risk.

Taking the square root of the result w^T*Cov *w is the final step to obtain the portfolio's standard deviation or volatility. It's the square root because the variance (or risk) of the portfolio is typically represented by the squared standard deviation.

The portfolio's risk is approximately 0.187 or 18.7%. This calculation demonstrates how linear algebra is used to determine the risk of a portfolio by considering the covariance relationships between assets and their respective weights in the portfolio.

 

(*) The covariance of an asset with itself, which is essentially measuring how an asset’s returns relate to its own returns, is equivalent to the variance of that asset. This relationship is represented on the diagonal of the covariance matrix.