Eigenvector and Eigenvalue simply explained

In quantitative finance, eigenvalues and eigenvectors are used in the analysis of financial markets, particularly in the study of portfolio theory and risk management. A practical example is in the application of Principal Component Analysis (PCA) to the covariance matrix of asset returns to identify the principal factors affecting portfolio variance.

In the equation A * v = λ * v, A is the matrix, v is the eigenvector, and λ is the eigenvalue. This equation tells us that when matrix A acts on eigenvector v, the output is the same vector v scaled by the eigenvalue λ.

Suppose an investment manager wants to reduce the dimensionality of a dataset of asset returns to identify the underlying factors that explain most of the variance in the returns of a portfolio. The portfolio contains a large number of assets, and the manager is interested in identifying the key sources of risk.

Imagine we have a portfolio with two assets, A and B. We have the following return data over five periods:

Returns:
Asset A: [5%, 10%, 15%, 10%, 5%]
Asset B: [10%, 20%, 10%, 5%, 0%]

We want to understand the risk structure of this portfolio by performing a Principal Component Analysis (PCA) on the covariance matrix of the returns.

The covariance matrix for assets A and B based on the above returns is:

| Var(A) Cov(A,B) |
Cov =
| Cov(A,B) Var(B) |


| 0.0025 0.00375 |
Cov = |
| 0.00375 0.00625 |

We solve the characteristic equation det(Cov - λI) = 0 for eigenvalues (λ).

For our matrix, the characteristic equation is:

| 0.0025 - λ 0.00375 |
| | = 0
| 0.00375 0.00625 - λ |

Let's say solving this gives us eigenvalues λ1 = 0.00875 and λ2 = 0.0000096875.

The corresponding eigenvectors (which are the principal components) might be:

For λ1 = 0.00875: Eigenvector1 = [1, 2]
For λ2 = 0.0000096875: Eigenvector2 = [-2, 1]

The eigenvectors are the directions of the axes where variance is observed in the data, and the eigenvalues quantify the amount of variance in those directions.

The first eigenvalue (λ1 = 0.00875) is larger and its eigenvector [1, 2] suggests that when asset A moves by some amount, asset B tends to move by twice that amount in the same direction. This principal component might be interpreted as a "market" factor that affects both assets.

The second eigenvector [-2, 1], corresponding to the small eigenvalue, represents a pattern of variance that is relatively insignificant in the overall risk structure of the portfolio. It may capture minor or diversifiable risks.

In this portfolio, the risk is predominantly in one dimension—represented by the first eigenvector. Asset B is more volatile and contributes more to the portfolio risk, as indicated by the eigenvector [1, 2].

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