Why did the data matrix break up with its old ways? To enter a new dimension of enlightenment!

All of us who’ve tried our best to analyze something would have found ourselves overwhelmed by the sheer number of factors present! Well, computers may make the task easier, but studying a more significant number of factors and variables still comes at the cost of space, time, and money! In this blog, we’ll explore the world of Principal Component Analysis, particularly in Risk Analytics, to make it easier wherein we’ll explore ways to cut down on the number of variables we’ll consider in our calculations using some interesting principles.

## Risk Factors

Let us dive into the various factor models generally applied to gauge potential returns on portfolios and thus choose the best way to distribute capital to make a profit.

## Types of Risk Factors

Risk factors are the variables that can affect the value of a financial instrument. They can be classified into two main categories: systematic risk factors and non-systematic risk factors.

**Systematic risk factors**affect all financial instruments, such as interest rates, inflation, and economic growth. These factors cannot be diversified away and are the primary drivers of market volatility.**Specific risk factors**are factors that affect only a specific financial instrument or group of instruments, such as the financial health of a company or the political stability of a country. These factors can be diversified by investing in a diversified portfolio of assets.

So, to measure the returns on any instrument we consider factors influencing our assets’ prices, such as Interest rates, Inflation, Economic growth, Exchange rates, Commodity prices, Political risk, Credit risk, etc.

## Limitations in the case of a large number of factors :

Analyzing the factors mentioned above is a good start for our risk analysis purposes; however, performing analysis when there is a presence of many features not only becomes computationally expensive but more difficult to draw conclusions.

To solve this, we may make use of some powerful methods that exist to solve this problem. In this blog, we’ll explore one such technique called PCA ( Principal Component Analysis).

Stocks and bonds are the major categories of risky assets, and whilst bond portfolios could be analyzed using regression-based factor models a much more powerful factor analysis for bond portfolios is based on principal component analysis.

~ Practical Financial Econometrics, by Carol Alexander

PCA, while useful for a wide variety of portfolios, is particularly advantageous for bond portfolios because:

1) They are generally highly diversified, making dimensionality reduction indispensable.

2) Comparatively higher correlation between risk factors.

Given its explicit benefits, this blog will focus on risk factors relevant to bond portfolios. But before implementing PCA, let’s dive deeper into the various risk factors generally involved in bond portfolios.

**1. Interest Rate Risk:**

Interest rate risk is one of the bond portfolios’ most significant risk factors. It refers to the sensitivity of bond prices to changes in interest rates. Bonds with longer maturities are generally more sensitive to interest rate movements. To quantify interest rate risk, you can consider factors such as:

- Duration: Duration measures the sensitivity of bond prices to changes in interest rates. It quantifies how long it takes to recover the bond’s price through coupon payments and principal repayment.

**2. Credit Risk:**

Credit risk, or default risk, is the risk that the bond issuer may default on interest or principal payments. To quantify credit risk, you can consider factors such as:

- Credit Ratings: Credit ratings assigned by rating agencies indicate the creditworthiness of bond issuers.
- Spread Over Benchmark: The yield spread of a bond over a benchmark (e.g., Treasury yield) can indicate its credit risk.

**3. Liquidity Risk:**

Liquidity risk refers to the risk of not being able to buy or sell a bond at the desired price due to a lack of market liquidity. To quantify liquidity risk, you can consider factors such as:

- Bid-Ask Spread: A wider bid-ask spread suggests lower liquidity.
- Average Trading Volume: Bonds with higher trading volumes are typically more liquid.

**4. Inflation Risk:**

Inflation risk, also known as purchasing power risk, is the risk that the purchasing power of future bond cash flows may be eroded by inflation. To quantify inflation risk, you can consider factors such as:

- Real Yield: The real yield of a bond (nominal yield minus expected inflation) provides an indication of its inflation-adjusted return.

Let us look at an example of yield vs. rate in the previous years to gauge the effects on inflation.

The aim of PCA is “dimensionality reduction.” This denotes that we are trying to reduce the number of features (risk factors) to a smaller number of important ones we can operate on more quickly. We aim to remove the redundant dimensions by focusing on the dimensions with the most amount of data.

Let’s take a moment here to explore what we mean by dimensions by thinking of it in a more geometric sense.

While it may not be humanly possible to imagine an N-dimensional space, an N-dimensional space is just a mathematical representation of such a possibility. As mentioned before, our goal to reduce dimensions can be accomplished by identifying directions containing “most information”. We will see later that this so-called information can be measured through the variance percentage of each component.

We can find these so-called most informative directions by figuring out the correlation between various dimensions and removing the strongly related ones.

Look at the below image to get a better idea:

## Measuring Correlation

We make a covariance matrix of standardized input features (input data) for this purpose.

Now that we have a mathematical representation of our data, we move further to try and identify “directions” (when seen in the dimensional sense) wherein we can get the most data.

We can do this by making use of eigenvectors! Well, we won’t get into the nitty-gritty of the mathematics behind why our principal components are the same as the eigenvectors of the covariance matrix.

We take the unit vector along which Mean Squared Error (MSE) is minimum as our PC, to ensure maximum information is retained along the chosen direction, as can be seen in the second diagram. It can be shown mathematically that minimizing MSE would maximize variance, along the vector in consideration.

Thus emerges the heuristic that variance is directly proportional and preserved, and this is intuitive since high-variance principal components would contain a significant amount of information because they capture the dominant patterns and variations in the data and in contrast, low-variance components represent noise or less important variations.

Now, let us explore this further by practically applying these concepts to real-life market data.