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Investing in stocks is like orchestrating a band. You don’t want all drummers or just vocalists — you need a mix to create a balanced music. Similarly, a good portfolio isn’t just about picking high-performing stocks; it’s about understanding how they move together. That’s where risk assessment comes in, and more importantly, where concepts like covariance and correlation become essential.

Understanding Portfolio Risk

Risk in a stock portfolio isn’t just about how volatile a single stock is. It’s about how all the stocks interact. You could have stocks that are individually risky, but together they form a stable portfolio because their ups and downs cancel each other out. This is why understanding the relationship between stocks is crucial.

Mathematically, portfolio risk is measured using standard deviation, variance, and the all-important covariance and correlation.

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Systematic vs. Unsystematic Risk

When analysing portfolio risk, it’s essential to distinguish between systematic and unsystematic risk.

• Systematic Risk: Also known as market risk, this is the risk inherent to the entire market or a particular segment. It includes factors like interest rate changes, inflation, recessions, and geopolitical events. This risk cannot be eliminated through diversification.

• Unsystematic Risk: Also called specific or idiosyncratic risk, this pertains to individual stocks or industries. It includes risks like a company’s poor earnings report, management changes, or industry downturns. This risk can be reduced through diversification.

What Risk Is Measured by Covariance and Correlation?

Covariance and correlation help in assessing unsystematic risk — the risk that can be mitigated by diversification. These metrics show how individual stocks move relative to each other, helping investors pick stocks that reduce overall volatility. While systematic risk remains no matter how diversified a portfolio is, covariance and correlation help minimize the impact of stock-specific risks by balancing assets that don’t move in sync.

How Many Stocks Should You Hold?

The goal of diversification is to reduce unsystematic risk. Studies suggest that holding around 20–30 well-selected stocks significantly reduces unsystematic risk, beyond which additional stocks offer diminishing benefits. However, systematic risk remains, as it impacts the market as a whole.

For an investor, this means:

1. Don’t over-diversify: Beyond a certain point, adding more stocks won’t reduce overall risk significantly but may dilute potential returns.

2. Focus on asset correlation: Instead of just adding stocks, choose those that don’t move in tandem to optimize risk reduction.

3. Accept that some risk is unavoidable: No matter how diversified your portfolio, systematic risk will always be present.

Covariance: The Relationship Between Stock Movements

Covariance measures how two stocks move relative to each other. If Stock A and Stock B tend to rise and fall together, they have a positive covariance. If one rises when the other falls, they have a negative covariance.

Covariance is calculated using the formula:

However, covariance alone doesn’t tell the full story — it depends on the scale of the stock prices. That’s why we need correlation.

Correlation: The Standardized Relationship

Correlation is covariance normalized by the standard deviations of both stocks. It ranges from -1 to 1:

• 1: The stocks move perfectly together.

• 0: No relationship.

• -1: They move in exactly opposite directions.

The correlation formula is:

Why Covariance and Correlation Matter for Portfolio Risk

In a portfolio of multiple stocks, risk isn’t just the sum of individual risks. A diversified portfolio aims to combine stocks that don’t move together, reducing overall risk. By assessing covariance and correlation, you can identify stocks that hedge each other’s risks, making your portfolio more stable.

Let’s see this in action with some Python code.

Python Example: Calculating Portfolio Risk

We’ll analyse four stocks — TCS, Infosys, ITC, and Reliance — to see how they interact and what that means for portfolio risk.

import numpy as np
import pandas as pd
import yfinance as yf

# Download stock data
stocks = ['TCS.NS', 'INFY.NS', 'ITC.NS', 'RELIANCE.NS']
data = yf.download(stocks, start='2023-01-01')['Close']

# Calculate daily returns
returns = data.pct_change().dropna()

# Covariance matrix
cov_matrix = returns.cov()
print("Covariance Matrix:\n", cov_matrix)

# Correlation matrix
corr_matrix = returns.corr()
print("\nCorrelation Matrix:\n", corr_matrix)

# Portfolio standard deviation assuming equal weights
weights = np.array([0.25, 0.25, 0.25, 0.25])  # Equal allocation to each stock
portfolio_variance = np.dot(weights.T, np.dot(cov_matrix, weights))
portfolio_std_dev = np.sqrt(portfolio_variance)

print("\nPortfolio Standard Deviation (Risk):", portfolio_std_dev)

The above snapshot shows the result of executing the python code. This risk in this case is ~0.0089. Also, observe that the stocks from same industry (here TCS and Infosys are from IT industry) have high correlation as compared to their correlation to ohter stocks from other industries

Interpreting the Results

• The covariance matrix shows how each stock moves in relation to the others.

• The correlation matrix helps us see which stocks are positively or negatively correlated.

• The portfolio standard deviation gives a numerical measure of overall risk.

How to Interpret Portfolio Standard Deviation

The portfolio standard deviation, as calculated in the Python example, represents the overall risk (volatility) of the portfolio. A higher value indicates greater fluctuations, meaning the portfolio is riskier. A lower value suggests a more stable investment.

For an investor, this number helps in:

1. Risk assessment: If the standard deviation is too high, the portfolio may be too volatile for conservative investors.

2. Portfolio optimisation: Investors can adjust stock weights or add assets with lower correlation to reduce overall risk.

3. Comparing investment choices: By comparing standard deviation across different portfolios, an investor can choose one that aligns with their risk tolerance.

Key Takeaways

1. Diversification is key: A mix of stocks with low or negative correlation reduces risk.

2. Covariance and correlation assess unsystematic risk, not systematic risk.

3. A lower portfolio standard deviation means a more stable investment.

4. Investors use standard deviation to gauge overall volatility and adjust their strategy accordingly.

So, next time you’re picking stocks, don’t just look at individual performances — check how they interact. A well-balanced band sounds better, and a well-diversified portfolio performs better in the long run.