# Portfolio Variance: Formula, Calculation, and Significance

Summary:

Portfolio variance is a crucial concept in modern portfolio theory (MPT) that assesses the risk of a diversified investment portfolio. It’s calculated by considering the individual variances and correlations of assets within the portfolio. This comprehensive article explores the definition, formula, calculation, and significance of portfolio variance in investment decision-making.

## Understanding portfolio variance

Portfolio variance is a fundamental metric for investors looking to build diversified portfolios that balance risk and return. It measures how the actual returns of a portfolio comprising various assets fluctuate over time. The key components of portfolio variance are:

### Weights and variances

The portfolio variance calculation takes into account the weights and variances of each asset within the portfolio. In simpler terms, it considers how much of each asset is included in the portfolio and how volatile each asset’s returns are.

### Correlations

Additionally, portfolio variance considers the correlations between pairs of assets in the portfolio. Correlation measures the degree to which the returns of two assets move in relation to each other. A lower correlation between assets results in a lower portfolio variance, which implies reduced overall risk.

Now, let’s delve deeper into the formula and calculation of portfolio variance:

## Formula and calculation of portfolio variance

The formula for calculating portfolio variance in a two-asset portfolio is as follows:

Portfolio Variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov₁,₂

Where:

• w₁ = the portfolio weight of the first asset
• w₂ = the portfolio weight of the second asset
• σ₁ = the standard deviation of the first asset
• σ₂ = the standard deviation of the second asset
• Cov₁,₂ = the covariance of the two assets, expressed as p(1,2)σ₁σ₂, where p(1,2) is the correlation coefficient between the two assets

It’s important to note that portfolio variance is equivalent to the portfolio standard deviation squared, which is a measure of risk.

As the number of assets in the portfolio increases, the number of terms in the variance calculation grows exponentially. For example, a three-asset portfolio has six terms, while a five-asset portfolio has 15 terms in the variance calculation. ## Portfolio variance and modern portfolio theory

Modern portfolio theory (MPT) provides a framework for constructing investment portfolios that aim to maximize returns while minimizing risk. MPT operates on the principle that rational investors seek to achieve an efficient frontier—a balance between risk and return.

Risk reduction in MPT portfolios is achieved by investing in non-correlated assets. Even assets that may be considered individually risky can contribute to reducing the overall risk of a portfolio when their movements are not strongly correlated. This reduced correlation can lower the variance of a portfolio.

In this context, an individual asset’s return is less important than its overall contribution to the portfolio’s risk, return, and diversification.

## Example of portfolio variance

Let’s illustrate portfolio variance with an example. Consider a portfolio comprising two stocks: Stock A and Stock B. Stock A has a value of \$50,000 and a standard deviation of 20%, while Stock B is worth \$100,000 with a standard deviation of 10%. The correlation between these stocks is 0.85. Calculate the portfolio variance as follows:

Variance = (33.3%² × 20%²) + (66.7%² × 10%²) + (2 × 33.3% × 20% × 66.7% × 10% × 0.85) = 1.64%

Variance, by itself, is a statistic that may not be easy to interpret. To gain a more realistic perspective of portfolio risk, analysts often calculate the standard deviation, which is the square root of variance. In this example, the square root of 1.64% is 12.81%.

## How is variance used in constructing a portfolio?

Most portfolio managers aim to minimize risk and maximize value, aligning with the principles of modern portfolio theory (MPT). The greater the variance in a portfolio, the higher the variance of the individual assets within it, and consequently, the greater the overall risk. Portfolio managers mitigate risk by including assets with low correlations, indicating minimal relationship in the movement of assets within the portfolio.

## Where does standard deviation fit in?

Most portfolio analysts focus on the standard deviation of the portfolio as a whole to get the best picture of the range of outcomes in the portfolio. Standard deviation is the square root of the variance and provides a more realistic look at the level of risk of the portfolio. The higher the standard deviation, the more volatile a portfolio is likely to be, and vice versa.

## Real-life application of portfolio variance

Understanding how portfolio variance works in practice is essential for investors. Let’s consider an example:

Imagine an investor has a portfolio consisting of three assets: Stocks X, Y, and Z. The portfolio allocation is as follows:

• Stock X: 40%
• Stock Y: 30%
• Stock Z: 30%

Now, we need to calculate the portfolio variance based on historical data:

Stock X has an annual standard deviation of 15%, Stock Y has a standard deviation of 20%, and Stock Z has a standard deviation of 25%. Additionally, the correlations are as follows:

• Correlation between X and Y: 0.6
• Correlation between X and Z: 0.3
• Correlation between Y and Z: 0.5

Let’s apply the formula to calculate the portfolio variance:

Portfolio Variance = (40%² × 15%²) + (30%² × 20%²) + (30%² × 25%²) + (2 × 40% × 30% × 0.6 × 15% × 20%) + (2 × 40% × 30% × 0.3 × 15% × 25%) + (2 × 30% × 25% × 0.5 × 20% × 25%)

After performing the calculations, we find the portfolio variance. Converting this variance into standard deviation will give us a clearer understanding of the risk associated with this portfolio.

## Benefits of diversification

Diversification plays a pivotal role in managing portfolio risk, and portfolio variance is a crucial tool in understanding how it works. Let’s explore the benefits of diversification in more detail:

### Reducing unsystematic risk

Unsystematic risk refers to the risk associated with individual assets in a portfolio. By combining assets with low or negative correlations, investors can reduce this risk. For instance, if one stock in a portfolio underperforms, another may compensate for the loss, minimizing the impact of individual asset fluctuations.

Diversification allows investors to achieve a more favorable risk-return trade-off. By spreading investments across different asset classes, industries, or geographical regions, investors can potentially maintain their expected returns while reducing overall portfolio risk. This approach aligns with the core principles of modern portfolio theory (MPT).

WEIGH THE RISKS AND BENEFITS

Here is a list of the benefits and drawbacks to consider when it comes to portfolio variance.

##### Pros
• Measures portfolio risk comprehensively
• Allows for diversification to manage risk
• Integral to modern portfolio theory
##### Cons
• Complexity increases with more assets
• Requires accurate data and calculations
• Does not account for extreme events (fat tails)

## The bottom line

Variance is a vital statistical measure for assessing the volatility and risk of both individual securities and a portfolio as a whole. However, the main figure to focus on is the portfolio’s standard deviation, which is the square root of the portfolio’s variance. A higher standard deviation indicates greater portfolio risk, while a lower standard deviation implies less risk.

Portfolio managers use variance and standard deviation to make informed investment decisions, adjusting their holdings to include assets with low correlations to manage and potentially reduce portfolio risk.

As you explore the world of investments, understanding portfolio variance and its significance in risk management is a key step towards becoming a knowledgeable investor.

### What is the portfolio variance formula?

The portfolio variance formula is a mathematical equation used to calculate the risk or volatility of a diversified investment portfolio. It takes into account the weights, variances, and correlations of individual assets within the portfolio to assess overall risk.

### How is portfolio variance calculated?

Portfolio variance is calculated by multiplying the square of each asset’s weight in the portfolio by its corresponding variance and then summing up these values. Additionally, the formula includes terms that account for the covariance between pairs of assets in the portfolio.

### Why is portfolio variance important for investors?

Portfolio variance is important for investors because it provides a quantitative measure of the risk associated with a portfolio. It helps investors understand how the returns of their investments may fluctuate over time and assists in making informed decisions to balance risk and return.

### What does a high portfolio variance indicate?

A high portfolio variance indicates that the individual assets within the portfolio have significant variations in returns, leading to greater overall portfolio risk. This suggests that the portfolio may experience larger fluctuations in value, which can be a concern for risk-averse investors.

### How can investors use portfolio variance in their decision-making?

Investors can use portfolio variance to make decisions about asset allocation and diversification. By analyzing portfolio variance, investors can adjust their holdings to include assets with low correlations, which can help reduce overall portfolio risk while aiming to achieve their desired level of return.

### Is there a relationship between portfolio variance and standard deviation?

Yes, there is a relationship between portfolio variance and standard deviation. Portfolio variance is equal to the square of the portfolio’s standard deviation. Standard deviation is often used as a more intuitive measure of risk because it represents the volatility of the portfolio’s returns in percentage terms.

### What are the limitations of using portfolio variance?

While portfolio variance is a valuable tool for risk assessment, it has limitations. It assumes that asset returns are normally distributed, which may not always be the case in reality. Additionally, it does not account for extreme events (fat tails) that can impact portfolio performance.

### How can investors interpret the results of portfolio variance?

Investors can interpret the results of portfolio variance by considering their risk tolerance and investment goals. A lower portfolio variance suggests lower risk, while a higher variance indicates higher risk. Investors should aim to strike a balance between risk and return that aligns with their financial objectives.

## Key takeaways

• Portfolio variance assesses the risk of a diversified investment portfolio.
• It considers the weights, variances, and correlations of assets within the portfolio.
• Modern portfolio theory (MPT) emphasizes the importance of portfolio variance in managing risk.
• Standard deviation, derived from variance, is a key measure of portfolio risk.
• Portfolio managers use variance and standard deviation to optimize risk-return trade-offs.
###### View article sources
1. 1 Portfolio mean and variance – Columbia University
2. Chapter 7 Portfolio Theory – Caltech
3. Handout 6: Portfolio Variance with Many Risky Assets – Wharton Finance