**Portfolio Variance Formula (Table of Contents)**

## What is Portfolio Variance?

The term “portfolio variance” refers to a statistical value of modern investment theory that helps in the measurement of the dispersion of average returns of a portfolio from its mean. In short, it determines the total risk of the portfolio. It can be derived based on a weighted average of individual variance and mutual covariance.

### Portfolio Variance Formula

Mathematically, the portfolio variance formula consisting of two assets is represented as,

**Portfolio Variance Formula = w**

_{1}^{2 }***ơ**

_{1}^{2}+ w_{2}^{2}_{ }***ơ**

_{2}^{2}+ 2 ***ρ**

_{1,2}* w_{1 }* w_{2}***ơ**

_{1}***ơ**

_{2}where,

- w
_{i}= Portfolio weight of asset i - ơ
_{i}^{2}= Individual variance of asset i - ρ
_{i,j}**=**Correlation between asset i and asset j

Again, the variance can be further extended to a portfolio of more no. of assets, for instance, a 3-asset portfolio can be represented as,

**Portfolio variance formula = w _{1}^{2}_{ }* **

**ơ**

_{1}^{2}+ w_{2}^{2}_{ }***ơ**

_{2}^{2}+ w_{3}^{2}_{ }***ơ**

_{3}^{2}+ 2 ***ρ**

_{1,2}* w_{1 }* w_{2}***ơ**

_{1}***ơ**

_{2}+ 2 ***ρ**

_{2,3}* w_{2 }* w_{3}***ơ**

_{2}***ơ**

_{3}+ 2 ***ρ**

_{3,1}* w_{3 }* w_{1}***ơ**

_{3}***ơ**

_{1}### Explanation of the Portfolio Variance Formula

The portfolio variance formula of a particular portfolio can be derived by using the following steps:

**Step 1:** Firstly, determine the weight of each asset in the overall portfolio and it is calculated by dividing the asset value by the total value of the portfolio. The weight of the i^{th} asset is denoted by w_{i}.

**Step 2:** Next, determine the standard deviation of each asset and it is computed on the basis of the mean and actual return of each asset. The standard deviation of the i^{th} asset is denoted by ơ_{i}. The square of the standard deviation is variance i.e. ơ_{i}^{2}.

**Step 3:** Next, determine the correlation among the assets and it basically captures the movement of each asset relative to another asset. The correlation is denoted by ρ.

**Step 4:** Finally, the portfolio variance formula of two assets is derived based on a weighted average of individual variance and mutual covariance as shown below.

**Portfolio Variance formula = w _{1 }* **

**ơ**

_{1}^{2}+ w_{2 }***ơ**

_{2}^{2}+ 2 ***ρ**

_{1,2}* w_{1 }* w_{2}***ơ**

_{1}***ơ**

_{2}### Example of Portfolio Variance Formula (with Excel Template)

**Let us take the example of a portfolio that consists of two stocks. The value of stock A is $60,000 and its standard deviation is 15%, while the value of stock B is $90,000 and its standard deviation is 10%. There is a correlation of 0.85 between the two stocks. Determine the variance.**

Given,

- The standard deviation of stock A, ơ
_{A}= 15% - The standard deviation of stock B, ơ
_{B}= 10%

Correlation, ρ_{A,B} = 0.85

Below is data for calculation of portfolio variance of two stocks.

Weightage of Stock A, w_{A} = $60,000 / ($60,000 + $90,000) * 100%

Weightage of Stock A = 40% or **0.40**

Weightage of Stock B, w_{B} = $90,000 / ($60,000 + $90,000) * 100%

Weightage of Stock B= 60% or **0.60**

Therefore, the portfolio variance calculation will be as follows,

Variance = w_{A}^{2} * ơ_{A}^{2} + w_{B}^{2} _{ }* ơ_{B}^{2} + 2 * ρ_{A,B} * w_{A }* w_{B} * ơ_{A} * ơ_{B}

_{ }= 0.4^2* (0.15)^{2} + 0.6 ^2* (0.10)^{2} + 2 * 0.85 * 0.4 * 0.6 * 0.15 * 0.10

Therefore, the variance is **1.33%.**

### Relevance and Use

One of the most striking features of portfolio var is the fact that its value is derived on the basis of the weighted average of the individual variances of each of the assets adjusted by their covariances. This indicates that the overall variance is lesser than a simple weighted average of the individual variances of each stock in the portfolio. It is to be noted that a portfolio with securities having a lower correlation among themselves, end up with a lower portfolio variance.

The understanding of the portfolio variance formula is also important as it finds application in the Modern Portfolio Theory which is built on the basic assumption that normal investors intend to maximize their returns while minimizing the risk, such as variance. An investor usually pursues what is called an efficient frontier, and it is the lowest level of risk or volatility at which the investor can achieve its target return. Most often, investors would invest in uncorrelated assets to lower the risk as per Modern Portfolio Theory.

There are cases where assets that might be risky individually can eventually lower the variance of a portfolio because such an investment is likely to rise when other investments fall. As such this reduced correlation can help in reducing the variance of a hypothetical portfolio. Usually, the risk level of a portfolio is gauged using the standard deviation, which is calculated as the square root of the variance. The variance is expected to remain high when the data points are far away from the mean, which eventually results in a higher overall level of risk in the portfolio, as well.

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