# Variance of a portfolio – What it is, definition and concept | 2023

The variance of a portfolio is a way of determining the overall risk of a set of assets, using a weighted average of their individual variance and their mutual covariance.

Therefore, what we calculate is the intensity with which prices can vary on an average. But instead of doing it with each asset in the portfolio, now we are interested in a value that encompasses them all, to have an overall vision.

For example, we have two assets, one with low risk (variance) and the other high. In addition, we have verified that they are related, when the price of one rises, the other also rises. Therefore, what we do is calculate the risk of the entire portfolio, combining both assets.

## The variance of a portfolio and the risk

The variance and the standard deviation are two very useful statistical indicators in investments and what they do is measure the risk of these. In this way, if prices vary sharply, the risk of losing is very high, and conversely, stable prices imply low risk.

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In this case, we are interested in the variance of the assets that make up that portfolio. For example, stocks and public debt, whose risks are very different. Also, there may be a correlation between them that needs to be taken into account. The simple formula for two assets would be:

wi represents the portfolio weight of asset i and the same with wj and asset j. Delta squared is the variance of each asset and delta is the standard deviation; finally, pij is the linear correlation coefficient of both assets.

The reason for multiplying the correlation coefficient by two is because we are correlating two assets, A and B. In the example with three assets (A, B, C), we will see that we have to perform the operation three times, which are the possible correlations : AB, BC and AC.

As we can see, what we do is take into account their individual average volatility and also the joint one. Without forgetting the weight that each one represents over the total portfolio, that is, the weighting that we must do, as well as the correlation between them.

## Example of variance of a portfolio

Let’s look at a simple example. We have two assets, A and B. The first (A) assumes an investment of \$40,000 with a standard deviation of 10%. The second (B) of 60,000 dollars and with a standard deviation of 15%. In turn, the correlation coefficient is 0.65. The weight of each asset is calculated by dividing 40,000 and 60,000 by the \$100,000 of the total investment.

Remembering the formula, taking into account that we must operate in so much for one, wA is 0.4 (40%) and wB is 0.6 (60%). On the other hand, the variance of A and B is calculated by squaring 0.10 (10%) and 0.15 (15%). The pAB value is 0.65 (65%) and the standard deviation of A and B is 0.10 and 0.15.

The table and the numerical formula would be those that appear in the figure. Keep in mind that the value obtained is multiplied by 100 to express it in percentages.

The value of the variance is 1.44%. and the higher it is, the higher the overall risk. In this way, we can compare it with others and make decisions about them, if necessary, depending on our investor profile. that is, if we have more or less risk aversion.

We can do the same for the variance of a portfolio in case of three assets. The calculations must include the three weights, the three variances and three correlation coefficients: that of AB, that of BC and that of AC. In this case, the table and formula would be as follows: