# Residual variance formula in portfolio

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. Again, the variance can be further extended to a portfolio of more no. 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 square of the standard deviation is variance i. Step 3: Next, determine the correlation among the assets and it basically captures the movement of each asset relative to another asset.

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. There is a correlation of 0. Determine the variance. 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. This has been a guide to Portfolio Variance Formula. Here we discuss the calculation of Portfolio Variance along with the practical example and downloadable excel sheet. You can learn more about accounting from the following articles —.

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What is Portfolio Variance? Popular Course in this category. View Course.In probability theorythe law of total variance  or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law states that if X and Y are random variables on the same probability spaceand the variance of Y is finite, then.

## FRM Formula Sheet

In language perhaps better known to statisticians than to probability theorists, the two terms are the "unexplained" and the "explained" components of the variance respectively cf.

In actuarial sciencespecifically credibility theorythe first component is called the expected value of the process variance EVPV and the second is called the variance of the hypothetical means VHM.

Note that the conditional expected value E Y X is a random variable in its own right, whose value depends on the value of X. Similar comments apply to the conditional variance. In this formula, the first component is the expectation of the conditional variance; the other two rows are the variance of the conditional expectation. The law of total variance can be proved using the law of total expectation. The following formula shows how to apply the general, measure theoretic variance decomposition formula  to stochastic dynamic systems.

Let Y t be the value of a system variable at time t. The collections need not be disjoint. The decomposition is not unique. It depends on the order of the conditioning in the sequential decomposition.

One example of this situation is when XY have a bivariate normal Gaussian distribution. When E Y X has a Gaussian distribution and is an invertible function of Xor Y itself has a marginal Gaussian distribution, this explained component of variation sets a lower bound on the mutual information : . For higher cumulantsa generalization exists. See law of total cumulance. From Wikipedia, the free encyclopedia. In Casualty Actuarial Society ed.

### Single-index model

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### Portfolio Variance Formula

It refers to the total returns of the portfolio over a particular period of time. The portfolio variance formula is used widely in the modern portfolio theory. The portfolio variance formula is measured by the squaring the weights of the individual stocks in the portfolio and then multiplying it by the standard deviation of the individual assets in the portfolio and also squaring it. The numbers are then added by the covariance of the individual assets multiplied by two, also multiplied by the weights of each stock, also multiplying by a correlation between the different stocks present in the portfolio. Hence, the formula can be summarised as.

The standard deviation of the assets is 2. The correlation coefficient between A and B is 0. The standard deviation of A and B are 0.

We further have information that the correlation between the two stocks is 0. The portfolio variance formula is calculated by using the following steps Step 1: First, the weight of the individual stocks present in the portfolio is being calculated by dividing the value of that particular stock by the total value of the portfolio.

Step 3: The standard deviation of the stock from the mean is then calculated by first calculating the mean of the portfolio and then subtracting the return of that individual stock from the mean return of the portfolio. This has been a guide to Portfolio Variance Formula.

Here we discuss How to Calculate Portfolio Variance along with practical examples. We also provide downloadable excel template. You may also look at the following articles to learn more —. Forgot Password? This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy.

If you want to know more or withdraw your consent to all or some of the cookies, please refer to the cookie policy. By closing this banner, scrolling this page, clicking a link or continuing to browse otherwise, you agree to our Privacy Policy. Portfolio Variance Formula. Popular Course in this category. Course Price View Course. Login details for this Free course will be emailed to you. Free Investment Banking Course.Portfolio variance is a measure of the dispersion of returns of a portfolio.

It is the aggregate of the actual returns of a given portfolio over a set period of time. Portfolio variance is calculated using the standard deviation of each security in the portfolio and the correlation between securities in the portfolio. Modern portfolio theory MPT states that portfolio variance can be reduced by selecting securities with low or negative correlations in which to invest, such as stocks and bonds.

To calculate the portfolio variance of securities in a portfolio, multiply the squared weight of each security by the corresponding variance of the security and add two multiplied by the weighted average of the securities multiplied by the covariance between the securities. To calculate the variance of a portfolio with two assets, multiply the square of the weighting of the first asset by the variance of the asset and add it to the square of the weight of the second asset multiplied by the variance of the second asset.

Next, add the resulting value to two multiplied by the weights of the first and second assets multiplied by the covariance of the two assets. For example, assume you have a portfolio containing two assets, stock in Company A and stock in Company B. The wise investor seeks an efficient frontier.

That's the lowest level of risk at which a target return can be achieved. The correlation between the two assets is 2. To calculate the covariance of the assets, multiply the square root of the variance of Company A's stock by the square root of the variance of Company B's stock.

The resulting covariance is 0. The resulting portfolio variance is 0. Modern portfolio theory is a framework for constructing an investment portfolio. MPT takes as its central premise the idea that rational investors want to maximize returns while minimizing risk, sometimes measured using volatility. Therefore, investors seek what is called an efficient frontieror the lowest level of risk and volatility at which a target return can be achieved.

Following MPT, risk can be lowered in a portfolio by investing in non-correlated assets. That is, an investment that might be considered risky on its own can actually lower the overall risk of a portfolio because it tends to rise when other investments fall.

This reduced correlation can reduce the variance of a theoretical portfolio. In this sense, an individual investment's return is less important than its overall contribution to the portfolio in terms of risk, return, and diversification. The level of risk in a portfolio is often measured using standard deviationwhich is calculated as the square root of the variance. If data points are far away from the mean, the variance is high and the overall level of risk in the portfolio is high as well.

Standard deviation is a key measure of risk used by portfolio managersfinancial advisors, and institutional investors. Asset managers routinely include standard deviation in their performance reports. Portfolio Management. Technical Analysis.In statistics, residual variance is another name for unexplained variation, the sum of squares of differences between the y-value of each ordered pair on the regression line and each corresponding predicted y-value; it is generally used to calculate the standard error of estimate.

In other words, residual variance helps us confirm how well the regression line that we constructed fits the actual dataset. The smaller the variance, the more accurate the predictions are. Review your given dataset and create a two-column table depicting corresponding x and y values. You may use a pen and paper, a table in a Word document, or an Excel spreadsheet.

Start with the lowest given x-value and continue in ascending order. Create the equation of the regression line based your dataset. Find the slope m and the y-intercept b and input results into the equation:.

Review the resulting equation of the regression line. Calculate the unexplained deviation for each ordered pair xi, yi. The regression line expresses the best possible prediction of y, given x, but most of the time there is a variation of data points around the regression line. Calculate the residual variance.

Elina VanNatta started writing professionally in for various websites, including GuppyWeightLoss. She has more than five years of experience in the financial services industry and more than 10 years of experience in sales and marketing. She completed part of her higher education in Russia, attended DeVry University and earned a Bachelor of Science in marketing management from Western Governors University. Mathematik image by bbroianigo from Fotolia. Photo Credits Mathematik image by bbroianigo from Fotolia.What is the expected return for a single asset whose return is generated by a factor model?

The answer conforms nicely with intuition -- each uncertain term in the factor model equation can simply be replaced with its expected value. Thus, if:. This equation will continue to hold the left side value must equal the right side value and there will be one such equation for each possible scenario.

Next, add together the equations for all S possible scenarios. S, we have:. By definition, the first sum is the asset's expected return, the next m sums are the expected returns of the factors, and the last term is the expected residual return. If the expected residual return is represented by a ithe equation can be written as:.

This rather tortuous proof suffices for other situations involving linear functions. The expected value of a variable that is a linear function of other variables will itself be a linear function of the expected values of the variables in question, using the same constants here, b ij values.

The equation for an asset's expected return could be used to compute the expected return on each asset, one at a time. However, it is far more efficient to generalize it so that the entire vector of asset expected returns can be computed in one operation. This is straightforward. With respect to expected returns, it would appear that the use of a factor model has actually increased the number of required estimates.

In this approach, for N assets the Analyst needs N estimates of a i plus estimates of the expected values of the M factors. While this is true, there are at least some cases in which it is reasonable to assume that each asset has the same expected residual return, or that each such expected residual return is related in a simple way to the asset's factor sensitivities.

In such cases, the number of estimates required to specify asset expected returns may be considerably smaller than N. Even when this is not so, a small increase in the size of the task of estimating expected values is a reasonable price to pay for the substantial decreases in the magnitude the task of estimating risks, as the next section shows.

To determine the relationship between factor characteristics and those of an asset it is useful to re-examine the nature of covariance. Put in future terms, the covariance of asset i with asset j is the expected value of the product of 1 the deviation of asset i's return from its mean and 2 the deviation of asset j's return from its mean:.

Calculating covariance matrix using Excel

In words: the covariance of a variable with a constant times another variable equals the constant times the covariance of the two variables. The definition of covariance also implies that if r i ,r jand r k are returns:. In words: the covariance of a variable with the sum of two variables equals the sum of its covariances with the two variables.

Clearly, a similar statement holds for the relationship between the covariance of a variable with the difference between two other variables. Now, consider the covariance between two assets i and jwhere the returns of each are determined by a factor model. To keep notation to a minimum, let there be two factors. The relationships are thus:.

Substituting the right-hand sides of the equations, we have:. Using the relationships derived earlier, the right-hand side of this equation can be re-written as the sum of nine covariances, since there are three terms in each component. However, some of these will equal zero. The maintained assumptions of the factor model are that each residual is uncorrelated with that of any other variable, and that each residual is uncorrelated with each of the factors.

Including only the terms that could be non-zero, and dropping the tildes gives:. A small amount of reflection on this derivation will indicate that the matrix version of the formula is as applicable if there are more than two factors as it is if there are two. More impressively, the formula can be generalized to compute the entire covariance matrix for asset returns.

As before, let:.

## Law of total variance

For problems involving many securities N and relatively few factors m the number of potentially different estimated variables those on the right-hand side of the equation can be very much smaller than the number of asset covariances on the left-hand side of the equation. For each of the covariances matrices C and CF we count only the elements on and below the diagonal, since once those are known, the remainder can be filled in.

Taking this into account, the numbers of values to be determined for each component are:.