## Market Risk Metrics – Portfolio Volatility

*in*Optimization

# Portfolio Volatility

The riskiness of a given portfolio may be gauged by the riskiness of the instruments that make up the portfolio. However the portfolio risk or volatility of portfolio returns is not necessarily equal to the sum of each instrument’s risk as given by their respective volatility. This is because the instruments within a portfolio may not be independent of each other which would mean that the impact of correlations would need to be factored into the calculation of portfolio volatility.

Volatility for a portfolio may be calculated using the statistical formula for the variance of the sum of two or more random variables which is then square rooted. Alternatively the volatility for a portfolio may be calculated based on the weighted average return series calculated for the portfolio. Both of these methodologies are discussed below.

**Portfolio Volatility by formula **

For a portfolio comprising of three assets X, Y and Z having weights in the portfolio of a, b and c respectively the portfolio volatility is given by the following formula:

Variance (X) = Variance in asset X’s returns, i.e. X’s returns volatility squared (?_{x}^{2})

Variance (Y) = Variance in asset Y’s returns, i.e. Y’s returns volatility squared (?_{y}^{2})

Variance (Z) = Variance in asset Z’s returns, i.e. Z’s returns volatility squared (?_{Z}^{2})

_{Cor_xy} = correlation coefficient of X and Y returns

_{Cor_xz} = correlation coefficient of X and Z returns

_{Cor_yz} = correlation coefficient of Y and Z returns

If the assets in the portfolio are independent of each other the correlation coefficient terms in the equation above would be zero. If the assets in portfolio are perfectly correlated with each other then the volatility of the portfolio would simply be the weight average sum of the asset return volatility.

Let us consider the following example:

Our portfolio consists of the following two stocks ABC and XYZ, whose weights within the portfolio are 89% and 11% respectively. The stock price data for month of January 2009, our period of analysis is given below:

Next, calculate the daily return time series from the price data. The daily returns are calculated by taking the natural logarithm of the ratio of successive (consecutive) prices:

The resulting return series for each stock are given below:

We then calculate the variance in daily returns of the stocks using the EXCEL function VAR() applied to the each stocks return series in turn. The calculation methodology for variance using VAR() has been discussed in an earlier post:

Market Risk Metrics – Beta with respect to market indices

The variance for stock ABC works out to 0.141% whereas the variance for stock XYZ works out to 0.578%. The standard deviation or daily volatilities equal to the square root of these variances. They are 3.76% and 7.60% for stocks ABC and XYZ respectively.

Next we calculate the correlation coefficient for Stock ABC and XYZ returns. The calculation methodology for correlation coefficient has been reviewed in the following post:

Correlation- Correlation coefficient

The correlation coefficient for Stocks ABC and XYZ returns is 0.64014.

Using the formula given above we can now calculated the portfolio volatility:

Portfolio volatility = Root(89%^{2}×0.141%+11%^{2}×0.578%+2×89%×11%×0.64014×3.76%×7.60%)=3.93%. Note that this is daily portfolio volatility.

**Portfolio volatility by weighted average returns**

As you can see this formula method for calculating portfolio volatility can get pretty cumbersome as the number of instruments within the portfolio increases. This is because one is required to calculate the variance for each instrument within the portfolio as well as the correlation coefficients for each pair of instruments.

A simpler and more practical way of handling this calculation is to first derive a weighted average return series for the portfolio. The weighted average return series for the portfolio is determined for each time point for which data is available and is calculated as Sum (**weight×return)** across all instruments in the portfolio.

This is illustrated for our two asset sample portfolio below:

The weighted average portfolio return on 2-Jan-2009 has been calculated as follows:

Weighted Return for Portfolio on 2-Jan-2009 = Weight of Stock ABC in portfolio × Return of Stock ABC on 2-Jan-2009 + Weight of Stock XYZ in portfolio × Return of Stock XYZ on 2-Jan-2009 = 89%× (-1.19%) + 11% × (-1.65%) = -1.24%.

Once the weighted average return series for the portfolio has been calculated it can be treated just like any other return time series calculated for individual instruments. The daily portfolio volatility is then calculated using EXCEL’s STDEV() function applied to the weighted average return series of the portfolio. The calculation methodology for daily volatility using the STDEV() function was discussed in the post:

Market Risk Metrics- Sharpe and Treynor Ratios

The portfolio volatility calculated using this methodology works out to 3.93%, i.e. the same result as that derived using the statistical formula methodology. Again it should be noted that we have calculated the daily volatility here. In order to calculate portfolio volatility for a J-day period the daily volatility for the portfolio may be scaled using the following formula:

J-day volatility = Daily volatility ×Root(J)

*Like this post – check out the new book – Portfolio Optimization Models in Excel, Revised Edition – 227 pages, Excel templates and dataset included.*