## Portfolio VaR – Variance Covariance Approach using the Short Cut technique – PROOF!!

*in*Value at Risk

# Variance CoVariance VaR Shortcut approach

Portfolio VaR is a very important measure for assessing the market risk inherent in the entire portfolio of an entity. It is a measure whose calculation is often linked to heart burn because the risk manager envisions the very labor-intensive construction of the variance covariance matrix. In our courses on Value at Risk, Calculating Value at Risk & Portfolio VaR, we propose a remedy that should provide the user with some level of comfort- a short cut approach, introduced by Columbia University Business School’s Professor Mark Broadie, to the matrix using a weighted average series of portfolio returns.

However, it is human nature to question a doctor’s prescription; to seek a second opinion, and we’ve had a number of people ask us for proof of whether our short cut more efficient, practical and convenient version of calculating portfolio VaR really does give the portfolio VaR derived using the traditional Variance Covariance matrix. Or were the results simply coincidental, mathematical magic per se?!

The PROOF, lies in the very familiar statistical equation:

**Variance (aX+bY) = a ^{2}Variance(X)+b^{2}Variance(Y)+2abCovariance(X,Y)**

The square root of variance is standard deviation which, as you know, in Value at Risk terminology is volatility, the edifice of the Simple Moving Average Variance Covariance (SMA VCV) Approach to calculation of the metric.

The traditional Variance Covariance Approach methodology employs the construction of the infamous variance covariance matrix which in statistical equation terms is denoted by the right hand side (RHS) of the above equation- a conglomeration of squared weights, individual asset return variances and covariances between pairs of variables.

Our short cut approach focuses of oft-forgot left hand side (LHS) of the equation, i.e. the Variance of the Weighted Average Sum of Variables. If the Weighted Average Sum of Variables, aX+bY= Z then all we need is the Variance of Z. In terms of the value at risk calculation the variables are the daily return series for each asset in the portfolio; the weighted average sum of variables, i.e. Z, is the weighted average sum of daily return series; Z is therefore the portfolio return series. And therefore by calculating the Variance of Z, the weighted daily return series, square rooting the result and applying the appropriate multiplier factor representing the confidence level and holding period we arrive at the simple moving average variance covariance VaR result.

Low and behold the proof of our short cut approach…it is truly equal to the SMA VCV VaR using the traditional variance covariance methodology.

It should be noted however that if you are applying the EXCEL functions of VAR() and COVAR() to calculate the variances and covariance respectively there will be a slight difference in results obtained from the traditional and efficient methods. The error lies with the traditional approach as there is an inconsistency between the Variance and Covariance formulas underlying the EXCEL functions. The COVAR() formula in EXCEL uses a sample size of n in the divisor whereas VAR() employs a sample size of n-1. A simple adjustment may be made to COVAR() prior to use in the RHS of the equation above to remove this discrepancy, specifically:

Adjusted COVAR() =COVAR()*n/(n-1).

Alternatively, instead of the RHS given above we could use the following:

**a ^{2}Variance(X)+b^{2}Variance(Y)+2abCorrelation(X,Y)StandardDeviation(X)StandardDeviation(Y)**

*[Recall statistically Correlation(X,Y) = Covariance(X,Y)/StandardDeviation(X)StandardDeviation(Y)]*

In EXCEL the CORREL() function is given as follows:

This implicitly assumes consistency between the variance and covariance formulas, as the divisors of each cancel out. Using CORREL() instead of COVAR() removes the discrepancy between results obtained using the traditional approach to SMA VCV Value-at-Risk and results derived using the short cut approach.

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