Estimating Portfolio Volatility for Variance-Covariance VaR

Summary

The document explains how the variance-covariance approach estimates portfolio volatility for value at risk. It requires historical returns for each asset to estimate the variance-covariance matrix, then combines that matrix with portfolio weights to calculate portfolio variance. Under the method’s normal-return assumption, the resulting volatility and mean support a VaR estimate.

The answer notes that this approach is not necessarily easier to implement than historical VaR, which can simulate portfolio returns from observed history. Its practical advantage may be speed for portfolios such as equities: once the covariance matrix is available, portfolio variance follows from a matrix calculation. The discussion is brief and does not provide a full VaR calculation, compare estimation windows, or assess how well normality fits real returns. The efficiency point is therefore a computational consideration, not evidence that the method produces more accurate risk estimates.

Key ideas

• Historical returns for each asset are needed to estimate the covariance matrix.
• Portfolio variance is calculated from portfolio weights and the covariance matrix.
• The normality assumption allows VaR to be based on the portfolio mean and standard deviation.
• Matrix-based variance calculations can be faster for some portfolios than repeated historical repricing.

Tags

# Variance-Covariance VaR: how to get the volatility?


# Variance-Covariance VaR: how to get the volatility?












Because the variance-covariance VaR assumes that the returns are normally distributed, in theory it is easy to get VaR by simply finding the mean and the volatility (standard deviation) of the portolio returns.

The volatility of the portfolio is retrieved by building a variance-covariance matrix. However how do you get the volatility of each asset? Do you need to get historical returns for each asset? If that is the case then how is it easier to implement than historical VaR where full history is needed to reprice the portfolio?

## Answer by Tim Wilding (score 0, accepted)

https://quant.stackexchange.com/a/42434

Yes, you would need historical returns for each asset to calculate a variance-covariance matrix.

I am not sure that people think using the variance-covariance matrix is "easier". As you point out, using historical returns makes it very simple to simulate the portfolio returns. On the other hand, there are various practical considerations that might make using a variance-covariance matrix more efficient. For example, the calculation of VaR is likely to be much quicker for equity portfolios since it just involves using simple matrix multiplication to calculate the variance ($=x'Vx$).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.