Basic Statistics for Risk Management in Banks and Financial Institutions by Arindam Bandyopadhyay;

Author:Arindam Bandyopadhyay;
Language: eng
Format: epub
Publisher: OUP Premium
Published: 2022-06-15T00:00:00+00:00

Eq. 5.11

This can be expressed in a general mathematical form:

Eq. 5.11a

Note that σ is capturing the standard deviation of each assets i and j and ρ is representing the correlation between assets i and j.

The above equation can also written in matrix form:

where S is the standard deviation of assets in the pool matrix; R is the covariance or correlation; and ST is the transpose of standard deviation matrix.

This gives us algebraic expression for variance. Thus, a variance–covariance matrix is a square matrix that contains the variances in the diagonal elements and covariances in the off-diagonal elements. Note that here weights can be adjusted accordingly if standard deviation of returns is estimated as percentage.

When the portfolio has multiple assets and multiple sub-segments, we get a better result through matrix solution to estimate portfolio volatility using variance–covariance matrix. Several statistical packages like Palisade, @Risk can accommodate this matrix pattern and give us portfolio variance which can be further used in estimating VaR and also for measuring portfolio efficiency. Monte Carlo simulation can be run using random numbers to find portfolio profit or loss to obtain VaR. The correlation coefficients can be specified in @RISK using the ‘define correlation’ function in the toolbar. Note that estimation portfolio volatility enables a risk manager to compare expected return with volatility of its portfolio of instruments. Many portfolio optimization models are based on this analysis.

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