Chapter 10 - Calculating the Variance-Covariance Matrix PDF

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CHAPTER 10 – CALCULATING THE VARIANCE - COVARIANCE MATRIX

Overview In order to calculate efficient portfolios, we must be able to compute the variance-covariance matrix from return data for stocks. The most obvious calculation is the sample variancecovariance matrix: This is the matrix computed directly from the historic returns. While the sample variance-covariance matrix may appear to be an obvious choice, a large literature recognizes that it may not be the best estimate of variances and covariances due to its often unrealistic parameters and from its inability to predict. This chapter makes heavy use of the array functions Transpose( ) and MMult( ) as well as some other “home-grown” array functions.

Computing the Sample Variance-Covariance Matrix

Using the Excel function 𝒍𝒏(𝑷𝒕 ⁄𝑷𝒕&𝟏 ) we can compute monthly returns (example on page 253).

A VBA Function to Compute the Variance-Covariance Matrix To automate this procedure, we write a VBA function that compute the variance-covariance matrix using the Excel function Covariance.S, When excel functions with periods, such as Covariance.S, are used in VBA, the period becomes an underscore: Covariance_S. The VBA computes Covariance.S for every entry of the variance-covariance matrix.

Should We Divide by M–1 or by M Excel 2010 and later Covariance.S

Comments Sample covariance, divides by M–1

When used in VBA Application. WorksheetFunction. Covariance_S

Covariance.P

Population covariance, divides by M

Application. WorksheetFunction. Covariance_P

Var.S

Sample variance

Application. WorksheetFunction Var_S

Var.P

Population variance

Application. WorksheetFunction. Var_P

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CHAPTER 10 – CALCULATING THE VARIANCE - COVARIANCE MATRIX

The Correlation Matrix Using the Excel function {CorrMatrix} we can compute the correlation matrix of the returns. Another version of the correlation matrix, which only computes the upper half is {CorrMatrixTriangular( ) }

Computing the Global Minimum Variance Portfolio (GMVP) The two most prominent uses of the variance-covariance matrix are to find the global minimum variance portfolio (GMVP) and to find efficient portfolios. The particular fascination of the minimum variance portfolio is, that it is the only portfolio on the efficient frontier whose computation does not require the asset expected returns.

Four Alternatives to the Sample Variance-Covariance Matrix 1. The Single-index model assumes that the only sources of variance risk are the market variance and the betas of the assets. 2. The Constant correlation model assumes the correlation between all asset is constant, so that 𝜎+, = 𝜌/𝜎+ /𝜎, 3. Shrinkage methods assume that the variance-covariance matrix is a convex combination of the sample variance-covariance and a matrix with variances on the diagonal and zeros elsewhere. 4. Option methods use options to derive the standard deviations of returns for the assets.

Alternatives to the Sample Variance-Covariance: The Single-Index Model (SIM) The single-index model (SIM) began as an attempt to simplify some of the computational complexities of calculating the variance-covariance matrix. The basic assumption of the SIM is that the returns of each asset can be linearly regressed on a market index x: 𝑟+ = 𝛼+ + 𝛽+ 𝑟4 + 𝜀+ Where the correlation between & and £, is zero. Given this assumption, it is easy to establish the following two facts: • 𝐸(𝑟+ ) = 𝛼+ + 𝛽+ 𝐸(𝑟4 ) •

𝜎+, = 7

𝛽+ 𝛽, 𝜎48 𝑤ℎ𝑒𝑛 /𝑖 ≠ 𝑗 /////////////// / 8 𝑤ℎ𝑒𝑛/𝑖 = 𝑗 𝜎+

Essentially the SIM involves changes in the estimates of the covariances, but not the sample variance. We can automate the procedure for computing the SIM by writing some VBA code.

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CHAPTER 10 – CALCULATING THE VARIANCE - COVARIANCE MATRIX

Alternatives to the Sample Variance-Covariance: Constant Correlation The constant correlation model of Elton and Gruber (1973) computes the variance-covariance matrix to assuming that the variances of the asset returns are the sample returns, but that the covariances are all related by the same correlation coefficient, which is generally taken to be the average correlation coefficient of the assets in question. Since 𝐶𝑜𝑣C𝑟+ , 𝑟,E = 𝜎+, = 𝜌+, 𝜎+ 𝜎, , this means that in the constant correlation model: 𝜎+, = F

𝑤ℎ𝑒𝑛/𝑖 = 𝑗 𝜎++ = 𝜎+8 /////////////// / 𝜎+, = 𝜌𝜎+ 𝜎, 𝑤ℎ𝑒𝑛/𝑖 ≠ 𝑗

Alternatives to the Sample Variance-Covariance: Shrinkage Methods A third class of methods of estimating the variance-covariance matrix has recently achieved popularity. So-called shrinkage methods assume that the variance-covariance matrix is a convex combination of the sample covariance matrix and some other matrix: 𝑆ℎ𝑟𝑖𝑛𝑘𝑎𝑔𝑒/𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 − 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒/𝑚𝑎𝑡𝑟𝑖𝑥 = /𝜆 ∙ 𝑆𝑎𝑚𝑝𝑙𝑒/𝑣𝑎𝑟 − 𝑐𝑜𝑣 + ( 1 + 𝜆 ) ∙ 𝑂𝑡ℎ𝑒𝑟𝑀𝑎𝑡𝑟𝑖𝑥

There is little theory about choosing the proper shrinkage estimator. Our suggestion is to choose a shrinkage operator 𝜆 so that the GMVP is wholly positive.

Using Option Information to Compute the Variance Matrix Another way to compute the variance matrix is to use the information from the options market. We use the implied volatility for each of the stocks from their at-the-money call options and then compute the variance matrix using constant correlation: 8 𝑤ℎ𝑒𝑛 /𝑖 = 𝑗 𝜎+,+WXY+Z[ /////////////// / 𝜎+, = F 𝜌𝜎+,+WXY+Z[ 𝜎,,+WXY+Z[ 𝑤ℎ𝑒𝑛/𝑖 ≠ 𝑗

Excel function: CallVolatility We can then use the implied volatilities as the basis for a constant correlation variancecovariance matrix. Excel Function: ImpliedVolVarCov

Which Method to Compute the Variance-Covariance Matrix? • The sample variance-covariance • The single-index model • The constant correlation approach • Shrinkage methods • Implied volatility-based variance-covariance matrices The choice of how to compute the variance-covariance matrix is largely a question of how you view capital markets. If you strongly believe that the past predicts the future, then perhaps your choice should be to use the sample varcov matrix . Our preference, however, is to use an option-based volatility model with a changing correlation: In “normal’’ times we would use a “normal” correlation of between 0.2 an 0.3; in times of crisis, we would use a much higher correlation, say, 𝜌 = 0.5 − 0.6.

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