FINANCIAL MODELLING (9)

• Suppose we have return data for N assets over M periods. Writing the return of asset i in period t as rit, we write the mean return of asset i as
1 ri M
r11 r1 r r A Matrixof Excess Returns 12 1 ... r1M r1 r21 r2 r22 r2 ... r2 M r2 ... rN 1 rN ... rN 2 rN ... ... ... rNM rN
Computing the Global Minimum Variance Portfolio
• The mean µGMVP of the GMVP and variance σ2 of the GMVP are given by • The following spreadsheet presents the GMVP in action, in the six stock portfolio.
Computing the Global Minimum Variance Portfolio
• The two most prominent uses of the variance-covariance matrix are to find the global minimum variance portfolio (GMVP) and to find efficient portfolios. • Both uses illustrate the problematics of working with sample data, and show why there’s been a movement away from the sample variance-covariance matrix
S 1[ E (r ) c] EfficientPortfolio Sum {S 1[ E (r ) c]}
Computing an Efficient Frontier
• Row 16 of the spreadsheet shows the market values of the individual stocks on the last date of the data (beginning of January 2004). • Row 17 computes the proportions of a portfolio composed of the market weights of each asset. • The difference between the market-weighted portfolio and the ‘efficient’ portfolio is striking. • Note that the market weights assume that the stock’s values are in equilibrium, so are a good “naïve” choice for initial portfolio values.
– Computed directly from historical returns
• However, it has been recognised that this estimator may not be the most appropriate
– Unrealistic parameters (e.g. spurious correlations) – Inability to make accurate predictions
Alternative Methods for Computing the Variance-Covariance Matrix
• We can use a VBA function to calculate the sample variance-covariance matrix.
Function VarCovar(rng As Range) As Variant Dim i As Integer Dim j As Integer Dim numCols As Integer numCols = rng.Columns.Count numRows = rng.Rows.Count Dim matrix() As Double ReDim matrix(numCols - 1, numCols - 1) For i = 1 To numCols For j = 1 To numCols matrix(i - 1, j - 1) = Application.WorksheetFunction.Covar(rng.Columns(i), rng.Columns(j)) _ * numRows / (numRows - 1) Next j Next i VarCovar = matrix End Function
Alternative Methods for Computing the Variance-Covariance Matrix
• Another way to calculate the variancecovariance matrix uses to Excel’s OFFSET function. • OFFSET allows you to define a block of cells relative to some initial cell. • Thus, for example OFFSET(initial cells, rows, columns) refers to a block of cells of the same size as the initial cells, but rows and columns over from the initial cells. • This can be incorporated with some clever naming of the row and column headers.
µGMVP = xGMVP.E(r), σ2GMVP = xGMVP .S. xTGMVP.
Computing an Efficient Frontier
• We have seen (last lecture) that we can compute the efficient frontier by solving the equation
Computing the Global Minimum Variance Portfolio
• Suppose there are N assets having a variancecovariance matrix S. The GMVP is the portfolio x = {x1, x2, …, xN} that has the lowest variance among all feasible portfolios. • The minimum variance portfolio is defined by:
FINANCIAL MODELLING
Lecture 12 Constrained Portfolio Optimization
Calculating the VarianceCovariance Matrix
• In order to be able 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
Calculating the Sample VarianceCovariance Matrix
• Columns of matrix A subtract the mean asset return from the individual asset returns. • The transpose of the matrix is
• Shrinkage approaches are a simple, effectபைடு நூலகம்ve way of dealing with inaccurate Variance-Covariance matrices
Calculating the Sample VarianceCovariance Matrix
r ,
t 1 it
M
i 1,2,...,N
• Then the covariance of the return of asset i with asset j is calculated as
1 M ij Cov(i, j ) (rit ri ) (rjt rj ), i,j 1,2,..., N M 1 t 1
Calculating the Sample VarianceCovariance Matrix
• The matrix of these covariances (which includes, of course, the variances when i=j) is the sample variance-covariance matrix. • Our problem is to calculate these covariances efficiently. • Define the excess return matrix to be
1.S 1 xGMVP {xGMVP ,1 , xGMVP , 2 ,..., xGMVP , N } 1.S 1.1T Where 1 = {1,1, …1} • The particular fascination with the minimum variance portfolio is that it is the only portfolio on the efficient frontier that does not require asset’s expected returns for its computation.
– There are two short positions, and two very large positions. – This is a potentially objectionable feature of computing the GMVP with the sample variance-covariance matrix. – It is not credible that an investor seeking minimum variance would put 61 per cent of his portfolio in GE and 100 per cent in IBM, financing these positions with a short of 77 per cent in BA and 10 per cent in MSFT. – The noncredible positions have led investors to seek more effective methods of computing such portfolios.
r11 r1 r r 21 2 AT ... rN 1 rN 1 r12 r1 r22 r2 ... rN 2 rN r1M r ... r2 M r2 ... ... ... rNM rN ...
• Multiplying AT by A and dividing by M – 1 gives the sample variance-covariance matrix. • S = [σij] = ATA / (M -1)