Suppose you want to identify an efficient set of portfolios that minimize the variance of the difference in returns with respect to a given target portfolio, subject to a given expected excess return. The mean and standard deviation of this excess return are often called the active return and active risk, respectively. Active risk is sometimes referred to as the tracking error. Since the objective is to track a given target portfolio as closely as possible, the resulting set of portfolios is sometimes referred to as the tracking error efficient frontier.
Specifically, assume that the target portfolio is expressed as an index weight vector, such that the index return series may be expressed as a linear combination of the available assets. This example illustrates how to construct a frontier that minimizes the active risk (tracking error) subject to attaining a given level of return. That is, it computes the tracking error efficient frontier.
One way to construct the tracking error efficient frontier is to explicitly form the target return series and subtract it from the return series of the individual assets. In this manner, you specify the expected mean and covariance of the active returns, and compute the efficient frontier subject to the usual portfolio constraints.
This example works directly with the mean and covariance of the absolute (unadjusted) returns but converts the constraints from the usual absolute weight format to active weight format.
Consider a portfolio of five assets with the following expected returns, standard deviations, and correlation matrix based on absolute weekly asset returns.
NumAssets = 5; ExpReturn = [0.2074 0.1971 0.2669 0.1323 0.2535]/100; Sigmas = [2.6570 3.6297 3.9916 2.7145 2.6133]/100; Correlations = [1.0000 0.6092 0.6321 0.5833 0.7304 0.6092 1.0000 0.8504 0.8038 0.7176 0.6321 0.8504 1.0000 0.7723 0.7236 0.5833 0.8038 0.7723 1.0000 0.7225 0.7304 0.7176 0.7236 0.7225 1.0000];
Convert the correlations and standard deviations to a covariance
ExpCovariance = corr2cov(Sigmas, Correlations);
Next, assume that the target index portfolio is an equally-weighted portfolio formed from the five assets. Note that the sum of index weights equals 1, satisfying the standard full investment budget equality constraint.
Index = ones(NumAssets, 1)/NumAssets;
Generate an asset constraint matrix using
portcons. The constraint matrix
expressed in absolute format (unadjusted for the index), and is formatted
[A b], corresponding to constraints
of the form
A*w <= b.
Each row of
AbsConSet corresponds to a constraint,
and each column corresponds to an asset. Allow no short-selling and
full investment in each asset (lower and upper bounds of each asset
are 0 and 1, respectively). In particular, note that the first two
rows correspond to the budget equality constraint; the remaining rows
correspond to the upper/lower investment bounds.
AbsConSet = portcons('PortValue', 1, NumAssets, ... 'AssetLims', zeros(NumAssets,1), ones(NumAssets,1));
Now transform the absolute constraints to active constraints
ActiveConSet = abs2active(AbsConSet, Index);
An examination of the absolute and active constraint matrices
reveals that they are differ only in the last column (the columns
corresponding to the
A*w <= b).
ans = 1.0000 0 -1.0000 0 1.0000 0.8000 1.0000 0.8000 1.0000 0.8000 1.0000 0.8000 1.0000 0.8000 0 0.2000 0 0.2000 0 0.2000 0 0.2000 0 0.2000
In particular, note that the sum-to-one absolute budget constraint becomes a sum-to-zero active budget constraint. The general transformation is as follows:
Now construct and plot the tracking error efficient frontier with 21 portfolios.
[ActiveRisk, ActiveReturn, ActiveWeights] = ... portopt(ExpReturn,ExpCovariance, 21, , ActiveConSet); ActiveRisk = real(ActiveRisk); plot(ActiveRisk*100, ActiveReturn*100, 'blue') grid('on') xlabel('Active Risk (Standard Deviation in Percent)') ylabel('Active Return (Percent)') title('Tracking Error Efficient Frontier')
Warning: In a future release, portopt will no longer accept ConSet or varargin arguments. 'It will only solve the portfolio problem for long-only fully-invested portfolios. To solve more general problems, use the Portfolio object. See the release notes for details, including examples to make the conversion. > In portopt at 83 In frontcon at 231
Of particular interest is the lower-left portfolio along the
frontier. This zero-risk/zero-return portfolio has a practical economic
significance. It represents a full investment in the index portfolio
itself. Note that each tracking error efficient portfolio (each row
in the array
ActiveWeights) satisfies the active
budget constraint, and thus represents portfolio investment allocations
with respect to the index portfolio. To convert these allocations
to absolute investment allocations, add the index to each efficient
AbsoluteWeights = ActiveWeights + repmat(Index', 21, 1);