Methods and Apparatus for Generating Purified Minimum Risk Portfolios
The quantitative construction of investment portfolios of securities such as stocks, bonds, or the like using optimization is addressed. More specifically, during optimization, constraints on non-target factor exposures are automatically converted to constraints on the exposure of the projections of the non-target factors that are orthogonal to a specified target factor. The target factor may be the implied alpha of a reference portfolio, such as a traditional minimum risk portfolio. Such constraints may be utilized to produce portfolios with superior performance to those produced with traditional factor exposure constraints.
1 . A computer-implemented method of constructing a portfolio comprising:
electronically receiving and storing by a programmed computer a set of N potential investments;
electronically receiving and storing by the programmed computer a risk model that predicts the asset-asset covariance of all pairs of assets in the N-dimensional universe of potential investments;
electronically receiving and storing by the programmed computer a set of one or more N-dimensional vectors of non-target factors scores for each of the possible investments;
electronically receiving and storing by the programmed computer an N-dimensional vector of reference portfolio holdings;
computing the implied alpha of the reference portfolio using the risk model;
linearly resealing the implied alpha;
determining projections of the non-target factor scores that are orthogonal to the linearly resealed implied alpha;
electronically receiving and storing by the programmed computer an optimization problem for determining an N-dimensional vector of investment allocations;
computing an optimal investment allocation vector for the optimization problem with upper and lower bound constraints for the exposures to the projections of the non-target factor scores to the linearly resealed, orthogonal implied alpha; and
electronically outputting the optimal investment allocation vector using an output device.
2 . The method of claim 1 in which the non-target factor scores are factors from a factor risk model.
3 . The method of claim 1 in which the optimized portfolios are determined at distinct historical times to simulate the performance of the optimized portfolio over time.
4 . The method of claim 1 in which the reference portfolio is determined by a second optimization problem that minimizes the predicted risk of the portfolio.
5 . A computer-implemented method of constructing a portfolio comprising:
electronically receiving and storing by a programmed computer a set of N potential investments;
electronically receiving and storing by the programmed computer a risk model that predicts the asset-asset covariance of all pairs of assets in the N-dimensional universe of potential investments;
electronically receiving and storing by the programmed computer a set of one or more N-dimensional vectors of non-target factors scores for each of the possible investments;
electronically receiving and storing by the programmed computer an N-dimensional vector of target factor scores for each of the possible investments;
determining a reference portfolio for the target factor by defining the reference portfolio investment allocation using the factor scores for each potential investment;
computing an implied alpha of the reference portfolio using the risk model;
linearly resealing the implied alpha;
determining projections of the non-target factor scores that are orthogonal to the linearly resealed implied alpha;
electronically receiving and storing by the programmed computer for an optimization problem for determining an N-dimensional vector of investment allocations;
computing an optimal investment allocation vector for the optimization problem with upper and lower bound constraints on the exposures to the projections of the non-target factor scores to the linearly resealed, orthogonal implied alpha; and
electronically outputting the optimal investment allocation vector using an output device.
6 . The method of claim 5 in which the non-target factor scores are factors from a factor risk model.
7 . The method of claim 5 in which the optimized portfolios are determined at distinct historical times to simulate the performance of the optimized portfolio over time.
8 . The method of claim 5 in which the reference portfolio minimizes the risk predicted by the risk model;
9 . A computer-based method of constructing an index for an exchange traded fund based on a purified factor portfolio comprising:
electronically receiving and storing by a programmed computer a set of N potential investments;
electronically receiving and storing by the programmed computer a risk model that predicts the asset-asset covariance of all pairs of assets in the N-dimensional universe of potential investments;
electronically receiving and storing by the programmed computer a set of one or more N-dimensional vectors of non-target factors scores for each of the possible investments;
electronically receiving and storing by a programmed computer an N-dimensional vector representing a reference portfolio of weights for each possible investment;
computing the implied alpha of the reference portfolio using the risk model;
linearly resealing the implied alpha;
determining projections of the non-target factor scores that are orthogonal to the linearly resealed implied alpha;
computing an optimal investment allocation vector that minimizes the predicted tracking error between the optimal allocation and the reference portfolio and minimizes the absolute active exposures of the portfolio to the projections of the non-target factor scores to the linearly resealed, orthogonal implied alpha;
electronically outputting the optimal investment allocation vector using an output device; and
utilizing the optimal investment allocation vector as the index for an exchange traded fund.
10 . The method of claim 9 in which the non-target factor scores are factors from a factor risk model.
11 . The method of claim 9 in which the reference portfolio minimizes the risk predicted by the risk model;
12 . A computer-implemented system of constructing a portfolio comprising:
a memory for storing data for a set of N potential investments;
a processor executing software to retrieve data for a risk model that predicts the asset-asset covariance of all pairs of assets in the N-dimensional universe of potential investments;
a processor executing software to retrieve data for a set of one or more N-dimensional vectors of non-target factors scores for each of the possible investments;
a processor executing software to retrieve data for an N-dimensional vector of reference portfolio holdings;
computing on the processor executing software the implied alpha of the reference portfolio using the risk model;
computing on the processor executing software a linear resealing of the implied alpha;
a processor executing software to retrieve data for an optimization problem for determining an N-dimensional vector of investment allocations;
computing on the processor executing software the projections of the non-target factor scores that are orthogonal to the linearly resealed implied alpha;
computing on the processor executing software an optimal investment allocation vector for the optimization problem with upper and lower bound constraints for the exposures to the non-target factor, linearly resealed, orthogonal implied alpha; and
computing on the processor an electronic output representing the optimal investment allocation vector.
13 . The system of claim 12 in which the non-target factor scores are factors from a factor risk model.
14 . The system of claim 12 in which the optimized portfolios are determined at distinct historical times to simulate the performance of the optimized portfolio over time.
15 . The system of claim 12 in which the optimization problem minimizes the predicted risk of optimized portfolio.