InvestmentMath

In this post, I introduce an additive method to perturb the market index. This method takes a typically centered score vector and adds it to the vector of portfolio weights. Nothing guarantees that the perturbed vector lies in the simplex. To bring the perturbed vector back into the simplex, a simple method consists in projecting it on the simplex using the standard euclidean distance. As it turns out, euclidean orthogonal projection on the simplex can be efficiently carried out using a simple algorithm which consists in applying a specific translation to the original vector while thresholding short positions to zero. This classic result is proven. Finally the multiplicative and additive are contrasted using the example of stocks exclusions.

In this post, I start looking at a multiplicative method. Multiplicative methods are best understood by working with pseudo-numbers of shares in the projective orthant. The market pseudo-numbers of shares are perturbed in proportion to a score built from stock characteristics. The market weights are shifted proportionately and then radially projected on the simplex. I detail the case where the stock characteristic is a valuation ratio, where the method has a natural interpretation. Numerous important practical indices arise from this setup such as the equal-weight index, the diversity-weighted index and fundamental ones. Value indices based on the multiplicative method generalize the fundamental-index construction.

This series of posts discusses methods relevant for equity index construction. Although portfolio construction typically subsumes index construction, there is a specific flavour to index construction: index rules need to be relatively simple and transparent. They are fully disclosed by index providers and implemented in a mechanical way. This narrows the possibilities. Posts in this series will therefore concentrate on simple and transparent rules to build equity portfolios. It will cover some existing constructions as well as new ones, with special attention brought to the conceptualization behind each approach.

This is the solution of the exam for the 2018 ensae course.

This is the exam for the 2018 ensae course.

This is the exam for the 2017 ensae course.

This post describes the martingale method in continuous time. It should be read after this one. It illustrates the method on a simple example within the Black and Scholes framework where an investor maximizes the utility of terminal wealth. In this context, the martingale method allows to spell out how optimal terminal wealth depends on the unique stochastic discount factor, or alternatively, how it is obtained as a transformation of the stock return. The transformation hinges on the shape of the utility function. The case of a CRRA utility function is fully spelled out. The results obtained through dynamic programming here are recovered although the martingale method does not easily uncover the trading policy that generates the optimal terminal wealth.

This post collects constrained optimzation results for reference.

This is the solution of the exam for the 2016 ensae course.

This is the exam for the 2016 ensae course.