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13

Short of having a 'reasonable' predictive model for expected returns and the covariance matrix, there are a couple lines of attack. Shrinkage estimators (via Bayesian inference or Stein-class of estimators) Robust portfolio optimization Michaud's Resampled Efficient Frontier Imposing norm constraints on portfolio weights Naively, shrinkage methods ...


10

Both answers from Shane and Vishal Belsare make sense and detail different models. In my experience, I have never been satisfied by a unique model since the majority of papers out there can be split in two categories: Those that predict the mean component of the problem. Those that predict the variance component of the problem. The ideal (to read ...


9

Hey, it's early days yet. After all it is still called MODERN portfolio theory. I think there are two main issues and they are both really cultural: 1) specifying alphas 2) wild results Alphas I agree with Gappy that alphas are the key thing you need to have effectiveness (unless you are doing minimum variance). Having a vector of expected returns is ...


7

You raise a very important point, which unfortunately doesn't have a simple answer. Black-Litterman addresses the allocation problem by allowing you to provide a prior within a bayesian framework. It doesn't really tell you how to produce the prior itself. But more importantly, it doesn't address the fundamental problem: it's difficult to accurately ...


6

I believe there are several ways you can tackle your problems. First, you mentioned that your perform several optimizations. One solution that comes to mind instead of speeding up the optimization itself is to perform the optimizations in parallel, so you could look at Mathwork's Parallel Computing Toolbox. Second, providing the optimizer with a good ...


6

Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ...


5

I think the original reference of mean-variance portfolios being “error maximizing portfolios” is: Michaud, R. (1989). “The Markowitz Optimization Enigma: Is Optimization Optimal?” Financial Analysts Journal 45(1), 31–42. The reason is that even small changes in the estimated means can result in huge changes in the whole portfolio structure. Have a ...


4

The unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\} $$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w $$ setting it to zero, and solving for $w$. This gives ...


4

With respect to issue one, it can be simpler to consider the case where the constraint on the expected return is an equality. In that case, the first problem can be transformed to Minimize with respect to $\left\{ x,\lambda_{1},\lambda_{2}\right\} $: $x'\Sigma x + \lambda_{1} (\mu'x - r) + \lambda_{2} (1'x - 1)$ by the technique of Lagrangian multipliers, ...


3

There are many papers on this subject (try googling portfolio optimization skewness kurtosis) that can describe the assumptions of including skewness and kurtosis in a utility function (if that's what you're interested in). I would highlight two main points. Mean-variance optimization does not make an assumption of normality. Assume returns are ...


3

Typical risk aversion levels lie between one and ten. See pages 11f. in the following paper: Preferences by Andrew Ang


3

In robust optimization, the true return is not known, we just have a prior $\alpha$ and you have to take into account a possible misestimate which can lower the true return. This is done under the assumption that the posterior return will be within the prior return $\alpha$ plus minus the error being in some $\sigma$-interval. Now a try for a more formal ...


3

The answer to the original question is simple: the Chopra-Ziemba paper is highly flawed and unreliable. Note that the framework is in-sample and based on a utility function. It has nothing to do with out-of-sample behavior of the mean vs. the covariance in an optimization. Estimation error grows linearly in the mean but quadratically in the covariance. At ...


3

The biggest problem with mean-variance optimization is that the sensitivity of the estimated covariance matrix. Mean variance optimization assumes that one "knows" the covariance of each asset with every other asset, or that the covariance matrix is constant. Without this assumption the MVO framework is not tractable. Axioma and others do a lot more than ...


3

There is a great deal of misinformation and out-of-date information on this site. Many of the references in this discussion and elsewhere have serious research flaws. The Michaud efficient frontier was invented and patented by Robert Michaud and Richard Michaud, U.S. patent # 6,003,018. The alternatives discussed here are not patented nor in many cases ...


3

In addition to the points above, I'd say that asset managers also have to bear two things in mind that limit their ability to "properly" optimise their portfolios: 1) general restrictions on asset allocation (regulatory, contractual, common-sense) 2) transaction cost In my experience, asset managers do use a variety of optimisation techniques occasionally ...


2

Assuming those are arithmetic returns and covariances at the horizon, calculate a $9\times1$ vector containing the betas with respect to the world index using the covariance matrix, call it $\beta$. The covariance resulting from the world index can be described as $\beta\sigma_{world}^{2}\beta'$. The matrix ...


2

One of the most salient empirical examples of "error maximization" is provided by Chopra and Ziemba (1993): Chopra, Vijay K., and William T. Ziemba. 1993. “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice.” Journal of Portfolio Management, vol. 19, no. 2 (Winter):6–11. The authors compare the performance of ...


2

You should have a look at chapter 8 (p. 261ff.) of Hedge Fund Market Wizards by Jack D. Schwager Excerpt from there (but it is much more detailed in the book): Perhaps the most potent risk control Platt employs in BlueCrest’s discretionary strategy is maintaining an extremely tight rein on what a trader can lose before capital is withdrawn. A mere 3 ...


2

Check out following link. In page 23 you'll find the derivation. http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf


1

Here is the full math proof. Let g be the GMV portfolio and p be another asset. We have: $$ \begin{align*} Cov(x_g, x_p) &= E[{w_g}^T (x- \overline{x}) {(x- \overline{x})}^Tw_p]\\ &= {w_g}^TE[(x- \overline{x}) {(x- \overline{x})}^T]w_p\\ &= {w_g}^T\Sigma w_p \\ &= (\displaystyle\frac{{i}^T {\Sigma}^{-1}}{C})\Sigma w_p\\ &= ...


1

The risk aversion coefficient is also referred to as the Arrow-Pratt risk aversion index. When λ is small (i.e., the aversion to risk is low), the pen- alty from the contribution of the portfolio risk is also small, leading to more risky portfolios. Conversely, when λ is large, portfolios with more exposures to risk become more highly penalized. If we ...


1

Before answering your questions directly i would like to briefly restate the idea of the resampled efficent frontier: One of the problem with classical mean variance optimization is (even if the multivariate normal assumption holds) that you cant estimate $\mu$ and $\Omega$ (which is usually denoted as $\Sigma$) exactly. Thats why you incur estimation ...


1

I'd suggest checking out MOSEK, I used it at my last firm (medium frequency stat arb) and I also know it's used at another large hedge fund.


1

Let $\mu$ and $\Sigma$ be the expected mean and covariance matrices for a mean-variance optimization. For a standard, unconstrained, utility-based optimization, it can be shown that the optimal weights will equal $$ w=\frac{1}{\lambda}\Sigma^{-1}\mu $$ where $\lambda$ is an arbitrary risk aversion coefficient. In order to measure the sensitivity of the ...


1

Try this: http://en.wikipedia.org/wiki/Rate_of_return


1

One approach which I've encountered in practice is Optimal risk budgeting (ORB). This method is similar to Black Litterman in the sense that it uses active investor views as a starting point. The mean variance optimization is then restricted to those assets for which an active investor view is available, and the allocation is calculated with the constraint ...



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