12

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 $$w_{mv,unc}\equiv\frac{1}{\lambda}\...


12

it doesn't require normality. What it requires is that the investor's decisions are determined by mean and variance. A normal distribution is determined by mean and variance, so if you assume joint normality then there is no point in the investor being interested in anything else. (we try to discuss assumptions thoroughly in our book, Introduction to ...


8

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 ...


8

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 ...


7

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


6

Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision. In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to "0" if the ...


6

As i understand your question you are confused as to why the expected parabola-shape of the frontier is not depicted clearly. If you want to see the shape more clearly you can do one of two things: Increase the number of random portfolios. As this numbers goes to infinity you will eventually plot all possible portfolio combinations, and your efficient ...


5

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 ...


5

Typical risk aversion levels lie between one and ten. See pages 11f. in the following paper: Preferences by Andrew Ang EDIT: Unfortunately the paper doesn't seem to be available online anymore. The final source is the following book: Asset Management: A Systematic Approach to Factor Investing (Financial Management Association Survey and Synthesis) 1st ...


5

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 ...


4

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 ...


4

The formula is $$ \mu = \lambda CX $$ in your notation. You find it in many places, e.g. here. The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if investors are unconstrained (long/short not long only). It is intuitive as it says that given the weighting the return expectation increases with risk aversion ...


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, ...


4

It is well known that the MV-optimal portfolio has some very bad properties in practice: Backtesting: The MV portfolio performs very bad in backtesting applications Diversification: The MV portfolio tends to invest all funds into the best asset (highest sharpe ratio) of the past, leading to very low diversification. Non-Normality: Return distributions are ...


4

You can use the package quadprog and define everything yourself. Code can look like this: library(quadprog) Sigma = cov(data) mu = mean(data) Amat_in # define constraints here bvec_in # define rhs of constraints here solve.QP( Dmat = 2*Sigma, dvec = mu, meq=0,Amat=Amat_in,bvec=bvec_in) EDIT: Yes, and reading the documentation we see that portfolio.optim(...


4

This is wrong! Notice that $dX_t=\mu(t,X_t)dt + \sigma(t,X_t)dW$ is a shorthand for $$\int_0^tdX_s = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s$$ Integrating: $$X_t-X_0 = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s \text{ (eq.1)} $$ If we take expectations, remembering that $\mathbb{E}[\int_0^t\sigma(s,X_s)dW_s]=0$, we have $$\mathbb{E}[...


4

Imagine a scenario where a beta neutral portfolio comprised being long one very high beta stock and short many low beta stocks. Such a portfolio clearly has extreme concentration of risk. Additionally imposing a 'dollar neutral' constraint, would help to spread the weights more evenly over all the stocks. A further observation is that measuring true 'beta'...


4

In a quick and easy first step you could add $L_1$-regularization to the Markowitz problem. That is, you add a term $\lambda ||w||_1$ to the goal function of your optimization problem (where $w$ are the allocation weights to be optimized). The $L_1$-regularization, which is often termed LASSO in the statistics community, will give you sparse solutions of ...


4

Something to perhaps realize is that your two problems may not be as different as you think if $\lambda$ is an ad-hoc parameter. For any solution to your 2nd problem (where $\theta > 1$), there exists a $\lambda$ for problem 1 which gives you the same solution as problem 2. Example Let $f$ and $g$ be convex functions and let $\mathbf{x}$ denote a ...


4

All that's going on here are essentially consequences of a linear pricing function. That asset prices should be linear in their payoffs makes intuitive economic sense: the value of a basket of payoffs is the sum of the basket contents. An assumption that the pricing function is linear is sometimes referred to as the law of one price. Quick review Let $f$ ...


4

Expected returns are very difficult to estimate reliably without incurring estimation error as found out by Merton (1980) "On estimating the expected return on the market". This is why estimating volatility/the covariance matrix has become the default approach in the mean-variance model because volatility is easier to predict than returns. Even the global ...


3

As indicated in my comment, the function mvFrontier in the development version of the NMOF package may help you. (Disclosure: I am the package maintainer.) You may get the latest version from GitHub. Some remarks, first on correlation: an efficient frontier shows portfolio risk, typically volatility, compared with portfolio return. Portfolio volatility is ...


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 distributed ...


3

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 mean-...


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

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 ...


3

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 ...


3

There is one minor mistake: If you compute sum(mean.var) you'll obtain $-1$ instead of $1$. So it should be mean.var<-xt/sum(xt) in order to ensure that the weights sum up to one. The remainder is correct. Incorporating a risk aversion parameter into the framework requires the solution to the minVar problem (See for example here). Therefore, dividing ...


3

I believe the question to be too vague to be a good interview question. If you want to do Mean Variance Optimization (MVO) it's hard to see the point of Monte Carlo simulation. One of the good thing of MVO is its analytic tractability. Clearly, the topic is not widely discussed as this Google Search has this question as the first result (I was in incognito ...


3

Nothing is "wrong," in the sense that your findings are out of line, but there is a very deep issue that is wrong. I have written a set of papers on this. Since you are not a student, but someone trying to use this, I will explain in a lightweight manner what is wrong. There are three main branches of statistics. In order of discovery they are the ...


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