14

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 ...


13

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}\...


10

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 ...


8

The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years: The standard model (Markowitz, 1952, 1959) originally only considered: Constrained model where short sales are forbidden Only risky assets considered for investment (no risk-free asset) Scenarios that the mean-variance model ...


7

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


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

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 ...


6

Two separate cases were identified by R.C. Merton in 1972: In the economically more relevant case, where $r_f < b/c$, efficient portfolios are combinations of a long position in [the tangency] portfolio M and lending or borrowing at the risk–free rate. In the case where $r_f > b/c$, efficient portfolios are generated by short (or zero) positions in ...


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 ...


5

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'...


5

The Risk Parity portfolio will be equal weighted if the assets have uniform correlation and equal variance. This would be the case for the shrunk covariance matrix if the shrinkage coefficient used equals unity. In sklearn, you can check the shrinkage coefficient for the Ledoit-Wolf shrinkage after fitting it from the instance's .shrinkage_ attribute. If the ...


5

Positive definite matrix $A$ is defined as $x^TAx > 0$ for all vectors $x$. Since a term $w^T\Sigma w$ in Markowitz (and other models as well) expresses variance in returns, it is a measure of dispersion. Any measure of dispersion has to be positive (or maybe zero but it is a case where there is no uncertainty and hence no risk). Negative dispersion is ...


5

To supplement the other answer, yes there are optimization reasons for the covariance matrix being symmetric positive definite (SPD). All positive definite matrices are invertible and its inverse is also positive definite. This guarantees a unique global minimum in a quadratic optimization problem (MVO). Lots of material available on the topic: https://www....


4

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 ...


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

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 ...


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 ...


4

The "hedging theory of investment" (which I first heard about from R. C. Merton) says you should invest not for returns but to hedge your liabilities. LDI (Liability Driven Investment) is one name for it. So for example a pension fund should hedge pension liabilities. A university endowment should hedge the cost of producing education, which might entail ...


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 ...


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