11

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 ...


5

The weak EMH states that it is impossible to earn an excess return given publicly known information such as past prices. Clearly, different securities have different expected returns. For example: the bond and the stock of one company or a security that generates twice the return of another one. This difference in expected return is explained by a ...


5

To solve this constraint minimization problem, first form the Lagrangian Function \begin{align} L(w,\lambda_1,\lambda_2)=w'\Sigma w + \lambda_1(w'\boldsymbol{\mu}-m) + \lambda_2 (w'\boldsymbol{1}-1). \end{align} The first order conditions for a minimum are then given by \begin{align} \frac{\delta L(w,\lambda_1,\lambda_2)}{\delta w}&=2 \Sigma w + \...


5

I prefer to interpret the mean-variance frontier as a consequence of linear algebra as developed in Hansen and Richard (1987) and discussed in Cochrane (2005). In brief: The space of returns is a hyperplane in the vector space of payoffs. The set of returns on the mean-variance frontier is a line in the space of returns. Any two distinct points on a line ...


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

The Efficient Market Hypothesis (EMH) states that you cannot beat the market on a risk-adjusted basis by looking at past prices. You can certainly earn higher returns than the market if you take on more risk (by leveraging, for example). Modern Portfolio Theory allows you to construct portfolios that are efficient. According to this theory, you still cannot ...


4

This is easier to explain with an example. Let's say there is no risk free rate as your question seems to imply. Also assume there are two portfolios $\phi_p$ and $\phi_q$ with the following characteristics: Both lie on the frontier; The first one has an expected return of 0.04 and standard deviation of 0.2 The second one has an expected return of 0.10 ...


3

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 ...


3

The problem lies in the definition of risk. It seems that in the cited paper, the authors treat risk as a concept connected with the uncertainty of the out-of-sample performance of the portfolio. In that way portfolios constructed using the proposed robust estimators would be what they call minimum-risk portfolios. Contrasted with minimum-variance ...


3

Risk is a broader concept than variance. That paper is specifically focused on robust estimators (i.e., estimators that are less sensitive to outliers) of dispersion. A robust estimator of dispersion is not the same thing as variance (which may be a dispersion parameter for some classes of distributions). Nevertheless, these robust estimators could be used ...


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

Broadly speaking, as you probably already know, there are 2 approaches to estimating large covariance matrices: 1) Shrinkage Methods like Ledoit-Wolf that try to reduce the noise in a large matrix (N by N) that has been estimated using the conventional method. 2) Factor Models of Covariance as described in for example Connor Korajczik 2007 that assume that ...


3

In a linear regression approach you do the following: $$ (X \beta - y)^2 \rightarrow Min $$ thus you try to predict something. Your objective is quadratic. You usually add constraints on $\sum \beta_i^2$ or $\sum |\beta_i|$. Without constraints the estimator is: $$ \hat{\beta} = (X^T X)^{-1} X^T y, $$ where $X^T y$ has to do with the covariance of $X$ and $...


3

You cannot eliminate the dependence of a solution on the risk aversion parameter (which this author confusingly calls $\lambda$). Perhaps a source of confusion? Typically $\lambda$ is used to denote a Lagrange multiplier in Lagrangian optimization, but the author is using $\lambda$ as a risk tolerance parameter. (In your other linked question, $\lambda$ ...


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

EMH says that one can not earn excess return using some information. This is known as joint-hypothesis problem: to test for market efficiency one have to determine first what is "normal" market return, i.e. what type of information is normally priced by the market. Usually to test for EMH they use CAPM or 3-factor Fama-French model (which is a kind of CAPM-...


2

Let $s_1 = r_1 -r_f$ and $s_2 =r_2-r_f$. Then, this is the maximization problem: \begin{align*} & \ \max_{w_1, w_2} SR = \frac{\mu_p}{\sigma_p}, \, \mbox{ subject to}\\ \mu_p = & \ w_1 s_1 + w_2 s_2,\\ \sigma_p^2 = & \ \sigma^2\big(w_1^2 + w_2^2 + 2 w_1 w_2 \rho\big),\\ 1 = & \ w_1+w_2. \end{align*} By certain substitution, we convert the ...


2

There is no generic solution. However, the KKT conditions are of the forms \begin{align*} \begin{cases} Qy + \lambda_1 \mu +\lambda_2 Py = 0,\\ \mu^T y = 1,\\ y^TPy \leq k^2 \sigma^2,\\ \lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0. \end{cases} \end{align*} Here, the condition $$\lambda_2 \big( y^TPy - k^2 \sigma^2\big) = 0 $$ means that two cases need to ...


2

That's the way you apply. Usually you get the closest number of shares possible. However, if you use that strategy you are very likely to underperform the market. Check table 3 on this paper for the Out of sample performance of the Markowitz strategy. Over their sample the Sharpe Ratio is 0.07 whereas a simple naive strategy 1/N yielded 0.18.


2

Out-of-sample is basically impossible to predict means. Second moments are much easier. You can take a look at this post: Estimating $\mu$ - only increasing $T$ improves estimate? Only with infinite $T$ you would be able to correctly estimate $\mu$. So theoretically your procedure could be correct if means are time-varying, but out of sample I bet your ...


2

In literature you'll find many approaches to compute the variance. As mentioned already, the standard ideas are to use MLE, Shrinkage on the Covariance Matrix (Ledoit, Wolf), Shrinkage on the inverse of the Covariance Matrix (Kourtis,Dotsis) which makes sense as in fact the inverse of the Covariance Matrix determines the shape of the efficient frontier. ...


2

First of all I’ll work with column vectors because I find it easier than with row vectors as you did. I guess it’s a little bit easier if we modify your first equation a little bit. Notice that is really the first order condition of the following lagrangian: $$L(w, \lambda, \delta)= \frac{1}{2}{\bf w^TCw} - \lambda({\bf w^Tm} - \mu_v) - \delta({\bf w^Tu} - 1)...


2

evolve the stock to the requisite time horizon using some model. Get its value and that of the options on it. Compute the returns implied by these. Store this vector. Do this many times. Compute the implied covariance matrix of these vectors of returns.


2

It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the standard $\sum w_i = 1$ constraint). Once he devised a computer algorithm [the Critical Line Method] for solving this problem (he was a math programming whiz) he seems ...


2

This is not a complete answer, just a few pointers regarding your code (can't post it as a comment, as I'm a new member, too): This part can be omitted, as you have specified PReturn before: if (abs(PVol - TargetVol) < 0.005) { PReturn <- Weights %*% t(EReturns) } In you optimization you don't specify any bounds on the weights. In particular, we can ...


2

The comments above re all the entries of $\mu$ not being the same is true, but can be removed if you make the 2x2 determinant in question $\ge 0$ instead of $> 0$. The commenters know this of course. The answer to your question can be obtained by an application of the Cauchy-Schwartz inequality along with knowledge that a symmetric positive definite ...


2

The constraints $$ w \le b_u $$ and $$ b_l \le w \Leftrightarrow-w \ge - b_l $$ can all be handled using the Kuhn–Tucker conditions. Numerical solvers exist for these linear constraints too (e.g. this is in R). See als this. However, with the lower bound you often want the optomizer to choose some assets to be zero and if greater zero then greater than some ...


2

I guess it amounts to saying that you want to exclude the case when the optimal portfolio $w_*$ is such that $\mu'w_{*}>m$. Notice that, given that $\Sigma$ is positive definite, you can choose another portfolio $w_{**}=w_{*}-1\epsilon$, with $\epsilon>0$ and small enough, such that $\mu'w_{**}=\mu'w_* - \mu'1\epsilon>m$, but clearly $w_{**} {'}\...


2

There is a recent a paper recently using a population test of all CRSP data from 1925-2013 as a test of whether a mean and a variance exist versus they do not exist. It overwhelmingly excluded mean-variance finance as not possible. It is also a population study so for mean-variance to be valid, there would have to be radically different behavior before and ...


2

Portfolio optimization techniques, such as those defined under Modern Portfolio Theory (MPT), are mildly predicated on the assumption of joint normality. Even though there will be a set of portfolio weights which minimizes variance regardless of the underlying distributions, correlation is only a complete measure of association if the joint multivariate ...


Only top voted, non community-wiki answers of a minimum length are eligible