16

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 ...


9

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

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 ...


6

In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), others (including Markowitz himself) thought this was not realistic (he thought if you have 1 dollar you cannot both buy 1 dollar worth of stock and also short 1 ...


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 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

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 ...


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

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 $...


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 ...


4

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, $\lambda$ ...


4

No, even if returns were perfectly normal (it really doesn't matter whether mean is zero and standard deviation is 1 - they can be anything), it wouldn't ensure that markowitz would perform well out of sample. The reason is because even if data is normally distributed it is hard to estimate means of returns. The standard error for an estimate of a mean like ...


4

If you're asking how to use or modify the closed-form analytical solution you showed, consisting of the building blocks $A$, $B$ and $C$, as derived by Merton in 1972, you can't. That is intended for solving the unconstrained portfolio only. There is no analytical solution for the constrained portfolio because of frictions with the non-negativity ...


4

This has already been dealt with multiple times. As @Dom explains, the purpose is to simplify partial derivatives. For exposition's sake, assume there are only two assets with weight vector $\omega=(\omega_1,\omega_2)$, then we seek to minimize a function of the form: $$f(\omega_1,\omega_2)=\frac{1}{2}\left(\omega_1^2\sigma_2^2+\omega_2^2\sigma_2^2+2\omega_1\...


4

Diversification is key. The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance across the asset universe. The setup Without loss of generality, let us assume there exist two assets $a$ and $b$ with variance $\sigma_a^2=\alpha^2<\sigma_b^2=...


3

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 ...


3

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 ...


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

Yes, two different set of returns can lead to the same weights (so you won't be able to prove the opposite). Also, the term "unique solution" means something different than how you used it. Taking $p$ and $Q$ as given, the mapping from $\boldsymbol{\mu}$ to solutions $\mathbf{x}^*$ is not injective I'll give a simple counterexample that shows the mapping ...


3

The tangent portfolio is the optimal portfolio of risk assets. So under the Modern portfolio theory world, all investors will buy and hold this portfolio if they want to make an investment in risky assets. The point where indifference curve and the CAL meets tells how much of your wealth would be invested in (1) the risk-free asset and (2) the optimal ...


3

Yes there are two ways to solve the tangency portfolio: closed-form analytical solution optimization problem (maximization of the Sharpe ratio) The closed-form analytical solution you incorrectly wrote is actually $$w_{\text{tan}} = \frac{\Sigma^{-1} \left(\mu - r_f \cdot \iota\right)}{\iota^{\prime}\Sigma^{-1}\left(\mu - r_f \cdot \iota\right)}$$ This is ...


3

You can use Lagrangian only with equal type constaints. There are inqualities in your problem, namely $w \ge 0$ and $w \le 1$. Hence Lagrange method cannot be employed here. According to tags you added to the question, you are solving Markowitz optimization problem. One of its formulation (maximizing profit and minimizing risk at the same time) is $$ f = \...


3

$\lambda$ is independent of the maximum sharpe ratio. The maximum sharpe ratio portfolio will give you a combination of the risk free asset and the tangency portfolio. Then your risk aversion just makes you choose the combination between these two assets. See picture below. The blue line is the efficient frontier with short-sales allowed. The red-curve is ...


3

Let's derive a possible approach from utility theory. Our investor is risk averse and exhibits CARA utility using an exponential utility function with risk aversion parameter $\gamma>0$ (risk averse agent): $$u(x)=\frac{1-e^{-\gamma x}}{\gamma}$$ A 3rd order Taylor series expansion around $x=0$ yields \begin{align} u(x)\approx& x - \frac{1}{2}\gamma ...


3

Examples of statistical measures to compare extreme rebalancing: Below, I have provided some examples of statistical measures, that compare the extreme re-balancing of different portfolios. They do not show how the concentration is allocated, only if the allocation is extreme. Many of the measures can be found in the empirical portfolio section of Patton et ...


3

Your first argmax is actually defined up to a constant multiplier: $argmax\left(\frac{\mu^Tw}{\sqrt{w^T\Sigma w}}\right)=\lambda\Sigma^{-1}\mu$, where $\lambda$ is an arbitrary portfolio size scale. In general, maximizing a scale-invariant ratio of the form $f(w)/g(w)$ can be done in conditional terms: $max(f)$ subject to $g=const$, or, using a Lagrange ...


3

Let $f(N) = \frac{1}{2} N (N + 1)$ then $f(50) = 1275$. A year has approximately 255 trading days. So you need at least 1275 / 255 = 5 years. I believe the rule above is used in practice but I think the text is not quite correct (which surprises me, maybe I should have a ☕). If the returns are IID, 51 observations ought to be enough, see the proof in this ...


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