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Why does the Markowitz mean-variance model require the assumption of normality?

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 ...
Mark Joshi's user avatar
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11 votes
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What does the concept "standard Markowitz approach" include?

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 ...
develarist's user avatar
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8 votes

Closed-form analytical solution for the variance of the minimum-variance portfolio?

A few more steps beyond your last equation gives the answer. With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have $$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
RRL's user avatar
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7 votes
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Can a capital market line have a negative slope?

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] ...
nbbo2's user avatar
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7 votes
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Why does portfolio optimization require a positive-definite covariance matrix?

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 ...
Quantoisseur's user avatar
7 votes

Contribution of an asset's variance to portfolio variance

In this answer, I am assuming that you want to keep correlations constant. To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written ...
Kermittfrog's user avatar
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7 votes
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Why isn't the asset with minimum variance given a 100% portfolio weight?

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 ...
Kermittfrog's user avatar
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6 votes
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Dollar-Neutral in addition to Market-Neutral?

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. ...
Yugmorf's user avatar
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6 votes
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Generalized Mean Variance Portfolio

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 ...
Matthew Gunn's user avatar
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6 votes
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Efficient frontier doesn't look good

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: ...
jthg's user avatar
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6 votes
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Ledoit/Wolf covariance shrinkage in risk-parity optimisation

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 ...
MGL's user avatar
  • 516
6 votes

Why does portfolio optimization require a positive-definite covariance matrix?

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 ...
Martin Vesely's user avatar
5 votes
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Closed-form analytical solution for the variance of the minimum-variance portfolio?

Let \begin{align} a&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}\\ b&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu}\\ c&\equiv \boldsymbol{\mu}^T\mathbf{\Sigma}^{-1}\...
Kermittfrog's user avatar
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5 votes
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Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations

Let's stick with the nomenclature in the literature and let $\gamma$ denote the decision maker's risk aversion coefficient. The optimization problem is $$ \max_{\mathrm{w}} \mathrm{w}^T\mathrm{\mu}-\...
Kermittfrog's user avatar
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5 votes
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Maximum skewness portfolio solution derived from its Lagrangean formulation

Unfortunately, there exist no closed form for this. The Lagrangean reads $$ L(w,\lambda)=w^TM_3(w\otimes w)-\lambda(w^T\mathbf{1}-1) $$ with first order conditions $$ \begin{align} \frac{\partial L }{\...
Kermittfrog's user avatar
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5 votes
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Mean-Variance Portfolio Axis Description

Suppose you have a risk-free security R and a risky security B. A portfolio with a 0.50, 0.50 combination will have a standard deviation of $0.5 \sigma_B$, but a variance of $0.25 \sigma_B^2$. So if ...
nbbo2's user avatar
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5 votes
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Is this quadratic form the Sharpe ratio?

Perhaps this is helpful. Look at my answer to a related question to follow my notation better. $$ \begin{align*}a &\equiv \sum_i \sum_j V_{ij} \mu_i \quad \quad \text{(in Merton paper)}\\ &= \...
Matthew Gunn's user avatar
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5 votes

Utility Theory and Mean Variance Analysis

Theoretically: no. For most practical purposes: yes; given that risks are small risks, see these lecture notes on p76. Belows's the background and one example showing you why you can run into ...
Kermittfrog's user avatar
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5 votes

Covariance Between Two Frontier Portfolios

Let $\Sigma$ denote the covariance matrix of our asset universe, $\mu$ is the vector of expected returns. Further, $\mathbb{1}$ is a vector of ones. Let's identify the vector of the minimum variance ...
Kermittfrog's user avatar
  • 6,663
5 votes

What is the mathematical difference between Mean-Variance Optimization and CAPM?

The CAPM is an asset pricing model, while mean-variance optimization is a type of optimization. These are objects from two different categories. When you characterize the CAPM as an optimization ...
Richard Hardy's user avatar
4 votes
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Economic intuition behind pricing cash flow

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 ...
Matthew Gunn's user avatar
  • 6,954
4 votes

Is this realized "efficient" frontier reasonable?

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 ...
Dave Harris's user avatar
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4 votes
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Mean Variance portfolio optimisation (Long Only) CVXPY including cardinality constraint

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 ...
davidhigh's user avatar
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4 votes
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Methods for superior estimates of returns in m.v. portfolio optimization

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 ...
develarist's user avatar
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4 votes

Tangency portfolio with two additional constraints so that portfolio weights are unconstrained

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 ...
develarist's user avatar
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4 votes
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Monte Carlo (resampling) in m.v. portfolio optimization

There might be some differences in how we define things, but there should be only one set of assumptions (i.e., for each asset, there should be only one expected return and expected volatility). Your ...
Helin's user avatar
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4 votes
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Prove that the portfolio that maximizes utility lies on the efficient frontier

The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e. $\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(...
Attack68's user avatar
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4 votes

Mean-Variance optimization with no short selling

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 ...
Martin Vesely's user avatar
4 votes

Any portfolio models not based on asset return moments?

Remember that asset returns are there because of the expected utility theory. More precisely, as long as you can assume a "reasonable" expected utility function to be approximated by a quadratic ...
LePiddu's user avatar
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4 votes
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Any portfolio models not based on asset return moments?

The answer is sort of. I am going to provide you a history of the mathematics so that you will understand why this discussion is challenging to have in economics. Also, you are probably going to ...
Dave Harris's user avatar
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