16 votes
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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 ...
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  • 6,763
10 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 ...
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  • 2,835
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] ...
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  • 9,587
6 votes
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Portfolio Optimization sum of weights constraint with short selling

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), ...
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  • 9,587
5 votes

Does Modern Portfolio Theory align with EMH?

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 ...
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  • 7,662
5 votes
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Derivation of the efficient frontier set (markowitz problem)

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). \...
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  • 177
5 votes
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Linear Regression vs Mean Variance Optimization

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 \...
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  • 13.3k
5 votes
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Markowitz; risky asset frontier w/o risk free asset

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 ...
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  • 6,354
4 votes
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Why did Markowitz not derive an equation for the efficient frontier?

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 ...
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  • 9,097
4 votes

How to perform portfolio optimization with user-defined expected return and variances using R?

You can use the package quadprog and define everything yourself. Code can look like this: ...
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4 votes
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Markowitz Mean-Variance Implied Returns

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 ...
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  • 13.3k
4 votes

Does Modern Portfolio Theory align with EMH?

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 ...
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  • 10.9k
4 votes

Backtest Results needed to Model Validate my Modern Portfolio Theory model

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-...
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  • 4,105
4 votes

Markowitz; risky asset frontier w/o risk free asset

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 ...
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  • 6,810
4 votes

What does risk tolerance represent for utility-maximizing optimization with linear constraints?

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 ...
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  • 6,354
4 votes
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Do normal returns make the mean-variance portfolio model perform properly?

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 ...
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4 votes
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Closed-form analytical solution for Markowitz efficient portfolio without short-selling

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 ...
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  • 2,835
4 votes

Why is there a $\frac{1}{2}$ in front of the portfolio variance formula?

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=(\...
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4 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 ...
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  • 5,773
4 votes
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Markowitz portfolio with factor/position constraints

Generally speaking, as long as we can accommodate the optimization using quadratic programming, we are still within the realm of Markowitz optimization: $$ \begin{align} \min&\quad w^Tq+\frac{1}{2}...
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  • 5,773
4 votes
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Is it possible to construct an efficient frontier without the mean?

The Markowitz efficient frontier maps the trade-off between risk (volatility or variance) and (expected) return. As such, there exists no way to construct the frontier without resorting to expected ...
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  • 5,773
3 votes
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Portfolio Optimisation/Covariance Estimation on a large scale

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 ...
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  • 9,097
3 votes
Accepted

Markowitz optimization - can two sets of returns produce the same set of weights?

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 $p$...
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  • 6,354
3 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 ...
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  • 2,835
3 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 ...
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3 votes
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how to relate risk aversion and sharpe ratio in optimisation

$\lambda$ 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 ...
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  • 6,810
3 votes
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How to add the effect of skewness in the portfolio optimisation objective function?

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 ...
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  • 5,773
3 votes
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Statistical methodology for proving the stability in time of asset allocation weights

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 ...
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  • 3,723
3 votes
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Maximum Sharpe ratio and mean-variance optimization

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. ...
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3 votes

Number of Observations for Non-Singular Covariance Matrix Estimation

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 ...
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