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Questions tagged [markowitz]

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23 views

Mean-cVaR model: How can one include transaction cost

$$ \min \delta CVaR - (1-\delta) \sum_i^{n} \mu_i x_i \\ \sum x_i = \sum x^{old}_i \\ Losses(s) = \sum x_i - \sum_i^{n} (R(s,i))x_i \\ VaRDev(s) = Losses(s) - VaR \\ CVaR = VaR + \frac{\sum_s^{} ...
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1answer
55 views

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

Referencing Wei Jiao (2003) p. 8, formula (1.12), for $Ax = b$ set of linear constraints in a portfolio, the solution for the optimum weights to maximize the utility is: $$w^* = \Sigma^{-1}A^T \left( ...
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1answer
74 views

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

The portfolio optimization problem I have in mind is a minimum variance optimization with positive weights, formulated as below: I am trying to show that the solution is unique, specifically in the ...
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1answer
52 views

Markowitz expected return time

This is perhaps a rather silly question for the more experienced people in the community but it has been puzzling my mind for a while. Let's say we have a portfolio of 10.000 dollar. We will apply ...
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32 views

Markowitz w/ riskless asset & CAPM

If risk free rate ($R_0$) is bigger than expected return on minimum variance portfolio ($\bar{\mu}$), so $R_0>\bar{\mu}$. I.e. the tanget portfolio is on the risky inefficient portfolio frontier ...
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2answers
137 views

Markowitz; risky asset frontier w/o risk free asset

What is the intuition behind the "spanning" result in the following statement? For a fixed pair of distinct frontier portfolios $\phi_p$ and $\phi_q$, any frontier portfolio $\phi$ can be obtained ...
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0answers
48 views

Portfolio optimization under two constraints

I hope here is he correct place to ask my question. I am trying to develop a portfolio strategy with three assets (one of which is risk free). For this I need to determine a vector of weights (w) and ...
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1answer
223 views

Linear Regression vs Mean Variance Optimization

Assume I have n signals, which I would like to linearly weight and combine to form an aggregate signal. Two possible ways of doing this based on historical data are:...
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23 views

Equivalence between Order of Optimization for Mean Variance

When calculating mean variance portfolios, the general strategy is to minimize variance subject to expected return being higher than some benchmark $\mu$. An alternative approach to this problem ...
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1answer
44 views

What price data should I used when making minimum mean variance portfolio, optimal risky portfolio and efficient frontier using Markowitz? [closed]

I need to make optimal risky portfolio, minimum variance portfolio and efficient frontier using Markowitz . But i don't know whether to used close price data or adjusted data. If i'm using adjusted ...
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1answer
148 views

Solving a system of two equations with non-convex matrix multiplication for MV optimization

Scenario: I am trying to do a variation of the MV optimization for a portfolio. In this instance, I already have a vector of mean returns ($\mu$), a vector of ones, a covariance matrix ($\Sigma$), and ...
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140 views

Optimisation problem with bid-ask spread

I want to optimise a static portfolio with a holding period of 90 days given 10 tradable assets. The assets are quoted in bid and ask prices. I want to minimise the risk measured by standard deviation ...
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1answer
215 views

Portfolio Optimisation/Covariance Estimation on a large scale

When using Markowitz Portfolio Theory, e.g. for finding an optimal portfolio composition, one needs to have estimates of the returns, but most importantly of the covariance matrix. If our universe of ...
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2answers
2k views

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

Given $N$ assets, the Markowitz mean-variance model requires expected returns, expected variances and a $N \times N$ covariance matrix. The joint distribution is fully defined by these measures. ...
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2answers
161 views

Backtest Results needed to Model Validate my Modern Portfolio Theory model

this is my 1st post, and I hope someone can help me! I have been searching for a week now without any luck I have built a Portfolio Allocation model based on Modern Portfolio Theory (MPT). I now need ...
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0answers
113 views

Solution for Markowitz problem with Safety-First Ratio

What is the solution for the following markowitz unconstrained problem? The sum of the entries of the weights vector $w$ should always be requied to sum one? Or if we use the risk-free asset it can be ...
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1answer
71 views

Show that two formulations of Markowitz problem are equivalent

I would like to solve (as mathematically and formally as possible) that the following Markowitz problems are equivalent. The big point is: I want to show that it is equivalent to constrain the return ...
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1answer
405 views

Derivation of the efficient frontier set (markowitz problem)

I would like to find a Derivation of the efficient frontier set for the markowitz problem:
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1answer
148 views

Solving a Markowitz problem with restrictions (lower and upper bound) to the weights vector

I would like to find a step by step solutionfor the following Markowitx problem. It is a standard markowitz problem. The unique detail (wich is why I am posting this question here) is that there is a ...
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1answer
149 views

Prove that a determinant in markowitz method derivation is greater than zero

I want to prove that the following determinant, that appears in the markowitz method of portfolio allocation is greater than zero. ($\mu$ is the vector of returns and $\sum$ is the covariance matrix)
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1answer
115 views

Return $\mu$ and volatility $\sigma$ for tangency portfolio of DOW30 too large?

I am calculating GMV and TAN mu and sigma as well as weights using the straightforward derivations, such as: \begin{equation} \mu_{gmv}=\frac{\mathbf{1}'\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}}{\...
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1answer
385 views

R optimization using OPTIM

I have a covariance matrix and vector of expected returns as my inputs. I have used optim to solve for the weights that maximize the portfolio's return/volatility. I like optim as you can create your ...
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404 views

Sharpe Ratio Optimization CPLEX Matlab

I am trying to determine the maximum Sharpe ratio in my MATLAB code; however we are forced to use CPLEX from IBM. My problem is I don't know who to translate my optimization problem to the correct ...
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1answer
680 views

optimal portfolio with different lending and borrowing rates

I have 4 risky securities (have returns and var-cov matrix for monthly data), and I can lend at 1% per annum, but borrow at 5% per annum. If i wish to obtain the s.d. of 5%, what is the optimal ...
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1answer
317 views

Why did Markowitz not derive an equation for the efficient frontier?

Currently, I´m studying portfolio management and portfolio selection. The founder of the MPT is Harry Markowitz, of course. But reading his famous article from 1952 and his book from 1959 (actually, I ...
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2answers
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Risk contribution of part of a portfolio

Is it quantitatively sound to say that if I have assets $x, y,$ and $z$ in a portfolio, and that the total variance of the portfolio is defined as $\sigma_p ^2 = w_x^2\sigma_x^2 + w_y^2\sigma_y^2 +...
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0answers
51 views

Periodicity justification in Markowitz optimization

Monthly returns seems to be the industry standard for everything. Markowitz used monthly returns in his original paper on mean-variance optimization, the efficient frontier, etc. Did he ever provide ...
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1answer
103 views

Mean-Var optimisation of Monte Carlo simulated model

I have a problem which involves optimisation of a portfolio containing one stock and multiple call options written on it, with the same maturity and different strikes. In order to use optimisation ...
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0answers
199 views

Mean-Variance Optimization Techniques with Multiple Asset Classes

Why does it make sense to use single-period Markowitz mean-variance optimization techniques when we're trying to figure out asset allocation across multiple asset classes (bonds, stocks, REITs, etc)? ...
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2answers
2k views

Formula for Optimal Portfolio of 2 Assets when No Shorting Allowed?

I am looking for a formula to calculate the weights of two risky assets that produce the optimal portfolio (i.e highest Sharpe ratio). So far I have found the following formula from a website of ...
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0answers
98 views

Relation between mean and variance of a portfolio in modern portfolio theory:

I hope that this is the right place to ask my question! Let a market with $N\ge1$ risky assets and denote by $(R_i,i=1,\cdots, N)$ their returns and $R$ the vector of these $N$ returns. In addition, ...
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0answers
101 views

Should the number of Markowitz Optimization steps be counted as backtest trials?

I'm backtesting a strategy that involves monthly investments in a few stocks out of a given set, that is, each month some of the stocks are shortlisted from an index and a long position is taken in ...
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1answer
253 views

dynamic Markowitz portfolio

Let's take 4 assets, whose values are known during a period of time of 2 years. Then I calculate the expected returns for each of these 4 assets thanks to these 2 years - historical data. I deduce the ...
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169 views

Behaviour of out of sample efficient frontier

I am comparing the efficient frontier of a set of portfolios that are in and out of sample. The first period is from 1991-01-03 until 1992-10-03 and the second one from 1992-10-03 until 1994-03-03. I ...
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2answers
339 views

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

I found some functions for Markowitz mean variance portfolio optimization in R such as portfolio.optim in tseries package. ...
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0answers
203 views

Finding mean vector and covariance matrix for annual returns given quarterly returns

I am currently trying to calculate a vector for the mean annual returns of 4 different asset classes along with their 4x4 covariance matrix in excel. However, I am having problems since the data I ...
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2answers
111 views

Computing $\gamma$ and $\mu$ at the efficient frontier

Consider the condition which the weights of any portfolio belonging to the efficient frontier satisfy: \begin{equation} \gamma\boldsymbol{wC} = \boldsymbol{m} - \mu\boldsymbol{u}\end{equation} ...
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1answer
247 views

Full Kelly portfolios having same weights as tangency portfolios

I'm currently comparing empirically the differences between Markowitz and Kelly portfolios. I calculated the Kelly weights for monthly return observations over 10 years for a sample of 50 stocks from ...
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182 views

Definition of sharpe ratio maximising and variance minimising portfolios

In this paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2226985, in the derivation of the mean variance efficient portfolio using lagrangians in the appendix, on page 29, the two portfolios ...
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1answer
80 views

Understanding portfolio weights and purchasing stock in modern portfolio theory

Recently I've been learning about the markowitz algorithm. It's pretty interesting, but I'm curious how we apply this in practice. Lets say I have some optimal portfolio: $R_p = x_aR_a + x_bR_b$ ...
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1answer
353 views

Matlab Portfolio Optimization with bid ask spread

I'm trying to find the optimal portfolio of options and stock which minimizes the standard deviation of the portfolio returns but also taking into consideration the bid and ask prices of the assets. ...
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1answer
1k views

Sharpe Maximization under Quadratic Constraints

When doing Sharpe optimization $$ \max_x \frac{\mu^T x}{\sqrt{x^T Q x}} $$ there is a common trick (section 5.2) used to put the problem in convex form. You add a variable $\kappa$ such that $x = ...
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198 views

Residual Covariance Matrix, and MVO for Residual Variance and Alpha

My overall goal is to find an efficient frontier using QP in terms of $\alpha$ and residual variance ($\omega^2$) for a portfolio $P$ given a benchmark $B$. We know the equation for residual variance ...
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2answers
280 views

Modern portfolio theory in practice

I am wondering about the Markowitz theory of portfolio construction in practice. Hence, if one wants to know the efficient frontier, what variances can one use. The only method that I can think is the ...
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0answers
402 views

Tangent portfolio weights without short sales?

Consider a mean-variance investor in a world with a risk-free asset. Let $R_f>0$ be the return of the risk-free asset, $\mathbb{E}(R_i)>R_f$ the expected return of the risky asset $i$ and $SD(...
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1answer
521 views

Markowitz Mean-Variance Implied Returns

What is the closed form solution for the following inverse Markowitz problem? Given a mean-variance optimized fully invested portfolio $X$, a risk aversion parameter $\lambda$ and a var-covar ...
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1answer
1k views

short-sale constraint with nonpositive-definite matrix in portfolio optimization

I need help about portfolio optimization in R. I have inverted matrix and I want to use it as an input in portfolio optimization. It was non-positive definite before I have handled it. In portfolio ...
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3answers
923 views

Finding Expression for Optimal Markowitz Weights

So there are two assets with return rates $r_1$ and $r_2$ which have identical variances and a correlation coefficient $p$. The risk free rate is $r_f$. I need to find an expression for the optimal ...
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4answers
2k views

Does Modern Portfolio Theory align with EMH?

I came to the conclusion that in literature Markowitz' Portfolio Theory is believed to be compliant with the Efficient Market Hypothesis. The weakest form states that the current price fully ...
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1answer
618 views

What are the assumptions of portfolio optimisation with higher moments?

I was wondering whether there are a set of assumptions for portfolio optimisation with higher moments (including kurtosis and skewness) as there are for regular mean-variance optimisation?