# Questions tagged [markowitz]

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### 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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### Tangency portfolio with two additional constraints

I know that the formula for determining the weights of the Tangency portfolio is given as $w_{tan}$ = $\frac{\Sigma \mu}{\iota^{\prime}\Sigma\mu }$, but I was wondering how to derive the weights in ...
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### Where to get MSCI World Index constituents (+ weights)

Where can I download The MSCI World index constituents and their weights (daily update) The current prices of the constituents plus three years of EOD (or weekly) history Corporate actions of the ...
261 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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### Optimal Portfolio from Efficient Frontier

I found this code on plotly site, using CVXOPT to find the efficient frontier, and then, the optimal Portfolio. The optimal function is ...
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### Max allowable return in Markowitz model

The Markowitz model solves the following problem: The portfolio with the smallest variance among attainable portfolios with expected return µV. Here we have to choose µV to get the optimal portfolio ...
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### Estimated betas and optimal portfolio

I ran a regression on 20 assets to estimate their beta with different methods. I would like to see the differences of these estimation differences in terms of mean-variance optimal portfolio. How can ...
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### How/Why Markowitz model is normatitive while CAPM positive?

I've tried economic books but they only give this "should/is" explanations and I still cannot see how it applies to MPT. On the other hand, almost every paper and book gives these adjectives before ...
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### Risk/Return Paradox in Markowitz Optimization?

It is possible that from the efficient frontier obtained varying the "lambda" parameter of the risk-appetite coefficient, in the Mean Variance Parametric Quadratic programming problem, it results that ...
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### fPortfolio specify our constraints for efficientPortfolio [closed]

I am working on the library fPortfolio in R and I have a question. How can we fix for a portfolio the sum of weights equal to 1 ? When I study the code, I see that we cannot choose which constraints ...
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### 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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### 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 ...
78 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 ...
189 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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### 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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### 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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### 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 ...
829 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\sigma_p ^2 = w_x^2\sigma_x^2 + w_y^2\sigma_y^2 +... 1answer 530 views ### R optimization using OPTIM I have a covariance matrix and vector of expected returns as my inputs. I have used optim to solve for the weights that maximize the portfolio's return/volatility. I like optim as you can create your ... 1answer 406 views ### Why did Markowitz not derive an equation for the efficient frontier? Currently, I´m studying portfolio management and portfolio selection. The founder of the MPT is Harry Markowitz, of course. But reading his famous article from 1952 and his book from 1959 (actually, I ... 0answers 59 views ### Periodicity justification in Markowitz optimization Monthly returns seems to be the industry standard for everything. Markowitz used monthly returns in his original paper on mean-variance optimization, the efficient frontier, etc. Did he ever provide ... 1answer 122 views ### Mean-Var optimisation of Monte Carlo simulated model I have a problem which involves optimisation of a portfolio containing one stock and multiple call options written on it, with the same maturity and different strikes. In order to use optimisation ... 0answers 240 views ### Mean-Variance Optimization Techniques with Multiple Asset Classes Why does it make sense to use single-period Markowitz mean-variance optimization techniques when we're trying to figure out asset allocation across multiple asset classes (bonds, stocks, REITs, etc)? ... 0answers 110 views ### Relation between mean and variance of a portfolio in modern portfolio theory: I hope that this is the right place to ask my question! Let a market with$N\ge1$risky assets and denote by$(R_i,i=1,\cdots, N)$their returns and$R$the vector of these$N$returns. In addition, ... 0answers 121 views ### Should the number of Markowitz Optimization steps be counted as backtest trials? I'm backtesting a strategy that involves monthly investments in a few stocks out of a given set, that is, each month some of the stocks are shortlisted from an index and a long position is taken in ... 2answers 114 views ### Computing$\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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### 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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### 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 ...