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How to prove that the feasible set of a two-asset portfolio is a hyperbola?

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book does not give a complete proof, and I did not find a ...
bokabokaboka's user avatar
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37 views

GARCH for Mean Variance Optimization

I am currently trying to carry out a mean variance optimisation, with the implementation of GARCH. I'm not sure if this is going to make complete sense as my understanding of GARCH is limited. In the ...
FraserM2000's user avatar
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41 views

Scaling returns to use PCA?

Many machine learning techniques perform better, if the data is preprocessed - either by normalization (MaxMin Scaler) or standardization (Standard Scaler). But that comes with a lack of ...
Marlon Brando's user avatar
1 vote
1 answer
94 views

Calculating marginal risk contribution of FX for foreign asset portfolio

I am a European investor investing in US equities. My US equities portfolio returns in EUR can be broken down into (1) equities returns in USD terms, and (2) USDEUR spot currency returns. Using the ...
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Discuss how you would allocate your budget between the two assets if their correlation is 1, 0, or -1

An asset A is expected to yield a $2\%$ return with a standard deviation of $1\%$, and another asset B is expected to yield a $1\%$ return with a standard deviation of $1\%$. Discuss how you would ...
LeFo's user avatar
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1 answer
148 views

Calculation of break-even correlation for diversification effect in N-assets case?

I'm thinking about a generalization of the following case: for 2 assets, there is a diversification effect as soon as i obtain a positive weight for the minimum-variance portfolio in the asset with ...
T123's user avatar
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76 views

Backtesting on factor model residual returns

I've heard in quantitative equity strategies, people backtest signals on residual returns. How does this work in practice? Do people find signals that forecast residual returns and then run the full ...
Michael's user avatar
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0 answers
66 views

Robust estimates of variance covariance matrix

I am looking for help from other people with experience creating variance covariance matrix that have enough predictive power to actually lower portfolio volatility out of sample. Using real world ...
helloimgeorgia's user avatar
1 vote
1 answer
284 views

PCA for portfolio optimization (Markowitz)

Suppose that I've used the spectral theorem of linear algebra to completely decompose the covariance matrix. I now know the largest and smallest eigenvalue, which corresponds to the largest and ...
Marlon Brando's user avatar
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60 views

How to change the covariance matrix for a parallel-shift of the efficient frontier?

I'm trying to obtain a parallel shift in my efficient frontier based on the Merton 1972-parameters. As i think a picture tells you more than 1000 words here is what i tried: The setting of my problem ...
T123's user avatar
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78 views

How to construct the behavioral efficient frontier

I just stumbled across an interesting chart in Meir Statman's book "Finance for Normal People" where he introduces his behavioral portfolio theory. There, he also provides the following ...
T123's user avatar
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2 votes
1 answer
364 views

How to solve for the optimal portfolio weight with target variance?

I'm confused a bit with the following problem: As far as i understand, the following problem where $$\min_{w} \omega^{T}\Sigma\omega$$ $$\textrm{s.t.}\hspace{0.5cm} \omega^{T}\mu=E$$ $$ \omega^{T}\...
T123's user avatar
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6 votes
1 answer
821 views

Markowitz Eigenvalues & PCA

I came across this passage in a book about PCA and denoising of Markowitz: But eigenvalues that are important from risk perspective are least important ones from portfolio optimization perspective. ...
Markowitz's user avatar
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1 answer
112 views

Markowitz Optimization with 2 assets

Suppose there are only two risky assets and we want to optimize our portfolio. Constraints are that we have a minimum return $\overline{r}$ and we can only invest $w_1 + w_2 = 1$. Is it possible that ...
Options's user avatar
1 vote
1 answer
168 views

How do asset prices behave in a single-period and multi-period model?

When we talk about the single-period CAPM, the return in a particular period t can be defined as $(P_t - P_{t-1})/P_{t-1}$. Investors plan at t-1 and get a payoff at t. After this period, the same ...
lkonoplev's user avatar
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117 views

For mean-variance portfolio optimization, shouldn't all the allocations sum to 1?

Reading a paper by Black and Litterman, I'm having trouble understanding the set of valid allocations in which we're trying to optimize expected returns. In Table III, the authors show two portfolios ...
Arthur Santana's user avatar
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1 answer
163 views

Is it possible to construct an efficient frontier without the mean?

If we assume the estimator for a sample mean is biased and if the optimal portfolio weights vary with the estimated mean, is there a way (similar to the zero beta portfolio approach wrt the risk free ...
T123's user avatar
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4 votes
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Why does the mean term have a higher effect than the covariance term in MV optimization? [closed]

I am trying to use the mean-variance (MV) optimization framework. When I change the mean term using future-ground-truth return (I am not supposed to do so), it has a higher effect on the MV ...
randy's user avatar
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0 answers
172 views

Why do we use half of the risk in objective function of markowitz portfolio theory

In some documents I have seen objective function of markowitz portfolio theory is as follows. minimize 1/2 * w'Σw where w is weights Σ is covariance matrix I could ...
Validus Oculus's user avatar
3 votes
1 answer
244 views

Markowitz portfolio with factor/position constraints

General Markowitz-style optimization (problem objective of $w^T \mu - \lambda w^T \Sigma w$) yields simple optimal weights policy $w \propto \Sigma^{-1} \mu$. However, I would like to add a series of ...
Michael Clinton's user avatar
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160 views

How to show that the normalized portfolio given by the Markowitz portfolio optimization model is efficient?

How can I show this? It looks relatively easy but am not sure if I am doing it right.
Ivan Lee's user avatar
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Error in eqsumW[2, -1] : subscript out of bounds - SolveRsocp

I am working on a research project where the goal is to get the highest possible return with different portfolios at a given risk rate. For this I would like to use the function "SolveRsocp" ...
Georg C's user avatar
3 votes
1 answer
264 views

Weights don't add up to one Markowitz Portfolio Model

I have recently implemented MPT in Python, however, when I allow negative weights (short selling), they do not add up to one. Isn't it suppose not to happen? On the other hand, when I don't allow, ...
Mr. N's user avatar
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1 vote
2 answers
373 views

Time-complexity of Markowitz portfolio optimization

What is the time-complexity of Markowitz mean-variance portfolio optimization (MVO)? I am unable to find any clear explanation of this on the internet and in academic papers. These are my questions: ...
questiondude's user avatar
1 vote
1 answer
622 views

Monte Carlo vs. Block Bootstrapping vs. Bootstrapping

Because I can fit e.g. ~25 distributions via empirical cumulative distribution fitting to correlated data (including stable dist.), and then simulate the original data based on correlation (covariance)...
user avatar
0 votes
2 answers
716 views

Number of Observations for Non-Singular Covariance Matrix Estimation

Marcos López de Prado writes the following in his book Advances in Financial Machine Learning: In general, we need at least \frac{1}{2} N (N+1) independent and ...
Nick's user avatar
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1 vote
1 answer
1k views

Maximum Sharpe ratio and mean-variance optimization

I want to understand why this holds: $argmax_w ( \frac{\mu^T w}{\sqrt{w^T\Sigma w}})=\Sigma^{-1}\mu $ I just found this post: Derivation of the tangency (maximum Sharpe Ratio) portfolio in Markowitz ...
Valentin's user avatar
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1 answer
194 views

Geometry of Efficient Frontier of Portfolios

I have been reading about Portfolio Theory, and though the algebra of it seems quite intuitive, I am having a hard time understanding it's geometry. For the sake of simplicity, I will only talk about ...
Aditya Kulkarni's user avatar
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1 answer
158 views

Portfolio Optimization constrained to maximum N% of short selling portfolio weights

For mean-variance portfolio optimization with short-selling allowed, but restricted to a certain percentage of the portfolio weights (lets assume N), we can constrain it in the follwoing way: (from j=...
Joquim's user avatar
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1 vote
0 answers
119 views

Herfindahl-Hirsch-Index for FX Portfolios

Assume we have an FX portfolio with $n$ currency pairs, such that $w_i$ is the weight of currency pair $i \in \{1, \dots, n\}$ in the portfolio, and $\sum_{i=1}^n w_i = 1$. All pairs have USD as one ...
Nick's user avatar
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Construct portfolio with assets having expected negative returns

I have been asked to select a n stocks among N stocks, to construct a portfolio. Some of them have have negative weekly returns on average. If I want to select these n stocks by constructing an '...
randomwalker's user avatar
0 votes
1 answer
380 views

Mixed-integer programming approach for index tracking

Suppose you currently own a portfolio of eight stocks. Using the Markowitz model, you computed the optimal mean/variance portfolio. The weights of these two portfolios are shown in the following table:...
statwoman's user avatar
  • 123
2 votes
2 answers
203 views

Statistical methodology for proving the stability in time of asset allocation weights

I am comparing the set of weights obtained by the classical Markowitz allocation process with those of another asset allocation technique I have devised. Markowitz's weights are unstable, as the ...
Vitomir's user avatar
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1 vote
1 answer
115 views

How to deal with securities that has short historical data when performing mean-variance portfolio analysis?

I am trying calculate expected return and risk (stdev) based on historical data using Mean Variance Analysis framework. Let's say the portfolio has 10 stocks, 9 of them have more than 10 years history ...
Xiaowan Wen's user avatar
2 votes
1 answer
98 views

Optimal Portfolio Formulation

I'm currently studying Luenberg's Article "Projection Pricing" (Jrl of Optimization Theory and Applications, Vol. 109, No. 1, pp. 1–25, April 2001) and there is a claim that I can't prove. ...
Felipe Teti's user avatar
2 votes
2 answers
381 views

State-of-the-art MVO methods?

I learned Markowit'z mean-variance optimzation in school. Now I've been googling a bit, and to my surprise, Markowit'z is STILL being used by most people, AFAIK. Are there really not some state-of-the-...
shudup's user avatar
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1 vote
1 answer
126 views

Optimal Portfolios with Skewed and Heavy-Tailed Distributions

I am learning about portfolio theory and been using Markowitz. I wondered, however, if I can use distributional and asymmetric information of the returns to solve the problem. For instance, I have a ...
PitPartizan's user avatar
-1 votes
1 answer
263 views

Portfolio variance $<=$ weighted average of individual variances [closed]

In portfolio theory, I often (with some justifications but the message is the same) come across the following statement: "The most important quality of portfolio variance is that its value is a ...
MathStat2718's user avatar
0 votes
1 answer
182 views

Mean estimate in portfolio optimization (Markowitz) [duplicate]

The Markowitz mean-variance portfolio optimization problem is to find the optimal allocation, $w_{optimal}$ by solving: \begin{equation} w = \mathrm{argmax} \ \mu_{t}^Tw - \frac{\gamma}{2}w^{T}\...
MathStat2718's user avatar
0 votes
2 answers
193 views

Why isn't the asset with minimum variance given a 100% portfolio weight? [closed]

The maximum expected return portfolio is the one that assigns a 100% weight to the asset with the highest expected return amongst all assets under consideration. Shouldn't then the asset with the ...
develarist's user avatar
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1 vote
2 answers
928 views

Standard deviation formula with Short selling- Markowitz model

I have 2 fast quastions. Before I begin I want to show you that I found minus before SD of bills in the book Principles of corporate finance(1.screen). I know SD of bills is zero and minus in this ...
Miroslav Holub's user avatar
1 vote
2 answers
148 views

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

Can someone explain me where $\frac12$ came from in the expression $\frac{1}{2} \omega'\Sigma \omega$? That is the expression to be minimized io order to get the minimal variance portfolio (with also ...
Fabio's user avatar
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1 vote
1 answer
2k views

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

In a portfolio without risk-free assets I know that the efficient portfolio si given by: $\omega=\frac{1}{BC-A^2}[\mu(C\Sigma^{-1}R-A\Sigma^{-1}\mathbb{1})+B\Sigma^{-1}\mathbb{1}-A\Sigma^{-1}R]$, ...
Fabio's user avatar
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3 votes
2 answers
477 views

How to add the effect of skewness in the portfolio optimisation objective function?

I have the following risk adjusted portfolio which I optimise, where gamma is the risk return trade off, $r$ are the returns and $C$ is the covariance matrix which considers scenarios, so it is not ...
Luigi87's user avatar
  • 326
0 votes
0 answers
75 views

Optimizing Portfolio Return by Targeting Variance

I understand Markowitz and targeting returns to minimize our variance. I know this optimization problem well and its constraints. However when the reverse scenario is to be considered I get very ...
eruiz's user avatar
  • 19
4 votes
1 answer
3k views

Portfolio Optimization sum of weights constraint with short selling

For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint. In most resources I've seen, such as https://www.coursera.org/learn/...
Astro Boy's user avatar
4 votes
1 answer
528 views

Do normal returns make the mean-variance portfolio model perform properly?

The Markowitz mean-variance model is known to suffer from estimation error due to financial returns not meeting the assumptions of a normal distribution, providing portfolio weights that underperform ...
develarist's user avatar
  • 3,000
1 vote
1 answer
1k views

how to relate risk aversion and sharpe ratio in optimisation

I am trying to optimise the following: U(w)=w′μ−λ/2w′Σw which is the typical risk aversion problem. I would like to set lambda in order to have the max sharpe but I cannot find in literature what is ...
Luigi87's user avatar
  • 326
2 votes
0 answers
384 views

Transform this non-linear portfolio optimization problem into a quadratic optimization problem

I have a portfolio optimization problem similar to this question here, with a V-shape transaction costs such that we pay a fee proportionally to the sum of absolute rebalancing: $$TC(\omega) = \frac{1}...
JejeBelfort's user avatar
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1 vote
0 answers
112 views

Can simple risk management outperform portfolio optimization like this, or is there most likely an error?

I am using a simulation approach to compare the performances achievable by simple risk management and portfolio optimization for portfolio selection. My problem is that my results indicate that simple ...
Ali Mustafa's user avatar