Questions tagged [mean-variance]
The mean-variance tag has no usage guidance.
148
questions
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127 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)...
1
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1answer
40 views
Mean-Variance Portfolio Axis Description
I'm currently looking into the mean-variance approach to portfolio theory and I wonder, why the standard deviation $\sigma$ is graphed on the x-axis and not the variance $\sigma^2$ as a measure of ...
0
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1answer
56 views
Proof that mean-variance opportunity set is closed
In the book Financial Economics (2010) by Hens and Rieger, on page 101 we find the following Lemma 3.1: If we have finitely many assets, the minimum-variance opportunity set is closed and connected.
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92 views
Criticism against mean-variance analysis
Are there any good review papers or books about the criticism against mean-variance analysis?
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1answer
50 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=...
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0answers
95 views
Black-Litterman for quant portfolio
I have seen a lot of research around the Black-Litterman approach and I think theoretically, it is a nice framework. However, it appears that its main strength is from a practitioner's point of view, ...
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0answers
38 views
CVXOPT quadratic programming mean variance example
Trying to learn how to use CVXOPT to do quant finance optimization.
For the example given on page https://cvxopt.org/userguide/coneprog.html#quadratic-programming . I feel confused how this "S&...
0
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1answer
108 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:...
2
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2answers
161 views
Deriving the risk-aversion coefficient
By considering the parametrised formulation of the mean-variance criterion by Markowitz, the risk aversion coefficient $\lambda$ can be derived as follow.
As suggested by Arrow and Pratt, given the ...
2
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1answer
61 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.
...
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1answer
65 views
Covariance Matrix for asset returns [closed]
Hey guys I'm pretty new here, not sure how to code my question so I'll include a picture reference instead. I'm a bit confused on how the standard deviation of F (commodity price) would affect the ...
0
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1answer
218 views
Corner portfolios
This is more a theoretical problem rather than a technical one.
I am looking for a clear and rigorous definition of corner portfolios and I like to understand more precisely their relation with the ...
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0answers
41 views
Is this efficient frontier graph reasonable?
Hi. The above image is taken from https://www.newfrontieradvisors.com/media/1166/optimization-with-non-normal-resampling.pdf.
Is this a reasonable chart? The 4 different methods give 4 non-overlapping ...
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1answer
60 views
Should a stock with high return autocorrelation be weighted more heavily in a portfolio?
Some say the presence of autocorrelation (aka serial correlation) in a stock's financial return time series helps with forecasting its next-day movements, unlike a stock that has low serial ...
0
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1answer
78 views
Calculate weight of an asset
Suppose there are three assets, and the first asset has volatility 18%, the second asset has volatility 16%, and the third asset has volatility 16%. Suppose also that the first two assets' returns ...
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1answer
66 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 ...
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0answers
79 views
Prove norm $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ of min-variance portfolio $\leq$ max-Sharpe portfolio
The minimum-variance portfolio weight vector is
$$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$
whereas the maximum ...
2
votes
1answer
89 views
Downside deviation (semivariance) in m.v. portfolio optimization
Currently I am considering the downside deviation or semivariance in a m.v. optimization framework.
For this specific measure of risk I have found in papers different formulae. The majority of them ...
2
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1answer
242 views
Mean Absolute Deviation in m.v. portfolio optimization
I just read some articles about $MAD$ as a measure of risk in finance.
Is the following formulation a correct way to implement a $MAD$ portfolio optimization model which minimizes risk without ...
2
votes
1answer
105 views
Maximum skewness portfolio solution derived from its Lagrangean formulation
$$\arg \min_w \quad w^\top \Sigma w$$
\begin{align}\text{s.t.} \quad \mathbf{1}^\top w = 1 \end{align}
is the optimization problem for the minimum-variance portfolio weights, whose analytical solution,...
0
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2answers
109 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 ...
0
votes
1answer
71 views
Correlation between mean-variance efficient portfolios
If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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1answer
113 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
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0answers
20 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
3
votes
2answers
336 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
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0answers
48 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
3
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1answer
333 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
820 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
3answers
283 views
Do the minimum VaR and minimum ES portfolios lie on the mean-variance efficient frontier?
The mean-variance efficient frontier holds the minimum variance portfolio, but in the graph above it shows that the minimum VaR (Value-at-Risk) and minimum ES (CVaR) portfolios (expected shortfall/...
0
votes
1answer
66 views
Maximum return portfolio using linear programming with quadratic constraints
In the maximum return portfolio problem formulation above,
is $A=\mu^\top \Sigma^{-1} \mu$?
What is $b$ equal to, and
is the second constraint required? An inequality constraint for target portfolio ...
0
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0answers
42 views
Symbol for the feasible set of portfolios in mean-variance analysis?
When we optimize some mean-variance efficient portfolio, it lies on the efficient frontier (blue line) which is considered superior to the feasible set of portfolios. The feasible set (red dots), on ...
2
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1answer
146 views
Is quadratic programming used to maximize portfolio skewness and kurtosis?
Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$
because the objective function coincides ...
1
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3answers
170 views
Do portfolio mean and portfolio variance have probability distributions?
If $X$ is a $T\times N$ matrix of multivariate asset returns,
and $w$ is some optimal portfolio weight vector,
then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
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votes
1answer
114 views
Should portfolios have zero or negative correlation between assets? [closed]
Is it more optimal to have a portfolio whose assets are negatively
correlated? (I am not requiring all assets to be negatively correlated in this case, nor (-1) perfectly negative correlation either. ...
3
votes
1answer
136 views
Why does the likelihood of corner solutions in portfolios increase as the number of assets grows?
A three- asset portfolio doesn't seem prone to generating corner solutions, which are very high allocations to one of the assets and $0$ to the others. Instead, when the number of assets is low, these ...
0
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1answer
91 views
Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?
The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
1
vote
1answer
109 views
Mean-EVaR efficient frontier
Entropic Value-at-Risk (EVaR) is an alternative and more efficient risk measure than conditional Value-at-Risk (CVaR). EVaR serves as an upper bound to both VaR and CVaR.
Below is a graph of the mean-...
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2answers
284 views
Cover's universal portfolio vs. Markowitz's mean-variance model
Cover's universal portfolio maximizes the wealth growth rate
Markowitz's mean-variance model minimizes portfolio variance
Both allocate assets based on historical returns.
How do these two models ...
0
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2answers
581 views
Why is portfolio optimization a convex problem if variance is concave?
Variance is concave, so portfolio risk must be too.
The mean-variance model employs quadratic programming to optimize (minimize) portfolio risk. My understanding is that quadratic programming requires ...
0
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0answers
86 views
Isn't portfolio optimization basically just feature selection?
Statistical learning has a large assortment of tools for conducting feature selection such as PCA analysis, ridge regression, LASSO, SVM and almost every other machine learning algorithm.
In portfolio ...
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2answers
453 views
Why does portfolio optimization require a positive-definite covariance matrix?
Why does the portfolio optimization mean-variance model require the covariance matrix to be positive-definite? Does this requirement have to do with the need to be able to invert the matrix during ...
0
votes
1answer
124 views
Is non-linear correlation an issue in portfolio optimization?
Portfolio weights are linear combinations of assets. How can it be true then for there to be, and how can someone prove that there is any, non-linear correlation issues in portfolio optimization? Is ...
0
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3answers
149 views
Are mean-variance efficient portfolio weights random variables with probability distributions?
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
2
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0answers
226 views
James-Stein estimator for superior estimates of returns in m.v. portfolio optimization
I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an ...
1
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0answers
48 views
Selecting the best characteristic portfolio per rebalance date
An investor typically decides a portfolio objective and sticks with that objective for every rebalance date in the portfolio's life. Common characteristic portfolios that the investor chooses are:
...
3
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1answer
351 views
Ledoit/Wolf covariance shrinkage in risk-parity optimisation
This is more of a theoretical question.
I have been working on some mean-variance / Black-Litterman models and played around with Ledoit/Wolf's covariance shrinkage method (sklearn function in Python)....
4
votes
1answer
281 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 ...
1
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3answers
504 views
Any portfolio theories not based on asset returns?
For data, the mean-variance model for portfolio optimization uses asset returns to minimize portfolio risk (covariance matrix), which is asset returns volatility, and sometimes simultaneously ...
1
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1answer
295 views
How do i find the covariance between two portfolios?
I know that the formula for covariance is
But this is for two securities. How do I find the covariance between two portfolios? more specifically between the global minimum variance (GMV) and the mean-...
4
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3answers
389 views
Any portfolio models not based on asset return moments?
The mean-variance model for portfolio optimization minimizes portfolio risk (covariance matrix), which is the second statistical moment of multivariate asset returns, and sometimes simultaneously ...