Questions tagged [mean-variance]
The mean-variance tag has no usage guidance.
139
questions
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35 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 ...
0
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1answer
48 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
45 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 ...
-1
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1answer
49 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 ...
1
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0answers
71 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
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1answer
73 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
votes
1answer
170 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
89 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
97 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
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1answer
57 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
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
-2
votes
1answer
78 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
0
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0answers
20 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
3
votes
2answers
195 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
1
vote
0answers
36 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
2
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1answer
88 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
438 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
3answers
181 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
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1answer
43 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
30 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
votes
1answer
123 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
168 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 ...
-2
votes
1answer
94 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
95 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
76 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 ...
0
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0answers
66 views
Minimum variance portfolio's analytical solution, but assuming $t$-distribution
$$ \boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}} $$
is the well known closed-form analytical solution to the minimum ...
1
vote
1answer
69 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-...
0
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1answer
173 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
votes
2answers
303 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
votes
0answers
58 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 ...
-1
votes
2answers
274 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
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1answer
114 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
136 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
169 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
44 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
votes
1answer
217 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
191 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
vote
3answers
492 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
123 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-...
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0answers
19 views
Risk aversion coefficients for multi objective quadratic programming
I am solving different quadratic programming optimizations in a setting that involves the usual mean variance problem plus other objectives. My concern is how to choose the many so-called risk ...
4
votes
3answers
334 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 ...
1
vote
1answer
230 views
Is this methodology for finding the minimum variance portfolio with no short-selling sound?
I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult,...
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0answers
50 views
Mean variance portfolio - alternative formulations
From this lecture on YouTube the lecturer states that there are three ways to form the mean variance portfolio (minimize variance for a given return, maximize return for a given variance, maximize a "...
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26 views
Calculating R* in a two-asset world
In chapter 5 of John Cochrane's Asset pricing, we derive a state-space interpretation of the mean variance frontier by defining $R^*$ and $R^{e*}$. A little forward, we have this formulation:
$$R^* = \...
2
votes
1answer
341 views
Mean-Variance optimization with no short selling
I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem:
$$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$
So ...
6
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0answers
786 views
How good is the inverse-volatility portfolio?
Heuristic portfolio construction techniques include the equally-weighted portfolio (1/N) and the inverse volatility portfolio (IVP), which is based on the low-volatility effect. They can be assembled ...
4
votes
1answer
907 views
Can a capital market line have a negative slope?
I am struggling to interpret my mean-variance / efficient frontier / capital market line results.
I have no issues calculating the efficient frontier. However, I do increase the risk-free rate from ...
0
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1answer
112 views
How to stress test a correlation matrix
As part of a mean variance portfolio task, I am calculating portfolio risk and optimal allocations between assets given required level of return. Input: expected returns, volatility and correlation ...
0
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1answer
130 views
What if all the weights are negative in mean-variance optimization during a crisis?
Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization.
But it contradicts ...
0
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1answer
188 views
Fixes of quadratic utility when probability of decreasing utility is large
In finance and specifically portfolio theory, a popular utility function is quadratic utility
$$
u(x)=x-\frac{\lambda}{2}(x-\mu_X)^2
$$
where $x$ is wealth and $\lambda$ is the parameter of risk ...
1
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0answers
86 views
Mean Semivariance Optimization VS PMPT
Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as:
$(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$
where $B$ is the benchmark rate, $R_{i}$...
