Questions tagged [mean-variance]
The mean-variance tag has no usage guidance.
134
questions
-1
votes
0answers
63 views
Conic programming for portfolio optimization instead of quadratic programming: Typo in source?
The benefit of using a cone constraint like
$$\sqrt{\left(w[1]^{2}+w[2]^{2}\right)^{2}}<r$$ compared to a non-cone
constraint like $$\sqrt{\left(w[1]^{2}-w[2]^{2}\right)^{2}}<r$$ is
that the ...
0
votes
1answer
51 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
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
-1
votes
0answers
39 views
How to prove that the portfolio mean $\mu_P$ is biased in the presence of skewness in the portfolio return distribution?
Portfolio returns are a vector that have a statistical distribution, which in turn has moments.
$$s_P = w^\top M_3 (w\otimes w)$$
is the formula for portfolio skewness, where $M_3$ is the coskewness ...
-2
votes
1answer
64 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
0
votes
0answers
17 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
2
votes
2answers
145 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
1
vote
0answers
34 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
2
votes
1answer
50 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
288 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
2answers
110 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
41 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
votes
0answers
21 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
108 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
vote
3answers
163 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
87 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
85 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
votes
1answer
63 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
votes
0answers
58 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
0answers
41 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
votes
1answer
121 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
180 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
44 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
202 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
109 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
votes
3answers
134 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
votes
0answers
139 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
vote
0answers
40 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
131 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
151 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
484 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
vote
1answer
90 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-...
0
votes
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
297 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
118 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,...
0
votes
0answers
46 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 "...
0
votes
0answers
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:
$$R^* = \...
2
votes
1answer
183 views
Mean-Variance optimization with no short selling
I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem:
$$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$
So ...
6
votes
0answers
512 views
How good is the inverse-volatility portfolio?
Heuristic portfolio construction techniques include the equally-weighted portfolio (1/N) and the inverse volatility portfolio (IVP), which is based on the low-volatility effect. They can be assembled ...
4
votes
1answer
683 views
Can a capital market line have a negative slope?
I am struggling to interpret my mean-variance / efficient frontier / capital market line results.
I have no issues calculating the efficient frontier. However, I do increase the risk-free rate from ...
0
votes
1answer
101 views
How to stress test a correlation matrix
As part of a mean variance portfolio task, I am calculating portfolio risk and optimal allocations between assets given required level of return. Input: expected returns, volatility and correlation ...
0
votes
0answers
137 views
Finding the Efficient Frontier in Tensorflow
I have two assets A and B. Asset A has an expected return of 0.92% with a standard deviation of 2.10. Asset B has an expected return of 1.39% and a standard deviation of 2.20. I want to find the set ...
0
votes
0answers
48 views
How to obtain tangency portfolio of the resampled efficient frontier in MATLAB?
I have generated the resampled frontier according to Michaud's approach. In order to compare it with the classical mean variance approach I want to invest in the respective tangency portfolios. While ...
0
votes
1answer
87 views
What if all the weights are negative in mean-variance optimization during a crisis?
Usually the constraint is that all weights sum up to 1. But in a crisis when all assets are falling in prices, intuitively, all the weights should be negative in the optimization.
But it contradicts ...
0
votes
1answer
145 views
Fixes of quadratic utility when probability of decreasing utility is large
In finance and specifically portfolio theory, a popular utility function is quadratic utility
$$
u(x)=x-\frac{\lambda}{2}(x-\mu_X)^2
$$
where $x$ is wealth and $\lambda$ is the parameter of risk ...
1
vote
0answers
78 views
Mean Semivariance Optimization VS PMPT
Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as:
$(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$
where $B$ is the benchmark rate, $R_{i}$...
1
vote
1answer
58 views
Surface plots of the mean-variance efficient frontier
3d surface plots contain an X, Y and Z axis. For the mean-variance efficient frontier:
X axis is portfolio volatility ($\sigma_p$)
Y axis is portfolio expected return ($\mu_p$)
any ideas for what ...
3
votes
2answers
228 views
What does the concept “standard Markowitz approach” include?
Does "standard Markowitz approach" include only mean-variance approach or does it also include other approach such as minimum-variance approach?
4
votes
6answers
273 views
Is a more robust Covariance estimation possible?
I'm working on a mean-variance optimization problem, but instead of financial securities I'm choosing a 'portfolio' of N athletes. It is a 1-period optimization problem over one generic statistic ...
2
votes
1answer
50 views
ARMA moments proof
Consider a standard ARMA(1,1) process such as
$$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$
where $u_t$ is i.i.d. $u_t \sim N(0,\sigma^2)$. I know how to derive mean and variance with stationary ...
0
votes
1answer
93 views
Prove that the portfolio that maximizes utility lies on the efficient frontier
When maximizing mean-variance utility in a portfolio optimization framework
$max \{R - \lambda \sigma ^2\}$
where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is ...
