Questions tagged [covariance-matrix]

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How to create three variable system + test the hypothesis that the VAR residuals from two variables' equations can be treated as "structural" errors

I am currently doing an econometrics assignment and am completely stumped on a question. I have screenshotted the question and pasted below. Both questions are to be answered on EViews; having looked ...
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1answer
68 views

Sample Variance of Portfolio

Let $w$ denote a vector of portfolio weights, $r_i$ denote the $i$th return vector, $\Sigma$ denote the Covariance matrix of $r_i$ and let $\hat{\Sigma}$ denote the sample covariance matrix of $r_i$. ...
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1answer
119 views

Why does Hierarchical Risk Parity ignore the clusters generated?

I am currently working through the Hierarchical Risk Parity algorithm (Lopez de Prado (2016) link ) and trying to understand each of the steps. I have completed the step of creating the clusters, and ...
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1answer
75 views

Express the covariance in terms of the standard deviations and correlations

I really need help on this problem. Any suggestion is greatly appreciated! Suppose there are $n$ assets with $n\times n $ covariance matrix $C=SRS$, where $S$ is a matrix with standard deviations $\...
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2answers
117 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 ...
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38 views

Generate a Fractional Gaussian Noise

I am trying to simulate a Fractional Gaussian Noise using Fast Fourrier algorithm.However,I couldn't even if I could retrieve my original covariance matrix such : $E\left[ X(t)X(s) \right] = \frac{1}{...
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38 views

How to use Partial Correlations in Portfolio Construction

In learning more about precision matrices and partial correlations, I've begun wondering (very generally) how these statistical measurements could be used in portfolio construction. More broadly, can ...
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1answer
185 views

Optimise the Sharpe ratio of a portfolio of uncorrelated assets

Given a portfolio of $n$ assets, mean returns vector $\mu$, covariance matrix $K$, one can calculate the portfolio weights $w^*$ that maximise the portfolio Sharpe ratio, by solving: $$w^*=\text{...
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1answer
116 views

Ledoit & Wolf (2003) shrinkage approach

http://www.ledoit.net/honey.pdf In the case where you have one sample covariance matrix S and an optimal shrinkage parameter and you want to estimate the covariance matrix resulting from the Ledoit ...
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58 views

Covariance matrix for risk factors of a FX Forward contract

Does it make sense to calculate log-returns of interest rates from a zero curve? Context: I'm trying to build a variance-covariance matrix for the risk factors of a USDBRL FX Forward maturing in 1 ...
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1answer
59 views

Sub-portfolio correlation

I am trying to reduce correlation matrices into sub portfolios. For example, I have a covariance matrix $\Sigma$ and weight-vector $w$ of two line items which I blend together into a sub-portfolio $\...
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1answer
73 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 ...
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161 views

What is the difference between np.cov(array) and array.cov()?

I'm trying to find a covariance matrix, so when i use returns.cov() on my returns variable, I get a good result. Unfortunately, when i want to use ...
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1answer
291 views

Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?

What I want to do is the following: Let's say I have two assets 1 and 2, and have a 2x2 covariance matrix. Then I have two portfolios A and B made of weights from assets 1 and 2. What I would like to ...
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1answer
61 views

Odd Result from Computing Correlation Matrix from Kalman Filter Posteriori Covariance Estimate

I am using a Kalman Filter to estimate the return dynamics of a forwards curve on a particular commodity. My state space is the initial forwards values, and an initial guess of the drift functions for ...
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1answer
207 views

For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$? [closed]

From multivariate asset returns $X$, we can calculate the sample covariance matrix $\Sigma$. The definition of (any) portfolio variance is $w^\top \Sigma w$, where $w$ are portfolio weights. If $X w$ ...
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1answer
131 views

Covariance matrix for multiple assets - Second attempt

Ok, on the advice of administration I open a new question, hoping that in this way it becomes clearer. Like I said before, I am trying to understand how the authors of this (page 76) and this (page ...
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63 views

PCA on covariance matrix with weights on the columns?

I'm reading two papers by Mark Kritzman on two indicators (turbulence proxied by the Mahalanobis distance and absorption ratio which is basically the ratio of the variance captured by the top 20% PCA ...
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1answer
274 views

Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$

How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to $$w_A^\top\Sigma w_B$$ where $w_i$ is a unique portfolio weight vector, and $\...
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1answer
168 views

Covariance matrix for historical series w/ different start and end dates

I am trying to compute the variance-covariance matrix of my portfolio composed by some shares of different companies. I would select a time horizon of two years but for some shares of one company I ...
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0answers
36 views

Can the covariance matrix be represented as a scalar or something similarly small, instead of a large pair-wise grid?

The covariance matrix tabulates pair-wise interactions between variables (assets) one-at-a-time into a grid, which can quickly become large as the number of assets included in a portfolio, for example,...
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1answer
238 views

Inverse Covariance Matrix Transformation from CAPM

Beginning with the CAPM model we have (with a risk free rate of 0%): $r_i=\beta_i (r_m)+\varepsilon_i$ with $\varepsilon_i$ the diversifiable risks per assets The variance matrix: $\Omega = \beta'\...
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1answer
128 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. ...
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1answer
59 views

Covariance matrix using world stocks [closed]

What is the best way to compute a covariance matrix of daily stock returns made up of international stocks. Knowing that the world markets are not trading simultaneously. This matrix could then be ...
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75 views

Covariance/correlation matrix from data with missing data points

I have a data set with index fund quotes, and I'm trying to compute the efficient portfolio frontier for it. But some data points are missing. In some cases there are few funds that trading even on ...
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1answer
100 views

Double objective in portfolio optimization

Is there anything infeasible or ethically wrong about optimizing portfolios like this? $$\min_w \enspace w' \Sigma w + w' C w$$ where $\Sigma$ is the asset return covariance matrix, and $C$ is the ...
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3answers
184 views

Interpretation and units of a covariance element in portfolio risk

Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and ...
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2answers
150 views

Which is more ill-conditioned, the asset correlation matrix or covariance matrix?

If i have a matrix of multivariate asset returns for $N$ stocks, and i compute from it the covariance matrix and then the correlation matrix, can I always know which of the two will have the higher ...
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38 views

Covariance of Individual Return and Portfolio Return

Hi guys, Is it possible to get the covariance between the individual return and portfolio return given the correlation matrix, volatility matrix, weights matrix and return matrix? I know how to get ...
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2answers
220 views

Meaning of an identity matrix for the covariance in portfolio optimization

Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This ...
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2answers
558 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 ...
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3answers
356 views

Simulating covariance matrices with nonzero correlation

How would you simulate a covariance matrix of 1,000 stocks where each pair has nonzero correlation? I have literally no idea how to start with this. Any suggestions?
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1answer
117 views

Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$

I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$. Is it an assumption or a general fact based on theoretical concepts?
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1answer
113 views

Effective Time Length of Exponentially Weighted Covariance Matrix Estimate

In [1] Pafka, Potters and Kondor mention the following in section 2: In contrast, if this covariance matrix estimate is used for portfolio optimization (i.e. for selecting the portfolio in a ...
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1answer
713 views

Variance attribution calculation from a covariance matrix

Say I have a portfolio with two assets with weights $(x, y)$, and the covariance matrix of the two asset is $((a, r)(r, b))$. Then the total portfolio variance would be $x^2a+2xyr+y^2b$. It is easy to ...
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1answer
448 views

Computing covariance matrix with historical data

I have been reading Active Portfolio Management by Grinold and Khan. In the chapter about risk, they mention, "The third elementary model relies on historical variances and covariances. This ...
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1answer
207 views

is it possible to get minimum variance line having only covariance matrix?

Hey I have covariance matrix: $$C=\begin{pmatrix} 0,01 & 0.01 & 0\\ \\ 0.01 & 0,02 & -0.01 \\ \\ 0 & -0.01 & 0,03 \end{pmatrix}$$ So the variance of porfolio is: $$\...
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1answer
77 views

Calculating covariance from three variances

I have been asked to look to refactor some code. There is a line shown below: $\text{implied covariance} = -\frac{(\text{var}_1 - \text{var}_2 - \text{var}_3)} {2}$, where $\text{var}_1$ is the ...
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2answers
215 views

Estimate covariance matrix using prices

We generally estimate the covariance matrix of assets using their returns instead of prices. Why is that the case? I can think of two possible reasons and would appreciate comments/feedback regarding ...
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2answers
1k views

Creating a Covariance Matrix

Lets say that you have the correlation of x,y and you have the standard deviations of x and y , how would you then find the covariance of x,y using the correlation of x,y and and the standard ...
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1answer
132 views

Demonstration of the Schweinler-Wigner Orthogonalization procedure

Can anyone give me a practical demonstration of the Schweinler-Wigner Orthogonalization procedure? The steps of performing it or possibly a code snippet. The Schweinler-Wigner Orthogonalization ...
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6answers
392 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 ...
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2answers
693 views

Volatility and weights of a portfolio whose value is negative

How do you calculate the one day standard deviation (in dollars) for a portfolio that is short $30,000? How do you calculate the weightings to use? I already have the necessary covariance matrix.
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2answers
1k views

Why annualizing sampled covariance matrix changes stock weight vector?

Question While optimizing a portfolio using 'Global Minimum Variance' (GMV) method, I found that annualizing a sampled covariance matrix makes a difference in stock weight vector. Q1. Why ...
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1answer
94 views

Covariance matrix from GJR-GARCH?

I am implementing a AR(1)-GJR-GARCH(1,1) model to some asset returns, and I would need to have a covariance matrix but I struggle to see how I can compute one from the model I used? I know I can have ...
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2answers
1k views

Widely accepted methods for coming up with the co-variance matrix of assets?

Question What are the widely accepted ways for coming up with co-variance matrix of assets after the Markowitz's modern portfolio theory? Question explained in more detail After Modern portfolio ...
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0answers
2k views

Barra covariance matrix construction

I am trying to replicate the covariance matrix used by Barra risk models. All Barra models have half life parameters for volatilities and correlations (e.g. if the half life for volatlity is 90 days, ...
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1answer
471 views

Variance-Covariance VaR: how to get the volatility?

Because the variance-covariance VaR assumes that the returns are normally distributed, in theory it is easy to get VaR by simply finding the mean and the volatility (standard deviation) of the ...
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0answers
141 views

Black Litterman - numerical instability

I am trying to work out the formula for the posterior mean in Black Litterman's model assuming 100% confidence : Ref: https://corporate.morningstar.com/ib/documents/MethodologyDocuments/IBBAssociates/...
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1answer
740 views

beginner portfolio statistics - annualized volatility of multi-asset portfolio

Sorry for the dumb question, but I wanted to make sure my understanding of what I read and compiled was correct! I am trying to calculate the variance-covariance matrix, and annualized volatility of a ...