Questions tagged [covariance-matrix]
The covariance-matrix tag has no usage guidance.
114
questions
15
votes
1
answer
2k
views
Optimization: Factor model versus asset-by-asset model
In portfolio management one often has to solve problems of the quadratic form
$$
w^T \Sigma w + w^T c \rightarrow \min_{\omega}
$$
with portfolio weights $w \in \mathbb{R}^N$ a constant $c \in \mathbb{...
- 13.6k
15
votes
1
answer
969
views
Covariance estimation: shrinkage, random matrix theory, what else?
Shrinkage was much en-vogue before random matrix theory (RMT) took everybody's attention in covariance matrix estimation, however the latter also showed its limits. A plethora of other estimators has ...
- 1,543
11
votes
2
answers
4k
views
What is the total correlation between assets in a portfolio?
Suppose I have portfolio with 10 assets, each one of them with a weight of 10% from the total portfolio (equally weighted).
It's well known how to measure from historical prices->returns a variance-...
- 257
10
votes
2
answers
2k
views
Hierarchical Risk Parity with allocation constraints?
In the really interesting paper by Marcos Lopez de Prado a variation of risk parity is applied whereby the underlying assets of the portfolio are first split in 'correlation clusters' and the ...
- 441
9
votes
0
answers
4k
views
Explanation or implementation of Ledoit-Wolf estimator (without math packages)
I have calculated weights of selected assets in a market-neutral portfolio (presumably with min variance) using PCA and simple data covariance matrix.
The question is :
It is obvious that Cov Matrix ...
- 415
8
votes
3
answers
6k
views
Why does the Markowitz mean-variance model require the assumption of normality?
Given $N$ assets, the Markowitz mean-variance model requires expected returns, expected variances and a $N \times N$ covariance matrix. The joint distribution is fully defined by these measures.
...
- 233
7
votes
2
answers
2k
views
Practical implementation of Libor Market Model
I am trying to implement a project about the BGM model, suggested in the book "The Concepts and Practice of mathematical finance" by Mark Joshi.
My question is related to the forward volatility ...
- 463
7
votes
4
answers
860
views
How to treat large (5K-10K) non-positive-definite (particularly near-singular) covariance matrices for Cholesky decomposition?
I have a very large covariance matrix (around 10000x10000) of returns, which is constructed using a sample size of 1000 for 10000 variables. My goal is to perform a (good-looking) Cholesky ...
- 71
6
votes
1
answer
473
views
How can I use a more efficient volatility estimator to improve the co-variance matrix?
Using mean-variance, I need to estimate a co-variance matrix $\Sigma$ to obtain the best weights in my portfolio.
However, there are other ways to compute the volatility $\sigma$ than historical ...
- 217
6
votes
2
answers
565
views
PCA for stand alone equity VaR
I am trying to compute equity VaR, forex VaR and total VaR on an international portfolio (10 stocks x 4 countries). Since I am not interested in the risk disaggregation among diffrent countries I was ...
- 61
6
votes
0
answers
139
views
How can one quantify the incremental value of better covariance matrix modeling in portfolio optimization?
Let's say we have two estimators of the covariance matrix, $\hat{C}_1$ and $\hat{C}_2$, and the latter is an improvement on the former.
Is there any measure of the improvement that can be sensibly ...
- 1,150
6
votes
0
answers
129
views
Clustered vs. GMM-based standard errors: which ones to use in asset pricing?
Consider estimating an asset pricing model such as the CAPM or a multifactor model using monthly data. Petersen (2009) section "Asset pricing application" suggests use of standard errors ...
- 2,638
5
votes
2
answers
1k
views
Black-Litterman: Why should the views be independent of each other?
This question relates to this question.
In the Black-Litterman framework views of inverstors on the market are modelled.
These views have a covariance-matrix $\Omega$.
I always found it quite ...
- 13.6k
4
votes
6
answers
478
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 ...
- 169
4
votes
2
answers
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 ...
- 530
4
votes
1
answer
1k
views
Interpreting Eigenvalues of Co-variance Matrix
Im working on market reaction to events and I'm using the co-variance matrix to do this. In this paper the author writes
It has been known for some time that the largest eigenvalue (λ1) contains ...
- 143
4
votes
2
answers
6k
views
Does one use the covariance or correlation matrix in cholesky decomposition to generate correlated samples
Can we interchangeably use Cholesky decomposition of covariance and correlation matrix to generate simulations? If not, in which situations do we use one or the other and why?
Thanks in advance.
- 83
4
votes
1
answer
439
views
Portfolio with lots of subportfolios
An account manager has $N$ distinct, equally-sized pots of money, which will be used to make $N$ distinct subportfolios, each of which is drawn from a slightly different (but potentially overlapping) ...
- 43
4
votes
1
answer
1k
views
Portfolio Optimization - n risky assets
I'm currently implementing a CAPM model in Excel:
A portfolio of n risky assets when n=6 (in this case)
A riskless borrowing rate of 8% and riskless lending rate of 3%
I'm given the expected return ...
4
votes
0
answers
79
views
Evaluating estimate of covariance matrix
I am testing out different methods / shrinkages to estimate a covariance matrix and I am wondering what is the best method of comparing the estimated covariance matrix to the true covariance matrix (...
- 41
4
votes
0
answers
5k
views
Annualize a covariance matrix?
I am attempting to find the annualized covariance between assets in a portfolio but I only have daily data. So how do I annualize the covariance matrix between these assets?
4
votes
0
answers
1k
views
Variance of a portfolio based on log-returns
Modern Portfolio Theory Optimization Problem is based on expected linear returns and covariances of linear returns.
That's said, variance and expected return of a portfolio based on linear returns r ...
- 51
3
votes
2
answers
476
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 ...
- 392
3
votes
1
answer
405
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'\...
- 446
3
votes
2
answers
2k
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 ...
- 309
3
votes
3
answers
2k
views
Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix
I used ten year daily data for 407 stocks and computed the daily and monthly covariance matrices. Since I have more variables than observations for the monthly matrix, I wasn't surprised to find the ...
- 95
3
votes
2
answers
3k
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 ...
- 530
3
votes
1
answer
3k
views
How to compute the variance of a Long-Short Equity Portfolio?
I am calculating the historical portfolio variance of various long-short equity portfolios. For simplicity, assume the portfolio is long stock A with weight 1.0 and short stock B with weight -0.5. ...
- 143
3
votes
1
answer
111
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 $r_i$.
...
- 175
3
votes
2
answers
604
views
Filtering smallest eigenvalues
In Risk Budgeting and Diversification Based on Optimized Uncorrelated Factors [1], which introduces minimum torsion bets, Meucci gives an example involving the computation of covariance matrices on ...
- 1,736
3
votes
2
answers
972
views
How to get Multivariate Betas from an Estimated EWMA co variance Matrix?
I have a portfolio of 4 assets. I also have returns for 3 indices. I want to get the multivariate betas for these 4 assets-based on these assets. I only have the 7 x 7 covariance matrix estimated by a ...
- 31
3
votes
0
answers
322
views
Residual Covariance Matrix, and MVO for Residual Variance and Alpha
My overall goal is to find an efficient frontier using QP in terms of $\alpha$ and residual variance ($\omega^2$) for a portfolio $P$ given a benchmark $B$.
We know the equation for residual variance ...
- 541
2
votes
1
answer
359
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 ...
- 185
2
votes
3
answers
595
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?
- 2,472
2
votes
4
answers
3k
views
Semi-variance/Downside Risk, what about the rest of the covariance matrix?
I just bumped into a rather interesting article from wikipedia :
http://en.wikipedia.org/wiki/Downside_risk
where they define the semi-variance also called Downside risk, which bascially only ...
- 189
2
votes
1
answer
1k
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 ...
- 145
2
votes
2
answers
4k
views
Variance Matrix with 'nan' values
I am trying to optimize a simple portfolio using several random weights and choosing the best. When the number of assets is large I get a covariance matrix with 'nan' values because some asset pairs ...
- 23
2
votes
1
answer
181
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?
- 23
2
votes
1
answer
84
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 ...
- 259
2
votes
1
answer
133
views
Transform raw forecasts into orthogonal forecasts
I am trying to combine multiple forecasts on each of N assets in line with Grinold and Kahn's methodology, taken from Active Portfolio Management, 2nd ed. On p.311, they suggest transforming the raw ...
2
votes
1
answer
224
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 ...
- 21
2
votes
3
answers
314
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 ...
- 2,980
2
votes
2
answers
1k
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.
- 31
2
votes
1
answer
1k
views
"Adding" risk-free asset to covariance matrix after the fact
Given a covariance matrix that was calculated from the returns of a number of risky assets.
Is there a way to "add" a risk-free asset to the covariance matrix without calculating its covariance with ...
- 65
2
votes
1
answer
110
views
Calculating Portfolios Covariance via Bilinearity with Log or Simple Returns
I'm wanting to calculate the covariance between two portfolios $A$ and $B$ which are allocated to assets $X_i$ (where $i \in \left[1, 2, \cdots, N \right]$) with weights $\vec{w_A}$ and $\vec{w_B}$, ...
- 118
2
votes
1
answer
138
views
Bayes Stein Porfolio Implementation
From this paper from Jorion.
Has anyone implemented this? How is the Covariance matrix estimated? It needs to estimate also the conditional distribution of the returns?
Best
- 165
2
votes
1
answer
2k
views
short-sale constraint with nonpositive-definite matrix in portfolio optimization
I need help about portfolio optimization in R. I have inverted matrix and I want to use it as an input in portfolio optimization. It was non-positive definite before I have handled it. In portfolio ...
- 31
2
votes
0
answers
39
views
What does a non-stochastic limiting shrinkage function mean?
I'm reading the paper "The Power of (Non-)Linear Shrinking: A Review and Guide to Covariance Matrix Estimation" by Ledoit and Wolf (2020). When a function that is used to transform the ...
2
votes
0
answers
64
views
Covariance Matrix: Calculating Error [duplicate]
I have a sample covariance matrix that is non positive-semi definite (due to missing data points). I am looking at a number of techniques to 'fix' my covariance matrix and make it positive semi-...
- 81
2
votes
0
answers
74
views
Reduced rank / matrix factorisation techniques and their uses in portfolio optimisation?
I am interested in reduced rank / matrix factorisation techniques and their uses across finance and portfolio optimisation. For example, PCA might be used to reduce the number of components you are ...
- 81