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Questions tagged [covariance-matrix]

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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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80 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 ...
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29 views

VECM Correlation Matrix

I am trying to implement the modified information share price discovery method desribed by Lien. They start by estimating a VECM of the form where and and it's Vector Moving Average ...
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32 views

How to compute the portfolio risk when weights are negative?

In QMiF (p. 239) , the variance of a portfolio is defined as: V(R) = w'Vw = w'DCDw = x'Cx Does this formula hold if the weights are negative (i.e., short)? For example, if I have a 5x5 covariance ...
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Why is my Covariance matrix not positive definite?

I'm trying to do PCA on historic forward rates. I'm using forward rates from the Bank of England going from Jan 2015 through end of May 2018. I calculate the differences in the rates from one day to ...
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216 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 ...
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1answer
49 views

Variance covariance matrix - number of periods required

Hi I am reviewing the example of Barra risk model in the following document page 23 there is the statement: "Estimating a covariance matrix for, say, 3,000 stocks requires data for at least 3,...
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1answer
128 views

Which program for a DCC-MIDAS model?

for a thesis research, I plan to use a DCC-MIDAS model. The program I was working with (STATA) is not able to run this. Do you have any suggestions as to which program is best for this analysis? ...
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141 views

How to calculate semi-covariance matrix?

I am struggling find a reasonable approach to calculate semi-covariance matrix, focused on downside risk. I've went through several papers, Sortino's book and Estrada's research as well. However, I ...
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559 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 ...
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1answer
146 views

Solving a system of two equations with non-convex matrix multiplication for MV optimization

Scenario: I am trying to do a variation of the MV optimization for a portfolio. In this instance, I already have a vector of mean returns ($\mu$), a vector of ones, a covariance matrix ($\Sigma$), and ...
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71 views

Correcting Inverse Covariance matrix for high correlation of returns

I am currently working on a portfolio maximization problem where I am assessing the economic value of a model that predicts bond excess returns (from 1 year maturity to 15 years) vs a naive model that ...
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1answer
74 views

Covariance Interest Rate Risk Time Series

Apologies in advance if this question has been asked already. I am estimating basis risk for different term points in the curve. Imagine i have three time series (1-month, 3-month, 1-year). I ...
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1answer
292 views

principal component analysis on non stationary data

I read that since stock prices are non-stationary it does not make sense to take their covariance. So I took the log returns of stocks, computed covariance matrix, took the top few eigen vectors that ...
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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-...
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1answer
190 views

Portfolio Optimisation/Covariance Estimation on a large scale

When using Markowitz Portfolio Theory, e.g. for finding an optimal portfolio composition, one needs to have estimates of the returns, but most importantly of the covariance matrix. If our universe of ...
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57 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 ...
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2answers
468 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 ...
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1answer
282 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 ...
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1answer
723 views

What is the difference between the Single Index Model and Multi-Index Models in computing the variance-covariance matrix of stock returns?

Would be very grateful for some help in comparing the single index model with other multi-index models in computing the variance-covariance matrix.
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2k 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. ...
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1k 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-...
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2answers
59 views

Explanations regarding Minimum Variance Portfolio

I am sorry in advance if this question seems a bit stupid but during my class my lecturer said that: "The traditional estimator of the variance-covariance matrix is the sample covariance. However ...
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1answer
149 views

Prove that a determinant in markowitz method derivation is greater than zero

I want to prove that the following determinant, that appears in the markowitz method of portfolio allocation is greater than zero. ($\mu$ is the vector of returns and $\sum$ is the covariance matrix)
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1answer
502 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. ...
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1answer
56 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 ...
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3answers
286 views

Generating a random covariance matrix with variances in range

I would like to generate a random covariance matrix with variances in certain range. How can it be done? (In R if possible) I tried to generate a lower triangular matrix $L$ where the diagonal $D = ...
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2k 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?
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313 views

How do I apply the decay factor in VaR using pandas

I have a variance/co-variance matrix calculated for 4 assets: ...
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1answer
103 views

How to adjust corporate actions for VaR

I am using variance co variance matrix for calculating the VaR. Now if the some corporate action comes in between like stock split, resulting a huge VaR number on that particular day as the volatility ...
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1answer
270 views

Average Correlation

We're given a spreadsheet with a correlation matrix for four stocks. Then there is a calculation for average correlation, but I don't know how it's derived. $$=\left(\operatorname{Average}(C14:F17)-\...
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1answer
217 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 ...
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2answers
390 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 ...
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457 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 ...
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1answer
379 views

Historical Scenario analysis for stress testing

I am doing historical scenario analysis in order to calculate stressed VAR for which I have taken 2007-2008 US crisis. I have two question in this regard:- 1) As we have to take prices for stocks ...
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1answer
637 views

Variance covariance matrix for a portfolio containing bonds also with other asset classes

What should we take for a bond or a zero coupon bond in order to make a variance covariance matrix? For example:- Equities - we take the market price Cash - we take the spot rates Bonds - Do we take ...
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1answer
236 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) ...
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2answers
726 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 ...
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632 views

Correlation Between 2 Portfolios

I have a set of assets, n. I'm trying to find the correlation between 2 portfolios, say x and y, where x is nested in, or, a sub-set of y. That is, x is a portfolio based on a sub-set of n, while y is ...
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190 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 ...
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3answers
1k 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 ...
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1answer
97 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
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2answers
624 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 ...
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2answers
454 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 ...
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2answers
547 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 ...
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1answer
1k 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 ...
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1answer
704 views

PCA on term structure of interest rates

Interest rate time series seems to be non-stationary whenever test is performed But covariance or correlation matrix is derived from term structure time series which are non stationary and later PCA ...
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1answer
665 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 ...
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1answer
874 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 ...
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4k 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.