# Questions tagged [covariance]

A measure of the degree of linear association between a pair of random variables.

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### In practice, how many days are used to estimate the covariance matrix of factor returns?

Let's say we have a factor model with $N$ factors. I understand that the unbiased estimator of the covariance matrix $\Sigma_f$ is: $$\Sigma_f = \frac{1}{n-1} X^T X$$ where $X$ is a matrix of daily ...
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### How to prove that the feasible set of a two-asset portfolio is a hyperbola?

The question comes from ‘Mathematics for Finance: An Introduction to Financial Engineering’ by Marek Capiński (Author), Tomasz Zastawniak. The book does not give a complete proof, and I did not find a ...
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### GARCH for Mean Variance Optimization

I am currently trying to carry out a mean variance optimisation, with the implementation of GARCH. I'm not sure if this is going to make complete sense as my understanding of GARCH is limited. In the ...
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### How to prove the inequality for the standard deviation of a linear combination of two random variables

The variance of the linear combination V of random variables X₁ and X₂ is given by the following formula: $$\sigma_{V}^{2} = s^{2} \sigma_{1}^{2}+(1-s)^2 \sigma_{2}^{2}+2 s(1-s) c_{12}$$ where s and ...
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### Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?

Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP. Is there any relationship between the covariance(EUR/USD, ...
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### 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 ...
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### Why do we need the covariance when calculating portfolio VaR?

I was recently learning about value at risk and how to calculate it, and one of the steps was to calculate the covariance of the returns of the securities making up the portofolio. This makes sense ...
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### Covariance Shrinkage - Am I getting the right variances?

I am looking into a quite simple task: shrinking the sample covariance matrix of a minor sample of monthly returns data on 5 different assets. I am using Python to process my data and have been using ...
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### Covariance of two Brownian Motions

During revision, I came across the following question in a past paper: Suppose $(B_t, t\geq0)$ is a standard Brownian motion. Compute for $0<s<t$ the covariance $$cov(tB_{3t}-B_{2t}+5, B_s-1).$$ ...
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### Show that $\text{Cov}[Z_t,Z_{t+h}]=\text{Cov}[Z_s,Z_{s+h}].$

Problem: If $X\sim\text{WN}(\mu,\sigma^2).$ Let then $Z$ be the process defined by $$Z_t=\sum_{i=0}^na_iX_{t-i}$$ for some coefficients $a_1,...,a_n\in\mathbb{R}$ with ...
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### Regression of stochastic integral on Wiener process

This question is a follow-up from the following: conditional expectation of stochastic integral so I won't repeat myself regarding assumptions and notation. Using Brownian bridge approach, we know ...
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### Disjoint covariance matrix estimation

I have always estimated correlations and variances disjointly and later combine them to construct covariance matrices. Specifically, variances are estimated in a univariate setting (only using the ...
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### How to reduce a covariance matrix after clustering?

I have an N = 100 covariance matrix. I am clustering the covariance matrix say into 5 clusters. How can I compute the reduced ...
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### 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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### 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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### Interpreting factor coefficients when correlation flips

I am looking at mainly value and growth factor coefficients of a fund during the recent Covid market “crisis”. I have found that said fund had a negative coefficient to value at the start of 2020 (let’...
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### Help Setting a Monte Carlo Simulation

I am trying to replicate the steps of the Barras, Scaillet, Wermer(2010) paper for a Monte-Carlo Simulation. More specifically the steps in Appendix B.1 (Attached image). I have so far done the ...
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### Covariance AR(2) Process [closed]

I am not sure what the formula is for the covariance of an AR(2) process, described by $X_t - \mu = \phi_1(X_{t-1} - \mu) + \phi_2(X_{t-2} -\mu ) + \epsilon_t$ where $\mu$ denoted the process mean ...
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If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is $$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\... • 3,080 0 votes 1 answer 208 views ### Update sample covariance matrix I would like to update a covariance matrix \mathbf{R}_T with a new incoming sample at time T+1, i.e. I would like a rank-1 update of the form \frac{1}{T+1} [T \mathbf{R}_T + \mathbf{x}_{T+1}\... • 101 1 vote 1 answer 320 views ### Covariance of mean-reverting Vasicek process? I am dealing with a mean-reverting Vasicek process defined as: $$S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t$$ I want to ... 2 votes 3 answers 423 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 ... • 3,080 0 votes 2 answers 2k 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 ... • 3,080 0 votes 0 answers 118 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 ... • 255 1 vote 2 answers 443 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 ... • 3,080 0 votes 2 answers 1k views ### What do large weights above 1 in a portfolio represent? If I have a portfolio consisting of weights -12,11,3,-2,5,-5, I know that negative weights correspond to shorting but what do these large weights represent? I thought the weights are the proportion of ... 0 votes 0 answers 119 views ### Can I build an efficient frontier using matrix algebra? If i have a vector of expected returns A, a covariance matrix C and a vector of the corresponding weights W for each investment, is it possible to generate the efficient frontier with vector ... 2 votes 3 answers 678 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? • 2,562 2 votes 1 answer 210 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 1 vote 1 answer 105 views ### Covariation of Ito semimartingales If we have two Ito semimartingales over [0,T]:$$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$What is the relationship between$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
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$C$ is a $N\times N$ covariance matrix of stock returns. Assuming $w$ is a vector of positions in each asset, the total variance of the portfolio is $$w^TCw$$ The contribution to total variance of the ...