Questions tagged [covariance]

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

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Portfolio optimization - Correlation risk stress testing - DCP

I have a script based on Python/CVXPY trying to define the portfolio with the maximum expected return, given some risk constraints. I would like to introduce a constraint that limits correlation risk. ...
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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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Shrinkage of the Sample Covariance matrix, theory

is there any theory behind the covariance matrix shrinkage paper, why it works? I am talking about this stats exchange thread
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Covariance of the product of log normal process and normal procces

I tried to compute the following covariance : $$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$ where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $ My idea was to ...
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Using the MMULT TRANSPOSE Formula for Ex-Ante Tracking Error

I understand that the most widely used ex-ante tracking error expression is $\sqrt{w^TSw}$. This is very similar to the calculation of Portfolio Variance and/or Standard Deviation. The conceptual ...
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Covariance of ARCH(2) model

I am having problems solving the following exercise: The solution is the following: I understand we are calculating E(r^2t) and E(r^2tr^2t-1) because they are part of the covariance formula, and ...
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Find k of n assets that "minimize" the correlation matrix

I'm trying to find an efficient way to select $k$ from $n$ risky assets that are the least correlated with each other. I know that I can perform a brute-force search of all $k$-sized combinations of ...
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Association between a random variable and Radon-Nikodym derivative

Suppose that $X$ is a random variable and $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the Radon-Nikodym derivative. The quantity under consideration is as follows: \begin{equation} Cov(X, \frac{d\mathbb{Q}}{...
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Show that the following result holds true for the variance of the return of a portfolio of shares

Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by: $$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
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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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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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Covariance between integral of brownian motion and brownian motion

Let $$ I = \int_0^1W_tdt, $$ where $W_t$ is a Brownian motion. From Integral of Brownian motion w.r.t. time we have that $$ \mathbb{E}[I]=0, $$ by Fubini's theorem. And that $$ \mathbb{V}\text{ar}[I] =...
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Show that $\text{Cov}[X_r,X_s]=\text{Cov}[X_{r+h},X_{s+h}]$ for $X_t=a+bZ_t+cZ_{t-2}.$

Problem: Let $\{Zt\}$ be a sequence of independent normal random variables, each with mean $0$ and variance $\sigma^2$, and let $a$, $b$, and $c$ be constants. Is $X_t=a+bZ_t+cZ_{t-2}$ a (weakly) ...
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Help understanding the step $\sum_{j=0}^n\sum_{k=0}^ng_jg_k\text{Cov}(\epsilon_{n-1},\epsilon_{n+h-k})=\sum_{j=0}^ng_j^2+h\sigma^2$

Given is that $\epsilon_n$ is a white noise process with $\text{Var}(\epsilon_n)=\sigma^2$ and that $g_j\in\mathbb{R}$. There is a step in my lecture notes that I don't get. It says the following $$\...
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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 \begin{equation} Z_t=\sum_{i=0}^na_iX_{t-i} \end{equation} 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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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{\...
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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}\...
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Covariance of mean-reverting Vasicek process?

I am dealing with a mean-reverting Vasicek process defined as: \begin{equation} S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t \end{equation} I want to ...
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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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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Decomposition of Contribution to Variance

$C$ 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 ...
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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-...
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3 votes
1 answer
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Turning a covariance sum into an integral

I am reading Lorenzo's Bergomi's book Stochastic Volatility Modeling, and I have come to this passage. I just would like to understand the derivation between the first and the second equality. I ...
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$n$-day ahead forecast for asymmetric DCC-GARCH model

I am working on forecasting covariances with the use of MGARCH models. I was wondering if anyone knows how to implement a n-day ahead forecast of the aDCC (asymmetric DCC) model in R. The ...
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covariance matrix in the CAPM model

I'm running a simulation for a 5 asset portfolio, calculating the optimal weights of each asset both with the statistical model (i.e. single index) and with the CAPM. my question is: how do you ...
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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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Decomposing Co-variance of Two Assets Terminal Prices into Forward measures

Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} ...
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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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Is there a differentiable formulation of Prado's HRP algo?

I get the sense the algo is just a graph representation the dependency structure. Am wondering if there is anything written on learning the weights by optimizing some parameters rather than a forward ...
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2 answers
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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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6 answers
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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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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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One day standard deviation of a portfolio (long/short, different scalars)

I am attempting to calculate the expected one-day standard deviation of a portfolio in dollars. In other words, I am looking for the following: "I expect my portfolio to move _______ dollars on ...
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Questions about beta, correlation, and covariance

Currently, I calculate beta, correlation, and covariance measures using daily log normal returns of Security A and Benchmark A. What would it mean if I were to use daily log normal excess returns in ...
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