# Questions tagged [covariance]

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

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### Reliability of R Package on Covariance Matrix Shrinkage

I recently used a R package CovTools in R with the command CovEst.2003LW(X), where X is your sample covariance matrix as an input, to compute the shrunk covariance matrix (an estimate that is closest ...
Let $y_t = \sqrt{h_t} \epsilon_t$ where $\epsilon_t\overset{ iid}{\sim} N(0,1)$ $h_t = \alpha_0 +\alpha_1 y_{t-1}^2+\beta_1 h_{t-1}$ with $\alpha_0>0, \alpha_1>0, \beta_1<1,\alpha_1+\beta_1&... 0 votes 1 answer 61 views ### Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B? Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP. Is there any relationship between the covariance(EUR/USD, ... 6 votes 0 answers 138 views ### How can one quantify the incremental value of better covariance matrix modeling in portfolio optimization? Let's say we have two estimators of the covariance matrix,$\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 ... 0 votes 2 answers 148 views ### 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 ... 0 votes 1 answer 99 views ### Covariance Matrix of Correlated Random Variable Suppose I know or have estimated the covariance matrix for one random variable (for example an asset) and have: $$\begin{bmatrix} <\text{spot, spot}> & <\text{atmv, spot}> \\ <\... 0 votes 0 answers 79 views ### "fix" a sample covariance matrix which is not positive semidefinite by using daily returns instead of monthly In the portfolio optimization problem at hand, one of the constraints is that the tracking error should not be greater than \gamma. The constraint is therefore: (\textbf{x}-\textbf{w})^\mathrm{T}\... 0 votes 0 answers 106 views ### Estimating covariance with intraday data I have intraday (30 min) data for a number of stocks, and I would like to calculate the covariance matrix of returns. For the purpose of calculating the covariance matrix, is it better/more correct to ... 3 votes 1 answer 275 views ### Covariance Between Two Frontier Portfolios Based on the definitions of A, B, C, and D in "An Analytic Derivation Of The Efficient Portfolio Frontier" by Robert Merton (1972), how can I prove the following in a line-by-line derivation?... -1 votes 1 answer 694 views ### Daily vs Monthly vs. other return for volatility calculation? I thought I read/heard somewhere that annualized volatility, using monthly returns vs daily returns is usually lower. With that said, I can't seem to find any papers on this. Does anyone have any ... 3 votes 1 answer 111 views ### Sample Variance of Portfolio Let w denote a vector of portfolio weights, r_i denote the ith return vector, \Sigma denote the Covariance matrix of r_i and let \hat{\Sigma} denote the sample covariance matrix of r_i. ... 1 vote 1 answer 447 views ### Shrinkage of the Sample Covariance matrix, theory is there any theory behind the covariance matrix shrinkage paper, why it works? I am talking about this stats exchange thread 6 votes 2 answers 158 views ### Covariance of the product of log normal process and normal procces I tried to compute the following covariance :$$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$where W^1_t and W^2_t are Brownian motions such that dW_t^1dW_t^2=\rho dt My idea was to ... 3 votes 2 answers 580 views ### Find k of n assets that "minimize" the correlation matrix I'm trying to find an efficient way to select k from n risky assets that are the least correlated with each other. I know that I can perform a brute-force search of all k-sized combinations of ... 1 vote 0 answers 143 views ### Association between a random variable and Radon-Nikodym derivative Suppose that X is a random variable and \frac{d\mathbb{Q}}{d\mathbb{P}} is the Radon-Nikodym derivative. The quantity under consideration is as follows: \begin{equation} Cov(X, \frac{d\mathbb{Q}}{... 0 votes 1 answer 61 views ### Show that the following result holds true for the variance of the return of a portfolio of shares Start with a portfolio p of n shares, each with weight x_i = \dfrac{1}{n} (for i ranging from 1 to n, discretely). Its return is given by:$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\... 0 votes 1 answer 834 views ### 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 ... 4 votes 1 answer 2k views ### 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 $$cov(tB_{3t}-B_{2t}+5, B_s-1).$$ ... 3 votes 2 answers 799 views ### 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 $$\mathbb{V}\text{ar}[I] =... 0 votes 1 answer 70 views ### Show that \text{Cov}[X_r,X_s]=\text{Cov}[X_{r+h},X_{s+h}] for X_t=a+bZ_t+cZ_{t-2}. Problem: Let \{Zt\} be a sequence of independent normal random variables, each with mean 0 and variance \sigma^2, and let a, b, and c be constants. Is X_t=a+bZ_t+cZ_{t-2} a (weakly) ... 2 votes 1 answer 86 views ### Help understanding the step \sum_{j=0}^n\sum_{k=0}^ng_jg_k\text{Cov}(\epsilon_{n-1},\epsilon_{n+h-k})=\sum_{j=0}^ng_j^2+h\sigma^2 Given is that \epsilon_n is a white noise process with \text{Var}(\epsilon_n)=\sigma^2 and that g_j\in\mathbb{R}. There is a step in my lecture notes that I don't get. It says the following$$\... 1 vote 1 answer 106 views ### 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 ... 1 vote 1 answer 82 views ### 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 ... 1 vote 0 answers 50 views ### 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 ... 2 votes 0 answers 73 views ### 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 ... 0 votes 0 answers 459 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 ... 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 ... 0 votes 0 answers 22 views ### 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’... 1 vote 1 answer 120 views ### 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 ... 1 vote 1 answer 117 views ### 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 ... 0 votes 1 answer 124 views ### 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{\... 0 votes 1 answer 141 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}\... 1 vote 1 answer 274 views ### 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 ... 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 ... 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 ... 0 votes 0 answers 90 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 ... 1 vote 2 answers 369 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 ... 0 votes 2 answers 643 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 108 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 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 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? 1 vote 1 answer 100 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 \... 0 votes 0 answers 599 views ### 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 ... 1 vote 1 answer 796 views ### 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-... 3 votes 1 answer 144 views ### 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 ... 1 vote 2 answers 238 views ###$n$-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 ... 1 vote 0 answers 584 views ### 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 ... 0 votes 1 answer 855 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 ... 0 votes 1 answer 116 views ### 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} ...
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 ...