# Questions tagged [covariance]

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

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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 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Cleansing covariance matrices via Random matrix theory

I am exploring de-noising and cleansing of covariance matrices via Random Matrix Theory. RMT is a competitor to shrinkage methods of covariance estimation. There are various methods expressed usually ...
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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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### Why the weight vector of 'global minimum variance' the 'eigenvector' with the minimum eigenvalue?

Question Why is it the case that the weight vector of the global minimum variance portfolio the eigenvector of the covariance matrix with the smallest eigenvalue? Question with more details I ...
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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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### How to efficiently get covariance matrices from a rolling window in Matlab?

I'am trying to produce a rolling window to estimate a covariance matrix using a for-loop. I have my returns under the variable returns_sec and I have 260 ...
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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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### 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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### target market correlation for long / short equity portfolio

Given a long / short equity portfolio, I want to have some net long market exposure. My portfolio volatility is fixed to a target, so I don't think it makes sense to target market beta. I think I ...
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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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### 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 ...