Questions tagged [covariance]

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

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Ranking assets by covariance vs correlation

After every change in portfolio (i.e. every trade), I need to calculate a price for each asset in my portfolio. But the calculator is slow. So I want to order the sequence of price updates, from the ...
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138 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 ...
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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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114 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 ...
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65 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 ...
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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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171 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?
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83 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?
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50 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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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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63 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-...
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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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51 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 ...
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102 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 ...
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38 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} ...
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Trading Bands for Tactical Asset Allocation

I am currently working through Portfolio Construction and Risk Budgeting, 4th edition, by Dr. Bernd Scherer. In chapter 12.6 the following methodology to construct trading bands around a tactical ...
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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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150 views

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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Mean-Variance Optimization - Appropriate to use covariance of daily returns if strategy involves fixed hold of 3 months?

I'm implementing the strategy outlined Quantitative Momentum (Gray & Vogel), which basically involves investing each quarter in a set of "high-quality momentum stocks" for a fixed 3-month hold ...
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260 views

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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146 views

Covariance - Negative Portfolio

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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Covariance time frequency

I have rolling 3-year returns for an asset and a benchmark. I want to compare the covariance of the asset and benchmark, should I use the covariance of the rolling 3-year returns or the covariance ...
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71 views

Covariance matrix from GJR-GARCH?

I am implementing a AR(1)-GJR-GARCH(1,1) model to some asset returns, and I would need to have a covariance matrix but I struggle to see how I can compute one from the model I used? I know I can have ...
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191 views

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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492 views

Widely accepted methods for coming up with the co-variance matrix of assets?

Question What are the widely accepted ways for coming up with co-variance matrix of assets after the Markowitz's modern portfolio theory? Question explained in more detail After Modern portfolio ...
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58 views

Generate scenarios of multiple related parameters

Assume I have three industry datasets: interest rates, inflation and unemployment. Data contains information of last ten years and it's monthly. Now, I would like to create N possible scenarios of ...
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Why are thousand-ish-factor vendor risk models not extremely overfit and inaccurate?

Many vendor risk models have many hundreds, or even thousands of factors (many of which are highly correlated with each other). Underlying all these risk models is some sort of covariance matrix in ...
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159 views

Prove ρ(X,Z) = ρ

The covariance of two random variables $X$ and $Y$ is defined by: $$\mathrm{Cov}(X,Y)= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
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Co-variance of Portfolio A with Portfolio B

I'm trying to calculate the correlation between two separate portfolios. I've used A*COV(AB)*B to calculate the co-variance of each portfolio where: A = Array ...
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Is Ledoit-Wolf Shrinkage with a Constant Correlation Prior Reasonable for a Stock/Bond Mix?

I've been looking into Ledoit-Wolf shrinkage but I've found the papers concentrate on large numbers of assets that tend to all be highly correlated. Often a universe of large cap stocks. I'm ...
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Covariance and Beta: can anyone explain this calculation?

Let us consider a simple equity portfolio that has exposures to only two factors: 0.5 exposure to value and 0.8 exposure to momentum. Let us assume that the volatilities of the two factors are 3% for ...
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406 views

How to calculate the covariance between two stochastic integrals?

How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\...
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267 views

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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100 views

What are the units of the variance of returns?

I am a little confused about the units of the variance of returns. One way to compute that would be to look at the units of returns- $$r=\frac{1}{\Delta t}\ln\frac{P(t+\Delta t)}{P(t)}=\text{...
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427 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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1k views

Negative variance?

Using the formula w*Cov*t(w) I can generate a negative portfolio variance. What are the implications of a negative variance? Should I just assume it's zero? A negative variance is troublesome ...
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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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CAPM model as a regression

The CAPM model states that the returns of a stock are- $r_s=r_f+\beta (r_m-r_f)+\varepsilon_s$ The $\beta$ defined above is then calculated as $\frac{cov(r_s,r_m)}{var(r_m)}$. My question is ...
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Finding a minimum variance portfolio when using a regulariser?

I am aware that the minimum variance portfolio of a market with $n$ securities can be shown to be: \begin{equation} w^* = (1^T_n\Sigma^{-1}1_n)^{-1}\Sigma^{-1}1_n, \\ s.t. \ \ 1^T_nw = 1 \end{...
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Factor Models: uncorrelated errors don't impact covariances of assets

This question stems from time series factor models (e.g., CAPM, Fama-French, etc.), but is a broader idea. I am trying to comprehend how adding noise to a time series (e.g., error/residual from a ...
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208 views

Parametric VaR of a portfolio of a stock and an option on that stock

I understand how to calculate the parametric VaR of a stock and an option separately. But I don't understand how one can calculate the VaR of a portfolio of a stock and an option on that stock using ...
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130 views

Interest rate risk using copulas

In order to simulate an interest rate yield curve, can I just estimate a covariance matrix of historical key rate data, simulate with a normal copula, spline my simulated key rates, then price my ...
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452 views

Correlation of asset X with a portfolio of asset Y and Z [closed]

I have three assets and a covariance matrix. How do I calculate the correlation of asset X with a portfolio that includes assets Y and Z? For example, assume I want to calculate the correlation of ...
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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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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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Deriving Cox, Ingersoll and Ross expression for the relationship between forwards and futures, how do they conclude a specific step?

I'm trying to derive a specific relationship about the relationship between forwards and futures from "The relationship between forward and futures prices", written 1981 by Cox, Ingersoll and Ross (...