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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Distribution of sample covariance times inverse covariance times sample covariance
I want to understand the distribution of the random variable:
$$S_n = \frac{1}{n^2} 1'\hat \Sigma \Sigma ^{-1} \hat \Sigma 1$$.
1 is a vector of ones of size n, and the variance is of size nxn. $\hat \...
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Scaling returns to use PCA?
Many machine learning techniques perform better, if the data is preprocessed - either by normalization (MaxMin Scaler) or standardization (Standard Scaler). But that comes with a lack of ...
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Standard deviation of large equal-weighted portfolios
Say I've got a portfolio of shares with the following parameters: Let $n$ be the number of shares in the portfolio, let $\bar\sigma$ be the average standard deviation (volatility/risk) for each share, ...
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Discuss how you would allocate your budget between the two assets if their correlation is 1, 0, or -1
An asset A is expected to yield a $2\%$ return with a standard deviation of $1\%$, and another asset B is expected to yield a $1\%$ return with a standard deviation of $1\%$.
Discuss how you would ...
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covariance between squared returns and past returns
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&...
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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 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}> \\
<\...
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"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}\...
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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 ...
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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?...
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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 ...
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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)$$
where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $
My idea was to ...
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Find k of n assets that "minimize" the correlation matrix
I'm trying to find an efficient way to select $k$ from $n$ risky assets that are the least correlated with each other. I know that I can perform a brute-force search of all $k$-sized combinations of ...
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Association between a random variable and Radon-Nikodym derivative
Suppose that $X$ is a random variable and $\frac{d\mathbb{Q}}{d\mathbb{P}}$ is the Radon-Nikodym derivative. The quantity under consideration is as follows:
\begin{equation}
Cov(X, \frac{d\mathbb{Q}}{...
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Show that the following result holds true for the variance of the return of a portfolio of shares
Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by:
$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
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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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