Deriving Variance and Covariance from Factor Loadings
Summary
The document explains how factor loadings determine the variances and covariance of two modeled risk factors. Each factor is expressed as a mean plus a weighted sum of underlying random variables. When those underlying variables are independent with unit variance, a factor’s variance is the sum of the squared loadings, and the covariance between factors is the sum of the products of their corresponding loadings. Correlation then follows by dividing covariance by the product of the factors’ standard deviations.
The answer frames these calculations through the general covariance-matrix rule for linear combinations and applies it to a two-factor example. It also points to Cholesky decomposition as related background. The unit-variance and independence assumptions are essential to the stated equations; if the underlying variables have a different covariance matrix, the general matrix formula must be used instead.
Key ideas
- A linear combination’s variance is obtained from its weights and the covariance matrix of its inputs.
- With independent unit-variance inputs, the variance equals the sum of squared loadings.
- The covariance between two combinations equals the sum of products of their aligned loadings.
- Correlation is covariance divided by the product of the standard deviations.
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# Question about equations and risk factors.
# Question about equations and risk factors.
Say I have two risk factors $X_1$ and $X_2$. Standard deviation for $X_1$ is $\sigma_1$ and $\sigma_2$ for $X_2$. Furthermore, $X_1$ has a mean of $\mu_1$ and $X_2$ has a mean of $\mu_2$. Correlation between $X_1$ and $X_2$ is $\rho$.
The system is as follows:
$$\begin{eqnarray} X_1 & = & \mu_1 + \lambda_{11} U_1 \\ X_2 & = & \mu_2 + \lambda_{21} U_1 + \lambda_{22} U_2 \end{eqnarray}$$
My book reads as follows:
"Accordingly: $\lambda_{11}=\sigma_1$ (1) $\lambda_{21}^2 + \lambda_{22}^2= \sigma_2^2$ (2) $\lambda_{21} \lambda_{11}= \rho \sigma_1 \sigma_2$ (3)"
I don't understand how they work out lines (2) and (3).
Can anyone please help?
## Answer by shabbychef (score 4, accepted)
https://quant.stackexchange.com/a/1553
If $\Sigma$ is the variance/covariance matrix of random variables $U_1, U_2, \ldots U_n$, and $V = c + w_1 U_1 + \ldots + w_n U_n$, where $c$ is a constant, and we let $\mathbf{w}$ be the vector with the 'weights' $w_1, w_2, \ldots, w_n$, then the variance of $V$ is equal to $\mathbf{w}^{\top}\Sigma\mathbf{w}$. Moreover, if $T$ is another random variable described by weights vector $\mathbf{x}$, then the variance of $T$ is $\mathbf{x}^{\top}\Sigma\mathbf{x}$, and the covariance of $V$ and $T$ is equal to $\mathbf{x}^{\top}\Sigma\mathbf{w}$.
In your problem, $$ \Sigma = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$ and you are looking at weights vectors $[\lambda_{11}\,0]$ and $[\lambda_{21}\,\lambda_{22}]$, thus the variance of $X_1$ is $\lambda_{11}^2$, which is the first equation, and the variance of $X_2$ is $\lambda_{21}^2 + \lambda_{22}^2$, which is the second equation. Computing the covariance of $X_1$ and $X_2$ gives the third equation.
## Answer by SpeedBoots (score 3)
https://quant.stackexchange.com/a/1258
You should offer more details [I assume U1 and U2 are N(0,1)] but I think you should read this: http://en.wikipedia.org/wiki/Cholesky_decompositionShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.