I am trying to generate monthly stock data using a one-factor model:
$$R_{a,t} = \alpha + B*R_{b,t}+\epsilon_{t}$$
The description says:
$R_{a,t}$ is the excess asset returns vector, $\alpha$ is the mispricing coefficients vector, $B$ is the factor loadings matrix, $R_{b,t}$ is the vector of excess returns on the factor portfolios, $R_{b}-N(\mu_{b},\sigma_{b})$, and $\epsilon_{t}$ is the vector of noise, $\epsilon - N(0,\sum_{e})$, which is independent with respect to the factor portfolios.
For our simulations, we assume that the risk-free rate follows a normal distribution, with an annual average of 2% and a standard deviation of 2%. We assume that there is only one factor (K=1), whose annual excess return has an annual average of 8% and a standard deviation of 16%. The mispricing $\alpha$ is set to zero and the factor loadings, B, are evenly spread between 0.5 and 1.5. Finally, the variance-covariance matrix of noise, $\sum_{\epsilon}$, is assumed to be diagonal, with elements drawn from a uniform distribution with support [0.10,0.30], so that the cross-sectional average annual idiosyncratic volatility is 20%.
Using the information provided here I try to generate the data:
alpha <- 0 #mispricing index is set to 0
B <- matrix(runif(1000,min=0.5,max=1),100,10) #factor loadings matrix is evenly spread between 0.5 and 1.5
R <- rnorm(100,mean=8/12,sd=16/sqrt(12)) #factor with annual excess return of 8% and standard deviation of 16%
epsilon <- rnorm(100, mean=0,sd=runif(10,min=0.1,max=0.30)) #error term with mean 0 and standard deviation drawn from a uniform distribtion
Then I generate the data:
data <- alpha + B*R + epsilon
My question is: am I interpreting this description correctly?
