Determining Errors in Monte Carlo Simulation

Question

I was wondering if anyone could throw light on possible errors associated with Monte Carlo sampling. I seem to be getting values that are slightly different each time despite running my model for 500,000 iterations and I was wondering how I could take account of this.

Answer 1

If you are for example using a linear combination of the 65 distributions, one can look at the variance of the linear combination: If we let $\bar X_i $ be the average of $m$ simulations of the random variable $X_i$ (in this case $m$ is 500 000) and you have a linear combination $$ c_1 \bar X_1 + c_2 \bar X_2 + \cdots +c_n \bar X_n $$ ($n$ is 65 in Your case) then $$ Var(c_1 \bar X_1 + c_2 \bar X_2 + \cdots +c_n \bar X_n ) = \sum^n c_i^2Var(\bar X_i) = \sum^n c_i^2 \frac{\sigma^2_i}{m}$$ where $\sigma_i$ is the standard deviation of $X_i$. If for example all coefficients $c_i$ are $1$ and all variances are equal, this becomes $$\frac{n \sigma^2}{m}$$ so as you see, the variance increases linearly with the number of contributing variables.