Simulation with Multiple Variables from Independent to correlated Variable

[sudeepdoon]

Hi David,

Referring to the uploaded excel sheet for the topic, i.e. 2-b-4. Here the values in the C and D columns are for the two random variable as specified by Jorion. The concluding line in the book says that the covariance of these variable should be equal to the correlation coefficient of the two variable. Now when I use COVAR(C7:C38,D738) to calculate the covariance, this is not equal to D3 which is the asumed correlation.
whats wrong here?

 

[David Harper CFA FRM]

Hi sudeep -

First please note that since correlation = covariance/[(stdDev)(stdDev)], Jorion presumably only asserts that covariance = correlation under the special case that variance = 1.0. So he refers to standard normal variables ~ N(0,1). So your test might better be CORREL() although it will not matter if the variables are standard normal.

As to why your (ex ante) empiral results differs from the input, that is "sampling variation." There is no reason to expect the observed correlation from the simulated sample to match the input. Rather, we expect that as the number of trials increases toward infinity, we should observe convergence.

It is like using =NORMSINV(RAND())

This function simulates a standard normal variable (mean = 0 , variance = 1). But if you run it 10 or 100 times and then compute the mean and variance of that *sample*, you will likely get a mean different than 0 and a variance different than 1. It paralles the parametric VaR (i.e., using the formula) versus monte carlo VaR (using the simulated distribution).

It's still true in this case, that the input to the MCS models the correlation of rho, but that is like an expected correlation, the realized correlation will be infected by sampling variation

David

append: we can connect this to Gujarati and the essence of his statistics. There is one true population but many samples and the samples vary; in this case, i am a bit loose with terms maybe but the model (the formula) is like the "population" - it has correlation parameter of rho. Only the 1! Now run Monte Carlo and each time your are creating a new sample. Each sample produces its own correlation "statistic" and owing to the randomness, these "sample statistic" rhos will fluctuate (presumably they are unbiased and will fluctuate around the true rho).