Monte Carlo financial simulation

[hotvette]

Let's say I want to simulate a 30-year investment return scenario by running n simulations (e.g. n = 1000) using a normal distribution with mean x% and standard deviation y%.

My first approach was to generate exactly n sets of 30 samples from N~(x,y) but I realized that for any given set of 30 samples the sample average isn't necessarily close to x%. Wouldn't a more valid approach be to run a sufficient number of scenarios to obtain n sets, each of which has a sample average within a pre-determined tolerance of x? It seems to me the answer should be yes.

I've seen results of financial monte carlo simulators that are offered by well known financial institutions, but nobody I talk to seems to know the details of how it is done.

 

[mfb]

but I realized that for any given set of 30 samples the sample average isn't necessarily close to x%

Good, that's what you expect from 30 independent samples.
If you expect the years to be correlated (and in general they will be) you first have to model this correlation.

Wouldn't a more valid approach be to run a sufficient number of scenarios to obtain n sets, each of which has a sample average within a pre-determined tolerance of x? It seems to me the answer should be yes.

Not if you want to study what happens to the investment.

 

[BWV]

If you want to calculate percentiles of 30 year annualized returns from a lognormally distributed return with mean m (log return) and vol sigma, it can be done in closed form like this in Excel:

=EXP(m+NORMSINV(percentile)*(sigma/SQRT(number of years)))-1

There is no appreciable year by year autocorrelation in financial market returns, so that assumption is good, however returns have fatter negative tails than reflected in a lognormal distribution, but this is a good approximation