# How to perform a simulation of stock investment, capturing the variance?

I am simulating an individual, who invests throughout his lifetime in stocks and bonds. Bonds have fixed returns $$r_f = 11\%$$. Stocks are highly volatile and have returns $$\mu = 22 \%$$ and standard deviation $$\sigma = 36 \%$$.

(i) Now, if I do a Monte Carlo on N repetitions, I will end up "averaging out" the variance and come at investment with the average of $$22 \%$$ return, i.e. as if stocks are as riskless as bonds. This would result in a conclusion, that everyone should invest in stocks, because "on average" they have no risk, and higher returns than bonds.

(ii) If I study one random draw with predetermined seed, specifically chosen to capture ups and downs of the stock movements, it feels like I'm only analyzing just one random case, and my paper will not be considered serious.

(iii) If I do three simulations for $$r_1 = \mu + \sigma$$ , $$r_2 = \mu$$, and $$r_3 = \mu - \sigma$$, all of the scenarios seem extreme to me - either crazily high returns, or crazily low returns, not how it happens in the real life.

So, am I wrong? Which approach should I pursue for the most comprehensive analysis? I personally like the (ii).

Thank you very much.

A statistics professor advised me to use the option (iii) --- draw three lines, one average and two dashed confidence intervals. Likewise, he told me to construct three tables corresponding to these three scenarios.

He told me, that this is the general practice in modeling random series.

I believe this answers my question.