Evaluate Mean and Standard Deviation for Asseet Allocation of SPY and Cash

Assume cash has a mean return of 0, return standard deviation of 0 and has no corration with SPY, in theroy, with x percent of cash ans 1-x percent of y, the portfolio mean return is (1-x) times SPY mean return and standard devation of (1-x) times SPY standard deviation. Mathematical proof is in this blog post.

SPY has a daily mean return of 0.046% and daily return standard deviation of 1.179%. Derivation is in this blog post.

Then, for example, a portfolio of 50% cash and 50% SPY will have a daily mean return of 0.046%/2= 0.023% and daily return standard deviation of 1.179%*/2 = 0.5895%.

I want to test the above statistial theory by runing simulation.

Simulator

SPY and Cash have the above mentioned daily mean return and daily return standard deviation. This simulator will generate SPY pirces base on geometric Brownian Motion that stock prices follow. Then it calculate portfolio value and its daily rerun data. Finally, the porfolio daily returns will be transformed into the mean and the standard deviation for comparing with the theoretical mean and standard deviation.

Portfolio Initail Value: $1000
SPY ratio: %
Cash ratio: %

• Theoretical Mean: 0.0230%
• Theoretical Standard Deviation: 0.5895%
• Simulated Mean: %
• Simulated Standard Deviation: %