Question

Running a Monte Carlo simulation to calculate the probability that the daily return from S&P will be > 5%. We will assume that the historical S&P daily return follows a normal distribution with an average daily return of 0.03 (%) and a standard deviation of 0.97 (%).

To begin we will generate 100 random samples from the normal distribution. For the generated samples we will calculate the mean, standard deviation, and probability of occurrence where the simulation result is greater than 5%.

Repeat simulator for the two cases where the number of simulations/samples is equal to 1000 and 1000. For each case record the mean, standard deviation, and probability.

Answer 1

As required, the program was written (using R) and the output for simuklation numbers 100,1000,10000,100000 and 500000 are given below.

However, probabilities are estimated as zero. This is natural as 99.73% of the observations of N(,.03,.97) distribution lies within the 3 sigma limits (.03-3*.97,.03+3*.97)= (-2.88,2.94) and hence chance of getting a sample more than 5(%) is almost nil, which is reflected in the simulation results also.

Simulation number= 100 Mean= 0.1233831 SD= 0.8791227 Probability= 0

Simulation number= 1000 Mean= -0.00416643 SD= 1.003485 Probability= 0

Simulation number= 10000 Mean= 0.03263355 SD= 0.9741533 Probability= 0
Simulation number= 100000 Mean= 0.02595217 SD= 0.9707109 Probability= 0

Simulation number=500000 Mean= 0.02999542 SD= 0.9696211 Probability= 0

R Program

nsim=500000
x=rnorm(nsim,.03,.97)
m=mean(x)
s=sd(x)
prob=sum(x>5)/nsim
cat("\n","Simulation number=", nsim,"Mean=",m, "SD=",s, "Probability=",prob)