How do i set up the simulation model to answer the following questions? Lynn Rog

Question

How do i set up the  simulation model to answer the following questions?

Lynn Rogers (who just turned 30) currently earns $60,000 per year. At the end of each calendar year, she plans to invest 10% of her annual income in a tax-deferred retirement account. Lynn expects her salary to grow between 0% and 8% each year, following a discrete uniform distribution between these two rates. Based on historical market returns, she expects the tax-deferred account to return between –5% and 20% in any given year, following a continuous uniform distribution between these two rates. Use N replications of a simulation model to answer each of the following questions.

What is the probability that Lynn will have in excess of $1 million in this account when she turns 60 (i.e., in 30 years)?

If Lynn wants this probability to be over 95%, what should be her savings rate each year?

Explanation / Answer

Simluated using R (code below) for N = 10,000 replications:

N <- 10000
savings <- rep(0, N)
savings_rate <- 10
for (j in 1:N)
{
income <- 60000
income_growth <- runif(30, 0, 8)
investment_return <- runif(30, -5, 20)
  
for (i in 1:30)
{
investment <- income * savings_rate / 100
savings[j] = (savings[j] + investment) * (1 + investment_return[i]/100)
income <- income * ( 1 + income_growth[i]/100)
}
}

# p = Probability that Lynn will have in excess of $1 million in 30 years
p = sum(savings>1000000)/N

(a)

Probability that Lynn will have in excess of $1 million in this account in 30 years = 0.47 or 47%

(b)

To find the savings rate for this probability to be over 95%, the simulation can be run by varying the savings rate manually and finding the minimum savings rate which gives the probability of more than 95% every time.

This savings rate comes to be 15.4%

So, if Lynn wants this probability to be over 95%, her savings rate each year should be 15.4%

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