COMPLETE IN R$,$ should not be done in chat GPT$/$use complex code

A life insurance company is pricing a new policy to sell to a group of $45-$year$-$old male non$-$smokers. They determine that the probability that a member of this group will die X years from the day they purchase the policy can be modeled with a Weibull distribution with shape parameter $4\mathrm{.}6$ and scale parameter $39,$ measured in years. The term of the policy is $20$ years. At the end of every month, policyholders $($PH$)$ are expected to pay a premium of $$120.$ If a PH in good standing dies during the term of the policy, his beneficiaries receive a benefit of $$1,000,000$ at the end of that month. Every month there is a $0\mathrm{.}3\%$ chance that the policy holder will let the policy lapse $($i$.$e$.$ he will permanently stop paying premiums and forfeit his right to the benefit$).$ The insurance company calculates cost of funds using a rate of $6\mathrm{.}5\%,$ compounding monthly.

Hint: The formula for the present value of n payments of size A at the end of each period with a per period interest rate of r is given by

PV$=$A$*(1-(1+$r$)^(-$n$))/$r

$PV=A**\frac{1-(1+r{)}^{-n}}{r}$

a$)(2$ pts$)$ Create a Monte Carlo simulation with $10,000$ trials of the above scenario to calculate the

net present value of cash flows between the company and one policyholder.

i$)$ pts$)$ Create a data frame with one row for each trial, one column for time until death

$($measured in months$)$ and another for number of months before the PH would let the policy

lapse. Note: you should be able to choose a distribution for the lapse column from among

those we discussed in class.

ii$)(0\mathrm{.}6$ pts$)$ Mutate the data frame from $($i$).$ Create new columns to calculate for each trial:

Whether or not the insurance company owes a benefit;

For how many months does the $PH$ pay premiums;

The NPV of the payments by the PH;

The NPV of the benefit from the insurance company $($or $0$ if they owe no benefit$)$;

The difference between the two NPVs$.$ This is the net NPV for the trial.

iii$)(0\mathrm{.}2$ pts$)$ Create a histogram describing the net NPV of the company. How would you

characterize the distribution?

iv$)(0\mathrm{.}2$ pts$)$ According to your simulation, what are the mean and standard deviation of the net

NPV$?$ On average, is the company making a profit?

v$)(0\mathrm{.}2$ pts$)$ Provide a $95\%$ confidence interval for the mean of the net NPV$.$ Interpret the result.

vi$)(0\mathrm{.}2$ pts$)$ How many iterations would be necessary to provide a $99\%$ confidence interval with

a half width of $$200?$

vii$)(0\mathrm{.}2$ pts$)$ The company can be $90\%$ certain their net NPV will be at least $x.$ Solve for $x.$

The company can be $99\%$ certain their net NPV will be at least $y.$ Solve for $y.$