Your son has just gotten his driver's license and your bank account is about to

nitraiddQ 2021-09-30 Answered
Your son has just gotten his driver's license and your bank account is about to get a little (a lot?) lighter. To help mitigate the impact on your finances, you have opted for a high deductible auto insurance policy. The deductible on your policy is $4000, which means that you will pay the first $4000 of any damages and then the insurance will cover the rest. Because of this, if there is an accident for which the damages are less than $4000, you aren't even going to file a claim with the insurance company. The less they know, the better? You will just pay it out of pocket. You believe that any accident will result in a damage amount which is normally distributed with a mean of $4500 and a standard deviation of $1500. The value of your son's car is only $7500, so that is the upper bound on the damage amount because in that case, you can junk the car and buy a different one for $7500. The lower bound on damages is obviously $0. The probability of an automobile accident this year is 7.5%.
Build a Monte Carlo simulation model to show your out of pocket expenses in this situation. If there is no accident, then there is no out of pocket expense.
Analyze the results of the 1000 iterations to find the following as a percentage of the 1000 interactions:
1. How often a claim was filed (damage met deductible).
2. How often you ended up buying a different car.
Also, calculate your expected out of pocket expense.
For these three questions, put a cell reference in Cells B4:B6 to wherever you have calculated those values in your spreadsheet so that I don't have to hunt for them.

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Expert Answer

hajavaF
Answered 2021-10-01 Author has 14495 answers

Step 1
This is a good exercise to understand how to simulate (a) Binomial and (b) Normal distributions. The data comes really nicely and gives some tangible results about the number of times a claim is actually made, i.e. the amount exceeds $4000 in order to get an insurance value, the percentage of these claims as the total of all incidence when a damage was made, the average insurance amount etc.
So the work is actually excel based and cannot be pasted here. I am giving a walkthrough of the process followed to simulate the data so you can understand how it works.
Step 2
First, we use the Inverse Binomial distribution function on excel, to generate 100 incidences with the parameter \(\displaystyle{p}={0.075}\). The exact input command is =BINOM.INV(1,0.075,RAND()). This generates 1000 values of 0 and 1, with proportions of 1's being very close to 0.075.
Second, we do a similar simulation of the Normal distribution, again 1000 times, with the command =NORM.INV(RAND(),4500,1500). We get a nicely laid out set of values.
The third step is to multiply, along each row, the value obtained from the above two distributions. What that does is, give us a sample output of the number of times a claim is expected to be made, with those values following the mean and standard deviation as defined earlier.
Now we shall have a certain proportion of those 1000 rows having non-zero values. This is important data because it tells us how many times some damage was borne by the person, irrespective of whether a claim ensued or not.
Lastly what we now need to do is look at only those values in this column that exceed 4500, and find their value post deductible. So we do the command =MAX(0, D3-5000) on the damage made, where D3 is the cell reference for the damage.
Voila!! You are all set to do summary statistics on the values obtained. Here is a sample output of the simulation
\(\begin{array}{|c|c|} \hline \text{Summary Statistic} & \text{Value} \\ \hline \text{No of Claim Incidence} & 76 \\ \hline \text{No of Claims Made}&40\\ \hline \text{Claims Made as % of incidence}&52.63\%\\ \hline \text{Average Amount of Claim Post Deductible}&1094.626732\\ \hline \end{array}\)
These values are random on excel, so on the file you download, they will change every time you click anywhere on the sheet. In case you need to report a fixed set of values, just copy the entire sheet, go to a new tab and use Paste with Values only.

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