2022/4/23 20:03 Quiz: Homework 4: Benchmarking Randomness
Homework 4: Benchmarking Randomness
Started: Apr 23 at 6:09pm
Quiz Instructions
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In our last Excel assignment for the semester, you are going to use simulation analysis to help benchmark the potential for random variation for a company. Please read below for the description and then answer the questions.
The situation: You work for an insurance company that has underwriters who decide which commercial insurance customers to select based on a combination of some set underwriting rules established by the company and their own judgment based on a range of factors about whether or not a particular customers would be a reasonable risk for the company to take on. Your company evaluates underwriting performance by comparing the underwriters’ experience using a couple of metrics that are based on the extent to which the customers that underwriter has accepted experience losses. The director of underwriting is responsible for doing performance evaluations based on the metrics that are calculated for each underwriter. Part of her goal in the process is to try to gauge whether each underwriter is systematically selecting better or worse risks when using their judgement. However, she also realizes that there will be some underlying randomness behind the performance metrics. Even if underwriters had no difference in their ability to select clients, some would have better experiences and some worse each year. What she has asked you to do is help her get a benchmark for what the random variation could look like so she has a better sense of what type of results would be “unusual” to see if underwriters were simply performing “at expectations”.
The performance metrics:
The primary performance metric used to track underwriter performance is the “per client loss” (PCL). The PCL is simple the average size of claims across the clients in that underwriter’s book of business, including clients who had no loss. So for example, suppose an underwriter had 2 clients and one had no loss (loss = $0) and the other had a loss of $30,000, then the underwriter’s PCL for that period would be $15,000.
They also use two other secondary metrics: a) the share of the underwriter’s clients that had any loss and b) the average severity of the losses for the underwriter’s clients who had losses. For the example above, where one client had $0 and one had $30,000 these would be a) 50% and b) $30,000, respectively.
Finally, they have a “performance problem flag” that gets set to one if any of the following thresholds on the performance metrics are crossed: a) PCL >= $6,000; b) share with loss >= 20%; c) average severity among clients with a loss >= $55,000.
Actuarial benchmarks:
The director has previously asked your unit to create some actuarial benchmarks for loss experiences for the average client at your company. The team has concluded that on average we should expect that clients would have a 10% chance of having a loss each period and that this chance is independent across the clients. They have also concluded that when a loss happens, the distribution of the severity of the loss can be well represented by a Lognormal distribution with mean = $35,000 and standard deviation of $15,000. So the expected value of the PCL is 10%*$35,000 = $3,500.
The ask of you for this simulation:
The director has asked you to take the actuarial benchmarks and simulate for her the range of performance metrics she might observe if her underwriters were really all just performing at the actuarial benchmarks and had no underlying differences in their ability to select clients. The idea is that this will help her know what ranges of performance metrics would show up commonly just due to background random chance in the mix of clients that have losses. She can then
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2022/4/23 20:03 Quiz: Homework 4: Benchmarking Randomness
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use that to help her consider how “extreme” a particular observed performance metric for one of her underwriters really is.
One complicating factor is that underwriters have different numbers of clients in their book of business. This variation is based largely on how long the underwriter has been with the company. The underwriters with the fewest clients have 40 clients and those with the most have as many as 250 clients. She has a vague sense that performance metrics should be more variable for underwriters with fewer clients, but she isn’t sure how big a difference it should make.
She would like the following statistics from you from a simulation with at least 2,000 iterations under the assumption that all underwriters’ clients are experiencing losses based on the actuarial benchmarks:
What is the range of PCL that she could expect underwriters to fall into 95% of the time (2.5% of time PCL below and 2.5% of time PCL above) for underwriters with 40 clients? How about for underwriters with 250 clients?
What is the range of the metric “share of clients with losses” she should expect underwriters to show 95% of the time with 40 clients? with 250 clients?
What is the range of the metric “average severity of losses among clients with losses” she should expect underwriters to show 95% of the time with 40 clients? with 250 clients?
What share of the time would underwriters with 40 clients trigger the “performance problem flag”? How about for 250 clients?
Question 1 20 pts
Please upload here a neatly formatted table showing your results for the ranges of the performance measures requested by the director. Specifically, your table should have three rows showing: PCL, share with losses, and average severity among clients with losses. The columns should show the lower bound and upper bound for the range we would expect to see 95% of the time (2.5% below and 2.5% above) for an underwriter with 40 clients and for an underwriter with 250 clients. So you should have 4 different columns of numbers in the table. Make sure to put a title for your table that helps to make it clear what the table is showing. Also put a text note right below your table that gives the basic information used in the simulation: (e.g., “Note: These results are based on a simulation of performance metrics for underwriters assuming that clients experience losses with probability …”)
I want you to again try to save this table as an image file (e.g., .png). You may want to do this by making or copying your table into powerpoint and then saving the powerpoint slide as an image file. Upload that table image here as an image upload in your answer. If you are unable to get the image to upload, you can instead upload an excel file with your table in the file upload box given below this. In that case, make a note here to look at that upload file.
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Question 2 0 pts
Use this upload for your table of results if you are unable to get the image file to load in the previous question.
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Question 3 5 pts
Please describe in words how the benchmark ranges differ (or don’t differ) for underwriters with 40 clients vs. those with 250 clients. Make sure to touch on whether the differences are “sizeable” from the perspective of how they might affect the way the director should think about evaluations for underwriters with more or less clients. Briefly explain why the results do or don’t differ between underwriters with 40 vs. 250 clients.
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2022/4/23 20:03 Quiz: Homework 4: Benchmarking Randomness
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Question 4 2 pts
What is the probability that an underwriter with 40 clients would trigger the current “performance problem flag” just do to randomness if that underwriter’s clients were all experiencing losses based on the actuarial benchmark distributions?
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Question 5 2 pts
What is the probability that an underwriter with 250 clients would trigger the current “performance problem flag” just do to randomness if that underwriter’s clients were all experiencing losses based on the actuarial benchmark distributions?
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Question 6 10 pts
Make a specific recommendation for a new way to define the “performance problem flag” that you would recommend based on the analysis you have done. Explain your reasoning for why this would be a more useful way to flag potential performance problems than the current flag. Note that the director uses the performance problem flag as one input into evaluations and it is not that the flag triggers an automatic penalty/reward.
There is no one right answer here and there are different ways you could take this. We are looking for whether you can give a sensible and thoughtful recommendation to the director based on your analysis and reflection about this.
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2022/4/23 20:03 Quiz: Homework 4: Benchmarking Randomness
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Question 7 5 pts
Please upload your completed spreadsheet model here.
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