Fisher.CaseStudy

Reading: Fisher, G. et al, "Individual Risk Study Note," CAS Study Note, Version 3, October 2019. Case Study.

Synopsis: The case study is meant to help you learn some of the concepts from the Fisher paper.

Study Tips

The case study doesn't have great instructions so we've added commentary and helpful formulas for each step in this wiki companion article.

Estimated study time: 2 Hours (not including subsequent review time)

BattleTable

Based on past exams, the main things you need to know (in rough order of importance) are:

• fact A...
• fact B...

reference	part (a)	part (b)	part (c)	part (d)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)
E (2018.Spring #1)

In Plain English!

Alice: "If you haven't already, download the latest version of the case study from the CAS website here: Fisher Case Study. Now let's open up the case study and get started".

Step 1

• Create a new column called "Natural Log". The values in this column are the natural log of the loss amount (=LN(loss)).
• Estimate μ as the average of the Natural Log column. Similarly, estimate σ as the population standard deviation of the Natural Log column (Excel =StDev.P(column))
• Fill in the summary matrix below

	Fitted	Empirical
Mean
Standard Deviation

• Empirical values are found by taking the average and population standard deviation of the loss column respectively.
• Fitted values use the parameter estimates in the following formulas from Bahnemann Chapter 2.4 for a lognormal distribution, X.
  • [math]E[X]=e^{\mu+\frac{\sigma^2}{2}}[/math]
  • [math]Var(X)=\left(e^{\sigma^2}-1\right)\cdot e^{2\mu+\sigma^2}[/math]

Alice: "Me again, you did remember to skim through Bahnemann Chapters 1 - 4 even though they're only recommended reading on the syllabus didn't you..."

Step 2

Here you need to calculate the limited expected value at each of the given limits before you can find the increased limit factors and excess ratios.

Like in the exam, you're given the limited expected value formula for the lognormal distribution since it's fairly complex. As the formula is long it can be helpful to use three "helper columns" in Excel, one each for [math]e^{\mu +\frac{\sigma^2}{2}}[/math], [math]\Phi\left(\frac{\ln x-\mu-\sigma^2}{\sigma}\right)[/math], and [math]x\left\{1-\Phi\left(\frac{\ln x - \mu}{\sigma}\right)\right\}[/math] which can be combined to get the limited expected value.

The standard normal distribution, Φ, in Excel is NORM.DIST(x,0,1,TRUE) where the 0,1 specifies the standard normal distribution and TRUE indicates we want the cumulative probability.

For the unlimited limited expected value, note the second and third helper columns converge to 1 as x goes to positive infinity.

Once you have the limited expected values, the increased limit factors are the ratio of the limit to the base limit, i.e. rebase the limited expected values to be 1.000 at 100,000.

The Excess Ratio is defined as [math]1-\frac{\mbox{Limited expected value at limit}}{\mbox{Unlimited limited expected value}}[/math].

Step 3

Step a:
We're given the expected loss amount and need to calculate the expected reported limited loss for each of the historical periods. This is done by first using the same expected loss for each policy period and de-trending and de-developing it to move the expected loss to the historical policy period. That is, we divide out the loss trend and LDF. Then multiply by the limited loss as a percentage of total loss.

Step b:
You're given the individual losses used in the experience rating. For each policy period cap any losses that excess the individual claims cap of 100,000. Keep the full amount of the losses under the cap. Sum the results by policy period to get the actual limited losses.

Step c:
Sum across the policy periods to get total actual and expected limited losses. Calculate the ratio of the actual divided by the expected losses.

We need a suitable complement of credibility. One option is assuming actual losses = expected losses, i.e. the ratio we just calculated is 1.000. Apply the standard credibility formula with the given credibility to get the credibility weighted ratio. This is the experience modification.

Multiply the experience modification by the expected experience to get the experience modified expected losses.

Step 4

We're pricing a large deductible policy, so the premium should cover only the losses in excess of the per-occurrence deductible plus expenses and profit.

Using the experience modified expected losses from Step 3, multiply by the Excess Ratio corresponding to the 100,000 deductible per-occurrence found in Step 2. This is the experience modified expected excess loss.

Next calculate the loss adjustment expense, remembering that we pay LAE on all losses not just the excess losses.

Calculate the premium using the standard approach from Exam 5. That is Premium = [(experience modified expected excess loss)*(1 + UW Profit %) + (Fixed Expenses) + LAE ] / [1 - (Premium Tax %) - (Commission %)].

Step 5

First bring forward the experience modified expected loss and the experience modified expected limited loss. The difference of these is the experience modified expected excess loss.