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].