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: 40 Hours (not including subsequent review time)

BattleTable

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

• Calculate the premium (or components of premium) for a large dollar deductible (LDD) plan.
• Calculate the premium for a retrospectively rated plan.
• Calculate limited expected values.
• Know when to de-trend and de-develop losses.

Questions from the Fall 2019 exam are held out for practice purposes. (They are included in the CAS practice exam.)

reference	part (a)	part (b)	part (c)	part (d)
E (2018.Fall #10)	Expected Reported Loss - calculate	Experience Mod - calculate
E (2018.Fall #12)	Retro & LDD Premiums - calculate & compare	Retro & LDD Premiums - justify selection	Retro & LDD Premiums - propose changes
E (2016.Fall #15)	WC Policy Deductible - calculate	Large Deductible Plan - Fisher.OtherLSPlans
E (2016.Fall #16)	LDD Policy Premiums - calculate	LDD Plans - recommend option
E (2015.Fall #14)	LDD Premium - calculate	Lee Diagrams - Fisher.Visualization
E (2015.Fall #18)	Expected Loss Cost - calculate
E (2015.Fall #19)	LDD Premium - calculate credit risk	Large Deductible Plan - Fisher.OtherLSPlans
E (2014.Fall #17)	Retro Balanced Premium - calculate	Retro Basic Premium - calculate	Retro Premium Parameters - describe relationships	LDD & Excess WC Policies Fisher.RiskSharing
E (2014.Fall #19)	LDD Premium - calculate	Risk Sharing Plans - Fisher.OtherLSPlans	Excess WC Premiums - calculate	Risk Sharing Plans - Fisher.OtherLSPlans
E (2012.Fall #23)	Retro Basic Premium - calculate

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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 [math]\Phi[/math] terms in 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 exceed 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 excess loss. The difference of these is the experience modified expected limited loss, L, which is the expected loss after accounting for the per-occurrence limit/deductible, i.e. the aggregate excess. It is the part of the loss the insured is responsible for.

Setting that aside for the moment, calculate the loss conversion factor, c, as 1 + (LAE as a percentage of loss), and the tax multiplier, T, as 1 / [1 - premium tax % - commission %].

The basic premium formula is B = (experience modified excess loss)* (UW Profit % + c) + Fixed Expenses.

Now we apply the retrospective premium formula [math]R=(B+c\cdot L)\cdot T[/math] in a couple of different actual limited loss situations. Remember, L is the experience modified expected limited loss.

First we suppose the actual limited losses equal L. We calculate the retrospective premium using the value of L found at the start of this step.

Next, suppose the actual limited losses equal 0.5L and calculate the retrospective premium. Lastly, suppose the actual limited losses equal 2L and calculate the premium again.

Question: What are some implications of including commission in the tax multiplier?

Solution:

• In the retrospective rating formula, [math]B+cL[/math] is the net deductible premium before the application of the tax multiplier. So including commission in the tax multiplier makes it a percentage of the net deductible premium.
• If instead commission was included in the loss conversion factor, c, then it is a percentage of loss.
• If instead commission was included in the basic premium, B, then the commission is guaranteed and fixed (doesn't vary with the performance of the book).

Step 6

Here, we take a break from pricing the account to formulate a Limited Table M using the vertical slicing method.

Alice: "Wait! How do you know it's a Limited Table M?"

The instructions say there's "a deductible or loss limit of $100k". This means the insured is responsible for at most $100,000 for each occurrence. We're not told anything about an aggregate limit, so it must be a Limited Table M.

We're given a list of the annual losses for 500 policies which have been limited to $100,000 per-occurrence (some policies may have more than one loss). It's important to note the list has already been sorted into ascending order. If it hadn't then you need to sort it first.

We want to calculate the insurance charge at an entry ratio of 2.0.

First, in a new column, calculate the entry ratio for each loss using [math]r=\frac{\mbox{actual loss}}{\mbox{expected loss}}[/math] (Note we're given the expected limited losses). Then in the next column calculate the amount by which each entry ratio exceeds the entry ratio of 2. If an entry ratio doesn't exceed 2.0 then we give it a value of 0.

The insurance charge percentage is the sum of the amounts in excess of 2.0 divided by the number of risks.

We can then calculate the insurance charge as the product of the insurance charge percentage and the expected losses.

Alice: "Although they gave you the expected loss this time, in the exam you might not be so lucky... If that happens, use the average of the actual limited losses. You can check here this is the same as the given expected loss".

Step 7

Continuing our side excursion, we'll use the horizontal slicing method to approximate the insurance charge. Remember, the vertical method gets the exact answer but the horizontal method is supposed to be easier to use.

We're told to use an entry ratio step size of 0.1 and consider entry ratios from 0 to 4 inclusive. Start by making a column which contains these entry ratios.

Next, we need the number of risks over the current entry ratio. Create a new column and use a COUNTIF function to count the number of risks on the Step 6 worksheet which have a entry ratio greater than the current row.

Add a new column that expresses the number of risks over a given entry ratio as a percentage of all the risks. Then add yet another column that measures the difference between the current entry ratio and the next entry ratio (leave the last row blank in this column).

Finally, in a new column and working from the bottom of the table up, let the last row be zero and move up the table by setting it equal to the row afterwards plus the percentage of losses over the entry ratio multiplied by the entry ratio step.

Now you've built your Limited Table M using the horizontal slicing method.

To find the insurance charge at entry ratio 2.0, look up the entry ratio and take the value in the last column. Multiply this by the expected limited losses to get the insurance charge (or you can compare the insurance charge percentages instead).

You should notice the value obtained in this step isn't very close to the value you got in Step 6. This is because the entry ratio difference isn't sufficiently small in the Step 7 Limited Table M. In the solution, Fisher shows a refinement using an entry ratio step size of 0.01 and the result is much closer to the exact value found in Step 6.

Step 8

Part a: We now assume there is also an aggregate deductible limit and find the insurance charge by building off of the earlier steps.

First, carry forward the expected limited loss from Step 5. Next, calculate the ratio of the aggregate deductible limit to the expected limited loss; this is the entry ratio. Lastly, multiply this by the insurance charge percentage found in the refined Limited Table M in the Step 7 Answer worksheet.

Part b: Instead of treating the per-occurrence limit as a deductible, we'll now think of it as a retrospective rating plan with a per-occurrence limit and assume the maximum ratable loss is the same as the aggregate deductible in part a above. We'll also assume there is a minimum ratable loss. We want to find the net insurance charge that accounts for both the minimum and maximum ratable loss.

The net insurance charge is the expected limited loss multiplied by the difference between the insurance charge percentage and the insurance savings percentage. The expected limited loss is the same as in part a (from Step 5) because we didn't change the company or per-occurrence limit/deductible.

To get the insurance charge percentage, calculate the ratio of the maximum ratable loss to the expected limited loss and look up the corresponding φ value in the refined Limited Table M.

Similarly, to get the insurance savings percentage, calculate the ratio of the minimum ratable loss to the expected limited loss and look up the corresponding ψ value in the refined Limited Table M or calculate it using [math] \psi(r) = \phi(r) +r -1[/math]. Note the Fisher solution rounds the entry ratios to 1 decimal place. Really it should be rounded to 2 decimal places to match the accuracy of the Limited Table M. If you don't do this then technically you have a slight mismatch when you apply the formula to calculate [math]\psi(r)[/math].

Lastly, take the difference and multiply this by the expected limited loss.

Step 9

In this step we'll use the Step 8 part a answer to determine the premium for the large deductible policy with an aggregate deductible.

Carry forward the total expected loss and modified expected loss from Step 4. The total expected loss is needed to calculate the loss adjustment expenses because these apply to all losses.

Bring forward the expected aggregate excess loss which is the insurance charge found in Step 8 part a.

Use the standard approach to determining premiums from Exam 5. [math]\textrm{Premium}=\frac{(\textrm{Expected} \, \textrm{Per-occurrence} \, \textrm{Excess}\, \textrm{Loss} + \textrm{Expected} \, \textrm{Agg.} \, \textrm{Excess}\, \textrm{Loss})\cdot (1 + \textrm{UW}\, \textrm{Profit}\, \%)+\textrm{Fixed}\, \textrm{exp.} +\textrm{LAE}}{ 1- \textrm{Prem.} \, \textrm{Tax} \, \% - \textrm{Commission} \, \%}[/math].

Now bring forward the premium for the policy without the aggregate deductible from Step 4 and compute the difference between it and the premium for the policy with the aggregate deductible.

Similarly, apply the premium formula above to the expected aggregate excess losses only.

You should notice these two quantities are the same amount.

Step 10

Here we'll use the result from Step 8 part b to update the premium from Step 5 to reflect the maximum and minimum ratable losses.

First, collect the Loss Conversion Factor, c and the Tax Multiplier, T as in Step 5. Second, bring forward the modified loss experience from Step 5 (really Step 4) and the limited expected loss (from Step 5).

Now form the basic premium using the following formula [math]\textrm{Basic}\,\textrm{Prem.} = (\textrm{Expected}\, \textrm{Per-occurrence}\,\textrm{excess}\,\textrm{loss} + \textrm{net}\, \textrm{ins.}\, \textrm{charge})\cdot(c+ \textrm{UW}\,\textrm{Profit}\,\%) + \textrm{Fixed}\,\textrm{exp.}[/math]

Then apply the retrospective rating formula, [math]R=(B+cL)\cdot T[/math], in each of the three scenarios given. Here, L, is the limited expected loss.

Alice: "It's important to be careful when applying the limited expected loss in each scenario as we now have a minimum and maximum ratable loss. If the scenario produces a limited actual loss less than the minimum ratable loss, use the minimum ratable loss instead in the formula. Likewise, if the scenario has a limited actual loss greater than the maximum ratable loss then use the maximum ratable loss in the formula".

Observations

• Bringing forward the first scenario premium from Step 5 (i.e. without the maximum and minimum ratable loss conditions), we see the difference between this and the corresponding premium in Step 10 is just the net insurance charge loaded for LAE, profit, commission and premium tax.
• When the limited actual losses are half the expected limited losses then the Step 10 premium is higher than the Step 5 premium because the insured is paying for the right to cap the aggregate losses if needed.
• When the limited actual losses are twice the expected limited losses then the Step 10 premium is lower than the Step 5 premium because the actual losses were capped.

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