Fisher.Visualization

Reading: Fisher, G. et al, "Individual Risk Study Note," CAS Study Note, Version 3, October 2019. Chapter 3. Section 2

Synopsis: To follow...

Study Tips

...your insights... To follow...

Estimated study time: x mins, or y hrs, or n₁-n₂ days, or 1 week,... (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!

When working with per-occurrence limits and aggregate limits it can be helpful to visualize them. A widely used approach to visualizing them was developed in a paper by Yoong-Sin Lee .

Alice: "Lee's paper isn't on the syllabus itself, but sometimes with tricky material it helps to refer to the original source."

The graphs described in the paper are usually referred to as "Lee diagrams". A Lee Diagram has the cumulative claims (count or percentage of loss distribution) on the horizontal axis, and the severity or aggregate loss (the "size") on the vertical axis. Let's look at this in more detail.

Let n be the number of losses, let the loss sizes be x₁, ..., x_q and assume they're ordered such that x₁ < x₂ < ... < x_q. Let the associated loss frequencies be given by n₁, ..., n_q thus n = n₁ + ... + n_q.

Figure 1 below (insert LeeDiagramsFig1.png) shows the more traditional way of viewing a sample of 10 claims. Notice how in the second part of the figure we've ordered the claims by loss size and added the incremental and cumulative counts. So here x₁ = 1, x₂ = 2, ..., x₁₅ = 15 and n₁ = 2, n₂ = 0, ..., n₁₅ = 2, and n = n₁ + ... + n₁₅ = 10.

When we produce a diagram like Figure 1 we are asking ourselves: "For a fixed size of loss, how many claims (count or percentage) are smaller than this loss size?"

To draw a Lee diagram we need to turn this question around and ask: "What is the size of loss that k% of claims are smaller than?"

Figure 2 below shows a Lee diagram of the ten claims used in Figure 1 above. To sketch this, order the losses (the x_i's) from smallest to largest. Let k be either the cumulative claim count or the percentage of claims, in the latter case the horizontal axis will be scaled between 0 and 1. For each value of k find the value q such that [math]\displaystyle\sum_{i=1}^{q-1} n_i \leq k \lt \displaystyle\sum_{i=1}^q n_i[/math]. Plot the point (k, x_q).

For example, there are 10 claims in Figure 1. Letting k = 5 we see that the cumulative claim count column goes directly from 2 to 6, which means q = 3. This means we plot the point (5, x₃) = (5, 3) on the Lee diagram. Since we're dealing with discrete data we get a step function, with the next value being (6, 5) as the next claim size is 5 and when we reach that we've incurred six claims.

Insert LeeDiagramsFig2.png

Notice in Figure 2 above that the green shaded area is a rectangle of height 3 and width 4. This corresponds to the incremental number of claims for claim size three. So an easier way of remembering to draw a Lee diagram is to order the claims by size and then draw successive rectangles whose height is the claim size and width is the number of claims of that size. Lastly, observe the area of the green shaded rectangle is the same as the total cost of all claims of size 3.

Figure 3 below generalizes this idea to the continuous case where a small change in cumulative probability, dF(x), corresponds to losses of size between x and x + δx.