Exposure Curves

It doesn't tell you anything about total losses. The joy of loss curves is that IF you can reasonably estimate the total loss cost, you can use this to estimate losses for a layer.

It doesn't. (A diagonal exposure curve would mean that if you increase the limit by a certain proportion, the expected loss cost would also increase by the same proportion. But it is extremely unlikely that this would happen across the board, I mean at all points of the loss distribution.)

Let's say you have 3 losses, loss A = 3000, loss B = 2000 and loss C = 1000. The average of these (ie the expected loss) is 2000.
Now let's say you impose a limit of 2000. Loss A is now reduced to 2000 (because the insurer won't pay more than the limit). Loss B and loss C are still 2000 and 1000 respectively. So the average of these (ie the limited expected value, the LEV) is 1,667.
Now let's say you reduce the limit still further, to 1000. The insurer will only pay the first 1000 of each claim. So the limited expected value is now 1000.

In summary then, we have:

• limit 1000: LEV = 1000
• limit 2000: LEV = 1667
• no limit: EV = 2000

So you can see the LEV is an increasing function (or more specifically, a non-decreasing function) of the limit, and it is increasing at a decreasing rate. This means the exposure curve can't lie below the diagonal ... if it did, these two conditions wouldn't hold for all points on the curve.

Well, since the loss curve is a function of the LEV, it must also have the same shape.

 

Ah I see, the confusion comes from your terminology. The straight line doesn't "define total losses" as you said, rather a straight line exposure curve can only occur when the only type of claim you can get is a total loss claim.

Let's use an argument similar to the one I gave you earlier: Let's say a total loss is 3000, and again you have 3 losses, but this time they are all total losses. So of course loss A = 3000, loss B = 3000 and loss C = 3000. The average of these is 3000.
Now let's say you impose a limit of 2000. All losses are now reduced to 2000 and the LEV is therefore 2000. Similarly, if you reduce the limit still further, to 1000, the insurer will only pay 1000 on every claim and the limited expected value is now 1000.

In summary then, if all losses are total losses, we have:

• limit 1000: LEV = 1000
• limit 2000: LEV = 2000
• limit 3000: LEV = 3000

and again, the exposure curve will follow a similar shape, so we have a straight line diagonal.

So what's the moral of this story?

Well, if a high proportion of your loss cost is made up of very large losses (including some total losses, say), then you get a curve that is very close to the main diagonal. Conversely, the more attritional losses you have, the more the exposure curve will rise above the main diagonal. (Clearly then, a portfolio is more attractive if the exposure curve is very "curved".)

 

The curve gives you G(0.5). G(0.5) is about 0.95 for curve A and 0.75 for curve B. y is on the x-axis, and G(y) is on the y axis - maybe that's where you're getting confused?