Cummins page 29 - What is the loss?

[great3981]

On page 29 of Cummins, there is a preliminary graph of the high risk and low risk demand curves, the optimal policy for each if the company can classify, and the pooled policy when the company cannot classify. The paper then says that the high risks will purchase full coverage and the low risks will cut back on coverage and thus a loss will occur.

From the high risks, the company collects a lower premium than it pays out in losses and suffers a loss denoted by the dot-shaded on the graph. From the low risks, the company collects a higher premium than it pays out in losses and makes a gain denoted in the stripe-shaded area of the graph. Overall, the loss is larger than the gain.

My question is this: what is the significance of striped area on the outside of the demand curve? The graph shades the area between the quantity demanded by low risks under classification, the pooled rate, and the demand curve of the low risks. Is this really extra profit made off the low risk policies or is it a mistake that this section is shaded?

[Sen.]

I think this area is meaningless. Not sure why it is shaded.

[kingston]

What mathematical proofs are excluded from the syllabus? Are they referring to the mathematical formulas or the derivatives they calculate? For example is equation 3.2 excluded, or is the derivation excluded or both?

[ReserveRage]

I haven't seen one formal proof in the entire paper. I'm hoping by "proofs" they mean all these unnecessarily complicated formulas.

[AnActualActuary]

Originally Posted by great3981

On page 29 of Cummins, there is a preliminary graph of the high risk and low risk demand curves, the optimal policy for each if the company can classify, and the pooled policy when the company cannot classify. The paper then says that the high risks will purchase full coverage and the low risks will cut back on coverage and thus a loss will occur.

From the high risks, the company collects a lower premium than it pays out in losses and suffers a loss denoted by the dot-shaded on the graph. From the low risks, the company collects a higher premium than it pays out in losses and makes a gain denoted in the stripe-shaded area of the graph. Overall, the loss is larger than the gain.

My question is this: what is the significance of striped area on the outside of the demand curve? The graph shades the area between the quantity demanded by low risks under classification, the pooled rate, and the demand curve of the low risks. Is this really extra profit made off the low risk policies or is it a mistake that this section is shaded?

Yes it is profits. It is saying if the low risk indifference curve rises above the pooling fair premium line, a company will be able to offer a policy below the low risk indifference curve and above the *pooled* fair premium line. This is good for both parties, as a policy is being offered below the low risk difference curve, which makes consumers happy, yet it is also above the *pooled* fair premium line, which allows the company to profit no matter who they sell to (since it is a *pooled* fair premium line).

Note however, once other companies learn of this profitable policy, they will start to offer it as well. This will cause competetion, and force the policy offering to So.

Make sense?

[great3981]

Originally Posted by AnActualActuary

Yes it is profits. It is saying if the low risk indifference curve rises above the pooling fair premium line, a company will be able to offer a policy below the low risk indifference curve and above the *pooled* fair premium line. This is good for both parties, as a policy is being offered below the low risk difference curve, which makes consumers happy, yet it is also above the *pooled* fair premium line, which allows the company to profit no matter who they sell to (since it is a *pooled* fair premium line).

Note however, once other companies learn of this profitable policy, they will start to offer it as well. This will cause competetion, and force the policy offering to So.

Make sense?

I am pretty sure that you refer to a graph later on in the paper, perhaps page 39 or 42.

As for the graph on page 29, I conclude that this area is shaded in error. The low risks have premiums of the mean premium × low quantity but have losses at low premium × low quantity. The shaded region between the two rates times the low quantitiy is the gain from low insureds. This area is smaller than the loss from high insureds, and thus there is a loss at the pooled rate.

[frank_exams]

Originally Posted by great3981

My question is this: what is the significance of striped area on the outside of the demand curve? The graph shades the area between the quantity demanded by low risks under classification, the pooled rate, and the demand curve of the low risks. Is this really extra profit made off the low risk policies or is it a mistake that this section is shaded?

It's probably a mistake. Although perhaps they meant to shade it differently: on p.30 they go on to say, if the low-risks buy full coverage, Q*, at the pooled rate then the plan would be financially sound. If this were the case, that rectangular region around the curve would be shaded in.

Originally Posted by ReserveRage

I haven't seen one formal proof in the entire paper. I'm hoping by "proofs" they mean all these unnecessarily complicated formulas.

Hey, RR, sorry to hear you're sitting this one out, but I feel ya. I took 2007 off (well, except for exam 7, which was brutal).

Yeah, that's somewhat problematic. In fact, there are some questionable statements like the one on p.30 about this result generalizing to the case where low- and high-risks are not equal in the population.

Take a look at the graph on p. 29 (Fig 3-1) again. Imagine now that there are an immense amount of high risks in comparison to the low risks. This would shift the pooled rate up and Q_H(p) to the right (or Q_L(p) to the left, depending on how you want to think about it).

Now I ask: Is it possible for the diagonally-shaded rectangle to be larger than the dot-shaded one? They assert no (by saying the result generalizes), but it's not clear to me from just using the graph.

Best,
Frank

[think888]

Originally Posted by frank_exams

... there are some questionable statements like the one on p.30 about this result generalizing to the case where low- and high-risks are not equal in the population.

Take a look at the graph on p. 29 (Fig 3-1) again. Imagine now that there are an immense amount of high risks in comparison to the low risks. This would shift the pooled rate up and Q_H(p) to the right (or Q_L(p) to the left, depending on how you want to think about it).

Now I ask: Is it possible for the diagonally-shaded rectangle to be larger than the dot-shaded one? They assert no (by saying the result generalizes), but it's not clear to me from just using the graph.

Not clear to me, either. I would think you would need to specify how Q_L and Q_H actually depend on the distribution. See image below.

[Sen.]

Originally Posted by think888

Not clear to me, either. I would think you would need to specify how Q_L and Q_H actually depend on the distribution. See image below.

Nice graph, think. For the graph on p29, the author tends to draw theta bar as the mid point b/w theta H and theta L. But later in the paper, theta bar is defined as the weighted average of high and low rates. This proves that Frank's concern is legitimate indeed. And think's graph shows it visually.