Course 3 November 2000 #39

[Still Confused]

For a new car dealer commencing business:

1. The dealer has neither assets nor liabilities
2. In exchange for continuous payments at the rate of expected sales, the car dealer receives cars on demand from the manufacturer
3. Car sales occur in accordance with a compound Poisson process at the rate of 12 cars per month.
4. The price of each car sold is either 20000 or 30000 with equal probability.
5. The dealer is in a positive position when cumulative sales exceed cumulative payments to the car manufacturer.

Calculate the probability that the dealer will ever be in a positive position.

Now, the answer is 0, because the relative security loading, theta, is 0. Therefore, the probability of ruin = 1 and the probability of success = 0.

I understand that this is the long term answer. But, couldn't the dealer sell more than 12 cars of $30,000 each the first month? He would then be in a positive position and the probability associated with this is not 0.

Unless I am missing something, it seems that the solution is not consistent with the question.

[Macroman]

The answer given at the SOA site and also at Klein's HTP site is E 1.0. This would mean that the company is certain to be in a positive position at some point.

If this were a traditional ruin problem I would say that ruin occurred at t=0 because there was no surplus. With such a problem the company ceases operation at the point where surplus = 0. Our car dealer in this example is being allowed to operate in a deficit position, unusual for this type of problem.

[Still Confused]

I went to the HTP site. I guess what he is saying it that this is the opposite of a ruin problem so Prob[Ruin] in an insurance problem = Prob[being in a positive position in this problem]

Thanks, I should have just gone to his site in the beginning.

[boognish]

What if the dealer never sells a car, yet pays the manufacturer? That would cause ruin, thus never put them in a positive position right?

[Still Confused]

Klein's solution starts out with 'this is a tricky question'.

It seems to me, that if the dealer doesn't sell a car immediately, he is in ruin.

[boognish]

thats what i say! any clarification?

[Macroman]

That's why Klein does not treat this as a ruin problem. It's kind of the opposite of a ruin problem...asking whether the company would ever be able to report that it is profitable...even slightly...

[tesla_styx]

If the dealer doesnt sell his car immediately is irrelevant... the question asks for the probability that at some point on an infinite time horizon, the dealer has sold more than he has paid. Since the dealer pays a Premium equal to the expected value of Losses (loss to the owner - parent company, not the dealer), security loading equals 0. therefore the probability of ruin for the parent company is 1. The question does not ask about the dealer being in ruin... in fact ruin for the dealer is undefined. So even if he pays out 1 million dollars b4 selling a single car, it doesn't matter... he in a sense has an infinite supply of income.

[Macroman]

how about an infinite credit line?

[Agtuary]

The problem never says anything about the car dealer not being able to operate in a negative position.

If you want, you can look at it from the manufacturers point of view. It receives monewy continously and must pay out cars. Assuming it has no other source of revenue or expense, what is the probability it will ever have a negative profit?

[Toonces]

The dealer starts with no assets, so essentially, if you thought of this as a typical ruin problem, the dealer would be ruined the second he opened shop, since he'd have an immediate payment to make. This wouldn't make for a very interesting problem, nor would part two of the problem be relevant, since it would be impossible not to be immediately ruined. Therefore, you have to assume that this dealer can operate from a negative position. You can also assume that, since the question asks the probability that the owner is "ever" in a positive position, implying that there will be times that owner will not be in a positive position.

[boognish]

I can't believe they asked this question!!

You start assuming things it only makes an ass out of u and me!!

[Agtuary]

I don't see anything wrong with this question. It is certainly realistic to start in a negative position, many businesses are started with loans, hence in debt. It also tests your understanding of a particular concept. All problems would be easy if they consitently test concepts in the same manner year after year, but you don't have to understand the underlying principles to get them correct. I know, that's how I passed high school.

If you are presented with a situation where the concept is viable, but you don't really know how to apply the concept because you never really learned it in the first place, then the exam process has taught you nothing. It becomes a worthless process with no meaning except to weed out the individuals who are unwilling to participate.

[boognish]

I agree with asking questions that test a deep understanding of the material.

I disagree with your assesment of this problem, but alas who cares. This problem is in the past, and we all know they won't ask something similar.

Just so you know you have contradicted yourself:

You state its not uncommon for a business to operate in the red(i agree), but would any manufacture of cars set up a business of this nature (other then Saucy Wench automotive)

Good Luck!!

[Macroman]

Certainly no dealer would. This is a zero profit model.

I'm not sure that actual car sales arrangements are all that different, but I think there must be (1.) a better link to sales (2.) some provision that would introduce profits for the car dealer.

[Agtuary]

I'm sorry if you think I contradicted myself because I said this problem was realistic. I should have stated that the idea that a business can operate in the red for a short period of time is realistic. This specific point is what people get stuck on when attempting this problem. In this cash the incorrect assumption that appears to be common for this thread is that negative cash implies ruin and therefore the car dealer goes out of business.

I am trying to stress the fact that I do not think this problem was unfair. More difficult than other problems, maybe, but I think it was a fair test of the concept.

[Macroman]

Really, I don't think there was anything difficult about this problem at all. It requires the exam taker to recognize a simple fact. This problem is not likely to reappear.

We have a concept of ruin to be used on course #3. This does not mean that it will be used on every problem. What the question actually asks is the complement to the ruin problem.

The important thing here is to read the problem. I think ruin problems will either (1) clearly state some conditions under which the company can no longer operate or (2) say that the company is an insurance company or insurance fund in which case ruin would occur if there is no positive surplus (implied). If the company is a car dealer or a long distance company or pizza shop ruin would not be a factor unless specifically stated otherwise.