Stanard

(Summary of arguments in first part of the paper) The paper says:

"It would be great if we could find a reserving technique that always gave the exact answer." (What is the "exact answer"? Answer: A reserving technique that gives the exact answer is a reserve indication that exactly equals the future unpaid losses.)

But we can’t find such a technique. Question: Why not? Answer: Loss emergence is a random process. (That is, there is some randomness in the process, not that the whole technique is a random draw.)

Compare the lines from page 126, first full paragraph:

A perfect reserve estimation procedure for accident year "a" would be a function R_a such that R_a(K) = E(L_a). However, K is also a random variable, so fulfilling this condition is not possible, except by change.

So we change our goal. We want to find a reserving procedure that is unbiased. What does "unbiased" mean? Answer: A biased reserving procedure is one whose average indication is too high or too low.

Be careful. You might say: "Okay, I understand this. An example of a biased reserving procedure is one that 60% of the time gives too high an indication and 40% of the time gives too low an indication." No, that’s not correct. You substituted "median" for mean." In fact, Stanard notes this explicitly, and he remarks on page 132:

An interesting result from the older version of the model is that the median prediction error for method 1 was usually negative – that means that in over half of the cases method 1 under-predicted the actual (simulated) results, but a few cases of large over-predictions made the mean prediction error (the bias) positive.

Continue with the next line on page 126:

We can, however, hope that R_a(K) is an unbiased indicator of E(L_a), that is, that E(R_a(K)) = E(L_a).

In fact, the general assumption among actuaries is that our reserving procedures are unbiased. That is not the case, says Stanard.

There is also a second desirable quality of reserving techniques. We want a reserving technique that has as little variability as possible. Question: What do we mean by this? Answer: Suppose we have two reserving procedures, both of which are unbiased. For each reserving procedure, we look at the difference between the reserve indication and the actual unpaid losses (which we know with hindsight many years later). We call this difference the "error." One reserving procedure has an average error equal to 10% of the actual unpaid losses. The other reserving procedure has an average error equal to 5% of the actual unpaid losses. We would prefer the second procedure, since it has a smaller average error.

Stanard uses a slightly different technique, since he minimizes the average squared error. This makes sense. For an unbiased procedure, the average error is 0%. What we would really like is the average absolute error. But the average absolute error is used less often than the average squared error (which also makes all the errors positive), for reasons of mathematical tractability. Stanard says:

We would also like R(K) to be close to E(L), on the average. One common way of expressing this is to minimize E((R(K) – E(L))²), the mean square error, which for an unbiased estimator is equivalent to minimizing Var(R(K)).

How should we proceed? You might say: "Let’s find the reserving procedure that is unbiased and that has minimum variance." This is an operations research "programming" problem: We want to minimize the variance with the constraint that the procedure be unbiased.

We can’t do this. There are an infinite number of possible reserving procedures, and there are an infinite number of possible loss scenarios. Instead, let’s pick four common reserving procedures and see how these four work.

The four procedures are (1) chain ladder, (2) Bornhuetter-Ferguson, (3) Cape Cod, and (4) Bühlmann. All these methods have "paid loss" versions and "reported loss" versions. We use the "reported loss" version.

We’ve got a problem. The chain ladder method can be used in a pure formula version, with no subjective judgment. But for Bornhuetter-Ferguson, we must choose an expected loss ratio. We want to eliminate subjective judgment from the simulation test. So we use a modification of Bornhuetter-Ferguson that derives the expected loss ratio from the data.

[Technical note: the chain ladder method does not give any indication for the most recent accident year. This is mentioned in the text and it is reflected in the exhibits.]

Armed with our four reserving procedures, we apply them to our company’s loss triangles, get indications, wait ten years, and compare the indications with the actual unpaid losses.

Well, this is not going to work. A minor problem is that we have to wait ten years. The real problem is that loss development has a large random element. A result in one individual case doesn’t tell us much about the expected result.

Okay, armed with our four reserving procedures, we visit the neighboring 1,000 companies and apply them to loss triangles. We get indications and we wait ten years . . . Oh, forget it. This is not going to work either.

Rather, we must simulate. This paper is a simulation test, so let’s be very clear about the simulation. Each iteration of the simulation must show two things:

1. It must show the ultimate losses.

2. It must show the loss triangle at the present time.

We repeat: the loss triangle at the present time is the reported loss triangle. We are using reported loss methods, not paid loss methods.

Let’s start with A, the ultimate losses. We must simulate ultimate losses. Well, we have many accident years. Do we simulate them as a group, or do we simulate them separately?

Answer: We simulate separately for each accident year.

Question: There are relationships between the losses of different accident years. If we had $10 million of losses in accident year X, we expect about $10 million of losses in accident year X+1, adjusted for monetary inflation and changes in exposure.

Stanard says: Yes, I know that. We are going to do all this, with various scenarios. For the moment, to explain what we are doing, let us assume no changes in exposure and no monetary inflation. We have the same expected ultimate losses in each accident year.

Remember: this is just heuristic. Stanard does show the complex scenarios, with monetary inflation and several other items. The explanation here focuses on the second scenario, which Stanard calls the "base scenario." Once you understand the base scenario, the others follow smoothly.

How should we simulate ultimate losses? Answer: We simulate separately loss frequency (claim count) and loss severity. This part we know from the early exams (modeling and simulation and loss distributions). We simulate loss frequency in individual insureds by a Poisson distribution. For a group of insureds, we use a negative binomial distribution.

True, says Stanard, but that’s too complex, and we want a simple paper. Instead of a negative binomial distribution, let’s use a normal distribution. Look at footnote 9:

The normal distribution was chosen as a good approximation for the negative binomial, which is more difficult to simulate. Also, N was restricted to be greater than zero.

What type of normal distribution do we use? Answer: We use a normal distribution with a mean of 40 claims and a variance of 60 claims-squared. Question: This seems strange. At most companies, each accident year has 10,000 claims, not 40 claims. Answer: Stanard is not concerned with company reserving. Stanard is CEO of a major reinsurer. He is concerned about individual risk rating for large insureds. Every so often, he talks about rating. He is using the reserve estimate to individually rate a large risk.

Okay, we have the simulated claim counts. Now we need simulated claim severities. We use a lognormal distribution; that’s standard. We choose a lognormal with a mean of $10,400 per claim and a variance of ($34,800)².

Question: Is that little "2" up there a footnote? Answer: No, variance is in "dollars-squared."

What about the loss cost trend? How do we adjust these numbers for an expected trend in average severity? Stanard does this, and he does this reasonably well. For now, let’s skip the trend piece. He uses a procedure by Butsic, which you will find confusing. It is similar to McClenahan’s procedure in his Exam 6 paper. That probably doesn’t help you. We come back to this at the seminar; let’s first understand the structure of this paper.

We simulate for all the accident years in our model. Question: How many accident years are there in the model? Answer: There are six accident years. See page 127, first line in section III: "The computer generates six accident years . . ."

Question: "Why do the summations go up to 4 or 5, not up to 6?" Answer: The summations start with accident year 0, not accident year 1. The last accident year is accident year 5.

"But many of the summations only go through accident year 4; how come?" Answer: the last accident year has 0 years of development so far. We can use Bornhuetter-Ferguson to estimate the expected losses for this year; we can’t use a chain ladder technique.

We simulate the ultimate losses for each accident year in each iteration. But that’s not sufficient; we also need the loss triangle at the present time. Now we have several more distributions to simulate. If we had to work out the mathematics, it would be a pain. But we don’t have to worry about the mathematics here; we just have to know how the loss triangles are simulated.

We must simulate three more items to get the loss triangles:

1. We already simulated claim counts for each accident year. For each claim, we must know when the claim occurred.

2. Given the date that a claim occurred, we must simulate whether it has been reported yet. We will use a distribution of "reporting lag"; that is, the time lag between the occurrence of the claim and the reporting of the claim.

3. For each claim that has been reported, we must simulate whether it has been paid yet. We use a distribution of "payment lag"; that is, the time lag between the reporting of the claim and the payment of the claim.

Now we have enough to start the model. Let us review how we do the simulation. First we simulate the number of claims in each accident year. Suppose we have 40 claims for a given accident year. For each claim we simulate the ultimate severity. That tells us the ultimate losses for all our claims.

To get the reported losses at the present time, we first simulate – for each claim – the month in which it occurs. We assume that claims occur evenly over the year, so we use a uniform distribution from month 0 (January) to month 11 (December).

We now know when the claim occurred; we don’t know if it has been reported yet. So we simulate the lag between the occurrence of the claim and the reporting of the claim. We use an exponential distribution to model the reporting lag. The exponential distribution has a mean of 18 months; that is, on average, claims are reported 18 months after their occurrence. If you want to know more about exponential distributions for claim reporting, look at McClenahan’s paper.

We now know if the claim has been reported. Be careful here. Suppose the valuation date for our reserve analysis is December 31, 20XX. If the accident year is 20XX – 1, and the month of occurrence is September, and the simulated reporting lag is 17 months, this claim has not been reported by the valuation date. If the accident year is 20XX – 2, and everything else is the same, then the claim has been reported by the valuation date.

If the claim has been reported already, we must also know whether it has been paid yet. The claims that we are concerned with are those that have been reported but have not yet been paid. Those claims form our reserve indication. We use a similar procedure. We simulate the payment lag, or the time lag from the reporting month to the payment month. We use an exponential distribution with a mean of 12 months.

Now we are all done. We have simulated the ultimate losses and we have simulated the loss triangle at the current valuation date.

We have two problems with this simple scenario. A lot of reserving errors are caused by monetary inflation of claim costs. We have ignored inflation so far. Fine; this overview is simplistic. You will have to deal with inflation, which complicates the model in various ways.

The second problem is case reserve accuracy. The reserving actuary has two tasks. First, some claims have not yet emerged (true IBNR). We have dealt with this in the simulation model. Second, some case reserves are not accurate. We have not yet dealt with this in the simulation model.

So Stanard adds another variable to deal with case reserve accuracy. He calls it "V" (probably stands for "variability"). He discusses V on the top half of page 129. Because this section is tough, let’s get technical. Each reported loss falls into one of three categories, shown in the middle of page 129, called k_i(a,t).

We have the loss triangle for each iteration and we know the ultimate losses for that iteration. We apply the four reserving procedures to the loss triangle, and we compute the differences from the ultimate losses.

If you have checked the four reserving methods, you have noticed that none of them has a tail factor. You say: "It must be that this simulation does not any development past four years or so." But that’s not so. We are using exponential distributions for claim reporting lag and claim payment lag, and these go on forever.

Standard knows this. Stanard is a brilliant actuary and a very successful businessman. His paper is well written, and he covers all the major points. He says the following:

All the reserving procedures get me only to the end of the triangle period. So instead of looking at the difference between the reserve indication and the actual unpaid losses, we’ll look at the difference between the reserve indication and the new reported losses by the end of the triangle period.

That’s the English of it. Read Stanard’s words in the first full paragraph on page 132:

We would expect any rating technique based on known data to (on the average) under-predict by the expected amount of development between the most mature known data amount and ultimate E(K_a4 – L_a).

Translation: If we don’t use a tail, we under-reserve by the amount of tail development. Note that Stanard says: "rating technique" and"predict." He is doing individual risk rating, not large company reserving.

Therefore, each of the expected prediction errors has been adjusted by this amount, so the exhibits actually show E(R_a – L_a) – E(K_a4 – L_a) = E(R_a – K_a4).

Translation: The prediction errors in the exhibits are all reduced by the amount of the tail development.

That is, we do not expect the estimation techniques to be able to predict beyond the triangle.

Translation: None of the reserving procedures use a tail factor.