The "Special Cases"
Brosius develops a "least-squares" loss reserving method. The most likely exam problems are determining bulk reserves by the least squares method. As an introduction, Brosius discusses three "special cases" of the least-squares method (pages 2-4):
the link ratio method (or the chain-ladder method),
the budgeted loss method (think of the "percent of premium" method in the Fisher and Lester paper), and
Bornhuetter-Ferguson method (or the expected loss method).
See question 19 from the Fall 1998 exam for an example of what you are expected to know. The question tests if you can connect the reasoning in the paper to a Bornhuetter-Ferguson example. In addition, Brosius connects these three methods to the three potential answers to Hugh White’s question in White’s discussion of the Bornhuetter-Ferguson paper (page 4 of Brosius).
Bornhuetter and Ferguson give several criteria which call for the use of their method (versus a chain-ladder method). Their criteria are vague. Brosius gives a more rigorous statistical test, which looks at variances and covariances; see below.
The Least-Squares Method
Know the basic formula for the least-squares method if you are given a set of empirical data. This is on page 3 of the Brosius paper, and it is a likely exam question. It has been asked already, and it will be asked in different guises every two or three years. (See, for example, question 42 from the Fall 1998 exam.)
Page 3 gives you the formula and the intuition. Page 11 develops the formula from a more rigorous statistical approach. Pages 16-18 show a full example, which proceeds accident year by accident year. An exam problem may be based on any of these illustrations.
Know the formulas for the "a" and "b" parameters. Sometimes the exam problem will give you the formulas; sometimes it won’t. To save yourself grief, memorize them. Know also the two extreme cases:
When "b" is zero (i.e., "x" and "y" are uncorrelated), the least-squares estimate is the budgeted loss estimate.
When "a" is zero, the least-squares estimate is the link ratio (chain ladder) estimate.
Some candidates ask: "What exactly are ‘x’ and ‘y’?" On your first pass through this reading, this may be unclear. Answer: In the introduction to the paper, "x" and "y" are the sets of figures in any two columns of the loss triangle. Thus, "x" could be incurred losses at 36 months of development and "y" could be incurred losses at 48 months of development.
As Brosius develops the more rigorous theory, he uses "x" as the reported losses (or reported claim counts) at a given duration and "y" as the ultimate losses (or ultimate claim counts). Brosius uses an accident year by accident year method, not a development period by development period method.
For instance, suppose that we have losses valued at December 31, 1999. For accident year 1995, losses valued at December 31, 1999, are losses valued at 60 months. For developing accident year 1995 losses, "x" is 60 months and "y" is ultimate. The empirical data for determining the "a" and "b" parameters are accident years 1994 and prior.
For accident year 1996, losses valued at December 31, 1999, are losses valued at 48 months. Thus, for developing accident year 1996 losses, "x" is 48 months and "y" is ultimate. The empirical data for determining the "a" and "b" parameters are accident years 1995 and prior.
The "a" and "b" parameters will change with each accident year in Brosius’s final example or for each development period in Brosius’s first (intuitive) example. This makes sense. For instance, at early development periods, a budgeted loss method may work better than a link ratio method, particularly for lines of business with slow loss reporting patterns. At later development periods, a link ratio method may work better. Brosius shows this result in his final example. He uses the same type of least-squares method for each accident year, but the "a" and "b" parameters change from accident year to accident year.
Review question 51 from the Fall 1996 exam and question 42 from the Fall 1998 exam. Do not think that if you can solve these past exam problem, you understand the Brosius paper. These are the simplest Brosius problems that one can devise. There are many other – more complex – potential exam problems that will be asked on the coming exams. See the Brosius study aid for discussions of these two exam problems.
Bayesian Analysis
On pages 5 and 6, Brosius presents "a simple model." The model is heuristic; it is "designed for simplicity and not realism." It is also perfect for examination problems. The examiner will change Brosius’s illustration ever so slightly – just enough that you must redo the Bayesian analysis.
Read carefully the "simple model" on pages 5-6. Now solve the practice problem below:
The number of claim incurred each year is a random variable Y which is 0, 1, or 2 with probabilities 25%, 50%, and 25%, respectively.
If there is a claim, there is a 50% chance that it will be reported by year end.
If "x" claims have been reported by year end, what is the expected number of claims outstanding?
The only change made to the illustration in the Brosius paper is the distribution of the random variable Y. We could also change the probability that a claim will be reported by year end. Make up other practice problems for yourself where you change the 50% chance to a 30% chance or an 80% chance.
The solution method is the same as in the Brosius paper, though there are more "cases." We illustrate the method with the case of P(Y=0 | X=0). In English, this means: "If no claims were reported by year end, what is the probability that there were no claims incurred?" You must still translate this into the expected number of claims outstanding, which Brosius terms R(x), but this is easy. If there are no claims reported and there were no claims incurred, then the number of claims outstanding is zero.
Bayes’s Theorem tells us that:
P(Y=0 | X=0)
= P(Y=0) P(X=0 | Y=0)
÷ [ P(Y=0) P(X=0 | Y=0) + P(Y=1) P(X=0 | Y=1) + P(Y=2) P(X=0 | Y=2) ]
= (25% * 100%) ÷ [ (25% * 100%) + (50% * 50%) + (25% * 25%) ]
= (0.25) ÷ (0.25 + 0.25 + 0.0625) = 44.4%.
You must continue the Bayesian analysis for the other cases as well. Do this in conjunction with the "simple model" in the paper, following Brosius’s procedure.
Continuous Distributions
Brosius gives more realistic models with continuous distributions: a Poisson-Binomial example on pages 6-8 and a Negative Binomial-Binomial example on pages 8 and 9. The Poisson-Binomial illustration is especially amenable to examination problems, since the general formulas are simple and the results are intuitive. Read pages 6 through 9 (both continuous examples), and then solve the following practice problem.
If “x” claims have been reported by year end, what the expected number of claims outstanding? |
We could use Bayesian analysis to solve this practice problem. We don’t have to, though, since Brosius has already solved the problem for us: both by an intuitive explanation and by a general formula.
For the intuitive solution, Brosius says (see page 7, "The Short Way"):
Break up the claims into two parts: (i) claims that have been incurred and that have been reported by year end and (ii) claims that have been incurred but that have not yet been reported by year end. The total number of claims incurred is Poisson distributed with mean 5. Of the claims incurred, 40% have been reported by year end. This implies that (i) the number of claims that have been incurred and that have been reported by year end is Poisson distributed with mean 2 and (ii) the number of claims that have been incurred but that have not yet been reported by year end is Poisson distributed with mean 3. The expected number of claims outstanding is therefore 3.
Once the intuition is clear to you, memorize the formulas in the middle of page 8. The practice problem asks for R(x), which equals "m(1–d)" = 5(1 – 40%) = 3.
Review Question 28 from the Fall 1996 Part 7 exam; see the Brosius study aid for a full discussion.
Bayesian Credibility and Development Formula 1
Memorize development formula 1 on the top of page 11, and understand the intuition behind the three cases (listed 1, 2, and 3, as the "answers" to White’s question). For the intuition, think of extreme cases; this will help you remember the principles; see the Brosius study aid for details.
Bühlmann Credibility and Development Formula 2
Memorize development formula 2 on page 13. This formula makes for wonderful exam questions. In fact, Brosius gives six illustrations of the use of this formula to solve loss reserving problems (the six bullet points on pages 13 and 14).
Note carefully what Brosius is doing, since the examiner will probably replicate one of Brosius’s examples. Prepare for an exam question of the following form:
The first half of the exam problem may give you a distribution and ask you to determine the credibility to be assigned to the ink ratio estimate and the credibility to be assigned to the budgeted loss estimate. For many distributions, the mathematics is manageable. The credibility equals VHM ÷ (VHM + EVPV). The examiner will probably give you a distribution for which Brosius tells you what VHM is and what EVPV is. [The examiner is not going to give you a Weibull distribution and ask you to calculate VHM and EVPV.] Know the formulas for VHM and for EVPV for the three distributions used by Brosius in these bullet points: the binomial, the Poisson, and the negative binomial.
The second half of the exam problem may give you a value for the parameter "d" and ask you to calculate L(x) or R(x). The calculation is from development formula 2, and Brosius shows numerous illustrations in his bullet points.