New England Actuarial Seminars

Actuaries in Wonderland


The Cathedral

The small hand-written notice on the math department bulletin board caught Alice’s eye:

Best Career in America!

Prerequisites: Excellent Exam-Taking Skills and Unlimited Endurance

Alice had always done well on tests, and she loved her numbers. Besides, graduation was fast approaching, and she desperately needed a job. There was nothing else on the notice, except one additional line neatly printed at the bottom:

Further information available at 107½ West 110 Street

"Why, that’s not far from here at all," thought Alice, though she would have much preferred a telephone number.

It was a cold winter afternoon, and in her haste, Alice left her coat behind. Nonetheless, the brisk walk and the excitement of a new career kept her warm as she made her way toward the address on the notice.

At 107 West 110 Street, Alice found a small cafe, with a sign hanging inside the front door: "Open: 6:00 pm to 11:00 pm." A second-hand bookstore stood at 109 West 110 Street, with volumes on Musicology and Semiotics in the front window. In between was just the dull, brown brick of the adjoining wall.

"Oh darn!" said Alice. "There are no half numbers for addresses, of course, and that’s why there was no phone number. What a cruel joke!" It was very cold now, and Alice remembered that she had left her gloves in her coat pockets.

"I wish I could get a cup of hot chocolate now," thought Alice, as she shivered. The only warm air that she could feel was drifting up from a grate on the sidewalk in front of her. Alice sat down on the grate, and she put her head down on her knees.

As she looked down, she could make out the words in tiny block letters on the edge of the grate:

107½ West 110 Street – Come In

"How strange," thought Alice. She knew never to enter grates on city blocks, but it was so cold on the sidewalk, and the air drifting up from the grate made her fingers tingle. So she got up, pulled open the grate, and peered in.

There were stairs leading down, but there was no handrail and no lights. Alice descended carefully. Soon the steps began getting shorter, as if they were made for smaller people, and Alice had to walk ever more slowly to keep her balance.

It was a cylindrical stairwell, and as the stairs turned, there was less light from the outside. Alice found it harder and harder to keep her balance upon the ever narrowing stairs, until she slipped and –

Sliding softly down the now smooth stairwell, Alice kept descending slowly. A bright light far below illuminated the walls about her. Alice could make out pencils and rulers and calculators interspersed among books and papers and binders and manuals, all of which were strewn on shelves on the now illuminated walls, shimmering and flickering in the bright light. Along one wall were long thin eels in triplicate swaying in a languid, integral waltz. Along the opposite wall, in an enormous book, were scores of tiny Greek footsoldiers, holding hands in a never ending equation that was forever running off the edge of the page.

The light ahead seemed to shine ever more fiercely. Alice descended into a vast cathedral, with rows of devout worshippers, kneeling before a diamond ark of blinding light, and intensely murmuring in a low, uniform whisper: "Arate . . . arate . . . arate . . . arate . . ."

Principles

Beside the ark stood a majestic priest dressed in a white linen tunic and a flowing purple robe, holding aloft a silver baton, and leading the chant of the worshippers. "My," thought Alice, "this is indeed a wonderful career," and she advanced up the center aisle toward the ark.

"Young woman!" thundered the majestic priest, "Where do you think you are going?"

Alice stopped and stammered: "I saw a notice, Sir –"

"Have you no respect?" thundered the priest again, "How dare you approach the Ark of Sacred Principles?"

"Sacred Principles?" said Alice. "What are these sacred principles?"

"Do you not hear the chant of the candidates?" replied the majestic priest. He twirled the baton, and the murmur of the worshipers echoed through the cathedral: "Arate . . . arate . . . arate . . . . arate . . ." He turned back to Alice: "They are memorizing the sacred principles."

"I have never heard of these sacred principles," said Alice. "What are they about?"

"Very well," said the priest. "Tell me, what is a rate?"

Alice looked puzzled. "A rate? What do you mean?"

"Clearly an unprepared candidate," said the priest. "An insurance rate. What is a rate?"

"That’s easy," answered Alice. "A rate is the price you pay for insurance. Just last week, in fact, when I had to buy auto insurance, my agent looked up the rate in his manual –"

"Untutored lass!" thundered the priest. "No points – no points at all!"

"No points?" asked Alice.

"No points; not even partial credit. A rate is not that at all. If it were so easy, we would be no better than underwriters. As every learned actuary knows, a rate is an estimate of the expected value of future costs. That is the first sacred principle."

"Now that’s silly," said Alice. "The word ‘rate’ means ‘price per unit’ or ‘charge per unit.’ What if this price is not an estimate of the expected value of future costs? You can’t just take a word and make it mean whatever you want it to mean."

"When we use a word," said the majestic priest, "it means just what we choose it to mean – neither more nor less."

"The question is," said Alice, "whether you can cause simple words to have such convoluted meanings when every underwriter follows the dictionary definition."

"The question is," corrected the priest, "who is to be master – the actuary or the underwriter."

Alice felt reprimanded. "Still, that’s a great deal to make a simple word mean," she said in a hushed tone.

"When we make a word do a lot of work like that," huffed the majestic priest, "we always pay it extra."

Alice gazed around the cathedral, feeling the intensity of the young worshippers, their eyes focused on the papers in front of them.

"Is that the only sacred principle in the ark?" asked Alice thoughtfully. "It seems an awfully small principle for such a large and expensive ark."

"That is just the first of the principles. Tell me, for what does a rate provide?"

This all sounded a bit absurd to Alice, but the diamond ark sparkled so brilliantly that she figured it might be worthwhile to try out this strange jargon. So she smiled and said: "A rate provides for the expected value of future costs."

"No, no, no points at all!" cried the priest. "The principle is: A rate provides for all costs associated with the transfer of risk. So try again, young woman, as the third sacred principle teaches us, for what does a rate provide?"

"OK," said Alice. "OK. A rate provides for all costs associated with the transfer of risk."

"No points at all, none at all!" screamed the priest, as the white linen tunic heaved up and down. "A rate provides for the costs associated with an individual risk transfer."

"Isn’t that what I just said?" asked Alice.

"Not at all," said the priest. "You recited the second sacred principle, but you were asked about the third sacred principle. You left out the word ‘individual.’ No points at all."

"You mean every time you add a word you have a new sacred principle?" mused Alice.

"We don’t just add words," sneered the priest. "Each word of the Sacred Principles is chosen after months of deliberation by the Board of Exalted Standards."

"Aren’t there any principles that say something meaningful?" asked Alice. "People are always complaining that insurance rates are too high, or that insurance companies discriminate among consumers. Shouldn’t the sacred principles say something about that?"

"Of course," came the reply. "Tell me, young lady, when is a rate reasonable and not excessive, inadequate, or unfairly discriminatory?"

"Now that’s a fair question," exclaimed Alice happily. "A rate is excessive if it is too high, it is inadequate if it is too low, it is unfairly discriminatory if it sets unequal prices for consumers of similar risk –"

"No, no, not even partial credit!" thundered the priest. "You have learned nothing, nothing at all, even after you have been told the first three sacred principles. Rather, you must say: A rate is reasonable and not excessive, inadequate, or unfairly discriminatory if it is an actuarially sound estimate of the expected value of all future costs associated with an individual risk transfer."

"I think I’m beginning to understand," said Alice. "First you make the nouns mean gibberish. Then you take the adjectives and you make them mean the same gibberish. How much did you have to pay these words?"

"Careful, lass," replied the stone-faced priest, as a hush fell over the cathedral. "You may be tried and sentenced for that by the Arbitrary Tribunal for Condemnation and Despair."

Considerations

"This still seems sparse for such a spacious ark," said Alice. "Moreover, these principles seem awfully abstract. Don’t you have anything more specific?"

"Of course," said the priest. "The Ark of Sacred Principles contains eighteen essential considerations for ratemaking, covering every aspect of the actuary’s task. For instance, what consideration should be given to investment and other income?"

"Now that’s a serious question," thought Alice. "After all, people have argued about the proper consideration to be given to investment income for 75 years now, and there is still so much dissension." She wasn’t sure, though, how to begin her answer; there were so many ways to start.

"Well, here goes," thought Alice, and she began: "Investment and other income may be considered by means of a cash flow model, an internal rate of return model, or an options pricing model . . ."

"No, no, all wrong, all wrong," said the priest. "Try again."

"Of course," thought Alice, "the interest rate. One has to start with the interest rate." She began again: "One can define the assumed interest rate as the net investment yield of the company, a risk-free Treasury rate, or a risk-adjusted rate . . ."

"No, no, no credit at all," said the priest. "You are terribly ill-prepared for the exam. The correct answer is that The contribution of investment and other income should be considered. But we’ll give you another try. How should the effect of reinsurance arrangements be considered?"

"I think I’m beginning to catch on," thought Alice, and she said: "Consideration should be given to the effect of reinsurance arrangements."

"Perfect!" cried the priest. "Full credit, full credit! It seems that you have been studying after all."

"Well," said Alice, "it’s really not that hard once you get the hang of it."

"We’ll give you one more," said the priest. "What about classification plans?"

This one seemed a bit more difficult, but Alice decided to give it a try: "Consideration should be given to the effect of classification plans."

"No, no, that’s terrible, all wrong!" screamed the priest. "Rather the consideration is as follows: A properly defined classification plan enables the development of actuarially sound rates."

Alice thought hard, but she could make no sense of the sentence. She could find the verb of the sentence, the subject, and the object, but she had no idea what the sentence was trying to say. She didn’t want to offend the majestic priest, so she said only: "Was this consideration also made by the Board of Exalted Standards?"

"Of course," replied the priest. "After months of deliberation, compromise, and consensus."

"Compromise and consensus?" said Alice. "I guess this was all that they could agree on with regard to classification plans."

Credibility

Alice felt disappointed. The Ark still sparkled, but its luster had dimmed. The Sacred Principles seemed like word games without substance, like glass jewelry that shone from afar but was a penny trinket up close. Surely there was more to the Sacred Principles – something that inspired the devotion of the candidates.

It was getting late, and Alice wished to be back on campus by dinner time, so she asked directly: "What is the best part of these Sacred Principles?"

"The best part?" repeated the Priest. "Why, that must be the section on Credibility."

"Credibility?" asked Alice. "What is that?"

"Credibility is the crown jewel of actuarial theory," answered the Priest, "the most enduring contribution to our science. Credibility theory is what differentiates the simple underwriter from the pure actuary. Many years ago, before the revelations of Bühlmann, people thought of credibility as a measure of the predictive value that the actuary attaches to a particular body of data. Now we know that this was the error of our forefathers; after all, credibility is not an attribute of the body of data itself. Rather, Bayesian theory teaches us that credibility measures the relative predictive power of one set of data compared with another. A data set has no inherent credibility; its credibility depends on what else we have for making rates. If we have no other information, we may assign a high credibility value to the data set. If we have other information that is an equally good or a better predictor of future costs, we may assign a low credibility value to the first data set."

Alice was pleased. Here was good theory, reflecting the brilliance of the Ark. "Go on," she said.

"In those dark ages," continued the Priest, "people made meaningless statements about credibility such as ‘A group should be large enough to be statistically reliable.’ Our perspective now is that we take all the data that are relevant and we assign them credibility weights according to their relative accuracy in predicting future loss costs. We no longer speak about data sets being ‘statistically reliable,’ even if we knew what that meant."

"Are these candidates taught the proper understanding of credibility?" asked Alice admiringly.

"Of course," replied the Priest. "Early on in their careers, before they even approach the Sacred Principles, they must learn the fundamentals of Bayesian credibility."

"I would love to hear how the Sacred Principles summarize what you have told me," said Alice.

The Priest held up the silver baton, and a hush fell over the candidates. "From the Consideration on Credibility," he intoned: "Credibility is a measure of the predictive value that the actuary attaches to a particular body of data. . . . A group should be large enough to be statistically credible. . . ."

Alice was silent for a long moment; then she turned and walked slowly back to the stairwell leading up to the sidewalk and back to campus. As she ascended, she heard again the constant murmur of the candidates: "Arate . . . arate . . . arate . . . arate . . ."

"It is a beautiful Ark," she thought. "I wish them well."