# Actuarial Exams

by Vault Careers | March 31, 2009

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Schooling, in some ways, is simply preparatory to the real measure of an actuary's skills -- the exams for associateship and fellowship in the CAS or the SOA. The exams given and titles bestowed by these two professional organizations are the standard by which actuaries are measured in the U.S. and Canada. Being an associate or fellow means a great deal of respect amongst your peers, and top-notch salaries and job opportunities from employers.

At lower levels, of course, the exams are nothing less than your doorway into the career. You must pass at least one, and preferably the first two, actuarial exams before you go to work for a major company after college.

Exam 1/P

Exam 1/P, required by both the SOA and CAS, is a basic test of the candidate's knowledge of probability mathematics and its use in quantitative analysis of risk. It's a three-hour, multiple-choice test that nonetheless requires an undergraduate-level knowledge of probability theories, calculus and, at the very least, a basic idea of how it might be used in real-world business setting.

Candidates taking Exam 1/P will be expected to know general probability theories, univariate probability distributions and multivariate probability distributions, including the bivariate normal.

Most undergraduate courses in probability mathematics will adequately prepare students for this test. Beginning courses in risk management and statistics will also help, as will a basic knowledge of finance and/or business.

The SOA calls this Exam P, while the CAS labels it Exam 1.

Sample questions for Exam 1/P:

1) An auto insurance company has 10,000 policyholders. Each policyholder is classified as:

(i) young or old;
(ii) male or female; and
(iii) married or single.

Of these policyholders, 3,000 are young, 4,600 are male and 7,000 are married. The policyholders can also be classified as 1,320 young males, 3,010 married males and 1,400 young married persons. Finally, 600 of the policyholders are young married males. How many of the company's policyholders are young, female and single?

(A) 280
(B) 423
(C) 486
(D) 880
(E) 896

4) The monthly profit of Company I can be modeled by a continuous random variable with density function f. Company II has a monthly profit that is twice that of Company I. Determine the probability density function of the monthly profit of Company II.

(A) = f (x/2)
(B) f (x/2)
(C) 2 f (x/2)
(D) 2 f (x)
(E) 2 f (2x)

5) A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 2 percent, independent of all other tourists. Each ticket costs \$50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay \$100 (ticket cost + \$50 penalty) to the tourist. What is the expected revenue of the tour operator?

(A) \$935
(B) \$950
(C) \$967
(D) \$976
(E) \$985

Exam 2/FM

Exam 2/FM (the FM stands for financial mathematics) tests the candidate's knowledge of the application of mathematics in finance. Both actuarial societies require this exam, and most employers will want to see it successfully completed before hiring you. Before taking this test, you should know the fundamental concepts of financial mathematics and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in areas such as valuation, budgeting, reserves and other basic corporate accounting functions.

The three main areas tested in this exam are:

Definitions of key terms in financial mathematics. These include inflation, rates of interest (and variations thereof), term structure of interest rates, equivalent measures of interest, future value, current value, net present value, and stocks, bonds and other investment instruments. You should not only be able to define the term, but know how it applies to a given situation and provide the proper equations for it.

Key procedures of financial mathematics. This includes determining equivalent measures of interest, discounting, accumulating, determining yield rates, estimating the rate of return on a fund and amortization. These procedures will be applied to a variety of real-world examples, from corporate accounting to mutual fund returns.

Definitions of key terms of modern financial analysis. While likely more intuitive at this stage, you'll be required to have a basic grasp of yield curves, spot rates, forward rates, duration, convexity, immunization and short sales. You'll be asked to manage basic computations dealing with these concepts.

In other words, while Exam 1/P tested your grasp of advanced mathematics and probability, this is more of a test of your financial knowledge and your ability to apply your mathematical skills to the business world. Undergraduate courses in financial mathematics will be critical to your success on this test. Note that this test does not require you to apply probability mechanics to real-world business situations. That will come in a later course and exam.

The CAS calls this Exam 2, while the SOA uses the FM acronym.

Sample questions for Exam 2/FM

1) John borrows \$10,000 for 10 years at an annual effective interest rate of 10 percent. He can repay this loan using the amortization method with payments of \$1,627.45 at the end of each year. Instead, John repays the \$10,000 using a sinking fund that pays an annual effective interest rate of 14 percent. The deposits to the sinking fund are equal to \$1,627.45 minus the interest on the loan and are made at the end of each year for 10 years. Determine the balance in the sinking fund immediately after repayment of the loan.

(A) \$2,130
(B) \$2,180
(C) \$2,230
(D) \$2,300
(E) \$2,370

3) A perpetuity-immediate pays \$100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8 percent greater than the preceding payment. The annual effective rate of interest is 8 percent. Calculate X.

(A) \$54
(B) \$64
(C) \$74
(D) \$84
(E) \$94

4) David can receive one of the following two payment streams: (i) \$100 at time 0, \$200 at time n, and \$300 at time 2n (ii) \$600 at time 10.

At an annual effective interest rate of i, the present values of the two streams are equal. Given vn = 0.76, determine i.

(A) 3.5 percent
(B) 4.0 percent
(C) 4.5 percent
(D) 5.0 percent
(E) 5.5 percent

5) Jose and Chris each sell a different stock short for the same price. For each investor, the margin requirement is 50 percent and interest on the margin debt is paid at an annual effective rate of 6 percent. Each investor buys back his stock one year later at a price of \$760. Jose's stock paid a dividend of \$32 at the end of the year while Chris' stock paid no dividends. During the one-year period, Chris' return on the short sale is i, which is twice the return earned by Jose. Calculate i.

(A) 12 percent
(B) 16 percent
(C) 18 percent
(D) 20 percent
(E) 24 percent

Exam M

Exam M, administered solely by the SOA for SOA associateship, is a four-hour, multiple-choice exam on basic actuarial models. The exam tests both your knowledge of the theoretical basis of actuarial models as well as the application of those models to insurance and other financial risk. This assumes you have the knowledge of probability mathematics that allowed you to pass Exam 1/P. You'll need thorough understanding of calculus, probability and interest theory.

You'll have to understand the basic meaning of the term "model" in an actuarial context, as well as how and why models are used. You'll also need to know their limits and how the results of using these models can help to make business decisions.

As you enter this test, you'll be given a variety of tables to use in answering these questions. They'll include values for the standard normal distribution, illustrative life tables and abridged inventories of discrete and continuous probability distributions. These tables are commonly used in study materials obtainable from the SOA, but don't bring your own tables with you -- they'll be taken at the door. If you have your own shortcuts or notes that you commonly use with these tools, try to have them memorized, and then write them on the tables provided before you start answering questions.

Exam M will be the first real test of your actuarial science skills. You'll be expected to use survival and severity models, frequency models, compound (aggregate) models and life contingencies. You'll also need to apply these models to a variety of real-life insurance problems.

Sample questions from Exam M

1) A health plan implements an incentive to physicians to control hospitalization under which the physicians will be paid a bonus B equal to c times the amount by which total hospital claims are under 400 (0d c d1). The effect the incentive plan will have on underlying hospital claims is modeled by assuming that the new total hospital claims will follow a two-parameter Pareto distribution with 1 = 2 and 8 = 300. E(B) = 100. Calculate c.

(A) 0.44
(B) 0.48
(C) 0.52
(D) 0.56
(E) 0.60

2) You are given:

(i) Losses follow an exponential distribution with the same mean in all years.
(ii) The loss elimination ratio this year is 70 percent.
(iii) The ordinary deductible for the coming year is 4/3 of the current deductible.

Compute the loss elimination ratio for the coming year.

(A) 70 percent
(B) 75 percent
(C) 80 percent
(D) 85 percent
(E) 90 percent

3) Insurance agent Hunt N. Quotum will receive no annual bonus if the ratio of incurred losses to earned premiums for his book of business is 60 percent or more for the year. If the ratio is less than 60 percent, Quotum's bonus will be a percentage of his earned premium equal to 15 percent of the difference between his ratio and 60 percent. His annual earned premium is \$800,000. Incurred losses are distributed according to the Pareto distribution, with 8 = \$500,000 and 1 = 2. Calculate the expected value of Quotum's bonus.

(A) \$13,000
(B) \$17,000
(C) \$24,000
(D) \$29,000
(E) \$35,000

4) A fund is established by collecting an amount P from each of 100 independent lives at age 70. The fund will pay the following benefits:

" \$10, payable at the end of the year of death, for those who die before age 72, or
" P, payable at age 72, to those who survive.

You are given: Mortality follows the Illustrative Life Table. (i) i = 0.08

Calculate P, using the equivalence principle.
(A) \$2.33
(B) \$2.38
(C) \$3.02
(D) \$3.07
(E) 3.55

5) A dam is proposed for a river that is currently used for salmon breeding. You have modeled:

(i) For each hour the dam is opened, the number of salmon that will pass through and reach the breeding grounds has a distribution with mean 100 and variance 900.
(ii) The number of eggs released by each salmon has a distribution with mean of 5 and variance of 5.
(iii) The number of salmon going through the dam each hour it is open and the numbers of eggs released by the salmon are independent.

Using the normal approximation for the aggregate number of eggs released, determine the least number of whole hours the dam should be left open so the probability that 10,000 eggs will be released is greater than 95 percent.

(A) 20
(B) 23
(C) 26
(D) 29
(E) 32

Exam 4/C

Exam 4/C, the CAS and SOA names for the exam, respectively, covers the construction and evaluation of actuarial models. This is a four-hour, multiple-choice examination that tests your ability not to simply use established actuarial models (as covered in Exam M), but your skill in selecting or even creating your own models and applying them to real-world situations.

This exam will require you to analyze business data, find or create a suitable model to solve a problem, and provide some measures of confidence for decisions based on the modeled data. In other words, this test will have you put your money where your model is.

The exam will test your abilities in the following areas:

• Construction of empirical models using the Kaplan-Meier estimator, the Nelson-Aalen estimator and kernel density estimators.

• Construction and selection of parametric models, including the use of Bayesian procedures, and testing of those parametric models using the Kolmogorov-Smirnoff test, the Anderson-Darling test and the chi-square goodness-of-fit test.
• Full credibility assessments of selected models and data derived from them.
• Your understanding of interpolation and smoothing of data.
• Simulations and their roles in selection and testing of models and applications in the context of actuarial science.

Like Exam M, the examination staff will provide you with the necessary tables to take the test, including standard normal distribution, chi-square distribution, and abridged inventories of discrete and continuous probability distributions. Again, if there are shortcuts or notes you tend to use, memorize them and write them down on the tables provided, since you won't be able to bring your own tables to the exam.

Sample questions from Exam 4/C

1) You study five lives to estimate the time from the onset of a disease to death. The times to death are: 2 3 3 3 7

Using a triangular kernel with bandwidth 2, estimate the density function at 2.5.

(A) 8/40
(B) 12/40
(C) 14/40
(D) 16/40
(E) 17/40

3) You are given the following about 100 insurance policies in a study of time to policy surrender:

(i) The study was designed in such a way that for every policy that was surrendered, a new policy was added, meaning that the risk set, rj , is always equal to 100.
(ii) Policies are surrendered only at the end of a policy year.
(iii) The number of policies surrendered at the end of each policy year was observed to be:

1 at the end of the first policy year
2 at the end of the second policy year
3 at the end of the third policy year
n at the end of the nth policy year

(iv) The Nelson-Aalen empirical estimate of the cumulative distribution function at time n, FF (n) , is 0.542.

What is the value of n?(A) 8
(B) 9
(C) 10
(D) 11
(E) 12

4) You observe the following five ground-up claims from a data set that is truncated from below at 100:125 150 165 175 250

You fit a ground-up exponential distribution using maximum likelihood estimation.

Determine the mean of the fitted distribution.

(A) 73
(B) 100
(C) 125
(D) 156
(E) 173

5) A survival study gave (1.63, 2.55) as the 95 percent linear confidence interval for the cumulative hazard function H(t0). Calculate the 95 percent log-transformed confidence interval for H(t0).

(A) (0.49, 0.94)
(B) (0.84, 3.34)
(C) (1.58, 2.60)
(D) (1.68, 2.50)
(E) (1.68, 2.60)

Exam for Course 6 (Finance and Investments)

This SOA-only course is designed to test actuaries' knowledge of a variety of investment vehicles and how actuarial science may be applied to the field. The test is a five-hour multiple-choice and essay exam. Those taking this course should, in addition to completing SOA's Course 6 offerings, have a good working knowledge of capital markets, investments, applications of derivatives and the principles of portfolio management.

The objectives of this exam are deceptively simple: to make sure you know how risk and reward interact in the capital markets. Yet this isn't Jim Cramer's show -- you have to back this up with actuarial science while explaining the risks investors face.

Before you take this test, here's what you should know:

• Various investments, as well as their risks and returns. This includes not only stocks, bonds, real estate and guaranteed investment contracts (GICs), but also the risk and reward characteristics of the various marketplaces.

• The basic fundamentals of modern portfolio theory. This includes various pricing models, including the Markowitz Portfolio Selection model, and the three versions of the efficient market hypothesis.

• Options. The options trade is where actuaries can make a real impact. Be prepared to discuss puts and calls, swaps, forwards, interest rate caps and no-arbitrage pricing models. Also be prepared to determine the value of cash flow streams with embedded options.

• The application of interest rate risk management and effective duration. Be prepared to discuss immunization in this context, as well as compute an effective duration measure using option-adjusted spread analysis.

• Principles of asset liability management, and how they can be applied to portfolio construction and management for institutional investors. Be prepared for real-world applications.

As you can see, you need to really know the workings of our capital markets. If you've not really touched on this material before -- say, because you've spent the vast majority of your time in insurance -- make sure you take the course from the SOA and read all of the suggested materials. If there's training or hands-on experience you can get from your company, take it.

Of course, this isn't a necessary course for associateship or fellowship in the SOA, but it proves that you've expanded your skill set beyond "traditional" actuarial applications and into the financial markets. As traditional insurance companies diversify both their offerings and their business mix, this exam becomes more critical for the extension of your career into the upper echelons of the field.

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