Soa Exam Models For Life Contingencies Sample Questions Solutions

SOA Exam Models for Life Contingencies: Sample Questions and Solutions

The Society of Actuaries (SOA) Exam on Life Contingencies is a rigorous test requiring a deep understanding of actuarial mathematics and its applications to life insurance and pensions. This exam covers a wide range of topics, including life tables, survival models, actuarial present values, annuities, and insurance. This article provides a comprehensive overview of the exam, focusing on sample questions and detailed solutions to illustrate key concepts. Mastering these concepts is crucial for success. We will explore various question types, highlighting different problem-solving techniques and emphasizing the importance of understanding the underlying principles.

Understanding the SOA Exam Structure and Content

Before delving into sample questions, let's briefly review the structure and content of the SOA Life Contingencies exam. The exam typically consists of multiple-choice questions, testing your knowledge of various aspects of life contingencies. Key areas covered include:

1. Life Tables and Survival Models

This section focuses on understanding and interpreting life tables, calculating survival probabilities, and applying various survival models (like the Gompertz, Makeham, and Weibull models). You'll need to be proficient in calculating probabilities of survival, death, and other related events within specified timeframes.

Sample Question 1:

A life table shows that l_x = 1000 and l_{x+1} = 990. Calculate the probability that a life aged x will die within the next year.

Solution:

The probability of death within the next year is given by:

q_x = (l_x - l_{x+1}) / l_x = (1000 - 990) / 1000 = 0.01

Therefore, the probability that a life aged x will die within the next year is 1%.

2. Actuarial Present Values (APV)

This is a core concept in life contingencies. You will be tested on your ability to calculate the present value of various life insurance and annuity contracts, considering the time value of money and the probability of survival and death.

Sample Question 2:

A whole life annuity-due pays $100 per year, starting immediately, to a life aged 65. The effective annual interest rate is 5%, and the life table shows that e_65 = 15. Approximate the actuarial present value (APV) of this annuity using the formula based on the curtate expectation of life.

Solution:

The approximate APV of a whole life annuity-due is given by:

APV ≈ 100 * (1 + 0.05 * e_65) = 100 * (1 + 0.05 * 15) = 100 * 1.75 = $1750

3. Annuities

This section covers various types of annuities, including whole life annuities, temporary annuities, annuities-due, annuities-immediate, and deferred annuities. You will be expected to calculate their present values under different scenarios and interest rates.

Sample Question 3:

Calculate the present value of a 10-year temporary life annuity-immediate of $100 per year for a life aged 40, given an effective annual interest rate of 6% and the following probabilities of survival:

_10p_40 = 0.90

Solution:

The present value of a temporary life annuity-immediate is calculated as:

PV = 100 * v * _10p_40 * a_{10|} = 100 * (1/1.06) * 0.90 * Σ (1/1.06)^k for k=0 to 9

We'd need to calculate the present value of an annuity-immediate for 10 years at 6%, then multiply by _10p_40. This requires using the formula for the present value of an annuity-immediate: a_{n} = (1 - v^n) / i, where i is the effective interest rate.

4. Insurance

This section examines various life insurance contracts, including term insurance, whole life insurance, and endowment insurance. You will need to calculate their APVs and understand their characteristics.

Sample Question 4:

What is the actuarial present value of a 20-year term insurance of $100,000 for a life aged 35, given an effective annual interest rate of 4% and the probability of death within 20 years given by _20q_35 = 0.15?

Solution:

The APV of a 20-year term insurance is given by:

APV = 100000 * v^{k} * _k q_{35}, where k is the year of death. In this case, we don't have the distribution of deaths throughout the 20 years, so we'll use the probability of death within the 20 years and approximate the APV by discounting the death benefit.

APV ≈ 100000 * (1/1.04)^10 * _20q_35 ≈ 100000 * 0.675564 * 0.15 ≈ $10133.46 (Note: This is a simplification, a more precise calculation would require the probabilities of death in each of the 20 years).

Advanced Topics and Problem-Solving Techniques

The exam also covers more advanced topics, requiring a deeper understanding of actuarial mathematics and problem-solving skills. These include:

• Multiple-decrement models: These models deal with situations where multiple causes of decrement (e.g., death, withdrawal, disability) exist simultaneously.
• Multiple life functions: This involves analyzing probabilities and present values related to the survival or death of multiple lives.
• Continuous annuities and insurances: These calculations utilize continuous interest and force of mortality.
• Select and ultimate life tables: These tables account for the fact that mortality rates may vary depending on how long a person has been insured.

Sample Question 5 (Multiple-Decrement):

A group of 1000 individuals are subject to two decrements: death (d) and withdrawal (w). The probabilities are: q_x^d = 0.02, q_x^w = 0.05. Assuming the decrements are independent, what is the probability that an individual will die within the year?

Solution:

The probability that an individual will die within the year, given two independent decrements, is approximately:

q_x^d * (1-q_x^w) = 0.02 * (1-0.05) = 0.02 * 0.95 = 0.019

Tips for Exam Success

Preparing for the SOA Life Contingencies exam requires a dedicated and structured approach. Here are some tips to maximize your chances of success:

• Thorough understanding of concepts: Memorization alone won't suffice. Focus on understanding the underlying principles and logic behind the formulas.
• Practice, practice, practice: Solve numerous problems from various sources, including past exams and practice materials. Analyze your mistakes and identify areas where you need improvement.
• Use a variety of resources: Supplement your textbook with additional resources, such as study manuals and online materials.
• Develop problem-solving strategies: Learn to approach problems systematically and efficiently. Break down complex problems into smaller, manageable steps.
• Time management: Practice solving problems under timed conditions to improve your speed and efficiency.

This comprehensive overview provides a foundation for understanding the SOA exam's scope. Remember that consistent study, problem-solving, and a deep understanding of the concepts are key to success. The sample questions and solutions illustrate several crucial areas, but thorough preparation using a broad range of practice questions is essential. Good luck!