Actuary Certification Actuary Certification Life Contingencies 2 — Questions and Answers
Question 1: A life annuity-due of 1 per year pays at the beginning of each year while a life (x) survives. Its APV is denoted:
- ax
- äx (Correct answer)
- Ax
- ex
Correct answer: äx
äx (a-umlaut-x) is the actuarial present value of a life annuity-due, with payments at the start of each year the life survives.
Question 2: The complete expectation of life e°x is defined as:
- E[K(x)]
- E[T(x)] (Correct answer)
- 1/μ(x)
- lx / l0
Correct answer: E[T(x)]
The complete expectation of life e°x equals E[T(x)], the expected value of the complete future lifetime T(x).
Question 3: For a select-and-ultimate life table, the select period represents:
- The period after which all underwriting effects have worn off
- The first years after underwriting when mortality differs from the ultimate table (Correct answer)
- The age at which the life was selected
- The duration of the insurance policy
Correct answer: The first years after underwriting when mortality differs from the ultimate table
The select period is the duration immediately following underwriting during which recently selected lives have different (usually lower) mortality than the ultimate rates.
Question 4: The variance of the present value of a whole life insurance Āx can be expressed using the second moment as:
- ²Āx − (Āx)² (Correct answer)
- ²Āx − Āx
- (Āx)² − ²Āx
- Āx(1 − Āx)
Correct answer: ²Āx − (Āx)²
Var(Z) = E[Z²] − (E[Z])² = ²Āx − (Āx)², where ²Āx is evaluated at double the force of interest.
Question 5: A term life insurance pays a death benefit only if the insured dies within n years. Its APV for (x) is written as:
- Āx:n|
- n|Ä€x
- A¹x:n| (Correct answer)
- äx:n|
Correct answer: A¹x:n|
A¹x:n| (with superscript 1 over x) denotes the n-year term insurance APV, where the superscript 1 indicates the benefit is paid only on the life's death within n years.
Question 6: Which relationship correctly connects the whole life insurance APV and the endowment insurance APV?
- Āx = Āx:n| + n|Āx
- Āx = A¹x:n| + Ax:n|1
- Āx = A¹x:n| + nEx · Āx+n (Correct answer)
- Āx = äx · δ
Correct answer: Āx = A¹x:n| + nEx · Āx+n
A whole life can be split into an n-year term (A¹x:n|) plus an n-year deferred whole life (nEx · Āx+n), which together cover all possible death times.
Actuary Certification Actuary Certification Life Contingencies 2
A life annuity-due of 1 per year pays at the beginning of each year while a life (x) survives.
Its APV is denoted: