Contents
Exam Overview
- Duration: 3 hours
- Questions: 30 multiple-choice questions
- Topics: Life Insurance, Life Annuities, and Net Premium Reserves
Survival Models & Life Tables
Survival Function
- S(x) = 1 – F(x)
- S(x) = T_x/l_0
- Where T_x is the number of survivors to age x
Force of Mortality
- μ_x = -[d/dx(ln(S(x)))]
- μ_x = f(x)/S(x)
- μ_x = -S'(x)/S(x)
Life Table Functions
- l_x = number of lives at age x
- d_x = number of deaths between age x and x+1
- p_x = l_(x+1)/l_x = probability of survival from age x to x+1
- q_x = d_x/l_x = probability of death between age x and x+1
- m_x = d_x/L_x = central death rate
- e_x = T_x/l_x = complete expectation of life at age x
Fractional Age Assumptions
- UDD (Uniform Distribution of Deaths):
q_(x+t) = t × q_x - CFM (Constant Force of Mortality):
μ_(x+t) = μ_x - Balducci Assumption:
p_(x+t) = (1-t × q_x)/(1-t)
Present Value Random Variables
Insurance Benefits
Whole Life Insurance:
- A_x = E[v^K]
- Where K is curtate-future-lifetime
- v is discount factor (1/(1+i))
Term Life Insurance:
- A^1_(x:n) = E[v^K] for K ≤ n
- Zero otherwise
Pure Endowment:
- A_(x:n) = v^n × n_p_x
Endowment Insurance:
- A_(x:n) = A^1_(x:n) + A_(x:n)
Life Annuities
Whole Life Annuity-due:
- ä_x = Σ[k=0 to ∞] v^k × k_p_x
Term Annuity-due:
- ä_(x:n) = Σ[k=0 to n-1] v^k × k_p_x
Deferred Annuity-due:
- {m|}ä_x = v^m × m_p_x × ä(x+m)
Continuous Life Annuity:
- a̅_x = ∫[0 to ∞] v^t × t_p_x dt
Net Premium Reserves
Net Premium Calculations
Whole Life:
- P(A_x) = A_x/ä_x
Term Insurance:
- P(A^1_(x:n)) = A^1_(x:n)/ä_(x:n)
Endowment:
- P(A_(x:n)) = A_(x:n)/ä_(x:n)
Recursion Formulas
Prospective Reserve:
- tV = A(x+t) – P × ä_(x+t)
Retrospective Reserve:
- tV = P × s̈(x:t) – A^1_(x:t)
Premium Difference Reserve
- tV = A(x+t:n-t) – (P_1 × ä_(x+t:n-t))
Multiple Life Functions
Joint Life Status
- t_p_(xy) = t_p_x × t_p_y
- μ_(xy) = μ_x + μ_y
- T_(xy) = min(T_x, T_y)
Last Survivor Status
- t_p_(x̅y̅) = 1 – (1 – t_p_x)(1 – t_p_y)
- T_(x̅y̅) = max(T_x, T_y)
Contingent Benefits
- Insurance: A_(xy) = E[v^(T_xy)]
- Annuity: ä(xy) = E[ä(T_xy)]
Multiple Decrement Models
Service Table Functions
- p_x^(τ) = probability of survival in all modes
- q_x^(j) = probability of decrement by cause j
- μ_x^(j) = force of decrement for cause j
- Total force: μ_x = Σ[j] μ_x^(j)
Associated Single Decrement Tables
- q_x^(j’) = decrement probability in absence of other modes
- Relationship: q_x^(j) = q_x^(j’) × (1 – 0.5 × Σ[k≠j] q_x^(k’))
Select & Ultimate Life Tables
Select Functions
- [s]q_x = mortality rate for lives age x, selected s years ago
- [s]p_x = survival probability for lives age x, selected s years ago
- [s]l_x = number of lives surviving to age x, from selection at age x-s
Pension Mathematics
Service Tables
- w_x = withdrawal rate at age x
- r_x = retirement rate at age x
- d_x = death rate at age x
Benefit Accrual
- B(t) = k × t × S(t)
- Where k = benefit accrual rate
- S(t) = salary at time t
Important Relationships
- Life Table Relationships:
- p_x + q_x = 1
- d_x = l_x × q_x
- l_(x+1) = l_x × p_x
- Present Value Relationships:
- äx = 1 + v × p_x × ä(x+1)
- A_x = 1 – d × ä_x
- A_(x:n) = 1 – d × ä_(x:n)
- Reserve Relationships:
- tV = A(x+t) – P × ä_(x+t)
- tV = _t+1V × v × p(x+t) + A_(x+t) – P
Study Tips
- Formula Memorization:
- Focus on fundamental relationships
- Derive complex formulas from basic ones
- Practice with numerical examples
- Key Concepts to Master:
- Life table calculations
- Present value calculations
- Reserve calculations
- Multiple life statuses
- Multiple decrement theory
- Calculator Tips:
- Store common formulas in calculator memory
- Practice efficient data entry
- Double-check decimal places
Remember: This exam focuses heavily on understanding and applying these formulas in practical scenarios. Make sure you can not only recall formulas but also understand when and how to use them.
