Table of Contents #

1. Exam Overview
2. Survival Models & Life Tables
3. Present Value Random Variables
4. Net Premium Reserves
5. Multiple Life Functions
6. Multiple Decrement Models
7. Select & Ultimate Life Tables
8. Pension Mathematics
9. Key Formulas Quick Reference
10. Common Approximations
11. Study Strategy & Tips

Exam Overview #

Duration: 3 hours
Format: 30 multiple-choice questions
Core Topics: Life Insurance Mathematics, Life Annuities, and Net Premium Reserves
Passing Score: Typically 60-70% (varies by sitting)
Calculator: SOA-approved financial calculator required

Key Exam Focus Areas #

• 40% - Life insurance and annuity present values
• 35% - Net premium reserves and policy values
• 15% - Multiple life and multiple decrement models
• 10% - Select mortality and pension mathematics

Survival Models & Life Tables #

Fundamental Survival Functions #

Survival Function:

• $S(x) = 1 - F(x)$ (complement of distribution function)
• $S(x) = \frac{l_x}{l_0}$ (ratio of survivors)
• $S(x) = P(T_x > 0)$ (probability of survival beyond age x)

Probability Density Function:

• $f(x) = -S’(x) = \frac{d}{dx}[1-S(x)]$
• Represents the “density” of deaths at age x

Force of Mortality (Hazard Rate) #

Definition and Formulas:

• $\mu_x = -\frac{d}{dx}[\ln S(x)] = -\frac{S’(x)}{S(x)} = \frac{f(x)}{S(x)}$
• $\mu_x = \lim_{h \to 0} \frac{P(x < T \leq x+h | T > x)}{h}$

Key Properties:

• $S(x) = \exp\left(-\int_0^x \mu_t dt\right)$
• $_{t}p_x = \exp\left(-\int_x^{x+t} \mu_s ds\right)$

Life Table Functions #

Basic Functions:

• $l_x$ = number of lives surviving to exact age x
• $d_x$ = number of deaths between ages x and x+1
• $q_x = \frac{d_x}{l_x}$ = probability of death within one year
• $p_x = \frac{l_{x+1}}{l_x} = 1 - q_x$ = probability of survival for one year

Extended Probabilities:

• ${t}p_x = \frac{l{x+t}}{l_x}$ = probability of surviving t years from age x
• $_{t}q_x = 1 - _{t}p_x$ = probability of dying within t years from age x
• $_{t|s}q_x = _{t}p_x \cdot {s}q{x+t}$ = deferred mortality probability

Person-Years and Central Rates:

• $L_x = \int_0^1 l_{x+t} dt$ = person-years lived between ages x and x+1
• $T_x = \sum_{k=0}^{\infty} L_{x+k}$ = total person-years lived beyond age x
• $m_x = \frac{d_x}{L_x}$ = central death rate
• $e_x = \frac{T_x}{l_x}$ = complete expectation of life at age x

Fractional Age Assumptions #

Uniform Distribution of Deaths (UDD):

• $_{s}q_x = s \cdot q_x$ for $0 \leq s \leq 1$
• $l_{x+s} = l_x(1 - s \cdot q_x)$
• $\mu_{x+s} = \frac{q_x}{1 - s \cdot q_x}$

Constant Force of Mortality (CFM):

• $\mu_{x+s} = \mu_x = -\ln(p_x)$ for $0 \leq s \leq 1$
• $_{s}p_x = (p_x)^s$
• $_{s}q_x = 1 - (p_x)^s$

Balducci Assumption:

• $_{s}p_x = \frac{1 - s \cdot q_x}{1 - s \cdot q_x + s}$
• Commonly used for policy reserves between valuation dates

Present Value Random Variables #

Life Insurance Benefits #

Whole Life Insurance:

• Benefit of 1 paid at end of year of death
• $Z = v^{K+1}$ where K is curtate-future-lifetime
• $A_x = E[Z] = \sum_{k=0}^{\infty} v^{k+1} \cdot {k}p_x \cdot q{x+k}$
• Variance: $^2A_x - (A_x)^2$ where $^2A_x = \sum v^{2(k+1)} \cdot {k}p_x \cdot q{x+k}$

n-Year Term Life Insurance:

• $A^1_{x:\overline{n|}} = \sum_{k=0}^{n-1} v^{k+1} \cdot {k}p_x \cdot q{x+k}$
• Benefit paid only if death occurs within n years

Pure Endowment:

• Benefit of 1 paid only if survival to age x+n
• $A_{x:\overline{n|}} = v^n \cdot _{n}p_x$

Endowment Insurance:

• Combination of term insurance and pure endowment
• $A_{x:n} = A^1_{x:\overline{n|}} + A_{x:\overline{n|}}$

Deferred Insurance:

• $_{m|}A_x = v^m \cdot {m}p_x \cdot A{x+m}$
• Insurance beginning after m-year deferral period

Continuous Insurance Models #

Continuous Whole Life:

• $\bar{A}_x = \int_0^{\infty} v^t \cdot {t}p_x \cdot \mu{x+t} dt$
• $\bar{A}_x = \frac{\delta}{\delta + \mu} A_x$ (under constant force)

Relationship between Discrete and Continuous:

• $\bar{A}_x = \frac{i}{\delta} A_x$ (approximate, under UDD)
• $A_x = \frac{\delta}{i} \bar{A}_x$ (approximate, under UDD)

Life Annuities #

Whole Life Annuity-Due:

• Payments of 1 at beginning of each year while alive
• $\ddot{a}x = \sum{k=0}^{\infty} v^k \cdot _{k}p_x$
• $\ddot{a}x = 1 + v \cdot p_x \cdot \ddot{a}{x+1}$ (recursive formula)

Temporary Annuity-Due:

• $\ddot{a}{x:\overline{n|}} = \sum{k=0}^{n-1} v^k \cdot _{k}p_x$
• $\ddot{a}_{x:\overline{n|}} = \ddot{a}_x - v^n \cdot {n}p_x \cdot \ddot{a}{x+n}$

Deferred Annuity-Due:

• $_{m|}\ddot{a}_x = v^m \cdot {m}p_x \cdot \ddot{a}{x+m}$
• $_{m|}\ddot{a}_x = \ddot{a}x - \ddot{a}{x:\overline{m|}}$

Annuity-Immediate:

• $a_x = \sum_{k=1}^{\infty} v^k \cdot _{k}p_x = \ddot{a}_x - 1$

Continuous Life Annuity:

• $\bar{a}_x = \int_0^{\infty} v^t \cdot _{t}p_x dt$
• $\bar{a}_x = \frac{\ddot{a}_x - A_x}{d}$ (exact relationship)

Key Relationships #

Fundamental Identity:

• $A_x + d \cdot \ddot{a}_x = 1$ (for whole life)
• $A_{x:\overline{n|}} + d \cdot \ddot{a}_{x:\overline{n|}} = 1 - v^n \cdot _{n}p_x$

Woolhouse Formula:

• For m-thly payments: $\ddot{a}_x^{(m)} \approx \ddot{a}_x - \frac{m-1}{2m} + \frac{m^2-1}{12m^2} \delta$

Net Premium Reserves #

Premium Calculation Principles #

Net Single Premium:

• Present value of future benefits equals present value of future premiums
• Whole Life: $A_x = P(\ddot{a}_x)$ where P is net annual premium

Net Annual Premium Formulas:

• Whole Life: $P(A_x) = \frac{A_x}{\ddot{a}_x}$
• n-Year Endowment: $P(A_{x:\overline{n|}}) = \frac{A_{x:\overline{n|}}}{\ddot{a}_{x:\overline{n|}}}$
• Term Insurance: $P(A^1_{x:\overline{n|}}) = \frac{A^1_{x:\overline{n|}}}{\ddot{a}_{x:\overline{n|}}}$

Limited Payment Life Insurance:

• h-payment whole life: $P = \frac{A_x}{\ddot{a}_{x:\overline{h|}}}$

Reserve Calculation Methods #

Prospective Method:

• Future benefits minus future premiums
• $tV = A{x+t} - P \cdot \ddot{a}_{x+t}$ (whole life)
• $tV = A{x+t:\overline{n-t|}} - P \cdot \ddot{a}_{x+t:\overline{n-t|}}$ (endowment)

Retrospective Method:

• Accumulated premiums minus accumulated cost of insurance
• $tV = P \cdot \ddot{s}{x:\overline{t|}} - A^1_{x:\overline{t|}}$
• Where $\ddot{s}_{x:\overline{t|}}$ is accumulated value of temporary annuity-due

Recursive Formula:

• $_tV = (t{-1}V + P)(1+i) - q{x+t-1} - {t-1}V \cdot p{x+t-1}$
• ${t+1}V = \frac{tV + P - q{x+t}}{p{x+t}}$

Premium Difference Formula #

For policies with different premium structures:

• $_tV = _tV^{(1)} - tV^{(2)} = (P_2 - P_1) \cdot \ddot{a}{x+t}$
• Where superscripts (1) and (2) refer to different premium schedules

Modified Reserve Methods #

Full Preliminary Term Method:

• First-year reserve set to zero
• Modified premium for first year equals term insurance premium
• Subsequent years use higher level premium

Commissioner’s Reserve Valuation Method (CRVM):

• Statutory reserve method with modified first-year premiums
• Prevents excessive first-year reserves

Multiple Life Functions #

Joint Life Status #

Notation: $(xy)$ represents joint life of persons aged x and y

Basic Probabilities:

• Assuming independence: ${t}p{xy} = _{t}p_x \cdot _{t}p_y$
• ${t}q{xy} = 1 - {t}p{xy} = 1 - _{t}p_x \cdot _{t}p_y$
• Force of mortality: $\mu_{xy} = \mu_x + \mu_y$

Insurance and Annuities:

• $A_{xy} = \int_0^{\infty} v^t \cdot {t}p{xy} \cdot \mu_{xy} dt$
• $\ddot{a}{xy} = \sum{k=0}^{\infty} v^k \cdot {k}p{xy}$

Last Survivor Status #

Notation: $\overline{xy}$ represents last survivor of x and y

Basic Probabilities:

• ${t}q{\overline{xy}} = _{t}q_x \cdot _{t}q_y$
• ${t}p{\overline{xy}} = 1 - {t}q{\overline{xy}} = _{t}p_x + _{t}p_y - _{t}p_x \cdot _{t}p_y$

Special Cases:

• $\ddot{a}_{\overline{xy}} = \ddot{a}_x + \ddot{a}y - \ddot{a}{xy}$
• $A_{\overline{xy}} = A_x + A_y - A_{xy}$

Contingent Insurance #

Contingent on First Death:

• Benefit paid on death of x, provided y survives
• $A^1_{xy} = \frac{A_x - A_{xy}}{1 + A_y - A_{xy}}$ (under constant forces)

Reversionary Annuity:

• Annuity to y contingent on prior death of x
• $\ddot{a}_{x|y} = \ddot{a}y - \ddot{a}{xy}$

Multiple Decrement Models #

Service Table Notation #

Decrements:

• $\mu_x^{(j)}$ = force of decrement j at age x
• $q_x^{(j)}$ = probability of decrement j between ages x and x+1
• Total force: $\mu_x^{(\tau)} = \sum_{j} \mu_x^{(j)}$

Survival Probabilities:

• $p_x^{(\tau)} = \exp\left(-\int_x^{x+1} \mu_s^{(\tau)} ds\right)$
• $q_x^{(\tau)} = 1 - p_x^{(\tau)}$

Associated Single Decrement Tables #

Independence Assumption:

• $q_x^{(j)’} = \frac{q_x^{(j)}}{1 - \frac{1}{2}\sum_{k \neq j} q_x^{(k)}}$ (approximate)
• Where primed quantities represent single decrement rates

Pension Applications #

Service Table Functions:

• Active lives: $l_x^{(a)}$
• Retired lives: $l_x^{(r)}$
• Withdrawal rate: $w_x$
• Disability rate: $i_x$
• Mortality rate: $q_x^{(m)}$

Select & Ultimate Life Tables #

Select Period Mortality #

Notation:

• $[x]$ = select age x (immediately after underwriting)
• $[x]+t$ = attained age after t years since selection at age x
• Select period typically 1-15 years

Key Functions:

• $l_{[x]}$ = number selected at age x
• $l_{[x]+t}$ = survivors t years after selection
• $q_{[x]+t}$ = mortality rate for select group
• Ultimate rates: $q_{x+s}$ for ages beyond select period

Select Life Insurance and Annuities #

Select Premiums:

• Generally lower due to favorable select mortality
• $P([x]) = \frac{A_{[x]}}{\ddot{a}_{[x]}}$

Select Reserves:

• $tV{[x]} = A_{[x]+t} - P \cdot \ddot{a}_{[x]+t}$
• May use select or ultimate rates depending on context

Pension Mathematics #

Service Tables and Decrements #

Multiple Decrement Environment:

• Death: $q_x^{(d)}$
• Withdrawal: $q_x^{(w)}$
• Retirement: $q_x^{(r)}$
• Disability: $q_x^{(i)}$

Benefit Formulas #

Final Average Pay:

• $B = k \times \text{service years} \times \text{final average salary}$
• $k$ typically 1.5% - 2.5% per year of service

Career Average Pay:

• Benefits based on revalued career earnings
• May include indexation factors

Actuarial Cost Methods #

Unit Credit Method:

• Allocates costs based on benefit accrual
• Normal cost = present value of benefits earned in current year

Entry Age Normal:

• Level percentage of pay throughout career
• Smooths contribution patterns

Key Formulas Quick Reference #

Survival and Mortality #

• $\mu_x = -\frac{d}{dx}[\ln S(x)] = \frac{f(x)}{S(x)}$
• $_{t}p_x = \exp\left(-\int_x^{x+t} \mu_s ds\right)$
• $q_x + p_x = 1$

Insurance Present Values #

• $A_x = \sum_{k=0}^{\infty} v^{k+1} \cdot {k}p_x \cdot q{x+k}$
• $A_{x:\overline{n|}} = A^1_{x:\overline{n|}} + A_{x:\overline{n|}}$
• $\bar{A}_x = \frac{i}{\delta} A_x$ (UDD approximation)

Annuity Present Values #

• $\ddot{a}x = \sum{k=0}^{\infty} v^k \cdot _{k}p_x$
• $A_x + d \cdot \ddot{a}_x = 1$
• $\ddot{a}x = 1 + v \cdot p_x \cdot \ddot{a}{x+1}$

Premiums and Reserves #

• $P = \frac{\text{PV Benefits}}{\text{PV Premium Annuity}}$
• $_tV = \text{PV Future Benefits} - \text{PV Future Premiums}$
• ${t+1}V = \frac{tV + P - q{x+t}}{p{x+t}}$

Common Approximations #

Force of Interest Conversions #

• $\delta \approx i$ for small interest rates
• $d = \frac{i}{1+i} \approx i - i^2$ for small i

UDD Approximations #

• $\mu_{x+s} = \frac{q_x}{1-sq_x}$ for $0 \leq s < 1$
• $L_x = l_x(1 - \frac{1}{2}q_x)$

Woolhouse Formula Applications #

• Monthly: $\ddot{a}_x^{(12)} \approx \ddot{a}_x - \frac{11}{24} + \frac{143\delta}{1728}$
• Continuous: $\bar{a}_x \approx \ddot{a}_x - \frac{1}{2} + \frac{\delta}{12}$

Study Strategy & Tips #

Formula Mastery #

1. Derive, Don’t Memorize: Understand how complex formulas derive from basic relationships
2. Practice Numerical Examples: Work through calculations by hand before using calculator
3. Focus on Relationships: Master connections between insurance, annuities, and reserves
4. Pattern Recognition: Learn to identify which formula applies to each problem type

Calculation Efficiency #

1. Calculator Programming: Store frequently used formulas in calculator memory
2. Intermediate Values: Save intermediate calculations to avoid rework
3. Decimal Precision: Pay attention to rounding requirements (typically 5 decimal places)
4. Sign Conventions: Be consistent with timing of payments and benefits

Common Exam Traps #

1. Beginning vs. End of Year: Carefully distinguish annuity-due vs. annuity-immediate
2. Select vs. Ultimate: Check whether problem uses select or ultimate mortality
3. Policy Year vs. Calendar Year: Understand timing of reserve calculations
4. Independence Assumptions: Multiple life problems may or may not assume independence

Problem-Solving Approach #

1. Identify the Model: What type of insurance/annuity is being described?
2. Check Assumptions: What mortality assumptions are in play?
3. Set Up Equations: Write out the fundamental equation before substituting numbers
4. Verify Reasonableness: Do the results make intuitive sense?

Final Week Preparation #

1. Formula Sheet Practice: Create and memorize your own condensed formula sheet
2. Speed Drills: Practice standard calculations for speed and accuracy
3. Past Exams: Work through released questions under timed conditions
4. Weak Areas: Focus remaining time on your identified weak spots

Calculator Strategy #

• Memory Functions: Use STO/RCL for intermediate values
• Built-in Functions: Master NPV, IRR, and TVM functions
• Double-Check: Always verify critical intermediate calculations
• Backup Method: Know how to calculate key values manually if needed

Remember: The FAM-L exam tests both conceptual understanding and computational ability. Success requires mastering both the underlying actuarial principles and the mechanical aspects of formula application.