📊Actuarial Mathematics Unit 4 – Life Contingencies & Survival Models

Key Concepts and Definitions

• Life contingencies involve financial transactions dependent on the occurrence of human life events (death, survival, disability)
• Survival models mathematically describe the probability of an individual surviving to a certain age or for a specified period
  • Based on mortality data and assumptions about future mortality improvements
• Life tables summarize the survival probabilities for a group of individuals at different ages
  • Cohort life tables follow a specific group of individuals over time
  • Period life tables represent mortality experience during a specific time period
• Actuarial present value represents the expected value of future payments, discounted for interest and the probability of payment
• Commutation functions simplify the calculation of actuarial present values by combining mortality and interest factors
• Valuation date is the point in time at which actuarial liabilities and reserves are calculated
• Equivalence principle states that the actuarial present value of future benefits should equal the actuarial present value of future premiums

Probability Theory Foundations

• Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1
• Conditional probability is the probability of an event A occurring, given that another event B has already occurred, denoted as P(A|B)
• Independence of events means that the occurrence of one event does not affect the probability of another event occurring
• Random variables assign numerical values to the outcomes of a random experiment
  • Discrete random variables have countable outcomes (number of claims)
  • Continuous random variables have an infinite number of possible outcomes within a range (claim amount)
• Probability density functions (PDFs) describe the probability distribution of a continuous random variable
• Cumulative distribution functions (CDFs) give the probability that a random variable is less than or equal to a specific value
• Expected value is the average value of a random variable, weighted by its probabilities
• Variance and standard deviation measure the dispersion of a random variable around its expected value

Survival Models and Life Tables

• Survival function S(x) represents the probability that an individual aged x will survive to a future age
  • S(x) = 1 - F(x), where F(x) is the cumulative distribution function of the future lifetime random variable
• Probability of dying between ages x and x+t is denoted as $_tq_x$ and can be calculated as $_tq_x = \frac{l_{x+t}}{l_x}$
• Force of mortality $\mu(x)$ represents the instantaneous rate of mortality at age x
  • Related to the survival function by $\mu(x) = -\frac{d}{dx} \ln S(x)$
• Curtate future lifetime $K_x$ is the random variable representing the number of complete years lived by an individual aged x until death
• Complete or aggregate life tables contain mortality information for each age from 0 to the maximum attainable age
• Abridged or select life tables provide mortality data for specific age ranges or selected ages
• Graduation techniques smooth and interpolate life table data to obtain values for intermediate ages

Mortality Laws and Distributions

• Gompertz's law states that the force of mortality increases exponentially with age, $\mu(x) = Bc^x$
  • B and c are constants determined from mortality data
• Makeham's law extends Gompertz's law by adding an age-independent component, $\mu(x) = A + Bc^x$
• Weibull distribution is a continuous probability distribution used to model survival times and failure rates
  • Probability density function: $f(x) = \frac{\alpha}{\beta} (\frac{x}{\beta})^{\alpha-1} e^{-(x/\beta)^\alpha}$, where $\alpha$ is the shape parameter and $\beta$ is the scale parameter
• Gamma distribution is another continuous probability distribution used in survival analysis
  • Probability density function: $f(x) = \frac{1}{\Gamma(\alpha) \theta^\alpha} x^{\alpha-1} e^{-x/\theta}$, where $\alpha$ is the shape parameter and $\theta$ is the scale parameter
• Heligman-Pollard model is a parametric model that describes the pattern of mortality across different age ranges
  • Combines three terms representing childhood, accident hump, and adult mortality
• Lee-Carter model is a stochastic mortality model that captures both age and time-dependent changes in mortality rates

Present Value Random Variables

• Present value random variable $Z$ represents the present value of future payments dependent on the survival of an individual
  • $Z = v^K$, where $v$ is the discount factor and $K$ is the curtate future lifetime random variable
• Actuarial present value $A_x$ is the expected value of the present value random variable for an individual aged x
  • $A_x = E[Z] = \sum_{k=0}^\infty v^k {}_kp_x q_{x+k}$
• Variance of the present value random variable measures the dispersion of the present value of future payments around the actuarial present value
• Higher moments of the present value random variable, such as skewness and kurtosis, provide additional information about the distribution of the present value of future payments
• Shared present value random variable represents the present value of payments that depend on the joint survival of multiple individuals
• Conditional present value random variable represents the present value of payments given the occurrence of a specific event (e.g., survival to a certain age)

Life Insurance Models

• Whole life insurance provides a death benefit upon the insured's death, regardless of when it occurs
  • Actuarial present value: $A_x = \sum_{k=0}^\infty v^{k+1} {}_kp_x q_{x+k}$
• Term life insurance provides a death benefit if the insured dies within a specified term
  • Actuarial present value for an n-year term policy: $A^1_{x:\bar{n}|} = \sum_{k=0}^{n-1} v^{k+1} {}_kp_x q_{x+k}$
• Endowment insurance combines a term life insurance with a pure endowment, providing a benefit upon the insured's death or survival to the end of the term
  • Actuarial present value for an n-year endowment policy: $A_{x:\bar{n}|} = \sum_{k=0}^{n-1} v^{k+1} {}_kp_x q_{x+k} + v^n {}_np_x$
• Deferred life insurance provides coverage starting at a future date, often used in retirement planning
• Increasing and decreasing life insurance have death benefits that change over time according to a predetermined schedule
• Participating life insurance policies allow policyholders to share in the insurer's profits through dividends or bonus additions

Life Annuity Models

• Life annuity is a series of payments made continuously or at fixed intervals for as long as the annuitant is alive
• Whole life annuity provides payments for the entire lifetime of the annuitant
  • Actuarial present value: $a_x = \sum_{k=0}^\infty v^k {}_kp_x$
• Temporary life annuity provides payments for a fixed term or until the annuitant's death, whichever occurs first
  • Actuarial present value for an n-year temporary annuity: $a_{x:\bar{n}|} = \sum_{k=0}^{n-1} v^k {}_kp_x$
• Deferred life annuity starts payments at a future date, often used in retirement planning
  • Actuarial present value for an n-year deferred whole life annuity: ${}_{n|}a_x = \sum_{k=n}^\infty v^k {}_kp_x$
• Guaranteed annuities provide payments for a minimum guaranteed period, even if the annuitant dies before the end of the period
• Increasing and decreasing annuities have payment amounts that change over time according to a predetermined schedule
• Joint and survivor annuities provide payments as long as at least one of the annuitants is alive

Premium Calculations and Reserves

• Net premium is the portion of the premium used to cover the expected cost of benefits, calculated using actuarial principles
  • Whole life insurance net annual premium: $P_x = \frac{A_x}{\ddot{a}_x}$, where $\ddot{a}_x$ is the actuarial present value of a life annuity-due
• Gross premium includes the net premium and additional amounts for expenses, profits, and contingencies
• Equivalence principle states that the actuarial present value of future benefits should equal the actuarial present value of future premiums
• Prospective reserve is the difference between the actuarial present value of future benefits and the actuarial present value of future premiums
  • Prospective reserve at time t for a whole life policy: $_tV_x = A_{x+t} - P_x \ddot{a}_{x+t}$
• Retrospective reserve is the accumulated value of past premiums less the accumulated value of past benefits
  • Retrospective reserve at time t for a whole life policy: $_tV_x = (P_x \ddot{a}_x - A_x)(1+i)^t$
• Premium conversion allows for the calculation of equivalent premiums payable at different frequencies (e.g., annual to monthly)
• Modified reserve systems, such as the commissioners reserve valuation method (CRVM), are used to calculate statutory reserves for regulatory purposes

Multiple Life Functions

• Joint life status refers to the condition that all individuals in a group are alive
  • Joint life probability: ${}_tp_{xy} = {}_tp_x \times {}_tp_y$, assuming independent lives
• Last survivor status refers to the condition that at least one individual in a group is alive
  • Last survivor probability: ${}_tp_{\overline{xy}} = 1 - {}_tq_{\overline{xy}} = 1 - {}_tq_x \times {}_tq_y$, assuming independent lives
• Multiple life annuities provide payments as long as at least one of the annuitants is alive
  • Actuarial present value of a joint life annuity: $a_{xy} = \sum_{k=0}^\infty v^k {}_kp_{xy}$
• Multiple life insurance policies provide death benefits upon the death of one or more insured individuals
  • Actuarial present value of a joint life insurance: $A_{xy} = \sum_{k=0}^\infty v^{k+1} {}_kp_{xy} q_{xy}$
• Contingent probabilities and actuarial present values depend on the survival or death of one life, given the survival or death of another life
  • Probability of (x) surviving t years, given (y) survives: ${}_tp_x^y = \frac{{}_tp_{xy}}{{}_tp_y}$
• Common shock models account for the possibility that multiple lives may experience a common event simultaneously

Multiple Decrement Models

• Multiple decrement models consider the presence of multiple causes of decrement (e.g., death, disability, withdrawal)
• Decrement probabilities $_tq_x^{(j)}$ represent the probability of exiting the population due to cause j between ages x and x+t
• Overall survival probability in a multiple decrement model: ${}_tp_x = \prod_{j=1}^m (1 - {}_tq_x^{(j)})$
• Associated single decrement tables isolate the effect of a single decrement cause while assuming other causes do not exist
• Actuarial present values in multiple decrement models incorporate the probabilities of different decrement causes
  • Actuarial present value of a whole life insurance in a double decrement model (death and disability): $A_x = \sum_{k=0}^\infty v^{k+1} ({}_kp_x^{(\tau)} q_{x+k}^{(1)} + {}_kp_x^{(\tau)} q_{x+k}^{(2)})$
• Transition intensities $\mu_{ij}(x)$ represent the instantaneous rate of transition from state i to state j at age x
• Kolmogorov forward equations describe the evolution of state probabilities over time in a Markov model

Practical Applications and Case Studies

• Pricing and reserving for life insurance and annuity products
  • Determine appropriate premium rates and reserve levels based on mortality assumptions and investment returns
• Pension plan valuation and funding
  • Calculate actuarial liabilities, normal costs, and required contributions for defined benefit pension plans
• Social insurance programs (e.g., Social Security, Medicare)
  • Assess the long-term financial sustainability and propose reforms based on demographic and economic projections
• Risk management and reinsurance
  • Evaluate and mitigate mortality risk exposure through risk-sharing arrangements and reinsurance contracts
• Longevity risk and securitization
  • Develop financial instruments and markets to transfer and manage longevity risk, such as longevity bonds and swaps
• Mortality experience studies and assumption setting
  • Analyze historical mortality data to develop and update mortality assumptions for pricing and valuation purposes
• Regulatory compliance and reporting
  • Ensure compliance with statutory reserve requirements, capital adequacy standards, and financial reporting guidelines
• Actuarial software and modeling
  • Utilize specialized software tools and programming languages (e.g., R, Python) to build and analyze complex actuarial models