Concepts of survival models

Welcome to Survival Models!

Hello there! Welcome to one of the most fundamental chapters in your CS2 journey. In this chapter, we learn how to model the time until an "event" happens. In actuarial science, we usually focus on human life and death, but these principles apply equally to machine component failure, loan default, or claims run-off. Don't worry if the math looks intimidating—we will break it down step by step.

The Future Lifetime Random Variable

Consider a life aged exactly x today. Because the future lifetime is unknown, we represent the remaining duration of life as a continuous random variable, denoted by \( T_x \).

• \( T_x \) is the future lifetime of a person aged \( x \). For instance, if a life aged 30 survives to age 75 and dies, then \( T_{30} = 45 \).
• The age at death is the random variable \( X = x + T_x \).

The Consistency Condition

A life aged \( x \) will reach age \( x + t + s \) if and only if they first survive to age \( x + t \) and then survive an additional \( s \) years from age \( x + t \). This gives rise to the fundamental consistency condition linking lifetime random variables across ages:
\( P(T_x > t + s) = P(T_x > t) \cdot P(T_x > t + s \mid T_x > t) = P(T_x > t) \cdot P(T_{x+t} > s) \)
In actuarial notation, this is expressed compactly as:
\( {}_{t+s}p_x = {}_tp_x \cdot {}_sp_{x+t} \)

Survival, Distribution, and Density Functions

Actuarial mathematics uses specialised notation to describe the distribution of \( T_x \):

1. The Survival Function: \( S_x(t) \) or \( {}_tp_x \)

The survival function gives the probability that a life aged \( x \) survives at least \( t \) years:
\( {}_tp_x = S_x(t) = P(T_x > t) \)
• At \( t = 0 \), \( {}_0p_x = 1 \).
• As \( t \to \infty \), \( {}_tp_x \to 0 \).
• When \( t = 1 \), we write \( p_x \) rather than \( {}_1p_x \).

2. The Cumulative Distribution Function: \( F_x(t) \) or \( {}_tq_x \)

This is the complement of survival—the probability that a life aged \( x \) dies within \( t \) years:
\( {}_tq_x = F_x(t) = P(T_x \le t) = 1 - {}_tp_x \)
When \( t = 1 \), we write \( q_x \).

3. The Probability Density Function: \( f_x(t) \)

Differentiating the distribution function yields the probability density function (PDF) of future lifetime:
\( f_x(t) = \frac{d}{dt} F_x(t) = -\frac{d}{dt} {}_tp_x \)

The Force of Mortality (Hazard Rate): \( \mu_{x+t} \)

The force of mortality at age \( x + t \), denoted \( \mu_{x+t} \), is the instantaneous rate of death for a person surviving to that age:
\( \mu_{x+t} = \lim_{dt \to 0^+} \frac{P(t < T_x \le t + dt \mid T_x > t)}{dt} = \frac{f_x(t)}{{}_tp_x} = -\frac{1}{{}_tp_x}\frac{d}{dt} {}_tp_x = -\frac{d}{dt} \ln({}_tp_x) \)

Integrating both sides from \( 0 \) to \( t \) gives the fundamental relation:
\( {}_tp_x = \exp\left( -\int_0^t \mu_{x+s}\,ds \right) \)

Integral Formulae for Probabilities and Density

From these relations, we obtain key integral expressions tested regularly:
• Density function: \( f_x(t) = {}_tp_x \mu_{x+t} \)
• Probability of death: \( {}_tq_x = \int_0^t f_x(s)\,ds = \int_0^t {}_sp_x \mu_{x+s}\,ds \)
• Probability of dying between ages \( x + t \) and \( x + t + u \): \( {}_{t|u}q_x = {}_tp_x - {}_{t+u}p_x = \int_t^{t+u} {}_sp_x \mu_{x+s}\,ds \)

Named Laws of Mortality

Before standard computational tables were common, actuaries devised parametric models to describe how the force of mortality evolves with age. The syllabus requires you to know two classical laws:

• Gompertz Law: Assumes the force of mortality increases exponentially with age due to aging/senescence:
\( \mu_x = B c^x \quad (B > 0, c > 1) \)
• Makeham Law: Adds an age-independent baseline hazard representing accidental deaths:
\( \mu_x = A + B c^x \quad (A \ge -B, B > 0, c > 1) \)

Curtate Future Lifetime: \( K_x \)

In life insurance and pension schemes, benefits and premiums are frequently paid annually. Actuaries therefore define the curtate future lifetime, \( K_x \), as the number of complete years lived by a life aged \( x \) before death:
\( K_x = \lfloor T_x \rfloor \)
Unlike \( T_x \), \( K_x \) is a discrete random variable taking non-negative integer values \( k \in \{0, 1, 2, \dots\} \).

Probability Function of \( K_x \)

The probability that a life aged \( x \) completes exactly \( k \) full years of life and dies in the \( (k+1) \)-th year is:
\( P(K_x = k) = P(k \le T_x < k + 1) = {}_kp_x - {}_{k+1}p_x = {}_kp_x \cdot q_{x+k} = {}_k|q_x \)

Expectations and Variances of Future Lifetime

1. Complete Expectation of Life: \( \mathring{e}_x \)

The complete expectation of life is the expected value of continuous future lifetime \( T_x \):
\( \mathring{e}_x = E[T_x] = \int_0^\infty t f_x(t)\,dt = \int_0^\infty {}_tp_x\,dt \)
The second moment is:
\( E[T_x^2] = \int_0^\infty t^2 f_x(t)\,dt = 2 \int_0^\infty t\,{}_tp_x\,dt \)
The variance is obtained via \( \text{Var}(T_x) = E[T_x^2] - (\mathring{e}_x)^2 \).

2. Curtate Expectation of Life: \( e_x \)

The curtate expectation of life is the expected value of \( K_x \):
\( e_x = E[K_x] = \sum_{k=0}^\infty k P(K_x = k) = \sum_{k=1}^\infty {}_kp_x \)
The second moment is:
\( E[K_x^2] = \sum_{k=1}^\infty k^2 P(K_x = k) = \sum_{k=1}^\infty (2k - 1)\,{}_kp_x \)
The variance is \( \text{Var}(K_x) = E[K_x^2] - (e_x)^2 \).

3. Relationship Between \( \mathring{e}_x \) and \( e_x \)

Assuming that deaths are distributed uniformly over each year of age (UDD), deaths occur on average halfway through the year of age, yielding the standard approximation:
\( \mathring{e}_x \approx e_x + \frac{1}{2} \)

Two-State Single-Decrement Model vs. Random Lifetime Model

A survival model can be conceptualised in two distinct mathematical ways:

• Random Lifetime Model: Formulates survival as the time \( T_x \) until death as a continuous random variable on \( [0, \infty) \), characterized by its survival function \( S_x(t) \) or hazard function \( \mu_{x+t} \).
• Two-State Markov Model: Formulates the process as a continuous-time Markov process \( \{Y_t\}_{t \ge 0} \) with state space \( S = \{\text{Alive}, \text{Dead}\} \), where State Dead is an absorbing state and the transition intensity from Alive to Dead at age \( x + t \) is \( \mu_{x+t} \).

Comparison: Both formulations yield identical survival and mortality probabilities: \( P(Y_t = \text{Alive} \mid Y_0 = \text{Alive}) = {}_tp_x = \exp\left(-\int_0^t \mu_{x+s}\,ds\right) \). However, the two-state Markov framework readily extends to multi-state models (such as Healthy–Sick–Dead, disability, or critical illness) and multiple decrements, whereas the random lifetime model focuses directly on the distribution of duration in a single decrement.