Week 6 Detailed Learning Outcomes

Actuarial Practice #

n/a #

Actuarial Techniques #

Survival function of a new born life #

The distribution of time to death of a newborn \(T(0)\) is $$F(x) = \Pr[T(0)\leq x],$$ with complement called “survival function” $$s(x) = \Pr[T(0) > x],$$ with properties:

1. \(s(0) = 1\);
2. \(s(\infty) = 0\);
3. \(s(x) \downarrow x\).

Survival function of a life aged \((x)\) #

With actuarial notation, the probability of \((x)\) to survive another \(t\) years is $$_tp_x = \Pr[T(x) > t] = \frac{s(x+t)}{s(x)}$$ with complement (probability of death over that same period) $$_tq_x = \Pr[T(x) \leq t] = 1- \frac{s(x+t)}{s(x)}.$$ Loooking forward, $$\Pr[t < T(x) \leq t+r] = {_tp}_x - {_tp}_{x+r} = {_{t+r}q}_x - {_tq}_x.$$

The force of mortality \(\mu_x\) #

The force of mortality provides the infinitesimal likelihood for \((x)\) to die in the next instant. $$\mu_x = \lim_{h \rightarrow \infty} \frac{1}{h} \Pr[x<T(0)\leq x+h | T(0) > x] = \lim_{h \rightarrow \infty} \frac{1}{h} \Pr[T(x) \leq h].$$ Properties:

1. Relationship with survival function $$\mu_x = -\frac{s'(x)}{s(x)} = -(log[s(x)])'$$ Also, for \(\mu_x\) is given, $$s(x) = e^{- \int_0^x \mu_y dy} = \Pr[T(0) > x]$$ and $${_tp}_x = \Pr[T(x) >t] =\frac{s(x+t)}{s(x)} = e^{- \int_0^t \mu_{x+r} dr} = e^{- \int_x^{x+t} \mu_{r} dr}.$$
2. As an approximation, $$\Pr[T(x) \leq h] = _nq_x \approx h \cdot \mu_X$$ when \(h\) is very small.

Life table and life table functions #

1. The radix \(l_0\) is often set to be 10,000 or 100,000.
2. \(l_x\) is the expected number of survivors to age \(x\) out of \(l_0\) initial number of inviduals (radix). Note that \(l_x\) can also be defined on non-negative integers. We have $$l_x = l_0 s(x).$$
3. The expected number of deaths at age \(x\) is $$d_x = l_x - l_{x+1}$$
4. The probability of death over the next year for an individual aged \(x\) exactly is $$q_x = \frac{d_x}{l_x}$$
5. The associated one year survival probability is $$p_x = 1 - q_x = \frac{l_{x+1}}{l_x}.$$

Some additional results:

1. If \(x\) and \(t\) are integers, then $$_tp_x = p_x p_{x+1}...p_{x+t-1}.$$
2. Probabilities as functions of \(l_x\): $$_tp_x = \frac{l_{x+t}}{l_x},$$ $$_tq_x = \frac{l_x-l_{x+t}}{l_x} =\frac{d_x+d_{x+1}+...+d_{x+t-1}}{l_x},$$

Aggregate functions \(L_x\) and \(T_x\) #

We have:

1. The number of people in the population who are aged \(x\) last birthday, i.e. actual age lies in \((x, x+1)\), is $$L_x = \int_0^1 l_{x+t} dt.$$ As an approximation, $$L_x \approx l_x - \frac{1}{2}d_x.$$.
2. The total population aged \(x\) or above is $$T_x = L_x + L_{x+1}+...+L_{x+k} +... = \int_0^\infty l_{x+t} dt =T_{x+1} + L_x.$$
3. \(T_x\), \(L_x\) are typically tabulated in the life table, too.
4. Applications include: $$\overset{\circ }{e}_x=\frac{T_x}{l_x}. $$

Stationary population #

1. The life table can be interpreted as a stationary population with some assumptions.
2. You should understand how the interpretation works, and make calculations similar to those made in lectures.

Period vs Cohort tables #

1. Period tables describe mortality of a population in a given year (so different individuals of different age, all in the same calendar year). So there are potentially different period tables for different calendar years of death.
2. Cohort tables look at the mortality of people born in the same calendar year – they track mortality of a given cohort. So there are potentially different cohort tables for different calendar years of birth.