Week 7 Learning Outcomes

Week 7 Detailed Learning Outcomes

Actuarial Practice #

n/a #

Actuarial Techniques #

  1. Explain why and how mortality can change from one table to another, from one group to another.
  2. Explain what we mean by “rectangularisation” of the survival function.
  3. Describe typical characteristics of mortality rates (shape, trends)

Mathematical models of mortality #

  1. Gompertz Law of mortality: $$\mu_x = Bc^x \Rightarrow {_tp}_x = \exp\left\{-\frac{B}{\log C} (c^x-1)\right\}.$$ Paremeters \(B\) and \(c\) in Gompertz Law can be found if two values for \(\mu_x\) are given.
  2. Makeham’s Law of mortality: $$\mu_x = A + Bc^x,\quad A,B>0 , \; c>1.$$ Parameters \(A\), \(B\), and \(c\) in Makeham’s Law can be found if three values for \(\mu_x\) are given.

The relationship between \(T(x)\) and \(K(x)\) #

  1. \(K(x) = [T(x)]\): integer part of \(T(x)\)
  2. \(_tp_x = \frac{S(x+t)}{S(x)} = Pr(T(x) > t)\)
  3. \(Pr(K(x) = k) = _kp_x - _{k+1}p_x\)
  4. \(e_x = E(K(x)) = \sum_{k=1}^{\infty} {_k p_x} = \sum_{k=1}^{\infty} \frac{l_{x+k}}{l_x}\)
  5. \(\stackrel{o}{e}_x \approx e_x +\frac{1}{2}\), as \(T_x \approx K(x) + \frac{1}{2}\)

Mesures of fertility #

Crude Birth Rate ( \(\text{CBR}\) ) #

$$ \text{CBR} = 1000 \times \frac{\text{Number of live births}}{\text{Population size (in the middle of a year)}}$$

General Fertility Rate ( \(\text{GFR}\) ) #

$$ \text{GFR} = 1000 \times \frac{\text{Number of live births}}{\text{Number of women aged [15,49]}}>\text{CBR}$$

Age Specific Fertility Rate ( \(\text{ASFR}_x\) ) #

$$\text{ASFR}_x = 1,\!000 \times \frac{\text{Number of live births by women aged [x,x+1)}}{\text{Number of women aged [x, x+1)}}$$

Total Fertility Rate ( \(\text{TFR}\) ) #

$$\text{TFR} = \sum_{x=15}^{49} \text{ASFR}_x$$

Comparison of \(\text{TFR}\) with \(\text{GFR}\) #

We have $$\text{ASFR}_x = 1,\!000 \times \frac{b_x}{w_x}$$ and hence $$\text{TFR} = 1,\!000 \times \sum_{x=15}^{49} \frac{b_x}{w_x} = \sum_{x=15}^{49} ASFR_x$$ which means \begin{eqnarray*} \text{GFR} &=& 1,\!000 \times \frac{\sum_{x=15}^{49} b_x}{\sum_{x=15}^{49} w_x} \\ &=& 1,\!000 \sum_{x=15}^{49} \frac{b_x}{w_{15}+w_{16}+...+w_{49}} \\ &=& 1,\!000 \times \sum_{x=15}^{49} \frac{w_x}{w_{15}+w_{16}+...+w_{49}} \frac{b_x}{w_x} \\ &=& \sum_{x=15}^{49} \theta_x \text{ASFR}_x, \end{eqnarray*} (a weighted average of \(ASFR_x\)), where \(\theta_x = \frac{w_x}{w_{15}+...+w_{49}}\) (percentage of women aged \(x\) to total women at reproductive age).

\(\text{ASFR}_x\) over an age group (e.g. 15-19 following the previous table) #

\begin{align*} \text{ASFR} =& 1,\!000 \frac{\text{Number of live births from women aged 15-19}}{\text{Number of women aged 15-19}} \\ =& 1,\!000 \frac{b_{15}+b_{16}+b_{17}+b_{18}+b_{19}}{w_{15}+w_{16}+w_{17}+w_{18}+w_{19}} \\ =& 1,\!000 \sum_{x=15}^{19} \tau_x ASFR_x, \end{align*} where $$\tau_x = \frac{w_x}{w_{15}+w_{16}+w_{17}+w_{18}+w_{19}}, \quad x = 15,16,17,18,19.$$

Gross Reproduction Rate ( \(\text{GRR}\) ) #

  1. We have $$\text{ASFR}_x^f = 1,\!000 \times \frac{\text{Number of live female births}}{\text{Number of women aged [x,x+1)}}.$$ which then leads to $$\text{GRR} = \sum_{x=15}^{49} \text{ASFR}_x^f.$$
  2. The sex ratio at birth is typically 1.05. Hence, we can use $$ASFR_x^f \approx \frac{1}{2.05} ASFR_x , \quad x= 15, 16,..., 49.$$ as an approximation.
  3. To sustain population we typically require (replacement level fertility rate) $$TFR \geq 2.1 \Longleftrightarrow GRR \geq \frac{2.1}{2.05} = 1.024 \;\text{(approx)}.$$

Net Reproduction Rate ( \(\text{NRR}\) ) #

  1. We have $$\text{NRR} = \sum_{x=15}^{49} ASFR_x^f \ \frac{l_{x+\frac{1}{2}}}{l_0}<\text{GRR}.$$
  2. One can use the approximation $$l_{x+\frac{1}{2}} \approx \frac{l_x+l_{x+1}}{2}$$
  3. We typically need \(\text{NRR} \geq 1\) to sustain population.

Fertility vs Reproduction Rates #

  1. Explain the difference between fertility and reproduction rates.
  2. Contrast both in developing vs developed countries.

Population projections #

  1. Polynomial Model: for given \(P_0\) $$P_t = P_0 + \sum_{i=1}^n a_i t^i,$$ where \(a_i\), \(i=1,\ldots,n\), are the \(n\) required parameters.
  2. Linear Model (special case of polynomial model): for given \(P_0\) $$P_t = P_0 + at = P_0(1+bt), \;\text{where} b=\frac{a}{P_0}$$ is the simple annual growth rate. Note, $$P_{t_2} = P_{t_1} + a(t_2-t_1), \ t_1< t_2.$$
  3. Geometric Growth Model: for given \(P_0\) $$P_t = P_0 (1+r)^t,$$ where \(r>0\) is the compound rate of growth per unit time. Note, \(P_{t_2} = P_{t_1}(1+t)^{t_2-t_1}, \ t_1< t_2\)
  4. Logistic Model: for given \(P_0\) $$P_t = \frac{1}{A+Be^{-rt}}, $$ where \(A>0\), \(B>0\), and \(r>0\) are required parameters. Note, we have \begin{eqnarray*} P_0 &=& \frac{1}{A+B}\;\text{and}\\ P_\infty &=& \frac{1}{A}. \end{eqnarray*}
Calendar 10 September 2021