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: μx=Bcxtpx=exp{BlogC(cx1)}. Paremeters B and c in Gompertz Law can be found if two values for μx are given.
  2. Makeham’s Law of mortality: μx=A+Bcx,A,B>0,c>1. Parameters A, B, and c in Makeham’s Law can be found if three values for μx are given.

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

  1. K(x)=[T(x)]: integer part of T(x)
  2. tpx=S(x+t)S(x)=Pr(T(x)>t)
  3. Pr(K(x)=k)=kpxk+1px
  4. ex=E(K(x))=k=1kpx=k=1lx+klx
  5. eoxex+12, as TxK(x)+12

Mesures of fertility #

Crude Birth Rate ( CBR ) #

CBR=1000×Number of live birthsPopulation size (in the middle of a year)

General Fertility Rate ( GFR ) #

GFR=1000×Number of live birthsNumber of women aged [15,49]>CBR

Age Specific Fertility Rate ( ASFRx ) #

ASFRx=1,000×Number of live births by women aged [x,x+1)Number of women aged [x, x+1)

Total Fertility Rate ( TFR ) #

TFR=x=1549ASFRx

Comparison of TFR with GFR #

We have ASFRx=1,000×bxwx and hence TFR=1,000×x=1549bxwx=x=1549ASFRx which means GFR=1,000×x=1549bxx=1549wx=1,000x=1549bxw15+w16+...+w49=1,000×x=1549wxw15+w16+...+w49bxwx=x=1549θxASFRx, (a weighted average of ASFRx), where θx=wxw15+...+w49 (percentage of women aged x to total women at reproductive age).

ASFRx over an age group (e.g. 15-19 following the previous table) #

ASFR=1,000Number of live births from women aged 15-19Number of women aged 15-19=1,000b15+b16+b17+b18+b19w15+w16+w17+w18+w19=1,000x=1519τxASFRx, where τx=wxw15+w16+w17+w18+w19,x=15,16,17,18,19.

Gross Reproduction Rate ( GRR ) #

  1. We have ASFRxf=1,000×Number of live female birthsNumber of women aged [x,x+1). which then leads to GRR=x=1549ASFRxf.
  2. The sex ratio at birth is typically 1.05. Hence, we can use ASFRxf12.05ASFRx,x=15,16,...,49. as an approximation.
  3. To sustain population we typically require (replacement level fertility rate) TFR2.1GRR2.12.05=1.024(approx).

Net Reproduction Rate ( NRR ) #

  1. We have NRR=x=1549ASFRxf lx+12l0<GRR.
  2. One can use the approximation lx+12lx+lx+12
  3. We typically need NRR1 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 P0 Pt=P0+i=1naiti, where ai, i=1,,n, are the n required parameters.
  2. Linear Model (special case of polynomial model): for given P0 Pt=P0+at=P0(1+bt),whereb=aP0 is the simple annual growth rate. Note, Pt2=Pt1+a(t2t1), t1<t2.
  3. Geometric Growth Model: for given P0 Pt=P0(1+r)t, where r>0 is the compound rate of growth per unit time. Note, Pt2=Pt1(1+t)t2t1, t1<t2
  4. Logistic Model: for given P0 Pt=1A+Bert, where A>0, B>0, and r>0 are required parameters. Note, we have P0=1A+BandP=1A.