







































AUSTRALIAN POPULATION STUDIES   2021 | Volume 5 | Issue 1 | pages 77-100 

 

© Wilson & Rees 2021. Published under the Creative Commons Attribution-NonCommercial licence 3.0 Australia (CC BY-NC 
3.0 AU). Journal website: www.australianpopulationstudies.org 

Introductory Guide 

A brief guide to producing a national 
population projection 

Tom Wilson*  The University of Melbourne 

Philip Rees  The University of Leeds 

* Corresponding author. Email: wilson.t1@unimelb.edu.au. Melbourne School of Population and 

Global Health, The University of Melbourne, Melbourne, Vic 3010, Australia. 

Paper received 10 February 2021; accepted 23 March 2021; published 31 May 2021 

Abstract 

Background 

There are surprisingly few resources available which offer an introductory guide to preparing a 

national population projection using a cohort-component model. Many demography textbooks cover 

projections quite briefly, and many academic papers on projections focus on advanced technical 

issues. 

Aims 

The aim of this paper is to provide a short and accessible guide to producing a national-scale 

population projection using the cohort-component model. 

Data and methods 

The paper describes the cohort-component model from a population accounting perspective, 

presents all the necessary projection calculations, and covers the key steps which form part of the 

projections preparation process – from gathering input data to validating outputs. An accompanying 

Excel workbook implements the model and contains example projections for Australia. 

Conclusions 

Calculating a national population projection using a cohort-component model involves fairly simple 

algebra, but the broader projections preparation process is more complex, and requires careful 

consideration and judgement.  

Key words 

Population projections; cohort-component model; population accounting; Australia 

  

 

http://www.australianpopulationstudies.org/
mailto:wilson.t1@unimelb.edu.au


78 Wilson & Rees  Australian Population Studies 5 (1) 2021 

1. Introduction 

National statistical offices and some international agencies, such as the United Nations, produce 

national population projections on a regular basis. Projections are used for many purposes, including 

calculating the amount of additional housing needed in coming years, estimating future aged care 

demand, assessing the fiscal implications for government of future population growth and ageing 

(e.g., Australian Government 2015), and contributing to national debates about population change. 

Researchers also produce national population projections from time to time in order to understand 

demographic change, explore alternative demographic scenarios, or demonstrate methodological 

refinements, among other reasons (e.g., Lomax et al. 2019; Wilson et al. 2021). 

Surprisingly, the guidance available on how to create national population projections is quite limited. 

Many academic papers and books on projections, as would be expected, report the latest research 

and usually focus on advanced technical aspects of projections. There are relatively few handbooks 

or websites which provide introductory advice on how to create population projections. Examples of 

resources which provide guidance include a manual on projections published by the United Nations 

(1989), the UNFPA/IUSSP web-based demography course which includes a section on projections 

(Timaeus 2015), and manuals for specific population projection software products (e.g., Shorter et al. 

1995; Stover and Kirmeyer 2007; US Census Bureau 2019); and the Practitioner’s Guide by Smith et 

al. (2013) is a good example for subnational projections. Most demography textbooks do include a 

section on the principal model used to create national population projections – the cohort-

component model – though the description is often brief. And the model usually includes only net 

international migration (which has some limitations) rather than separate immigration and 

emigration flows. 

The aim of this paper is to present a short practical guide to producing a national population 

projection. Our goal is to offer an accessible introduction to projection modelling which avoids 

complex algebra but includes sufficient mathematical detail so that readers can produce their own 

projections if desired. We present the cohort-component model from a population accounting 

perspective because it offers an intuitive and transparent way of expressing the model. This 

approach is based on the well-known population accounting (or balancing) equation, and involves 

projecting the demographic components of change as the product of occurrence/exposure rates and 

populations-at-risk. Our description is less directly connected to the life table than some other 

presentations of the cohort-component model, and we do not express the model in matrix algebra as 

many researchers do1. Our projection equations also differ somewhat from those used by the 

Australian Bureau of Statistics for their population projections (ABS 1999). Primarily, the description 

in this paper is most relevant to countries with good statistical systems and reasonable quality 

demographic data. 

We begin with a few preliminary terms and concepts (section 2) and then give an overview of the 

projections process (section 3). The cohort-component model forms the focus of section 4, while 

section 5 covers the preparation of necessary input data and assumptions. In section 6 we briefly 

describe how projection outputs can be checked for plausibility and consistency, before a few 

 
1 Readers interested in matrix arrangements of the cohort-component model are recommended to consult 
Keyfitz and Caswell (2005: chapter 3) or Rogers (1995: chapter 3). 



Australian Population Studies 5 (1) 2021 Wilson & Rees  79 

concluding remarks are given in section 7. Accompanying this paper is an Excel workbook which 

incorporates a cohort-component model for national population projections, illustrated with 

example projections for Australia. It is available from 

https://doi.org/10.6084/m9.figshare.13565222.v3.  

2. Preliminaries 

Before preparing any national-scale population projections, it is very useful to have a reasonable 

knowledge of: 

• projections terminology, 

• life tables,  

• the Lexis diagram, and 

• population accounting. 

We also assume knowledge of basic demographic concepts and terms, which are usually covered in 

the initial chapters of introductory demography textbooks (e.g., Preston et al. 2001; Rowland 2003). 

2.1. Projections terminology 

Box 1 defines terms population projections terms used in this paper. 

Box 1: Definitions 

Jump-off year: the year from which the projections ‘jump off’ into the demographic future. It is 

sometimes also known as the launch year. 

Population estimate: the population for a past point in time based on information available for that 

time. 

Population projection: a calculation of population beyond the jump-off year based on selected 

assumptions about the future drivers of population change (commonly, fertility, mortality and 

migration). 

Population forecast: a population projection regarded as the most likely outcome; a prediction. 

Projection horizon: the time period between the jump-off year and the final year of the projections. 

Projection interval: the time interval between each projected population (usually 1 or 5 years). 

Projection assumptions: data inputs to the projection calculations which usually include the assumed 

future paths of fertility, mortality, and migration (and sometimes other variables). More broadly 

defined, projection assumptions also implicitly or explicitly include qualitative elements, such as 

expectations about future immigration policy. 

Population-at-risk: The number of person-years lived in a projection interval by a particular 

population exposed to the ‘risk’ of a demographic event. It is often approximated as the mean of the 

start and end populations of a projection interval. 

(Occurrence/exposure) rate: The ratio of the number of demographic events occurring divided by 

the population exposed to the risk of that event (population-at-risk). 

Source: Wilson (2019); authors’ definitions 

https://doi.org/10.6084/m9.figshare.13565222.v3


80 Wilson & Rees  Australian Population Studies 5 (1) 2021 

2.2. The life table 

A life table is a model of a cohort’s mortality experience over its lifetime, but it is often calculated for 

a particular year or group of years and therefore describes a synthetic cohort. Although we do not 

make extensive use of life table measures in this paper, it is helpful to understand the calculations 

behind, and meaning of, nLx, the life table stationary population for each age group. In life table 

notation, 𝑥 is the exact starting age of an age group and 𝑛 is the width of the age group in years. The 

life table is covered comprehensively in demography textbooks, so is not described here. Readers 

unfamiliar with the life table are recommended to consult an appropriate source (e.g. Kinter 2004; 

Preston et al. 2001: chapter 3; Rowland 2003: chapter 8; Sloggett 2015). We return to life tables in 

section 5 on input data preparation. 

2.3. The Lexis diagram (or age-time plan) 

The Lexis diagram consists of a graph with age increasing up the vertical axis and time progressing 

along the horizontal axis. It provides a very useful way of precisely locating the age, time and cohort 

extent of demographic stocks and flows in population projections (and, more generally, any 

demographic analysis). Longer discussions on the Lexis diagram can be found in Calot (2015), Pressat 

(2008: chapter 2), and Rau (2018). 

Figure 1 illustrates some key demographic variables. If the Lexis diagram is used to illustrate the 

demographic history of just a handful of individuals then individual lifelines can be drawn, and 

demographic events (e.g. death, migration, giving birth) marked at points on those lines (Figure 1, 

upper diagram). An individual’s lifeline starts on the horizontal axis and proceeds at a 45º angle 

because as each year passes, the person ages by 1 year. Symbols are placed on lifelines to denote 

demographic events, such as giving birth, migration, and eventually, death. 

However, once there are more than just a few lifelines the diagram becomes cluttered, and it 

becomes necessary to use thick lines to denote all the lifelines crossing a point in time (or exact age) 

and shapes to represent all demographic events occurring within a particular age-time space. For 

example, in the lower diagram in Figure 1 the thick vertical lines represent the population of the 

cohort aged 1 year old last birthday at time 𝑡 and aged 2 years at time 𝑡 + 1. The parallelogram 

shaded grey defines the period-cohort age-time space which refers to demographic events 

experienced by the cohort over the one year period. The shaded square depicts demographic events 

defined by period and age. The thick lines at the left and right sides of the square represent 

populations of that age at the start and end of the time interval. The horizontal thick blue line 

denotes babies born during a year, and the shaded triangle is the age-period-cohort space 

representing demographic events occurring to this cohort of newly-born babies in that year. 

2.4. Population accounting 

Population accounts connect population stocks at two points in time with the demographic 

components which add to, and deplete, the population (Rees 1985; Rees and Willekens 1986). At the 

national scale, the population at time 𝑡 + 1 equals the population at time 𝑡 plus births, minus deaths, 

plus immigration, minus emigration: 

𝑃(𝑡 + 1) = 𝑃(𝑡) + 𝐵(𝑡, 𝑡 + 1) − 𝐷(𝑡, 𝑡 + 1) + 𝐼(𝑡, 𝑡 + 1) − 𝐸(𝑡, 𝑡 + 1)   (1) 



Australian Population Studies 5 (1) 2021 Wilson & Rees  81 

 
3 

        

 
2 

 
  

        

Age 
1 

        

 
0 

        

    Time →     

 
3 

        

 
2 

 
  

        

Age 
1 

        

 
0 

        

    Time →     

Figure 1: The age-time location of lifelines and demographic stocks and flows  

 

where: 

𝑃 = population 

𝑡 = a point in time 

𝑡 + 1= a point in time one year later 

𝑡, 𝑡 + 1= the one year interval starting at t and ending at t+1 

𝐵 = births 

𝐷 = deaths 

𝐼 = immigration 

𝐸 = emigration 

migration 

period-age period-cohort 

age-period-cohort 

death 

population 

population 

t                 t+1  births 

population population 

population 



82 Wilson & Rees  Australian Population Studies 5 (1) 2021 

This is the demographic accounting, or balancing, equation for a national population. Strictly, it refers 

to what are known as movement population accounts in which the migration flows are counts of 

moves across the national border. An alternative population accounting equation in which the 

migration components refer to counts of migrants (not moves) uses data on people living in different 

countries at time 𝑡 and time 𝑡 + 1 (as measured by a census). This transition accounts perspective is 

more complex and is not considered in this paper2. 

As an example, consider the population accounts for Australia over the 12 month period from mid-

2019 to mid-2020 (ABS 2020a). The start-of-interval population 𝑃(𝑡)  was 25,365,745, the number of 

births 304,101, deaths 166,974, immigration 486,757, and emigration 302,588. The end-of-interval 

population 𝑃(𝑡 + 1) was therefore: 

= 25,365,745 + 304,101 − 166,974 + 486,757 − 302,588 

= 25,687,041. 

3. The process for producing a national population projection 

The population projections process involves a number of steps, starting with gathering data, checking 

data and making adjustments if necessary, preparing projection assumptions, entering data into your 

projection program, calculating the projections, and then checking outputs. 

Step 1: Obtain the jump-off populations and all data needed to prepare fertility, mortality and 

migration assumptions 

All projections begin with the jump-off population, which is usually the most up-to-date set of 

population estimates published by the national statistical office. At the national scale, projections of 

fertility and mortality are often created by models which work best with many decades of data. The 

extent of historical time series required will depend on the precise fertility and mortality projection 

approaches chosen (discussed in section 5 later). Immigration and emigration data are also ideally 

obtained for the last few decades – or at the very least total immigration and emigration flows for 

several decades, and by age and sex for the most recent years. All projections begin with the jump-

off population, which is usually the most up-to-date set of population estimates published by the 

national statistical office. At the national scale, projections of fertility and mortality are often created 

by models which work best with many decades of data. The extent of historical time series required 

will depend on the precise fertility and mortality projection approaches chosen (discussed in section 

5 later). Immigration and emigration data are also ideally obtained for the last few decades – or at 

the very least total immigration and emigration flows for several decades, and by age and sex for the 

most recent years.  

Step 2: Check that the input data are good quality and that recent population accounts are 

consistent, and carry out adjustments if necessary 

Past fertility and mortality rates by age are usefully graphed to check for any obvious data problems. 

The national statistical office which published the data may also have accompanying notes or reports 

on data quality. In most countries with well-developed statistical systems, the quality and coverage 

 
2 Interested readers are recommended to consult Rees (1986) or Rees and Wilson (1977). 



Australian Population Studies 5 (1) 2021 Wilson & Rees  83 

of births and deaths data are normally good (though not perfect). In Australia, the quality of 

international migration data is very good, but in some countries it is less reliable. A high-level check 

of population accounting consistency can be made by comparing past 

• recorded net international migration (calculated as immigration minus emigration), with  

• residual net migration (calculated as population growth minus recorded natural change). 

A more detailed check of accounting consistency involves comparing residual net migration with 

recorded net migration by sex and cohort for a recent period to ensure that the age profiles of 

immigration and emigration are reasonable. Does period-cohort  

• recorded net migration (immigration minus emigration) equal 

• residual net migration (calculated as cohort population change plus period-cohort deaths)? 

If there are large discrepancies, then there is a problem. If the population estimates are deemed 

more reliable than the migration data, then either or both immigration and emigration values should 

be adjusted.  

Step 3: Prepare assumptions on the expected future direction of fertility, mortality and migration 

Once data quality checks have been completed, and data adjustments made if necessary, then 

assumptions on the future of fertility, mortality and migration can be prepared. Section 5 provides an 

overview of various approaches and methods which can be used. 

Step 4: Enter all input data into the projection program/spreadsheet 

The next stage involves loading all the input data into computer code or a workbook for producing 

population projections. Even with highly automated processes there is still potential for errors to 

creep in (e.g. using old assumptions; male and female populations being mixed up), so it is useful to 

double check all the input data has been entered, and entered in the correct locations. 

Step 5: Calculate the projections 

Then the population projections are calculated using the cohort-component model projection 

equations described next in section 4. The Excel workbook accompanying this paper can be used, or 

you may wish to write your own code. 

Step 6: Check outputs for plausibility and consistency, and revise assumptions if necessary 

An important part of the projection process is checking that the output projections are plausible and 

internally consistent. Unfortunately, it is quite possible, even with carefully prepared projection 

assumptions, for the cohort-component model to generate implausible or suboptimal outcomes. 

These outcomes can be caused by data quality issues, random noise in input data, errors in 

projection assumptions, mistakes in loading assumptions data into the projection 

program/spreadsheet, projection model coding errors, and the lack of mechanisms in the cohort-

component model to maintain sensible relationships between certain aspects of the projections. For 

example, it might be the case that sex ratios become increasingly unlikely over the projection 

horizon. Section 6 covers some of the key checks which should be made to projections output. A 

further check on the input assumptions and projection model can be made by creating a test run set 



84 Wilson & Rees  Australian Population Studies 5 (1) 2021 

of projections from a few years ago up to the latest population estimate, and comparing the 

‘projected’ populations and components with recorded data.  

4. The cohort-component model 

4.1. Overview 

The cohort-component model projects the population broken down by cohort by adding and 

subtracting the demographic components of change. It is usually also disaggregated by sex. The Lexis 

diagram in Figure 2 illustrates the age-time spaces of the cohort-specific population accounting 

equation which lies at the heart of the model. The population of a cohort at the start of the time 

interval, 𝑡, is depleted by deaths and emigration during the interval, but supplemented by 

immigration. After taking into account these component changes, we have the cohort population size 

at the end of the time interval, 𝑡 + 1. 

 
 
 
Final cohort: 

  

 
                          immigration 

  End-of-interval  
 population 
 aged 101+ 

                            Start-of-interval  
                                   population 
                                    aged 100+ 

  
       emigration & deaths 
 

 
 
 
              

  

            Age 
                            Immigration 
 
Most cohorts: 

  End-of-interval  
 cohort population 
 aged a+1 

                            Start-of-interval  
                         cohort population 
                                         aged a 

  
        emigration & deaths 
 

   

                                Immigration 
 
First cohort: 

  End-of-interval  
 cohort population 
 aged 0 

  
Births 

 
Time → 

              emigration & deaths 

Figure 2: Defining features of the cohort-component population projection model 

Notes. Population stocks are shown by thick black lines, and births by the thick blue line. The age-time extent of 

immigration, emigration and deaths for the cohorts shown is depicted by the grey shaded areas. 

age ∞ 

t                    t+1 



Australian Population Studies 5 (1) 2021 Wilson & Rees  85 

The middle part of the diagram shows the age-time spaces of stocks and flows for all cohorts except 

the very oldest and youngest. The cohort aged 𝑎 at the start of the projection interval ages to 

become aged 𝑎 + 1 by the end, and during the year experiences population losses through deaths 

and emigration, and gains from immigration. The lower part of the diagram illustrates the newly-

born infant cohort where the ‘starting’ population is the number of births which occur during the 

year. The upper section of the diagram illustrates the slightly different situation at the highest ages. 

The elongated extent of this part of the diagram indicates the absence of an upper age limit. So, for 

example, if the oldest open-ended age group in the projections is 100+, then this part of the diagram 

illustrates the population aged 100+ at time 𝑡 ageing to become the 101+ population at time 𝑡 + 1. 

There are three main stages to the cohort-component model calculations: 

• Stage 1: project cohorts already alive at the start of the projection interval 

• Stage 2: project the number of births 

• Stage 3: project newly-born babies to age 0 at the end of the projection interval 

which we describe in turn below. 

4.2. Stage 1: Project existing cohorts 

In the first stage, all cohort populations alive at the start of the projection interval are projected. This 

refers to all cohorts except for the cohort of babies born during the interval. The cohort-specific 

population accounting equation may be written as: 

𝑃𝑠,𝑎+1(𝑡 + 1) =  𝑃𝑠,𝑎(𝑡) −  𝐷𝑠,𝑎→𝑎+1 − 𝐸𝑠,𝑎→𝑎+1 + 𝐼𝑠,𝑎→𝑎+1     (2) 

where: 

𝑠 = sex  

𝑎 = age group  

𝑎 + 1 = age group one year older than 𝑎 

𝑎 → 𝑎 + 1 = the period-cohort aged 𝑎 at the start of the time interval which shifts to age 𝑎 + 1 by 

the end (as shown by the shaded parallelogram in the middle of Figure 2). 

Time labels 𝑡, 𝑡 + 1 have been omitted from the components in the equation to avoid clutter.  

The accounting equation above is converted to a projection equation by substituting deaths and 

emigration flows for occurrence/exposure rates multiplied by populations-at-risk (Rees 1984; 

Willekens & Drewe 1984): 

𝑃𝑠,𝑎+1(𝑡 + 1) =  𝑃𝑠,𝑎(𝑡) −  𝑑𝑠,𝑎→𝑎+1  
1

2
(𝑃𝑠,𝑎(𝑡) + 𝑃𝑠,𝑎+1(𝑡 + 1)) 

−𝑒𝑠,𝑎→𝑎+1  1

2
(𝑃𝑠,𝑎(𝑡) + 𝑃𝑠,𝑎+1(𝑡 + 1)) + 𝐼𝑠,𝑎→𝑎+1   (3) 

where: 

𝑑 = death rate 

𝑒 = emigration rate. 



86 Wilson & Rees  Australian Population Studies 5 (1) 2021 

The population-at-risk is the number of person-years the cohort population is exposed to the risk of 

death or emigration over the projection interval. It is approximated by the mean of the start- and 

end-of-interval populations. For the moment, we leave aside the issue of preparing the various 

assumed age-specific rates and return to this in section 5 on input data and assumptions. 

In the cohort-component model described here, immigration is projected directly as immigration 

numbers3. It would be possible in theory to project immigration as the product of immigration rates 

and the rest of the world’s population, but we would then have to include the rest of the world in the 

projections to calculate the required population-at-risk for sending migrants to the country we are 

producing projections for. In any case, immigration to many countries is strongly shaped by the 

migration policies and regulations of that country. For this reason, it is often projected directly as 

immigration flows. Some cohort-component models use net international migration, but we advise 

against this wherever possible. See Box 2 for a brief note on this issue. 

By re-arranging the projection equation, the end-of-interval populations can be removed from the 

right-hand side (Willekens and Drewe 1984). First, multiplying out the terms in brackets gives: 

𝑃𝑠,𝑎+1(𝑡 + 1) =  𝑃𝑠,𝑎(𝑡) −  1

2
 𝑑𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎(𝑡) − 1

2
 𝑑𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎+1(𝑡 + 1) 

− 1
2

 𝑒𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎(𝑡) − 1
2

 𝑒𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎+1(𝑡 + 1) + 𝐼𝑠,𝑎→𝑎+1. 

Then those terms involving the end-of-interval populations are all moved to the left-hand side of the 

equation: 

𝑃𝑠,𝑎(𝑡 + 1) + 1

2
 𝑑𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎+1(𝑡 + 1) + 1

2
 𝑒𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎+1(𝑡 + 1) 

=  𝑃𝑠,𝑎(𝑡) −  1
2

 𝑑𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎(𝑡) −  1
2

 𝑒𝑠,𝑎→𝑎+1 𝑃𝑠,𝑎(𝑡) + 𝐼𝑠,𝑎→𝑎+1. 

This can be simplified to: 

(1 + 1

2
 𝑑𝑠,𝑎→𝑎+1 + 1

2
 𝑒𝑠,𝑎→𝑎+1) 𝑃𝑠,𝑎+1(𝑡 + 1)

= (1 −  1

2
 𝑑𝑠,𝑎→𝑎+1 − 1

2
 𝑒𝑠,𝑎→𝑎+1) 𝑃𝑠,𝑎(𝑡) + 𝐼𝑠,𝑎→𝑎+1 

and then re-arranged so that just the end-of-interval population remains on the left-hand side of the 

equation: 

𝑃𝑠,𝑎+1(𝑡 + 1) =
(1− 1

2
 𝑑𝑠,𝑎→𝑎+1−1

2
 𝑒𝑠,𝑎→𝑎+1)

(1+1
2

 𝑑𝑠,𝑎→𝑎+1+1
2

 𝑒𝑠,𝑎→𝑎+1)
𝑃𝑠,𝑎(𝑡) +

𝐼𝑠,𝑎→𝑎+1

(1+1
2

 𝑑𝑠,𝑎→𝑎+1+1
2

 𝑒𝑠,𝑎→𝑎+1)
   (4) 

To illustrate the calculations, we use the example of the female cohort aged 45 years in mid-2019 

and 46 in mid-2020. If the population in 2019 was 164,940, the period-cohort death rate 0.001127, 

the emigration rate 0.005916, and the immigration flow 1,693.3, then the projected population in 

2020 is: 

 
3 Alternative ways of handling international migration in the cohort-component are often applied. See also 
Bijak (2012), Rees (2015), and Vandresse (2015). 



Australian Population Studies 5 (1) 2021 Wilson & Rees  87 

=
(1 −  1

2
 0.001127 − 1

2
 0.005916)

(1 + 1
2

 0.001127 + 1
2

 0.005916)
164,940 +

1,693.3

(1 + 1
2

 0.001127 + 1
2

 0.005916)
  

= 165,469.84 

4.3. Stage 2: Project the number of births 

In the second stage, births are projected as the product of age-specific fertility rates and female 

populations-at-risk, and then summed over all childbearing age groups (usually ages 15 to 49). The 

total number of projected births is calculated as: 

𝐵 = ∑ (𝑏𝑎  
1

2
[𝑃𝑓,𝑎(𝑡) + 𝑃𝑓,𝑎(𝑡 + 1)])𝑎        (5) 

where: 

𝑏 = age-specific fertility rate 

𝑓 = female population. 

Note how the calculations make use of end-of-interval female populations which were calculated in 

the first stage. Age-specific fertility rates are period-age (shown by the shaded square in Figure 1) 

and not period-cohort because this is the form in which age-specific births and fertility statistics are 

often provided5. 

As an example, consider females aged 30. If the age-specific fertility rate for women aged 30 is 

0.112436 and the start- and end-of-interval populations of women aged 30 are 193,557 and 198,656 

respectively, then the projected number of births born to women in this age group is: 

= 0.112436  
1

2
 [193,557 + 198,656] 

= 22,049.4 

Projected births from each age group of women are summed to obtain total births, and they are then 

divided into male and female births using the assumed sex ratio at birth: 

𝐵𝑓 = 𝐵 
100

100 + 𝑆𝑅𝐵
 

𝐵𝑚 = 𝐵 
𝑆𝑅𝐵

100 + 𝑆𝑅𝐵
 

where: 

𝑆𝑅𝐵 = sex ratio at birth, the number of males born for every 100 females. 

Typically, this lies between 103 and 107 male births per 100 female births, with a mean of about 105 

(Ritchie 2019). 

 
4 In reality, there cannot be fractional population numbers, of course. But during the calculations, fractions are 
usually retained, and rounding applied to the final projection outputs at the end of the calculation process. 
5 It would be possible to project births using period-cohort fertility rates and populations-at-risk, but it isn’t 
necessary for projecting total numbers of births. 



88 Wilson & Rees  Australian Population Studies 5 (1) 2021 

As an example, assume the total number of projected births is 304,365 and the SRB is 105.7 males 

per 100 females. Then the number of female babies is: 

= 304,365 
100

100+105.7
  

= 147,965.5  

and the number of male babies: 

= 304,365 
105.7

100+105.7
  

= 156,399.5 

4.4. Stage 3: Project the infant cohort 

The third stage involves projecting the cohort of newly-born babies forwards to obtain the 

population aged 0 at the end of the projection interval. The population accounting equation includes 

the number of births as the ‘initial’ population: 

𝑃𝑠,0(𝑡 + 1) =  𝐵𝑠 −  𝐷𝑠,𝑏→0 − 𝐸𝑠,𝑏→0 + 𝐼𝑠,𝑏→0      (6) 

where: 

𝑏 → 0 = the infant cohort born during the projection interval and aged 0 at the end (illustrated by 

the shaded triangle at the bottom of Figure 2). 

For the projection equation, deaths and emigration flows are replaced by rates multiplied by the 

population-at-risk: 

𝑃𝑠,0(𝑡 + 1) =  𝐵𝑠 −  𝑑𝑠,𝑏→0   
1
2

 𝑃𝑠,0(𝑡 + 1) −  𝑒𝑠,𝑏→0 1
2

 𝑃𝑠,0(𝑡 + 1) + 𝐼𝑠,𝑏→0   (7) 

The population-at-risk is approximated as half the end-of-interval population. This provides a 

reasonable estimate of the person-years lived by the newly-born cohort where there is low infant 

mortality because the end-of-interval population consists of nearly all members of the new cohort 

who were born, and they were part of the population for an average of half the time interval. 

With re-arranging, this projection equation can be simplified to: 

𝑃𝑠,0(𝑡 + 1)  +  1

2
  𝑑𝑠,𝑏→0 𝑃𝑠,0(𝑡 + 1)  +  1

2
  𝑒𝑠,𝑏→0 𝑃𝑠,0(𝑡 + 1)  =  𝐵𝑠 + 𝐼𝑠,𝑏→0 

then: 

(1 + 1

2
  𝑑𝑠,𝑏→0 + 1

2
  𝑒𝑠,𝑏→0)𝑃𝑠,0(𝑡 + 1)  =  𝐵𝑠 + 𝐼𝑠,𝑏→0 

and: 

𝑃𝑠,0(𝑡 + 1)  =  
1

(1+1
2

  𝑑𝑠,𝑏→0+1
2

  𝑒𝑠,𝑏→0)
𝐵𝑠  +   

𝐼𝑠,𝑏→0

(1+1
2

  𝑑𝑠,𝑏→0+1
2

  𝑒𝑠,𝑏→0)
    (8) 

To illustrate the calculations, we take the female cohort born during 2019-20 which ends the 

projection interval in mid-2020 aged 0. If there are 147,965.5 births, a death rate of 0.005207, an 



Australian Population Studies 5 (1) 2021 Wilson & Rees  89 

emigration rate of 0.012508, and an immigration flow of 1,865.8, then the end-of-interval population 

aged 0 is: 

=  
1

(1 + 1
2

  0.005207 + 1
2

  0.012508)
147,965.5 +  

1,865.8

(1 + 1
2

  0.005207 + 1
2

  0.012508)
  

= 148,515.8 

4.5. Prepare for the next projection interval 

The end-of-interval populations then become the start-of-interval populations for the next projection 

interval. This requires summing the end-of-interval populations for the two oldest cohorts. For 

example, if the oldest age group in the projections output is 100+, then the projected end-of-interval 

populations at age 100 and age 101+ must be summed to obtain the 100+ start-of-interval 

population for the next projection interval. 

𝑃𝑠,100+(𝑡 + 1) = 𝑃𝑠,100(𝑡 + 1) + 𝑃𝑠,101+(𝑡 + 1)      (9) 

The projection calculations are then repeated for as many years ahead as required. 

 

Box 2: Why is it best to avoid projecting international migration as net migration?  

Projecting international migration as net migration numbers is appealing because the calculations are 

simple and there are minimal data requirements. In fact, net migration can be calculated from a 

recent year as the remaining cohort population change once deaths have been taken into account, so 

no actual migration data are needed. However, there are both conceptual and practical limitations of 

modelling net migration. Conceptually, there is no actual demographic process of net migration, only 

immigration flows into a country and emigration flows out. In addition, by using net migration 

numbers, we remove the direct relationship between the age pattern of migration and the 

population age structure. By using emigration rates, emigration is projected as a function of the 

population-at-risk in each period-cohort, so that the age pattern of emigration is influenced by 

population age structure. From a practical perspective, net migration age patterns are much more 

variable over time than immigration and emigration age patterns, and are shaped by the relative 

amounts of immigration and emigration. It is therefore difficult to adjust the age profile of net 

migration over the course of the projection horizon (in response to migration policy changes or 

global pandemics, for example). Simply scaling the net migration age profile up and down is unlikely 

to give a realistic outcome. In addition, in some extreme circumstances, negative net migration held 

constant over long periods can yield ‘negative’ populations. 

 

4.6. Strengths and weaknesses 

The cohort-component model is widely seen as the ‘gold standard’ method of population projection, 

and it possesses many excellent features. But it also has some weaknesses which are worth keeping 

in mind. Some of the strengths and weaknesses are listed in Table 1 below (see also Burch 2018). 

  



90 Wilson & Rees  Australian Population Studies 5 (1) 2021 

Table 1: Some of the strengths and weaknesses of the cohort-component model 

Strengths Weaknesses 

Good representation of actual population 
processes. It includes the effects of population 
age structure and age-specific rates 

Assumes a homogenous population, with no 
allowance for the changing internal composition 

Can be accurate if good assumptions about 
fertility, mortality and migration are chosen 

Can be highly inaccurate if poor assumptions 
about fertility, mortality and migration are chosen 

Provides age-disaggregated projections which are 
useful for a wide variety of planning, policy, 
budgeting, and service delivery purposes 

Requires a non-trivial amount of input data; and 
assumptions about future fertility, mortality and 
migration are not intrinsic to the model and have 
to be prepared separately 

Provides the starting point for many derived 
projections (e.g., labour force; living arrangement; 
school enrolments) 

Lack of in-built mechanism for ensuring 
consistency between male and female projections 

Useful for understanding population change     
(e.g., by creating alternative future scenarios; or 
applying to past periods) 

Immigrants to the population are assumed to take 
on the fertility, mortality and emigration rates 
assumed for the whole population 

Simple mathematics. Basic versions can be 
implemented in a spreadsheet 

Births are produced by women only, rather than 
men and women. 

 

4.6. The Excel workbook  

The Excel workbook accompanying this paper (National projection model example.xlsx) contains an 

example set of projections for Australia using the cohort-component model equations presented 

above. It includes the following worksheets. 

Notes: A short introduction to the layout and workings of the Excel workbook. 

JumpOff: The jump-off populations should be supplied in this sheet. 

Fertility: This is where the fertility assumptions are placed. To aid assumption-setting, fertility 

assumptions are provided separately for the level (TFR) and age profiles (age-specific rates) of 

fertility. The age-specific fertility rates are scaled to match the TFR further down the sheet. This 

approach allows you to make slight adjustments to the TFR assumptions without having to re-

calculate the age-specific rates. 

Mortality: This is where the projected 1Lx life table values, used to calculate death rates by period-

cohort, are placed. 

Migration: Like fertility, migration assumptions are split into level and age profile components to aid 

assumption-setting. For immigration, both the annual total immigration intake and immigration 

numbers by period-cohort are required. For emigration, the assumed Gross Migraproduction Rate 

(GMR) (Rogers et al. 1978) for the overall level of emigration, and emigration rates by period-cohort, 

are needed. The GMR is simply the sum of all age-specific rates, much like the Total Fertility Rate. 

Further down the sheet the supplied immigration numbers and emigration rates are scaled to sum 

to, respectively, the total immigration and emigration GMR assumptions. 



Australian Population Studies 5 (1) 2021 Wilson & Rees  91 

Projections: The projection calculations are carried out in this sheet using equations 4, 5 and 8. Scroll 

down to find projected births, deaths, and emigration flows by sex and period-cohort. 

Population: The final projected populations are set out in this sheet. 

AccountsSummary: Summary population accounts for each single year projection interval are 

provided in this sheet (start-of-interval populations, births, deaths, immigration, emigration, and 

end-of-interval populations).  

5. Input data and projection assumptions 

The cohort-component model requires the following input data: 

• Jump-off populations by sex and age group 

• Future fertility rates by age of mother 

• Future mortality rates by sex and period-cohort 

• Future emigration rates by sex and period-cohort 

• Future immigration numbers by sex and period-cohort 

5.1. Jump-off populations 

Jump-off population estimates by sex and single years of age are needed to provide the starting point 

of the projections. Ideally the age disaggregation would extend up to an age high enough so that 

there are relatively few people in the highest open-ended age group. Usually the jump-off 

populations would be the most recently published population estimates available from the national 

statistical office. 

5.2. Fertility 

Age-specific fertility rates are required for each year of the projection horizon. Sometimes age-

specific rates are projected separately from the Total Fertility Rate, and then adjusted to be 

consistent with the TFR if necessary. Three broad approaches are suggested here: 

1) Assume the average TFR and age-specific fertility rates of recent years remain constant into the 

future. If fertility has remained approximately constant in recent years, and there are no obvious 

reasons for assuming change, this could be a reasonable assumption. 

2) Use fertility assumptions prepared by others, such as the national statistical office (e.g. ABS 2018), 

the United Nations Population Division (2019), or researchers who have prepared projections and 

make the data available (e.g. McDonald 2020). 

3) Prepare your own projections. This option does come with some challenges, however. There is no 

single fertility projection method widely regarded as the best option. And, despite considerable 

amounts of research over many decades to understand fertility better, it remains notoriously hard 

to predict. 

Some fertility projections are produced from extrapolative models, some from explanatory models 

linked to variables such as unemployment rates, some projections are created by trending from 

recent fertility to a long-run target, while others are based on expert opinions. An overview of 

fertility projection methods used in practice can be found in Gleditsch and Syse (2020), while reviews 



92 Wilson & Rees  Australian Population Studies 5 (1) 2021 

of the academic literature are included in Booth (2006), Shang and Booth (2020), Hilton et al. (2019), 

and Bohk-Ewald et al. (2018) ,though this last paper focuses largely on cohort fertility projections. In 

practice, fertility assumptions are often created by drawing on a variety of models and approaches.  

The TFR assumptions for Australia in the example Excel workbook accompanying this paper were 

prepared qualitatively, but were informed by other projections (McDonald 2020), research on the 

impacts of major economic shocks to fertility (e.g. Matysiak et al. 2020; Sobotka et al. 2011), and the 

declining fertility trend observed in Australia in recent years (ABS 2020b). A long-run TFR of 1.70 was 

assumed, with a short-run dip to allow for the economic impacts of COVID-19. The age profile of 

fertility rates was projected by extrapolating the parameters of the Peristera and Kostaki (2007) 

model of fertility, which fits well to the Australian fertility rate age profile. 

5.3. Mortality 

Death rates by sex and period-cohort are required for every year of the projection horizon. Unlike 

fertility, the assumption of holding rates constant into the future is rarely a good choice for mortality 

because it tends to decline over time. Therefore, two broad approaches are recommended here. 

1) Use mortality assumptions prepared by others, such a national statistical office. 

2) Prepare your own mortality projections. National mortality forecasting has received the greatest 

amount of attention in demographic forecasting research and there is no shortage of methods 

(Booth and Tickle 2008; Ediev 2020; Terblanche 2015) and software (e.g., Hyndman et al. 2019; 

Sevcikova et al. 2019) to choose from.  

Generally, mortality trends are fairly stable over time, making it easier to forecast than fertility and 

migration (though in some countries COVID-19 has now interrupted those trends). Many of the 

mortality forecasting methods and software packages devised over the last couple of decades have 

been shown to produce quite accurate forecasts (e.g. Terblanche 2016). 

The mortality assumptions in the Excel workbook accompanying this paper were created using 

Ediev’s (2008) extrapolative mortality model. Age-specific death rates (in period-age form) were 

projected, and then life tables created. The Excel workbook contains projected 1Lx values which are 

used to calculate period-cohort death rates. Because these values comprise the life table stationary 

population, the difference between populations at adjacent ages can be interpreted as the number 

of period-cohort deaths, shown by the shaded parallelogram in the Lexis diagram in Figure 3. Note 

that the life table populations and deaths exactly match the age-time locations shown for the cohort-

component model earlier (middle of Figure 2). Death rates are calculated with these deaths in the 

numerator and the mean of the two populations as the population-at-risk: 

𝑑𝑎→𝑎+1 =
𝐿𝑥1 − 𝐿𝑥+11

1
2

( 𝐿𝑥1 + 𝐿𝑥+11 )
   

where 𝑥 is the exact starting age of age group 𝑎. The sex subscript is omitted in the equation to 

prevent clutter. 

  



Australian Population Studies 5 (1) 2021 Wilson & Rees  93 

 
 
                                          
                                        Age 

   
 1Lx+1 

 
                                               1Lx 
 

  

 
 

Time →  

Figure 3: Life table data used to calculate period-cohort death rates 

For newly-born infants and at the very highest ages, death rates are calculated slightly differently: 

𝑑𝑏→0 =
𝑙0− 𝐿01

1
2

 𝐿01
  

𝑑100+→101+ =
𝐿100∞ − 𝐿101∞

1
2

( 𝐿100∞ + 𝐿101∞ )
  

where 

𝑙0 = the radix (or number of births) in the life table population. 

These deaths and populations exactly match the age-time locations as indicated for the youngest and 

oldest cohorts in Figure 2. 

5.4. International migration 

International migration is hard to forecast. It tends to exhibit quite volatile trends in volume over 

time in response to changes of policy (at least for immigration), macroeconomic conditions, and 

major world events. And for some countries (though not Australia), international migration data 

suffers from quality and coverage problems. Fortunately, the age profiles of immigration and 

emigration rates tend to be far more stable. Our suggested approaches to migration forecasting are 

like those for fertility: 

1) Assume the trends of recent years remain constant into the future. In the absence of clear 

changes in trends in immigration and emigration, this can be a reasonable assumption. 

2) Use migration assumptions prepared by others (e.g. ABS 2018). 

3) Prepare your own migration projections. This can be a challenging task. 

Unlike fertility and mortality, there are relatively few available projection methods and software 

packages to choose from and, unfortunately, no single method has been found to produce 

consistently accurate migration projections. Projection approaches include extrapolative models, 

explanatory models, trending to a target, expert panels, and what might best be described as a 

“forecaster’s judgement” approach that is not based on any formal model or process. This last 

approach qualitatively takes into account recent migration trends, perceptions of factors driving 

those trends, broad economic trends, policy influences (including immigration program plans), and 

expert opinions, among other things. It is only loosely connected to theory and modelling – but that 

is not necessarily a weakness given that theory and modelling have not yet delivered a robust 

migration projection method. 



94 Wilson & Rees  Australian Population Studies 5 (1) 2021 

If a more formal process or model is to be used, what are the options? A few examples are given 

here. One option is to apply time series or econometric models. Examples of these types of models 

can be found in Bijak et al. (2019), de Beer (2008), and Raymer and Wiśniowski (2018). Statistics 

Norway uses a model based on neoclassical economic theory to project various immigration streams 

(Cappelen et al. 2015). Explanatory variables include relative income, Norwegian unemployment 

rates, and the size of the migrant population. A similar explanatory model of migration has been 

devised for use in Belgium’s population projections (Vandresse 2015). Another approach is to engage 

expert panels who are actively involved in the assumption formulation process (e.g. Dion et al. 2020; 

Lutz 2009). The expert panel considers and discusses a wide range of factors which might increase or 

decrease migration (and other demographic variables) and then suggests future migration levels or 

distributions. Whichever methods and approaches are selected, many authors have recommended 

dividing international migration flows into different streams (e.g. permanent immigrants, temporary 

workers, foreign students, etc.) and creating projections by focusing on the different drivers of those 

flows (e.g. de Beer 2008; Wilson 2017a). 

For the Excel workbook, immigration and emigration assumptions for Australia were prepared 

separately for the headline indicators of total immigration flows and the emigration Gross 

Migraproduction Rate (GMR), and the age profiles of immigration and emigration rates. Given that 

the shapes of immigration and emigration rate age profiles tend to change slowly over time, they 

were assumed to remain unchanged in the projections. In contrast, the total volumes of immigration 

and emigration can change markedly from year to year. Given the disruption to international 

migration created by the COVID-19 pandemic and the closure of the Australian border, we took note 

of the projected migration flows from the federal government’s Centre for Population in formulating 

our assumptions over the next few years (Centre for Population, 2020). A large drop in both 

immigration and emigration was assumed in the short term, reflecting the border restrictions 

introduced in response to the pandemic. For further into the future, we assumed total immigration 

would resume its long-run gradual upward trend and that the emigration GMR would return to its 

previous level and remain constant. 

6. Checking outputs 

The final part of the projections process involves checking the projections output for plausibility, 

consistency, and any possible errors with input data preparation or projections calculations. This is a 

form of exploratory data analysis for projections data. What constitutes ‘plausible’ is not easy to 

define, but projections outputs should look roughly in line with expectations given the chosen input 

assumptions. Checks of plausibility may not provide a precise answer. Instead, they may only suggest 

that assumptions would benefit from some refinement, or that certain data inputs may need 

attention. Unfortunately, just because projections seem plausible does not mean that they will turn 

out to be accurate, but it should at least mean they will be free of obvious errors or unbelievable 

assumptions. 

Total population. Check that the projected total population trend looks plausible in light of historical 

data by creating a graph of past and projected total populations (and/or total population growth 

rates). Figure 4 shows an example. In most circumstances there should be no discontinuity or sudden 



Australian Population Studies 5 (1) 2021 Wilson & Rees  95 

changes in trend, unless there is clearly good reason for it (e.g., the impact of a pandemic; a sudden 

immigration policy change). 

 

Figure 4: Example graph of total population to check projection outputs 

Source: ABS (2020a); authors’ projections 

Demographic components. Do the projected demographic components of change look plausible in 

light of historical trends? Again, it is useful to create a graph showing past and projected births, 

deaths, immigration and emigration. Births and deaths normally show relatively smooth trends over 

time, while immigration and emigration are often more variable.  

Population age-sex structure. Does the evolution of the age-sex structure of the population look 

plausible? It is useful to create population pyramids, or at least line graphs of the population by age, 

over time. If a population has been ageing in the past, it is likely to continue to age in the future. 

Does the number of 0 year olds seem consistent with the fertility assumptions? Does the sex ratio of 

the total population (males per 100 females) change slowly over time and not differ from the initial 

value too much? Does the sex ratio by age show a pattern of values above 100 in the childhood and 

younger adult ages followed by declines with increasing age? Does it change slowly over time? If 

male and female life expectancy is projected to converge in the future, does the sex ratio at the older 

ages increase? Does the relative size of the young adult population seem about right when 

considering the migration assumptions? 

Others’ projections. It can be useful to compare your projections with those produced by others, 

such as projections produced by a national statistical office. Unless your projections have radically 

different assumptions, the broad shape of the outputs is likely to be similar, at least in the initial 

years of the projection horizon. But differences with other projections should not necessarily be 

interpreted as problems with your projection. If you have a higher fertility assumption, for example, 

you would expect a larger projected childhood population. 

For a more detailed checklist see ‘A checklist for reviewing draft population projections’ (Wilson 

2017b). 

 



96 Wilson & Rees  Australian Population Studies 5 (1) 2021 

7. Conclusions 

This paper has presented a brief guide to producing a national population projection with a cohort-

component model. Several stages in the projections preparation process were described, starting 

with data gathering and assumptions preparation, and finishing with projection calculations and 

output checking. We described the cohort-component model from a population accounting 

perspective, which derives projection equations directly from simple cohort-specific population 

accounting equations. And we created an Excel workbook to accompany this paper which 

implements the cohort-component model for Australia. 

The paper covered many of the key elements of the projections preparation process, but it has not 

been possible to discuss all relevant issues in one short paper. We said little about the smoothing of 

age profiles of fertility, mortality and migration rates to remove noise, though noise is generally 

minimal at the national scale. This topic is covered by Preston et al. (2001: chapter 9), de Beer (2011, 

2012), and Wilson (2020). We didn’t go into much detail about population accounting reconciliation 

because accounting discrepancies are usually not a major problem at the national scale, nor did we 

consider forecast uncertainty and error (on this, see Keilman 2018, 2020, and for an Australian 

example of probabilistic forecasts, see Bell et al. 2011). We also did not cover any topics which 

become relevant once the projections have been finalised, such as the best ways of communicating 

projections to stakeholders (Johnstone 2015). 

It is important to note that while preparing a population projection may initially seem like a mostly 

quantitative and objective exercise, there are actually many judgements which must be made as part 

of the process. While some may seem like minor technical decisions, they require careful 

consideration because they might substantially impact the projection outputs. For example, decisions 

are made about input data adjustments, about which fertility, mortality, and migration projection 

methods and assumptions to use, and whether the outputs seem plausible and reasonable. 

Mastering the calculations for population projections is relatively easy; making good decisions 

throughout the projections process is more challenging and is developed with practice. 

 

Excel workbook 

The Excel workbook containing a cohort-component model and example population projections for 

Australia may be downloaded from https://doi.org/10.6084/m9.figshare.13565222.v3. 

Acknowledgements 

Funding support for TW from the Australian Research Council Centre of Excellence in Population 

Ageing Research is gratefully acknowledged (project number CE1101029).  

Helpful comments on an earlier draft of this paper were kindly provided by Ahana Banerjee Sen, 

Rosabella Borsellino, Dagmara Laukova, and the anonymous reviewers. 

  

https://doi.org/10.6084/m9.figshare.13565222.v3


Australian Population Studies 5 (1) 2021 Wilson & Rees  97 

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