A stochastic forecast for the Croatian pension system

Tomaš, Ivan

doi:10.3326/pse.44.1.2.

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A stochastic forecast for the Croatian pension system*

Ivan Tomaš*

Ivan Tomaš

0000-0002-6598-9601

Correspondence
itomas97@gmail.com

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https://doi.org/10.3326/pse.44.1.2

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A stochastic forecast for the Croatian pension system*

Ivan Tomaš

https://doi.org/10.3326/pse.44.1.2 | Published online: March 3, 2020

Figure 1
Total population in Croatia (in millions)

Source: Eurostat.

Figure 2
Total population in Croatia in two different age groups (in millions)

Source: Eurostat.

Figure 3
Total population in Croatia in two different age groups (in millions)

Source: Eurostat.

Figure 4
Net migration in Croatia (in thousands)

Source: Eurostat.

Table 1
Dickey-Fuller test statistics results for random walk and random walk with a drift

Age	Test Statistics	Test Statistics
Age	∆Y_t= α₁ Y_t-₁+ ε_t	∆Y_t= α₀ + α₁ Y_t-₁+ ε_t
15-19	τ = -1.9585	τ = -3.9106	ф = 5.6557
20-24	τ = -1.6152	τ = -4.2537	ф = 9.0580
25-29	τ = -2.1451	τ = -2.4060	ф = 2.8946
30-34	τ = -1.3703	τ = -2.1818	ф = 2.3866
35-39	τ = -0.7783	τ = -3.5498	ф = 6.6255
40-44	τ = -0.9005	τ = -3.0782	ф = 5.1171
45-49	τ = -3.5818	τ = -4.1482	ф = 8.6203

Source: Author’s calculations.

Table 2
Dickey-Fuller statistics and confidence intervals for the data of 16 observations

Model	Hypothesis	Test Statistics	Confidence intervals
Model	Hypothesis	Test Statistics	95%	99%
∆Y_t= α₁ Y_t-₁+ ε_t	α₁= 0	τ	-1.95	-2.66
∆Y_t= α₀ + α₁ Y_t-₁+ ε_t	α₁= 0	τ	-3.00	-3.75
	α₀ = α₁ = 0	ф	5.18	7.88

Source: Author’s calculations

Table 3
Parameters in the Lee-Carter model

Age	Parameter
Age	α₁	*b_i*
15-19	-5.11	-0.33
20-24	-3.55	-0.35
25-29	-3.03	-0.09
30-34	-3.21	0.20
35-39	-4.09	0.38
40-44	-5.78	0.42
45-49	-8.85	0.64

Source: Author’s calculations

Table 4
Dickey-Fuller statistics for newborns per person index

Test Statistics	Test Statistics
∆k_t = α₁ k_t-₁+ ε_t	∆k_t = α₀ + α₁ k_t_-1+ ε_t
τ = -2.4176	τ = -5,0459	ф = 12.8369

Source: Author’s calculations.

Figure 5
Autocorrelation function (ACF) for newborns per person index

Note: The dotted line is used as a boundary for statistical significance. Everything that is above the dotted line is statistically significant.

Source: Author’s calculations.

Figure 6
Partial autocorrelation function (PACF) for newborns per person index

Source: Author’s calculations.

Table 5
AR (1) process of newborns per person index

Estimated equation	R²	s	*E(k_t)*
k_t= 0.0528 + 0.9639 * k_t_-1+ ε_t	0.8635	0.1975	1.4598

Source: Author’s calculations.

Figure 7
Stochastic simulation results for newborns per person index

Source: Author’s calculations.

Figure 8
Stochastic simulation results for newborns per person rates, age group: 15-19

Source: Author’s calculations.

Figure 9
Stochastic simulation results for newborns per person rates, age group: 30-34

Source: Author’s calculations.

Figure 10
Stochastic simulation of total newborns per person rates, time horizon: 2017-2060

Source: Author’s calculations.

Table 6
Parameters for mortality index in the Lee-Carter model

Age	Parameter
Age	a₁	*b_i*
0-1	-5.29	-0.28
1-4	-8.44	-0.28
5-9	-8.99	-0.19
10-14	-8.94	-0.20
15-19	-7.83	-0.38
20-24	-7.43	-0.37
25-29	-7.40	-0.29
30-34	-7.20	-0.26
35-39	-6.80	-0.21
40-44	-6.32	-0.27
45-49	-5.73	-0.24
50-54	-5.19	-0.20
55-59	-4.75	-0.13
60-64	-4.33	-0.09
65-69	-3.91	-0.14
70-74	-3.43	-0.20
75-79	-2.89	-0.17
80+	-2.01	-0.08

Source: Author’s calculations.

Table 7
Dickey-Fuller statistics for mortality index

Test Statistics	Test Statistics
∆K_t = α₁ k_t-₁+ ε_t	∆k_t = α₀ + α₁ k_t_-1+ ε_t
τ = -1.1768	τ = -2.3118	ф = 2.6724

Source: Author’s calculations.

Figure 11
Mortality index over time

Source: Author’s calculations.

Figure 12
Mortality rates in the USA for the period of 1930-2017

Source: Mortality.org (2019).

Table 8
Dickey-Fuller statistics for mortality index set as a model with linear trend

Test Statistics	Test Statistics
∆K_t = α₁ K_t-₁+ ε_t	∆K_t = α₀ + α₁ K_t_-1+ ε_t
τ = -2.4707	τ = -2.3118	ф = 2.6724

Source: Author’s calculations.

Table 9
Estimated parameters for mortality index

Coefficients	Estimate	Standard error	t-value	Pr(>\|t\|)
b₀	-1.2012	0.0746	-16.11	~0
b₁	0.1335	0.0073	18.34	~0

Source: Author’s calculations.

Figure 13
Mortality index simulations

Source: Author’s calculations.

Figure 14
Simulated results for mortality rates for the first cohort: newborns

Source: Author’s calculations.

Figure 15
Simulated results for the mortality index for group aged 45-49

Source: Author’s calculations.

Figure 16
Stochastic forecast of the Croatian population from 2018 to 2060

Note: Generated by the Leslie stochastic matrix.

Source: Author’s calculations.

Figure 17
Histogram of the Croatian population in 2060 forecast by using the stochastic method on the Leslie model

Source: Author’s calculations.

Figure 18
Stochastic forecast of the population aged 65+ from 2018 to 2060

Source: Author’s calculations.

Figure 19
Stochastic forecast of the population aged 15-65 from 2018 to 2060

Source: Author’s calculations.

Table 10
Proportion of specific age population in cohorts 60-64 and 65-69

Age	Weights in cohort
62	0.1996
63	0.1961
64	0.1925
65	0.2086
66	0.2039
67	0.2002

Source: Author’s calculations.

Table 11
Analysis of new pensioners

Ratio of new male pensioners to men aged 65	Ratio of new female pensioners to women aged 62	Women aged 62	Men aged 65	New female pensioners	New male pensioners	New pensioners	Year
0.92	0.85	31,654	26,229	26,930	24,145	51,075	2018
0.93	0.84	30,862	24,572	25,874	22,760	48,634	2017
0.95	0.84	30,644	25,518	25,868	24,127	49,995	2016
0.92	0.88	30,352	24,754	26,582	22,674	49,256	2015
1.06	0.97	28,119	23,014	27,223	24,303	51,526	2014
0.96	0.90	29,867	22,642	26,963	21,798	48,761	2013
1.00	0.88	30,076	21,014	26,513	20,946	47,459	2012
1.40	0.99	28,196	16,052	27,948	22,507	50,455	2011
1.41	1.28	27,957	17,636	35,835	24,834	60,669	2010
1.25			19,540	32,591	24,379	56,970	2009
1.12			22,004	29,798	24,588	54,386	2008
1.10			20,769	28,801	22,744	51,545	2007
1.02			22,387	28,255	22,901	51,156	2006

Source: Eurostat; HZMO; Author’s calculations

Figure 20
PACF for the ratio of new male pensioners to male population aged 65

Source: Author’s calculations.

Figure 21
PACF for new female pensioners to female population aged 62

Source: Author’s calculations.

Table 12
AR (1) process for ratio of new female/male pensioners to female/male population aged 62/65

Gender	Estimated equation	R²	s	*E(y_t)*
Female	Y_t = 0.1629 + 0.8242 * Y_t_-1 + ε_t	0.64	0.1421	0.93
Male	Y_t = 0.350618 + 0.67092 * Y_t_-1 + ε_t	0.42	0.1437	1.07

Source: Author’s calculations

Figure 22
Stochastic forecast of new male pensioners ratio from 2018 to 2060

Source: Author’s calculations.

Figure 23
Stochastic forecast of new female pensioners ratio from 2018 to 2060

Source: Author’s calculations.

Figure 24
Stochastic forecast of the number of pensioners from 2018 to 2060

Source: Author’s calculations.

Figure 25
Distribution histogram for the number of pensioners in 2060

Source: Author’s calculations.

Figure 26
Labour force to working age population

Source: CBS (2019); Eurostat; Author’s calculations.

Figure 27
PACF of labour force to working age population ratio

Source: Author’s calculations.

Table 13
AR (1) process of labour force to working age population ratio

Estimated equation	R²	s	*E(Y_t)*
Y_t = 0.1093 + 0.8278 * Y_t_-1 + ε_t	0.71	0.0152	0.63

Source: Author’s calculations

Figure 28
Stochastic forecast of labour force ratio from 2018 to 2060

Source: Author’s calculations.

Figure 29
Historical unemployment rate in Croatia

Source: CBS.

Figure 30
Forecast of the future unemployment rate: Scenario 1

Source: Author’s calculations.

Figure 31
Stochastic forecast of the employed population: Scenario 1

Source: Author’s calculations.

Figure 32
Stochastic forecast of the employed population: Scenario 2

Source: Author’s calculations.

Figure 33
Stochastic forecast of the employed population: Scenario 3

Source: Author’s calculations.

Table 14
Average gross salary and growth rates

Year	Average gross salary (in HRK)	Growth rates (in %)
2001	5,061	3.9
2002	5,366	6.0
2003	5,623	4.8
2004	5,985	6.4
2005	6,248	4.4
2006	6,634	6.2
2007	7,047	6.2
2008	7,544	7.1
2009	7,711	2.2
2010	7,679	-0.4
2011	7,796	1.5
2012	7,875	1.0
2013	7,939	0.8
2014	7,951	0.2
2015	8,055	1.3
2016	7,752	-3.8
2017	8,055	3.9
2018	8,448	4.9

Source: Narodne novine, author’s calculations

Figure 34
PACF of growth rates in wages

Source: Author’s calculations.

Table 15
AR (1) process for growth rates in wages

Estimated equation	R²	s	*E(Y_t)*
Y_t = 0.0143 + 0.5218 * Y_t_-1 + ε_t	0.28	0.0264	0.0299

Source: Author’s calculations.

Figure 35
Stochastic forecast for growth rates in wages

Source: Author’s calculations.

Figure 36
Stochastic forecast of future wages

Source: Author’s calculations.

Figure 37
Histogram of wage distribution in 2060

Source: Author’s calculations.

Table 16
Average pension by year and percentage in gross salary

Year	Average pension (in HRK)	Percentage in gross salary (in %)
2001	1,672	33
2002	1,720	32
2003	1,801	32
2004	1,848	31
2005	1,904	30
2006	2,000	30
2007	1,999	28
2008	2,169	29
2009	2,164	28
2010	2,151	28
2011	2,184	28
2012	2,502	32
2013	2,417	30
2014	2,423	30
2015	2,437	30
2016	2,525	33
2017	2,439	30

Source: HZMO; Author’s calculations

Figure 38
Stochastic forecast of government net cash flow from the pension system: Scenario 1

Source: Author’s calculations.

Figure 39
Stochastic forecast of government net cash flow from the pension system: Scenario 2

Source: Author’s calculations.

Figure 40
Stochastic forecast of government net cash flow from the pension system: Scenario 3

Source: Author’s calculations.

Figure 41
Expected present value of government future cash flow generated from the pension system: Scenario 3

Source: Author’s calculations.

Figure 42
Stochastic forecast of government net cash flow with the assumption that pensioners decide to use a combined pension scheme after 2030. Unemployment rate from scenario 1 is used

Source: Author’s calculations.

Figure 43
Stochastic forecast of government net cash flow with the assumption that pensioners decide to use a combined pension scheme after 2030. Unemployment rate from scenario 3 is used

Source: Author’s calculations.

Figure 44
Stochastic forecast of government net cash flow with the assumption that the second pillar gains enough competitiveness to cover 50% of expenses after 2040. Scenario 1 unemployment rate is used

Source: Author’s calculations.

Figure 45
Stochastic forecast of government net cash flow with the assumption that the second pillar gain enough competitiveness to cover 50% of expenses and extra assumption that pension amounts to 35% of gross salary between 2030 and 2040 and 40% of gross salary after 2040. Scenario 1 of unemployment rate is used

Source: Author’s calculations.

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