Calibration of the Model

page 69

page 70

Income Groups – Contribution Rates and Real Increase in Income

The model uses a base case where it considers the average Norwegian who starts full-time employment at the age of 25 and retires at the age of 67. It is a sensible assumption to begin in a full-time job with defined-contribution schemes at the age of 25. This person has a yearly income at 𝑡 = 0 amounted to NOK 500.000 and saves 10% in defined-contribution each year. An annual income of NOK 500.000 is reasonable, as this was the average of men and women in 2017 (SSB, 2, 2018).

To test the robustness of the results, the model also comprehends two other income groups, low-income and high-income. The low-income group has a yearly income of NOK 100.000 and saves 5% in defined-contribution each year. The high-income group has a yearly income of NOK 1.300.000, and the model assumes that higher income levels indicate a higher contribution rate. Therefore, the contribution rate of the high-income group is assumed to be 17% per annum.

There are three different levels of increase in income. These represent the increase in income for low, middle and high-income groups. Statistics from 2000-2017 show that occupations like entertainment and culture are low-income occupations. Entertainment and culture have had an average nominal increase of 3,6%. For the middle-income group, the model uses the average of all occupations from the same research, which gives an average nominal increase of 3,79%. The oil and gas sector counts as high-income occupations, and the average nominal increase the last 17 years is 4,1%. With a historical inflation rate of 3,14%, the real increases of income levels are respectively 0,45%, 0,63% and 0,93% (see Appendix E) (SSB, 3, 2018).

Management Fees

The total fees are defined as the fees for the first 42 saving years, and the model uses the five firms’ own fees for the different time-periods. The fees for estimating the contribution profiles for 100% equities, 100% bonds, and the theories, Malkiel, Cocco et al., and Markowitz, are an average of the fees of the seven largest firms’ fees, where 𝑓̅₄₂= 0,997%.

These estimates are based on data from the different pension insurance companies and can be found in Appendix B.

page 71

Risk-Free Rate

Following the Department of Finance, Mork and the central bank of Norway, the yearly return on bonds should lie around 0% to 1%. The Norwegian 10-year government bond has since 2000 had a real average of 0,6%, discounted by the historical inflation. The estimation can be found in Appendix D. As the real bond return is expected to be low, the model therefore assumes that 0,5% is a realistic yearly real risk-free rate.

Inflation

Inflation is determined as the rate at which the general level of prices for goods and services is rising, and the historical coefficient of Norway is estimated to 3,14% (Jordà, Schularick &

Taylor, 2017). The historical coefficient is based on the Norwegian consumer price index, dated back to 1872. It is chosen to use this dataset in the estimation of the historical inflation because it covers the same time-period as the dataset from Dimson, Marsh and Staunton.

The estimation can be found in Appendix D. Nonetheless, the inflation should not be determined based on historical data, instead be an indicator of the general increase the next years. In 2018, the central bank of Norway evaluated the inflation goal to 2%, which will be used in the model. According to the central bank of Norway, the inflation rate will create a sustainable production and employment rate (Norges Bank, 2018).

5.3.2 Determination of Coefficients in the Utility Functions

Expected Utility

The coefficient of risk aversion determines the certainty equivalent for a rational investor.

Assuming an investment with two outcomes, NOK 50.000 and NOK 100.000 that has an expected utility of NOK 75.000. The certainty equivalent of the investment varies with CRRA, outlined in Table 5-5.

Table 5-5: Certainty equivalents of different levels of risk aversion.

𝜸 0 1 2 5 10 20 30

𝑪𝑬 75.000 70.711 66.667 58.566 53.991 51.858 51.209

page 72

The table above illustrates that a rational investor, with a risk aversion of 10, will settle for NOK 53.991 even though the expected utility of the payoff is NOK. 75.000. Gandelman and Hernàndez-Murillo (2014) estimated the coefficient of relative risk aversion for 75 countries. Their analysis suggests that the coefficient varies closely around 1, and determines the coefficient in Norway to 1,16. However, in 2006 Schroyen and Aarbu (2017) conducted a survey asking about what makes people feel safe and secure and what situations they fear most. This resulted in a sample of average risk aversion coefficients ranging from 3,8 (Norway) to 10,2 (Chile). The model assumes that people are risk averse towards pension savings and decide to use the coefficient of risk aversion of 3,8.

Prospect Theory

Following Tversky and Kahneman (1992), the estimations of the coefficients in prospect theory are problematic. If the coefficients associated with the theory are not constrained, the number of estimated parameters for each subject will be too large. Tversky and Kahneman (1992) estimated the value function for gains (𝛼) and losses (𝛽) to 0,88, by diminishing sensitivity. Further, they denote the loss aversion (𝜆) to 2,25, and the convexity (𝛾) and change in value (𝛿) to respectively 0,61 and 0,69. However, in 2011 Rieger, Wang and Hens (2011) estimated cumulative prospect theory parameters for the median in each country. In Norway, they estimated the values to 0,55 (𝛼), 1,00 (𝛽), 0,55 (𝛾) and 1,83 (𝜆).

The analysis will use the coefficients from both Tversky and Kahneman, and Rieger, Wang and Hens to get a satisfying basis of comparison. Further, two reference points will be used in the estimations. The first reference point is the average total pension payout of an investment in 100% bonds. The second reference point is 66% of the expected final income at the date of retirement, as this is what employees in the public sector use as a standard when selecting contribution profile. All employees in municipal and public sector will receive, with certainty, 66% of final income from municipal pension funds if they do not change pension profile to private insurers. This means that if the yearly pension of contribution schemes are less than the expected final income, there will be no incentive to change contribution profile (Andreassen, 2014). The expected gain and loss from an uncertain outcome is profoundly influenced by the point of reference in prospect theory.

Therefore, the model includes two reference points to validate the results.

page 73