EL APRENDIZAJE COOPERATIVO: UN PROCESO DE INCLUSIÓN EN LA ESO

An investment portfolio was not deemed to be an appropriate investment portfolio in a retirement funds default investment strategy based only on its ability to maximise a final retirement benefit. This study also considered risk measures when evaluating each investment portfolio. The risk measures that were implemented in this study to evaluate risk, were measures of dispersion, the probability of reaching a target, the probability of shortfall, value-at-risk and conditional value-at-risk.

3.4.3.1 Measures of dispersion

This study first assessed the riskiness of each investment portfolio by assessing the common measures of dispersion. The reason measures of dispersion were assessed was to describe the risk inherent in the distribution of final retirement benefits. The common measures of dispersion are: range, variance, standard deviation, interquartile range and the coefficient of variation.

The range was calculated by finding the difference between the highest final retirement benefit and the lowest final retirement benefit in the distribution. The range indicated what

46 the difference was between the best and worst case scenarios in a distribution. The larger the range, the more risk inherent in the distribution and vice versa (Reilly & Brown, 2012). The variance was calculated by first calculating the mean of the distribution and then deducting the mean from each value in the distribution. The results were then squared and added together. The product of the squares was divided by the number of data points. The variance indicates the average deviation from the mean of the distribution (Reilly & Brown, 2012).

The standard deviation was calculated by calculating the square root of the variance. The standard deviation indicates the amount of the volatility and risk present in a distribution. The standard deviation indicates how much a final retirement benefit will vary from the mean final retirement benefit in a distribution (Reilly & Brown, 2012).

The interquartile range was calculated by finding the difference between the 25th percentile of the distribution and the 75th percentile of the distribution. The interquartile range is a better measure of risk when compared to the range, as the interquartile range only looks at the middle of the distribution and looks at the risk inherent in the middle of the distribution (Reilly & Brown, 2012).

The coefficient of variation was calculated by dividing the standard deviation of the distribution by the mean of the distribution. The coefficient of variation measures the amount of risk that is taken for each unit of return earned. A lower ratio would imply a better risk- return trade-off (Reilly & Brown, 2012).

3.4.3.2 Probability of reaching a final retirement benefit target

Each investment portfolio was analysed based on its ability to reach or surpass a final retirement benefit target for each of the 10 000 different iterations. The final retirement benefit target was based on the assumption that the hypothetical member required 75% of their final annual salary at retirement. It was also assumed that the hypothetical member would live 20 years after retiring, and that their annual income requirements would increase with the average inflation rate of 5.18%. It was also assumed that the hypothetical member was able to earn a return of 8.5% per annum. Using all of the assumptions above, the targeted final retirement benefit that the hypothetical member required at the date of retirement was calculated to be R16 061 463.95. The 10 investment portfolios were then analysed against the R16 061 463.95 to assess their ability of reaching or surpassing the final retirement benefit target.

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3.4.3.3 Shortfall probability

The shortfall probability measure measured the probability that a specific investment portfolio would underperform when compared to other investment portfolios. This study compared the final retirement benefits for each investment portfolio against the life-stage portfolios for each iteration. The final retirement benefit of the life-stage portfolios that fell below the final retirement benefit of the other investment portfolios, was added up and then divided by 10 000 to determine the probability of shortfall. The reason that the shortfall probability was calculated was to determine the likelihood that the life-stage portfolios would underperform compared to the other investment portfolios. Basu et al. (2011) also conducted research on retirement funds that used shortfall probability measures.

3.4.3.4 Value-at-risk

Value-at-risk (VaR) is a risk measure that indicates the worst loss that an investor could expect from an asset or a portfolio of assets over an investment horizon at a certain confidence level (Crouhy et al., 2006). It is also added that VaR is the probability of losses suffered from holding an asset, or a portfolio of assets, exceeding a specified proportion (Brealey et al., 2011).

The distribution of final retirement benefits for each investment portfolio was analysed using VaR at the 95% and 99% confidence intervals. The 95% VaR generates an estimate that is considered to be the minimum final retirement benefit that a member could receive, with a 5% probability that a member could receive a final retirement benefit that is lower. Similarly, the 99% VaR generates an estimate that is considered to be the minimum final retirement benefit that a member could receive, with a 1% probability that a member could receive a final retirement benefit that is lower. Basu and Drew (2010) as well as Basu et al. (2011) used VaR as a risk measure when conducting research on retirement funds.

The reason that VaR was used as a risk measure in this study was to provide minimum final retirement benefits for each investment portfolio in order to make a comparison for a hypothetical member.

There are three approaches to calculating VaR. These approaches are: the historical simulation, analytic variance-covariance and Monte Carlo simulation approaches (Crouhy et al., 2006; Matz, 2005). There are advantages and disadvantages of using the Monte Carlo simulation approach to calculate VaR (Crouhy et al., 2006). These advantages and disadvantages are listed next.

48  It can be applied to any distribution of risk factors

 It can be used to model complex portfolios

 It allows the calculation of confidence intervals for VaR

 It allows stress testing and sensitivity analyses to be performed

The disadvantages of using Monte Carlo simulation approach to calculate VaR are:  The outliers are not incorporated into the distribution

 It is computer and time intensive

There are two major flaws with VaR. Firstly, VaR disregards any loss beyond the VaR level and secondly, VaR is not a coherent risk measure because it is not sub additive (Resti & Sironi, 2007; Yamai & Yoshiba, 2005; Artzner, Delbaen, Eber & Heath, 1999). VaR is only able to indicate that there is a probability that a final retirement benefit could fall below a certain value if the worst case scenario were to occur. Unfortunately, VaR is unable to indicate what the value of the final retirement benefit would be if the worst case scenario were to occur.

A more coherent risk measure was required in order to indicate what the value below VaR would be. The conditional value-at-risk (CVaR) was developed to have a risk measure that would indicate what the value below VaR would be (Rockafeller & Uryasev, 2000; Artzner et al., 1999).

3.4.3.5 Conditional value-at-risk

CVaR is generally considered to be a better measure of risk when compared to VaR. The CVaR calculates the average final retirement benefits that fall below the VaR. For example, the VaR at the 95% confidence interval generates an estimate that is considered to be the minimum final retirement benefit that a member could receive, with the 5% probability that the final retirement benefit a member receives would be less than VaR. The CVaR calculates what the average value of the final retirement benefit would be if the 5% probability occurred and the final retirement benefit dropped below VaR. The 95% CVaR and 99% CVaR were investigated in this study. The CVaR was used as a risk measure in this study in order to investigate the value of the final retirement benefit that a hypothetical member would have received at retirement if the worst case scenario were to occur. Basu and Drew (2010) as well as Basu et al. (2011) conducted research on retirement funds and used CVaR as a risk measure when conducting the research.

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