1
Do Life-Cycle Strategies Perform Better Than Constant
Proportion Asset Mix Strategies in Defined Contribution
Plans ?
The Dutch case
MASTER THESIS
University of Groningen Faculty of Economics and Business
MSc Finance
Author: Ane Piet Strikwerda Student number: S1951017 Mail: apstrikwerda@gmail.com
Supervisor: Dr. B.A. Boonstra
Abstract
Life-cycle investment strategies are very popular among Defined Contribution (DC) plans. The most important allocation characteristic of a life-cycle strategy is the shift away from equities in favor of bonds and cash as the DC plan participant approaches the retirement date. In this paper it is investigated whether life-cycle strategies perform better than constant proportion asset mix strategies. A simulation and a historical analysis is used to compare the performance of life-cycle and constant proportion asset mix strategies in the long term. The results show there is no evidence that life-cycle strategies perform better than constant proportion asset mix strategies. These results are in line with most recent literature on the performance of life-cycle strategies.
JEL Classification: D31; G11
2 I. Introduction
Since 2011, a new category of retirement plan providers is introduced in the Netherlands: the Premium Pension Institution (PPI). A PPI is designed to host Defined Contribution (DC) pension plans (DNB, 2012). In a DC plan the contribution is certain and the entitlement benefit at retirement is a direct result of the investment returns. Participants thus face an uncertain payoff.
Since participants bear the risk of the investments in a DC plan, often diverse investment strategies are offered to choose from. Bodie and Treassard (2007) note however, that many participants do not know enough about investing in order to choose rationally between the alternative investment strategies, and moreover it is found too time-consuming. As a result, most participants accept the default investment strategy as offered by their plan (see, e.g., Choi et al. (2003)). Indeed, for participants who do not choose between the various investment strategies offered, Dutch PPI’s offer a default investment strategy in which the participant is automatically enrolled.
The default investment strategies of Dutch PPI’s can all be characterized as ‘life-cycle’1
strategies (corporate websites2
, 2012). Life-cycle strategies are characterized by a shift in investment weights, from aggressive to more conservative, as the target retirement date of a participant approaches. The popularity of life-cycle strategies is often attributed to the fact that these offer a ‘set it and forget it’ solution by automatically modifying the portfolio allocation in tune with the participants’ changing capacity to bear risk (Basu and Drew, 2009).
Academic interest and research related to life-cycle strategies is relatively new. Classical theory states that the optimal investment portfolio is age-invariant. A rational investor should hold the same fraction of the portfolio in risky assets at all ages when risky returns are serial uncorrelated and there is no labor income (Samuelson (1969) and Merton (1969)). The main question regarding life-cycle strategies is therefore, whether these strategies perform better than simple constant proportion asset mix strategies.
Since all default investment strategies of the PPI’s are life-cycle strategies, and a large part of PPI participants choose to depend on the default investment strategy, it is
1 Life-cycle strategies are also known as target-date strategies. For reasons of consistency, in this paper the term life-cycle is used.
3 important to know whether life-cycle strategies lead to more wealth at retirement as compared to constant proportion strategies.
By comparing life-cycle strategies with constant proportion asset mix strategies, an answer is given to the following question:
How do life-cycle allocation strategies perform relative to constant proportion asset mix strategies in DC plans?
To answer this question, both a historical and a simulation analysis is performed, comparing 20 different investment strategies. The performance of the investment strategies is measured by the replacement rate. The replacement rate measures to what extend the assets at retirement are able to cover the liabilities. It relates the actual to the desired yearly retirement income, and thus, measures the participants ability to keep up the same standard of living at retirement.
The most important risk a participant faces is the risk of not reaching the same standard of living at retirement. The risk of the strategies is therefore measured by the failure rate, which is the probability of not obtaining the same standard of living at retirement. Furthermore by comparing the lower tails of the strategies it can be analyzed to which extend the strategies are failing in the case of adverse outcomes. The mean replacement rates of the life-cycle and constant proportion strategies are furthermore compared with a simple t-test. By using long term return series for the different asset classes, it is calculated which replacement rates would have been obtained by the different strategies historically. Using long term series, economic cycles and outliers are included in the return data, which improves the quality of the analysis. For reasons of long term historical data availability, the analysis is applied by using a United States return data set.
5 II. Context and Literature
Context
As a result of demographic and economic developments there has been a shift towards DC plans in developed countries over the last century (see, e.g., Ponds and van Riel, 2006). This shift did also take place in the Netherlands. Relatively speaking, DC plans have been the fastest growing pension plan in the Netherlands over the last 13 years (see Table 1). Although only 6.4 percent of the Dutch pension plan participants were enrolled in a DC plan in the year 2011, with respect to the year 1998, the increase in size cannot be ignored.
Table 1: Total pension fund and DC plan participants in the Netherlands (x 1,000).
Total DC (% DC)
1998 4,822 40 0.8 2011 5,841 376 6.4
Source: DNB(2012).
The Premium Pension Institution (PPI) is a new type of retirement plan provider. A PPI is a company designed to host DC plans. A PPI is thus nothing more than a facilitator of DC plans. Until recently, a DC plan had to be executed by an insurance company in the Netherlands. These DC plans where opaque and often it was not clear which part of the contribution was actually invested.
A PPI is designed to facilitate participants to accumulate pension capital. At retirement date the pension capital is paid out as lump sum to the participant. If a participant passes away before retirement date, the capital is redistributed to the other participants of the PPI. Furthermore, a PPI is, in contrast with an insurance company as DC facilitator, not allowed to take risks. This implies that a PPI may, for example, not guarantee pay-outs at retirement, since this may lead to longevity risk.
6
All PPI’s offer a default investment strategy in which participants automatically enroll if there is not made a choice between the alternatives. Although there is not yet information about the investment strategy of Pensional, all the other PPI’s have published their investment policy and strategy. A comparison between the default investment strategies shows that all PPI’s implement life-cycle strategies (DNB, corporate websites, 2012). This implies that every PPI participant that accepts the default investment strategy has a DC plan with a life-cycle strategy.
The default investment strategy of the PPI is of particular interest since research has shown that most of the DC participants accept this strategy passively. Choi et al. (2003) find that most American employees accept the default investment strategy when they are enrolled in a DC plan. In their sample, around 80 percent accepted the default investment strategy. Beshears et al. (2006) show similar results; they find that around two-third of the employees who are automatically enrolled in a DC plan have all of their assets invested in a default investment strategy. Studies in Australia and Sweden on the investment choice of DC plan participants confirm the hypothesis that most of the participants accept the default investment strategy. For example, the Cronqvist and Thaler (2004) Sweden study shows that since 2003 only 10 percent of the new DC participants actually made a choice between the optional strategies. In Australia, about two-thirds of all retirement plan assets are invested in default investment strategies (Australian Prudential Regulatory Authority (APRA), 2005). The inertia among plan participants is also manifested in the reluctance to change the allocation of their assets through time. (Ameriks and Zeldes, 2004).
Table 2: Premium Pension Institutions in the Netherlands.
Name PPI Organisator/Manager License since
1 Be Frank BinckBank, DeltaLloyd 6-24-2011
2 Robeco Smart Pension Robeco 8-9-2011
3 Essentie Pensioen NN, ING, AZL 12-23-2011
4 Brand New Day ASR 12-6-2011
5 Het Nieuwe Pensioen Goudse verzekeringen 12-30-2011
6 TKP-PPI TKP 12-30-2011
7 Pensional APG, ABNAMRO 1-30-2012
8 AEGON Pensioenabonnement AEGON 2-16-2012
7 Literature
The economic justification for the life-cycle strategies and it’s shift away from equity over time is driven by the fact that young people have next to financial wealth another source of wealth; human capital. Human capital can be seen as present discounted value of expected future earnings (Viceira, 2007). Thus, as participants age, the value of human capital declines (there are less years left to earn labor income) while the financial wealth grows. When a participant is young, and has thus a relatively sizeable amount of human capital, the retirement portfolio is diversified towards other risky assets. Obviously, changes in human capital therefore leads to a continuously changing asset mix (Bodie et al., 1991). In particular, life-cycle strategies account for the decreasing human capital of participants.
Life-cycle strategies did become very popular in DC plans over the last decade (see, e.g., Porterba (2006), Basu and Drew (2009)). Life-cycle strategies are intuitively attractive and are widely recommended by financial advisors. One of the most given arguments is that by gradually switching investments from stocks to less-volatile assets over time, life-cycle strategies aim to lessen the probability of an adverse investment outcome as the retirement date nears (Basu and Drew, 2009).
Poterba et al. (2006) suggest that the popularity of life-cycle strategies may have been the catalyst of research that tries to uncover reasons why a participant chooses to reduce her equity exposure as she ages. This is questioned since classical literature states that a rational investor should hold the same fraction of the portfolio in risky assets at all ages when risky returns are serial uncorrelated and there is no labor income (Samuelson (1969) and Merton (1969)). Poterba et al. (2006) note that the empirical evidence on age-specific patterns in household asset allocation suggests at best weak reductions in equity exposure as households age.
8 certain assumptions, the expected utility derived from an all equity allocation is greater than that from more conservative strategies over long accumulation periods. Hickman et al. (2001) use a Monte Carlo simulation to examine the relative performance of bills, bonds, and stocks over various holding periods. They conclude that for longer holding periods, there are significant and large penalties for not pursuing risky investments, i.e., equities.
Basu and Drew (2006) performed a stochastic simulation based on the default investment strategies of 20 Australian DC plans and compared these strategies with each other, and with a 100 percent equity strategy. Of the 20 default investment strategies there were 3 with life-cycle characteristics; the others were age-invariant strategies. Although it is mentioned that life-cycle strategies are gaining popularity, most Australian pension funds did not offer these type of strategy back in 2006. The age-invariant strategy with the lowest equity exposure in their sample is 67 percent, all strategies can therefore be characterized as aggressive. The results indicate that in most occasions a 100 percent stock strategy offers the largest upside potential and lowest downside risk as compared to the strategies under consideration. Since there are only three life-cycle strategies in the simulation they do not draw general conclusions on life-cycle strategies.
Using a Monte Carlo simulation Schleef and Eisinger (2007) find that life-cycle strategies that reduce equity allocations over time, fail to increase the likelihood of reaching a targeted portfolio value as compared with age invariant asset allocation models. They compare age-invariant asset allocation strategies with life-cycle strategies that decrease their exposure to equities linearly. More recently, Antolin et al. (2010) also assessed the relative performance of life-cycle strategies, using a stochastic model. They conclude that strategies that maintain a constant, high exposure to risky assets during most of the accumulation period lead to the best performance. These so called, ‘last period decreasing’ strategies provide higher expected benefits for the given level of risk than other life-cycle strategies. Scheuenstuhl et al. (2010) show that life-cycle strategies and constant asset mix strategies are located along a curve resembling an efficient frontier. Thus, they conclude that the suitability of a specific plan depends on an individual’s risk bearing preferences.
9 not wise to blindly adopt life-cycle strategies as DC investment strategy, since most of them have a predetermined mechanistic allocation rule regardless of the actual accumulation in the retirement account. As a result of the portfolio size effect, implying that the growth in value of accumulated wealth takes for a large part place in the later years, a conservative portfolio can destroy significant value in the last years. The argument used in life-cycle strategies to preserve accumulated wealth is therefore only valid if there is already sufficient wealth accumulated.
Almost all recent literature investigating life-cycle strategies conclude that it is questionable whether life-cycle strategies add value compared to constant asset mix strategies. However, as all PPI’s in the Netherlands conduct a form of life-cycle asset allocation strategy (as a default investment strategy), the following hypothesis is tested:
H0: Life-cycle strategies lead to the same risk adjusted wealth outcomes as constant proportion asset mix strategies.
10 III. Methodology
In a DC plan, the wealth that is accumulated at retirement determines the success of an asset allocation strategy. Therefore, by using a historical analysis and a simulation model, wealth outcomes at retirement of different investment strategies are compared in order to answer the question whether life-cycle strategies lead to significant other wealth outcomes than constant proportion asset mix strategies.
As wealth outcome, the level of retirement income is often measured in relation to the last income before retirement. This gives a good indication whether the same standard of living can be obtained at retirement. Wealth outcomes at retirement are therefore measured in terms of replacement rate (RR). The replacement rate measures to what extend the assets cover the liabilities and is obtained by dividing the yearly retirement income (A) by the desired yearly retirement income (L),
𝑅𝑅 = 𝐴/𝐿 . (1)
As the replacement rate is the ratio of assets and liabilities, it can be seen that a replacement rate of less than 1 leads to failure in obtaining the desired yearly retirement income. The most important risk that a DC plan participant faces when retiring is that he or she cannot obtain the same standard of living as before. The investment strategies are therefore evaluated on their inability to obtain the replacement rate of 1. This is represented by the failure rate, which measures the number of outcomes which do not lead to the desired replacement ratio of 1 in relation to the total number of outcomes.
The portfolio value at retirement and the last earned salary are simulated using a Monte Carlo method. The portfolio value is dependent on the yearly contribution and the investment results. The contribution is a percentage of the yearly salary, which is assumed to grow in line with inflation, and the investment universe consists of the asset classes equities, government bonds and cash equivalents. All asset classes and inflation are assumed to follow a Geometric Brownian motion.
11 been able to invest in a DC pension scheme and earned the investment returns that actually occurred.
Liabilities
As a proxy of the income that can guarantee the desired standard of living at retirment, a percentage of the last earned salary (𝑆𝐴𝐿𝑟) before retirement is taken. The
desired percentage (DP) multiplied by the last earned salary gives the yearly pension liability (L) as result,
𝐿 = 𝑆𝐴𝐿𝑟∗ 𝐷𝑃. (2)
The last earned salary is calculated in the following manner: the simulation starts with a given initial salary (𝑆𝐴𝐿0) and under the assumption that the salary grows in line
with inflation a path,
𝑆𝐴𝐿𝑡= 𝑆𝐴𝐿𝑡−1∗ 𝑔𝑆𝐴𝐿, (4)
is calculated, where the growth in salary (𝑔𝑆𝐴𝐿) equals,
𝑔𝑆𝐴𝐿= exp [(𝜇−
𝜎2
2) + 𝜎 ∗ 𝜀] , (5)
with 𝜇 and 𝜎 to represent the mean and the standard deviation of the inflation rate respectively. Salary uncertainty is modeled by the random variable 𝜀. Equation (4) is a direct representation of the Geometric Brownian motion (see Appendix A).
Assets
The yearly retirement income is obtained by dividing the total assets at retirement (portfolio value at retirement (𝑃𝑉𝑟)) by the life expectancy (LE),
𝐴 = 𝑃𝑉𝑟/𝐿𝐸. (3)
12 𝐶𝑡= 𝑆𝐴𝐿𝑡∗ 𝐶𝑅, (6)
where CR represents the contribution rate.
The portfolio value at the end of each year is a result of the investment returns on the different asset classes and the contribution. For example, at the end of the first year the portfolio value equals the contribution of the first year. At the end of the second year the portfolio value is determined by the investment result on the contribution of the first year plus the contribution of the second year. Equation (7) gives a mathematical representation of the portfolio value at t,
𝑃𝑉𝑡= 𝑃𝑉𝑡−1∗ 𝑅𝑃𝑉+ 𝐶𝑡 , (7)
where the portfolio return (𝑅𝑃𝑉) is the weighted sum of the return rates of the different asset
classes,
𝑅𝑃𝑉= 𝑋𝑆∗ 𝑅𝑆+ 𝑋𝐵∗ 𝑅𝐵+ 𝑋𝑇 ∗ 𝑅𝑇. (8)
The rates of return on stocks (𝑅𝑆), bonds (𝑅𝐵) and T-bills (𝑅𝑇) are also calculated with the
equation representing salary growth,
𝑅𝑆= exp [(𝜇−
𝜎2
2) + 𝜎 ∗ 𝜀] . (9)
13 Investment strategies
So, how are the assets invested according to life-cycle theory? The most simple rule for a life-cycle asset allocation strategy that is followed by DC plans is the 100 – age rule (Schmit & Post, 2012). The proportion of the portfolio invested in risky assets is 100 percent minus the age of the individual. When an individual’s working life starts at the age of 25, assuming that he or she will work until the age of 65, the proportion of risky assets decreases from an initial 75 percent to 35 percent.
The second commonly used rule is in many ways similar to the 100-age rule, but shows a step wise linear decrease glide-path (Schmit & Post, 2012). These two simple rules are used, although with different initial allocations to equities.
The initial allocation is defining the aggressiveness of the strategy. In this paper three linear decreasing life-cycle strategies are used in the analysis; an aggressive, intermediate and conservative strategy.
Graph 1: Linear decrease strategy; portfolio composition during the accumulation period.
For the step-wise linear decrease strategies there are also three investment styles used (aggressive, intermediate and conservative). Graph 1 and 2 give respectively a graphical representation of the linear decrease and step-wise linear decrease strategies.
14 Graph 2: Step-wise linear decrease strategy; portfolio composition during the accumulation period.
Since previous research (Antolin et al., 2010) has indicated that a decreasing allocation to equity, only in the last part of the accumulation period outperforms other life-cycle strategies, there are also three strategies composed that let the equity exposure decrease in the last period of the accumulation period. Although slightly different in style and composition, none of the last period decreasing strategies can be characterized as conservative. Graph 3 shows the glide path of a last period decreasing strategy.
Graph 3: Last period decrease strategy; portfolio allocation during the accumulation period.
15 Next to the life-cycle asset allocation strategies, there will also be asset allocation strategies with a constant, yearly weighted, asset mix. Here again, there will be an aggressive, intermediate and conservative strategy. There will also be an all equity, all bond and all cash strategy. This brings the total to 20 strategies.
The strategies are divided into three categories; conservative, intermediate and aggressive. Since the life-cycle strategies are changing the asset allocation through time, the weighted equity exposure determines in which category a strategy belongs. The weighted equity exposure is measured assuming a portfolio in which the contribution is accumulated through time without being invested and without salary growth. As a result of the accumulation of contributions, the value of the portfolio will be much higher in the latter years as compared to the first years. In this manner, independent of the salary growth and investment returns, the weight of the portfolio in the last ten years before retirement determines more than 40 percent of total portfolio weight during the accumulation period. The weights of every year in the accumulation period are given in Appendix C.
Performance measurement and comparison
To compare the constant proportion strategies with the life-cycle strategies the replacement rate outcomes of the strategies are compared. The mean replacement rates of the strategies are analyzed with a t-test. With the t-probability it is shown whether the replacement rates (RR) of the constant proportion strategies differ significantly from the life-cycle strategies, 𝑡 =𝑅𝑅̅̅̅̅̅̅ − 𝑅𝑅𝑠1 ̅̅̅̅̅̅𝑠2 √𝜎𝑠12 𝑛𝑠1+ 𝜎𝑠22 𝑛𝑠2 (10)
where S1 and S2 represent investment strategy 1 and 2, 𝜎 represents the standard deviation and n is the amount of observations.
16 able to obtain the desired yearly retirement income. The constant proportion strategies are compared with the life-cycle strategies with respect to ability of obtaining the desired yearly retirement income. The most straightforward approach to compare the strategies on their ability to obtain the desired yearly retirement income is done by calculating the failure rate. The failure rate is the probability that a strategy does not obtain the desired yearly retirement income and is used as risk measure.
Although the probability of failure is an important risk measure, in case of failure it is also important to know how much the replacement rate deviates from the desired replacement rate. A strategy with high failure rates may, for example, have very low deviations from the desired yearly retirement income, while a strategy with low failure rates may have extreme negative outcomes in case of failure. As a result of this, the strategy with high failure could be preferred if the deviations are taken into account. Taking the lower tails into account as second risk measure will therefore lead to a more robust conclusion. The lower tails are represented by the lower percentile (25th
, 15th
, 5th
and 1st
17
IV. Data
The investment universe of retirement contributions in this study consists of the three main asset categories: equities, bonds and cash equivalents. Moreover, inflation is needed to model the plans liabilities.
Long term return data is essential to create a reliable replacement rate distribution for long accumulation periods. Since there is a lack of available data on Dutch stock and bond returns over a long historical time frame a U.S. data set is considered. Dimson et al. (2002) show that returns on various asset classes have displayed broadly similar trends over the last century and returns are as a result of globalization becoming more and more correlated. Therefore, high quality foreign data instead of domestic return data is used. There are several reasons which have led to the choice of using U.S. return data. First the availability and accuracy of long term data is important. The second reason is that two stock market crunches are included in the period since 1928. To this end U.S. data is preferred.
The post WW2 period is characterized by large economic growth and it is therefore likely that stock returns over this period are biased upwards3 (Dimson et al., 2002). The
period 1928-2011 seems therefore more representative for the parameter set up in the simulation, since in this way two stock market crashes are included and the returns are potentially less biased to the upside. However, for reasons of robustness the model is also simulated with the characteristics of the period 1946-2011 as input. The historical analysis is also based on the period 1928-2011, where 1966 is the end of the first accumulation period that starts in 1927.
The data on Dutch inflation is obtained from the International Institute of Social History4. The data for treasury bond and T-bill returns are obtained from the Federal Reserve
database in St. Louis (FRED)5. The T-bill rate is a 3-month rate and the treasury bond is the
constant maturity 10-year bond, but the treasury bond returns includes coupon and price
3 Things turned out better than expected in the second half of the twentieth century: there was no third world-war, the cold war ended, the Berlin Wall fell. There was unprecedented growth in productivity and efficiency and extensive technological change. It is therefore likely that future returns from equities will be lower than those achieved in the last decades (Dimson et al., 2002).
18 appreciation. The exchange rate data and S&P total return data are obtained from Measuring Worth6
.
Since the pension contributions are invested in U.S. equities, bonds and cash equivalents, there has to be accounted for dollar/Euro (guilders) exchange rates. With the historical analysis the yearly contribution in Dutch guilders/Euro’s is converted to U.S. dollars at the exchange rate at year-end. Assuming the pension contributions will be held on a U.S. retirement account, after the period of 40 years, the accumulated pension in U.S. dollars is converted back to Dutch guilders/Euros at the exchange rate of that moment. For the simulation the yearly returns of the asset classes are on forehand corrected by the yearly percentage change in exchange rate between U.S. dollars and Dutch guilders/Euros. This since the accumulation periods of 40 years are random Table 3 shows the return characteristics.
Table 3: Annual return (%) characteristics adjusted for exchange rates.
1928-2011 1946-2011
Geo Arith St. Dev Geo Arith St. Dev Inflation 3.4 3.5 3.9 3.8 3.8 2.7 Stocks 8.5 11.1 22.9 9.5 11.6 20.4 Bonds 4.6 5.3 12.6 4.8 5.6 12.5 T-bills 3.1 3.7 10.5 3.6 4.1 9.7
Source: FRED, Measuring worth , International Institute of Social History (2012).
In simulating the returns also correlation is accounted for. The historical correlations as given in table 4 are used input and by means of a Cholesky decomposition the correlations are simulated. Next to the historical correlations, both period inputs are also simulated assuming zero correlation.
Table 4: Correlation characteristics based on annual return data.
1928-2011 1946-2011
Inflation Stock Bonds T-bonds Inflation Stock Bonds T-bonds
Inflation 1.00 1.00
Stocks 0.11 1.00 -0.03 1.00
Bonds 0.19 0.37 1.00 0.09 0.37 1.00
T-bills 0.29 0.46 0.81 1.00 0.23 0.52 0.76 1.00
Source: FRED, Measuring worth , International Institute of Social History (2012).
19 In the simulation 10,000 salary paths are calculated with corresponding equity, bond and T-bill paths of 40 years. For every investment strategy this will lead to 10,000 replacement rates. The resulting replacement rates are compared using the risk measures.
The replacement rate is obtained after a contribution and accumulation period of 40 years; the participant enters the plan at age 25 and retires at age 65. Furthermore it is assumed that the life expectancy at retirement date is 20 years, which implies the total DC plan wealth (i.e. portfolio value) at retirement should be divided by 20 to get the yearly pay-out. These characteristics are standard in retirement saving research.
In general, as a proxy of income that can guarantee the same standard of living, Dutch pension funds aim at 70 percent with respect to the last earned salary (Ponds and van Riel, 2006). The 70 percent is mainly a result of tax exemptions at retirement. The 70 percent should be obtained by a combination of the basic flat-rate pension that every individual receives in the Netherlands7
(Dutch General Old Age Pension Law, AOW) and the DC plan. It is assumed that the AOW yields a replacement rate of 35 percent (the argumentation and calculations performed in defining the AOW replacement rate can be found in Appendix A). The yearly pension liability is thus given by multiplying the last earned salary by 35 as desired percentage.
The percentage of the yearly salary that is paid to the DC plan is defined as the contribution rate. The average percentage of the Dutch national income that is paid to occupational pension funds was 12.6 percent in 2009 (CBS, 2012) and is used as the contribution rate in this paper. With the contribution rate set at 12.6 percent, the final outcome of the different simulated scenarios can give a good indication of the likelihood, under current contribution policies, that a (average) participant obtains his or her desired retirement return at retirement.
7 Every inhabitant of the Netherlands receives 2 percent AOW for every year of residence in the Netherlands
20
V. Results
Historical analysis
With the historical analysis, the replacement rate of the different strategies are compared on the basis of actual historical accumulation periods of 40 years. Graph 4 shows the year on year replacement that would have been obtained if the pension contributions were allocated to a single asset class over a period of 40 years.
The graph shows that, no matter in which year an individual retires in the period 1966-2011, an all equity strategy always results in higher pension benefits respective to an all bond or T-bill strategy. Furthermore, with an all equity strategy the minimum replacement rate of 1 is historically always obtained. A participant with an all equity strategy, retiring in the year with the lowest all equity replacement rate, 1978, would still have a replacement rate 1.6, which is 60 percent more than the desired replacement rate.
The all bond and all T-bill strategy would not obtain a replacement rate of 1 in 28 and 42 of the 46 years respectively. The failure rates over the historical period (1966-2011) of the all bond and all T-bill strategies are therefore 61 and 91 percent respectively. It should be noted that an all bond strategy would lead to a replacement rate of minimal 1 in the most recent years and would not fail once since 1995.
A reason for the increasing replacement rate of an all bond strategy can be found in graph 5, which shows the 40 year rolling return on equities, bonds and T-bills. It can be seen that since the beginning of the eighties the 40 year rolling bond returns have exhibited an
1 2 3 4 5 6 7 8 1 9 6 6 1 9 6 8 1 9 7 0 1 9 7 2 1 9 7 4 1 9 7 6 1 9 7 8 1 9 8 0 1 9 8 2 1 9 8 4 1 9 8 6 1 9 8 8 1 9 9 0 1 9 9 2 1 9 9 4 1 9 9 6 1 9 9 8 2 0 0 0 2 0 0 2 2 0 0 4 2 0 0 6 2 0 0 8 2 0 1 0 Re p la ce m e n t Ra tio Year of retirement
Graph 4: Historical replacement rates after accumulation periods 40 year
21 upward trend. An all bond strategy expiring in 2009, invested for 40 years in bonds, would have an average return of more than 4 percent.
In table 5 the life-cycle strategies are compared with the constant proportion strategies on the basis of the t-probability.
Table 5: Historical replacement rate comparison of constant proportion asset mix strategies with life-cycle strategies on the basis of the t-probability.
-4% -2% 0% 2% 4% 6% 8% 10% Av er a g e re tu rn Year
Graph 5: 40 year average, rolling return of equities, bonds and T-Bills
Equities T-bills Bonds
Category Strategy XE Mean Median St. dev. Constant
Agr Constant Int Constant Con ALL EQUITY 100% 3.2 2.6 1.6 2% 0% 0% Agressive LAST 10 88% 3.0 2.7 1.3 6% 0% 0% CONSTANT AGR 81% 2.5 2.2 1.1 100% 0% 0% LINEAR AGR 74% 2.4 2.3 0.9 70% 0% 0% SW LINEAR AGR 71% 2.5 2.5 0.9 95% 0% 0% LAST 15 78% 2.7 2.6 1.0 36% 0% 0% LAST 20 69% 2.4 2.3 0.8 41% 0% 0%
Intermediate CONSTANT INT 56% 1.9 1.7 0.7 0% 100% 0%
BEFRANK 53% 1.9 1.7 0.7 0% 80% 0% TKP 52% 1.9 1.8 0.6 0% 72% 0% LINEAR INT 49% 1.8 1.7 0.7 0% 63% 0% AEGON 49% 1.8 1.6 0.7 0% 71% 0% SW LINEAR INT 46% 1.8 1.6 0.6 0% 51% 0% Conservative ROBECO 37% 1.6 1.4 0.6 0% 3% 14% CONSTANT CON 31% 1.4 1.3 0.6 0% 0% 100% NN 25% 1.3 1.1 0.5 0% 0% 33% LINEAR CON 24% 1.3 1.2 0.6 0% 0% 59% SW LINEAR CON 21% 1.3 1.1 0.5 0% 0% 46% ALL BOND 0% 1.0 0.9 0.5 0% 0% 0% ALL TBILL 0% 0.7 0.8 0.2 0% 0% 0%
22 The first column shows the different strategy categories. The strategies and their weighted equity exposure (XE) during the accumulation period are represented in the second and third column. It can, for example, be seen that the step-wise linear aggressive strategy has a weighted equity exposure of 71 percent, has had a historically mean replacement rate of 2.7, a median replacement rate of 2.6, and a standard deviation of 1.0.
In the last three columns the t-probability with respect to the constant proportion strategies are given. The step-wise linear strategy differs significantly from the intermediate and conservative constant proportion strategies with a probability of 0 percent, and does not have a significant other mean replacement ratio with respect to the aggressive constant proportion strategy.
The aggressive life-cycle strategies do not significantly differ from their constant proportion equivalent. This is also the case for the intermediate and conservative life-cycle strategies as compared to their constant proportion strategy. Although, when the constant proportion aggressive strategy is compared with the intermediate and conservative strategies they all do significantly differ. This also holds the other way around. This result is as expected, since the weights to the different asset classes differ significantly over an accumulation period of 40 years.
With the t-probabilities no conclusions can be drawn on the performance of the strategies between categories. Within the categories it can be concluded that the life-cycle strategies do not lead to significant different performance in terms of mean return as compared to their constant proportion counterparts.
23 Graph 6: Mean historical replacement rate with respect to the standard deviation.
Table 6 shows the historical failure rates and the lower percentile values of the different strategies. The aggressive constant proportion strategies would historically never lead to failure in obtaining the desired retirement return and would therefore lead to the best downside protection.
Table 6: Historical replacement rate comparison on the basis of failure rate and lower percentiles.
Category Strategy XE 25 15 5 1 Failure
ALL EQUITY 100% 2,0 1,8 1,7 1,6 0% Agressive LAST 10 88% 1,9 1,7 1,6 1,5 0% CONSTANT AGR 81% 1,7 1,5 1,3 1,2 0% LINEAR AGR 74% 1,6 1,5 1,3 1,3 0% SW LINEAR AGR 71% 1,7 1,5 1,5 1,4 0% LAST 15 78% 1,8 1,6 1,5 1,5 0% LAST 20 69% 1,6 1,5 1,4 1,4 0%
Intermediate CONSTANT INT 56% 1,2 1,1 0,8 0,8 11%
24 There is historically no conservative strategy that would never fail to obtain the minimum replacement rate. If the contribution would be solely depleted to cash equivalents (T-bills) there would historically be a probability of 91 percent that the strategy would not lead to the desired retirement return. Comparing the failure rates of the life-cycle strategies with the constant proportion strategies shows that the life-cycle strategies would not lead to better protection for failure.
Although the aggressive strategies have higher standard deviations than the more conservative strategies, even the most adverse outcome of the aggressive strategies lie above the desired retirement return ratio of one. This implies that taking more risk in terms of standard deviation, would historically lead to a lower risk of not obtaining the desired retirement return.
Taking the lower tails of the distribution as risk measure neither supports the argument that life-cycle provide better downside protection in the most adverse time paths. At the lower percentiles of the replacement rate distribution the life-cycle strategies do not lead to higher replacement rates as compared to the constant proportion asset mix strategies. The all equity strategy would historically lead to highest replacement rate (1.6) at the 1st
25 Simulation
Table 7 shows that, based on the simulated replacement rates, all strategies differ significant from the constant proportion strategies. Even the strategies in the same category differ significantly. The linear aggressive strategy, with a mean replacement rate of 3.3, shows a slightly different mean as compared to the constant aggressive strategy, with a mean of 3.4, but still differs significantly with a t-probability of 0 percent. This is a direct result of the large sample size. Therefore, no conclusions can be drawn on the basis of these outcomes. Table 7: Simulated replacement rate comparison of constant proportion asset mix strategies with
life-cycle strategies on the basis of the t-probability.
Table 8 shows the failure rate and the lower percentile outcomes of the simulation. Comparing the lower tail of the replacement rate distribution does not indicate that life-cycle strategies lead to better adverse outcome protection. Based on the 25th percentile values, the
life-cycle strategies do not lead to higher replacement rates. However, it can be noticed that the more aggressive strategies lead to higher replacement rates at the 25th percentile. This
still holds at the 15th
percentile, which shows that, even if an adverse outcome as that of the 15th percentile occurs, the plan participant will still be better of choosing a more aggressive
strategy.
Category Strategy XE Mean Median St. dev. Constant
Agr Constant Int Constant Con ALL EQUITY 100% 4.7 2.6 6.9 0.00 0.00 0.00 Agressive LAST 10 88% 4.1 2.4 5.6 0.00 0.00 0.00 CONSTANT AGR 81% 3.4 2.2 3.9 1.00 0.00 0.00 LINEAR AGR 74% 3.3 2.2 3.5 0.00 0.00 0.00 SW LINEAR AGR 71% 3.3 2.2 3.7 0.00 0.00 0.00 LAST 15 78% 3.6 2.3 4.4 0.00 0.00 0.00 LAST 20 69% 3.1 2.1 3.4 0.00 0.00 0.00
Intermediate CONSTANT INT 56% 2.4 1.8 1.9 0.00 1.00 0.00
26 Table 8: Simulated replacement rate comparison of constant proportion asset mix strategies with
life-cycle strategies on the basis of failure rate and lower percentiles
The replacement rates at the 1st and 5th percentile show that the life-cycle strategies
do not give a better protection for the occurrence of very rare outcomes than their constant mix counterparts. The lowest tail of the outcome distribution shows that the intermediate strategies lead to a slightly better replacement rate than their aggressive and conservative counterparts. Thus, for the best protection at extreme outcomes, an intermediate strategy should be followed. However, choosing a strategy that gives a little more protection in the most extreme outcomes, is accompanied by giving up a lot of upside potential in terms of replacement ratio’s. For example, the all equity strategy still leads to a replacement rate of 0.4 at the 1st
percentile, whereas the highest replacement rate value of all strategies at the first percentile is 0.5.
The failure rate shows the percentage of time paths which do not lead to the desired replacement rate of 1. Again, there is no major difference between constant mix strategies and life cycle strategies. With the more aggressive strategies, the probability of obtaining the desired replacement rate is the highest.
Category Strategy XE 25 15 5 1 Failure
ALL EQUITY 100% 1.3 1.0 0.6 0.4 16% Agressive LAST 10 88% 1.3 1.0 0.6 0.4 16% CONSTANT AGR 81% 1.3 1.0 0.6 0.4 16% LINEAR AGR 74% 1.3 1.0 0.7 0.4 15% SW LINEAR AGR 71% 1.3 1.0 0.7 0.5 15% LAST 15 78% 1.3 1.0 0.6 0.4 16% LAST 20 69% 1.2 1.0 0.6 0.5 16%
27 Appendix E shows the results of the simulation under different return characteristics and assuming no correlation. The outcomes of these simulations lead to the same conclusions.
In graph 7 the median return ratios are plotted against the 1st percentile replacement
rates of the different strategies. The plotted strategies resemble more or less an efficient frontier and the life cycle strategies do not deviate much from this line. This is a further confirmation that even in the most adverse outcomes the life cycle strategies do not give better protection than the constant asset mix strategies. The graph shows that all conservative strategies are inefficient based on the 1st percentile as risk measure and that the linear
intermediate strategy provides the highest replacement rate at the 1st percentile. Graph 7: Risk return tradeoff based on the 1st percentile and corresponding median return ratio.
28 Based on the results of the simulation there can be concluded that life-cycle strategies do not perform better than constant proportion asset mix strategies. It can be concluded that based on the historical analysis and the simulations that the more aggressive strategies have lower failure rates and higher replacement rates at the lower tails of the distribution.
Graph 8: Risk return tradeoff based on the failure rate with respect to the median return ratio.
A comparison of the historical and simulated replacement rates (table 6 and table 8) shows that the actual historical time-paths lead to similar median replacement rates as the simulated replacement rates. However, the mean replacement rates are much higher in the simulated case. This is accompanied by much higher standard deviations for the strategies with more overall equity exposure and fatter tails, implying lower values at the lowest percentiles of the distribution.
29 VI. Conclusion
This paper has evaluated the relative performance of life-cycle strategies in comparison with constant proportion asset mix strategies. Based on the results of the simulation and the historical analysis there is no evidence that life-cycle strategies in general perform better than simple constant proportion asset mix strategies. The results show that life-cycle strategies do not increase the probability of obtaining the desired retirement return. The H0, which states that constant proportion asset mix strategies lead to the same relative wealth outcomes as compared to life-cycle strategies, cannot be rejected based on the results. Constant proportion asset mix strategies have the same relative performance as life-cycle strategies. This answer the the research question: “How do life-cycle allocation strategies perform relative to constant proportion asset mix strategies in DC plans?”.
The results of the simulation are a confirmation of the findings in Porterba (2006), which concludes that life-cycle strategies are leading to similar retirement wealth outcomes as age invariant strategies. In fact, the results obtained are in line with most recent literature on long term asset allocation strategies and confirm the notion that risky assets should have a significant weight in long term asset strategies.
Based on the results it is unclear why all Dutch PPI’s choose a life-cycle strategy as the standard. Most recent literature indicates that life-cycle strategies do not increase the probability of achieving the desired retirement return. This paper does not clarify or justify the reason for the popularity of life-cycle strategies in DC plans. Further research is needed to explain the popularity of life-cycle strategies and the conservatism of the default strategies.
30 A one on one comparison of strategies is most of the times comparing apples and oranges, since the input differs, and the weighted exposure to equities is never exactly the same. Measuring the wealth outcomes to a relative benchmark that resembles the desired return, makes the strategies comparable to each other. Comparing the failure rate and lower tails of the replacement rate distribution gives a good representation of the risks that a participant faces. Although it is still hard to make definitive conclusions, by comparing 20 strategies there can be given a realistic view of how the strategies perform.
31 VII. Literature list
Antolin, P., S. Payet, and J. Yermo, (2010) Assessing Default Investment Strategies in Defined Contribution Plans, OECD Journal: Financial Market Trends, 1, 1-29 Australian Prudential Regulation Authority, (2005) Annual Superannuation Bulletin,
(APRA, Sydney)
Basu, A., and M. Drew, (2006) Appropriateness of Default Investment Options in Defined Contribution Plans: The Australian Evidence, MPRA Paper No. 3314
Basu, A., and M. Drew, (2009) Portfolio Size Effect in Retirement Accounts: What Does It Imply for Lifecycle Asset Allocation Funds?, The Journal of portfolio management, 35, 61-72
Beshears, J., J.J. Choi, D. Laibson, and B.C. Madrian, (2006) The Importance of default options for retirement savings outcomes: Evidence from the United States, National Bureau of Economic Research Working Paper No. 12009
Bikke, J., and P. Vlaar, (2006) Conditional indexation in defined benefit pension plans, DNB working paper available at http://www.dnb.nl/publicatie/publicaties-dnb/dnb-working-papers-reeks/dnb-working-papers/auto69826.jsp
Blake, D., A. Byrne, A. Cairns, and K. Dowd, (2004) The stakeholder pension lottery: An analysis of the default funds in UK stakeholder pension schemes, Pensions Institute Discussion Paper No. PI-0411.
Bodie, Z., and J. Treussard, (2007) Making Investment Choices as Simple as Possible, but Not Simpler, Financial Analysts Journal, 63, 42–47
Bodie, Z., R.C. Merton, and W.F. Samuelson, (1991) Labor supply flexibility and portfolio choice in a life-cycle model, Journal of Economic Dynamics and Control, 16;427– 449
Choi, J.J., D. Laibson, B.C. Madrian, and A. Metrick, (2003) "For better or for worse: Default effects and 401 (k) savings behavior, in David Wise, ed.: Perspectives in the Economics of Aging (University of Chicago Press, Chicago)
32 De Nederlandsche Bank, (2012) Premium Pension Institution (PPI), accessed 1 august
2012, available at http://www.toezicht.dnb.nl/en/2/51-202370.jsp
De Nederlandsche Bank, (2012) Premiepensioeninstellingen (PPI’s),De Nederlandsche Bank, accessed 1 august 2012,
<http://www.dnb.nl/binaries/Register%20Premiepensioeninstellingen _tcm46-254578.pdf>
Dimson, E., P. Marsh, and M. Staunton, 2002. Triumph of the Optimists: 101 Years of Global Investment Returns,1st
ed., Princeton: Princeton University Press Hickman, K., H. Hunter, J. Byrd, and W. Terpening, (2001) Life cycle investing, holding
periods and risk, Journal of Portfolio Management, 27,101-111. Hull, C., (2009) Options, Futures and other Derivatives, 7th
ed., London: Pearson education
Merton, R.C., (1969) Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case, Review of Economics and Statistics, 51, 247-257
Ponds, E.H.M., and B. Riel van, (2006) The recent evolution of pension funds in the Netherlands: The trend to hybrid DB-DC plans and beyond, working paper available at http://ssrn.com/abstract=964428
Poterba, J., J. Rauh, S. Venti, and D. Wise, (2006) Lifecycle asset allocation strategies and the distribution of 401 (k) retirement wealth, NBER working paper available at http://www.nber.org/papers/w11974
Samuelson, P., (1969) Lifetime Portfolio Selection by Dynamic Stochastic Programming, Review of Economics and Statistics, 51, 239-246
Scheuenstuhl, G., S. Blome, W. Mader, D. Karim, and T. Friederich, (2010) Assessing Investment Strategies in Defined Contribution Pension Plans under Various Payout Options, Risklab working paper available at
http://www.risklab.com/de/meta/publikationen/index.html
33 Schmit, J., and T. Post, (2012) Measuring the performance of life-cycle asset allocation,
Consumer Knowledge and Financial Decisions - Lifespan Perspectives, Series: International Series on Consumer Science, 285-302
34
VIII. Appendix
Appendix A: Derivation of equation 1 (source: Hull, 2009, p. 419-420) First, the geometric Brownian motion model for stock price behaviour, can be approximated by,
𝑆(𝑡 + ∆𝑡) − 𝑆(𝑡) = 𝜇 ∗ 𝑆(𝑡) ∗ ∆𝑡 + 𝜎 ∗ 𝑆(𝑡) ∗ 𝜀 ∗ √∆𝑡,
With 𝜀 being a Gaussian variable. Using Itô’s lemma it holds that,
𝑑𝑙𝑛 𝑆 = (𝜇 − 𝜎
2
2) ∗ 𝑑𝑡 + 𝜎 ∗ 𝑑𝑧,
with z being the standard Brownian motion. Hence, in approximation,
𝑙𝑛 𝑆 (𝑡 + ∆𝑡) − 𝑙𝑛 𝑆 (𝑡) = (𝜇 − 𝜎 2 2) ∗ ∆𝑡 + 𝜎 ∗ 𝜀 ∗ √∆𝑡, or equivalently, 𝑆(𝑡 + ∆𝑡) = 𝑆(𝑡) ∗ exp [(𝜇 − 𝜎 2 2) ∗ ∆𝑡 + 𝜎 ∗ 𝜀 ∗ √∆𝑡 ], in discrete formulation, 𝑆(𝑡) = 𝑆(𝑡 − 1) ∗ exp [(𝜇 − 𝜎 2 2) + 𝜎 ∗ 𝜀].
Thus for salary (SAL) it holds that,
𝑆𝐴𝐿(𝑡) = 𝑆𝐴𝐿(𝑡 − 1) ∗ exp [(𝜇 − 𝜎
2
2) + 𝜎∗ 𝜀]
35 Appendix B: Cholesky decomposition
To account for the correlation between the returns of the different variables (asset classes and inflation) the random elements (𝜀𝑆, 𝜀𝐵, 𝜀𝑇, 𝜀𝐼) are produced using a Cholesky
decomposition.
To perform a Cholesky decomposition there are produced independent random numbers for every variable under consideration, called (𝛾𝑆, 𝛾𝐵, 𝛾𝑇, 𝛾𝐼). If there would only
be two variables (𝜀𝑆, 𝜀𝐵) with a certain correlation (𝜌)then the following calculation would
be performed with the independent random numbers to get correlated random numbers: 𝜀𝑆= 𝛾𝑆,
𝜀𝐵= 𝜌 ∗ 𝛾𝑆+ 𝛾𝐵∗ √1 − 𝜌2.
Since there are 4 variables we need 4 correlated samples: 𝜀𝑆= 𝛼𝑆𝑆∗ 𝛾𝑆,
𝜀𝐵= 𝛼𝐵𝑆∗ 𝛾𝑆+ 𝛼𝐵𝐵∗ 𝛾𝐵,
𝜀𝑇= 𝛼𝑇𝑆∗ 𝛾𝑆+ 𝛼𝑇𝐵∗ 𝛾𝐵+ 𝛼𝑇𝑇∗ 𝛾𝑇,
𝜀𝐼= 𝛼𝐼𝑆∗ 𝛾𝑆+ 𝛼𝐼𝐵∗ 𝛾𝐵+ 𝛼𝐼𝑇∗ 𝛾𝑇+ 𝛼𝐼𝐼∗ 𝛾𝐼,
were the coefficients 𝛼 are chosen in such a way that the correlations and variance are correct. This can be done step by step as follows. Set 𝛼𝑆𝑆= 1; choose 𝛼𝐵𝑆 so that 𝛼𝐵𝑆∗ 𝛼𝑆𝑆
= 𝜌𝐵𝑆; choose 𝛼𝐵𝑆 so that 𝛼𝐵𝑆2 + 𝛼𝐵𝐵2 = 1; choose 𝛼𝑇𝑆 so that 𝛼𝑇𝑆∗ 𝛼𝑆𝑆 = 𝜌𝑇𝑆; choose 𝛼𝑇𝐵
so that 𝛼𝑇𝑆∗ 𝛼𝐵𝑆+𝛼𝑇𝐵∗ 𝛼𝐵𝐵 = 𝜌𝑇𝐵; choose 𝛼𝑇𝑇 so that 𝛼𝑇𝑆2 + 𝛼𝑇𝐵2 + 𝛼𝑇𝐵2 = 1; and so on.
36 Appendix C: Yearly asset allocation of the investment strategies
37 Appendix D: The AOW replacement rate
Table 9 represents the year on year income data obtained from the CBS, with (NLI) representing the National Labour Income, (TA) the total AOW pay-out, (NW) the net workforce and (R) the number of retirees. With these CBS statistics the following calculations are performed:
The average income is represented by (AI), which is the result of dividing the National Labour Income (NLI) by the Net Workforce (NW). The average AOW pay-out is represented by (AP), which is the result of dividing the total AOW pay-out (TA) by the number of retirees (R). Dividing the average income (AI) by the average AOW pay-out (AP) results in the AOW ratio.
The table shows clearly that the AOW replacement rate, as a percentage of the average income, has been an accurate predictor with an average of around 35 percent. It is therefore assumed that the AOW replacement rate is 35 percent.
38 Appendix E:
Table 10: Simulated replacement rate comparison of constant proportion asset mix strategies with life-cycle strategies on the basis of failure rate and lower percentiles. Return characteristics: 1928-2011,
assuming no correlation.
Table 11: Simulated replacement rate comparison of constant proportion asset mix strategies with life-cycle strategies on the basis of failure rate and lower percentiles. Return characteristics: 1946-2011
Category Strategy XE Mean Median St. dev. 25 15 5 1 Failur e ALL EQUITY 100% 4.8 2.6 7.4 1.3 0.9 0.6 0.3 17% Agressive LAST 10 88% 4.2 2.5 5.9 1.3 1.0 0.6 0.4 15% CONSTANT AGR 81% 3.5 2.3 4.0 1.4 1.0 0.7 0.4 14% LINEAR AGR 74% 3.4 2.3 3.6 1.4 1.1 0.7 0.5 13% SW LINEAR AGR 71% 3.4 2.3 3.7 1.3 1.1 0.7 0.5 13% LAST 15 78% 3.7 2.4 4.6 1.3 1.0 0.7 0.5 14% LAST 20 69% 3.2 2.2 3.5 1.3 1.0 0.7 0.5 14%
Intermediate CONSTANT INT 56% 2.4 1.9 1.8 1.3 1.0 0.7 0.5 13% BEFRANK 53% 2.4 1.8 2.1 1.2 0.9 0.7 0.5 18% TKP 52% 2.5 1.8 2.3 1.2 0.9 0.7 0.5 17% LINEAR INT 49% 2.3 1.9 1.6 1.3 1.0 0.7 0.5 14% AEGON 49% 2.3 1.9 1.7 1.2 1.0 0.7 0.5 14% SW LINEAR INT 46% 2.3 1.8 1.7 1.2 1.0 0.7 0.5 15% Conservative ROBECO 37% 2.0 1.6 1.3 1.1 0.9 0.7 0.5 18% CONSTANT CON 31% 1.7 1.5 0.9 1.1 0.9 0.7 0.5 21% NN 25% 1.6 1.3 0.9 1.0 0.8 0.6 0.4 28% LINEAR CON 24% 1.6 1.4 0.9 1.0 0.8 0.6 0.5 25% SW LINEAR CON 21% 3.5 2.3 4.0 1.4 1.0 0.7 0.4 14% ALL BOND 0% 1.1 1.0 0.6 0.7 0.6 0.4 0.3 53% ALL TBILL 0% 0.8 0.7 0.3 0.5 0.5 0.4 0.3 80% Percentiles of distribution
Category Strategy XE Mean Median St. dev. 25 15 5 1Failur e ALL EQUITY 100% 4.8 2.9 6.2 1.6 1.2 0.7 0.4 11% Agressive LAST 10 88% 4.3 2.7 5.0 1.5 1.2 0.7 0.5 11% CONSTANT AGR 81% 3.5 2.5 3.6 1.5 1.1 0.8 0.5 11% LINEAR AGR 74% 3.3 2.4 3.2 1.4 1.1 0.8 0.5 11% SW LINEAR AGR 71% 3.3 2.4 3.3 1.4 1.1 0.8 0.5 11% LAST 15 78% 3.7 2.5 4.0 1.5 1.1 0.7 0.5 11% LAST 20 69% 3.2 2.3 3.1 1.4 1.1 0.7 0.5 12%
39 Table 12: Simulated replacement rate comparison of constant proportion asset mix strategies with
life-cycle strategies on the basis of failure rate and lower percentiles. Return characteristics: 1946-2011, assuming no correlation
Category Strategy XE Mean Median St. dev. 25 15 5 1 Failure
ALL EQUITY 100% 4.8 2.9 6.1 1.6 1.2 0.7 0.4 11% Agressive LAST 10 88% 4.2 2.8 4.9 1.6 1.2 0.8 0.5 10% CONSTANT AGR 81% 3.5 2.5 3.3 1.5 1.2 0.8 0.5 9% LINEAR AGR 74% 3.3 2.5 3.0 1.5 1.2 0.8 0.6 9% SW LINEAR AGR 71% 3.3 2.4 3.1 1.5 1.2 0.8 0.6 9% LAST 15 78% 3.7 2.6 3.9 1.5 1.2 0.8 0.6 10% LAST 20 69% 3.2 2.3 2.9 1.5 1.2 0.8 0.6 9%
