PENSION INVESTMENT AT THE RISK OF NOT INVESTING
OPTIMALLY
—a study on the optimal ratio between investment and consumption—
Didier Quintius
Afstudeerscriptie voor
Bachelor Actuari¨ele Wetenschappen
Universiteit van Amsterdam
Faculteit Economie en Bedrijfskunde
Auteur: Didier Quintius
Studentnr.: 10985085
Email-adres: diquisr@gmail.com
Datum: 26 juni 2018
Statement of Originality
This document is written by Student Didier Quintius, who declares to take full responsibility for
the contents of this document. I declare that the text and the work presented in this document
are original and that no sources other than those mentioned in the text and its references have
been used in creating it. The Faculty of Economics and Business is responsible solely for the
Summary
The current pension system needs be urgently updated because of the increasing need for individual-ization. This paper studies what the optimal strategy should be for an individual pension investor. This strategy consists of optimal asset allocation and optimal consumption levels, which are calcu-lated via simulation. To clearly evaluate the results of the simulation, certain assumptions were made. Although these assumptions did prove useful in most of the evaluation, the assumptions made it hard to compare the optimal strategies to strategies used in practice. The time sensitivity of long-term investment proved to be fundamental to cause the optimal strategy to produce the highest utility levels.
Contents
1 Introduction 5
2 Theoretical background and Model specification 6 2.1 Interest and Maturity . . . 6 2.2 Consumption and utility . . . 7 2.3 Portfolio strategy . . . 10
3 Results 12
3.1 Optimal strategy . . . 13 3.2 Sub optimal strategies . . . 15 3.3 Utility and certainty equivalent . . . 16
4 Evaluation 17
4.1 Optimal Strategy . . . 17 4.2 Time Sensitivity . . . 18 4.3 Practical comparison . . . 19
1
Introduction
The Dutch pension world is changing. Life expectancy has increased and interest rates have signifi-cantly decreased over the past couple of years. These changes cause pension funds to evaluate their increasing liabilities and search for solutions to this crucial issue. Bovenberg and Nijman (2017) de-scribe this issue by discussing different pension contracts and combining the positive contributions of each contract to form a solution. They find that the legal liabilities of pension funds are not completely hedged and that employers stop forming a buffer of risk for pension plans, which conducts more risk to the individual. To meet the future consumption of this individual a specific investment portfolio is needed to prevent loss of wealth.
The occupational pension system forms the largest component of the pension income for most middle and high-class citizens. This system is also known as the second pillar of the Dutch pension system. Most of the pensions in this pillar are Defined Benefit (DB) pensions. Bovenberg and Nijman (2017) discuss that this system offers stable financial income at retirement because, as the name suggests, the investments are obtained over the life cycle to be able to pay out the previously defined pension income, meaning that the investments may vary but the payouts must stay constant. The system also eliminates longevity risk1 due to pooling2 and mandatory participation, but the effect of the mandatory participation is decreasing because of the increase of self-employment. Furthermore, it is common for an individual to work less than forty years or to change employer. These changes in the labor market cause the need for individualization of pension plans, which is only possible with a more actuarially fair pension system. Bovenberg and Nijman (2017) claim that alignment of pensions to the individual can significantly increase his well-being.
It is clear that a new system is needed, Bovenberg and Nijman state that the base of this new system can be the Defined Contribution pensions. This pension offers more opportunities to individualize the investment strategy. The liberty is obtained by offering the individual a choice to take more risks regarding investments and longevity. More flexibility in the amount of pension income is also provided by this pension. Another benefit of this pension is the ability to cover shocks. This allows for risky investments, but a yet constant pension income. Furthermore, they state that a life cycle portfolio, that lowers investment risks as the age of the individual, increases is an essential factor of this pension. Showing that the strategy of investment and consumption is essential to obtain a stable pension income. Optimizing the investment strategy can prevent significant losses.
The risks, that some individuals face, are also discussed in the study by Shuey and O’Rand (2004).
1 The risk of living longer than expected
2using the profit gained from those who die early to pay for the losses suffered from those who live longer
They discuss that by conducting investment risk to the individual, it is possible that the individual cannot acquire enough wealth to sustain a decent pension income. Longevity risk, labor income risk and health risk is all conducted to the individual. These problems can be solved by a pension system that allows for optimal investment for a specific individual.
The need for optimal life cycle portfolios grows as the individualization of the pension scheme increases. Participants require individualization of their pensions. This change increases the impor-tance of optimal portfolios. The amount of wealth that should be invested and the amount of wealth that should be consumed play an important role in such a portfolio. These amounts differ among the different stages in a life cycle. The next sections discuss the optimal portfolio in a life cycle and evaluate the possible losses that can be countered by investing optimally.
The next section provides the literary base for the theorems, assumptions and the model that is used. In the third section, the optimal strategy is calculated. This optimal strategy consists of a portfolio strategy and a consumption strategy. Where the portfolio strategy allows the individual to invest in a nominal bond3 and cash. Sub optimal strategies, and their different utility levels are also
shown in this section. The fourth section evaluates the results of the previous chapters and compares the different strategies. Lastly, the results are concluded in the fifth section.
2
Theoretical background and Model specification
As discussed above, this section covers the empirical part of the thesis, where different studies are compared to inform what the equations and the assumptions are based on and what studies agree and disagree with these statements. The equations, used to derive interest rate, maturity, optimal consumption in the life cycle, the composition of the portfolio and the utility levels are elaborated in this section . Apart from the model, this section discusses the specific alterations that are made to fit the goal of the study. Additional empirical evidence of the models is also discussed.
2.1
Interest and Maturity
The first part of the section provides the equations, used to simulate the results, and gives empirical evidence to the validity of these equations by comparing it to previous studies. Note that all of the equations are derived from the study by Brennan and Xia (2002), who conducted a similar study to find the optimal strategy for pension investment. The alterations that are made to their model are also elaborated below.
3 Meaning that the investor is able to buy a bond and after the defined amount of time receives a payout,
Equation (1) is used to simulate the rate of maturity, here r denotes the interest rate and θr a
constant that represents the sensitivity to shocks.
dM
M = −rdt + θrdzr (1)
dr = κ(¯r − r)dt + σrdzr (2)
Secondly, equation (2), simulates the interest rate. The equation is a mean reversion model4, a one factor model. This model only captures shocks that influence the short-term interest rate. Campbell and Viciera (1999) incorporated a two factor model, which is able to capture the shocks on the real and nominal interest rate. They explain that their model is in a reduced form. Expanding the model adds the possibility to analyze the essential sources of these shocks. The real and nominal interest rates are both AR(1) processes. They contain a relation with the previous interest rate. This also provides information about the different shocks of the interest rate.
Similarly Black, Derman and Toy (1990) analyze bond prices with a one factor model. They explain that although the accuracy of the model is heightened as the number of factors increases, the model becomes more complicated. The nature of the shocks in the interest rate is not essential when analyzing the life cycle of an individual. The focus in this study is to discuss the differences within the life cycle of the individual. Expanding the model can lead to unnecessary complication and may not provide beneficial information when merely evaluating portfolio strategies.
A parameter that may also cause complication is inflation, an intricate subject when regarding long-term investment, it is the first element of the study by Brennan and Xia (2002), which most of the equations are based on, that is disregarded. On the trading market there are no assets which bet against inflation resulting in difficulty when trying to hedge this risk. This difficulty causes pension funds to neglect inflation risk in practice. This does not suppress the substantial effects inflation may have on the pension income. Battocchio and Menoncin (2004) state that due to the inflation risk even risk-free assets become risky in theory. The additional convenience of omitting inflation combined with the absence of practical usage cause it to be excluded from the model.
2.2
Consumption and utility
Next equation (3) and (4), the consumption optimization, are discussed. They are characterized by two equations, first the maximization problem and secondly the constraint. Equation (3) uses the risk
aversion γ that specifies the amount of risk the individual is willing to take. An important assumption made is that the risk aversion of the investor is average (γ = 3). This level is chosen as average by Brennan and Xia (2002). Pension investment is an undertaking that effects all members of society and thus an average level of aversion to risk is chosen. To test the sensitivity to the risk aversion the optimal consumption levels are also simulated for a less risk averse and a more risk averse investor.
max C(s):t≤s≤TEt Z T t C(s)1−γ 1 − γ ds (3) s.t. Et Z T t Ms Mt C(s) ds = Wt (4)
The assumption is made that the individual at t = 0 works forty years and then dies after twenty years. Amounting to a total life cycle of sixty years, a generalized life cycle for a single individual. The consumption of this individual is optimized throughout the life cycle. Constraining this optimization, equation (4), is the fact that the discounted value of this consumption must be equal to the initial wealth, which is equal to the discounted value of the accumulated value of forty years of working.
Equations (3) and (4) optimize the consumption over a lifetime. Describing most retirees because they aim for stable income throughout their remaining life with a given amount of wealth. Brennan and Xia (2002) also use this strategy but additionally provide another optimization strategy. This strategy is constructed to optimize the consumption to obtain maximum wealth at the end of the horizon. At which this wealth is the only remaining income and therefore consumed till the time of death. After rewriting both strategies they conclude that both equations had similar results. Implying that the consumption strategy for a working individual is comparable to someone who is retired. Therefore approving using the same strategy to calculate the consumption during labor and retirement. The optimization of the consumption can be rewritten as:
C(s) = Ms Mt −1/γ Q−1Wt (5) Q = 1 RT t Ms Mt 1−1γ (6)
After the consumption is calculated, equation (4) can be used to calculate the utility level of the consumption strategy. Subsequently the certainty equivalent level can be determined by using
equation (7). The utility level is a parameter, that can be used to compare different cases equally. It shows how useful a certain decision can be for a certain individual. This is also the case for the certainty equivalent but this has an added feature of being easier to quantify, because it is equal to the amount that is yearly consumed on average.
CE = U(1 − γ) 60 1 1−γ (7)
As mentioned before the consumption level is optimized for the life cycle of an individual. This causes the goal function to be the same for every period in this life cycle. The individual strives to maximize his consumption over this life cycle. Andreasson, Shevchenko and Novikov (2017) found that the consumption level at retirement decreases as the individual ages, caused by the increasing mortality risk. They claim that as the possibility to die increases, individuals become more careful in consuming their remaining wealth.
The equations, used to optimize consumption, shown are also used by Zou, Chen and Wedge (2014). Their model contains a complete market without borrowing constraints. This is the same as the model of Brennan and Xia (2002), who also explained that the investor must decide for every period how much is consumed. These assumptions are based on the framework of Merton (1969). He discusses that consumption may differ over time and the portfolio choices correspondingly. His work is used by most studies regarding these subjects. Pension investment is of a similar form and therefore able to be calculated by comparable frameworks.
In contrast, Campbell and Viciera (1999) use a recursive equation to calculate the preferences of the investor. They claim that recursive preferences are beneficial when analyzing the difference between their investment strategy towards risk and their consumption strategy. These differences can aid when analyzing long-term investments because the amount of consumption is likely to change over time. This method allows the horizon of the life cycle to be infinity. When discussing pension investments, the end of the life cycle needs to be defined. A recursive equation can therefore not be used.
Although the end of life cycle needs to be defined, Cocco et al (2005) show that it does not need to be defined as a constant. They use an element of uncertainty described in Hubbard, Skinner and Zeldes (1995). The exact age of death in this study is assumed to be t = 60 without any uncertainty. This assumption can be validated by the fact that the age of death slightly influences the attitude towards risk and consumption. Assuming that the individual can not die is an assumption that decreases the validity of the study.
2.3
Portfolio strategy
The last part of the sections, elaborates on the equations that are involved when simulating the interest rate. More assumption are made and an expectation of the results with respect to these assumption is made. The effects of the risk aversion on the portfolio allocation is also featured. Giving an overview of the framework of the optimal investment strategy.
x = λr γσ2 + 1 −1 γ ˆ B B (8) ˆ B = Et h Ms Mt C(s) i Bs Et h RT t Ms Mt C(s) ds i (9) Bs= κ−1(1 − eκ(s−T )) (10)
Equation (8) describes the fraction of the portfolio that is invested in the nominal bond. The implemented parameters of the equation are calibrated to fit a model where the maximum share invested in bonds is equal to 1. Meaning that the individual is not allowed to lend money to invest in bonds. His wealth is either invested in bonds or held in the bank. In the study by Brennan and Xia (2002) the individual invested op to four times his wealth in stocks. To prevent this occurrence the parameters are calibrated to fit the lending constraint. An overview of the values of the parameters that are used in the model is provided in the appendix.
THe first part of equation (8) is characterized by a constant containing the risk aversion, the average excess interest(λ) and the volatility of the asset. This equation describes the proportion of the asset allocation that is invested unconditionally, as expected this part is higher if the excess interest increases, if the volatility decreases and if the risk aversion decreases. The first two features characterize an asset with a higher sharp ratio5 and the third an investor who is more likely to invest, which all increase the unconditional proportion invested in the bond.
The second part of the function includes the risk aversion again but mainly the two time sen-sitivities6 to risk, the speculative part of the equation. The ˆB is the consumption time sensitivity. It is a weighted average of the time sensitivity to the end of life, where the consumption forms the weights. The other time sensitivity is of the nominal bond. Dividing these two variables creates a factor of the individual’s preference towards investment and consumption for every stage of the life cycle. Therefore can be claimed that as the time sensitivity of the consumption decreases. Meaning
5average interest divided by the standard deviation
that the importance of the consumption decreases. The fraction that is invested into the nominal bond decreases. The opposite applies to the time sensitivity of the consumption. This factor is then multiplied by the the complement of the 1γ, which the first part of the equation was multiplied by, showing that a more risk averse investor invest less in the unconditional constant proportion and more into the speculative part.
Another parameter used in the equations is the age till death (T ). Instead of time till death, Brennan and Xia (2002) analyze the optimal investment strategy for different horizons7. The results of this strategy show that the share of the bonds is constant over the horizons and that the share of the risky assets increases to a certain point as the horizon increases. In a life cycle, that means that the older the individual is the lower the share of risky assets must be. This relates to the aforementioned decrease of consumption as the mortality risk rises. This occurrence is also caused by the fact that the amount of wealth decreases over time. Meaning that the available buffer decreases causing the attitude towards risky investments to change. In this study there are no risky assets, meaning that the nominal bond is the riskiest asset in the portfolio. Therefore it is expected to encounter the increasing trend of the risky assets in the bonds in this study.
The nominal bond is not the riskiest asset in the study by Brennan and Xia (2002). They use a model where the individual has four possible investments namely stocks, two nominal bonds and cash. The portfolio given by equation (8) consist of a single nominal bond with an arbitrary maturity and cash. The cash is placed in the bank and accumulates interest at the risk-free rate. When analyzing an individuals attitude towards risk, a portfolio with only two possible investment shows more distinct results than if the portfolio contains more investment options. Although it is not practical to assume that an individual only has an option to invest in a nominal bond, the results show clear differences in a portfolio during a life cycle.
None of the equations that are discussed contain a bequest motive, being derived from the model of Brennan and Xia (2002), who concentrated less on factors that may influence the investor. Conversely, Cocco, Gomes and Maenhout (2005) discuss subjects an individual may face during a life cycle while investing for his pension. The bequest motive is such a subject, where the individual not only strives toward stable pension income but also aims to provide income for his descendants when he dies. This motive would lower the total consumption of the individual. Cocco et al (2005) explain that it cannot be empirically shown that this motive is considered when investing in practice. The increased difficulty of expanding the model with a bequest motive and the lack of empirical evidence for its existence cause the bequest motive to be neglected.
Lastly, the arguments made in this section show that the goal is to find a model that can be clearly
interpreted. Additional factors that make the model more realistic but complicated to understand are omitted. Further, the model is also used by several other studies conducting the same type of research, which increases the confidence in the model. The model is employed in the next section simulating the results after which they can be evaluated and concluded.
3
Results
The practical part of the study is discussed in this next section. Theories and equations that are explained in the previous section are applied here. The different simulated variables are portrayed, first of all the interest and maturity rate are discussed and after that the optimal strategy and its adjustments are regarded according to the different values chosen for the constants and the equations used to simulate the results. These values are mostly based on the study by Brennan and Xia (2002), some were changed to fit the assumptions of this study. While regarding the optimal strategy the sensitivity to the risk aversion is shown. Next follows the portfolio strategies in the sub optimal cases and lastly the section is concluded with the utility levels and the certainty equivalent of each case. The section visualizes the theoretical information that is given in the previous section.
First of all the interest rate is simulated, this must be done first because it is indirectly used in every following simulation. Although figure (1) shows visual variation in the interest rate, it is clearly constant in approximation. This is mainly due to the mean reversion rate that is incorporated into equation (1), used to simulate the interest rate. The average interest rate is 1.2% and the average deviation is about 0.1%.
Secondly The maturity rate is simulated. Figure (2) shows the behaviour of the maturity. The maturity steeply decreases the first 20 years, after that the acceleration of the decrease is reduced to a point where the maturity seems nearly constant around 0.6. The maturity even seems to increase at the end of the life cycle. This is possible due to the increase of the significance of the white noise at the end of the life cycle, which causes unusual behaviour to become visible. The growth of this effect comes from the decrease of the maturity, which reduces the decrease caused by the interest rate.
Figure 2: Maturity
3.1
Optimal strategy
The steep decrease of the maturity rate has a direct effect on the consumption rate, which is the first part of the optimal solution. When calculating the levels of consumption the initial wealth needs to be determined. The assumption is made that the previous wealth is the discounted value of the total amount of wages an individual can obtain within forty years of working. Assuming constant yearly wages of one (1), the initial wealth is the sum of the first forty values of the maturity. Then a transformed form of the maturity, involving the risk aversion, is multiplied by a constant and the initial wealth to obtain the consumption levels, equation (5).
The values of the level of consumption seen in figure (3) correspond to the proportion of a years wage an individual consumes in a single year. Next figure (3) shows three curves, which correspond to the different levels of risk aversion. For all levels of risk aversion the individual starts by using around 65% of their salary. The ultimate level of consumption differs heavily among the different levels of risk aversion. The figure shows that the higher the risk aversion the lower the increase of the consumption level. This can be explained by the fact that riskier investment corresponds to higher returns which allows higher consumption throughout the life cycle.
Figure 3: Consumption
Secondly the portfolio strategy is calculated. The shares of the portfolio which are invested in bonds are calculated to optimally fit the consumption level. The portfolio shares follow a decreasing seemingly exponential trend. Again there are three curves shown, now in figure (4), resembling the different levels of risk aversion.
Figure 4: Portfolios
Figure (4) also shows three horizontal lines. These lines form the constant part of the shares of the bond portfolio. This means the proportion of the portfolio that is invested in the portfolio regardless of the part of the life cycle. This line is higher if the risk aversion is lower, which is expected because the lower the risk aversion the higher the proportion that is invested unconditionally. The proportion above the horizontal line forms the speculative part of the portfolio. In the beginning of the life cycle this is higher for more risk averse investors but decreases faster than less risk averse investors. This
can be translated to the fact that risk averse investors speculate more.
In the rest of the paper the average risk aversion (γ = 3) is chosen to compare to the other strategies. The consumption in this case still shows significant increase. The final level of consumption rises to 10% more than the previous yearly wage, this shows that the consumption level increases almost twofold. Also the portfolio level decreases gradually, investing merely 25% of the portfolio in bonds at the end of the life cycle.
3.2
Sub optimal strategies
To further compare the optimal strategy several sub optimal cases have been opted. The first is the constant strategy, where the individual invests a fixed proportion of the portfolio in bonds throughout the life cycle. This fixed share is the average of the shares of the optimal portfolio. Allowing for a comparison that shows the importance of the time sensitivity of an investment. Furthermore the second case that is discussed, is a linear strategy starting by investing fully in bonds and ending by investing nothing in bonds. This strategy shows another element to the importance of the time sensitivity.
Figure 5: Portfolio Time Sensitivity
Lastly from the insurance company Aegon a neutral investment strategy is chosen. This is done because the risk aversion level chosen in this study corresponds to that of an average risk taking investor. The strategy is then corrected for an investment of sixty years. The data from Aegon, retrieved from their website, implies to invest maximally the first years of investment. After a certain amount of time the individual begins to decrease and then slightly increase the investment following a quadratic equation. In this study the point for starting the decrease is chosen at year 35, because Aegon starts this decrease 25 years before the end of the investment.
Figure 6: Portfolio Aegon
3.3
Utility and certainty equivalent
The final way to compare the different risk aversions and investment strategies is by discussing their utility levels. This way the different cases can be compared equally. When an individual faces a choice between cases, it is most likely that the case with the highest utility level will be the most useful one. The levels are calculated by taking the sum of the different consumption rates transformed by equation (4). First the different levels of risk aversion are compared. It is clear to see that the utility level increases with the risk aversion, but because the risk aversion of an individual isn’t something they choose, it is not fair to compare these utility levels. What can be seen is that difference between utility levels decreases as the risk aversion increases.
γ 2 3 5
Utility level -70.2562 -47.3731 -45.2593
Figure 7: Utility risk aversion
Finally the utility levels are given for the different scenarios. The consumption rates of the sub optimal strategies are calculated by assuming that the wealth at t = 60 is equal to zero. Then is assumed that the proportions between the levels of consumption stay the same, but because the investment strategy is not optimal the total consumption must be lower. This lower consumption is calculated by multiplying the optimal consumption by a constant α. The new level of consumption is then lowered to fit the corresponding sub optimal strategy. With the new consumption levels the utility can be calculated and after this the utility is used to calculate the certainty equivalent of the different cases, equation (7). To show the loss of wealth when not investing at all a zero-strategy is opted, here the investor only receives interest from his bank account.
Strategy Utility level certainty equivalent Optimal -47.3731 0.7958 Constant -47.7171 0.7929 Linear -47.4535 0.7951 Zero -54.5786 0.7413 Aegon -47.3896 0.7956
Figure 8: Utility levels and Consumption
The results clearly show that the optimal solution gives the highest utility and therefore the highest certainty equivalent. The section discusses the two components of the strategy, the consumption and the investment, for the different cases. There is also shown what adjustments were made to the equations to fit the assumptions of the study so that the results can be evaluated correctly.
4
Evaluation
The following section discusses the results seen in the previous section. The optimal pension investment is discussed and compared to the sub optimal strategies. Time sensitivity is an important factor when deciding what strategy is optimal and is thus the main subject when comparing the different strategies. Lastly the strategy of Aegon is discussed, a strategy used in practice, concluding the evaluation of the results.
4.1
Optimal Strategy
The first reason to conduct this study is to find what the optimal strategy is. The previous section shows both parts of this strategy, the consumption and the investment. The consumption develops differently than is described in the theory. In the theory the consumption decreases at the end of the life cycle. The optimal solution of this study contains consumption that increases constantly, increasing when the risk aversion decreases, the consumption at the end of the life cycle, in the case of average risk aversion, is almost twice as large as the consumption at the beginning.
The risk aversion further shows that low risk aversion allows for higher investments and sub-sequently higher consumption. But this higher consumption does not translate into higher utility. Adversely the utility level increases as the risk aversion increases, meaning that an individual with a risky attitude needs substantially higher consumption levels than given in this case to be equivalently satisfied, but the risk aversion of an individual is rarely chosen. It is more likely a given characteristic,
thus it is not completely fair to compare the utility levels between risk aversions.
On the contrary the portfolio behaves exactly as expected. The shares of investment strategy behave like the stocks in the study by Brennan and Xia (2002), the shares decrease slowly the first years and at the end of the life cycle this decrease accelerates rapidly. In this study the bond is the only risky asset and thus follows the same trend as the riskiest asset in a portfolio where there are multiple investment options and different levels of risk. The risk aversion affects the optimal strategy differently then the consumption. When the risk aversion rises the unconditional investment share decreases because the investor is less likely to invest but the speculative part of the investment increases because the investor bases the amount to invest more on the conditions of the asset.
4.2
Time Sensitivity
Sub optimal solutions are discussed in the study to show the effects of different strategies on the wealth of an individual. The most extreme case is the zero-strategy, this case shows the importance of investing in bonds because of the significant loss of wealth that is suffered by not investing at all. This can be seen by the substantial difference between the utility level and the certainty equivalent of the optimal case and the zero-strategy.
Looking at the differences between the results of the other strategies, the time sensitivity of the investment seems to play an important role. Every sub optimal strategy shows this in a different way. When comparing the constant strategy to the optimal strategy, knowing that the mean of the shares of both strategies is the same, the optimal solution results in a higher utility. The reason for this is that in the time that the individual consumes less, he invests more. This shows that it is important to vary the amount you invest to the period in the life cycle. The decrease is also essential because this allows for higher profits and if these profits were to disappoint there is still enough time to change the portfolio in such a way that it can provide steady pension payments.
The other sub optimal solution is a linear approximation of the optimal solution. This strategy also has lower utility levels than the optimal solution but higher than the constant strategy. This shows that the decreasing trend is beneficial when investing long-term. Meaning that investing more in the beginning of the life cycle is profitable because the individual consumes less in that period so there is more money to invest. Also the comparison shows that although the linear strategy incorporates a decreasing trend, there is a more optimal way to invest. The exponential behaviour of the optimal strategy clearly allows for higher utility levels.
4.3
Practical comparison
The last sub optimal strategy is an actual strategy that is used by the insurance company Aegon. They chose this strategy because they want to cover the interest risk at the end of the life cycle. as mentioned before if an individual suffers large losses at the end of a life cycle he may not be able to receive sufficient pension payments for his remaining life. Covering interest risk is clearly implemented in a different way by Aegon than it is in the optimal solution, as can be seen in figure (6). Further in the graph can be seen that the shares of Aegon are constantly higher than that of the optimal strategy, but this does not amount in a higher utility level. Meaning that investing more does not directly result in higher levels of utility.
The fact that Aegons strategy to invest their pensions differs from the optimal strategy can also have to do with the assumptions made in this study. The strategy of Aegon is most likely very practical. The assumption made to create the optimal solutions may play a significant role in these differences. Omitting stocks is likely the most sensitive assumption that is made. This assumption prevents the direct comparison between the optimal solution and all practical strategies.
The results of the study were similar to that what was expected in the theory. Only the consump-tion was very different. The sub optimal soluconsump-tions proved to be insightful when analyzing the time sensitivity of a long-term investment and they allow for certain comparisons between different cases. Also the assumptions made in this study make it difficult to compare it to practical strategies.
5
Conclusion
This study highlights the importance of portfolio choices. In the life cycle of most people the pension investment is largest investment they conduct. This investment determines your future wealth and must therefore be handled properly. As the results show, sub optimal strategies decrease the certainty equivalent level. The consumption strategy is also discussed, it is as important for insuring future wealth as the investment strategy and must also be executed carefully.
Consumption showed to differ from the theory. The eventual levels of consumption also proved to be unrealistic with the consumption increasing with almost 100% over the life cycle. This increment was higher for lower levels of risk aversion and lower for higher levels. this is expected because higher levels of risk aversion correspond with lower levels of investment meaning less interest thus lower ultimate consumption.
The other half of the investment strategy is the portfolio strategy. The results showed that the shares of a portfolio for long-term investing should slowly decrease till the end of the life cycle where the decrease should be accelerated to cover interest risks. If this risk is not covered, it could lead to
irreversible losses of wealth. This behaviour was expected and can be found in other studies like that of Brennan and Xia (2002). The only difference is that in this study the behaviour of the bonds is comparable to the behaviour of the stocks. Stocks are mostly the riskiest asset in a portfolio but in this study the bonds are the riskiest asset therefore the trends are similar. These trends can still be found when using other levels of risk aversion. Noticeable is that for higher levels of risk aversion the initial share is higher but the portfolio decreases faster, thus ending lower than levels with lower risk aversion.
To find out what makes the strategy optimal, it was compared with different sub optimal strate-gies. When comparing these strategies the time sensitivity proved to be the best way to explain the differences. The value of investing the right amount at the right time could make a significant differ-ence in ultimate level of utility. Also investing at the beginning of the life cycle proves to be more lucrative because the individual consumes less in that period.
Further the optimal strategy was compared with the strategy of the insurance company Aegon. Comparing the optimal strategy to a real life strategy proved to be difficult because of the assumptions made in this study. An individual who invests at Aegon is allowed to invest in stocks, that is not possible in that portfolio and to assume that the individual would then chose the same strategy, substituting the stocks for bonds, is not realistic.
Another assumption made that influences the validity of the results is omitting the inflation. Inflation has a large influence of investments especially long-term investments. Although in this study the goal was to find and compare the optimal strategy in a basic format. It is questionable if this shape would not be quite different if certain assumptions were not made.
Not making the mentioned assumptions could prove to give more insights into the long-term investment strategy. It could also allow for more realistic results regarding the level of consumption. Further the theory explains different assumptions that were made that could be tested to influence the behaviour of an individual when investing for his pension. Assumptions on the bequest motive and the possibility to lend money to invest could have a significant effect on the results and would allow the optimal strategy to be compared to strategies used in practice.
Appendix
Parameters Calibrated value ¯ r 0.012 r0 0.012 κ 0.105 σr 0.042 λr -0.165 φr 0.17 T 60 tmaturity 10
References
Andreasson, J. G., Shevchenko, P. V., and Novikov, A. (2017). Optimal consumption, investment and housing with means-tested public pension in retirement. Insurance: Mathematics and Economics, 75(1), 32-47.
Battocchio, P., and Menoncin, F. (2004). Optimal pension management in a stochastic framework. Insurance: Mathematics and Economics, 34(1), 79-95.
Black, F., Derman, E., and Toy, W. (1990). A one-factor model of interest rates and its application to treasury bond options. Financial analysts journal, 46(1), 33-39.
Bovenberg, L., and Nijman, T. (2017). New Dutch pension contracts and lessons for other countries. Netspar Academic Seriess; Vol. DP 09/2017-014). Tilburg: NETSPAR.
Brennan, M. J., and Xia, Y. (2002). Dynamic asset allocation under inflation. The Journal of Finance, 57(3), 1201-1238.
Campbell, J. Y., and Viceira, L. M. (1999). Consumption and portfolio decisions when expected returns are time varying. The Quarterly Journal of Economics, 114(2), 433-495.
Choose the Life Cycle that suits you retrieved from https://www.aegonppi.nl/choose-life-cycle-suits-you
Cocco, J. F., Gomes, F. J., and Maenhout, P. J. (2005). Consumption and portfolio choice over the life cycle. The Review of Financial Studies, 18(2), 491-533.
Hubbard, R. G., Skinner, J., and Zeldes, S. P. (1995). Precautionary saving and social insurance. Journal of political Economy, 103(2), 360-399.
Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: the continuous time case. Rev. Econom. Statist. 51 (3), 247257.
Shuey, K. M., and O’Rand, A. M. (2004). New risks for workers: Pensions, labor markets, and gender. Annu. Rev. Sociol., 30(1), 453-477.
Zou, Z., Chen, S., and Wedge, L. (2014). Finite horizon consumption and portfolio decisions with stochastic hyperbolic discounting. Journal of Mathematical Economics, 52(1), 70-80.
