Default Investment Strategies and Life-styling

Transcription

1 Default Investment Strategies and Life-styling A Paper to the Society of Actuaries in Ireland Brendan Johnston David Kavanagh Dervla Tomlin Brian Woods

2 Table of Contents EXECUTIVE SUMMARY 2 CHAPTER 1 Introduction.. 6 CHAPTER 2 Legislative Background and Relevant Guidance for PRSA Default Investment Strategies... 8 CHAPTER 3 Common Default Investment Strategies. 10 CHAPTER 4 A Particular Approach to Evaluating Default Investment Strategies.. 12 CHAPTER 5 Var Justification for Life-Styling CHAPTER 6 Modern Finance Theory and Default Investment Strategies. 27 CHAPTER 7 Conclusions and Recommendations.. 44 REFERENCES 46 1

3 Executive Summary Purpose of this paper The purpose of this paper is to provide views and ideas that actuaries might find useful when designing a default investment strategy (DIS), when assessing the appropriateness of a particular DIS or when comparing DISs. The paper reviews the DISs currently available to Personal Retirement Savings Accounts (PRSAs). Because, the managed fund with life-styling is the most common DIS available in the market, the paper considers whether the inclusion of life-styling is appropriate for a DIS. Existing Guidance on Default Investment Strategies The Society of Actuaries in Ireland has issued guidance to actuaries assessing the proposed default investment strategy for PRSAs: The stated intention of the Department of Social and Family Affairs in having a requirement to provide a DIS is to reduce the difficulties sometimes encountered in setting up a defined contribution pension due to the financial inexperience of the potential contributor. The contributor should be informed that a DIS is not intended to be free from risk or volatility. The benefits ultimately provided by a PRSA product will be affected by investment conditions, the wider economic environment and the skill of the investment managers in exploiting them and the default investment strategy is unlikely to be able to fulfil the expectations of all contributors at all times. The reasonable expectations of a typical contributor are likely to be primarily determined by the communications received by the contributor from the PRSA provider, both prior to the completion of the contract and during the currency of the contract. The PRSA Actuary should therefore ensure that sufficient information is provided to enable the contributor to be aware of the principal features of the DIS. Such information should include a clear description of the asset classes in which the PRSAs will be invested, the ranges within which the allocations to each asset class would normally be expected to be, an explanation of the likely volatility of returns, and how the investment strategy might vary over the term of the contract. Unless the typical contributor s expectations are clearly likely to be otherwise, it would be expected that the purposes of savings for retirement should be mainly to generate an income over the period of retirement, by means of annuity purchase, rather than to generate a retirement fund for drawdown. It is reasonable, in the absence of any other information to assume that benefits will be taken at the time the state retirement pension would commence. The PRSA Actuary should consider whether adequate asset diversification is provided for in the strategy, whether concentration risk has been satisfactorily controlled and whether adequate arrangements are in place in relation to pricing, liquidity and cash holdings. (This paper does not consider diversification issues). 2

4 Approaches to Designing and Assessing DISs The optimal investment strategy for any individual would take into account a myriad of factors. Clearly, trustees or PRSA providers setting a DIS will not know all of the factors impacting on all of their contributors. Instead, the DIS must be set based on the objectives, requirements and attributes of a typical contributor. This paper does not attempt to design the best DIS, nor does it suggest one definitive approach to assessing the appropriateness of a DIS. Instead, it considers the issues and develops views and ideas using three different approaches. This paper uses very simple versions of investment models. Whilst we acknowledge the limitations of these models, we believe that they usefully highlight issues involved in setting a DIS. Chapter 4 sets out a pragmatic technique to assess the relative attractiveness of default investment strategies. It takes a prospective view based on a framework which sets out a typical target, an unacceptable outcome and an acceptable level of possibility of failure. Attempting to assess the relative merits of two default investments strategies may be seen as futile because the risk tolerance of an individual is not capable of unquestioned precise definition. However taking a number of characteristics from the projected distributions of outcomes gives a practical approach that may appeal to some actuaries. Chapter 4 highlights that any reasonable financial model for equity and salary growth will show very wide distributions of outcomes. The extent of the variability is probably not appreciated by contributors or financial planners. It is clearly an important point that needs to be communicated to encourage contributors to think clearly about both their investment and contribution plans. The conclusion of this chapter is that default investment strategies that make late switches to safer assets are shown to be superior but only if the individual is prepared to adjust contributions in the light of experience. The second approach to setting a DIS is based on the assumption that the contributor implicitly takes a VaR approach to decisions on asset allocation. Chapter 5 notes that it would appear that any life-styling DIS which finishes 100% in cash/bonds at retirement has implicitly adopted the chosen retirement age as the time horizon for formulating investment strategies with the objective to maximise this value subject to not taking undue risks. Chapter 5 concludes that all of the DISs observed in the marketplace can be justified by plausible assumptions on the equity risk premium, equity volatility and contributors risk tolerance, with perhaps the greatest sensitivity being the equity risk premium. Chapter 6 sets out the third approach based on modern finance theory. By using this approach and making a number of assumptions, it is possible to calculate the optimal allocation for a pension contributor at each point in time before retirement. 3

5 Although there has been a great deal of research into how risk-averse investors are, there is no consensus on the answer. The size of premium earned on risky assets relative to risk-free assets is also a matter of debate. Changing from one plausible assumption of level of risk-aversion or risk premium to another plausible assumption, the optimal allocation can change significantly. The examples in Chapter 6 take into account that the contributor s total assets consist not only of his pension, but also of his future salary, his future old-age pension, and some risk-free savings. Based on plausible assumptions, the optimal allocation of pension assets to risky assets decreases as the investor nears retirement. This is because the pension makes up an increasing proportion of the contributor's total assets, so he becomes less willing to take risks with it. However, knowing that the allocation to risky assets should decrease as the investor nears retirement is not enough - the level of risky-asset exposure is obviously even more important. These two items, and therefore, the best investment strategy for a particular contributor, will take into account his risk appetite and the risk associated with each asset. Chapter 6 concludes that as it is not possible to divine the typical contributor's risk appetite, there is no way of designing the most suitable investment strategy for him. Or, in other words, as it is not possible to divine the typical contributor's expectations, it is not possible to design a strategy that is consistent with fulfilling them. And if it were possible, the path-dependent nature of an optimal investment strategy would mean that pre-determined investment strategies could not fulfil their goal. Does this mean that it is impossible to design a DIS? This chapter concludes that it is not, but that the focus needs to change - rather than trying to shape the DIS according to unknowable expectations, the expectations should be shaped according to the known DIS. That is, the actuary should design a DIS that he thinks is reasonable and then ensure that the workings of the DIS, the thinking behind its design, and the risks associated with it should be clearly communicated to the contributor. Conclusions The broad shape of a managed fund with life-styling DIS can be justified as appropriate for a typical contributor, based on plausible assumptions about both the typical contributor and investment markets. However, it is not possible to accurately set these assumptions for the typical contributor. Hence, it is not possible to determine the ideal DIS. Even if the ideal DIS could be determined on a prospective basis, this does not guarantee that the typical contributor s expectations will be met. For a given contributor, there is only one set of actual investment outcomes and these will determine the return to this contributor. The contributor s circumstances and attitude to risk may also change over time and hence change his or her expectations. The extent of the variability in potential returns from a given DIS is probably not appreciated by clients or financial planners. 4

6 Recommendations We recommend that when designing a DIS, actuaries should set a strategy which they believe is reasonable to meet what they believe are the expectations and objectives of a typical contributor. We recommend that descriptions of a DIS should explain the strategy, the working of the DIS and also highlight its limitations and risks. The extent of the variability in potential returns should be communicated to encourage contributors to think clearly about both their investment and contribution plans. Contributors should be encouraged to regularly review their contribution rates and investment strategy based on their then current circumstances, attitude to risk and actual accumulated fund. We recommend that trustees and/or product providers communicate with contributors prior to the commencement of the life-style switching giving them the option to change the strategy and/or defer or accelerate switching. 5

7 Chapter 1 Introduction Most defined contribution pension schemes and Personal Retirement Savings Accounts (PRSAs) offer a default investment strategy (DIS) for those contributors who do not wish to make a specific investment choice. All standard PRSAs must offer a DIS. The purpose of this paper is: to review the DISs currently available to defined contribution pension schemes and PRSAs in the Irish market. to provide views and ideas that actuaries might find useful when designing a DIS, when assessing the appropriateness of a particular DIS or when comparing DISs. Because life-styling is a common feature in DISs, the paper considers whether the inclusion of life-styling is appropriate for a DIS. It is hoped that these ideas will assist actuaries in providing advice to trustees, sponsors and product providers. The optimal investment strategy for any individual would take into account a myriad of factors, including inter alia attitude to risk, family status, existence of other assets/income, whether ARF or annuity is more appropriate, flexibility as to choice of retirement age etc. and, of course, it is highly recommended that individuals regularly review their approach. Clearly, trustees or PRSA providers setting default investment strategies will not know all of the factors impacting on all of their contributors. Instead, the DIS must be set based on the objectives, requirements and attributes of a typical contributor. Chapter 2 outlines the guidance issued by the Society of Actuaries in Ireland to PRSA actuaries assessing the proposed default investment strategy for PRSAs. The points covered are relevant to all actuaries providing advice on DIS. Chapter 3 describes the DISs commonly available in the Irish market. The managed fund with life-styling is the most common DIS. While the description of the managed fund with life-styling has intuitive appeal, how can one prospectively assess the appropriateness of the DIS to meet the reasonable expectations of a typical contributor? Kevin Murphy considered related issues in his paper entitled It s the Outcome, Stupid! presented to the Society of Actuaries in Ireland in May In his paper, Kevin considered a case study of defined contribution investors. He considered the range of outcomes that an investor in a typical managed fund would receive, the range of outcomes that an investor in a more balanced portfolio would receive and the impact of life-styling on the outcomes. He noted that: The somewhat surprising result is that life-style doesn t reduce the number of bad outcomes. It does however reduce the average outcome but the probability of bad outcomes is much the same. 6

8 Life-styling is typically justified by the scenario of very strong equity returns followed by very poor equity returns. Life-styling is certainly a positive in that environment. However, less consideration is given to the alternative outcome which is the scenario of investors first experiencing poor equity returns and then, as they get close to retirement, quite positive equity returns. The movement into cash via the life-style process prevents them from gaining from that upside and essentially entraps the investor in their bad outcome. So the outcome charts indicate that the positive aspects of life-style have to be balanced against their negative aspects. Clearly, more work needs to be done here. The philosophy behind life-style is quite sensible, but outcome tables should help us to figure this out in a more detailed fashion. This paper gives further consideration to these issues. The paper does not attempt to design the best DIS, nor does it suggest one definitive approach to assessing the appropriateness of a DIS. Instead, it considers the issues and develops views and ideas using three different approaches. This paper uses very simple models of investment markets to illustrate the issues involved in setting and assessing a DIS. While we acknowledge the limitations of these models, we believe that they usefully highlight issues involved in setting a DIS. The frameworks described in later chapters could be adapted if more complex investment models were used. Note that because this paper uses simple investment models, it does not consider the advantages of asset diversification. The benefits of diversification have been addressed in other papers to the Society. The first approach to assessing a DIS takes a long-term prospective view. It is based on a framework which sets out a typical target, an unacceptable outcome and an acceptable level of possibility of failure. Having established this framework, various strategies are then tested to see which provides the optimal solution e.g. the lowest average cost over the contribution period. However, the long-term is made up of a series of short-terms and contributors circumstances and attitude to risk may change as they approach retirement. The second approach to setting a DIS is based on the assumption that the contributor implicitly takes a VaR approach to decisions on asset allocation. The third approach is based on modern finance theory. While the frameworks have been illustrated by the use of particular assumptions about typical contributors and investment markets, they can be adapted for use with other assumptions or philosophies. The final chapter sets out our conclusions and recommendations. We hope that actuaries will find the ideas in this paper interesting and useful. 7

9 Chapter 2 Legislative Background and Relevant Guidance for PRSA Default Investment Strategies The PRSA actuary has a specific regulatory role in assessing the appropriateness of the PRSA DIS. The Society of Actuaries in Ireland has issued relevant guidance for PRSA actuaries. This chapter outlines parts of the guidance which are relevant to all actuaries advising on the appropriateness of a DIS. The Pensions Act 1990 requires a PRSA provider to prepare a default investment strategy for each standard PRSA contract it offers. The Act requires the default investment strategy to adopt an investment profile for each PRSA product consistent with fulfilling the reasonable expectations of a typical contributor with respect to the said PRSA product for the purposes of making savings for retirement. The Act also requires that an application for approval of a PRSA product be accompanied by a certificate. The PRSA Actuary, along with two directors of the PRSA provider, must certify that in his or her opinion, the product covered by the certificate complies with the requirements of that Part of the Pensions Act and any regulations which relate to the investment of PRSA assets, including minimum requirements in relation to the default investment strategy prepared for the product. There are currently ten PRSA providers in the market offering sixty-two PRSA contracts approved by the Pensions Board. Twenty-seven standard PRSA contracts have been approved by the Pensions Board. 'ASP-PRSA-4: PRSA actuaries and Personal Retirement Savings Accounts Investment sets out guidance for PRSA actuaries in assessing the proposed default investment strategy. ASP-PRSA-4 notes: The stated intention of the Department of Social and Family Affairs in having a requirement to provide a default investment strategy is to reduce the difficulties sometimes encountered in setting up a defined contribution pension due to the financial inexperience of the potential contributor. The contributor should be informed that a default investment strategy is not intended to be free from risk or volatility. The benefits ultimately provided by a PRSA product will be affected by investment conditions, the wider economic environment and the skill of the investment managers in exploiting them and the default investment strategy is unlikely to be able to fulfil the expectations of all contributors at all times. It is understood that the Department of Social and Family Affairs would expect the reasonable expectations of a typical contributor to be assessed on a prospective basis. The actual outcome of the investment strategy will depend on changes in the value of, and income from, the assets held over the full duration of the contract. For each PRSA product, the reasonable expectations of a typical contributor with respect to the said PRSA product are likely to be primarily determined by the 8

10 communications received by the contributor from the PRSA provider, both prior to the completion of the contract and during the currency of the contract. The PRSA Actuary should therefore ensure that sufficient information is provided to enable the contributor to be aware of the principal features of the default investment strategy. In particular, such information should include a clear description of the asset classes in which the PRSAs will be invested, the ranges within which the allocations to each asset class would normally be expected to be, an explanation of the likely volatility of returns, and how the investment strategy might vary in accordance with the estimated outstanding duration until the contributor might be expected to receive benefits from the PRSA account. In assessing the nature of a typical contributor, a PRSA Actuary should take into account any particular features of the PRSA product and its intended method of distribution that might influence the characteristics of a typical contributor to the PRSA product, along with characteristics of the existing contributors where relevant. Unless the typical contributor s expectations are clearly likely to be otherwise, it would be expected that the purposes of savings for retirement should be mainly to generate an income over the period of retirement, by means of annuity purchase, rather than to generate a retirement fund for drawdown. The PRSA Operational Regulations requires the default investment strategy to take into account the estimated outstanding duration until the contributor might be expected to receive benefits that are to be payable from the PRSA account. In practice, PRSA contributors will have considerable flexibility over the time at which the benefits are to be taken. It is reasonable, in the absence of any other information specific to the intentions of the individual contributor, or of any communication to contributors, to assume that benefits will be taken at the time the state retirement pension would commence. In addition to satisfying himself or herself that the asset mix is consistent with fulfilling the reasonable expectations of a typical contributor the PRSA Actuary should take into account (among others) whether adequate asset diversification is provided for in the strategy, whether concentration risk has been satisfactorily controlled and whether adequate arrangements are in place in relation to pricing, liquidity and cash holdings. For each approved PRSA product, the PRSA actuary must be satisfied that adequate arrangements are in place to monitor the default investment strategy. 9

11 Chapter 3 Common Default Investment Strategies The most common default investment strategies available in the market fall into three broad categories: Asset Allocation in Pre-phasing period Phasing commences Asset Allocation at retirement Type 1 a Managed Fund 5 years to retirement 75% Fixed Interest 25% Cash Type 1 b Managed Fund more aggressive the longer the duration to retirement 5 years + to retirement 75% Fixed Interest 25% Cash Type 2 a Managed Fund 5 years to retirement 100% Fixed Interest Type 2 b Managed Fund more aggressive the longer the duration to retirement 5 years to retirement 100% Fixed Interest Type 3 Managed Fund more aggressive the longer the duration to retirement n/a Managed Fund balanced fund Types 1 and 2 default investment strategies are designed for people targeting an income at retirement, while Type 3 default investment strategies are designed for those targeting an Approved Retirement Fund. Types 1 and 2 can be broadly described as managed fund with life styling strategies. These are the most common DISs available in the market. Examples of how the strategies are described in marketing material are given below: The default investment strategy (DIS) is a service where we invest your builtup fund in the XYZ Fund until you are near retirement. We then switch your fund into more secure funds. So, it will help protect your pension fund from the ups and downs of the market as you get closer to retirement. It is intended to provide a suitable investment approach to meet the needs of a typical contributor for the purposes of making savings for retirement. The general principle of this strategy is to move from a high equity content to a high fixed interest content as retirement age approaches. A PRSA is generally a long-term investment and therefore in the early years the 10

12 investment strategy can be geared towards achieving high rates of growth through investment in assets such as equities (stocks and shares) and property. While these types of assets are more volatile than cash and fixedinterest securities, they have the potential to offer higher rates of growth...as retirement approaches, the asset mix of each XYZ fund is changed gradually in order to protect the pension that can be bought at retirement. This is achieved by investing predominantly in assets such as fixed interest securities whose value should move closely in line with the cost of purchasing a pension. If you are not comfortable making investment decisions, Default Investment Strategy (DIS) may be the route for you. DIS is an automatic mechanism that gradually transforms your pension fund from a higher-risk portfolio to a lowerrisk portfolio as you approach retirement. This protects you from the impact of a stock market crash prior to retirement. The DIS invests contributions in a PRSA fund with a reward/risk profile that is suitable to the number of years remaining to your selected retirement date. With a longer period to retirement, we believe that you should invest in funds with the potential for higher returns, even though these funds are more inherently risky. As you approach retirement, however, we recommend that your PRSA move into a more stable investment to protect the investment performance achieved to date. In the simplified models we use in later chapters, we have translated the contributor objectives that the DISs available in the market address: Maximising annuity purchase or Maximising cash free lump sum and annuity purchase Into the simpler objective of: Maximising cash value at retirement. Similarly, we have simplified the investment strategy into managed fund with life-styling into cash and fixed interest equity fund life-styling into cash. 11

13 Chapter 4 A Particular Approach to Evaluating Default Investment Strategies Introduction The purpose of this chapter is to develop a usable methodology for comparing default investment strategies. The methodology involves comparing the range of outcomes that are achieved by different default investment strategies on the basis of a particular distribution for future investment returns. It is not an argument for the merits of the particular distribution but a discussion on how, given a model for future returns, one might be able to make a rational comparison of differing default investment strategies. Attempting to assess the relative merits of two default investments strategies may be seen as futile because the risk tolerance of an individual is not capable of unquestioned precise definition. However taking a number of characteristics from the projected distributions of outcomes gives a practical approach that may appeal to some actuaries. Stochastic Modelling For the sake of illustration we assume that a contributor with 30 years to retirement is targeting a fund of 10 times final salary after the 30 year period. It is assumed that different assets have different expected return and variance of return. For ease we restrict the investments to equities and risk free investments. We assume that equities will each year outperform the risk free investment by a mean of 4% and standard deviation of 15%. The distribution is log normal with each year being independent. It is also assumed as a simplifying assumption that salary inflation exceeds the return on the risk free asset by 2% per annum. Once the distribution has been established projecting future scenarios is relatively simple by applying random numbers between 0 and 1 to the distribution function of returns to simulate a return in excess of the risk free investment for the year. 12

14 fixed cont % equity investment over 20 The table above shows the results of 1000 simulations of 30 year periods with a fixed contribution of 24.17% of salary each year ( the amount required to fund 10 times salary on a 2% gap between investment return and salary growth) being invested 100% in equities throughout the period. The histogram measures the distribution of the results. For example 52 of the 1000 cases had a score of 10.5 e.g. more than 10 and less than 11 times salary at the end of the projection. The variability on a 100% equity basis is very substantial. The higher values are open to question in that if returns are substantial over a 30 year period it is likely that inflation and salary levels would be affected. In assessing the cost of the strategy we look at the total contributions paid as a percentage of salary giving the same weight to each year. The cost of this strategy is thus assessed to be 7.25 times salary (30 times.2417). The expectation of return is 10 times salary with a modal score of 6.5 times salary. 13

15 fixed cont % annum shift to cash over over 20 The above graph shows the effect of a default investment strategy that moves to risk free investments over a 10 year period. The contribution is again fixed at 24.17% and hence the cost of this strategy is again There is significantly less variation in the return but also a significantly reduced expectation of approximately 8.5 times salary with the modal score being 6.5. variable cont with scale 15/25/40 100% equity over 20 The graph above shows the outcomes with 100% equity investment but with the contributions being recalculated through each year of simulation with a maximum contribution rate of 15% in the first 10 years 25% in the next 10 years and 40% in the 14

16 last 10 years. The distributions are much less dispersed. The cost reduces in this case to 6.91 times salary and has a tighter distribution than table 1. The expected return in this case is about 9 times salary with the modal score being 7.5. variable cont with scale 15/25/40 10% annum shift to cash over over 20 The fourth graph above shows a combination of the switching over the last 10 years and the year by year assessed contribution strategy. A substantial amount of the variability has been taken out of the distribution of final payouts. The cost of that strategy is 7.17 times salary. The expectation is approximately 8 times salary with a modal score of 8.5 times salary. 15

17 Which is the right strategy? fixed cont % equity investment fixed cont % annum shift to cash over variable cont with scale 15/25/40 10% annum shift to cash over variable cont with scale 15/25/40 100% equity over 20 The graph above takes the four alternative strategies together and shows the difficulty of making a decision on both investment and switching strategy. Many people would see the first strategy of fixed contribution of 24.17% and 100% equity as the most preferable given the long tails of upside. However it all depends on how one would value the possibility of upside of excess return against the downside of financial ruin. One could arrive at an answer by overlaying a utility weighting to the various outcomes, but it is difficult to arrive at a utility function that has universal applications as it seeks to measure the human emotional reaction to potential gains and losses. The method of comparison below relies on a crude utility to allow some reasonable conclusions. Interesting observations The example below shows two strategies: Both are based on the same constant contribution per annum with the investment strategy of the first being initially 100% equity investment with a gradual switch over the 10 years to retirement. The second is a constant proportion of 70% throughout. As can be seen from the chart below there is no real distinction between the two distributions 16

18 In the next graph we allow the contributions to vary as above and again look at two investment strategies 100% in the first years switching to risk free over ten years and the second a constant proportion of 85% equities (the change from 70% in the previous example to 85% was to ensure that the assessed cost of the two compared strategies remained broadly the same). There are substantially different distributions generated using a late switching strategy and an average strategy if the investor adjusts contributions in the light of ongoing experience. The time to retirement is an important factor for the investment strategy only if the investor is prepared to act on changing circumstances by altering 17

19 contributions. The combination of increasingly conservative investment and changing contributions allows for substantially less variable returns. The term to go is an important influence on an investor s investment strategy as the investor is increasingly running out of time to get back on track by altering contributions. Reducing the complexity of comparison In judging the success of a default investment strategy it is appropriate to look at two measures. The first is the likelihood of the strategy missing the target and the second is the expected cost of using the strategy relative to the expected cost of an unconstrained investment strategy. It is this that leads to the simple comparison method described below Likelihood of Missing the Target Before calculating the likelihood of missing the target one has to define the target and given that there is an infinite number of possible targets the process is simplified by identifying one simple target based on a reasonable aim for the typical investor. We keep to our aim of 10 times salary. The target may be judged to be hit if the outcome is greater than X times salary. Again the value X is a matter of subjective choice but one natural choice would be 8 times salary on the basis that it would not be a major disaster for someone targeting 10 to end up with that size of a fund. Another measure would be to look at a lower limit of 5 times salary as to aim for 10 and achieve 5 would be a significant failure of the funding plan. In the table below we use both 5 and 8 times as useful limits for our targets and compare the score. The numbers show the number of simulations that lead to a fund of less than 5 or 8 times salary. 18

20 As we showed above there is very little difference in C or E Although it might be arguable whether A or B is preferable it is clear that using the measurement criteria D is better than C 19

21 * Contribution recalculated each year based on a 2% gap between investment return and salary subject to a max of 15% in first ten years 25% in second ten years and 40% in third ten years Looking at this table H is definitely better than I and arguably the best of the ten strategies that have been listed. Conclusion While the projections are dependent on the underlying financial model, the methods above show how, for a given model of future returns, one might make rational comparisons of differing default investment strategies. Default investment strategies that make late switches to safer assets are shown to be superior but only if the member is prepared to adjust contributions in the light of experience. Any reasonable financial model for equity and salary growth will show very wide distributions of outcomes. The extent of the variability is probably not appreciated by clients or financial planners. It is clearly an important point that needs to be communicated to encourage investors to think clearly about both their investment and contribution plans. 20

22 Chapter 5 Var justification for life-styling Preamble Most Default Investment Strategies (DISs) involve a quite rapid transition from equities to bonds/cash in the run up to retirement, often referred to as life-styling. The correct investment strategy approaching retirement for a particular individual would take into account a myriad of factors, including inter alia attitude to risk, family status, existence of other assets/income, whether ARF or annuity is more appropriate, flexibility as to choice of retirement age etc. It is, of course, to be highly recommended that such individuals regularly review their approach. However, the fact that we are discussing DISs does allow us the luxury of dismissing many of these considerations for, by definition, it is the strategy to be followed by a typical contributor in default of a regular review. Thus it is not unreasonable to assume that such an individual will be taking 25% tax free lump sum in cash and using the balance to invest in an annuity, and that they will be highly dependent on their defined contribution plan or PRSA to augment first pillar retirement income. As such a person approaches retirement we can assume that the primary focus will be to maximise the retirement fund subject to not taking undue risk with its value at the point of retirement. If we also perceive the role of the provider/prsa actuary as quasi trustees of the PRSA then they too will be very much inclined to this same focus. Defining the DIS Objective A DIS objective might read as follows: to maximise the expected amount of the fund at retirement subject to not taking undue risk with that value This seemingly trivial statement contains one very significant element, namely that the focus is on a single point in the future the retirement age chosen at outset. This merits some further discussion. It could be argued that, at least at the psychological level, contributors are also concerned about more short term horizons especially with the communication of annual valuation statements. Indeed many DISs in the marketplace, by not being exclusively in equities a long way from retirement, appear to pander to this psychology. However, at a rational level the PRSAs and defined contribution plans are retirement plans and this is underpinned by the absence of access to the funds pre retirement. Therefore, we feel justified in ignoring any horizons pre retirement. More contentious is that we are also ignoring the post retirement horizons or even the ability to amend retirement age. Our justification for doing so is that we are considering a typical contributor and as explained in the preamble we have assumed that the typical contributor will in effect be liquidating the whole fund at retirement. We will attempt to recast our DIS objective in terms which will be suitable for numerical analysis. 21

23 For a start, we accept as a premise that maximising expected outcomes implies maximising the amount invested in equities. But investing in equities increases risk so we immediately recognise a conflict within our objective. Utility theory is a brave attempt to incorporate risk aversion and value in the same function, so that the objective reduces to maximising expected utility. This approach is explored further in a later chapter but it is not the approach adopted here. Instead, we will assume the contributor adopts (implicitly) an (X,Y) VaR 1 approach. By this we mean the contributor does not wish the risk of the final value being Y% less than could have been achieved on risk free investments to be greater than X%. Since we will be using the same lognormal shaped distribution, we can, without significant loss of generality, fix X and let Y be the only variable in the VaR analysis. For example, let s say the correct value of X, if we could ever assess such a thing, was 25%, by using X equal to 10% and adjusting Y accordingly we can represent the same risk constraint in any situation we may be analysing. This is akin to an origin shift in a co-ordinate system. We have chosen X equal to 10% for our analysis. We can now restate our DIS objective in a form which is getting close to being amenable to numerical analysis: to maximise investment in equities subject to a (10%,Y) VaR on the final value at retirement So we must now focus on Y and what it might be for the typical contributor and, in particular, how it might change with time. For example, is it reasonable to posit that Y should be a constant, at least in the run in to retirement? We have of course dispensed with most of the factors that Y might depend on for any particular individual by prescribing our typical contributor. Let us consider some possible residual dependencies. Path dependency: one could argue that if the pension plan is performing very well that we might be prepared to take more risk, i.e. increase Y. On the other hand one could argue that if we are ahead of target, now is the time to consolidate gains, i.e. reduce Y. Not surprisingly, none of the DISs in the marketplace has a path dependency. Accordingly we have assumed that Y does not vary with achieved performance. Time to retirement dependency (t): two such potential dependencies spring to mind. (a) with a long way to go to retirement we may take the view that any adverse experience can be compensated by increasing future contributions. This points to Y increasing with t. Clearly with 1 year to go there is not much solace to be gained by knowing that we can make good any shortfall by accessing other assets at retirement. Indeed, since we are considering the run in to retirement, we will assume that we have passed the stage when the fact that we can increase future contributions assuages in any material way the fear of adverse outcomes. We will go further and assume that future contributions are not material to our analysis in any event. Hence we assume that we are dealing with the investment of a lump sum or single premium. This makes the analysis considerably more tractable. 1 VaR Value at Risk 22

24 (b) In betting parlance, our fundamental premise is that investing in equities is a bias wager. Our expectation is for a positive return (compared to risk free) for taking on the risks. It seems reasonable then that the greater that expected positive return the greater the risk we should be prepared to take. Clearly the expected positive return increases with t so it would seem reasonable that Y should increase with t also. However, it could be argued that the relationship between risk aversion and reward expectation is much less direct for the typical contributor i.e. that the implied VaR is more of an absolute risk aversion statement than one which varies much with increased expected returns. In broad terms, the expected equity (log) out-performance increases linearly with t whilst the (log) standard deviation increases with t so the premium/risk trade off can be argued to increase with t/ t i.e. t. We will therefore consider two possibilities in our analysis, constant Y and Y varying with t. Methodology of the Analysis Before crunching any numbers we need an economic model. For the purposes of this chapter we will keep the model very simple. Equity performance relative to cash/bonds is assumed to be time independently lognormal with mean µ and variance σ 2. We will set σ equal to 15% per annum and consider two values for µ, viz. 3% per annum and 5%. We will make a further simplifying assumption that a fund which is α invested in equities will have a lognormal distribution of returns greater than cash with mean αµ and standard deviation ασ. Since we are dealing with lognormal distributions rather than normal ones, this assumption is not strictly speaking mathematically coherent, but it greatly eases the analysis and should not be materially in error. Mathematical purists may satisfy themselves by regarding our analysis as one which examines funds which do have an (αµ,ασ) lognormal distribution and that as a first approximation these are funds with α invested in equities. We also assume that α is only adjusted yearly. We recall that the only variable we have left in our criterion for deciding α(t) is Y(t). α(1) is dependent solely on Y(1). With 2 years to go to retirement, assuming we have preset Y(1), we know for sure what α(1) is going to be and so α(2) is determined from this value and Y(2) and so on for t=3 etc. Thus whilst α(t) is not path or past dependent it is in this sense future dependent. 23

25 Results Figure 1: Y=10%; µ=3% 90.0% 80.0% 70.0% % invested in Equities 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% Years to Retirement Let us remind ourselves that we have fixed X at 10% in all these illustrations. Thus the above graph illustrates what the maximum proportion invested in equities should be to ensure that the risk of underperforming risk free outcomes by 10% is not greater than 10%. With Y constant this is the fairly typical shape. With 1 year to go we have quite a high appetite for equity investment but this very fact weighs heavily on our risk awareness in the preceding years. We have to be quite a long way out before the equity risk premium persuades us to have equity exposures even as high as a typical managed fund. One should caution that these graphs are less appropriate as we get a long way from retirement as they are based on analysis of a lump sum investment whereas over the long term regular contributions and the ability to increase them become very relevant. Figure 2: Y=10%; µ=5% 120.0% 100.0% % invested in Equities 80.0% 60.0% 40.0% 20.0% 0.0% Years to Retirement By increasing µ to 5% we retain the same shape but transform the figures. The picture is somewhat closer to typical DIS strategies and we are forced to cap our equity investment at 100% for much of the duration of the pension plan. 24

26 Figure 3: Y=5%; µ=3% 45.0% 40.0% 35.0% % invested in Equities 30.0% 25.0% 20.0% 15.0% 10.0% 5.0% 0.0% Years to Retirement Figure 4: Y=5%; µ=5% 120.0% 100.0% % invested in Equities 80.0% 60.0% 40.0% 20.0% 0.0% Years to Retirement Figures 3&4 show the expected shifts from figures 1&2, when we reduce our risk tolerance to a constant Y=5%. Figure 5: Y=5% t; µ=3% 120.0% 100.0% % invested in Equities 80.0% 60.0% 40.0% 20.0% 0.0% Years to Retirement Letting Y vary with t gives a much better behaved graph and whilst similar to some DISs, the transition into cash/bonds starts much earlier if the equity risk premium is only 3%. 25

27 Figure 6: Y=5% t; µ=5% 120.0% 100.0% % invested in Equities 80.0% 60.0% 40.0% 20.0% 0.0% Years to Retirement Increasing µ to 5% rectifies that problem and provides a picture which is in remarkable agreement with a typical DIS. Conclusion It would appear that any life-styling DIS which finishes 100% in cash/bonds at retirement has implicitly adopted the chosen retirement age as the time horizon for formulating investment strategies with the objective to maximise this value subject to not taking undue risks. If we believe in the equity risk premium and that the expected reward increases compoundly with time whilst the risk increases at a lesser rate, which is true of typical lognormal models, then this underpins the life-styling approach of increasing equity exposure with time to retirement. The speed of that increase is highly dependent on the assumed level of the risk premium relative to the risk/volatility. All of the DISs observed in the marketplace can be justified by plausible assumptions on the equity risk premium, equity volatility and contributors risk tolerance. 26

28 Chapter 6 Modern Finance Theory and Default Investment Strategies This chapter applies modern finance theory to the problem of how to design a default investment strategy. Rather than assessing an investment strategy by examining the likelihood of a small number of discrete events, it applies a continuous utility function to the entire range of possible outcomes. Utility functions can describe how risk-averse a person is, and a few different definitions of risk-aversion are described below. Absolute risk-aversion Absolute risk-aversion (ARA) measures how a person's attitude to a particular monetary risk varies with changes in that person's wealth. For example, consider the table below showing the pay-off based on the toss of a fair coin: Result Pay-off Heads Loss of 100 Tails Gain of X A person's ARA could be measured by determining what the minimum value of X is for the person to take the wager. If X decreases as wealth increases, the person is said to have decreasing ARA (DARA); constant ARA (CARA) and increasing ARA (IARA) are defined similarly. The Arrow-Pratt measure of ARA introduced in Pratt (1964) and Arrow (1965) is given by the following formula: r ( w) ( w) ( w) u =, u where r( ) is the ARA, w is wealth, and u( ) is the person's utility function. Relative risk-aversion Relative risk-aversion (RRA) measures how a person's attitude to a particular proportion-of-wealth risk varies with changes in that person's wealth. For example, consider the table below showing the pay-off based on the toss of a fair coin: Result Heads Tails Pay-off Loss of 10% of wealth Gain of Y% of wealth A person's RRA could be measured by determining what the minimum value of Y is for the person to take the wager. If Y decreases as wealth increases, the person is said to have decreasing RRA (DRRA); constant RRA (CRRA) and increasing RRA (IRRA) are defined similarly. The Arrow-Pratt measure of RRA is given by the following formula: 27

29 R ( w) = w r( w) u ( w) ( w) w* * =, u where R( ) is the RRA. Economists generally agree that people exhibit DARA; that is, as a person's wealth increases, he looks for lower reward for taking on a particular monetary risk. Merton (1969) assumed that investors also had CRRA; that is, the potential reward required for taking on a specified chance of losing a particular proportion of his wealth does not depend on an investor's level of wealth. Arrow (1965), on the other hand, argued that investors exhibit behaviour consistent with IRRA (although still DARA). Simplified World In the following examples, it is assumed that all assets fall into one of two classes: risk-free assets and risky assets. The return on a risk-free asset in any period is always zero. The return on risky assets is uncertain, but all risky assets experience the same returns. It is also assumed that this is a world without taxation. Pension plans are assumed not to be able to short-sell assets. The typical investor being investigated is a 25-year-old in full employment. He has a salary of 30,000 and will retire at the age of 65 with an old-age pension of 10,000. He will live until the age of 80. Normal and CRRA What proportion of the sample investor's pension plan should he invest in risky assets? And how should this proportion change over time? To answer these questions using financial economics, an assumed utility function for the sample investor and an assumed distribution of risky-asset returns are needed. To keep things simple, the normal distribution of returns and a CRRA utility function will be used; therefore R(w) = R, a positive constant. The utility function is the following: u ( w) = 1 R w 1 1 R To confirm that an investor with this utility function exhibits decreasing absolute riskaversion and constant relative risk-aversion, the first two derivatives of the utility function with respect to wealth need to be calculated. u u ( w) = 1 R 1 ( 1 R) * w R 1 R 1 ( w) = R * w R = w The investor's ARA, r(w), is calculated as follows: r ( w) u u R 1 ( w) R * w R = = R ( w) w w = 28