Choosing and Using Utility Functions in Forming Portfolios This version: 22 September 2018 Geoffrey J. Warren College of Business and Economics The Australian National University Email:
[email protected] Phone: +61 411 241 091
Abstract Guidance is provided on how utility functions can be used to form portfolios in applied settings. Utility functions offer a means to encode investor objectives and preferences, allowing a score to be placed directly on outcomes, and then identify optimal portfolios by maximizing expected utility. The central theme is that utility functions should be tailored to the investor. How an appropriate utility function might be chosen and parameterized is discussed, and the concepts demonstrated for power utility and reference dependent utility. An approach is presented for using utility functions to identify optimal portfolios that is suitable for investors with long investment horizons spanning multiple periods, and can be applied by practitioners without resorting to dynamic optimization. The selection of utility functions and the modeling approach is illustrated for four investor types of differing investment objectives, risk aversion, horizons, and plans for drawing on and evaluating their portfolios.
Keywords: utility functions, portfolio construction, investment horizon JEL Codes: G11, G23
Acknowledgements: Many thanks go to those who provided input, some of which related to a preceding working paper titled “Investment Horizon, Risk Drivers and Portfolio Construction”. This includes: Sean Anthonisz, David Bell, Henk Berkman, Adam Butt, Colin Bowers, Douglas Bucknell, Richard Dunn, Don Ezra, John De Ravin, Barry Gillman, Graham Harman, Martin Hickling, Tim Hodgson, Gaurav Khemka, Geoff Kingston, Damian Lillicrap, Aaron Minney, Anna von Reibnitz, Michael Rice, Bob Schmidt and Susan Thorp. This input was integral to recasting and targeting the research. Thanks also to participants at the Institute of Actuaries Australian Insights Seminar on 8 February 2018; ANU Brown Bag Seminar on 25 May 2018; UNSW Australia Colloquium on Pension and Retirement Research on 2-3 July 2018; Q-group luncheon in Sydney on 15 August 2018; and Investment Management Research Conference at University of Technology Sydney on 20-21 September 2018.
1. Introduction Utility functions are familiar in the academic literature, but are not widely used for forming portfolios in practical settings. Mean-variance analysis remains the dominant approach employed by the investment industry, in part because it is familiar. However, two features limit its usefulness. First, variance is a measure of the variability of investment outcomes over a specific interval. The typical interval used is monthly, quarterly or yearly; although sometimes the horizon is extended to span multiple years. Either way, mean-variance analysis is designed to analyze portfolio return and risk over a discrete and often limited horizon. This is particularly problematic for long-term investors, especially those intending to draw on wealth over multiple periods and hence do not operate under one horizon. Second, the objectives of many investors are not well described by mean and variance. It is well-known that mean-variance analysis is valid under conditions of either a normal distribution or quadratic utility, neither of which are satisfied in practice. In particular, distributions of expected wealth are often positively skewed over longer horizons, to the extent that wealth is bounded below by zero, but on the upside can multiply many times over the long run. Further, the objectives of some investors involve generating wealth to satisfy a target such as achieving a real return, a required income stream, or sufficient assets to cover liabilities. In sum, mean-variance analysis often does not cut the mustard either because variance is not the appropriate measure of risk, or because there exists disconnect with underlying objectives. To make this point clear, consider an individual that is investing to generate income for retirement. Analyzing the mean and variance of their portfolio returns over some discrete time horizon is only vaguely relevant to their main concern, which is the stream of income they can draw over time. This paper investigates utility functions as an alternative. The underlying theme is that a wide range of investment situations can be effectively analyzed by modeling the evolution of total portfolio value – or accumulated wealth – over the horizon of concern, and using a utility function to ‘score’ the associated outcomes. By structuring the analysis to generate a distribution of outcomes that depends on portfolio weights, optimal portfolios can be identified through finding the asset mix that maximizes expected utility across all potential outcomes. Of course, there is nothing new in proposing the use of utility functions for forming portfolios in applied settings. Adler and Kritzman (2007) make a similar appeal. Notable examples where utility functions are used to evaluate portfolios in an applied context including Blake, Wright and Zhang (2013), Levy (2016) and Estrada and Kritzman (2018). Bell, Liu and Shao (2017a) put forward a “Member’s Default Utility Function” (MDUF)1 to assist in the design of retirement products. What this paper brings to the party is to set out how a utility function approach might be implemented by industry practitioners aiming to cater to the needs and wants of particular investors. This is done from two directions. First, the choice of utility function is discussed, focusing on how a utility function may be selected that reflects the investor’s circumstances. Second, an approach for the use of utility functions is presented that can be readily applied in practice, without delving into overly complex mathematical or simulation techniques. With regard to utility function choice, the key recommendation is that it should be tailored to the investor. This is essentially a subjective task. It may be approached by forming an understanding of the investor’s objectives and preferences over outcomes, and selecting a functional form and parameters to represent their situation. The process can be assisted by analyzing the proposed function to gauge its implications. It is acknowledged that some caution is required when deviating from the academic literature, which is typically very careful in its proposal and testing of utility functions. Nevertheless, the literature has reached no consensus over which utility function provides the ‘best’ description of individual behavior (see Starmer, 2000). This paper suggests turning the process on its head by starting with the investor, and then choosing a utility function to suit. It makes sense that utility functions are not one-size-fits-all, implying that the functional form is somewhat up for grabs. Further, this paper demonstrates that simply extracting a functional form and parameters from the academic literature without checking that the function is appropriate for the investor may have perverse implications. Choosing a utility function that is fit for purpose seems more important than seeking validation from an unsettled literature. This paper takes no stand on which is the ‘right’ utility function in any particular situation. Nevertheless, it demonstrates the process of choosing a utility function, and the importance of the choice, using power utility and reference dependent utility. Both are broadly used within the related finance literature, and together provide considerable flexibility to accommodate a wide range of investors. Power utility is a smooth function that is directly applied to the dollar value of the outcomes (e.g. wealth). It tends to impose a heavy penalty on the lower 1
See http://membersdefaultutilityfunction.com.au/. Page 1
tail of the distribution, especially at higher risk aversion coefficients (e.g. 4, and above). Power utility may be suitable for investors who are concerned with the raw magnitude of wealth and the outcomes it supports, while having high aversion to relatively low outcomes. Reference dependent utility is presented in two functional forms, one defined over the difference between the projected and target outcome, and the other over the ratio of outcometo-target. The difference form is similar to the value function under prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992), and reflects that used by authors such as Blake, Wright and Zhang (2013) and Levy (2016). The ratio form follows Tarlie (2017). Reference dependent utility functions are kinked around the target. They are typically applied in a manner that the penalty on losses is much heavier than the discount applied to gains, but with the penalty on losses not increasing exponentially as outcomes fall further below target. A reference dependent utility function may hence be more appropriate for investors with objectives that imply a target outcome, and where aversion to shortfall is relatively consistent across the realm of losses. The choice between the difference or ratio form can depend on whether the raw outcome units are meaningful, or whether evaluating percentage deviation from target is more in accordance with investor objectives. This paper also investigates the implications of utility function choice for portfolio formation, in particular how the preference for equities evolves with aspects such as horizon and distance from target. Power utility tends to constrain the extent to which equities are preferred as the horizon lengthens. This is because it can penalize equities heavily for the possibility of extreme wealth losses over longer periods, even if they occur with low probability: an attribute that aligns with the arguments of Samuelson (1971) and Merton and Samuelson (1974). Reference dependent utility tends to favor equities as the horizon lengthens. This reflects an interaction between the tendency to penalize the lower tail relatively less aggressively than power utility, and the fact that the higher returns for equities shift the distribution upwards as the horizon lengthens. The latter means that equities increase the probability of attaining or exceeding the target, meanwhile reducing the portion of the distribution below the target – notwithstanding their greater volatility (see Levy and Levy, 2017). Another implication of reference dependent objective functions is that the optimal equity weighting often emerges as u-shaped function depending on distance from target. Equities are preferred when below-target in a bid to secure higher returns and achieve the target. Equities are also preferred when well-above target because the prospect emerges of gaining even more wealth without greatly increasing the risk of shortfall. Fixed income features more strongly in the proximity of the target where it de-risks the portfolio and helps secure the target. A similar pattern can be seen in Blake et al. (2013). The use of utility functions is demonstrated under an approach for forming portfolios that can be readily applied in practice, and can accommodate investing to generate long-term income. The approach entails four steps: A) Choose a utility function – A ‘tailored’ utility function is selected and parameterized to represent the objectives and preferences of the investor. The utility function is used to evaluate withdrawals and terminal portfolio value in accordance with the evaluation plan. B) Specify a plan for withdrawal and evaluation – A pre-determined plan is specified for any withdrawals from the portfolio, as well as the combination of withdrawals and terminal portfolio value to be evaluated. The evaluation plan includes any quantities that depend on portfolio value and thus asset weights. Specifying the plan also entails setting the investment horizon and hence the analysis horizon. C) Project joint distributions of wealth outcomes for candidate assets – The assets to be considered are selected, and a joint distribution of expected wealth from investing in each asset is projected over the horizon. This is done by generating accumulated wealth outcomes for each asset associated with a range of paths (alternatively scenarios, or states). This includes allowing for wealth arising from the reinvestment of any cash flows arising from the asset, such as dividends, coupons or principal at maturity. D) Solve for optimal portfolio weights – Optimal portfolios are identified as follows. For each path, a series of portfolio values is generated by projecting forward, with closing portfolio value in each period estimated as a function of opening portfolio value, asset weights, the change in accumulated wealth associated with each asset from step C, and any withdrawals specified under step B. This delivers a series of withdrawals and portfolio values, with sequential estimation incorporating the effect of prior withdrawals hence accounting for sequencing effects. Within each simulated path, the withdrawals and/or terminal portfolio value are evaluated using the utility function defined under step A, thus delivering total utility for each path. Utility estimates are averaged (probability-weighted) across all paths to generate expected utility. The optimal asset weights are identified as those that maximize expected utility. Page 2
The above steps generate optimal asset weights over the horizon for the particular utility function, a predetermined plan for withdrawal and evaluation, and the joint asset distributions – as specified under steps A, B and C. The approach is relatively straightforward to apply, and may be handled within a spreadsheet package such as Microsoft Excel. Complexity is reduced by restricting the scope of analysis, in particular by pre-determining the withdrawal strategy and solving for static asset weights. Reducing complexity in this way means that the approach does not immediately deliver an optimal dynamic solution for the investment and withdrawal strategy. Section 6 addresses the issue of when dynamic analysis of some form might be appropriate. Illustrative analysis is presented for four representative investors. The first aim is to demonstrate the approach and its flexibility. The second aim is to highlight the importance of the choice of utility function, including its parameterization and the horizon over which it is applied. The analysis entails simulating wealth outcomes from investing in equities and 10-year bonds using two methods: random draws from a statistical distribution, and drawing on historical data. The four investor scenarios are summarized below: (i) Private investor – Optimal portfolios are estimated for private investors with the objective of accumulating
wealth over two time horizons of 3-years and 10-years, with end-horizon wealth evaluated using power utility. The analysis applies a range of risk aversion coefficients. A range of equity weights arise. (ii) Endowment fund – The fund aims to maintain the real value of withdrawals in perpetuity. This situation is
analyzed by evaluating real withdrawals plus terminal portfolio value over a ‘long- term’ horizon of 10-years. The terminal portfolio value represents the ‘corpus’ available to satisfy withdrawals beyond year-10. The analysis uses a reference dependent utility function based around the ratio of withdrawals and terminal portfolio value versus target, which is to maintain the value of both in real terms. The utility function is parameterized for moderate loss aversion. The optimal portfolio is heavily skewed towards equities. (iii) Defined benefit pension fund – This fund faces an obligation to make fixed nominal pension payments over
a 15-year period, but has a preference that any deficit in portfolio value versus the discounted value of these obligations is sustained no longer than 3-years. The fund thus evaluates the funding ratio at the end of year-3, using a reference dependent utility function that is parameterized for moderately high aversion to deficits. The analysis is undertaken at a range of initial funding ratios, revealing how the optimal equity weights are ushaped and sensitive to the proximity to target. (iv) Retired individual – The retiree is assumed to have invested in a retirement account with the aim of generating
income over a 30-year period based on a pre-determined withdrawal plan. They have a strong preference for a minimal level of income, but no bequest motive. The analysis evaluates the stream of withdrawals over 30years using a reference dependent utility function defined over the dollar difference between withdrawals and target income. A range of starting balances is considered, as well as two different parameterizations of the utility function, both of which reflect high aversion to shortfall in income versus target. Equity weights emerge as a u-shaped curve depending on the starting balance, with a profile that varies with the utility parameters. This paper is arranged as follows. Section 2 discusses utility functions; while Section 3 addresses their selection and parameterization. Section 4 details the simplified approach for forming portfolios. Section 5 illustrates the approach for the four representative investors, thus demonstrating the use of utility functions in specific applications. Section 6 considers static versus dynamic analysis. Section 7 concludes.
2. Utility Functions This section addresses the use of utility functions for the purpose of identifying optimal portfolios in an applied setting, emphasizing the importance of selecting a function that represents the objectives and preferences of the investor. The discussion commences by setting out the rationale for using utility functions. This is followed by an outline of the three utility functions examined, comparing their features, and characterizing how they interact with horizon. The selection and parameterization of an appropriate utility function is discussed in Section 3.
(i) Rational for Utility Functions The advantages of using utility functions to form portfolios largely relate to the fact that they provide a mechanism for directly evaluating all potential outcomes. They do this by attaching a ‘score’ to each point in the distribution Page 3
of outcomes. Expected utility can then be estimated as the probability-weighted value of these scores; and portfolios evaluated by comparing the overall expected utility scores they deliver. The main alternative involves evaluating portfolios based on metrics that summarize the distribution of outcomes. Mean-variance analysis is one such approach. Portfolios might be similarly evaluated using other metrics such as probability of shortfall, valueat-risk, conditional value-at risk, failure rates, and so on. A key advantage of utility-based approaches is that they capture all information about the range of potential outcomes, whereas using metrics may discard some relevant information. Adler and Kritzman (2007) argue that a utility-based approach (they call it ‘full-scale optimisation’) is more comprehensive and relevant than defining investor preferences over mean and variance. A related advantage is that utility functions can accommodate the analysis of distributions of any shape. Unlike mean-variance analysis, the distribution does not have be (approximately) normal for utility analysis to work. This attribute can be particularly relevant over longer horizons, when the distribution of portfolio outcomes is often positively skewed given that wealth is bounded by zero on the downside but unconstrained on the upside. This argument extends to analysis of option-like pay-offs. Rubinstein (1976) argues that specifying the form of the utility function is superior relative to imposing assumptions about asset returns under many applications. Utility-based approaches offer considerable flexibility. A variety of functions are available that may be designed to encapsulate a wide range of investor objectives and preferences. They can be designed to recognize and place appropriate scores on various parts of a distribution. Unlike shortfall measures, utility functions can directly give appropriate credit for the upside. And if there is a particular dislike for being in the lower tail of the distribution, the utility function may be specified to make sure lower tail outcomes are sufficiently penalized. Utility functions can also be applied to multi-period outcomes, given that utility scores can be added up over time as well as across time. This can be particularly helpful where the objective relates to a stream of withdrawals from the portfolio, such as generating income in retirement. Indeed, the pension industry worldwide faces the task of developing better retirement products that combine an investment strategy and a withdrawal strategy, most likely in a dynamic framework. It is hard to imagine how mean-variance techniques of the MPT tradition provide the required toolkit. Utility functions offer the analytical mechanics to move forward in this key area. Utility-based approaches are often criticized on the basis that choosing a utility function is difficult and subjective. This is true. However, analysis based around metrics requires a mechanism to trade off the expected outcome against a risk measure. This involves addressing investor preferences, and reflecting them in a risk aversion parameter, or some rule of thumb that dictates how return is traded off against risk. An underlying utility function may be implied in the process, even if not explicitly specified. Exercising some degree of subjective choice is unavoidable either way. The ultimate question is which approach is more effective at addressing the problem at hand. In many circumstances, a utility-based approach may work better than relying on metrics. Finally, utility-based approaches have broader uses than just portfolio optimization. Utility functions can be used to place an overall score on a proposed portfolio, strategy or action; with the aim of establishing if value is being added relative to the status quo or some other alternative. For instance, the potential addition of a new asset class to a portfolio might be evaluated in terms of whether it increases utility at the margin. Utility-based analysis can perform this function by summarizing the impact in a single number; similar to how portfolios or investment strategies may be compared based on their Sharpe ratios or information coefficients, for instance. Utility may be converted into economically meaningful measures, such as certainty equivalents, to aid comparisons if so desired.
(ii) Three Utility Functions No stand is taken on what is the ‘correct’ utility function. Indeed, the key message is that the utility function should be selected and parameterized to reflect investor circumstances. Three utility functions are selected for examination, being power utility and two variations of reference dependent utility. These three are chosen because they are widely used, are capable of addressing many practical situations, and suffice for the purpose of discussing and demonstrating the main concepts. A number of utility functions are available, with the general family known as hyperbolic absolute risk-aversion (HARA) utility2 and Epstein-Zin-Weil3 utility worth noting. 2
Merton (1971) notes that HARA utility is a rich class that accommodates absolute and relative risk aversion which can be either increasing, decreasing or constant in wealth; and includes the power, quadratic, exponential and log utility functions. 3 The utility function attributed to Epstein and Zin (1989) and Weil (1989) decouples the ‘elasticity of intertemporal substitution’ from risk aversion, allowing any preference for earlier resolution of uncertainty to be evaluated separately from Page 4
Power utility is broadly used within the academic literature, and sits within HARA family. Equation (1) describes the functional form:
𝑈
,
=
(
)
(If CRRA = 1, 𝑈
,
= 𝑙𝑛(𝑊 ), i.e. log utility) 4
(1)
where: UPU,t = power utility, period t Wt = wealth, period t CRRA = coefficient of relative risk aversion Power utility is defined over the level of the outcome being evaluated. Equation (1) specifies this outcome as ‘wealth’, but consumption and income levels are other common choices. Power utility assumes constant relative risk aversion, which means that the curvature of the function is the same regardless of the level of wealth.5 The implication is that, while power utility places a higher score on higher wealth levels, the manner in which a distribution of a particular shape is evaluated scales up. For instance, say utility scores are estimated for a particular spread of wealth outcomes, then all wealth outcomes are doubled. The utility scores at each point in the distribution will all improve by a common factor. For instance, the doubling of wealth leads to the corresponding (negative) utility scores being scaled by a factor of 0.5 for CRRA of 2, and by 0.125 for CRRA of 4. Power utility may thus be appropriate for an investor that is concerned with comparing outcomes based on the relative level and spread of the outcomes, but whose risk aversion does not change with the overall level of wealth. A reference dependent utility function evaluates outcomes relative to some reference level. Such functions may be more suitable for an investor with an objective that entails achieving some target. Various authors have proposed functions that define utility with reference to a wealth or return target, e.g. Hogan and Warren (1972), Fishburn (1977), Holthausen (1981), Stutzer (2003), Kahneman and Tversky (1979), Tversky and Kahneman (1992), Anthonisz (2012), among others. Further, experimental studies support the idea that many investors perceive risk as related to shortfall versus some reference point (see Unser, 2000; Veld and Veld-Merkoulova, 2008). A number of authors have used utility functions that reflect the value function component6 in the prospect theory of Kahneman and Tversky (1979) and Tversky and Kahneman (1992) in evaluating investment outcomes, e.g. Bierman (1998), Adler and Kritzman (2007), Kőszegi and Rabin (2007), Blake, Wright and Zhang (2013), Levy (2016), and Estrada and Kritzman (2018). This approach is followed here, noting that various other forms of reference dependent utility exist, including habit persistence (see for example: Campbell and Cochrane, 1999; Grishchenko, 2010) or the inclusion of floor levels (see Kingston and Thorp, 2005).7 A reference dependent utility function defined over the difference between projected wealth and target wealth is described by equation set (2); while one defined over the ratio is appears as equation set (3). The functional form of equation (2) is similar to the value function within prospect theory, with exception that it is generalized here by providing for a weighting parameter (γ) to be applied to gains above target. Under prospect theory, the parameter γ is set equal to one. The reference dependent utility function in ratio form follows Tarlie (2017). The reference point in both cases is the target outcome, denoted as wealth of W*. Note that both equations provide for the target to vary with time. The deviation between the projected and target wealth is moderated by curvature parameters (α, β), and multiplied by weighting parameters (γ, λ). Different curvature and weighting parameters may be applied in the realms of wealth above and below target, providing scope to discount gains and penalize losses to varying degrees of intensity and in a non-linear manner if desired. Both functions are positive when W>W*, negative when W W* (Gains) 𝛾((𝑊 − 𝑊 ∗ )∝ )
𝛾
𝑊 𝑊∗
Utility if W < W* (Losses) −𝜆 (𝑊 ∗ − 𝑊 )
∝
−1
𝜆
𝑊 𝑊∗
−1
Utility if W = W* (At target) 0
(2)
0
(3)
reference dependent utility in difference form, period t reference dependent utility in ratio form, period t wealth, period t target wealth, period t curvature parameter on gains curvature parameter on losses weighting parameter on gains weighting parameter on losses
Although reference dependent utility functions in their difference and ratio forms may appear superficially similar, there are some notable differences in their implications. The difference form defines utility over the absolute dollar value of wealth versus target. The ratio form estimates utility after a transformation where wealth outcomes are scaled by the target, and hence effectively defines utility over the percentage distance from target. Which form is appropriate will depend on investor objectives, and how they are best framed. Differing units mean that the same parameters have different implications under each functional form: this is largely related to the curvature parameters. An example may help explain. Consider two potential outcomes of wealth of 9 and 8 units relative to target of 10 units, and a curvature parameter for losses of 0.8. The difference form attaches a value of -1 unit and -1.74 units to each of these outcomes, whereas the ratio form attaches values of -0.081 and -0.164. The relative magnitude of the utility scores for the two loss outcomes differ under the two functions. Note that: -1.741/-1.000 = 1.741 < -2/-1 = 2 < -0.164/-0.081 = 2.025. This implies that a curvature less than one imposes a diminishing penalty rate on losses under the difference form, and an increasing penalty under the ratio form. The impact of a given curvature parameter is also symmetric around the target for the difference form, whereas it entails some non-linearity under the ratio form. One implication of incorporating a reference level is that the evaluation depends on where the overall distribution of outcomes sits with respect to the target. This is because the function is kinked, and hence the way that a given distribution is evaluated and scored is no longer independent of the level of wealth. As a given wealth distribution is scaled up further above the target, a diminishing portion of that distribution will be evaluated as a loss, while an increasing portion will be evaluated as a gain. This means that less of the distribution is attracting the penalty applied to losses, which is steeper than the discount that is applied to gains. This feature turns out to be important as the horizon lengthens and assets offer different levels of return, as will be demonstrated below. Finally, reference dependent utility functions of general type presented in equation (2) and equation (3) have some practical advantages. One is their flexibility. The four parameters facilitate applying a wide variety of discounts on gains and penalties on shortfalls, as befits the circumstances. Tarlie (2017) discusses how his function embeds a number of utility functions in the literature. In addition, utility estimates arrive in intuitive units, being either a dollar number or a percentage deviation from target. By contrast, power utility has just one parameter,8 and the utility numbers are largely uninterpretable. This feature of reference dependent utility is helpful in calibrating the parameters, and will be discussed further in the Section 3. Nevertheless, one needs to be careful not to overplay this feature, as utility functions operate only as a score to rank outcomes and hence portfolios. They need to be converted into a certainty equivalent or similar in order to become economically meaningful.
8
The general form of the HARA family of utility functions entails three parameters, and provides flexibility to manipulate absolute and relative risk aversion (see Merton, 1971, p389). Page 6
(iii) Functional Forms Compared: Power Utility versus Reference Dependent Utility Figure 1 presents two charts that visualize the broad functional form of power utility assuming CRRA of 4, and the reference dependent utility function in ratio form using the parameters of Blake et al. (2013). The charts span wealth outcomes ranging from 0.20 to 2.4 with target wealth set at 1.0 (allowing for the same x-axis). The aim is to characterize each function: the implications of parameter choice is investigated in Section 3. Figure 1 highlights that power utility function is a smooth curve, but is relatively harsh on outcomes in the lower end of the range. Indeed, power utility provides comparatively little differentiation for outcomes that are near or above wealth of 1, where the slope is comparatively gradual (at least for CRRA = 4). The reference dependent utility function is kinked around the target. This kink reflects loss aversion, with penalty applied to losses (shortfall) versus target being much greater than the discount that is applied to gains of an equivalent magnitude. However, the curvature implies far more ‘evenly-paced’ changes in utility scores as outcomes move away from reference point than under power utility. Most notably, while scores under power utility (for CRRA > 1) decline more than exponentially as wealth deceases, the reference dependent function declines in something much closer to a linear manner with the weighting parameter of 4.5 dominating. Figure 1: Comparison of Utility Functions (a) Power Utility
(b) Reference Dependent Utility - Ratio
0 -5 For CRRA = 4
-15 -20 -25 -30 -35 -40 -45 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 Wealth Level
Utility
Utility
-10
1.60 1.20 0.80 0.40 0.00 -0.40 -0.80 Risk Neutral -1.20 Reference -1.60 Dependent Utility -2.00 -2.40 -2.80 U(Gain)=(Wealth/Target)^0.44-1 -3.20 U(Loss)= -4.5((Wealth/Target)^0.88-1) -3.60 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 Wealth / Target
The key implication to draw from Figure 1 is that power utility has a propensity to penalize outcomes in the lower tail of a given distribution far more vigorously than reference dependent utility. Related to this, reference dependent utility can also tend to give more credit for gains above target. Of course, these are just tendencies, albeit ones that associate with commonly chosen parameter values. The extent to which these tendencies apply will depend on the actual parameter values. Another point to take from Figure 1 is the units in which utility is measured. The power utility scores are all negative, and the scale is in itself somewhat meaningless (and will change markedly with CRRA). Basically, power utility scores are uninterpretable. The reference dependent utility scores, however, arise as a more straightforward transformation of the outcome versus target. For example, a wealth-to-target outcome of 1.25 generates a utility score of 0.1, which might be interpreted as a 25% gain being treated like a 10% gain. Conversely, the score for a wealth-to-target outcome of 0.75 is -1.01, which might be interpreted as treating a 25% loss like a 101% loss. Although it is the relative shape of the curve around the target that actually matters when evaluating outcomes, the fact that the units are somewhat intuitive can assist in calibrating parameters to ensure that the relative scores accord with the preferences of the investor. Figure 2 sheds further light on how the two utility functions evaluate a common distribution of outcomes. The figure plots the density of utility for each function under a lognormal distribution. To form the series, a distribution of wealth outcomes is proposed that extends from 0.1 to 3.5 in intervals of 0.05. This is intended to represent the range of wealth outcomes that might arise when investing over a long horizon spanning multiple years. It is then Page 7
assumed that the range of assumed outcomes is generated by a lognormal distribution,9 and that the log of terminal wealth has a mean of 0 and standard deviation of 0.5.10 These assumptions support estimation of the probability of observing a wealth outcome within each 0.05 interval. For instance, the probability of observing a wealth outcome between 0.50 and 0.55 (45%-50% wealth decline) is estimated at 3.31%; while the probability of observing between 1.95 and 2.00 (near doubling of wealth) is 0.80%. Utility scores are then calculated for each interval under both the power utility and reference dependent utility functions that appear in Figure 1. Finally, the utility score for each interval is then multiplied by its probability, providing an estimate of the percentage contribution of each interval to expected utility under each utility function under a lognormal distribution. Figure 2: Utility Densities of Outcomes under a Lognormal Distribution 10%
Contribution to Expected Utility
9% 8%
Power Utility, CRRA = 4
7% 6% 5% 4% 3% 2%
Reference Dependent Utility - Ratio, with Blake et al (2013) parameters Certainty Equivalent Wealth: . . . . 0.72 Power Utility - - - 0.88 Reference Dependent
1% 0% 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50
Wealth vs $1 Base or Target
Consistent with the earlier contentions, Figure 2 reveals that power utility places far more weight on outcomes in the lower tail. It also affords little credit for outcomes in the upper tail. Meanwhile, the reference dependent utility function places weight on losses in a relatively more even fashion, with less skew towards weighting the lower tail. The reference dependent function also gives greater credit for gains relative to target. Finally, the reference dependent utility function places little weight on outcomes near the target wealth of 1.00. Figure 2 also reports certainty equivalent values of expected wealth, which stand at 0.72 for power utility and 0.88 for reference dependent utility. This confirms that a power utility investor would require considerably more compensation for risk of loss than a reference dependent investor, at least for the chosen parameters. Figure 1 and Figure 2 highlight the manner in which different utility functions may place different values on the equivalent wealth outcomes, and hence arrive at quite different conclusions in evaluating the same distribution. These differences matter most where candidate portfolios differ in exposure to equities, which offer higher returns but a wider distribution of outcomes. Indeed, reference dependent utility functions may favor equities to the extent that they place relatively lesser penalty on the lower tail, and give more credit for the upper part of the distribution where the gains reside. This aspect emerges when the interaction with horizon is considered, as discussed next.
(iv) Interaction with Horizon This subsection explains how different utility functions hold differing implications for optimal portfolio weights as horizon varies. Specifically, reference dependent utility functions tend to give rise to increasing preference for higher returning assets like equities as the horizon lengthens, despite their higher volatility. This does not occur under power utility, at least where the return or wealth distribution is independent and identically distributed (iid). It is well-known optimal portfolios vary with horizon when the distribution is not iid, such as where there is serial 9
This equates to compounding returns over multiple periods, with ln(1+Return(t)) independent and normally distributed. This standard deviation would arise from investing over 10-years with a yearly standard deviation of 15.8%, i.e. 0.50 = ((0.158*0.158)*10)^0.5. 10
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correlation or expected returns and risk are time-varying (see Samuelson, 1994; Barberis, 2000; Campbell and Viceira, 2005; Kritzman, 2015). The discussion below abstracts from this element by assuming iid returns. To place this issue in context, whether investors should hold more equities over the long term is a hotly debated topic. This debate often occurs under the banner of ‘time diversification’ (for an overview, refer: Thorley, 1995; Bennyhoff, 2009; Kritzman, 2015). However, the theoretical roots can be traced to discussions over the merits of ‘Kelly’ investing dating back to authors such as Kelly (1956), Latané (1959), Markowitz (1976), Samuelson (1971) and Merton and Samuelson (1974). The Kelly investing view has been interpreted as implying that assets with higher geometric returns should be increasingly favored as horizon lengthens, as such assets come to ‘almost stochastically dominate’ assets with lower geometric returns (Leshno and Levy, 2002). Nevertheless, there typically remain some states where the higher returning asset with higher variance delivers lower wealth (see Levy, 2009). As a consequence, preferences play a central role in where one stands on this debate – as pointed out by Ziemba (2015) and Levy and Levy (2017). Preferences determine the relative weight placed on tail losses for the more volatile asset, and thus whether equities might be preferred as horizon lengthens. Samuelson (1971) and Merton and Samuelson (1974) are credited with arguing that optimal portfolio weights do not change with horizon. Their case is based on the assumption of constant relative risk aversion preferences (power utility) as well as iid returns. This result relates to the variance of wealth growing proportionately with time. Other authors point out that constant weights need not hold under a reference dependent utility function (see Benartzi and Thaler, 1995; Bierman, 1998; Levy and Levy, 2017). This arises because the probability of shortfall versus the target evolves with horizon, depending in part on how the expected compound return compares with the rate at which the target grows. The action of compounding returns acts to ‘shift’ the overall level of the wealth distribution upwards. The trajectory of this shift depends on the level of expected returns (geometric mean). The overall shift in the distribution then interacts with manner in which gains and losses are asymmetrically evaluated under reference dependent utility functions, as was discussed in Section 2(iii). Figure 3 illustrates these aspects. The analysis is based on drawing 14 years of real returns for equities and bonds from a normal distribution, with a mean of 6% and a standard deviation of 16% for equities, and a mean of 1% and a standard deviation of 8% for bonds. The mean of 1% for 10-year bonds approximately represents real yields at the time of analysis, while the 6.0% mean for equities accords with an equity risk premium of 5.0% which broadly reflects historical experience. The left chart in Figure 3 plots optimal equity weights at horizons of 1-year to 14-years under power utility and reference dependent utility functions using the same parameters as Figure 1, assuming a 3% real return target to establish the reference point (W*). The optimal equity weight does not vary with horizon under power utility, apart from that arising from some variation in the random draws. This is exactly the case presented by Samuelson (1971) and Merton and Samuelson (1974). By contrast, equity weightings under the reference dependent utility function unambiguously increase with horizon, attaining 100% after 12-years. This is the case underpinning the work of Benartzi and Thaler (1995), Bierman (1998) and Levy and Levy (2017). Figure 3: Optimal Weights and Probability of Shortfall as Horizon Increases (a) Optimal Equity Weighting 100%
90%
90%
80%
80%
70%
Reference Dependent Utility
70% 60%
Power Utility
50%
(b) Probability of Shortfall
Bonds vs. Target Wealth Equities vs. Target Wealth
60%
Equities vs. Bonds
50% 40% 30%
40%
20%
30% 1
2
3
4
5
6 7 8 9 10 11 12 13 14 Horizon (Years)
1
2
3
4
5
6
7 8 9 10 11 12 13 14 Horizon (Years)
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The right chart in Figure 3 helps explain why equity weights increase with horizon under the reference dependent function. This chart plots the probability of shortfall versus target arising from both equities and bonds, as well as the probability of equities delivering lower wealth than bonds. The probability of equities delivering a belowtarget outcome drifts lower from 43% in year-1 to 34% in year-14. Meanwhile, that for bonds rises from 60% to 86%. Thus the proportion of the distribution falling into the loss area decreases with horizon for equities, while it increases for bonds. Further, the probability of equities underperforming bonds declines, reaching 20% in year14. This indicates the portion of the equity distribution that provides a worse outcome than bonds: extending the charts would eventually lead to the ‘almost stochastic dominance’ of equities over bonds (Leshno and Levy, 2002). What the right chart in Figure 3 does not reveal is a lower tail where equities do much worse than bonds. It is in this region that the relatively more extreme penalty imposed by power utility on the lower tail comes into play. In the case of power utility, this penalty is sufficiently high to generate the static equity weights seen in Figure 3(a), notwithstanding that equities do better than bonds over the bulk of the outcome distribution. Meanwhile, the reference dependent utility function not only places a more ‘measured’ penalty on this lower tail. It also gives more credit to the increasing slice of the distribution that is above the target as horizon lengthens. Hence the differing manner in which each function balances gains versus losses interacts with the manner in which the distribution of outcomes evolves over time to produce quite different optimal portfolio weights as horizon lengthens. The main takeaway is that the utility function is a key player in these results, underlining the importance of utility function selection given the potential implications for portfolio formation.
3. Utility Function Selection and Parameterization An appropriate utility function should faithfully describe the objectives and preferences of the investor. This section discusses how an appropriate utility function might be chosen. It starts by highlighting the dangers in picking a utility function ‘off the shelf’ for use in any specific investment context based on findings reported in the literature. This supports the key message that it is better to select the functional form and parameters that make sense given the context in which the utility function will be used. Guidance is then provided on how the task of choosing a utility function might be approached. (i) Dangers in Picking a Utility Function ‘Off the Shelf’ The literature contains various studies that aim to identify and parameterize utility functions. Such functions are specific to the circumstances under which they are estimated, and potentially unsuitable for a particular investor. Estimation typically arises from experimental studies, sometimes employing students as subjects. A classic and poignant example is the cumulative prospect theory of Tversky and Kahneman (1992). The discussion here will focus on the estimate from that study of the curvature parameter on losses of 0.88 for what is a reference dependent utility function in difference form.11 Applying a curvature parameter of less than one is consistent with imposing a proportionately lower penalty on losses as they increase in magnitude. For instance, the curvature parameter in isolation converts a $10,000 loss to a utility value of -$3,311.3 (= -($10,000^0.88)), meaning that the difference is ‘diluted’ to about 33% of the raw value. A loss of $20,000 is converted to a utility value of -$6,094.1, which is about 30% of the raw value. Thus a loss of $20,000 is treated as proportionately less painful than a loss of $10,000. (The heavy lifting in terms of penalizing losses is carried by the weighting parameter.) One might question whether a proportionately reducing penalty on losses is appropriate in all circumstances. The source of the 0.88 curvature parameter derives from the finding by Tversky and Kahneman (1992) of what they call “diminishing sensitivity” to gains and losses as outcomes move away from the reference point. It is highly questionable whether diminishing sensitivity to losses under simple choices made by students can be transferred to other contexts, such as an institutional investor aiming to use utility functions to identify optimal portfolios. In some situations, it may be appropriate for the penalty on losses to increase at an increasing rate as outcomes fall further below target because the investor suffers increasing pain. For example, an investor relying on drawing an income to meet living expenses could become increasingly destitute as their available income falls below target. Such a situation implies a curvature parameter in the realms of losses in excess of one. 11
The curvature parameter on losses of 0.88 has been adopted by authors such as Benartzi and Thaler (1995), Blake et al. (2013) and Levy (2016). The latter also investigates values of between 0.5 and 1 for this parameter. Levy comments: “In practice the loss aversion is presumably more profound than what is estimated in laboratory experiments” (page 1421). Page 10
To drive the point home about the need to closely evaluate any candidate utility function, Figure 4 presents the implications of applying two parameterizations appearing in the literature to both the difference and ratio forms of the reference dependent utility function, thus generating four functions. The first parameterization follows Tversky and Kahneman (1992), and includes curvature parameters of 0.88 on both gains and losses, and a weighting parameter of 2.25 on losses (1.0 on gains). The second parameterization follows Blake et al. (2013), and includes a curvature parameter of 0.44 on gains, a curvature parameter of 0.88 on losses, and a weighting parameter of 4.5 on losses (1.0 on gains). A target of $20,000 is assumed. Figure 4a plots utility values under the four functions for outcomes ranging from $0 (i.e. -$20,000 in difference form, and 0.0 in ratio form) up to $40,000 (i.e. +$20,000, and 2.0). Utility values under the difference form appear on the left scale; while those under the ratio form appear on the right scale. The chart is designed so the curves are directly comparable in terms of scales and hence the implications for evaluating outcomes. Figure 4b examines the relative scores placed on gains and losses under each function. Reported are the absolute score for a gain as a percentage of the absolute score for a loss of an equivalent magnitude. Numbers for power utility are also shown for comparison. The aim of Figure 4 is to highlight how the choice of both functional form and parameters can have important implications, rather than comment on which combination of parameters is “right”. Utility functions are used to rank prospects, so that the units are not directly meaningful in isolation. The main attribute to consider is the broad shape of the function, and what it implies for evaluating outcomes. From Figure 4a, it is apparent that the Tversky and Kahneman (1992) parameters impose a relatively modest penalty on losses relative to the discount applied on gains. Figure 4b confirms this by revealing that a gain is afforded a utility value with a magnitude of about 40% or so of the value of an equivalent loss.12 The parameters selected by Blake et al. (2013) are much more loss averse. There is also a notable difference in the implications of the same parameters under the two functional forms. Under the difference form (employed by Blake et al., 2013), an extremely heavy discount is imposed on gains which are afforded minimal credit. Figure 4b reveals that the magnitude of the utility score on a gain is less than 1% of the score attached to an equivalent loss. As a consequence, it implies that the investor is primarily concerned with minimizing the magnitude of losses versus target, without caring too much for the possibility of gains. Figure 4a reveals that applying the parameters of Blake et al. (2013) in the ratio form applies a heavier penalty to losses, but also a less aggressive discount to gains. Figure 4b reports that the ratio of utility scores for gains and losses of an equivalent magnitude is around 10% in this case. Figure 4b also highlights the heavy and increasing penalty imposed by power utility on outcomes in the lower part of the distribution.
10,000
1.0
5,000
0.5
0
0.0
-5,000
-0.5
-10,000
-1.0
-15,000
-1.5
-20,000
-2.0
-25,000 -30,000 -35,000 -40,000 -45,000 -20,000
-2.5
Tversky & Kahneman (1992) parameters - Difference Blake et al. (2013) parameters - Difference Tversky & Kahneman (1992) parameters - Ratio Blake et al. (2013) parameters - Ratio -15,000 -10,000
-5,000
0
5,000
10,000
15,000
-3.0
Utility - Ratio Form
Utility- Difference Form
Figure 4a: Comparison of Parameterizations under Reference Dependent Utility
-3.5 -4.0 -4.5 20,000
Wealth vs. Target
12
The ratio is exactly 44.4% in the difference form, as the same curvature parameter is applied to gains and losses and the weighting parameters is an additional linear transformation. The ratio is non-linear under the ratio form of the function, as taking ratio imposes a non-linear transformation on the outcomes before the utility function is applied. Page 11
Figure 4b: Ratio of Utility Values for Gains vs. Losses* Functional Form:
Difference Tversky & Blake et Kahneman (1992) al. (2013)
Parameters:
Ratio Tversky & Kahneman (1992)
Blake et al. (2013)
Power Utility CRRA = 4
Outcomes compared +$2,000 vs -$2,000 +$4,000 vs -$4,000
±10% ±20%
44.4% 44.4%
0.8% 0.6%
43.9% 43.4%
10.7% 10.4%
4.56% 2.47%
+$8,000 vs -$8,000 +$16,000 vs -$16,000
±40% ±80%
44.4% 44.4%
0.4% 0.3%
42.3% 39.8%
9.8% 8.7%
0.66% 0.01%
* Notes: Figure 4b reports the absolute value of the ratio of the utility values for gains and losses of equivalent magnitude, relative to target wealth of $20,000, as plotted in Figure 4a. The counterpart ratios reported for power utility are based on equivalent values for wealth in absolute terms, e.g. +$4,000 vs -$4,000 is the ratio of power utility values for wealth of $24,000 vs. $16,000.
The key message is that adopting a utility function and associated parameters simply because they appear in the academic literature may not be an effective approach. There is no guarantee that a utility function chosen in this way will provide a reasonable representation of the objectives and preferences of the investor. It is better to forfeit the comfort of being able to cite existing literature for validation, and aim to identify the utility function that is appropriate given the circumstances. (ii) Choosing a Utility Function Choosing a utility function is a large topic worthy of a paper in itself. General guidance is offered here on how the task might be approached. Section 5 illustrates the process by describing the selection of utility functions for four representative investors. The starting point is to form an understanding of the investor’s objectives and their preferences over outcomes. These findings are then reflected in a utility function that can be used to evaluate outcomes. Below are key aspects that should be considered as part of the process. Note that the ‘investor’ might be an individual, cohorts of individuals, or an organization. a) Which outcomes should utility be defined over – Utility might be defined over withdrawals from the portfolio (e.g. income, or consumption), end-period wealth (i.e. portfolio value), or some combination of the two. This aspect is discussed in Section 4 with regard to specifying a withdrawal and evaluation plan under the approach. b) Whether the concern is with raw levels, or some target – Concern with the level of withdrawals or wealth implies a function that directly evaluates outcomes, which includes power utility and other forms of the HARA class. If a target exists, then some kind of reference dependent utility function may be appropriate. c) Nature of any reference point – Reference points may take a variety of forms, including: target withdrawals and/or wealth; return targets (e.g. CPI-plus); performance delivered by a benchmark or peers; funding ratios; and capital ratios. A further issue is whether the reference point is time-varying, e.g. resetting as some form of habit persistence. The nature of any target, where one exists, needs to be reflected in the utility function. d) Investment horizon – The horizon over which utility is to be measured can be influential, as highlighted by the discussion over interaction with horizon in Section 2(iv). This aspect is also discussed within Section 4 as a component of the evaluation plan. Utility should be defined with respect to the time period of concern to the investor. It is quite possible that the evaluation horizon may differ from the period over which wealth will be deployed. For instance, investors with defined contribution pension plans might be characterized quite differently depending on whether they are concerned with their near-term balance, or the potential retirement income they can draw over the long run. e) Time preference – When a stream of outcomes is generated across time, the question arises as to whether the investor has a preference for attaining those outcomes earlier rather than later. If so, a time preference parameter may need to be incorporated within the utility function. f) Views about the value attached to the residual balance – In some circumstances, the value attributed to any residual balance may have distinctive attributes that need to be captured in the utility function. A classic example is a bequest motive, where the value placed on bequests may differ to that placed on withdrawals or wealth accruing to the investor themselves.
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g) Preferences over differing outcomes – A key issue for parameterization of the utility function is the value that the investor places on outcomes that are lower relative to higher within the distribution. It is helpful to think of this as a preference for ‘good’ relative to ‘bad’ outcomes, rather than as risk aversion. Understanding these preferences and encoding them in the utility function is perhaps the most challenging aspect. As discussed in Section 3(i), the key consideration is the relative scores that are placed on outcomes across the range. For instance, extreme aversion to being in the lower tail would suggest using a high CRRA under power utility; or relatively large curvature and/or weighting parameters on losses under reference dependent utility, probably in conjunction with a small curvature parameter on gains. Another issue is whether there exists a point of ruin. Examples might include where a retiree is left destitute if their income is too low or their retirement balance falls to zero, or where erosion of capital leads to bankruptcy for an organization. Under such cases, it may be appropriate to ensure that the utility function substantially ramps up the penalty on losses as the ruin point approaches,13 so that the optimization becomes dominated by avoiding ruin as the first priority. A number of techniques can assist in the process of selecting and parameterizing a utility function. Potential methods for teasing out investor objectives and preferences include: observing behaviors (revealed preference), choice experiments, surveys or questionnaires, and even open discussions such as at Board level or with the client. The process might be assisted by plotting and analyzing candidate utility function to gauge its implications, perhaps in line with the presentation appearing in Figure 4.
4. An Approach for Forming Portfolios This section details an approach for identifying optimal portfolios, and establishes the base for the illustrative analysis of Section 5. The approach entails four steps, which are discussed in subsections 4(i) to 4(iv) respectively: A) B) C) D)
Choose a utility function Specify a plan for withdrawal and evaluation Project joint distributions of accumulated wealth outcomes for the assets Solve for optimal portfolio weights
(i) Step A: Choose a Utility Function As the task of selecting and parameterizing a utility function was discussed in Section 3, only a few additional points are made here. The three forms of utility function described in Section 3 might suffice in most circumstances. They afford the flexibility to capture the objectives and preferences of many classes of investor; and have considerable recognition in the academic literature. They also offer the advantage of being time-additive, which simplifies the aggregation of utility over time and accommodates solving in a forward direction. Whereas it is open to use other utility functions, in many cases either power utility or reference dependent utility in one of its two forms may do the job effectively.
(ii) Step B: Specify a Plan for Withdrawal and Evaluation The approach requires specifying any withdrawals over the investment horizon, along with the combination of withdrawals and terminal portfolio value to be evaluated using the chosen utility function. Withdrawals constitute distributions from the portfolio (i.e. wealth) for consumption or redeployment. They must be specified in their entirety, noting that the sequence of portfolio values will depend on any prior withdrawals. The terminal portfolio value is the residual available at the end of a horizon. Terminal portfolio value need not be intended for consumption or redeployment, but should be evaluated nevertheless to the extent that residual wealth is considered of value to the investor. The utility function must be used to evaluate any amounts of value that are conditional on investment performance and hence asset weights. Withdrawals that are fixed in value may be excluded from the evaluation plan for the purpose of optimization if desired, as their utility value will be a constant. Nevertheless, including any fixed withdrawals in the calculations will do no harm, and may be useful if expected utility estimates are of interest in themselves. 13
The literature is often concerned that utility functions are continuous and differentiable. While this is required for analytical solutions, it is not necessary under numerical approaches that simply assign and aggregate scores attached to outcomes. Utility could even be specified as a schedule of scores, rather an explicit function, where numerical methods are used. Page 13
While a withdrawal plan might be specified in many ways, defining withdrawals as a percentage of wealth can be convenient from an analytical perspective as it ensures that portfolio value remains positive. Defining withdrawals as fixed amounts may lead to wealth falling to zero in certain circumstances. This then complicates the estimation of utility, which will hinge on the evaluation of either any residual left after meeting the planned fixed withdrawals, or the utility consequences of the portfolio value declining to zero before all planned withdrawals are satisfied. This does not invalidate the approach, but it requires giving careful consideration to how any residual portfolio value and/or zero withdrawals are evaluated under the utility function.14 To convey a sense for what is entailed in step B, some plans for withdrawal and evaluation are briefly described below. The first four dot points summarize the plans for the four representative investors examined in Section 5. The last two dot points address two other situations, also commenting on the utility function in each case. The private investor examined in Section 5(i) is assumed to be concerned only with their terminal portfolio value, and does not intend to make any withdrawals in the interim. Thus their plan is to evaluate the terminal portfolio value only (3-years and 10-year horizons are examined). The endowment fund of Section 5(ii) aims to make yearly withdrawals in perpetuity, based on a pre-determined percentage of portfolio value. Rather than evaluate a perpetual stream of withdrawals, their evaluation plan entails estimating utility values associated with the stream of withdrawals over an extended horizon (to year10), along with the terminal portfolio value. The latter is taken to represent the pot of wealth that remains to satisfy withdrawals beyond the forecast horizon. The defined benefit pension fund in Section 5(iii) is established to support a series of withdrawals over 15years that are fixed in value. Nevertheless, their prime concern is with the funding ratio, with a desire to avoid deficits lasting for more than 3-years. Accordingly, their plan is to evaluate the funding ratio at year-3. The fixed withdrawals occurring in the interim are not included in the evaluation plan; but still have an influence via their impact on residual asset value at the end of year-3. The retired individual modeled in Section 5(iv) plans to withdraw a pre-determined but time-varying percentage of portfolio value each period to consume. They are concerned only with income over the 30-years they expect to remain alive, and place no value on any bequest. Their plan is to evaluate the entire stream of withdrawals only. A retired individual with a bequest motive would include the terminal portfolio value in their evaluation plan. This would require estimating the utility associated with the terminal portfolio value, probably imposing a discount on bequests through applying a scaling factor. Bell, Liu and Shao (2017b) demonstrate this for power utility, following Lockwood (2014) in the treatment of residual portfolio value. Under reference dependent utility, the evaluation might be performed by setting a target terminal portfolio value of zero, so that any bequest is evaluated as a gain. Either the curvature parameter could be adjusted, or a weighting parameter on gains of less than 1 applied, in order to scale the utility score on the bequest in accordance with investor preferences. For most investment managers, a suitable plan might be to evaluate terminal portfolio value at the end of the horizon over which they expect to be evaluated by their end-investors. For instance, if investors are willing to afford the manager a 3-year period of grace, the portfolio value at year-3 might be evaluated. The fact that most mandates are based around benchmarks suggests a reference dependent utility function, with a target that reflects projected wealth from investing in the benchmark. Where the benchmark is an index, the target might refer to an index fund or ETF.15 Where the benchmark is a return target, the target becomes the terminal portfolio value consistent with achieving the target return. The target might be varied conditionally within each simulated path, providing a connection between the manager performance and the market state. For instance, a real return target of CPI+3% would be modeled by reference to projected inflation within each path. 14
A zero withdrawal becomes undefined (infinite) under power utility. This might be handled by specifying a minimum utility value under power utility, e.g. substitute the utility value for $1 for any withdrawal less than $1. Under reference dependent utility, the maximum negative utility value will be achieved at $0. For example, under reference dependent utility functions with a curvature parameter of 0.88 and weighting parameter of 4.5, a withdrawal of $0 versus a target of $20,000 gives a utility value of –$27,423 under the difference form, and -4.5 under the ratio form. Consideration might be given to whether this is an adequate utility penalty for running out of money. In any event, the aim is to ensure that an appropriate utility value emerges under states where the portfolio value and hence withdrawals decline to zero. 15 Another approach may be to conduct the analysis in benchmark-relative terms. Page 14
(iii) Step C: Project Joint Distributions of Accumulated Wealth Outcomes for the Assets The approach is based around modeling outcomes as a function of accumulated wealth. It is hence convenient to model assets in terms of the accumulated wealth they deliver, and aggregate these estimates into a portfolio value from which any withdrawals are extracted. Modeling accumulated wealth is also particularly helpful when projecting long-term outcomes over multiple periods. Potential methods for modeling the joint distribution of accumulated wealth arising from investing in a group of candidate assets include: Simulating from historical data, where observations are jointly drawn from the same period for each asset. Statistical models, such as vector auto-regressive or regime switching, which are specified in a way that
accounts for any inter-relationships between assets. Structural models that impose relationships between key variables and the assets, e.g. stochastic model of
Wilkie (1984), which is widely used in actuarial practice. Value-based models, such as plowback models for equities, with input variables shared across assets, e.g. return
on equity is connected to interest rates. Scenario analysis, e.g. see Gosling (2010).
Regardless of the method, the modeling process should satisfy two requirements. First, covariance between the assets should be embedded in the joint distribution of wealth outcomes. Under mean-variance analysis, this is achieved by specifying a covariance matrix. For multi-period analysis over longer periods, the modeling may be structured by simulating a series of joint paths, or specifying a range of scenarios, which act as ‘states’ to which each asset responds. Second, wealth generated from reinvestment should be accounted for appropriately. Reinvestment of cash flows generated by an asset account for an increasing portion of wealth as the investment horizon lengthens, making it important for long-term investors in particular to ensure reinvestment is modeled. The common approach is to form accumulation indices which assume distributed cash flows are reinvested back into the asset itself. While this may suffice in some circumstances, in others it may be more appropriate to explicitly model the reinvestment process. One example is modeling equities using a plowback model or similar, thus accounting for the marginal return on reinvestment. Another is explicitly modeling the reinvestment of coupons and principal in fixed income. The illustrative analysis in Section 5 simulates accumulated wealth for two assets using two basic methods, which suffices to illustrate the approach. The two assets include equities and 10-year bonds. Both are modeled on the assumption that dividends and coupons are reinvested back into the asset itself, such that accumulated wealth reflects total return indices as usually calculated. Joint distributions are generated in two ways: Random draws – A series of 10,000 paths of accumulated wealth spanning up to 30 years is formed by randomly drawing real returns from a normal distribution, with allowance for uncertainty over the expected equity return. For equities, the expected return of 6% is treated as a random variable, modeled by drawing different expected returns for each path from a normal distribution with a mean of 6% and a standard error of 2%. The standard deviation for yearly equity returns is 16%. For 10-year bonds, expected returns have a known mean of 1%, and a standard deviation of 8%. Apart from the random equity mean, these are the same assumptions underpinning Figure 3. Allowing for uncertainty over the expected equity return has the effect of causing the distribution of expected wealth to fan out as the horizon increases, relative to the iid case. The randomly drawn data assumes that equities and bond returns are uncorrelated. Historical data – Yearly data for US equity returns, long bond yields and inflation from 1871 to 2016 are taken from Robert Shiller’s website, and used to create a series of (overlapping) return paths over time spans of up to 10-years. In order to create a series of 10-year bond returns, it is assumed that a 10-year bond paying yearly coupons equal to the reported long bond yield is purchased at the end of each year. This bond is notionally sold as a 9-year bond16 at the end of the year, and the sale proceeds plus the coupon are reinvested into another 10year bond. The accumulated wealth series are formed in nominal terms, then divided by the inflation index during each path to generate a real wealth and hence a real return series. This method implicitly embeds the historical covariance between equities and bonds, as well as any serial correlation in returns. The method generates wealth paths that represent a total of 135 actual 10-year episodes that were experienced by investors. The mean real return (standard deviation) for the sample is 8.0% (18.0%) for equities, and 2.4% (7.9%) for 16
The reported long bond yield for that period is used in pricing the bond. Page 15
bonds, and the correlation between the two assets is 0.163 (0.045 in nominal terms). Another feature of the data is that bonds demonstrate positive serial correlation of 0.155, while equities is close to zero at 0.012.
(iv) Step D: Solve for Optimal Portfolio Weights Forming optimal portfolios requires generating a set of paths, each comprising a sequence of portfolio values and associated withdrawals over the horizon. The sequence is formed by conditioning within each path on the wealth arising from investing in each asset, the withdrawal plan, and asset weights. Closing portfolio values for each year are calculated by adjusting opening portfolio values for the change in wealth arising from investing in accordance with candidate asset weights, then deducting any withdrawals. By deducting withdrawals, the modeling process implicitly accounts for any sequencing effects. The utility function is used to evaluate withdrawals and the terminal portfolio value for each path in accordance with the evaluation plan. Utility estimates are averaged across paths to form an expected utility measure. Finally, the optimal portfolio is identified by locating the asset weights that maximize expected utility. Further detail on the modeling process now follows, including formulas. The initial aim is to estimate the accumulated portfolio value (equivalently total wealth, or assets) that is available to support withdrawals for a forecast period t=h within each simulated path p (i.e. state). Equation (4) is used to estimate the sequence of portfolio values in a forward direction, so that portfolio values incorporate withdrawals up until the beginning of period t. Equation (5) estimates the withdrawal for period t in situations where the withdrawal is determined as a percentage of the pre-withdrawal portfolio value. The handling of fixed withdrawals was discussed above under part B, and would amount to simply deducting a pre-determined amount for Wp,t.
𝑃𝑉 𝑊
, ,
= 𝑃𝑉 = 𝑐 𝑃𝑉
where: PVp,t = xe = Re,p,t = Rb,p,t = ct = Wp,t =
1+𝑥 𝑅
, ,
1+𝑥 𝑅
+ (1 − 𝑥 )𝑅
, ,
+ (1 − 𝑥 )𝑅
, ,
−𝑊
(4)
,
, ,
when Wp,t is a function of PVp,t (5)
portfolio value in path p for period t weight in equities equity return in path p for period t bond return in path p for period t percentage of portfolio value to be withdrawn in period t withdrawal in path p for period t
Withdrawals in each path and terminal portfolio value are ‘scored’ using the chosen utility function, and the scores aggregated to derive the utility associated with that path. This is straightforward where the withdrawal is equal to the terminal portfolio value at the end of the horizon, which gives rise to a single utility score for that path. Where a stream of withdrawals as well as possibly the terminal portfolio value are being evaluated, it becomes necessary to aggregate utility scores across time. Where utility is time-additive, the standard approach of simply summing discounted utility can be applied using equation (6). This equation allows for a time preference parameter, which may be required in instances where there is a preference for earlier withdrawals. Time preference is ignored by setting β=1. This assumption is made in Section 5, where the analysis is conducted in real terms.
𝑈(.), = ∑
𝛽 𝑈(.) (𝑊 , ) + 𝛽 𝑈(.) (𝑃𝑉 , )
(6)
where: U(.),p = aggregate utility for path p under (time-additive) utility function (.) β = time preference parameter Equation (6) assumes that utility may be summed across time within each path. This is usually valid for power utility, and for reference dependent utility in its difference form where the utility estimates represent a direct transformation of dollars versus target. However, it may not be valid under reference dependent utility functions in their ratio form, where the scaling by target has the potential to undermine the comparability of utility scores over time where similar percentage deviations from target represent different ‘value’ to the investor. For instance, a value of W/W* = 1.10 when W equals $11,000 and W* equals $10,000 may not be equivalent to the same estimate when W equals $1,100 and W* equals $1,000. Scaling may also be appropriate where the evaluation plan Page 16
entails a mix of withdrawals and terminal portfolio value with differing value implications.17 In such circumstances, one approach is to specify weights to permit the utility estimates to be aggregated in meaningful manner, as per equation (7):
𝑈
( ),
= ∑
𝑦𝑈
( ),
(7)
where: URDU(R),p = aggregate reference dependent utility in ratio form for path p yt = weight place on utility generated in for period t The weights (yt) would be chosen in accordance with the circumstances. The choice may often emerge naturally. If the withdrawal for each period is of equal importance, then simple addition will suffice. In many instances, it may be appropriate to weight by the respective target withdrawals (i.e. 𝑊 ∗ / ∑ 𝑊 ∗ ), or perhaps projected withdrawals (i.e. 𝑊 / ∑ 𝑊 ). Where time preference exists such that earlier withdrawals are considered more relevant, this can be accommodated by further adjusting the weights for a discount factor. In any event, the key aim is to aggregate the utility score in a manner that encapsulates the preferences of the investor. Judgment is arguably more important than mathematical precision. The utility estimates for each path are then weighted by probability to generate expected utility, as described by equation (8). In Section 5, expected utility is estimated by taking a simple average of utility across all paths on the assumption that each path is equally likely.
𝐸 [𝑈] = 𝑃𝑟 ∑
𝑈(.),
(8)
where: U(.),p = estimated utility in path p, estimated by either equation (6) or equation (7) Prp = probability of path p Finally, the portfolio weights are located that maximize the value of equation (8). In the two-asset example reflected in equation (4) and employed in Section 5, this amounts to solving for the optimal equity weight (xe). The analysis might be conducted in Microsoft Excel by using the Solver add-in to locate the optimal weights, providing that the problem is sufficiently well-behaved.
5. Illustration for Four Investors This section illustrates the choice and use of utility functions in forming portfolios by applying the approach to four representative investors. The aim is to demonstrate the process, and draw out some key themes.
(i) Private Investor The private investor is investing a ‘pot of wealth’ they want to grow over time, with no specific return or wealth target. They are averse to large losses that are sustained through to the end of their investment horizon, but are not overly concerned about volatility along the path. This situation implies using power utility to evaluate the terminal portfolio value at the end of the investment horizon. The opportunity is taken to demonstrate how optimal portfolios vary with horizon and risk aversion under such circumstances. This is done by generating results for two horizons of 3-years and 10-years, and across CRRA parameters ranging from 1 (log utility) to 8. Accumulated wealth associated with investing in equities and 1-year bonds is simulated using historical data from Shiller, as described in Section 4(iii). Figure 5 summarizes the setting, including the key inputs under step A to step C, and reporting the optimal equity weights under step D. Equity weights are also plotted in Figure 6.
17
Another approach is to apply a discount to terminal portfolio value directly within the utility function, similar to that described for bequests above in Section 2(ii). Page 17
Figure 5: Private Investor Setting Step A) Utility function
Input / Output
B) Plan for withdrawal and evaluation C) Assets and projection of wealth outcomes
D) Portfolio formation: optimal portfolios
Power utility CRRA from 1 (log utility) through to 8 investigated Aggregation of utility across time not required, given analysis focuses on terminal portfolio values Evaluation of terminal portfolio value Two horizons investigated: 3-years and 10-years Equities and 10-year bonds Simulations using 10-year historical real returns based on Shiller data, with the 10-year bond rolled over each year Equity weights CRRA 3-year horizon 10-year horizon 1 100.0% 100.0% 2 100.0% 100.0% 3 83.4% 98.1% 4 63.7% 79.1% 5 51.4% 66.6% 6 43.1% 57.6% 7 37.1% 50.9% 8 32.5% 45.6%
Figure 6: Optimal Portfolio Weights for the Private Investor Under Different Inputs 100% 90%
Equity Weight
80% 70% 60%
10-year horizon 50%
3-year horizon
40% 30%
1
2
3
4
CRRA
5
6
7
8
The optimal equity weights seem intuitive, ranging from 32.5% up to 100% depending on risk aversion and horizon. For this data set, equity weights are around 15% higher for the 10-year horizon relative to the 3-year horizon. This reflects serial correlation in the data, specifically the fact that real bond returns are persistent (serial correlation of 0.155, versus 0.012 for equities), making them riskier than equities over longer periods. The difference in serial correlation stems from 10-year bonds being more exposed to inflation innovations over longer holding periods, noting that inflation tends to be persistent. The equity weights are a higher but not out of line with what a financial adviser might recommend to a private investor in the situation.
(ii) Endowment Fund The endowment fund is investing a pool of capital (‘corpus’) intended to support philanthropic activities in perpetuity. Their objective is to maintain, if not increase, the real value of withdrawals over the long term. The Page 18
withdrawal rate is set at 5% of the portfolio value at the end of each year. The situation implies evaluating the entire stream of future withdrawals using a reference dependent utility function. Bearing in mind the perpetual nature of the fund, a practical solution is to project and evaluate withdrawals plus terminal portfolio value over a long horizon. The terminal portfolio value is taken to represent the residual pot of wealth available to support philanthropic activities beyond the forecast horizon. A forecast horizon of 10-year is selected, on the basis it is sufficiently long and accommodates the effective use of the historical Shiller data in simulating the asset return and hence accumulated wealth paths.18 The target is to maintain the value of both withdrawals over the forecast horizon and the terminal portfolio value in real terms. Given that the units differ between the withdrawals (a yearly amount) and the terminal portfolio value (larger pot of wealth), the reference dependent utility function in its ratio form makes as natural choice. The resulting utility values are aggregated through weighting by the real value of the target. This works out as a weight of 3.33% on each of the 10 withdrawals and 66.7% on the terminal portfolio. Parameterizing the utility function requires understanding the preferences of the key stakeholders, which possibly includes the Board, the donors, and the recipients. It is assumed that the fund has no commitment to deliver withdrawals of a certain value, but rather that the yearly 5% withdrawal (i.e. distribution) of portfolio value sets a budget to be allocated to philanthropic activities. The lack of firm commitment acts to reduce aversion to losses, as it limits the adverse implications of a reduction in portfolio value and hence withdrawals. This affords some latitude to pursue higher returns and hence greater capacity to support philanthropic activities, albeit at some risk of diminishing the corpus. The utility function is hence parameterized for moderate aversion to losses versus target, and to afford reasonable credit for gains. Figure 7 plots the utility function for the chosen parameters, which include curvature parameters of 0.50 on both gains and losses, and a weighting function of 3.0 on losses. For example, this curve implies a utility value of +0.12 for exceeding the target by +25%, and -0.40 for a shortfall of -25%. The curve also applies an increasing rate of penalty on losses and discount on gains, with utility values of +0.22 for exceeding the target by +50%, and -0.88 for a shortfall of -50%.
Utility
Figure 7: Utility Function for the Endowment Fund 0.50 0.25 0.00 -0.25 -0.50 -0.75 -1.00 -1.25 -1.50 -1.75 -2.00 -2.25 -2.50 -2.75 -3.00 0.00
0.20
0.40
0.60
0.80
1.00 1.20 Wealth / Target
1.40
1.60
1.80
2.00
Figure 8 summarizes the endowment fund setting. The optimal portfolio weight is 100% in equities under the setup. Basically, the analysis recommends that the endowment fund should run an aggressive strategy in pursuit of high long-term returns to increase the potential to ‘do more good’ over time. This result stems from the application of a reference dependent utility function with moderate risk aversion over a long horizon. The outcome could change if more conservative parameters are applied in the utility function, or the horizon is reduced. For instance, halving the curvature parameter on gains to 0.25, and doubling the parameters on losses to a curvature parameter of 1.0 and a weighting parameter of 6.0 generates an optimal equity weight of just under 90%. If these parameters 18
The 10-year horizon is a compromise. Extending the length of the data blocks drawn from the Shiller data acts to reduce sample size, and would not change the result. Page 19
are combined with an evaluation of distributions plus terminal portfolio value over a 3-year horizon, the optimal equity weight declines to 53%. A shortening of the horizon might be justified if stakeholders were sensitive to maintaining the value of the corpus through time, or if there was no flexibility to vary the distributions from the assumed 5% and maintaining the absolute value of withdrawals is required to support spending commitments. In any event, the analysis needs to be tailored to the situation of the investor. Figure 8: Endowment Fund Setting Step A) Utility function
B) Assets and projection of wealth outcomes
C) Plan for withdrawal and evaluation
D) Portfolio formation: optimal portfolios
Input / Output Reference dependent utility, ratio form Parameters: - Curvature on gains vs. target: 0.50 - Curvature on losses vs. target: 0.50 - Weighting on gains vs target: 1.0 - Weighting on losses vs. target: 3.0 - Time preference: 1.0 (no discounting) Target: Maintenance of the real value of withdrawals and portfolio value Aggregation: Utility values weighted by target (implies 70% weight on terminal portfolio value) Equities and 10-year bonds Simulations using 10-year historical real returns based on Shiller data, with the 10-year bond rolled over each year Yearly withdrawals equal to 5% of portfolio value Evaluation of withdrawals from year-1 to year-10 plus terminal portfolio value at year-10 (latter represents the wealth available to support withdrawals beyond year-10) Equity weight – 100%
(iii) Defined Benefit Fund (Liability-Driven Investing) The defined benefit fund is investing to satisfy a stream of pension payments that are fixed in nominal terms. The analysis is undertaken from the perspective of a fund sponsor who is concerned about the reported funding ratio. They have some tolerance for funding deficits providing they are sustained for a limited period, and would much prefer that deficits did not persist beyond 3-years. They are also keen to avoid making additional contributions to top-up the fund assets. The situation implies evaluating the funding ratio in year-3 using a reference dependent utility function in ratio form, parameterized for relatively high aversion in the realms of deficits along with a relatively modest value attached to surpluses. This is achieved through applying the parameters of Blake et al. (2013), which include a curvature parameter on surpluses of 0.44, a curvature parameter on deficits of 0.88, and a weighting parameter on deficits of 4.5. A feel for these parameters was provided in Section 3 (see Figure 4), noting that they deliver utility values for gains in the order of 10% of that for losses of equivalent magnitude. This implies a much stronger preference for avoiding a deficit that generating a surplus, which accords with the situation described. The pension payments and their valuation is modeled in a simple way, sufficient to draw out the key concepts. The liability is a fixed nominal pension payment (i.e. term annuity) spanning 15-years, valued by discounting at the prevailing 10-year bond rate.19 The analysis is based on the Shiller data, delivering a series of simulated paths for bond yields, asset returns and accumulated wealth over a 3-year horizon. To value the liability, the remaining 12 pension payments at the end of year-3 are discounted by the prevailing 10-year bond yield for that path. This establishes a path-specific value for the target, i.e. W*p,t=3. One consequence of this set-up is that the simulations embed a strong correlation between the liability and the 10-year bond, so that the bond plays the role as the lowrisk hedging asset for the liability. Meanwhile, equities offer a higher return but are mismatched with the liability. 19
The pension payments are set at a value ($0.094 per $1 of liability) so the discounted value of the payments is $1 at the bond yield assumed to prevail in period t=0 of 4.92%. The latter represents the average long bond yield in the Shiller dataset over the period of analysis. This establishes a baseline that dovetails with the historical data. Page 20
Evaluating the funding ratio at the end of year-3 bypasses the withdrawals made in year-1 through to year-3, which are fixed and make no difference to the optimization. These withdrawals nevertheless influence the terminal portfolio value at year-3, which is reduced by the value of withdrawals under the sequential modeling. Figure 9 summarizes the setting, Figure 10 plots the utility function, and Figure 11 plots the optimal equity weights. Figure 9: Defined Benefit Fund Setting Step A) Utility function
B) Plan for withdrawal and evaluation
C) Assets and projection of wealth outcomes
D) Portfolio formation: optimal portfolios
Input / Output Reference dependent utility, ratio form Parameters (follows Blake et al., 2013): - Curvature on gains vs. target: 0.44 - Curvature on losses vs. target: 0.88 - Weighting on gains vs target: 1.0 - Weighting on losses vs. target: 4.5 Target: Funding ratio of 1.00 Aggregation: Not required, given analysis focuses on terminal funding ratio at end of year-3 Evaluation of funding ratio at year-3, calculated as portfolio value divided by present value of liabilities by discounting at the 10-year bond in year-3 in each path Withdrawals in year-1 through year-3 are fixed. While not evaluated by the utility function, they impact on portfolio value at end year-3. Equities and 10-year bonds Simulations using 3-year historical nominal returns based on Shiller data, with the 10-year bond rolled over each year Optimal portfolio depends on initial funding ratio: Initial Funding Ratio Equity Weight 0.70 100.0% 0.80 91.9% 0.90 51.5% 1.00 15.8% 1.10 26.3% 1.20 51.3% 1.30 69.4% 1.40 82.9%
Figure 10: Utility Function for Defined Benefit Fund
Figure 11: Optimal Equity Weight for Defined Benefit Fund
0.5
100%
0.0
90% 80%
-1.0
70%
Optimal Equity Weight
-0.5
Utility
-1.5 -2.0 -2.5 -3.0 -3.5
60% 50% 40% 30% 20% 10%
-4.0 -4.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Funding Ratio
0% 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 Initial Funding Ratio
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The analysis is conducted across a range of initial funding ratios, to which the optimal equity weight is sensitive as evident in Figure 11. The mechanism is a follows. When the fund commences at a funding ratio of around 1.00 or just above, the optimal portfolio becomes heavily weighted in bonds in an attempt to secure the funding ratio, reflecting the hedge that bonds provide for the liability coupled with high aversion to deficits. As the initial funding ratio moves increasingly into deficit, more equities are favored as they increase the probability of reaching the target funding ratio of 1.00 by year-3. Meanwhile, bonds become unattractive as they only serve to lock-in the existing deficit. As the initial funding ratio moves further into surplus, the probability of falling back into deficit by year-3 reduces, and scope emerges to add equities to capture some of the upside. This should be a familiar pattern for those who have ventured into analysis of liability-driven investing.
(iv) Retired Individual The retired individual is modeled on an Australian setting, albeit very much simplified. It is assumed that the investor has just retired at 65 years of age, and has transferred their superannuation into a pension account (‘account-based pension’). They intend to draw on this account over a 30-year period until they turn 95, when they expect to die. Their withdrawal plan is to follow the minimum drawdown requirements set by the Australian Government, starting at 5% at age 66 and rising to 11% by age 91-95 (see Figure 14 for further details). Their target is to draw income of at least $24,506 in real terms over the 30-years, which reflects the ‘modest’ retirement standard estimated by the Australian Superannuation Fund Association (ASFA) as at September 2017. No allowance is made for the many other aspects faced by retirees, including the availability of the age pension, the existence of other assets outside their pension account, uncertainty over their longevity, or any bequest motive. Isolating the analysis from these influences supports drawing out how the choice of utility function might affect asset weights without too many confounding effects. The exclusion of the aged pension is an important deviation from practice, given it provides a risk-free hedge against both investment losses and longevity risk. The situation suggests evaluating the entire stream of withdrawals using a reference dependent utility function in difference form, while ignoring any terminal portfolio value.20 It also implies imposing a high penalty on shortfall versus target income, given that the retiree is relying on withdrawals from their account-based pension for living expenses. The opportunity is taken to contrast the results for two utility function parameterizations, one deemed ‘preferred’ and the other reflecting the parameters used by Blake et al. (2013) in a UK context. Figure 12 plots the two utility functions, Figure 13 plots the optimal equity weights, and Figure 14 summarizes the setting. Figure 12: Utility Functions for Retired Individual
100% 95% 90% 85%
Preferred parameters Blake et al. (2013) parameters
Equity Weight
80% 75% 70% 65% 60% 55%
Preferred parameters Blake et al. (2013) parameters
50% 45% 40% 200k 300k 400k 500k 600k 700k 800k 900k 1m 1.1m 1.2m Break 1.7m 2m 3m 4m 5m 6m
Utility
10,000 0 -10,000 -20,000 -30,000 -40,000 -50,000 -60,000 -70,000 -80,000 -90,000 -100,000 -110,000 -120,000 -130,000
Figure 13: Optimal Equity Weight for Retired Individual
Withdrawal less Target Withdrawal
Starting Balance
20
If the retiree placed a value on any bequest, then the terminal portfolio value would also need to be evaluated. Bell, Liu and Shao (2017a) provide an example of such an application. Page 22
Figure 12: Retired Individual Setting Step A) Utility function
B) Plan for withdrawal and evaluation
C) Assets and projection of wealth outcomes
D) Portfolio formation: optimal portfolios
Input / Output Reference dependent utility, difference form Parameters – two sets examined: Parameter Preferred Blake et al. (2013) Curvature – Gains 0.80 0.44 Curvature - Losses 1.10 0.88 Weighting - Gains 1.0 1.0 Weighting - Losses 2.0 4.5 Target: Withdrawal of $24,506 maintained in real terms; equivalent to AFSA Standard for a ‘modest’ lifestyle estimated for September, 2017 Aggregation: Utility summed across all 30-years, with time preference of 1.0 (no discounting) Withdrawals over 30-years between age 66 and age 95 are evaluated, and reflect the minimum drawdown rate for Australian retirees: Age Drawdown Age 65-75 5% Age 76-80 6% Age 81-85 7% Age 86-90 9% Age 91-95 11% Terminal portfolio value not evaluated (i.e. no bequest motive) Equities and 10-year bonds Simulations using 10,000 draws of real returns from lognormal distributions, including real equity returns with a random mean of 6% (standard error of 2%) and standard deviation of 18%, and real bond returns with a mean of 1% and a standard deviation of 8%. Optimal portfolio depends on initial balance and preference parameters (see Figure 14 for plot of full series of optimal weights): Selected Balances $400,000 and below $600,000 $800,000 $1.0 million $1.2 million $1.7m (Blake min. weight) $2.0 million $3.0 million $4.0 million $5.0 million $5.5m (Blake max. weight)
Preferred Parameters 100.0% 75% 69% 80% 100% 100% 100% 100% 100% 100% 100%
Blake et al. (2013) Parameters 100% 69% 56% 49% 46% 44% 47% 68% 83% 95% 100%
Relative to Blake et al. (2013), the preferred parameters are designed to give more credit for exceeding the target (the retiree is assumed to place some value on the possibility of a better lifestyle); but also impose a penalty on losses that increases in relative magnitude with shortfall. More credit is given to exceeding target by assuming a curvature parameter on gains of 0.80, versus 0.44 under Blake et al. (2013). The increasing penalty on shortfall versus target is achieved by assuming a curvature parameter on losses in excess of 1 (specifically 1.1), versus the 0.88 assumed by Blake et al. (2013). To ensure the penalty on losses is not overly large, this is coupled with a weighting parameter on losses of 2.0 versus the 4.5 assumed by Blake et al. (2013). The analysis is conducted across a range of starting balances, which turns out to be an important variable. The analysis is based around random draws of 10,000 paths for accumulated wealth with uncertainty over the expected equity returns (see Section 4(iii) for details). Figure 13 reveals that a u-shaped pattern emerges with respect to the level of starting balance, with the shape of the curve depending on the utility function parameters. Both the preferred and Blake et al. (2013) parameters lead Page 23
to 100% equity at balances of $400,000 and below, as equities are favored because their higher returns increase the odds of attaining the target. Optimal equity weights diverge considerably beyond that point. Under the preferred parameters, the optimal equity weight declines to 69% at a starting balance of $800,000, then returns to 100% at a balance of $1.2 million. Under the parameters of Blake et al. (2013), the equity weight declines more sharply and bottoms at just below 44% at a balance of $1.7 million. It then increases, reaching 100% at an (unlikely) balance of $5.5 million. The utility curves plotted in Figure 12 explain the weighting patterns seen in Figure 13. The Blake et al. (2013) parameters afford little credit for exceeding the target, meaning that the optimization is dominated by the outcomes in the realms of losses. This results in a much more extended u-shape, as a reluctance emerges to increase equity exposure until the balance is high enough to render only a small probability of falling into the loss region. In contrast, the preferred parameters give more credit to the possibility of exceeding the target, resulting in a greater propensity to take on equity risk. The Blake et al. (2013) parameters might be more appropriate for a retired individual with extreme aversion to shortfall and doesn’t care much for exceeding target. The preferred parameters might be more suitable for a retired individual that is willing to balance the downside against the upside.
6. Static versus Dynamic Analysis The approach offers a method for forming portfolios using utility functions that can be readily applied by practitioners with a good knowledge of spreadsheet packages such as Microsoft Excel, and who want to avoid complex mathematical or numerical methods. The main technique to reduce complexity is shutting down the dynamics. This is done by specifying a pre-determined withdrawal plan, and solving for static optimal asset weights that are assumed to be maintained throughout the forecast horizon. What is being precluded is the possibility that the investor might anticipate changing their future investment and/or withdrawal strategy in response to developments along the path. This accommodates a simply-structured analysis, including modeling in a forward direction. This section discusses potential extensions into dynamic analysis. Key questions are what dynamic elements might be included, and whether they would make any substantial difference to the results. Dynamics can be incorporated on a number of levels. The most advanced and potentially complex method assumes that the investment and withdrawal strategy will both be re-optimized in response to future developments. Such problems require full-scale dynamic programming that solves for the optimal asset weights and withdrawals, typically as a recursive problem, i.e. working backwards.21 Such methods require either closed-form mathematical solutions that constrain the analysis to make it tractable, or complex numerical techniques (for an example of the latter, see Butt and Khemka, 2015; Khemka and Butt, 2017). Cochrane (2014) suggests that few investors use such models, let alone understand them.22 A more limited approach entails specifying pre-determined rules for how either asset weights and/or withdrawals (and possibly contributions) might respond to investment returns along the path. This will not yield a fully optimal dynamic solution, but can still support an analysis with a relatively simple structure. For example, rules may be set for how either asset weights or withdrawals respond after low or high realizations of asset returns,23 or as a function of distance from target. Nevertheless, applying pre-determined rules can potentially make it necessary to iterate on joint sets of investment and withdrawal strategies to converge on a preferred solution that is not totally ad hoc. This can be cumbersome if a large ‘grid’ of possible strategy combinations needs to be evaluated. A variation on the above is to limit the withdrawal plan to a single or a few obvious alternatives, and focus on identifying an ‘acceptable’ dynamic investment strategy given the specified plan(s). The modeling task is thus reduced to locating the set of dynamic asset weights that maximize utility for a candidate plan, with the aim of identifying the preferred pair of investment strategy and withdrawal plan across a manageable number of iterations. This method may be suitable where it is not necessary to solve for the optimal withdrawal plan as an important choice variable, and it is acceptable to propose and evaluate a limited class of withdrawal strategies.
21
It is worth noting that utility functions such as that of Epstein and Zin (1989) and Weil (1989) require recursive solutions. Admitting dynamic portfolio management may give rise to positions aimed at hedging changes in the investment opportunity set, in the sense of Merton (1973). 23 The analysis in Section 5 assumes rebalancing back to target asset weights, which amounts to simple rule of this type. 22
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In any event, the question arises over whether dynamic analysis would lead to a substantially different asset allocation today, or whether the static weights arising under the approach in its basic form might suffice. The answer depends on the extent to which the optimal weights for today depend on what could happen tomorrow. Needless to say, the importance of this issue will vary with the circumstances. In many instances, it may be sufficient to conduct the analysis based on what is known today, and ignore the possible response to future developments. This approach does not imply the intent to stick with a static strategy regardless: it can be conducted on the expectation that the analysis will be re-run and the strategy re-adjusted each period. Cochrane (2014) implies this is what is currently done by most practitioners in any event. To illustrate how dynamic analysis might make only a modest difference, stochastic dynamic programming is applied to identify the optimal investment strategy for the defined benefit fund introduced in Section 5(iii).24 The analysis draws on the techniques used by Butt and Khemka (2015) and Khemka and Butt (2017). It involves using recursive analysis to create a grid of optimal equity weights for each period conditional on the realized funding ratio (treated as the ‘state’ variable) and the distribution of future expected asset returns.25 As in Section 5(iii), optimal equity weights at the beginning of year-1 are estimated for initial funding ratios in the range of 0.60 to 1.40. Utility estimates are also collected. Under the dynamic analysis, the year-1 equity weight is 3.1% greater on average than under the static analysis of Section 5(iii). While the equity weight differences are uneven across the range of initial funding ratios,26 most differences are relatively modest in magnitude and a similar u-shaped equity weighting function still emerges. Thus the extent to which the representative defined benefit fund might alter their current equity weights knowing they have the option to revise their future equity weights is not very substantial, notwithstanding a strong embedded relation between optimal equity weights and the funding ratio in this case. The utility gains average +0.028 (median +0.019),27 which indicates that the option to revise weights is of some modest value. Importantly, the tenor of the results does not change in any major way under the dynamic analysis. There is a trade-off between reducing complexity, and accepting what could be a sub-optimal solution. Whether some level of dynamic analysis is needed to properly address the problem at hand will depend on the circumstances. In particular, if it is important to jointly solve for the optimal investment and withdrawal strategy, then full-scale dynamic optimization may be necessary. If the withdrawal plan can be largely pre-determined, then the approach as presented in this paper may suffice, or may be extended to accommodate dynamics on a limited level. It is comforting that the illustrative analysis described in Section 5 generate portfolio weights that appear sensible and intuitive, and that applying dynamic analysis to the defined benefit fund led to modest changes. While it is always dangerous to generalize from specific cases, these results hint that the approach in its basic form may deliver acceptable results across a range of common situations.
7. Conclusion This paper establishes that utility-based analysis can provide an effective and flexible means for identifying optimal portfolios in practical settings. The investment industry has tended to rely on mean-variance analysis, in part because of tradition and familiarity with the approach given it is widely taught at university. However, meanvariance analysis is quite limiting. It implicitly makes strong assumptions about the objectives and preferences of the investor, while catering to a discrete and often short horizon. It is not very effective under situations where the horizon is long-term and involves a stream of withdrawals over time. Section 5 of this paper illustrates how analysis based around utility functions can be tailored to a wide variety of investors with differing objectives and preferences, and can generate some intuitive results. Further, the analysis was conducted entirely in Microsoft Excel, without having to resort to dynamic optimization. This is achieved by constraining the analysis, specifically by precluding dynamics related to the investment and withdrawal strategy. Of course, the simplifications made to 24
Recall that the withdrawals were fixed for this investor. Initially, a grid of optimal equity weights for year-3 is established by optimizing utility at the end of year-3, conditional on the year-2 funding ratio. The next step is to create a similar grid of equity weights for year-2 conditional on the realized year1 funding ratio, again optimizing utility at the end of year-3 assuming that the year-3 equity weights will follow the year-3 grid. The final step is to estimate the optimal equity weights at the beginning of year-1 to optimize utility at the end of year3, assuming that the equity weights for year-2 and year-3 will subsequently be updated in line with both grids. 26 The year-1 equity weight differences between the dynamic and statics were mainly positive and ranged up to about +8%, although the optimal dynamic equity weights were modestly lower for initial funding ratios in the 0.90-1.00 range. 27 Under the reference dependent utility function, this might be roughly interpreted as a risk-adjusted change in wealth. 25
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limit complexity may not be appropriate in all circumstances. For instance, dynamic optimization may be unavoidable if the aim is to locate an optimal dynamic withdrawal plan that responds to investment outcomes. Nevertheless, a more basic approach might suffice across a wide range of circumstances, especially where the withdrawal plan is known or limited to a few choices. A key contention is that utility functions should be selected and parameterized with a view to tailoring to the investor in question. Choosing an appropriate utility function is essentially a subjective exercise. There is a natural desire to rely on findings from the academic literature in selecting the functional form and parameterizing a utility function, in order to provide validation. The problem is that the utility functions and parameter estimates appearing in the academic literature could be quite unsuitable for the investor in question. A much better approach is to design the utility function to reflect the circumstances of the investor. Going through the process of understanding their objectives and preferences and encoding them in a utility unction may not only lead to more meaningful outputs, but invites addressing the question of what really matters to the investor along the way.
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