j o i joim · for how to invest, both when the investor’s port-folio is sufﬁciently funded or...

JOIMwww.joim.com

Journal Of Investment Management, Vol. 16, No. 3, (2018), pp. 1–27

© JOIM 2018

A NEW APPROACH TO GOALS-BASEDWEALTH MANAGEMENTSanjiv R. Dasa, Daniel Ostrova,

Anand Radhakrishnanb and Deep Srivastavb

We introduce a novel framework for goals-based wealth management (GBWM), where riskis understood as the probability of investors not attaining their goals, not just the standarddeviation of investor’s portfolios. Our framework is based on a foundation of developmentsin behavioral economics and finance and is consistent with modern portfolio theory. Usinga simple geometric analysis, we determine a specific portfolio that matches each individualinvestor’s stated goals. Our approach requires information from the investors about theirgoals, elicited in a clear manner that market research shows is superior to commoncurrent practices. This new approach can improve the communication between advisorsand clients and produce better advice for enabling clients to attain their goals with highprobability through the use of efficient portfolios.

1 Introduction

Traditionally, the financial industry, financialadvisors, and academics in finance have asso-ciated the notion of “risk” with the standarddeviation of an investor’s portfolio. Investors, onthe other hand, typically associate “risk” withthe likelihood of not attaining their goals. Thisdistinction is important: for example, decreas-ing standard deviation risk in an underfundedinvestor’s portfolio increases, as opposed todecreases, the risk of not attaining their goals.

aSanta Clara University, CA 95053, USA.bFranklin Templeton Investments, CA, USA.

We suggest that goals-based wealth management(GBWM) needs to incorporate viewing risk fromthe investor’s goals-based perspective, as well asthe traditional standard deviation perspective.

This has important implications for the portfo-lios that advisors propose to their clients, as wellas for how advisors and their clients communi-cate. In traditional financial planning, advisorslook to understand what an investor’s goals are,then they ask questions designed to determine theinvestor’s tolerance for portfolio standard devia-tion, which leads to advising the investor to adopta portfolio that has a mean and standard devia-tion corresponding to the investor’s risk appetite,

Third Quarter 2018 1

2 Sanjiv R. Das et al.

strictly from a standard deviation perspective. Inour GBWM approach, we also look to under-stand what an investor’s goals are, but then weseek to elicit what probability the investor wouldlike to maintain in attaining these goals. Thisis a different conversation, which benefits byusing language and ideas that are more naturalfor investors.

It also leads to different advice for the investor.In this paper, we show how to take the infor-mation from this conversation and map it to aspecific range of portfolios that will meet theinvestor’s goal-based specifications. In contrastwith traditional planning where a static portfo-lio is generally selected and maintained throughrebalancing, our method produces a portfolio thatwill move about on the Efficient Frontier, dynam-ically addressing changes to the market in orderto optimize the investor’s goals. The trajectory ofthis evolution will depend on the investor’s prefer-ences, which can be pre-selected both in the casewhen the portfolio is sufficiently funded and whenit becomes underfunded if the financial situationworsens sufficiently.

We emphasize that incorporating goals-basedwealth management does not mean abandoningrisk-based asset allocation. It is an overlay thatremains fully consistent with portfolio construc-tion based on modern portfolio theory. GBWMwill result in choosing portfolios that reside on theEfficient Frontier, so they will be optimal, but theyaccrue the additional benefit of being cognizant ofinvestor’s risk in not meeting their goals.

The GBWM approach in this paper has implica-tions for improving the relationship that advisorshave with their clients, as well as the outcomesthey can obtain for them. Investors are able tobenefit not only from individualized advice, butalso from being able to explicitly see the effectsof choices they understand on the probability oftheir outcomes. This delivers an experience for

investors that is more intuitive, transparent, andunderstandable, both in the initial set-up for theirinvestment and for later discussions between theinvestor and the advisor as market conditionsevolve.

This paper proceeds as follows: In Section 2 wedescribe the antecedent behavioral finance lit-erature and the recent practitioner research onGBWM. Section 3 discusses current practices,introduces nine properties that we believe shouldbe evidenced by GBWM, and also details eightgoals-based items of information that investorsneed to provide our GBWM process. Section 4presents a simple geometry for GBWM that formsthe technical underpinnings for our approach,leading to an investor-specific recommendationfor how to invest, both when the investor’s port-folio is sufficiently funded or when it becomesunderfunded. This new approach has implicationsfor asset allocation, fund selection, and improvingthe role of financial advisors. Finally, Section 5offers a concluding discussion.

2 Antecedent literature

Our approach in this paper is based on an exten-sive history of academic portfolio theory andbehavioral finance research, practitioner-basedresearch, and wealth management experience. Welook to combine and extend these different strandsinto a straightforward, but rigorous, approach.

The 2017 Nobel prize was awarded to RichardThaler for his work in behavioral economics. In itsdetailed scientific explanation1 of the award, theNobel Prize committee highlighted Thaler’s workon the “endowment effect”, i.e., the asymmet-ric valuation of assets by individuals, who valueitems more when they own them as opposed towhen they do not. This is related to loss aversionin Prospect Theory, see Kahneman and Tver-sky (1979). These psychological constructs areimportant underpinnings of mental accounting

Journal Of Investment Management Third Quarter 2018

A New Approach to Goals-Based Wealth Management 3

theory (Thaler, 1985, 1999), where people treatmoney with different risk–return preferences,depending on what use the money is to be put to,or whether their portfolios are performing poorlyor well (Shefrin and Statman, 1985). The conceptsand ideas in these seminal contributions formthe basis for the goals-based wealth managementapproach developed in this paper.

In its simplest form, goals-based wealth manage-ment can be defined as a process that focuseson helping investors realize their goals, bothshort-term and long-term, through a portfoliomanagement method primarily focused on reach-ing well-defined financial goals. One of the keyunderpinnings of GBWM is mental accountingtheory. Eliciting investor goals is a key part of theGBWM process, and is facilitated by breakingdown overall portfolio goals into sub-portfoliogoals using the ideas of mental accounts, wheredifferent goals are managed in different accounts,each aggregating into the overall portfolio. She-frin and Statman (2000) developed behavioralportfolio theory (BPT), and argued that investorsbehave as if they have multiple mental accounts,a version they titled BPT-MA (in contrast to sin-gle account investors, BPT-SA). Each mentalaccount portfolio has varying levels of aspira-tion, depending on the goals for the mentalaccount. These ideas naturally lead to portfo-lio optimization where investors are goal-seeking(aspirational), while remaining concerned aboutdownside risk in the light of their goals. Ratherthan trade-off risk versus return, investors trade-off goals versus safety, see Roy (1952). This leadsto normatively different statements of the portfo-lio problem than in the mean–variance theory ofMarkowitz (1952).

These ideas form the bedrock of goals-based port-folio construction, irrespective of whether goalsare articulated over single or multiple mentalaccounts. Interestingly, Das et al. (2010) showed

that, under specific technical assumptions, there isa mathematical linkage between mental account-ing theory (MAT) and mean–variance theory(MVT), arguing that there is a mapping from agoals-based portfolio to a portfolio on the mean–variance Efficient Frontier. This mathematicalreconcilement showed that GBWM is supportedby MVT, which also forms part of the basis forthe model described in this paper.

There is a growing practitioner literature ongoals-based wealth management. Nevins (2004)extended the mental accounting approach. Hecontended that traditional investment planningfails to recognize investor’s behavioral prefer-ences and biases, resulting in suboptimal portfolioperformance. He argued that traditional risk mea-sures do not fully capture market behavior andare of limited relevance to investors. His paperadvocated that, in addition to investor’s behav-ioral attributes, such as mental accounting andloss aversion, a goals-oriented approach helpsinvestors ameliorate their biases, such as over-confidence, hindsight bias, overreaction, beliefperseverance, and regret avoidance, all identifiedby researchers in behavioral finance.2 Comple-menting this work, Zwecher (2010) discusseshow risk management can be done more activelyand efficiently by demonstrating how a retirementportfolio that provides income, generates growth,and protects assets from disasters, can be cre-ated by adopting a bucketing (mental accounting)approach.

Brunel (2015) discussed the equal importance oftwo goals for an investor: being able to avoidnightmares while realizing dreams. Brunel’swork focussed on demonstrating how goals-basedwealth management can be achieved across mul-tiple time horizons for multiple life goals. Healso suggested how to map the language investorsuse in describing the importance of dreamsor the severity of nightmares into acceptable

Third Quarter 2018 Journal Of Investment Management


probabilities that the investor will realize suchdreams or avoid such nightmares.

Based on the research above, our approach inthis paper follows from the idea that investors arebetter able to articulate and discuss their goals,including safety criteria, than they are able tospecify a mean–variance trade-off and that work-ing with goals leads to portfolios that are betterdesigned to meet investor aspirations. Practition-ers have recognized the need for a goals-basedapproach, and this paper offers a framework andimplementation model for a single goal, showinghow this goals-based portfolio approach is man-aged over time in a manner that also fully supportstraditional asset-allocation approaches.

In the next section, we discuss how our approachdiffers from typical current approaches used bymany advisors.

3 Properties of the goals-based wealthmanagement (GBWM) approach forfinancial advising

As we see from the previous section, GBWMstrategies can offer a rich, normative structurewith many attractive properties. These propertieshave practical value in supporting a new approachto financial advising:

Property 1: Clarity. The financial goals of theinvestor should be clear.

Property 2: Customization. The advice given tothe investors should be individualized to cater totheir specific goals.

Property 3: Risk Specificity. The probability ofthe investors attaining (or not attaining) theirgoals should always be clear.

Property 4: Risk Compliance. The advice givento the investor should take into account theseprobabilities.

Current strategies that are intended to be goal-based usually satisfy the first property, but thethree other properties are often not met. Con-sider, for example, the following four commonapproaches currently taken by advisors lookingto be goal-based:

(1) The financial advisor helps investors to elab-orate their broad investment goals and sub-goals. They consider the future value of themoney for these goals adjusted for inflation.Money is then allocated to these goals. Thisapproach meets the first two properties, butnot the third or fourth.3

(2) The financial advisor helps investors cre-ate risk-based portfolios depending on howfar ahead in the future their goals are, andaccordingly adjusts their equity exposure. Forexample: 70% equity exposure for a goal 20years away, 60% for 15 years, and so on. Thisapproach meets the first property and may beexecuted in a way that meets the third prop-erty, but it does not meet Property 4, nor, sincethe advice in a given time frame is the sameregardless of the size of the goal wealth, doesit meet Property 2.

(3) After the advisor determines the investor’sfinancial goals, each goal is mapped to a spe-cific investment portfolio (e.g., a “must have”goal is mapped to the least volatile investmentportfolio, a long-term investment is mappedto a risk-adjusting glide path portfolio, etc.).Again, this approach meets the first propertyand may be executed in a way that meets thethird property, but it does not meet Property 4,nor, since the size of the goal wealth is nottaken into account, does it meet Property 2.

(4) In addition, a financial advisor may also offerexpense management solutions, enablingthem to advise investors on how to cutback expenses or streamline monthly bud-gets, incorporating present and future cashflows, so that investor can achieve certain



savings goals. While these are an importantpart of financial planning, the objective ofexpense management is different from theobjective of goals-based wealth management,so it is unsurprising that this approach meetsonly the first two properties, but not the thirdor fourth.

The first and the last of these example approachesare disconnected from the equities market. Thesecond and third approaches look at risk strictlyas the standard deviation of the portfolio, not theprobability of failing to achieve the investor’sgoals. Because risk is viewed by many advi-sors strictly as portfolio standard deviation andbecause investment success is often defined byadvisors as individual investments’ outperfor-mance of their benchmarks, instead of a portfoliomeeting its goal or goals, advisors often view thesuccess (or lack of success) of a portfolio in differ-ent ways from investors. The crucial componentsin GBWM of considering the probability that theinvestor’s goals will actually be met, encompass-ing all four properties, is by and large, missingin current practice. The GBWM process also hasto address “how” the goals will best be achieved.Investors need to know how close or far they are to

achieving their goals and whether or not they areprotected in the event of adverse market events.

Market surveys on outcome-oriented investmentresearch allow us to better understand and quan-tify this issue. For example, consider Figure 1,from a recent market survey of investors andadvisors,4 which shows that 72% of clients viewsuccess based on the overall performance oftheir portfolio, not the performance of individ-ual investments, whereas only 51% of advisorsbelieve that their clients view success primar-ily based on overall portfolio performance, asopposed to individual investment performance.

Furthermore, by focusing more on individualinvestments, instead of the overall goals of theportfolio, the advisor will gravitate toward lan-guage and ideas that are less clear to the client,as Figure 2, which comes from the same recentmarket survey, demonstrates. Talking with aclient in goals-based probability language suchas “Based on your current strategy, there is a90% chance that you will achieve your invest-ment goal.” is very clear to 49% of clients andeither very clear or quite clear to 92% of clients,whereas individual investment-oriented language

Figure 1 Responses of advisors and investors to a survey on investor’s portfolio performance.



Figure 2 Responses of investors to a survey on their goals.

like “Your U.S. equity investment has been out-performing its benchmark index.” is only veryclear to 29% of clients and either very clear orquite clear to 71% of clients.

This is a key observation: clients are more com-fortable with discussing and thinking about theirgoals and the probability of attaining them, andadvisors who adopt this goal-based language willbe better able to communicate with their clientsand also be in a better position to consider howto address their clients’ needs. Therefore, in addi-tion to the previous four stated properties, GBWMshould have the following property:

Property 5: Client-centered Communication.Clients should be asked for information abouttheir goals in terms that are clear to them, such asinvestment time frames, desired dollar amounts atthe end of these time frames, and desired probabil-ities of attaining these dollar amounts. Both theinitial and future interactions between advisorsand clients benefit by a method enabling discus-sions that can, if desired, exclusively use clear,relatable, goals-based language.

Discussing goals with investors involves under-standing their Target Wealth and the desiredprobability of attaining that wealth, which speaksto the aspiration-related goals of the investor(see Lopes, 1987). But it also involves under-standing their Loss Threshold Wealth and the

desired probability of not ending up below thisLoss Threshold Wealth, which speaks to the fear-related goals of the investor (Shefrin and Statman,2000). The advisor plays a crucial role in workingwith the investor to determine these probabilities.One example for how an advisor can help trans-late the investor’s qualitative desires into specificprobability values is by using the goals-basedframework of Brunel (2015). This framework,represented in Table 1, associates the investor’sneeds, wants, wishes, or dreams of achieving theirTarget Goal with associated probabilities, as wellas associating the investor’s nightmares, fears,

Table 1 Brunel’s Goal-Probability table can be usedto classify investor goals into different probabil-ity values, which can then be used in our GBWMframework.

Realize Avoid Success probability (%)

Dreams Concerns 505560

Wishes Worries 657075

Wants Fears 8085

Needs Nightmares 9095



worries, or concerns about ending below their lossthreshold with probabilities.

Within the GBWM approach that we develophere, Prospect Theory (Kahneman and Tversky,1979) is modified. Prospect Theory specifies asingle “reference point” of return, and investorsbehave as if they have different risk preferenceswhen they are above the reference point versuswhen they are below it. For example, in the caseof Shefrin and Statman (1985), the dispositioneffect implies that people hold onto losers toolong and sell winners too early. In our imple-mentation of GBWM, instead of one referencepoint, we have two: (i) a Target Wealth on thespectrum of upside outcomes and (ii) a ThresholdWealth for losses on the downside. The frame-work in this paper will present an implementationapproach for achieving investor goals, taking intoaccount the probabilities for Target Goals to bemet and the risk of falling below acceptable lossthresholds.

Qualitative research including discussions withdozens of financial advisors,5 many of whomuse one or more of the four common approachesdescribed above, uncover a number of obser-vations from the advisors concerning currentgoal-based strategies, such as the following:

(1) Although clients go through an elaboratefinancial planning process by answeringdetailed goals-based questionnaires, portfo-lio construction generally reverts to one ofa finite number of standard asset allocationmodels, and the allocation model chosen isdetermined primarily by the time horizon forachieving the client’s goal.

(2) In evaluating a portfolio’s performance, stan-dard financial industry performance indica-tors are used, such as excess returns, alpha,tracking error, R-squared values, and infor-mation ratios, which are only relevant for

comparing individual investments with theirbenchmarks.

(3) Once the investment strategy is developed,one investment may be replaced by a similarinvestment within a sector, with the assump-tion that the new investment would maintainthe same interrelationships with the otherinvestments in the portfolio.

(4) Underfunded portfolios are often ignored,resonating with an often heard criticism in thewealth and asset management industry thatfinancial advisors are “doctors who treat onlyhealthy patients.”

(5) Critical goals are often over-funded, consum-ing too much of the investor’s principal.

(6) Periodically, portfolios are adjusted to bringthem back to a target asset allocation, regard-less of market conditions.

(7) While advisors understand the concept andintuitions of goals-based wealth manage-ment, they do not have a frameworkto analyze its implications for portfolioconstruction.

While a GBWM method with the above fiveproperties addresses some of these observations,it does not address all of them. Therefore, weincorporate the following GBWM properties:

Property 6: Goal/State Specificity. Advice dis-cussed with the investors should be based oninformation regarding the investor’s goals and theoverall state of their portfolio, not, in general, theportfolio’s individual investment components.

Property 7: Portfolio Efficiency. Investors shouldalways be advised to invest in a portfolio that ison the Efficient Frontier.

Property 8: State Dependency. The advice givento investors (i.e., the specific location on theEfficient Frontier) should be affected by mar-ket changes and easy to understand investorpreferences, both in bull and bear markets.



Property 9: Rules for Rebalancing. The investor’sportfolio should be able to be updated automat-ically at regular intervals (e.g., annually, so asto avoid short-term capital gains) and manually,whenever the investor wishes.

GBWM Properties 4, 5, and 6 help address thefirst observation of the advisors in the focus group.GBWM Property 6 also addresses the secondobservation, whereas Property 7 addresses prob-lems that can arise from the third observation, andProperties 8 and 9 help address the fourth, fifth,and sixth observations.

In the next section we detail a portfolio processthat satisfies all nine of these GBWM propertiesand also addresses the seventh noted observationof advisors, by providing specific portfolio advicefor clients from their answers to simple, goals-based information, following GBWM Property5. Specifically, we first ask clients to specify thefollowing basic information:

(1) Their time frame (Investment Horizon).(2) The size of their initial investment (Initial

Wealth).(3) Their goal wealth (Target Wealth).(4) The probability they would like to main-

tain of reaching their goal wealth (TargetProbability).

(5) The wealth they would not want to end below(Loss Threshold Wealth).

(6) The probability they would like to main-tain of ending above the loss threshold (LossThreshold Probability).

Following GBWM Property 7, we will initiallyrequire that there is a range of portfolios (or atleast one portfolio) on the Efficient Frontier thatmeets all of these investor specifications. If theportfolio does well, we will refer to the portfo-lio as being in a “good” state for the investor.Should the market not do so well, including evolv-ing to cases where no investment on the frontier

meets all of the investor’s specifications, we willrefer to the investor’s state as “bad.” The specificsof these definitions will be given in the nextsection.

Investors may have different priorities in goodstates and in bad states, so we will ask themfor two final pieces of information to understandthese priorities:

(7) Their investment preferences in good states,and

(8) Their investment preferences in bad states.

For good states, this means having the investorspecify if they would like to (i) pursue a strat-egy that optimizes the probability of meeting theirtarget wealth, (ii) pursue a strategy that, subjectto conforming to their target and loss thresholdspecifications, optimizes their average portfo-lio returns, or (iii) pursue a strategy somewherebetween these two strategies.

For bad states, the choices are different, with theinvestor specifying if they would like to (i) pur-sue a strategy that optimizes the probability ofmeeting their Target Wealth, (ii) pursue a strategythat optimizes the probability of maintaining theirLoss Threshold Wealth, or (iii) pursue a strategysomewhere between these two strategies.

We show how to determine a portfolio using ourGBWM approach in the next section by just usingthese eight pieces of information.

4 Implementation geometry

4.1 Overview

Because our view of risk is connected to theprobability of not achieving goals, instead ofjust the volatility, σ, of an investor’s portfolio,the approach we take in this section will differsignificantly from current common approachesto goals-based investing. Our approach will



clearly connect how the probability of attain-ing an investor’s goals corresponds to a specificinterval of risk–return combinations on the Effi-cient Frontier. As we progress upward throughthis interval on the Efficient Frontier, the meanand volatility of the portfolio increase. The prob-ability of the investor attaining her goals alsochanges, but not in a way that has been pre-viously well understood. We will show, forexample, that the traditional notion of gravitatingtoward the least volatile portfolio in this inter-val generally does not correspond to the safestchoice for the investor. Again, this is becausethe notion of “safest” is defined by the probabil-ity of attaining their goal, whereas traditionallyit has been defined without this goals-basedperspective.

In Subsection 4.2, we explore the three buildingblocks needed to determine this interval on theEfficient Frontier that meets the investor’s speci-fications. Subsection 4.3 shows how to put thesebuilding blocks together to determine this inter-val and how, based on the client’s investmentpreferences in good and bad states, to determinethe specific point on the frontier that correspondsto the advice given to the client for how toinvest. Finally, in Subsection 4.4 we explain someinsights and observations about our approach, andin Subsection 4.5 we discuss some aspects of howto apply our approach in practice.

4.2 Three building blocks

There are three primary building blocks in ourapproach: (1) The Goal Region and its associ-ated Goal Probability Level Curves, (2) The LossThreshold, and (3) The Efficient Frontier of avail-able investments. Geometrically, each of thesewill correspond to regions or curves in the planewhere the risk, σ (i.e., portfolio volatility6) is onthe horizontal axis and the return, µ (i.e., portfolioexpected return) is on the vertical axis.

4.2.1 Goal Probability Level Curves (GPLCs)

Recalling the information we have collected, eachinvestor has specified an Investment Tenure, t,by which time they want their specified InitialWealth, W(0), to grow at least to their speci-fied Target Wealth, W(t), with a specified TargetProbability. Under the model of geometric Brow-nian motion, we know that future wealth, W̃(t),is given by

W̃(t) = W(0)e

(µ− σ2

2

)t+σ

√tZ, (1)

where Z is a standard normal random variable.Rearranging Equation (1) and replacing W̃(t)

with the Target Wealth, W(t), yields the keyrelationship

µ = 1

2σ2 + z0√

tσ + 1

tln

(W(t)

W(0)

), (2)

where z0 is defined so that the Target Probabil-ity equals �(z0), with �(z) being the cumulativedistribution function (CDF) for a standard normalrandom variable. Note that Equation (2) definesan upward curving (i.e., convex) parabolic rela-tionship between the expected return µ and thevolatility σ.

Any portfolio lying on or above this parabola inthe (σ, µ) (i.e., risk–return) plane will satisfy theinvestor’s stated goals, therefore, the region onor above this parabola is called the Goal Region.The parabola itself is the Boundary of the GoalRegion. If we replace z0 with a generic value z inthe above equation, we call the resulting parabolaa Goal Probability Level Curve (GPLC) corre-sponding to the probability that a normal randomvariable will take a value less than z. For exam-ple, an 80% Goal Probability Level Curve will bethe parabola corresponding to the set of all (σ, µ)

pairs where there is exactly an 80% chance thatthe investors will meet or exceed their goal (seeFigure 3).



Figure 3 A Goal Region example. The Goal Region (in light blue) in this figure corresponds to a base casewhere an investor has an Investment Tenure of 10 years, an Initial Wealth of $400,000, a Target Wealth of$500,000, and a Target Probability of 80%. The boundary of the Goal Region is the 80% Goal Probability LevelCurve (GPLC), which is defined as the curve for which any (σ, µ) pair above the curve corresponds to a portfoliowith an 80% or higher chance of the investors attaining their Target Wealth at the end of their Investment Tenure.

We note that Equation (2) above only dependson W(0) and W(t) through their ratio. The GoalRegion in Figure 3 corresponds to an increaseover an Investment Tenure of 10 years from anInitial Wealth, W(0), of $400,000 to a TargetWealth, W(10), of $500,000. This correspondsto a continuous rate of return of 2.23% per year.We retain the same Goal Region in Figure 3 forany other W(0) and W(10) pair that correspondsto this same 2.23% rate of return, such as, say,W(0) = $200,000 and W(10) = $250,000, sincethe ratio, W(10)/W(0), stays the same.

We also note that if we increase z, which increasesthe percentage for the GPLC, the parabolic curve

moves upward in the (σ, µ) plane everywhereexcept where the curve meets the µ axis. On theµ axis, σ = 0, so from Equation (2) we have that

µ = 1

tln

(W(t)

W(0)

), (3)

regardless of the value of z. In other words the Ini-tial Wealth, the Goal Wealth, and the InvestmentTenure determine the location at which all theGPLCs meet the µ axis, while increasing the per-centage for the GPLC moves the curve upward,except at this fixed point on the µ axis. By varyingz, we generate a family of GPLCs, as we show inFigure 4.



Figure 4 The dependence of Goal Probability Level Curves on Probability Levels. The top Goal ProbabilityLevel Curve (GPLC) in this figure is the 99% GPLC. Below that are the 95%, 90%, 80% (in dark blue, denotingthe base case), 70%, 60%, and 50% GPLCs. The Investment Tenure, Initial Wealth, and Target Wealth parametersfor these GPLCs are the same as in the base case given in Figure 3.

We can look at the effect on the GPLC of vary-ing other parameters from their base case values,such as the Target Wealth (see Figure 5) or theInvestment Tenure (see Figure 6).

4.2.2 Loss Threshold Curves (LTCs)

In addition to their Target Wealth and Target Prob-ability, investors also specify a Loss ThresholdWealth along with a Loss Threshold Probabil-ity. Recall that the Loss Threshold speaks to theinvestor’s fears, whereas the Target speaks to theinvestor’s dreams. The intent of each investor isto stay at least at or above their Loss ThresholdWealth at the end of their Investment Tenure with

a probability equal to or higher than the LossThreshold Probability.

The Loss Threshold Wealth, of course, is a smallernumber than the Target Wealth. Indeed, it maybe below or equal to the Initial Wealth, as wellas above it. The Loss Threshold Probability isalways higher than the Target Probability, how-ever, as a rule of thumb, values exceeding 99%may be problematic, because tail events abovethis level are hard to model accurately.

Mathematically, the Loss Threshold Curve (LTC)is characterized in the same way as the GoalProbability Level Curve. That is, the LTC is



Figure 5 The dependence of Goal Probability Level Curves on Target Wealth. The top GPLC corresponds to aTarget Wealth of $650,000, instead of the base case value of $500,000. Below that are the GPLCs as the TargetWealth is reduced to $600,000, $550,000, $500,000 (in dark blue, again denoting the base case), $450,000,$400,000, and $350,000. We note that these GPLCs are all parallel curves. The GPLC curve for a Target Wealthof $400,000 goes through the origin, since the Initial Wealth is also $400,000. As in Figure 4, all parametersother than the Target Wealth have their base case values, as given in Figure 3.

described by Equation (2) with the Loss Thresh-old Wealth substituted for W(t), and z0 chosen sothat the Loss Threshold Probability is equal to theprobability that a normal random variable takes avalue less than z0. Satisfying the Loss Thresholdrestriction means using a portfolio whose stan-dard deviation and expected value remain abovethe Loss Threshold Curve. That is, we would liketo have a portfolio in the (σ, µ) plane that lies inthe intersection of the Goal Region and the regionon or above the Loss Threshold Curve.

4.2.3 Efficient Frontier

We consider a portfolio with access to n assets. Tominimize the portfolio risk, σ, for a given return,µ, we ideally choose a combination of thesen assets that put us on the Efficient Frontier,defined by the formula:

σ =√

aµ2 + bµ + c, (4)

which shows the relationship between σ andµ traces out a hyperbola (see Figure 7). The



Figure 6 The dependence of Goal Probability Level Curves on Investment Tenure. The top GPLC correspondsto an Investment Tenure of 2 years, instead of the base case value of 10 years. Below that are the GPLCs asthe Investment Tenure is increased to 4 years, 7 years, 10 years (in dark blue for the base case), 15 years, 20years, and 30 years. We note that the GPLCs decrease at a slower pace as the Investment Tenure increases. Asin Figure 4, all parameters other than the Investment Tenure have their base case values, as given in Figure 3.

constants, a, b, and c are defined by M, whichis a vector of the n expected returns; O, which isa vector of n ones; and �, which is the covariancematrix of the n assets, via the following equations:

a = h��h

b = 2g��h

c = g��g,

where the vectors g and h are defined by

g = l�−1O − k�−1Mlm − k2

h = m�−1M − k�−1Olm − k2

,

and the scalars k, l, and m are defined by

k = M��−1O

l = M��−1M

m = O��−1O.

Imposing additional restrictions like only hold-ing long positions usually has only small effectson the Efficient Frontier, see Das et al. (2010),although it may impose a maximum value beyondwhich the hyperbolic curve is cut off.

We will only consider portfolios that are on our(potentially restricted) Efficient Frontier. Ideally,



Figure 7 An Efficient Frontier example. The hyperbola in the risk–return plane corresponding to an EfficientFrontier. This frontier was based off of 10 years of data for three assets: a low-risk money market fund, amedium-risk stock market index, and a higher-risk tech stock index. The full set of feasible portfolios lies onthe Efficient Frontier or to the right of it, but only portfolios that lie on the upper half of the Efficient Frontier(e.g., on the part of the green curve where µ ≥ 0.01 in the figure) should be used. These are the portfolios whereexpected return is maximized for a fixed amount of portfolio risk.

they will also be in the Goal Region and on orabove the Loss Threshold Curve (see Figure 8).Therefore, we ensure that optimal asset allo-cation is combined with goals-based portfoliooptimization.

4.3 Building blocks for a goals-based strategy

We now combine the Goal Region, the LossThreshold, and the Efficient Frontier into a spe-cific goals-based strategy for investors that ismean–variance efficient.

4.3.1 Relevant points on the Efficient Frontier

Because Goal Probability Level Curves (GPLCs)are upward curving (convex) parabolas and theEfficient Frontier curves downward (i.e., is con-cave), the value of z can be adjusted to determinethe unique GPLC that intersects the upper halfof the Efficient Frontier (i.e., the half of thefrontier where µ increases as σ increases) atexactly one location. We will call this unique(σ, µ) pair the Optimal Goal Probability Point.When the Optimal Goal Probability Point lies in



Figure 8 Feasible portfolios that satisfy the investor’s Target Goals and Loss Threshold. The golden regioncorresponds to portfolios that are feasible (since they are to the right of the Efficient Frontier in green), while alsosatisfying both the investor’s goals (by staying above the boundary of the Goal Region in dark blue) and theirloss threshold tolerance (by staying above the Loss Threshold Curve, shown in red). The red Loss ThresholdCurve (LTC) in this figure corresponds to a 95% chance that the initial portfolio wealth of $400,000 will beworth at least $300,000 at the end of the Investment Tenure of 10 years. The Efficient Frontier is the frontiergiven in Figure 7. The Goal Region is the base case given in Figure 3.

the interior of the Efficient Frontier (as opposedto an endpoint of the frontier, which may existif we prohibit short selling, for example), itcorresponds to the point of tangency betweenthe unique Goal Probability Level Curve andthe Efficient Frontier (see Figure 9). We showhow to use the method of Lagrange multipliersto explicitly compute this point of tangency inSubsection 4.3.4.

In general the boundary of the Goal Region will,at least initially, intersect the Efficient Frontierin two locations, which we will call the LowerGoal Point and the Upper Goal Point. Should

the boundary of the Goal Region dip so low thatit goes below the left endpoint of the upper half ofthe Efficient Frontier, we set the Lower Goal Pointto equal this left endpoint. A similar assignmentis made for the Upper Goal Point if the bound-ary dips below the right endpoint of a restrictedEfficient Frontier. If the Goal Region does notintersect the Efficient Frontier, the Lower andUpper Goal Points are not defined. When theyare defined, the Optimal Goal Probability Pointwill always lie between them or on them.

Next, we define the Loss Threshold Point to bethe point of intersection of the Loss Threshold



Figure 9 Optimizing the Probability of Investors Meeting their goal. The base case from Figure 3 correspondsto an 80% GPLC. We increase this percentage, as was done in Figure 4, until the GPLC intersects the EfficientFrontier at only a single tangent point. This single point is the Optimal Goal Probability Point, which correspondsto the highest percentage GPLC that is feasible for the investor. In this figure the Optimal Goal Probability Pointis located at (0.158, 0.0902). This is the black point on the graph, which is on the 86.6% GPLC.

Curve parabola with the upper half of the EfficientFrontier. In the less common case where the LTCintersects the upper half of the Efficient Frontierin two locations, the Loss Threshold Point is therightmost intersection point. As time evolves, theLTC may lie completely above the Efficient Fron-tier, in which case, the value of z0 in Equation (2)for the LTC is reduced until the parabola againintersects the Efficient Frontier at a unique loca-tion, which is then defined as the Loss ThresholdPoint.

Investors must specify inputs that are at leastinitially feasible, which means the LTC mustintersect the upper half of the Efficient Frontierand the Loss Threshold Point must be to the rightof the Lower Goal Point. This means that we havea portfolio that lies on the Efficient Frontier that

is in the Goal Region and on or above the LTC.This corresponds to a portfolio that lies on thepart of the Efficient Frontier that forms the upperboundary of the golden region in Figure 8.

Finally, we define the Halfway Point betweentwo points on the Efficient Frontier to be the pointon the frontier whose volatility, σ, is equal to theaverage of the two points’ volatilities.

4.3.2 Goals-based investing in good states

Good states are defined by the existence of thegolden region in Figure 8 and having the LossThreshold Point be to the right of the Opti-mal Goal Probability Point, where “to the right”means having higher µ and σ values on the upperhalf of the Efficient Frontier. Recall that each



investor had been asked for their investment pref-erence in good states by selecting one of the threeoptions: Option 1, optimizing the probability ofobtaining their goal; Option 2, optimizing theirexpected return subject to their other specifica-tions, like maintaining their Target Probability;or Option 3, a mix between these two options.

In the best case scenario, the Loss Threshold Pointis to the right of the Upper Goal Point. In this caseOption 1 corresponds to selecting the portfoliothat corresponds to the Optimal Goal Probabil-ity Point, Option 2 corresponds to selecting theUpper Goal Point, and Option 3, if a half andhalf mix is selected, corresponds to selecting theHalfway Point between Option 1 and Option 2(see Figure 10).

In the next best case scenario, the Loss ThresholdPoint lies between the Optimal Goal ProbabilityPoint and the Upper Goal Point. In this case, foreach of the three options, we use whichever ismore to the left: the same portfolio selected in thebest case scenario or the Loss Threshold Point.This enforces maintaining the Loss Threshold(see Figure 11).

4.3.3 Goals-based investing in bad states

When the Loss Threshold Point is to the left of theOptimal Goal Probability Point and/or the goldenregion in Figure 8 disappears, we are in a badstate. In bad states, the Loss Threshold Point maystill be to the right of the Optimal Goal ProbabilityPoint. This will only happen if the Goal Region

Figure 10 Options in the best case scenario. Option 1 maximizes the investor’s probability of achieving theirgoal. Option 2 maximizes expected return subject to maintaining the investor’s other specifications, like theirTarget Probability. Option 3 corresponds to a point on the frontier between the first two options. Option 1, theOptimal Goal Probability Point, (0.158, 0.0902), is in black. Option 2, the upper goal point, (0.424, 0.225), isin red. An example of Option 3 is the halfway point, (0.291, 0.158), which is in magenta, where the risk, σ, isthe average of the σ values for Option 1 and for Option 2.



Figure 11 Options in the next best case scenario. To conform to the investor’s loss threshold, the optionsfrom Figure 10 are restricted to remain on or above the red Loss Threshold Curve. Option 1, the Optimal GoalProbability Point, is still the black point at (0.158, 0.0902). Option 2 and Option 3, however, are restricted bythe LTC, so both now correspond to the loss threshold point, (0.249, 0.136), which is in magenta.

lies strictly above the Efficient Frontier, so thegolden region no longer exists. In this case, theportfolio at the Optimal Goal Probability Point isselected.

When the Loss Threshold Point is to the left of theOptimal Goal Probability Point, we use the invest-ment preference specified by each investor for badstates. Recall, this is one of the three options:Option 1, optimizing the probability of obtainingtheir goal; Option 2, optimizing maintaining theirLoss Threshold; or Option 3, a mix between thesetwo options.

Option 1 corresponds to selecting the portfoliothat corresponds to the Optimal Goal ProbabilityPoint, Option 2 corresponds to selecting the LossThreshold Point, and Option 3, should a half andhalf mix be selected, corresponds to selecting the

Halfway Point between Option 1 and Option 2.Note that for cases where the golden region inFigure 8 still exists, all the three options selecta portfolio within the golden region. These threeoptions are also well defined in cases where theLoss Threshold and/or the Goal Region does notintersect the Efficient Frontier. By using the Opti-mal Goal Probability Point, we maximize theinvestor’s likelihood of attaining their goal. Byusing the Loss Threshold point, we maximize thelikelihood of not violating the investor’s specifiedtolerance for underperformance (see Figure 12).

4.3.4 Computing the Optimal GoalProbability Point

Recall that to obtain the Optimal Goal ProbabilityPoint, we increase the value of z in Equation (2)until the GPLC that corresponds to this value of z



Figure 12 Options in bad states. The Loss Threshold Point is now to the left of the Optimal Goal ProbabilityPoint and/or the golden region in Figure 8 no longer exists. In this figure, both are the case. The investor hasselected one of the three options: Option 1, the Optimal Goal Probability Point, (0.158, 0.0902), which is inblack, corresponds to prioritizing attaining the investor’s Target Wealth at the expense of ignoring their LossThreshold. Note that the boundary of the Goal Region in this figure is above the Efficient Frontier, which meansthat the probability of attaining the Target Wealth, while optimized by Option 1, will still be below the TargetProbability. Option 2, the Loss Threshold Point, (0.0791, 0.0500), which is in red, corresponds to prioritizingthe investor’s Loss Threshold at the expense of ignoring the investor’s Target Wealth. In this case, the investor’sLoss Threshold Probability can be preserved, because the LTC still intersects the Efficient Frontier. Should theLTC lie above the Efficient Frontier and therefore not intersect it, we can still minimize the probability that theinvestor will end up below the Loss Threshold Wealth, although this minimized probability will now be belowthe Loss Threshold Probability. As before, Option 3 corresponds to a point on the frontier between the first twooptions. Assuming an equal mix of the other two options, we get the halfway point, (0.119, 0.0704), in magenta,where the risk, σ, is the average of the σ values for Option 1 and for Option 2.

intersects the upper half of the Efficient Frontier inonly a single location. This location is the OptimalGoal Probability Point.

To calculate the Optimal Goal Probability Point,we first rearrange Equation (2) to isolate z:

z(σ, µ) = 1

σ

((µ − σ2

2

) √t − 1√

tln

(W(t)

W(0)

)).

(5)

We look to optimize z subject to the restrictionthat we remain on the Efficient Frontier given inEquation (4), which is easily rearranged into theform

g(σ, µ) = aµ2 + bµ + c − σ2 = 0. (6)

This means employing the method of Lagrangemultipliers, namely, simultaneously solving

∇z(σ, µ) = λ∇g(σ, µ) (7)



for some scalar Lagrange multiplier, λ, along withthe Efficient Frontier restriction

g(σ, µ) = 0.

From Equation (7), using Equations (5) and (6),we have from differentiating with respect to σ that

− 1

σ2

(µ

√t − 1√

tln

(W(t)

W(0)

))−

√t

2= −2λσ

and from differentiating with respect to µ that√t

σ= λ(2aµ + b).

Combining these two equations so as to removeλ yields, after rearrangement,

2√

tσ2 =(

µ√

t − 1√t

ln

(W(t)

W(0)

)+

√t

2σ2

)

× (2aµ + b).

Now we use the Efficient Frontier restriction tosubstitute aµ2 + bµ + c for σ2, and, after rear-rangement, we obtain a third-degree polynomialequation, i.e., a cubic equation, for the value of µ:

c3µ3 + c2µ

2 + c1µ + c0 = 0, (8)

where

c3 = a2

c2 = 3ab

2

c1 = ac + b2

2− b − 2a

tln

(W(t)

W(0)

)

c0 = bc

2− 2c − b

tln

(W(t)

W(0)

).

In general, cubic Equations (8) have either onereal (and two complex) roots or three real roots.From the geometry of Equation (7), each of thereal roots in Equation (8) must correspond to loca-tions where a GPLC and the Efficient Frontier areat a tangent to each other. Since GPLCs are con-vex parabolas, when there is only one real root,

it must correspond to a point on the upper halfof the Efficient Frontier. Note that the upper halfof the Efficient Frontier corresponds to the regionwhere µ ≥ − b

2a, since the Efficient Frontier from

Equation (4) can be rewritten in the form

σ =√

a

(µ + b

2a

)2

− b2

4a+ c.

When there are three real roots, there must still beone root corresponding to a point in the upper halfof the frontier. The other two roots (which mayboth form a double root) must always correspondto points (or a point in the case of a double root)on the lower half of the Efficient Frontier (seeFigure 13).

Therefore, there will always be exactly one rootof our cubic equation where µ ≥ − b

2a. This root,

which may be found exactly using Cardano’s for-mula for cubic equations or approximated usinga numerical solver, must be the µ coordinate ofthe Optimal Goal Probability Point, and the cor-responding σ coordinate is then determined fromEquation (4): σ = √

aµ2 + bµ + c.

4.4 Observations concerning our goals-basedinvesting approach

The main new perspective and contribution inour algorithm is that it quantifies the probabil-ity of attaining an investor’s goal—be it growingthe worth of a portfolio above a Target Wealthor staying above a Loss Threshold—and clarifiesthe role these probabilities can play in a GBWMportfolio. We can not only generate the rangeof portfolios that satisfy the investor’s specifiedprobabilities for ending above their Target Wealthand for ending above their Loss Threshold Wealth(as in Figure 14), but also, we can generate thespecific portfolio that, in addition, best satisfiesthe investor’s preferences under both good statesand bad states.



Figure 13 Three points corresponding to three real roots of the cubic Equation (8) for µ. The three curves inlight blue are the only GPLCs that share a tangent line with the Efficient Frontier at their point of intersectionwith the frontier. The highest tangential intersection point, (0.158, 0.0902), in black, is the only tangentialintersection point on the upper half of the Efficient Frontier, meaning the half of the frontier where µ increasesas σ increases. There will always be exactly one tangential intersection point on the upper half of the EfficientFrontier, and this point is the Optimal Goal Probability Point. This point is on the GPLC that intersects theEfficient Frontier with the highest probability level. The other two tangential intersection points—here (0.0108,0.00829), in red, and (0.153, −0.0669), in magenta—either both exist, possibly as a double root, or neitherexists. When they both exist, they will always lie on the lower, irrelevant half of the Efficient Frontier, like wesee here.

Our analysis allows us to determine the answersto some questions that may not be intuitivelyclear. For example, it is intuitive to advisorsthat investors choosing a portfolio with too lit-tle risk and return will have a small probability ofattaining their financial goals. It is also clear thatincreasing an investor’s risk and return increasesthe potential for larger losses when losses occur.

But it is less clear whether or not increasing riskand return increases or decreases the actual prob-ability of the investors attaining their goals. Ouranalysis clearly answers this question: increasingan investor’s risk and return may initially increasethe probability of attaining the investor’s goal,but further increases will eventually diminish thisprobability to zero. And, of course, our analysis



Figure 14 Ranges of portfolios on the Efficient Frontier that satisfy different combinations of Target Goals andLoss Thresholds. The ranges on the Efficient Frontier here correspond to the upper boundary of the golden regionshown in Figure 8 under three different circumstances. Note that we specify rates of return, r, in this figure inplace of the Initial Wealth, W(0), and the Target Wealth/Loss Threshold Wealth, W(t), where r = 1

tln

(W(t)W(0)

).

goes further by finding the specific risk/returncombination on the Efficient Frontier that opti-mizes the probability of the investors attainingtheir Target Wealth, as well as the interval onthe frontier between the Lower and Upper GoalPoints that will attain the Target Wealth withat least the investor’s specified Target WealthProbability.

Note also that the Lower Goal Point is not rel-evant to any advice given to the investor. TheUpper Goal Point is important because it tells ushow much the investor is willing to increase therisk of not attaining their goal so that they canincrease their expected gains, but when the LossThreshold is not relevant, the algorithm will notchoose a portfolio between the Lower Goal Pointand the Optimal Goal Probability Point. This isbecause switching from the Optimal Goal Prob-ability Point to a portfolio between the LowerGoal Point and the Optimal Goal Probability Pointincreases, as opposed to decreases, the risk of the

investors not attaining their Target Wealth goal,and, at the same time, it decreases the investorexpected gains.

It is well established that most investors who donot consult financial advisors will sell in a bearmarket and buy in a bull market, despite the factthat they are “buying high and selling low.” Auto-matic rebalancing, including using Life Cyclefunds, can help ameliorate this by automaticallydoing some buying in bear markets and sell-ing in bull markets as it rebalances to its fixedpercentages of less risky and more risky assetclasses.

Does our approach automatically “buy low andsell high” as any investment strategy should? Theanswer, generally, is yes, and to a more signifi-cant degree than automatic rebalancing and LifeCycle funds do. For example, consider the case ofa bear market where the poor market conditionslead to a significant decrease in the portfolio’sworth. This corresponds to W(0) being reduced,



which increases the µ-intercept given in Equa-tion (3) for both GPLCs and LTCs and, in fact,creates a corresponding upward translation of theentire GPLC or LTC by this amount, as shownin Figure 5. If, for bad states, the investor hasselected Option 1, this will induce a shift to higherrisk–return portfolios on the Efficient Frontier.That is, the investor will be buying more riskystocks in a bear market, as they should. We notethat this is better than just staying at the same pointon the risk–return plane, as automatic rebalancingor Life Cycle funds do.

On the other hand, if the investor has selectedOption 2 for bad states, this will correspond tomoving to lower risk–return portfolios as longas the LTC continues to intersect the EfficientFrontier. But should performance continue toworsen to the point that the LTC lies strictlyabove the Efficient Frontier, the selected portfo-lio will then change course and begin to move tohigher risk–return locations. This makes sense,given the investor’s investment preference: in aslight downmarket, the investor can protect her-self from loss by moving away from risk, but in asignificant downmarket, the investor will switchto purchase more risky, but now even more inex-pensive, assets to optimize their probability of notending below their loss threshold.

There is extensive discussion of GBWM in thepractitioner literature, see for example Chabbra(2005). Our paper complements this by provid-ing a systematic mathematical exposition of theapproach. Next, we comment on some practicalconsiderations.

4.5 Running our approach in practice

Working with the Efficient Frontier has advan-tages from a practical standpoint, other than themain property that it minimizes portfolio volatil-ity for a given expected return. Some advisorsstrongly prefer to work with a small number of

funds, others with a large number. Using theEfficient Frontier can accommodate any numberof funds. Some companies use historical data todetermine the mean, standard deviation, and cor-relations between funds. Others use projectionsbased on current data or a combination of histor-ical data and future projections. All of these canbe accommodated. This is a strong point of theGBWM framework, i.e., that it is done in twostages. First, we pick a set of efficient portfo-lios that may be acceptable to the investors interms of their portfolio weights and their distri-butional properties, and any other constraints thatthe investors may require. Second, from this set,we then choose the portfolio that best meets theinvestor’s goals. Operationally also, this has theadvantage that the two steps may be performedby different teams in a wealth management firm,both applying their specific expertise.

Also, our model is flexible in two ways. First,although we have used geometric Brownianmotion to model returns in this paper, there isno problem with adapting our algorithm to a pre-ferred alternative model for returns. The onlyalteration to the algorithm that is needed is inthe computation of the Optimal Goal Probabil-ity Point, which would generally need to belocated numerically, instead of through the useof Lagrange multipliers. Second, we may over-lay options positions on the portfolio as well, asthese may be appropriate securities with whichto meet the Goal Probability and Loss Thresholdconstraints.7

Although the discussion of our algorithm has cen-tered on its ability to determine the probability ofattaining a Target Wealth or maintaining a LossThreshold Wealth, it is important to note that ouralgorithm can also determine a complete prob-ability distribution for the investor’s returns atany future time. That is, along with a recom-mended portfolio for the investor, a probability



distribution for the wealth of the portfolio couldeasily be generated for, say, every year until theend of the Investment Tenure.

Once the eight pieces of information listed at theend of Section 3 are gathered from the investor,our algorithm can be run once and the portfoliocan be maintained with what our algorithm rec-ommends, but we strongly suggest instead that thealgorithm should be rerun periodically as the con-ditions in the market, and therefore in the EfficientFrontier, change. This can be done automati-cally at regular intervals for investors who prefera hands-off approach. In addition, for hands-oninvestors, it can be done manually whenever theinvestor desires, including when the investorswant to change their goals or consider the effectof additional investments or withdrawals. The keything to note is that even though our algorithm isstatic, it can be updated dynamically, either toadapt to new market data or a change in the eightpieces of information the investor supplies to thealgorithm.

5 Conclusion

This paper presents a new approach to goals-based wealth management (GBWM) that focuseson understanding risk as the probability of notattaining the investor’s goals, not just the tra-ditional perspective of risk being the standarddeviation of the investor’s portfolio. Our GBWMapproach is a framework that recognizes investorgoals and offers a mathematical foundation foranalyzing portfolios, while remaining fully con-sistent with modern portfolio theory. It alsosatisfies nine desirable GBWM Properties listedin Section 3 that are generally not fully satisfiedin extant portfolio strategies.

Our approach has significant advantages for bothadvisors and their clients. It enables advisors togive individualized advice (GBWM Property 2)after ascertaining the goals of the investor in

a manner that is clearer (GBWM Property 1),because it uses language and ideas that marketresearch has shown are more natural and under-standable to investors (GBWM Property 5) thanthe language that advisors typically currently use.More specifically, the advisor works with theclient to determine eight pieces of information:the size of their initial investment, their timeframe, their goal wealth at the end of this timeframe, and the acceptable probability of attain-ing this goal wealth, as well as a Loss ThresholdWealth and an acceptable probability of not goingbelow this Loss Threshold Wealth.

For the final two pieces of information, theinvestor is asked to choose (1) in good stateswhere the portfolio is well funded, how muchthe investor prioritizes decreasing the risk of notattaining their goal wealth versus increasing theirexpected returns, and (2) in bad states wherethe portfolio is less well funded or has becomeunderfunded, how much the investor prioritizesdecreasing the risk of not attaining their goalwealth versus decreasing the risk of ending belowtheir loss threshold. For good states, since risk isdefined in terms of not attaining their goal insteadof the portfolio standard deviation, the mean-ing of this prioritization decision is more clearfor the investor. For bad states, the investor isbeing asked to prioritize between her dream ofattaining her goal wealth and her fear of end-ing below their Loss Threshold Wealth, which,again, involves more intuitive notions for theinvestor, and makes for interactions of greaterclarity between advisors and their clients.

Using just these eight items of goals-based infor-mation from the investor, this paper detailsan algorithm that determines a portfolio thatbest fits the investor’s individual specifications.This portfolio is formed by determining andanalyzing the parabolic geometry in the risk–return plane that corresponds to any specific



probability for attaining a goal, and then consid-ering the intersection of these parabolas with thehyperbolic geometry that corresponds to the Effi-cient Frontier. This produces a portfolio on theEfficient Frontier (GBWM Property 7), therebyminimizing unnecessary standard deviation risk,while also being able to calculate and mini-mize the risk of not attaining the investor’s goals(GBWM Properties 3 and 4). The algorithm’sportfolio recommendation is based on the overallstate of the portfolio, as opposed to the perfor-mance of the individual portfolio components(GBWM Property 6). Furthermore, the portfo-lio can be recalculated and rebalanced eitherautomatically or manually at any time (GBWMProperty 9) to adjust to changes in the marketor changes the investors wish to make to theeight pieces of information they have specified(GBWM Property 8).

Employing our approach allows advisors to helptheir clients in a number of ways that alternativeslike Life Cycle funds, which exhibit almost noneof these nine GBWM properties, are not designedto do. Furthermore, because the required input isso simple and the algorithm is designed to work inboth good and bad states, it can be used effectivelyboth for clients that are wealthy and clients thatare not wealthy.

We mention four future directions for improvingand expanding the GBWM approach explained inthis paper:

• Multiple goals: In this paper, we have restrictedour analysis to a single investment with onedream-oriented goal and one fear-oriented goalin the same time frame. In reality, of course,investors look at multiple time frames thatcorrespond to different goals like buying ahouse, sending children to college, and fundingretirement.

• Periodic investments and withdrawals: In thispaper, we have restricted our analysis to a

single initial investment and a single time framefor withdrawal. Ideally, we also want to modelthe effect of periodic investments, say, everyyear or with every paycheck, as well as peri-odic withdrawals to model common events likeincome withdrawal throughout retirement.

• Dynamic probability determination: In thispaper, we have a static model that can bedynamically updated, but it would be betterto have a dynamic model that would allowus to determine, or at least approximate viasimulation, the effects on forecasted proba-bilities that future portfolio reconfigurationswould induce. Under such dynamic models, itmay also be possible to employ dynamic pro-gramming approaches to help determine newoptimal portfolio strategies. This is the focusof a follow-up paper.

• Dynamic asset opportunity space: In this paper,we have assumed that the investable universeis stationary, i.e., the parameters of the pro-cesses driving the portfolio assets remain thesame throughout the life of the goals-basedportfolio. Generalizing to non-stationary assetprocesses can be accomplished in many ways,such as using multivariate GARCH models tomodel a changing return covariance matrix,coupled with a changing mean return vec-tor based on a stochastic equity premium. Adynamic asset space may also be modeled witha regime-switching model.

In conclusion, we have developed a new geome-try for goals-based wealth management that hasseveral new, appealing properties, and aims tominimize the risk of investors failing to meettheir goals, while remaining fully consistent withmean–variance portfolio theory. We believe thatthe new approach presented here will free advi-sors to be able to respond to the goals expressedby their clients in a more concrete, direct way thanhas been available previously, improving both therelationship between advisors and their clients



and the quality of the recommendations advisorsprovide to their clients.

Acknowledgments

Sanjiv R. Das is the William and Janice Terry Pro-fessor of Finance and Data Science in the LeaveySchool of Business at Santa Clara University.

Dan Ostrov is a Professor of Mathematics andComputer Science at Santa Clara University.

Anand Radhakrishnan is a Strategic BusinessConsultant within the Client Strategies and Ana-lytics group at Franklin Templeton Investments.

Deep Srivastav is a Senior Vice President andheads the Client Strategies and Analytics groupat Franklin Templeton Investments.

We wish to thank Jean Brunel, Arun Muralid-har, Smarpit Menon and Abhitej Obul Reddy forcomments on our article.

Notes1 The Sveriges Riksbank Prize in Economic Sciences in

Memory of Alfred Nobel 2017. https://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/2017/ad-vanced-economicsciences2017.pdf.

2 See the following excellent books that cover many ofthese biases: (i) Shefrin (2007, 2016) and (ii) Statman(2011). These ideas are embodied in a concept paper byChabbra (2005).

3 Advisors are mostly focused on long-term goals such as“not running out of money” or “keeping one’s home.”To this end, retirement is clearly the most encompass-ing goal, followed by retirement income planning. And,as a result, advisors say that their solutions are veryfocused on tax strategies (taxable versus tax-free versustax-deferred assets) and time horizons.

4 Franklin Templeton partnered with Hall and Partners toconduct this survey of 300 advisors and 503 investors inMay 2017.

5 Franklin Templeton partnered with aQity Research &Insights, Inc. to conduct qualitative research fromSeptember–December 2017. Independent qualitativeresearch was also conducted over the same period.

6 We will use the term “risk” to mean portfolio volatil-ity in this section, because this has been the historicaldefinition, although our approach, of course, continuesto focus on thinking about risk as the chance that theinvestors will not meet their goals, as well as the portfoliovolatility.

7 We note, however, that put and call options, as wellas futures contracts and most other derivatives, areprohibited in 401(k) plans. They are also prohibitedby many IRA custodians. The sole exception to thisis covered calls, which may be purchasable througha 401(k) brokerage link or through an IRA. (Seethe Investopedia articles http://tinyurl.com/y797jqpqand http://tinyurl.com/ycdust9u.) In accounts that areintended for the long-term goal of retirement whereoptions are available, the fact that options usually haveshort maturities creates rollover risk, as discussed inChabbra (2005).

References

Brunel, J. L. P. (2015). Goals-Based Wealth Management:An Integrated and Practical Approach to Changing theStructure of Wealth Advisory Practices, Wiley & Sons,New Jersey.

Chabbra, A. (2005). “Beyond Markowitz: A Compre-hensive Wealth Allocation Framework for IndividualInvestors,” Journal of Wealth Management, Spring,8–34.

Das, S. R., Markowitz, H., Scheid, J., and Statman, M.(2010). “Portfolio Optimization with Mental Accounts,”Journal of Financial and Quantitative Analysis 45(2),311–334.

Kahneman, D. and Tversky, A. (1979). “Prospect Theory:An Analysis of Decision Under Risk,” Econometrica 47,263–291.

Lopes, L. (1987). “Between Hope and Fear: The Psy-chology of Risk,” Advances in Experimental SocialPsychology 20, 255–295.

Markowitz, H. (1952). “Portfolio Selection,” Journal ofFinance 6, 77–91.

Nevins, D. (2004). “Goals-Based Investing: Integrat-ing Traditional Behavioral Finance,” Journal of WealthManagement 6, 23–48.

PwC Consulting Research Paper (2017). Asset &Wealth Management Revolution: Embracing ExponentialChange.

Roy, A. D. (1952). “Safety-First and the Holding ofAssets,”Econometrica 20, 431–449.



Shefrin, H. (2007). Beyond Greed and Fear: Understand-ing Behavioral Finance and the Psychology of Investing,Oxford University Press.

Shefrin, H. (2016). Behavioral Risk Management: Manag-ing the Psychology That Drives Decisions and InfluencesOperational Risk, Palgrave Macmillan.

Shefrin, H. M. and Statman, M. (1985). “The Disposition toSell Winners to Early and Ride Losers Too Long: Theoryand Evidence, Journal of Finance 40, 777–790.

Shefrin, H. M. and Statman. M. (2000). “Behavioral Port-folio Theory,” Journal of Financial and QuantitativeAnalysis 35(2), 127–151.

Statman, M. (2011). What Investors Really Want: KnowWhat Drives Investor Behavior and Make Smarter Finan-cial Decisions, McGraw-Hill.

Thaler, R. H. (1985). “Mental Accounting and ConsumerChoice,” Marketing Science 4, 199–214.

Thaler, R. H. (1999). “Mental Accounting Matters,” Jour-nal of Behavioral Decision Making 12, 183–206.

Zwecher, M. (2010). Retirement Portfolios: Theory, Con-struction, and Management, Wiley & Sons, New Jersey.

Keywords: Wealth management; goals-basedinvesting; goal probability; loss threshold; effi-cient frontier


j o i joim · for how to invest, both when the investor’s port-folio is sufﬁciently funded or...

Documents