What is a Sustainable Spending Rate? A Simple Answer (That Doesn t Require Simulation)

Transcription

1 Wha is a Susainable Spending Rae? A Simple Answer (Tha Doesn Require Simulaion) By: Moshe A. Milevsky, Ph.D. 1 Finance Professor, York Universiy Execuive Direcor, The IFID Cenre Torono, Canada Absrac A number of financial commenaors have emphasized he need for more research on susainable spending raes from diversified porfolios see, for example, Arno (2004) -- a opic which is of relevance o individual reirees as well as mos large foundaions and endowmens. Moivaed by his apparen gap in he lieraure, I provide a simple analyic formula for he probabiliy ha a porfolio earning a lognormal invesmen reurn subjec o a consan (aferinflaion) wihdrawal rae will be susainable over a random lifeime horizon. The formula parsimoniously meshes asse allocaion parameers, moraliy esimaes and spending raes wihou resoring o opaque and ofen irreproducible Mone Carlo simulaions. I demonsrae how he biological aging process can be mapped ino he mean and variance language of invesmen heory. In his framework, increasing he force of moraliy is equivalen o reducing porfolio variance and increasing porfolio reurns. Among he pracical insighs emanaing from his approach, I confirm ha a 65-year old reire wih a sochasic lifeime horizon faces a 10% chance of ruin if he/she consumes more han $4-per-$100 principal of an equiy-based porfolio wih an expeced real reurn of 7% and volailiy of 20%. The insighs obained from his aricle will hopefully inspire financial analyss o provide invesmen guidance o he oncoming wave of reired baby boomers on more han jus asse allocaion maers. 1 Dr. Milevsky can be reached via a milevsky@yorku.ca or via Tel: (416) x 3010 or Fax: (416) Hardcopy address: The IFID Cenre a he Fields Insiue, 222 College Sree, 2 nd Floor, Torono, Onario, M5T 3J1, Canada. The auhor would like o hank Jin Wang and Anna Abaimova for research assisance. Page 1 of 32

2 Reirees Don' Have o Be So Frugal: Here is a Case for Wihdrawing up o 6% a Year Jonahan Clemens, Wall Sree Journal, Page C1, November 17, 2004 Forge he radiional approach and insead plan on wihdrawing a fixed 5% of your porfolio's beginning-of-he-year value. Call i he ake-five sraegy Jonahan Clemens, Wall Sree Journal, Page C1, Ocober 12, 2003 Secion #1: MOTIVATION Reirees like endowmen and foundaion rusees -- share a similar dilemma. In addiion o he classical asse allocaion decision hey mus se an appropriae spending rae from heir invesmen fund ha will las forever (for endowmens) - or a leas during heir random fuure lifeime (for reirees.) Susainable wihdrawal and spending raes have been he focus of some academic research over he years. Bu, he opic has developed a renewed sense of urgency as a wave of Norh American baby boomers approaches reiremen and seeks wealh managemen guidance on wha s nex for heir IRA or 401(k) plan. Ye, for endowmens and foundaions, his opic has a 30-year hisory going back o a special session a he American Economics Associaion devoed o spending raes in which Tobin (1974) cauioned agains consuming anyhing oher han dividends and ineres income 2, which in oday s environmen doesn amoun o much. Around he same ime -- wihin he conex of a privae foundaion or endowmen -- Ennis and Williamson (1976) were he firs o joinly analyze an appropriae asse allocaion in conjuncion wih a given spending policy. More recenly, Alschuler (2002) has argued ha endowmens are acually oo singy and no spending enough, while Dybvig (1999) has 2 According o daa compiled by he Naional Associaion of College and Universiy Business Officers (NACUBO) endowmen survey, he median spending rae in 2003 was 5% of asses, wih he 10 h percenile being 4.0% and he 90 h percenile being 6.4% of asses. Page 2 of 32

3 discussed how asse allocaion can be used o proec a desired level of spending using a pseudo porfolio insurance scheme. In he parallel reiremen planning arena, many auhors such as Bengen (1994), Ho, Milevsky and Robinson (1994), Cooley, Hubbard and Walz (1998, 2003), as well as Pye (2000) and Ameriks, Veres and Warshawsky (2003) have run exensive compuer simulaions moivaed by he game of life simulaions envisioned by Markowiz (1991) -- o locae pruden spending raes. These resuls usually advocae wihdrawals in he range of 4% o 6% of iniial capial depending on age and asse allocaion. The problem wih hese and similar Mone Carlo based sudies is ha hey (i) are exremely difficul o replicae, (ii) are quie ime consuming o generae if done properly using he required number of simulaion, and (iii) provide very lile financial or pedagogical inuiion on he radeoff beween risk and reurn 3. Along hese lines, a recen aricle in he Financial Analyss Journal by Arno (2004) claims ha our indusry pays scan aenion o he concep of susainable spending which is key o effecive sraegic planning for corporae pensions, public pensions, foundaions and endowmens even for individuals Therefore, parially driven by he call for more research in his area, in his paper I address he issue from a differen and wha I believe is a novel -- perspecive. My pedagogical objecive is o shed ligh on he financial inuiion of spending raes by deriving a simple analyic relaionship beween spending and susainabiliy in a random environmen. Namely, I inroduce he analyic concep of a sochasic presen value (SPV) and use his o provide an expression for he probabiliy ha an iniial corpus (nes egg) will be depleed under a fixed consumpion rule, when boh raes of reurn and horizons are sochasic. I sress he dual uncerainy for reurns and horizons, which is somehing ha has no received much aenion in he porfolio managemen lieraure, as i perains o reirees. 3 I have run some case-sudy examples using he 6, or so, free web-based simulaors and have found a wide variaion beween he suggesed nes eggs needed o suppor a comforable reiremen. A similar heme which was misinerpreed as a criicism of he Mone Carlo mehod -- was echoed in a recen Bloomberg Wealh Manager aricle by Ed McCarhy (Dec2002/Jan2003), page Page 3 of 32

4 And, in conras o almos all oher papers and auhors ha have ackled his problem, I do no resor o any forward-looking Mone Carlo Simulaions o locae pruden spending raes. Raher, I base he analysis on he above-menioned SPV and a coninuous-ime approximaion under lognormal reurns and exponenial lifeimes. In he case of an invesor wih an infinie horizon (perpeual consumpion), his formula is exac. In he case of a random fuure lifeime, he formula is based on momen maching approximaions which arge he firs and second momen of he rue sochasic presen value. The resuls are remarkably accurae when compared agains more cosly and ime consuming simulaions. I also provide several numerical examples o demonsrae he versailiy of he closed-form expression for he sochasic presen value (SPV) in deermining susainable wihdrawal raes and heir respecive probabiliies. This formula can easily be implemened in Excel or any oher spreadshee using a variey of porfolio risk/reurn parameers, ages and wihdrawal raes, and reproduces resuls ha are wihin 5% of exensive Mone Carlo Simulaions. The remainder of his paper is organized as follows. Secion 2 cass he mahemaics of he susainable spending problem wihin he conex of a radiional presen value of fuure cashflows calculaion. Secion 3 provides a closed-form analyic expression for he probabiliy ha a given spending rae is susainable. Secion 4 provides exensive numerical examples over a variey of ages and spending raes. Secion 5 concludes he paper and a echnical appendix provides addiional sress-esing resuls on an alernaive invesmen reurn generaing process. Secion #2: STOCHASTIC PRESENT VALUE (SPV) of SPENDING Imagine ha you plan o inves your money in a porfolio earning {R%} per annum and you plan o consume a fixed real (afer inflaion) dollar each and every year unil some horizon denoed by {T}. If he horizon and invesmen rae of reurn are known wih absolue cerainy, he presen value (PV) of your consumpion a iniial ime zero would be compued via: Page 4 of 32

5 PV = T i= 1 1 (1 + R) i 1 (1 + R) = R T, (eq.1) which is he exbook formula for an ordinary simple annuiy, which should be familiar o all sudens of business and finance. Thus in a deerminisic world -- if you sar reiremen wih a nes egg greaer han he PV in equaion (eq.1) imes your desired consumpion, your money will las for he res of your life. If you have less han his amoun you will be ruined a some age prior o deah. For example, a an {R=7%} annual invesmen reurn and a {T=25} year horizon, he required nes egg is (he PV in (eq.1)) imes your real consumpion. If you have more han his sock of wealh a reiremen your plans are susainable. However, if you sar your reiremen years wih only 10 imes you desired real consumpion, hen you will run ou of money in precisely years an income gap of 7.2 years -- because he presen value of $1 annuiy for years a 7% ineres is $10. Noe ha as {T} goes o infiniy which we call he endowmen case he PV converges o he number {1/R}. A {R=0.07} he resuling PV is imes he desired consumpion. Of course, human beings have a random (and finie) lifespan and any exercise ha aemps o compue required presen values a reiremen mus accoun for his uncerainy. Table #1 and he corresponding Figure #1 illusrae he probabiliies of survival using moraliy esimaes from he U.S.-based Sociey of Acuaries. A 65 year-old female has a 34.8% chance of living o age 90. A 65 year-old male has a 23.7% chance of living o age 90. The probabiliies of survival decline in a roughly exponenial manner wih age, reaching close o zero someime beween ages 105 and 115 depending on he moraliy able, projecion mehod and gender. And, while he ofen quoed saisic for life expecancy is somewhere beween 78 and 82 years in he U.S., his is only relevan a he ime of birh. By he ime pensioners reach heir reiremen years, hey may be facing 25 o 30 more years wih subsanial probabiliy. From our reiremen spending perspecive, a 65-year-old migh live 20 more years or 30 more years or only 10 more years. How should his impac he wihdrawal rae? Page 5 of 32

6 Table #1 and Figure #1 Placed Here. Should a 65-year-old plan for he 75 h percenile, 95 h percenile of he end of he moraliy able? Wha {T} value should be used in (eq.1)? The same quesion applies o he invesmen reurn {R}. Wha is a reasonable number o use? The average real (afer inflaion) invesmen reurns from a broadly diversified porfolio of equiy during he las 75 years has been in he viciniy of 6% o 9% according o Ibboson Associaes (2004), bu he year-by-year numbers can vary widely. The correc approach, arguably, is no o guess, assume or ake poin-esimaes bu o acually accoun for his uncerainy wihin he model iself. Nobel laureae Bill Sharpe has amusingly called he (misleading) averaging approach wih fixed reurns and fixed daes of deah, financial planning in fanasy-land. So, in conras o he rivial deerminisic case -- where boh he horizon and he invesmen reurn are known wih cerainy -- when boh of hese variables are sochasic, he analogue o (eq.1) is a sochasic presen value (SPV) defined by: 1 1 SPV = ~ + ~ ~ (1 + R ) (1 + R )(1 + R = ~ T i i= 1 j= 1 1 ~ (1 + R ) j ) ~ T j= 1 1 ~ (1 + R ) j (eq.2) where he new variable {T ~ } denoes he random ime of deah (in years), and he new { R ~ } denoes he random invesmen reurn in year j. Wihou any loss of generaliy { T ~ = } is he infiniely lived endowmen or foundaion siuaion. Likewise, if he consumpion/wihdrawals ake place once per monh or once per week, he random variables { R ~ } and {T ~ } are adjused accordingly. And, if he reurn frequency is infiniesimal, he summaion sign in (eq.2) converges o an inegral, while he produc sign is convered ino a coninuous-ime diffusion process. j j Page 6 of 32

7 The inuiion behind (eq.2) is as follows. Looking forward, we mus sum-up a random number of erms in which each denominaor is also random. The firs iem discouns he firs year of consumpion a he firs year s random invesmen reurn. The second iem discouns he second year s consumpion (if he individual is sill alive) a he produc of he firs and second years random invesmen reurn, ec. The SPV defined by (eq.2) can be visualized in Figure #2. One can hink of he sochasic presen value as a random variable wih a probabiliy densiy funcion (PDF) ha depends on he risk/reurn parameers of he underlying invesmen generaing process as well as he random fuure lifeime. If we sar wih an iniial endowmen or nes egg of $20 and inend o consume $1 (afer-inflaion) per annum, he probabiliy of susainabiliy is equal o he probabiliy ha he SPV is less han $20. This corresponds o he area under he curve o he lef of he ray emanaing from $20 on he x-axis. The probabiliy of ruin is he area under he curve o he righ of he $20 ray. The precise shape and parameers governing he SPV depend on he invesmen and moraliy dynamics, bu he general picure is remarkably consisen and similar o Figure #2. This family of SPVs is defined over posiive numbers, is righ-skewed and is equal o zero, a zero. Figure #2 Placed Here. The four disinc curves in Figure #2 denoe differing random life-spans. In he firs, he (unisex) individual is 50 years old, in he second he/she is 60, in he hird - 65 and in he las one As he individual ages, he SPV of fuure (planned) consumpion shifs oward he lef relaive o he same $20 mark since he odds are ha $20 is enough o susain his sandard of living when saring a an older age. Now, we move on o our goal of obaining a closed-form expression for he disribuion of he SPV. I is quie common in financial economics (and especially opion pricing heory) o assume ha invesmen reurns are generaed by a LogNormal disribuion, a.k.a. he geomeric Brownian moion diffusion process. On a heoreical level his assumpion has many supporers - from Meron (1975) o Rubinsein (1991). Empirically, however, I admi Page 7 of 32

8 ha i does no fi high-frequency daa or observed reurns over all ime horizons. Ineresingly, hough, in a recen paper by Levy and Duchin (2004) he Log-Normal assumpion acually won many of he horse races when comparing plausible disribuions for hisorical reurns. Furhermore, many popular opimizers, asse allocaion models and ofen-quoed common advice are based on he classical Markowiz/Sharpe assumpions of Log-Normal reurns. Therefore, for he remainder of his paper I will follow his radiion and shif he discussion on he impac of alernaive assumpions o he appendix. Secion #3: ANALYTIC FORMULA: SUSTAINABLE SPENDING BACKGROUND: Before I come o he main par of he sory, I mus review hree imporan probabiliy disribuions ha play a criical role in he susainabiliy calculaions. The firs is he ubiquious Log-Normal (LN) disribuion, he second is he Exponenial Lifeime (EL) disribuion and he hird and final one is he perhaps lesser know -- Reciprocal Gamma (RG) disribuion. The connecion beween hese hree will become eviden in ime. Log-Normal Random Variable: The invesmen oal reurn denoed by { R } beween ime zero and ime, is said o be Log-Normally disribued wih parameers { µ, σ } if he expeced µ oal reurn is { E[ R ] = e }, he logarihmic volailiy is { E[ SD[ln[ R ]]] = σ } and he probabiliy law can be wrien as { Pr[ln[ ] < x] = N(( µ 0.5σ ), σ, x) }, where { N (.) } denoes he R cumulaive normal disribuion. For example, a muual fund or porfolio ha is expeced o earn an inflaion-adjused coninuously compounded reurn of { µ = 7%} per annum, wih a logarihmic volailiy of { σ = 20%} has a { N (0.05,0.20,0) = 40.13%} chance of earning a negaive reurn in any given year. Bu, if he expeced reurn is a more opimisic 10% per annum, he chances of losing money are reduced o { N (0.08,0.20,0) = 34.46%}. Noe ha while he expeced value of he Log-Normal random variable { R } is { e µ }, he median value (a.k.a. geomeric mean) is a lower { e 2 ( µ 0.5σ ) }. By definiion, he probabiliy a Log-Normal random variable is less han is median value is precisely 50%. The gap beween he Page 8 of 32

9 expeced value { e µ } and he median value { e proporional o he volailiy and increasing in ime. 2 ( µ 0.5σ ) } is always greaer han zero, Exponenial Lifeime Random Variable: The fuure lifeime random variable denoed by he leer T is said o be exponenially disribued wih moraliy rae { λ } if he probabiliy law for T can be wrien as: { Pr[ T > s] = e λs }. The expeced value of he Exponenial Lifeime random variable is equal o and denoed by { E [ T ] = 1/ λ } while he median value which is he 50% mark -- can be compued via: { Med [ T ] = ln[2]/ λ }. Noe ha he expeced value is greaer han he median value. For example, when { λ = } he probabiliy of living for a leas 25 more (0.05)(25) (0.05)(40) years is: { e = 28.65%}, and he probabiliy of living for 40 more years is: { e = 13.53%}. The expeced lifeime is { 1 / =20} years and he median lifeime { ln[ 2]/ =13.86} years. The exponenial assumpion is a very convenien one for fuure lifeime random variables. And, alhough human aging does no quie conform o an exponenial or consan force of moraliy assumpion, I will show ha for he purposes of esimaing a susainable spending rae, i does a remarkably good job of capuring he salien feaures. Reciprocal Gamma Random Variable: A random variable denoed by X is said o be Reciprocal Gamma disribued wih parameers { α, β } if he probabiliy law for X can be wrien as: α β Pr[ X < x] : = Γ( α) 0 x y ( α + 1) e ( 1/ yβ ) dy (eq.3) One need no be inimidaed by he somewha messy-looking inegral since knowledge of calculus is no required o acually use he formula. The cumulaive disribuion funcion (CDF) displayed in equaion (eq.3) plays he same role as he CDF of he Normal or Log- Normal disribuion, boh of which are now ubiquious in finance. The definiion of he Reciprocal Gamma random variable is such ha he probabiliy an RG random variable is greaer han x is equivalen o he probabiliy ha a Gamma random variable is less han { 1 / x }. The CDF of a Gamma random variable is available in all saisical packages -- even in Excel and hus should be easily accessible o mos readers. Page 9 of 32

10 Finally, he expeced (mean) value a.k.a. firs momen -- of he Reciprocal Gamma disribuion is { E [ X ] = ( β ( α 1)) } and he second momen is { E [ X ] = ( β ( α 1)( α 2)) }. For example, wihin he conex of his paper a ypical parameers pair is: { α = 5, β = }. In his case, he expeced value of he Reciprocal Gamma variable is { 1 /((0.03)(4)) =10}. The probabiliy he Reciprocal Gamma random variable is greaer han 8, for example, is 45.62%. In conras, if we increase {α } from a value of 3 o a value of 4, he relevan expeced value becomes { E [X ] =6.66} and he probabiliy is { Pr[ X > 8] =24.24%} THE MAIN RESULT: EXPONENTIAL RECIPROCAL GAMMA (ERG) Wih he mahemaical background behind us, my primary claim is ha if one is willing o assume Log-Normal reurns in a coninuous ime seing hen he sochasic presen value he one displayed graphically in Figure #2 is acually Reciprocal Gamma disribued in he limi. In oher words, he probabiliy ha he SPV is greaer han he iniial wealh or nes egg denoed by { w }, is he simple-looking: 2 2µ + 4λ σ + λ 1 Pr[ SPV > w] = GammaDis 1, (eq.4) 2 σ + λ 2 w where GammaDis( α, β.) denoes he cumulaive disribuion funcion (CDF) of he Gamma disribuion using he Microsof Excel noaion evaluaed a he parameer pair { α, β }. The familiar { µ, σ } are he expeced reurn and volailiy parameers from he invesmen porfolio and { λ } is he moraliy rae. The expeced value of he SPV based on he Reciprocal 2 1 Gamma represenaion is { ( µ σ + λ) }. For example, sar wih an invesmen (endowmen, nes egg) fund conaining $20 ha is invesed in an equiy fund ha is expeced o earn { µ = } per annum, wih a volailiy or sandard deviaion of { σ = } per annum. Assume ha a (unisex) 50-year-old wih a median fuure lifespan of 28.1 years -- according o Sociey of Acuaries moraliy ables -- inends on consuming $1 afer-inflaion per annum for he res of his or her life. Page 10 of 32

11 Recall ha if he median lifespan is 28.1 years, hen by definiion he probabiliy of survival for 28.1 years is exacly 50%, which implies ha our moraliy rae parameer is: { λ = ln[ 2]/ 28.1 = }. According o (eq.4) he so-called probabiliy of reiremen ruin, which is he probabiliy ha he sochasic presen value of $1 consumpion is greaer han $20, is approximaely 26.8%. In he language of Figure #2, if we evaluae he SPV a {w=20}, he area o he righ has a mass of unis. The area o he lef which is he probabiliy of susainabiliy has a mass of unis. Naurally, differen values of {w} will resul in differen ruin probabiliies. The more echnically inclined readers migh wan more han a formula. Indeed, a proof ha (eq.4) is he disribuion of he sochasic presen value is based on momen maching echniques and he Parial Differenial Equaion (PDEs) for he probabiliy of ruin. A varian of his resul can acually be raced back o a paper by Meron (1974), alhough i was never exploied in he conex of spending raes. For more deails, proofs and resricions, see Milevsky (1997) or Browne (1999) and he references conained herein. In addiion, he appendix conains a brief descripion of ess ha where conduced o sresses he formula in equaion (eq.4) under a variey of alernaive asse-reurn dynamics, and specifically he implicaions of assuming an exreme pure jump process. Overall, he formula survives he various limus ess provided he parameers are wihin he region of normal reiremen. The following secion displays exensive numerical resuls. Secion #4: DETAILED NUMERICAL EXAMPLES: A newly reired 65 year-old has a nes egg of $1,000,000 which mus provide income and las for he remainder of his individual s naural life. In addiion o expeced Social Securiy benefis of $14,000 per annum and a defined benefi (DB) pension from an old employer providing $16,000 per annum boh paymens adjused for inflaion each year he reiree esimaes he need for an addiional $60,000 from he invesmen porfolio. The $60,000 income will be coaxed from he million dollar porfolio via a sysemaic wihdrawal plan (a.k.a. SWiP) ha sells-off he required number of shares/unis each monh using a reverse dollarcos average (DCA) sraegy. All of hese numbers are prior o any income axes. Nor do I Page 11 of 32

12 disinguish beween ax shelered (IRA, 401k) plans versus axable plans, which are a differen se of imporan issues I do no address in his paper. Wha is imporan o noe is ha he $90,000 consumpion plan will be saisfied wih $30,000 from a de faco inflaion adjused life annuiy and he remaining $60,000 from a SWiP. In our previous lingo, I am ineresed in wheher he sochasic presen value (SPV) of he desired $60,000 income per annum is probabilisically less han he iniial nes egg of one million dollars. If his is he case, he sandard of living is susainable. If, however, he SPV of he consumpion plan is larger han one million dollars, he reiremen plan is deemed susainable and he individual will be ruined a some poin in heir life, unless hey reduce heir consumpion habis. The basic philosophy of his paper is ha he SPV is a random variable and he proper analysis comes down o probabiliies. Table #2a, #2b, #2c Placed Here Table #2 provides an exensive combinaion of consumpion/wihdrawal raes across various ages so readers can gauge he impac of hese facors on he ruin probabiliy. The firs column displays he reiremen age he second column displays he median age-a-deah based on acuarial moraliy ables and he hird column compues he implied hazard rae from his median value. Wih a { λ } value in hand and he { µ } and {σ } given in he lower-lef corner, he able evaluaes he SPV of various spending raes ranging from $2 o $10. The firs row wihin able #2a provides resuls in he case of a reiree who would like he spending o las forever and hence he median age a deah is infiniy which is also applicable o an endowmen or foundaion wih an infiniy horizon. The probabiliy of ruin ranges from a low of 15% ($2 spending) o a high of 92% ($10 spending) when he porfolio is invesed in an equiy-based porfolio ha is expeced o earn a (lognormal) invesmen reurn wih a mean value of { µ =7%} and a volailiy of {σ =20%} per annum. Back o our reiree, according o Table #2a, if he 65 year-old invess he one million dollar nes egg in he same equiy-based porfolio, he exac probabiliy of ruin i.e. he probabiliy he plan is no susainable is 25.3%. Roughly one ou of four reirees who adop his Page 12 of 32

13 reiremen consumpion plan will be forced o reduce heir sandard of living during reiremen. By exac probabiliy of ruin, I mean he oucome from discouning all fuure cash-flows using he correc (unisex) acuarial moraliy able saring a age 65. In he same able, jus above he exac 25.3% number, I lis he resuls using he ERG approximaion formula, which is based on an exponenial fuure lifeime implemened wihin equaion (eq.4). Noe he approximae answer is a slighly higher 26.2% probabiliy of ruin, relaive o he 25.3% under he exac mehod. The gap beween he exac and approximae number is less han 0.9% which provides addiional confidence in our ERG formula (eq.4) Now, I would argue ha regardless of wheher one uses he exac or he approximae mehodology, a 25% chance of reiremen ruin, which is only a 75% chance of success, should be unaccepable o mos reirees. Table #2a indicaes ha lowering he desired consumpion or spending plan by $10,000 o a $50,000 SWiP reduces he probabiliy of ruin o 16.8% (using he exac mehod) or 18.9% (using he approximaion). And, if he spending plan is furher reduced o $40,000 he probabiliy of ruin shrinks o 9.4% (exac) and 12.3% (approximae). The reiree and his or her financial planner or analys can deermine wheher hese odds are accepable vis a vis heir olerance for risk. In he oher direcion, if he same individual were o wihdraw (he enire) $90,000 from he million dollar porfolio using he 7% mean and 20% volailiy porfolio parameers he probabiliy of ruin would be 50.5% (exac) or 48.3% (approximae). To undersand he inuiion behind he numbers, noe ha he mean or expeced value of he 2 sochasic presen value (SPV) of $1 of real spending is: { 1/( µ σ + λ) }, where { µ } and {σ } are he invesmen parameers, while { λ } is he moraliy rae parameer induced by a given median fuure lifeime. For a 65 (unisex) year-old he median fuure lifeime is 18.9 years according o he RP2000 Sociey of Acuaries moraliy able. To ge he 50% probabiliy poin 18.9λ wih an exponenial disribuion, we mus solve he equaion { e = 0. 5}, which leads o { λ = ln[ 2]/18.9 = } as he implied rae of moraliy. Page 13 of 32

14 Now back o he mean value of he SPV, in he case of { µ =7%} and volailiy of {σ =20%}, his works ou o { 1 /( ) }, which is an average of $15 for he SPV per dollar of desired consumpion. Thus, if he reiree inends on spending $90,000 per annum, i should come as no surprise ha a nes-egg of only 11 imes his amoun is barely enough o give even odds. Noe ha he expeced value of he SPV decreases in { µ, λ } and increases in {σ }. While he impac of porfolio parameers should be obvious higher mean is good, higher volailiy is bad -- he benefi of a higher moraliy rae { λ } comes from reducing he anicipaed lifespan and hence he lengh of ime over which he wihdrawals are aken. Now, if he same individual were o delay reiring by five years, or more precisely, begin consuming from he nes egg a age 70, he same $60,000 consumpion plan would resul in a 17.6% probabiliy (exac) or 20.1% probabiliy (approximae) of ruin according o he same Table #2a. The increased susainabiliy of he same plan relaive o he roughly 25% probabiliy if his individual were o reire a age 65 is due o he reduced fuure lifespan and hence he lower sochasic presen value of consumpion. Think back o he expeced value of he consumpion plan. A age 70 he median fuure lifespan is only 14.6 years, which leads o a higher { λ = } and hence a lower value for E[SPV]. The reiree can sar reiremen wih less or can consume more. Table #2b and #2c provide resuls on alernaive porfolio invesmen parameers using he ERG approximaion from equaion (eq.4). In Table #2b I have reduced he expeced invesmen reurn from 7% o 5% bu lef he volailiy a 20%. In his case all he probabiliies are higher compared o Table #2a since a higher volailiy can only make hings worse. In Table #2c I have reduce he volailiy from 20% o 10% and kep he expeced reurn a 5%. For example, he 65 year-old wihdrawing $60,000 from a million dollar porfolio has 39.8% probabiliy of ruin under a { µ =5%} and {σ =20%} invesmen regime, compared o a 26.2% probabiliy of ruin under a { µ =7%} and {σ =20%} invesmen regime, which is obviously due o he 200 basis poin loss in reurns. Bu, when he { µ =5%} invesmen reurn is mached wih (a more reasonable) {σ =10%} volailiy, he probabiliy of ruin shrinks o 21% according o Table #2c. The inuiion once again comes down o he expeced value of he SPV of $1 2 spending: { 1/( µ σ + λ) }. When { µ =5%} and {σ =10%} he firs par of he denominaor is Page 14 of 32

15 0.04, bu when { µ =7%} and {σ =20%} he same erm is only 0.03, which ceeris paribus increases he SPV which lowers he susainable spending rae. Noe ha Table #2c does no provide uniformly lower probabiliies of ruin. For high levels of consumpion a more aggressive { µ = 0.07, σ = 0. 20} porfolio may lead o beer susainabiliy odds compared o he more conservaive { µ = 0.05, σ = } porfolio. One can hink of a number of ways in which o play wih his formula. For example, our main (eq.4) can be invered o solve for a safe rae for a given probabiliy of ruin olerance. This idea is akin o some recen applicaions of shorfall as a measure of risk in he conex of porfolio managemen. See Browne (1997, 1999) or Young (2004) for more examples of his concep in a dynamic conrol framework. Thus, if one desires a 90% probabiliy of susainabiliy when he median fuure lifeime is 15 years under a porfolio ha is projeced o earn { µ = } wih a sandard deviaion of { σ = }, hen he consumpion rae ha leads o a 10% probabiliy of ruin is $5.03 per annum. Along he same lines, he impac of he expeced reurn {µ } on he susainabiliy of a given wihdrawal sraegy can easily be sress esed. For example, a 85% desired probabiliy of susainabiliy he implied wihdrawal rae is $4.41 per annum when he expeced reurn is 7% (wih a 20% sandard deviaion). However, if we remove 100 basis poins from he equiy reurn so ha { µ = }, he same 85% forces a more conservaive $3.76 wihdrawal rae. Anoher ineresing insigh comes from examining he inerplay beween he parameers in our formula. If we reduce he fixed moraliy rae { λ } by 100 basis poins which increases he median fuure lifeime from { ln[ 2]/ λ } o ln[ 2]/( λ ) -- i has he probabiliy equivalen effec of increasing he porfolio reurn by 200 basis poins and increasing he porfolio variance by 100 basis poins. They boh lead o he same saisical resuls. Recall ha our { α, β } parameer argumens in (eq.4) can be expressed as a funcion of { µ + 2λ } 2 and { σ + λ }. Thus, having a longer lifespan (i.e. lower hazard rae) is inerchangeable wih decreasing he porfolio reurn and porfolio variance relaive o he baseline. In aggregae, however, his increases he probabiliy of ruin and reduces he probabiliy ha a given level of wealh is enough o susain reiremen spending. Page 15 of 32

16 Finally, i is imporan o sress ha in he { λ = 0 } -- infinie horizon -- case our resul is no an approximaion. I is a heorem ha he SPV is in fac Reciprocal Gamma disribued. For hose readers who remain unconvinced ha wha is effecively he sum of lognormals in (eq.4) can converge o he inverse of a Gamma disribuion, I urge you o simulae he SPV for a reasonably long horizon and conduc a KS goodness of fi es of he inverse of hese numbers agains he Gamma disribuion, wih he parameers given by 2 2 { α = (2µ + 4λ) /( σ + λ) 1, β = ( σ + λ) / 2}. As long as he volailiy parameer {σ} is no oo high relaive o he expeced reurn {µ}, we ge convergence of he relevan inegrand. Thus, i is only in he random lifespan where { λ > 0 } ha our resul is approximae, albei correc o wihin wo momens of he rue SPV densiy. To illusrae his graphically, Figure #3 provides a sylized illusraion -- under a 7% mean and 20% volailiy -- of he approximaion error from using he ERG formula based on an exponenial fuure lifeime when in fac he rue fuure lifeime random variable is more complicaed. Figure #3 Placed Here Figure #3 displays he reiremen ruin probabiliy a.k.a. he probabiliy he spending rae is no susainable saring a age 65 for a range of consumpion raes from $1 o $10 per original $100 nes egg. For low consumpion raes he ERG formula slighly overesimaes he probabiliy of ruin and hus gives a more pessimisic picure of he susainabiliy of spending. A higher consumpion raes he exac reiremen ruin probabiliy is higher han wha is claimed by he approximaion. Ye, noice he relaively small error gap beween he wo curves, which a heir wors is no more han 3% - 5%. The wo curves are a heir closes which implies ha he approximaion is a is bes when he spending raes is beween $5 o $7 per original $100, which coincidenly is precisely where he curren debae regarding susainable spending currenly resides. Secion #5: CONCLUSION AND NEXT STEPS A casual search on he Web reveals close o a dozen on-line calculaors -- mos sponsored by financial services companies -- ha purpor o compue a susainable wihdrawal rae (and Page 16 of 32

17 asse allocaion) for reirees by using Mone Carlo Simulaions. A number of hese calculaors are plagued by opaciy in he deails of heir sochasic generaing mehodology, and conduc an absurdly small number of simulaions when compared wih he ens of housands needed for convergence. Moreover, he uncerainy generaed by he randomness of human life is ofen ignored or alluded o ouside of he formal model. Indeed, he black-box and ime consuming naure of obaining resuls do lile o enhance a pedagogical undersanding of he wihdrawal or spending problem. The same issues are relevan in he endowmen business where rusees and oher decision-makers mus radeoff curren spending agains fuure growh. The disincion beween radiional Mone Carlo simulaions and he analyic echniques promoed in his paper is more han jus a quesion of academic ases and echniques 4. While Mone Carlo simulaions will coninue o have a legiimae and imporan role wihin he field of wealh managemen and reiremen planning, I believe ha a simple, easy o use and baseline formula can serve as a saniy check or a calibraion poin for more complicaed simulaions. A he risk of overselling, his is akin o having a Black-Scholes formula for he price of a call or pu opion when many of he underlying assumpions are quesionable -- which provides a deep undersanding of he embedded risk and reurn radeoffs and can live side by side wih more sophisicaed simulaion based opion-pricing models. For example, using he formula I find ha a (unisex) 65 year-old reiree who invess his/her porfolio in a marke ha is expeced o earn a real (afer-inflaion) 7% wih a volailiy of 20% and consumes $4 per-year per $100 of iniial porfolio value, will ge ruined 10 imes ou of 100. However, if he same reiree wihdraws a more aggressive $6 per $100, he probabiliy increases o abou 25% or one ime ou of four. This is clearly no susainable. As an upper bound, a reiree should be spending no more han { µ σ 2 + λ } percen of he iniial nes egg, where { µ } is he expeced reurn, {σ } is he volailiy and { λ = ln[2]/ m }, where m is a median fuure lifeime. This spending rae would be susainable on average bu no much beer. 4 See, for example, he recen Wall Sree Journal aricle (Augus 31, 2004) eniled: Tool Tells How Long Nes Egg Will Las, in which he reporer Kaja Whiehouse described he benefis of analyic PDE-based soluions over Mone Carlo simulaions. Page 17 of 32

18 Noe ha mos of hese numbers are in-line wih resuls from a variey of simulaion sudies for example he widely used Ibboson Associaes reiremen wealh simulaor -- albei produced by an insighful formula in a fracion of he ime. Fuure and ongoing research is examining he impac of income axes and opimal locaion decisions as well as well as he role of life annuiies in increasing he susainabiliy of a given spending rae. Our hero, of course, is he (Reciprocal) Gamma disribuion, which should ake is righful place beside he Log-Normal densiy in he panheon of probabiliy disribuions ha are of immediae relevance o financial praciioners and porfolio managers. References: Alschuler, G. (2000), Endowmen payou raes are oo singy, The Chronicle of Higher Educaion, March 31 Ameriks, J., M. Veres and M. Warshawsky (2003), Reiremen income ha lass a lifeime, Journal of Financial Planning, pg Arno, R.D. (2004), Edior s Corner: Susainable Spending in a Lower Reurn World, Financial Analyss Journal, Sepember/Ocober 2004, pg 6-9. Bengen, W.P. (1994), Deermining wihdrawal raes using hisorical daa, Journal of Financial Planning, Ocober 1994, Vol. 7(4), pg Browne, S. (1999), The risk and reward of minimizing shorfall probabiliy, Journal of Porfolio Managemen, Vol. 25(4), pg Baes, D. (2003) Empirical opion pricing: A rerospecion," Journal of Economerics, Vol. 116:1/2, Sepember/Ocober 2003, pg Page 18 of 32

19 Cooley, P.L., C.M. Hubbard, and D.T. Walz, Reiremen Spending: Choosing a Wihdrawal Rae Tha Is Susainable. Journal of he American Associaion of Individual Invesors, 20, 1 (1998): Cooley, P.L., C. M. Hubbard and D. T. Walz (2003), Does Inernaional Diversificaion Increase he Susainable Wihdrawal Raes from Reiremen Porfolios? Journal of Financial Planning, January 2003 Dybvig, P.H. (1999), Using asse allocaion o proec spending, Financial Analyss Journal, January/February, pg Ennis, R.M. and J.P. Williamson (1974), Spending policy for educaional endowmen, Research Publicaion Projec of The Common Fund. Hannon, D. and D. Hammond (2003), The looming crisis in endowmen spending, The Journal of Invesing, Fall 2003 Ho, K., M. Milevsky and C. Robinson (1994), How o avoid ouliving your money, Canadian Invesmen Review, Vol. 7(3), pg Ibboson Associaes. Socks, Bonds, Bills and Inflaion 2004 Yearbook. Chicago: Ibboson Associaes, Levy, H. and R. Duchin (2004), Asse Reurn Disribuions and he Invesmen Horizon, Journal of Porfolio Managemen, Vol. 30(3), pg Markowiz, H.M. (1991), Individual versus Insiuional Invesing, Financial Services Review, Vol. 1(1), pg Milevsky, M. (1997), The presen value of a sochasic perpeuiy and he Gamma disribuion, Insurance: Mahemaics and Economics, Vol. 20(3), pg Page 19 of 32

20 Meron, R. (1975), An asympoic heory of growh under uncerainy, Review of Economic Sudies, Vol. 42, pg Chaper 17 of Coninuous-Time Finance, Revised Ediion, Blackwell press. Orszag, M. (2002), Ruin in Reiremen: Running ou of money in reiremen programs, Wason Wya echnical paper RU06, London, UK. Pye, G. (2000), Susainable invesmen wihdrawals, Journal of Porfolio Managemen, Vol. 26(3), pg Rubinsein, M. (1991), Coninuously rebalanced invesmen sraegies, Journal of Porfolio Managemen, Fall, pg Ross, S.M. (1997), Simulaion, 2 nd Ed., Harcour Academic Press Tobin, J. (1974), Wha is permanen endowmen income? American Economic Review, Vol. 64(2), pg Young, V.R. (2004), Opimal invesmen sraegy o minimize he probabiliy of lifeime ruin, Norh American Acuarial Journal, Vol. 8(4), pg Page 20 of 32

21 Appendix 1: Alernaive Process for Invesmen Reurns The simulaions and analyic resuls wihin he main body of he paper are all predicaed on Log-Normally disribued porfolio reurns a.k.a. GBM diffusion -- under which (i) he mean and sandard deviaion of invesmen reurns compleely specify he reurn dynamics and (ii) he porfolio wealh process is coninuous in ime. In his appendix I conduc he same ruin probabiliy calculaions under various ages and spending raes, bu assuming he underlying invesmen reurns follow a pure disconinuous jump process. My objecive is o sress es he robusness of he main formula in equaion (eq.4). Indeed, he disconinuiy over ime a.k.a. jumps -- for boh individual socks and aggregae index values has been exensively researched in he empirical finance lieraure for over en years. In addiion, he Log-Normal assumpion has fallen ino somewha of disrepue amongs opion pricing specialiss who have documened ime-varying (i.e. non consan) volailiies for mos raded financial insrumens. For example, Baes (2003) develops and calibraes a diffusion-based model for opion pricing under which he reurn generaing process is a mixure of Brownian movemens and Poisson jumps. A large number of recen papers in he mahemaical finance lieraure have adoped a mixure approach. And, i is mos common in he derivaive securiies hedging arena. One migh herefore righfully expec o see a more sophisicaed model for securiy prices in a 21 s cenury paper. However, in his paper I am no ineresed in locaing he mos precise model for he fine srucure of invesmen reurns. Raher, I am ineresed in a reasonable approximaion for long-erm invesmen reurns o help shed ligh on a reasonable spending rae. This core issue I would like o address is wheher jumps and disconinuiies in marke prices have a meaningful impac on susainable spending raes implied from a model ha assumes a Log-Normal process. I am no ineresed in hedging derivaive securiies or compuing Value-a-Risk (VaR) over wo week horizons. Raher, I am concerned wih 20 o 30 year forecass where parameer uncerainy i.e. wha is he equiy premium going forward migh swamp he model uncerainy. I approach his quesion in he form of a classical horse race where I simulae wo disinc ime series, one being a pure diffusion and one being a pure jump process, and compue he ruin probabiliies for various spending raes. Page 21 of 32

22 To make his analysis meaningful or an apples o apples comparison -- I impose some srucure on he jump process. I force he firs wo momens of he simulaed pure jump process o equal he firs wo momens of a pure diffusive process. This momen maching procedure is ofen used in finance when he rue reurn generaing disribuion is unknown bu he momens are known wih reasonable cerainy. For example, we migh expec markes o earn a 7% per annum wih sandard deviaion of 20%, bu we are no sure abou he exac reurn generaing process. More specifically and for he purposes of his appendix we calibraed parameers for he simulaed jump process so ha he (real, afer inflaion) arihmeic mean reurn was 7% (geomeric mean reurn was 5%) and he volailiy of reurns was 20%. To formally se-up he horse race I sar wih a generic invesmen reurn process { S }, defined in an exponenial manner by he equaion: S 0 X = S e. (a.1) The process { X B X } in he exponen can eiher be he familiar and coninuous: : = g + σb, (a.2) or he purely disconinuous process: Y N ( c) X : = g + Y, (a.3) i= 1 i One can recognize he former process for { X } equaion (a.2) -- as he sandard Brownian moion plus a linear rend g. The laer process equaion (a.2) is a random sum { N (c) } of discree jumps of random size { Y }. The { N (c) } denoes a generic Poisson arrival process a a rae of c jumps per annum. The {} can be described heurisically as a process ha couns he oal number of jumps ha have occurred prior o ime. I sars a zero and hen increases by one uni eac random arrival ime. By definiion of he Poisson process, he ime beween jumps is exponenially disribued wih a mean ime of { 1 / c } years. Finally, he probabiliy densiy funcion for each jump { Y } is assumed o be: 1 ky 1 ky f ( y) : = ke I{ y< 0} + ke I{ y> 0}, (a.4) 2 2 Page 22 of 32

23 which is symmeric around zero bu exponenial-like in characer on boh sides of zero. Thus, he exponenial disribuion plays wo disinc roles in our pure-jump simulaion process. Firs, i deermines when jumps occur and second i deermines he magniude of hose same jumps. Of course, equaions (a.3) and (a.4) are one of many possible ways o generae and model jumps for he invesmen reurn process all of which are par of he Levy family for { X } and I have seleced his paricular parameerizaion for analyic convenience and is abiliy o capure he essence of jump processes in coninuous ime. Acual simulaion sample pahs for he process { Y X } are generaed in hree sages. Firs an exponenial random variable wih parameer c for example an average of 3 jumps per year - - generaes a jump ime. Then, a fair coin oss deermines wheher any given jump { Y } is posiive or negaive. Finally an exponenial random variable wih parameer k -- for example { k = 8% } per jump -- generaes a magniude. The exponenial disribuions can be simulaed using he Inverse Transform algorihm { ln[ U ]/ k } or { ln[ U ]/ c } applied o a uniformly disribued random variable U. See Ross (1997) for a deailed reference on sandard probabiliy simulaions. The hree-sep procedure is repeaed as simulaed calendar ime increases unil he end of he desired sample pah. Recall ha our objecive is o compue he probabiliy he porfolio value process defined by he sochasic differenial equaion (SDE): dw = ds d, (a.5) will hi a zero level (a.k.a. ruin) prior o deah using boh candidaes for he exponenial process { X }, bu ha hey share he same firs and second momens. Noe ha he complee symmery of he jump-process densiy in equaion (a.4) leads o a firs-momen: Y E[ X B ] = E[ X ] = g, (a.6) where he parameer g is he familiar geomeric mean reurn. The second momen condiion leads o: E[( X B Y ) ] = E[( X ) ] σ = (2c / k ) (a.7) Page 23 of 32

24 Thus, if we consruc wo disinc processes { B 2 2 X, X } wih parameers {g} and { σ = 2c / k } hey will share he firs wo and as a resul of symmery he hird as well logarihmic momens. Y For example, if we se c = 3 and k = 12.25, he second momen of he log-reurn process is 0.04, as per equaion (a.7), which hen momen maches o a geomeric Brownian moion wih volailiy {σ = 20%}. Along hese lines, Table 3 summarizes he resuls of he simulaions. The main insighs are as follows. A young ages he ERG approximaion which is based on diffusion processes -- provides very poor resuls compared o he rue ruin simulaions under a pure jump process. However, a higher ages he accuracy improves and wihin he spending ranges discussed in he paper, he formula is quie reliable. In sum, he reader should be careful no o read oo much ino discrepancies beween he wo numbers given he compleely differen price process underlying he calculaion, bu he qualiaive resuls should be noed. Page 24 of 32

25 Table #1 Condiional Probabiliy of Survival a Age 65 To Age Female Male % 92.2% % 81.3% % 65.9% % 45.5% % 23.7% % 7.7% % 1.4% Source: Sociey of Acuaries RP-2000 Table (wih full projecion) Capion: Table #1 illusraes he randomness of he reiree s invesmen horizon. A 65 yearold migh only survive for 10 years, or migh live for an addiional 30 years. This uncerainy should be incorporaed in any financial advice regarding spending raes. Page 25 of 32

26 Figure #1 100 The Probabiliy of Survival: Female Male Probabiliy Source: RP-2000, SoA To Age... Capion: The figure provides a graphical illusraion of he numbers in Table #1. The condiional probabiliy of living o any given age declines almos exponenially wih ime and evenually reaches zero. Page 26 of 32

27 Figure #2 0.1 Sochasic Presen Value (SPV) of Reiremen Consumpion SPV a Age 75 SPV a Age 65 SPV a Age 60 SPV a Age Probabiliy Densiy Funcion Susainable Ruin Curren Dollars (Nes Egg) Capion: When he ime horizon and he invesmen reurn in a classical cash-flow discouning calculaion are unknown, he presen value becomes a random variable wih he following general shape. Page 27 of 32

28 Figure #3: Age 65: Probabiliy Given Spending Rae is No Susainable: E[R] = 7%, SD[R] = 20% Exac Ruin Prob. ERG Ruin Prob. 60% 50% Probabiliy 40% 30% 20% 10% 0% $1.0 $1.4 $1.8 $2.2 $2.6 $3.0 $3.4 $3.8 $4.2 $4.6 $5.0 $5.4 $5.8 $6.2 $6.6 $7.0 $7.4 $7.8 $8.2 $8.6 $9.0 $9.4 $9.8 Spending Rae per $100 of Iniial Wealh Capion: How good is he approximaion? The exac ruin probabiliy is compued using he RP2000 moraliy able and hen compared o he ERG approximaion which is based on assuming an exponenial fuure lifeime wih he same median lifeime as he moraliy able. Page 28 of 32

29 Table #2a Capion: The able compares he resuls of he ERG approximaion presened in he paper agains he exac resuls using he correc moraliy able. For he exac soluion I used he Unisex (non-projeced) RP2000 moraliy able from he Sociey of Acuaries. For he approximae soluion I used an exponenial fuure lifeime assumpion mached o he RP2000 median age-a-deah. Page 29 of 32

30 Table #2b Capion: The able assumes a lower (5%) invesmen reurn and should be conrased wih he resuls in Table #2a where he expeced reurn was 7%. Page 30 of 32

31 Table #2c Capion: The able assumes a lower (5%) reurn and (10%) volailiy and should be conrased wih he resuls in Table #2b and #2a. Page 31 of 32

32 Table 3 Capion: The algorihm momen-maches he pure jump process o a lognormal densiy wih he same (logarihmic) mean and sandard deviaion. I hen simulaed lifeime ruin probabiliy using he jump process and compared wih he ruin probabiliy using he ERG approximaion. Page 32 of 32