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The Effect of Investment Horizons onRisk, Return and End-of-Period Wealth forMajor Asset Classes in Canada

Lakshman AllesCurtin University of Technology

George Athanassakos*The University of Western Ontario

AbstractThe objective of this paper is to investigate whether thecurrent practice among financial planners of recom-mending .stocks at an early age and progressively mov-ing into cash or bonds as retirement approaches wouldbe appropriate. We computed returns, risks and end-of-period wealth distributions of various Canadian assetclasses at increasing horizons between 1957 and 2003,based on the bootstrapping technique. Results show thatinvestment outcomes at short horizons can be quite dif-ferent from outcomes at longer horizons. Evidence isprovided in favour of time diversification, while the cur-rent market practice of life cycle investing is not fullysupported as stocks continue to exhibit more favourablerisk-return payoffs than other asset classes, even atshorter time intervals,

JEL Classifications: G11,G23.

Keywords; Time Diversification, Investment Horizons,End-of-Period Wealth. Relative Pertbrmance

ResumeCet article se propose d'etudier le bien-fond4 de la pra-tique actuelle qui consiste a recommander des actionsaux investi.sseurs dans leur jeunesse et I'argent liquideou les obligations lorsqu'Us approchent I'age de laretraite. Grace a la technique de bootstrapping, nouscalculons les retours sur invesrissement, les risques et ladistribution de richesse enfin de periode pour plusieurstypes d'actifs canadiens a horizons divers entre 1957 et2003. Les resultats presentent des differences impor-tantes entre les investissements a court terme et lesinvestissements a long terme. Les donnees disponiblessoutiennent I'idee de la diversification temporelle etrefutent partiellement la pratique actuelle du cycle devie d'investissement. Defait. les actions comportent tou-jours un profil risques-benefices plus favorable que lesautres types d 'actifs, meme pour des intervalles de tempsreduits.

Mots c\H : diversification temporelle, periode de cal-cul, fin de la periode de fortune, performance relative

The purpose of this study is to provide research, evi-dence, and answers with regard to the following ques-tion: How does the investment horizon affect investors'portfolio performance? To this end, the specific ques-tions we examined are as follows:

The first author acknowledges financial support received from theSchool of Business and Economics of Wilfrid Laurier University whilevisiting ihe School. A version of this paper was presented at the 2005Multinational Finance Conference in Athens. Greece. Commenis cf theconference participanls and, particularly, of the discussant ManfredFruhwirth, are gratefully acknowledged.

* Please address correspondence to George Athanassakos, RichardIvey School of Business, The University of Western Ontario, London,Ontario, Canada N6A 3K7, [email protected].

1. For different asset classes, how does the holding peri-od of retum and risk, measured by the standard devi-ation of returns, change as the investment horizon isgradually increased?

2. How does asset class portfolio efficiency measuredby the coefficient of variation and the Sharpe ratiochange as the horizon lengthens? Does this haveimplications for an optimal investment horizon wheninvesting in a particular asset class?

3. What is the distribution of ending period wealth wheninvesting in different asset classes as the investmenthorizon lengthens, and how does the probability ofending up with a shortfall in wealth change as theinvestment horizon is increased in each asset class?

© ASAC 2006 138

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EFFECT OF INVESTMENT HORIZONS ON RISK. RETURN AND END-OF-PERIOD WEALTH ALLES & ATHANASSAKOS

4. As the investment horizon lengthens, what is theprobability of one asset class outperforming another(i.e., stocks outperforming bonds)? What does thisimply for the preferences of investors to invest in oneasset class versus another?

The need for research to answer these questions hasbecome increasingly important primarily due to funda-mental changes that are taking place in the retirementplanning industry. For example, corporate pensionfunds, traditionally structured as defined benefit plansthat assured contributors of a predefmed benefit at thetime of retirement, are rapidly changing tbeir structure todefmed contribution plans (DCP) where decisions relat-ing to investments, asset allocations and risk manage-ment are thrust upon the plan contributors.

Long term investment planning requires an under-standing of the implications of investing over long hori-zons, particularly in respect to the conceptualization andmeasurement of risk. Yet. even among professionalsthere exists a considerable divergence of opinionsregarding the implications of investment horizon on risk.For example, Olsen and Khaki (1998) suggest tbat tbelack of closure on this topic is perhaps due to the profes-sion's failure to accept a common defmition of risk.They argue that investors should not only look at tbeconventional measures of risk, such as standard devia-tion of returns, but also view risk in terms of loss func-tions and the possibility of realizing returns below targetlevels.'

The usual argument advanced in life cycle invest-ment recommendations is that people should not jeopar-dize their pension funds as they approach retirement andsbould therefore switch towards less risky assets such ascash or govemment bonds. This is inconsistent witbmost investment models as the probability distribution ofterminal wealth is tbe same regardless of wben risk istaken. Merton (1969. 1971), for example, formulatedmodels taking into account that an investor's portfolioallocation is not influenced by his time horizon. Mer-ton's conclusions do not support current practice ofswitching from stocks into cash and bonds at the onset ofretirement. Equity exposure will never be zero in Mer-ton's models and hence the current practice of investingfor retirement would not be optimal and would reducethe value of retirement funds. Might it be possible thatfinancial planners and DCP trustees cboose solutionsthat at the same time limit the chance of being accusedof improper efforts to minimize their own risk at theexpense of their clients'/members' pension ftind value atretirement?

The empirical literature on the aspect of long bori-zon risk and retum analysis is scarce, possibly due to thelimited span of historical data available for analysis

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especially outside tbe US. To overcome tbe data limita-tion in examining long horizon investment decisions,some recent US-based studies bave adopted resamplingtechniques sucb as the bootstrap to extend the span ofactual datasets (e.g., Hickman, Hunter, Byrd, Beck &Terpening, 2001). Nevertheless, financial markets out-side the US, of wbicb Canada is an example, have notbeen subjected to such analysis.

There are distinct differences between the Canadianand US markets. It is widely believed tbat the Canadianhistorical investment experience is very different fromtbat of the US, where most of the studies on long termhorizon performance are based (Gluskin 2006). The dif-ferential taxation of investment income in Canada versustbat of the US and the heavier dependence of the Cana-dian market on commodities could play a role in tbisregard. Given that investment income is taxed moreheavily in Canada, investors should expect higherbefore-tax retums from Canadian than US investmentsas they look at after-tax retums when making investmentdecisions. This is particularly true as in the long run for-eign exchange differentials should not play an importantrole in determining retums. As a result, US holding peri-od returns should behave differently tban Canadian boid-ing period retums, especially over long term horizons.Moreover, as Canadian equity markets are dominated bythe fmancial and commodity sectors, a portfolio thatlooks like tbe Canadian market is not likely to be diver-sified.^

Hence, diversification is more essential in Canadathan the US (Damseil 2006). Tbe heavier weighting ofthe Canadian markets towards commodity stocks thantheir US counterparts sbould imply flatter long term per-formance of Canadian versus US equity markets vi&-h-vis such markets' short term performance. Because com-modity stocks belong to companies tbat over a typicalcycle will produce low retums on capita! and deficientstock market performance (DeCloet, 2006), if one staysinvested for the long term and does not try to time themarket, average retums will be about zero as there is notmuch growth in such stocks over the long run. As aresult, active managers have beaten the index more oftenin Canada than in the US (Gluskin, 2006).^ On tbe otherhand, the US market is based more on growth-orientedstocks and, as a result, one may expect higher long-termretums. However, tbe risk of the US market can be bigh-er as well. Given these two contradictory expectations, itwould be informative to compare tbe Canadian and USmarkets to determine which is more efficient.

The limited diversification ofthe Canadian marketin conjunction with the foreign content limitations (towbich Canadian RRSP and pension fund portfolios aresubjected) and the home bias demonstrated by Canadianinvestors make this study all the more important. For

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example, Ackert, Church, Tompkins and Zhang (2005)found that Canadian investors have a greater familiaritywith local and domestic securities than foreign and non-local firms and, as a result, tbey invest more in suchsecurities.

Moreover, Canadian results point towards the needto incorporate the autocorrelation structure of tbisstudy's asset classes in tbe resampling techniques used toextend the dataset. This is in contrast with US basedstudies, sucb as Hickman et al. (2001). who report tbatgenerally the autocorrelation coefficient levels of tbeasset classes they examined were low enough to obviatethe need to incorporate correlation structures in theirresampling procedures. Finally, unlike others investigat-ing a similar question, our study is not based on simula-tions (e.g., Butler and Domian, 1991), but ratber on actu-al retum data that we bootstrap.

We find that investment outcomes at short borizonscan be quite different from outcomes at longer borizons.While the current market practice of life cycle investingis not fully supported as stocks continue to exhibit morefavourable risk-return payoffs than other asset classes(even at shorter time intervals), evidence is provided infavour of time diversification. Moreover, we find that theprobability of a shortfall in end-of-period wealtbdecreases as the holding period lengthens. We also findthat higher risk asset classes outperform lower risk assetclasses and bave higher end-of-period wealth for longerholding periods. Also, investing in higher risk assetclasses will increase benefits monotonicaliy as the timehorizon increases.

Our study is the most comprehensive to examinehorizon period returns in tbe Canadian markets. Itshould be useful to otber studies of Canadian marketsthat examine asset allocation strategies designed to pre-vent people from outliving their money. Some studiesneed distributions of actual asset returns at various bori-zons to run simulations in order to determine probabili-ties of shortfall (e.g., Milevsky, Ho, and Robinson,1997; Ho, Milevsky, and Robinson, 1994a; and Ho,Milevsky and Robinson. 1994b). Due to a lack ofempirical data of the sort we analyzed in our study,these prior studies used retum approximations from avariety of different sources. This is likely to have limit-ed tbe consistency and generalizability of tbeir fmdingsand conclusions. Nevertbeless, their conclusions areconsistent with our own as they too found that tradi-tional advice from financial planners on life cycleinvesting is not always correct. Finally, our resuits areconsistent with Barberis (2000, p. 225) wbo found tbat"even after incorporating parameter uncertainty, there isenough predictability in retums to make investors allo-cate substantially more on stocks, the longer their hori-zon". They are also consistent with Cocbrane (1999),

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who showed that stock prices exhibit negative serialcorrelation or mean reversion, which contributes to timediversification resulting in conclusions consistent witbour own.

Tbe remainder of our paper is structured as follows:The next section discusses the data and return computa-tions. The following section highlights the method weemployed, and the subsequent presents the results. Thefinal section summarizes the findings and concludes thepaper. . . , .

Sample Data and Retum Computations

Data from January 1957 to December 2003 wereobtained from the Canadian Financial Markets ResearchCentre (CFMRC) data base. This data source inciudesstock index (Canadian Universe Equally Weighted [EWJand Value Weighted [VW]) total retum data, as well asrates of returns on indices of Long Term Government ofCanada Bonds ([TBONDSJ over 10 years) and 91-dayTreasury Bills (TBILLS).

The CFMRC Equal Weighted Index return is tbeaverage monthly return for all domestic common equi-ties in tbe CFMRC database. The CFMRC VW Indexreturn is the market value weighted average monthlyreturn for all domestic common equities in theCFMRC database. A security's market weight isdefined as its market value at the beginning of the cur-rent month (shares outstanding multiplied by closingprice on tbe last trading day in tbe previous month)divided by the market value of all securities includedin tbe index. Returns used in the above indexes arefully adjusted for distributions.** Tbe 91-day T-Billreturn is defined as the return on a 91 day T-bil! pur-chased at the end of last month and sold at the end ofthe current month. Long Term Govemment of Canada(GOC) Bond Return is defined as the retum on a longterm GOC bond witb an approximate term to maturityof 17 years purchased at the end of last month and soldat the end of the current month. More on the descrip-tions of these series and their construction can befound in Hatcb and White (1988).These rates ofreturns are discrete monthly returns. In carrying outour tests, we converted tbese returns into continuouslycompounded retums.

Tbe summary statistics of tbe sample data arereported in Table I. A comparison of tbe mean and tbestandard deviation of annualized monthly retums of tbeseries confirm the standard risk-retum trade-off thatwould be expected across these asset classes. The end-of-period wealth is computed as the ending value of aninvestment of $1 in each asset class witb monthly com-pounding.

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Table 1Sumtnary Results Based on Annualized Returns*

Sample Data from January 1957 to December 2003 (Number of Observations = 564)

Retum series Mean Standard deviation End-of-period wealth

EWVWTBONDSTBILLS

0.16280.09390.07480.0652

0.19550.15930.08870.0104

92.849853.987943.230337.6473

* Returns and standard deviations are the anntialized measures of the monthly retums and standard deviations.

Note:HW Continuously compounded retums of the CFMRC equal weighted index.VW Continuously compounded returns of the CFMRC value weighted index.TBONDS Continuou.sly compounded retums of ihe Long Term Govemment Bond index. , •'TBILLS Conlinuously compounded returns of the T-Bill index.

Method

To reach the objectives, we undertook a comparativeassessment of tbe retums, risks, and end-of-period wealtbdistributions of investments in major Canadian assetclasses (e.g.. large stocks, small stocks, long term gov-emment of Canada bonds and Treasury bills) for differentinvestment horizons. To this end, we used the historicalindex series for eacb of the above asset classes, comput-ed holding period retums for a series of investment hori-zons, and then bootstrapped these retums to get therespective distributions of retums and end-of-periodwealth for each horizon. From these distributions, wethen examined the means, standard deviations, efficiencyof performance measures (such as the coefficient of vari-ation and the Sharpe ratio'̂ ), as well as the end-of-periodwealth outcomes of the various investment borizons.

Second, we examined the comparative efficiency ofinvesting in different asset classes. We did this by com-puting the return differentials between pairs of assetclasses at different holding periods, applying a sitnilarmethod. The incremental risk-retum profile at differenthorizons indicate the relative benefits of investing in dif-ferent asset allocation strategies (i.e., large and smallstocks versus govemment of Canada bonds; large andsmall stocks versus T-bills; and govemment of Canadabonds versus T-bills) at different investment horizons.

Resampling Technique and Computation of HoldingPeriod Returns

The holding period returns for the chosen assetclasses were computed for holding periods of 1, 5, 10,15

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and 20 years. If we bad computed holding periods as onewould normally do by starting at the beginning of eachdata series and working our way towards the end of theseries, holding period retums would overlap but tbe cal-endar order of retums within the holding periods wouldbe maintained and, hence, the autocorrelation structureof tbe retums would be preserved. There are shortcom-ings with computing such overlapping holding periodretums. First, tbe holding period retum observations arenot independent. Second, the results will be influencedby tbe initial values of the investing periods, so that tberesults cannot be generalized. Third, there will beincreasingly fewer observations as the holding periodlengthens.

To overcome these limitations and provide a suffi-cient number of nonoverlapping holding periods neededto generate a distribution of retums (especially with longholding periods such as 15 or 20 years), we extended thedataset by resampling tbe available sample data using abootstrap approach.'' Samuelson (1963) and Bodie(1995) argue tbat time diversification is driven by tbenonindependence or mean reversion of stock retums.This implies that in the absence of mean reversion therewould be no time diversification benefits. The approachfollowed in tbis section corrects for the independenceassumption to see whether with this improvement inmethodology a reduction in risk is still achieved whenthe independence assumption is maintained.

To compute holding period retums with the boot-strap method, we started with the monthly continuouslycompounded retums for each series and randomlyresampled the monthly retums with replacements untilthe number of observations needed to compute a bold-

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Table 2Autocorrelations

EWVWTBONDSTBILLS

of Return Series

1

0.2320.0820.0900.892

(Monthly

2

0.016-0.034-0.0100.862

3

0.0700.0500.0380.820

Data From

Lags

4

-0.010-0.015-0.0400.799

January

5

0.0130.0460.0000.785

1957 to December 2003)

6

0.0240.041 ,0.0000.760

Ljung-Box Q statistic

116.90065.65056.100

9942.700

Significance level

0.0000.0300.1700.000

Note:EW CFMRC equal weighted index continuously compounded retum.VW CFMRC value weighted index conlinuously compounded return.TBONDS Long Term Govemment Bond index continuously compounded retum.TBILLS T-Bill index confinuotisly compounded retum.

ing period retum was obtained. By repeating thisprocess 1000 times, we obtained 1000 samples with ran-domly selected starting observations. Means and stan-dard deviations of the 1000 holding period returns foreach asset class and for each holding period were thencalculated. To provide a relative measure of retums andrisk, we calculated the coefficient of variation. Thecoefficient of variation measure for each investmenthorizon was computed as the mean of the continuouslycompounded retums of the KKX) bootstrapped samplesfor each holding period divided by the standard devia-tion ofthe 1000 holding period returns. Another mea-sure of relative retum and risk calculated is the Sharperatio. The Sharpe ratio is calculated similarly to thecoefficient of variation except that the Sharpe rationumerator includes an asset's mean excess holding peri-od retum over TBILLS.

To determine which computational method wouldbe more efficient with our data, we had to understandthe time series properties of the data. To this end. weestimated the autocorrelation structure of the retumseries for each of the asset classes examined in thisstudy. Table 2 reports the fmdings. First lag autocorre-lation coefficients were quite high in the case of theTBILLS and the EW series, with the Ljung-Box Q datastatistically significant at the 1% level. The first lagautocorrelation coefficient was also statistically signifi-cant for the VW series at the 5% level. This autocorre-lation coefficient was quite low and statisticallyinsignificant for the TBONDS series. Jn general, how-

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ever, these results suggested the need to incorporate theautocorrelation structure in the resampling techniquesadopted to extend the dataset. These results are in con-trast with US based studies, such as Hickman et al.(2001), who reported that, other than in TBILLS, theautocorrelation coefficient levels were low enough toobviate the need to incorporate correlation structures intheir resampling procedures.

Further, to apply the proper method when retum dif-ferentials were examined, we also had to examine thecross correlations between the series employed in thisstudy. That is, unless the asset classes used are indepen-dent from each other, it would be inappropriate to direct-ly compare their retum distributions. As can be seenfrom Table 3, the cross correlation coefficients show thatTBONDS versus TBILLS, EW versus TBONDS andEW versus VW are significantly correlated. This indi-cates that in the resampling procedures, a draw in a par-ticular series must be matched by draws in the otherseries at the same point in time if the interseries rela-tionships are to be preserved when the retum differen-tials are analyzed. Moreover, these high cross correla-tions motivate the investigation of retum differentialsamong asset classes.

In the next section, we provide our resuits using thebootstrap method. However, comparative results (notreported here but available from the authors uponrequest), were also obtained by computing retums, risk,and end-of-period wealth distributions using overlap-ping holding period retums.

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Table 3Cross Correlations of Series

(Monthly Data From January 1957 to December 2003)

Lags

VW and TBONDSTBONDS and TBILLSEW and TBILLSEW and VWEW and TBONDSVW and TBILLS

1

0.0980.116

-0.0200.2100.1400.005

2

0.0800.117

-0.0300.010

-0.1900.002

3

0.0790.100

-0.0300.070

-0.150-0.010

Ljung-Box Q statistic

55.490344.970

17.58091.510-0.0300.120

Significance level

0.1800.0000.9900.0000.0000.980

Note:EW Continuously compounded returns of the CFMRC equal weighted index.VW Continuously compounded retums ofthe CFMRC value weighted index.TBONDS Conlinuously compounded returns of the Long Term Govemment Bond index.TBILLS Continuously compounded retums ofthe T-Bill index.

Results

Risk and Retum Behaviour - Independent Retums

Table 4 reports holding period retums of the assetclasses in question with retums calculated by resamplingwith the bootstrapping technique. In the resampling pro-cedure, the number of monthly return observations weredrawn independently to compute the holding periodretums (e.g., 12 observations for a one year holding peri-od), and the retums are summed to give the holding peri-od retum. This was repeated 1000 times and the meanand standard deviation of the 1000 holding periodretums are reported in Panel A of Table 4. Panel B showsthe coefficients of variation and Panel C shows theSharpe ratios.

Jn Panel A, both total horizon retums and risksincreased monotonicaliy. While annualized monthlyretums increased with the holding period, standard devi-ations declined monotonicaliy as the holding periodlengthened. Despite employing a method to preserve theindependence assumption, the benefits of time diversifi-cation are still apparent, as evidenced by the reduction ofstandard deviation of retums with longer horizons. Con-ceming the behaviour of the coefficient of variation, irre-spective of the investment horizon, investment inTBILLS' is by far the most efficient investment strategyover the past 47 years in terms of risk and retum. Theother three asset classes performed similar to each otherwithout a clear "winner". The implication of this is that

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investing in TBILLS, for most risk-averse investors, maybe a better investment strategy than investing in any otherasset class, regardless of the investment horizon. This isa very surprising finding that requires further investiga-tion as it contradicts previous research and all previouslyreported asset model allocations and recommendations.Perhaps previously inappropriate sampling techniquesproduced inappropriate asset allocation recommenda-tions followed by financial planners (to the detriment oftheir clients' wealth). The Sharpe ratios reported in Table4, Panel C behaved as one would normally expect; name-ly, increasing as one moved from low to higher risk assetclasses and from short to longer holding periods.

While academics and practitioners tend to favour theSharpe ratio as a measure of relative performance, theSharpe ratio masks some interesting findings that the coef-ficient of variation reveals. One may have to calculate bothmeasures to obtain a complete and accurate picture of therelative risk-retum behaviour of various a.sset classes. Forexample, due to its extremely low standard deviation,TBILLS seem to have been a very efficient asset classbased on the coefficient of variation in terms of retums perunit of risk. This infonnation, however, would have beentotally missed if one had kxiked solely at the Sharpe ratio,which calculates risk premium per unit of risk.

Wealth Distributions '

If someone chooses to invest in a particular assetclass for a specific holding period, what would be the

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Table 4Returns

Panel A -

and Risks

Returns

of Asset Classes

' • ' .

at Different Holding Periods

(Based on 1000 Bootstrapped Samples)

1 year 5 years 10 years 15 years 20 years

Total horizon returnsEW Mean

standardVW Mean

standardTBONDS Mean

standardTBILLS Mean

standardAnnualized retumsEW

VW

TBONDS

TBILLS

MeanstandardMeanstandardMeanstandardMeanstandard

deviation

deviation

deviation

deviation

deviation

deviation

deviation

deviation

0.1650.2020.0960.1620.0750.0920.0640.011

0.1580.1960.0890.1620.0730.0840.0650.010

0.8070.4440.4730.3550.3840.2110.3230.023

0.1640.0900.0960.0740.0750.0400.0650.004

1.6060.5860.9260.3550.7580.2830.6500.033

0.1620.0580.0940.0480.0740.0280.0650.003

2.4950.731t.4130.6171.0950.3530.9750.037

0.1660.0480.0940.0410.0730.0230.0650.002

3.3000.8771.9110.7121.5160.4041.3000.047

0.1640.0440.0940.0360.0740.0200.0650.002

Panel B - Mean coefficient of variation of asset categories

Total horizon retumsEWVWTBONDSTBILLSAnnualized returnsEWVWTBONDSTBILLS

0.820.580.876.15

0.810.550.876.5

L871.34L92

13.57

1.821.291.87

16.25

2.681.892.78

19.54

2.791.962.64

21.67

3.412.293.10

26.35

3.462.293.17

32.50

3.722.654.08

28.59

3.732.613.70

32.50

Panel C - Mean Sharpe ratios of asset categories

Total horizon returnsEWVWTBONDSTBILLSAnnualized returnsEWVWTBONDSTBILLS

0.5000.1980.119D

0.4750.1480.0950

1.0900.4220.2890

1.1000.4190.2500

L63I0.5740.3820

1.6720.6040.3210

2.0790.7090.3930

2.1040.7070.3480

2.2810.8580.5340

2.2500.8060.4500

N o t e : . • • ,

EW Continuously compounded retums of the CFMRC equal weighted index. • «VW Continuously compounded retums of the CFMRC value weighted indexTBONDS Continuously compounded retums ofthe Long Term Govemment Bond index.TBILLS Continuously compounded returns of the T-Bill index.Data are for January 1957-December 2003. This Table reports holding period returns of the asset classes studied in this paper over L 5. 10, LS and 20year holding periods with retums calculated by resampling with the bootstrapping technique. The coefTicienl of variation measure for each investmenthorizon is computed as the mean of the continuously compounded returns of the 1000 bootstrapped samples for each holding peritKl divided by thestandard deviation of the 1000 holding period retums. The Sharpe ratio is calculated in a similar fashion a.s the coefficient of variation except that thenumerator now includes an asset's mean excess holding period retum over TBILLS,

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Table 5The Statistics of the Distribution

EWMeanStandard deviationMedianMinimumMaximumProbability of shortfall in wealth

VWMeanStandard deviationMedianMinimumMaximumProbability of shortfall in wealth

TBONDSMeanStandard deviationMedianMinimumMaximumProbability of shortfail in wealth

TBILLSMeanStandard deviationMedianMinimumMaximumProbability of shortfall in wealth

of Ending Period Wealth with Different Holding

(Based on 1000 Bootstrapped !

1 year

1.1680.1921.1710.4671.891

20%

1.0920.1551.0960.4531.667

26%

1.0820.0851.0810.7711.334

i8%

1.0650.0111.0651.036LlIO0%

5 years

1.8280.4421.8360.3943.1663%

1.4840.3601.4940.2662.5389%

1.3700.1921.3710.8071.9794%

1.3260.0241.3241.2551.4060%

samples)

10 years

2.6430.6122.6250.9064.9080%

1.9530.5031.9370.4153.5193%

1.7530.2701.7530.5902.5370%

1.6490.0331.6471.5411.7570%

Periods

15 years

3.4660.7813.4740.8995.36!0%

2.4190.6252.4230.4383.8471.50%

2.1230.3342.1241.0833.1460%

. • _ 1

1.9760.0411.9751.8202.1030%

20 years

4.2390.8704.2171.3186.6590%

2.8660.7052.8650.6184.9510%

2.5170.3722.5141.4903.5670%

2.3010.0462.3012.1312.4370%

Note:EW Continuou.sly compounded relums of the CFMRC equal weighted index.VW Continuously compounded returns of the CFMRC value weighted index.TBONDS Continuously compounded returns of the Long Tenii Govemment Bond index.TBILLS Continuously compounded returns of the T Bill index.Data are for January 1957 December 2{KK)3. This Table reports the summary statistics of the distribution of the end-of-period wealth for the 1000 boot-strapped samples used in this paper. Probability of shortfall in wealth refers to the probability of negative total retums after 1. 5, 10, 15 and 20 years,respectively.

expectation of wealth at the end of the holding period?To answer this question, we assumed an investment of$1 in each of the four asset classes considered in ourstudy and compared the characteristics of the distribu-tion of end-of-period wealth to changes in the distribu-

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tion of ending period wealth as the holding periods wereprogressively extended for each asset class. As earlier,we examined holding periods of 1, 5, 10, 15 and 20-years.

For each sample, to compute the end-of-period

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Figure 1Cumulative Probability Distributions of End-of-Period Wealth for the Equally Weighted (EW) and ValueWeighted (VW) Stock Indices, Treasury Bonds (TBONDS), and Treasury Bills (TBILLS) at Different Horizons

Panel A

1-Year Ending Period Wealth for Different AssetClasses

Panel B

•EW-VW1B0NDS

-TBILLS

&-Year Ending Period Wealth for Different AMetClasses

0.5 1 1.5

Woalth (dolfar4

Panel C


5 a- 0.2

0

-EW

TBONDS.TBILLS

Waalth (dollars)

Panel D


2 4

WeaWi (dollare)

Noles\The cumulative probability represents the probabiliiy of ending up with less than a specified amount of end-of-period wealth, or shortfall risk. Forexample, ihe TBILLS curve is entirely to the righi of an ending wealth value of $1, reflecting the absolute safety of recovering the capital by investingin that a.ssel class. The other classes ail have higher degrees of shortfall risk.

wealth of investing $ I in a given asset class, we summedthe continuously compounded retums obtained with theresampling method for the number of periods required tomake up the holding period in question, and added theoriginal $1. For example, the value of a 5 year end-of-period wealth is given by I plus the sum of 60 continu-ously compounded retums. The summary statistics ofthe distribution of the end-of-period wealth for the 1000bootstrapped samples are reported in Table 5.

Table 5 demonstrates that stocks have a greaterwealth potential than other asset classes. The expectedvalue of $1 in 10 years is 2.64 in EW versus $1.95 inVW, $1.75 in TBONDS and $1.64 in TBILLS. The tablealso shows patterns in risk characteristics. The probabil-ity of ending with a shortfall in wealth decreases as theholding period lengthens. For example, for the EW asset

146

class, the probability of a negative total retum is 20%after 1 year, but 3% after 5 years and 0% after 10 years.The only asset class that guarantees an investor no short-fall, regardless of the holding period, are TBILLS. Ourresults contrast with those of Hickman et al. (2001) inthat our wealth distributions seem to be more symmetricthan the distributions of wealth relative to US data. Theirregularities found in US data by Hickman et al. ledthem to conclude that the higher risk of end-of-periodwealth for small stocks was mostly on the upside andthus irrelevant. This does not appear to be the case in ourstudy, as the risk of end-of-period wealth distributionsfor stocks is more evenly distributed in the down andupside.^

To provide better insight into asset class perfor-mance. Panels A to D of Figure I plot the cumulative

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Table 6Statistics of the Distribution of the Differential in Ending Period Wealth between Asset Classes with DifferentHolding Periods (Based on 1000 Bootstrapped Samples)

1 yearMeanStandard deviationMedianMinimum01-Percentiie05-Percentile!O-Percentile90-Percentiie95-Percentile99-PercentileMaximum

5 yearsMeanStandard deviationMedianMinimum01 -Percentiie05-Percentile10-Percentile90-Percentile95-Percentile99-PercentileMaximum

10 yearsMeanStandard deviationMedianMinimum01-Percentile05-PercentilelO-Percentile90-Percentile95-Percentile99-PercentiIeMaximum

EW/VW

0.9111.1760.869

-2.320-1.580-0.851-0.5432.4433.0774.0534.608

4.2412.6584.211

-2.884-1.7990.0560.8687.5708.632

10.32713.458

8.1883.8978.303

-3.017-0.0201.6902.848

13.05114.55717.55921.170

VW/BONDS

0.3061.9140.319

-8.879-4.283-2.763-2.1212.7483.2894.1836.271

1.1894.2131.248

-12.894-9.400-5.762-3.7626.5307.890

10.34813.041

2.4445.8922.403

-17.706-11.664-7.186-5.1739.629

12.38015.24120.227

TBONDS/ VW/TBILLS

0.0541.0050.014

-3.130-2.204-1.394-1.1411.3421.7142.6403.340

0.6732.3340.756

-4.850-4.433-3.147-2.3903.6504.5255.8578.086

0.9203.4460.835

-8.062-6.582-4.868-3.4995.1776.7309.193

11.437

TBILLS

0.3601.9310.452

-7.689-4.071-2.830-2.1492.7973.4334.2755.905

1.8634.2302.162

-12.122-8.506-5.079-3.333l.Wl8.429

11.01415.514

3.3655.8003.681

-14.751-9.611-6.338-4.16110.66212.50016.06419.191

15 years .MeanStandard deviationMedianMinimum01-Percentiie05-Percentile10-Percentiie90-Percentile95-Percentile99-PercentiIeMaximum

20 yearsMeanStandard deviationMedianMinimum01-Percentile05-Percentiie10-Percentile90-Percentile95-Percentile99-PercentileMaximum

EW/VW

12.3934.812

12.171-1.1511.4044.9326.404

18.68120.33623.34426.658

» » • • !

16.9015.454

16.4143.0875.7008.662

10.04424.26126.57229.23336.715

VW/BONDS

3.1467.2033.618

-22.519-13.985-9.176-6.55212.54714.50119.08524.361

4.6118.5954.940

-19.451-15.942-9.880-7.38315.72917.73020.63926.126

,. ,

TBONDS/TBILLS

1.7183.8881.537

-9.039-7.307-4.471-3.1666.8607.841

11.68113.527

2.5512.3274.625

-10.571-8.802-5.145-3.4358.3299.520

12.83214.624

-

1 ;

, •}

'. '^

VW/TBILLS

4.8657.0534.994

-21.229-12.033-5.915-4.35013.61415.91219.64327.306

6.9388.7307.362

-21.666-13.956

-7.641-4.61117.83720.98725.46230.361

Note:EW Continuously compounded returns of the CFMRC equal weighted index.VW Continuously compounded retums of the CFMRC value weighted index.TBONDS Continuou.sly compounded returns of the Long Tenn Govemmetit Bond Ltidex. - '''."-TBILLS Continuously compounded retums of the T-Bill indexData are for January 1957-December 2003. This Table reports the characteristics of the distribution of the difference in total horizon retums obtained fromthe bootstrapped stimple for indicated pairs at 1, 5,10,15 and 20 year horizons. Positive values in this Table represent situations where the firet asset classoutperforms the second, and vice versa.

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) , 138-152


probability distributions of end-of-period wealth for thefour asset classes we examined at holding periods 1. 5.10 and 20 years respectively.'̂ The cumulative probabil-ity represents the probability of ending up with a lessthan specified amount of end-of-period wealth, or short-fall risk. In Figure 1, Panel A. the TBILLS curve isentirely to the right of an ending wealth value of $1,reflecting the absolute safety of recovering the capital byinvesting in that asset class. The other classes all havehigher degrees of shortfall risk. A noteworthy fact is thelower cumulative probability of recovering tbe initialcapital in tbe EW index in comparison to the VW indexasset class despite the bigher standard deviation ofretums of tbe EW index observed in Table 4. A compar-ison of Panels A to D of Figure 1 shows the remarkablyfavourable performance of tbe HW asset class as theinvesting borizon lengthens. At a 20 year borizon thecurve is almost entirely to the right of tbe curves of allother asset classes. This reflects this asset's dominanceover all other asset classes in the first order sense.Regardless of the investor's risk preferences, a firstorder stocbastic dominance wouid imply a preference forthat asset class. • . . i

Asset Class Retum Differentials

Investors wbo wish to invest in one asset class overanother would probably cboose tbe asset class that out-performs. To provide the information sought by suchinvestors, we investigated the relative performance ofasset classes by examining tbe differential in retumsbetween pairs of asset classes over different holdingperiods. We again applied the bootstrap technique todraw monthly retums from the data set and computedholding period retums and end-of-period wealth values.The differential in end-of-period wealth is defined as thedifference in total borizon retums of a holding periodbetween the two asset classes considered. To preservethe contemporaneous correlation structure across assetclasses, we drew retum observations from each assetclass at the same time we resampled. From these data,we then computed retum differentials and tbe differen-tials in total horizon retums for the different boldingperiods. Table 6 reports tbe cbaracteristies of the distrib-ution of difference in total horizon retums obtained withthe bootstrapped samples. Tbe pairs examined areEW/VW, VW/TBONDS, TBONDS/TBILLS andVW/TBILLS.

Positive values in this Table represent situationswbere tbe first asset class outperforms the second, andvice versa. The mean values are all positive, and indicatethat on average, the first class is the superior performerin all cases. In other words, moving to higher risk assetclasses as the time horizon increases benefits the

148

investors. Moreover, investing in bigher risk classesincreases benefits monotonicaliy as the time horizonincreases. Tbe percentiles of the distribution give anindication of the probabilities of the differentials beingabove or below certain values. For example, with oneyear holding periods, there is more tban a 10% probabil-ity of the second asset class outperforming tbe first in allthe pairs considered, Yet, tbere is a considerable cbangein the probability values for such out-performance as thebolding period lengthens from 1 to 20 years.

To examine tbe changes of the probabilities of theretum differentials and tbe possibilities of out-perfbr-mance, we computed tbe cumulative probabilities of theretum differentials of the asset pairs. Panels A to D ofFigure 2 plot tbe cumulative probability distributions ofthe paired differences among the four asset classes weexamined at holding periods of I, 5, 10, 15 and 20 years.Figure 2. Panel A clearly demonstrates the increasinglysuperior pertbrmance of the EW class relative to the VWclass at longer borizons. Panels B and C of Figure 2compare tbe VW performance with TBILLS andTBONDS respectively. The greater upside potential ofVW relative to these two asset classes as the horizonlengthens is quite evident. For example. Figure 2, PanelC shows that the cumulative probability of an endingwealth differential of zero or less drops from approxi-mately 0.5 to 0.25 as the investment horizon lengthensfrom 1 to 20 years. This means tbat tbe probability of theTBONDS outperforming VW declines quite dramatical-ly with increases in the investment horizon. This indi-cates Ibat stocks may become more attractive to risk-averse investors when longer investment borizons areconsidered. The same conclusion can be reached whenconsidering investing in TBONDS versus TBILLS as isshown in Figure 2, Panel D.

Summary and Conclusions

Summary Highlights

Tbe objective of tbis paper was to investigatewhether the current practice among financial planners ofrecommending a heavy investment in stocks at an earlyage and progressively moving into cash or bonds asretirement approaches would be appropriate given thataccording to most investment models moving away tromequities altogether would be suboptimal- To this end fourquestions were examined: First, how do holding periodretums and investment risk cbange at longer investmenthorizons? Second, what implications does this havetowards the efficiency of portfolios invested in differentasset classes, and does this bave implications for an opti-

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Figure 2Cumulative Probability Distributions of Paired Differences of Ending Period Wealth Among the Value-Weight-ed Stock Index (VW), Equally-Weighted Stock Index (EW), TBONDS, and TBILLS at Different Horizons

Panel A

Equally Weighted (EW) Lass Value Weighted (VW) DidlngPeriod Wealth al Different Horizons

-10 10 20

Wealth (dollar^

30 40

Panel B

Value Weighted (VW) Less Treasury Bills (TBILLS) EndingPeriod Wealth at Different Horizons

^—EW-VW(I)

.^EW-VW(5)

EW-VW(IO)

i EWVW(15)

'^—eW-VW(20)

l i= S

Pro

-20 0 20

Weaiti) (dollars)

i^—VW-TBB±S(1)VW-TBILLS(5)

VW-TB1LLS{1O)

; Wtf-TBILLS(15)

^^—VWTB!LLS{20)

Panel C

Value Weighted (VW) Less Treasury Bonds (TBONDS) EndingPeriod Wealth at Diftrsnt Horizons

-20 0 20

Wealth (doltara)

40

-VW-7BONDS(1)

-VW-TBONDS (5)

VW-TBONDS (10)

-VW-TBONDS (15)

-VW TBONDS (20)

Panel D

Traasury Bonds (TBONDS) Lees Treasury Bills (TBILLS) EndingPeriod Wealth at Nfferont Horizons

5 *TB0NDS-TBILLS(5)

TBONDS-TBILLS(IO)

TB0NDS-TBILLS{15)

>10 0 10

Weal th (do1l*r«)

20

Notes:This Figure plots the cumulative probability distributions of the differences between the EW and VW asset classes examined in this paper at holdingperiods of 1, 5, 10. IS and 20 years, respectively. This Figure cleariy demonstrates the increasingly supierior performance of ihe EW ciasss relative tothe VW class al longer hori/on-;.

mal investment horizon when investing in a particularasset class? Third, what is the distribution of endingwealth as the investment horizon and the probability ofending up with a shortfall in wealth change as the invest-ment horizon lengthens in each asset class, and what isthe probability of one asset class outperforming another?Fourth, what does this imply for the preferences ofinvestors to invest in one asset class versus another?

In answering these questions, we computed retumsand risks at increasing horizons for a number of Canadi-an asset classes between 1957 and 2003, based on thebootstrapping technique. Although traditionally fol-lowed, high correlations across asset classes would makeit inappropriate to compare the mean and standard devi-ation of the asset classes.

Based on this method, we examined the risk andretum properties and wealth distributions for four assetclasses, namely. large cap stocks, small cap stocks, long-

term govemment of Canada bonds and T-Bills. Resultsshowed that investment outcomes at short horizons canbe quite different from outcomes at longer horizons.Investment outcomes can also vary with the asset class.While the current market practice of life cycle investingis not fully supported as stocks continue in many casesto exhibit quite favourable risk-retum payoffs comparedto other asset classes, even at shorter time intervals, evi-dence was provided in favour of time diversification. Inthis respect, evidence supporting Merton's models isprovided in that while a lighter stock position may beappropriate as one approaches retirement, there is noneed to exit stocks completely.

It was observed that with longer investing horizons,portfolio efficiency increases in all asset classes. Sur-prisingly, T-Bills seemed to be the most efficient assetclass across all time horizons examined based on thecoefficient of variation. This information, however,

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would be totally masked if one looked only at the Sharperatios. From this we conclude that one needs to calculateboth the coefficient of variation and the Sharpe ratio toobtain a full and complete picture of the relative return-risk pertbrmance of various asset classes.

Investing in stocks, especially those classified assmall cap, has a greater end-of-period wealth potentialthan other asset classes. Interesting patterns in risk char-acteristics are also found. The probability of ending witha shortfall in wealth also decreases with the holding peri-od. While T-Bills and government of Canada long-termbonds may be safer asset classes in the short run, as theinvesting horizon lengthens, the stock classes examineddominate in return. However, despite the length of theinvestment horizon, the risk of stock classes sti!l remainsa consideration that may make some investors unwillingto commit to stocks even for longer time periods. Furtherresearch may be needed to examine this last point.

We also looked at period by period retum differen-tials among the asset classes examined. We found thathigher risk classes have higher end-of-period wealth asthe time horizon increases and will thus outperformlower risk asset classes. Moreover, investing in higherrisk classes will increase benefits monotonicaliy as thetime horizon increases. There is, however, a consider-able change in the probability values for such out-per-formance as the holding period lengthens from I to 20years.

Applied Implications

What are the conclusions financial planners canreach from this study and how can they use our fmdingsto advise clients, especially those with 5 years or lessbefore retirement? The main lesson learned is that whileit is unlikely for an investor to end up with a shortfall inend-of-period wealth by investing in equities (regardlessof the time horizon), the issue of asset allocationbecomes more important as an investor approachesretirement. Figure 2, Panel C, for example, shows thatthere is about a 50% probability of bonds outperformingequities for time horizons of 5 years or less. Given thesignificant probability that other asset classes may out-perform equities at short investment horizons, a financialplanner should thus advise clients nearing retirement tocarefully consider an asset allocation strategy thatincludes equities and bonds, as well as T-Bills. Our fmd-ings are consistent with Milevsky, et al. (1997) and Ho,et al. (1994a; 1994b) who concluded that the traditionaladvice fmancial planners give on life cycle investing isnot always correct. Our conclusions are also consistentwith Barberis (2000, p. 225) who noted that "even afterincorporating parameter uncertainty, there is enoughpredictability in retums to make investors allocate sub-

150

stantially more on stocks, the longer their horizon". Sim-ilarly, Cochrane (1999). showed that stock prices exhib-it negative serial correlation or mean reversion, whichcontributes to time diversification and is consistent withour conclusions.

Limitations

In considering past findings when making recommen-dations on future performance, we must understand thehistoric economic environment within which these pastfindings were realized. Booth (1995, 2(X)1) argued thatwhile equity markets have not changed much over time,the govemment bond market has changed significantly. Henoted that there is definitely a break in the historical gov-emment bond yields of 1957. Govemment bond yieldswere "basically flat from 1936 to about 1956". "From1956-1981, yields then on average increased (. . .) bondyields on average declined from 1981 -2000" (Booth, 2001,p. 6). Our sample period started in 1957 and spanned to2003: a period which should better reflect the future thanthe pre-1957 period of relative yield stability and bondmarkei iltiquidity. Yet even within the 1957-2003 period ofour sample, the govemment bond markets experienceddramatic changes with implications for bond yields (andretums). These changes were driven by: (a) the change inthe contact of US monetary policy in the late 7O's and sub-sequent changes in Canada that broke the back of inflationby targeting money supply rather than interest rates, and(b) the increased liquidity of the Canadian govemmentbond market which was prompted by the increased gov-emment deficits in the mid to late 7O's and the resultingincrease in govemment bond issuance.

As a result, our sample period contained a period oflow and rising yields and a period of high and fallingyields. The average behaviour of govemment bondyields from 1957 to 2003 could very well be used as aproxy of future developments. We would have been con-cemed if we were standing at the early 198O's and hadattempted to extrapolate into the future as yields hadbeen rising in a secular fashion up to that point and viceversa if we had not had the pre-1980 history and weretrying to anticipate the future by looking at the 1981-2003 government bond performance in a declining yieldenvironment. We feel more comfortable about the com-bination of rising and falling yields over our sample peri-od as a better environment from which to draw conclu-sions ahout the future. Hence, the conclusions drawn inthis paper should be considered, on average, a realisticreflection of the expected performance of equity andbond retums going forward.

Can our results be translated into practical invest-ment decisions given that the asset classes examinedfrom CFMRC are not directly tradable? An investor can

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Still invest in tradable asset classes that are highly corre-lated with those examined from the CFMRC database.Exchange traded funds (ETFs), such as Barclays' iUnitsS&P 60, iUnits S&P Mid Cap, and iUnits Govemment ofCanada 10 Year bonds, are all ETFs that mimic large andintermediate cap stock portfolios and long term govem-ment bonds. Additionally, there is the existence of a larg-er number of mutual funds that also mimic the seriesexamined in this study.

Directions for Future Research

As discussed earlier, our study should be useful toother research on Canadian markets examining assetallocation strategies designed to prevent people fromoutliving their money. Studies, such as those ofMilevsky, et al. (1997), Ho, et al. {1994a, 1994b) neededdistributions of actual asset returns at various horizons torun simulations in order to determine probabilities ofshortfall. However, due to their different approach, theyused retum approximations from a variety of diflerentsources which may have affected the consistency of theflndings and generalizability of the conclusions. Futureresearch should examine whether their conclusionschange under our study's holding period retums.

Moreover, our finding T-Bills to be the most effi-cient asset class across all time horizons based on thecoefficient of variation is surprising and needs to beinvestigated further as it contradicts previous researchand previously reported asset model allocations and rec-ommendations. It may also suggest that previouslyemployed sampling techniques may have produced inap-propriate asset allocation recommendations widely fol-lowed by flnancial planners.

Finally, another possible avenue for future researchis to examine whether understanding an investor's riskaversion at a given point in time and throughout his lifecan enable us to better identify appropriate asset alloca-tions as the investor's horizon lengthens.

Notes

1. Discussions along these lines have led to what is knownas the time diversification debate (e.g., Athanassakos,1997; Bodie, 1995; Kritzman, 1994; Samuelson, 1963).

2. The Canadian economy is considerably more heavilytaxed, more mature, less growth oriented, more regulated,smaller, less diversified and more risk averse than the US.If one accounts for the major banks and utilities orpipelines, that is, the more mature, more regulated andless growth oriented sectors, one will have accounted fora large part of the Canadian economy. The Canadian econ-omy is also heavily resource based with resource stocks

151

dominating the Toronto Stock Exchange (TSX). About25-30% of the TSX capitalization has heen in metals,minerals, and Gold stocks, making the TSX less diversi-fied and more exposed to the business cycle swings thanthe US market

3. For example, 36% of Canadian equity funds, as opposedto only 16% of US equity funds, have beaten the indexover a 10 year period ending January 31, 2006 (See Car-rick, 2006).

4. The EW index is a good proxy of small cap stocks and theVW index is a good proxy of large stocks. For example,for the period 198l-2{)03. the average January return forthe VW index is .014 (t-stat=1.4I) and for the EW Indexis .043 (t-stat=3.27). It is well known that the "Januaryeffect" is a small fimi effect (see Keim, 1983).

5. See Sharpe (1966).6. A description of the bootstrap resampling technique can

be found in Efron (1979).7. While this may be stirprising and counter-intuitive, it is

consistent with Hatch and White (1988) who discussextensively the CFMRC data-base and produce summarystatistics of the various data series in this data-base over amuch shorter period.

8. What may explain, to some extent, the discrepancy, is thatour distributions are generated with 1(XX) iterations, whilethose of Hickman et al (2(K)1) were generated with 500iterations.

9. Due to space limitations, the cumulative distributions ofend of period wealth for the four asset classes for the 15-year holding period are not shown but are available fromthe authors upon request.

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