problems with monte carlo simulation

ContributionsContributions

106 Journal of Financial Planning/November 2001

1980s and 1990s with the microcomputer,think again. The 1980s and 1990s are justhistory repeating itself. It is actually thestory of the universities in the mid 1960s,who were taking delivery of the firstmainframe computers that had the com-putational power to do something useful.These multi-million dollar machines werenot under the tree for most businesses, letalone for a small child. Only the universi-ties with their research and instructionalfunds could afford them. It was here thatall of the quantitative toys were playedwith and tested—including Monte Carlosimulation, which is the topic for thispaper.

It was 1964 when Hertz [1964] firstsuggested using Monte Carlo simulation inbusiness applications. This paper createdan explosion of usage in all business disci-

plines, including finance. It was covered intextbooks and in courses where professorswere almost giddy with their new toys.Everything was going great whenLewellen and Long [1972] provided thewakeup call. They stated that MonteCarlo simulation failed to provide perti-nent information and that the informationthat it did provide could just as easilycome from single point estimates.

Monte Carlo simulation requires thatthe analyst set up a mathematical model ofthe process. This setup can be very timeconsuming and provides the simulation avery low benefit-cost ratio. Philippatos[1973] notes that while some dynamicproperties can be obtained through simu-lation, there are other techniques that canachieve the same purpose. He concludesby advising that future converts should

N a w r o c k i

by David Nawrocki

Mr. Nawrocki is professor of finance, Villanova

University and director of research for The QInsight

Group. He received M.B.A. and Ph.D. degrees from

Pennsylvania State University. He can be reached at:

[email protected] .

“Those who cannot remember the past arecondemned to repeat it.”

George Santayana (1863–1952)

A New Toy, Again

It was a wonderful time. Everybodyhad brand new computers that weremore powerful than anything that had

come before. With the increase in compu-tational power came the new toys withwhich to play. Linear programming, non-linear programming, integer program-ming, goal programming, queuing theory,Box-Jenkins ARIMA models, Almon dis-tributed lags and Monte Carlo simulationwere among the many toys under the tree.However, just like the day after Christ-mas, some of the toys still worked, somewere ignored and some were broken. Andthe parents learned which toys not to buyin the future.

Unfortunately, the parents turn intograndparents with foggy memories, andthe children into parents who don’tremember the toys that broke. So severalgenerations later, the old toys are backunder the tree waiting to break again.

If this Christmas story sounds like the

Monte Carlo simulation has enjoyed a resurgence in financial

literature in recent years. This paper explores the reasons why

implementing Monte Carlo simulation is very difficult at best

and can lead to incorrect decisions at worst. The problem is

that the typical assumption set used in Monte Carlo simulation

assumes normal distributions and correlation coefficients of

zero, neither of which are typical in the world of financial

markets. It is important for planners to realize that these

assumptions can lead to problems with their analysis. Financial

planners will find that exploratory simulation provides

equivalent or better answers and is simpler to implement

without assumptions.

Finance and Monte Carlo Simulation


use the technique sparingly and, perhaps,only after everything else fails. Myers[1976] also agrees with the Lewellen andLong position, though he points out thatMonte Carlo simulation would be appro-priate if the analyst has no other idea howa variable may work.

Rubinstein [1981] echoes Myers’ senti-ments and develops a set of criteria to beused in deciding whether it is appropriateto use Monte Carlo simulation. MonteCarlo simulation is appropriate when• It is impossible or too expensive to

obtain data• The observed system is too complex• The analytical solution is difficult to

obtain• It is impossible or too costly to validate

the mathematical experimentA more recent study by Rees and Sut-

cliffe [1993] confirms the above criteria.Essentially, Monte Carlo simulation isuseful only when nothing else will work.It has proved to be useful in academicfinancial and statistical research, but onlywhen the data or the analytic solution isnot available. This is not the case in theinvestment decisions typically faced byfinancial planners. Financial market data isplentiful and cheap. Analytic models areavailable to quickly analyze the data andprovide the same or better answer thanMonte Carlo simulation. Chau and Nord-hauser [1995] provide a good overview ofarticles using Monte Carlo in accountingand capital budgeting research. Theyfound that Monte Carlo simulation hasbeen found useful wherever data are notavailable, but they found no articles sup-porting its use with financial marketreturns.

There was a recent article on MonteCarlo simulation in this journal by Abey-sekera and Rosenbloom [2000] where theanswer to the authors’ problem becameevident as soon as they stated theirassumptions for the Monte Carlo model.Obviously, there is no need to run themodel when we already know the answer.

Monte Carlo simulation is also the focalpoint for another recent article in thisjournal by Kautt and Hopewell [2000].Let’s apply the Rubinstein criteria to botharticles. The appropriate use of MonteCarlo simulation would require a “no”answer to the three questions in Table 1.Given that both data and analytic modelsare available, Monte Carlo simulation isnot appropriate in either article.

TABLE 1 HERE

Today, history is repeating itself withnifty spreadsheet add-ons that do all of theheavy lifting for Monte Carlo simulationand with the converts thrilled with theirnew toy. Rekenthaler [2000] states in hisMorningstar.com column that he sees anarticle every couple of weeks that explainshow the new technique of Monte Carlosimulation improves upon conventionalmath. After discussing the fact that MonteCarlo simulation has been around for awhile, Rekenthaler points out that you canarrive at the same answer with a mathe-matical formula and with Monte Carlosimulation, except that the formula isslightly more accurate and is quicker ingetting you the answer. He concludes thatMonte Carlo simulation would be appro-priate whenever the mathematical formulacannot be solved. Déjà vu all over again,only 25 years later.

Evensky [2001] gets to the heart of thematter—that is, risk versus uncertainty:“The problem is the confusion of risk with

uncertainty. Risk assumes knowledge ofthe distribution of future outcomes (i.e.,the input to the Monte Carlo simulation).Uncertainty or ambiguity describes aworld (our world) in which the shape andlocation of the distribution is open toquestion. Contrary to academic ortho-doxy, the distribution of U.S. stockmarket returns is far from normal.”

Evensky continues by examining theassumption set used by planners, whichincludes normal distributions and theexpected returns and variances for stocksand bonds. He cites a survey of experts fortheir forecast of future returns given thecurrent economic environment. Theirforecasts for stock returns ranged fromnegative returns to positive returns of 11percent for the next five to ten years,making it difficult to decide on the meanof the distribution for the Monte Carlosimulation. Evensky notes that MonteCarlo simulation is an effective way ofeducating people regarding the uncer-tainty of risks, but rather than reducinguncertainty, it increases the guessworkmanyfold because of its assumption set.The two articles that have appeared in thisjournal include both normal distributionsand zero correlations in their assumptionsets. It is important for planners to realizethat these assumption sets can lead toincorrect decisions and that the implemen-tation of Monte Carlo simulation is goingto take a great deal of care. It is not aneasy implementation.

At this point, Monte Carlo simulation


107Journal of Financial Planning/November 2001

N a w r o c k i ContributionsContributions

Rubinstein Criteria

Are data available?

Is analytic model available?

Is mathematicalexperiment validated?Notes: 1. Modern portfolio theory2. Geometric mean criterion. See Gressis, Hayya and Philippatos [1974].

Abeysekeraand Rosenbloom

Yes

Yes1

Yes

Kauttand Hopewell

Yes

Yes2

Yes

AppropriateUse ofMonte Carlo

No

No

No

T A B L E 1



is not generally used by financial planners[McCarthy, 2000] nor has there been astrong case put forward to encourage suchuse. One could accept the judgment of thenumerous authors cited above, but thereisn’t a clear picture as to why so manyauthors recommend against the use ofMonte Carlo simulation. The problem isthat it is very difficult to implement anassumption set that matches the real worldwhen using Monte Carlo simulation. Thepurpose of this paper is to discuss variousproblems that are inherent in the assump-tion set of a Monte Carlo application. Thenext section provides a short and rathernarrow description of simulation model-ing. Later sections present a description ofpotential problems with simulation andprovide an example demonstrating someof these problems.

Simulation Models

Philippatos [1973] provides a useful defini-tion of simulation. Simulation is the use ofa model to approximate the behavior of areal-world system within an artificial envi-ronment. The artificial environment iswhere the analyst attempts to model thereal-world system. In finance, we usuallyare working with an accounting model ofcash inflows and cash outflows. Therefore,a model could be a simple definitionalmodel:

Y = W + X (1)

We might also use linear models such as(2) or nonlinear models as in (3).

Y = a + bW + cX (2)

Y = a + bX + cX2 (3)

In all cases, Y is the result of themodel. It may be the answer to a problem,or it may be the forecast of the future. Inany event, we want to know Y. W and X

are known as exogenous variables becausetheir values are determined outside of themodel. We have to have values for W andX in order to get an answer for Y. Howthe values are determined for the exoge-nous variables, W and X, determines thetype of simulation model. Monte Carlo isjust one type of simulation used to gener-ate values for the exogenous variables.There are four general types of simula-tion.

Monte Carlo (random number genera-tor). This is used when it is not possible toobtain sampling data but we have someknowledge about the population. Actualsampling is either impossible or uneco-nomical to use in order to generate valuesfor the exogenous variables. Monte Carlosimulation could also be used to model theerror process in a regression relationship.In equation (4), the error term, e, isassumed to be normally distributed andcan be simulated using Monte Carlo.

Y = a + bW + cX + e (4)

Tactical (sensitivity analysis). Here westudy behavior of the model as changesare made in parameters and assumptions.For example, we would ask how changesin a, b and c in equations 2 and 3 affectthe results.

Strategic or exploratory. Exogenousvariables are changed to reflect certaincourses of action. This is the famous“what if” simulation technique. What ifwe implement this action? What if thathappens? This type of simulation gener-ally relies on a model built from historicaldata and sometimes may be called histori-cal simulation. However, it can be usedwith other types of models includingMonte Carlo simulation. The key is to setup a model of how the world works andthen test different policies or decisionsthrough the model to see what works.

Interactive. A human decision processdetermines the exogenous variables. Inaddition, an artificial intelligence or

mechanical decision process could deter-mine the exogenous variable. This type ofsimulation is typically played in real time.

A simulation model may include sev-eral simulation techniques. For example,playing a game of Monopoly is an exampleof Monte Carlo simulation and interactivesimulation. Rolling the dice is an exampleof the former. Deciding to buy Boardwalkis an example of the latter. Kautt andHopewell [2000] provide another examplein their paper, which combinesexploratory simulation and Monte Carlosimulation.

In finance, exploratory simulation isgenerally the most useful. It doesn’t createa large computational burden and is rela-tively easy to implement. The use of his-torical data provides as realistic a model ofreal-world behavior as can be achieved.Monte Carlo simulation is generally anoversimplification of the real world. Aspointed out by Evensky [2001], the prob-lem is with the assumption set used inMonte Carlo simulation. In the typicalLake Wobegon world of Monte Carlo sim-ulation, all distributions are normal and allcorrelations are zero. It doesn’t capture thecomplexity of interrelationships that arecontained in the historical data. MonteCarlo variables assume that the processesbeing studied are independent of eachother and that each value is a randomdraw from a distribution, or serially inde-pendent. Proponents of Monte Carlo sim-ulation point out that the available com-puter programs can handle dependentrelationships between exogenous variables.However, the problem is that the interre-lationships between two or more variablesare generally quite complex, and it is diffi-cult to determine the correct relationshipsand distributions.

Using Correlation Coefficientsto Measure InterrelationshipsBetween Variables

N a w r o c k iContributionsContributions

The relationship between two variablescan be described by a correlation coeffi-cient that tells how strong the relationshipis between two variables and whether therelationship is a positive or negative rela-tionship. However, more than one correla-tion is needed in order to provide a realis-tic model of the relationship. Figure 1demonstrates how a correlation is com-puted using paired observations of the twovariables over time. Since the relationship

is across the two variables, it is called across correlation. Using daily returns forthe Nasdaq composite and the S&P 500index from 1970 to 1994, the cross correla-tion is computed to be 0.79, which is a sig-nificant positive relationship.

FIGURE 1 HERE

At this point, Monte Carlo is still in thegame. Most spreadsheet add-in packages

can model this relationship. Unfortu-nately, there is still the relationship overtime or serial dependence. Rolling a fairpair of dice represents an independentprocess over time—that is the result of thecurrent roll cannot be used to predict thenext roll of the dice. When serial depend-ence occurs, the result of the current rollcan be used to predict the next roll. In thiscase, the dice will not be fair. Serialdependence is measured with a serial cor-relation. Note that in Figure 2, the obser-vation pairs used for the computation ofthe correlation coefficient is across time,not across the two variables. The two vari-ables each have their own serial correla-tion.

FIGURE 2 HERE

The Nasdaq has a significant serial corre-lation coefficient and therefore cannot bemodeled using an independent MonteCarlo simulation process. The S&P, onthe other hand, could be modeled usingMonte Carlo simulation, as it is not seri-ally correlated over time. Unfortunately,this isn’t the only serial dependence thatoccurs between two variables. Figure 3demonstrates the concept of a cross-serialcorrelation. A cross-serial correlation cap-tures a lagged correlation between the twovariables. Note that the observation pairsrepresent both a serial correlation and across correlation process.

FIGURE 3

Figure 3 demonstrates a cross-serial corre-lation between the Nasdaq and the S&Plagged one day. Note that there is signifi-cant cross-serial correlation in both caseswhere the S&P is lagged and when theNasdaq is lagged. By now, the modelrequired by the Monte Carlo simulation inorder to capture all of the cross, serial andcross-serial correlation coefficients is quitecomplex. To make matters worse, there isonly a one-day lag at this point. As we go



N a w r o c k i

Using daily returns, the correlation is 0.79

Wednesday ThursdayMonday Tuesday Friday

Computation of Cross Correlations Between S&P andNasdaq Indexes Using Observation Pairs—1970 to 1994

F I G U R E 1

Nasdaq 100 Index

S&P 500 Index

Using daily returns, the correlation is 0.79

• Pairwise observations are for a one-day lag• Serial correlation for Nasdaq = 0.13 (5.47)• Serial correlation for S&P 500 = 0.01 (0.62)T-test in Parentheses


Serial Correlations for Nasdaq and S&P Indexes—Daily Data 1970 to 1994

F I G U R E 2

Nasdaq 100 Index

S&P 500 Index

to longer lags such as a week, month,quarters and years, the model becomeseven more complex. Nonlinear Relationships

There is even more bad news. All of thecorrelation coefficients that we have justdescribed represent linear relationshipsbetween variables. They may be nonlin-ear, which is very likely with stock marketdata. Figure 4 provides a graphical view ofthe nonlinear serial correlation for theS&P 500 index computed from daily datafrom January 1970 to August 2000.

FIGURE 4 HERE

The linear serial correlation for theS&P 500 in this case (daily data from 1970to 2000) is not very impressive (0.09).However, there is a clear nonlinear pat-tern to the graph. Generally with stockmarket data, the linear serial correlation isclose to zero, but at the same time there issignificant nonlinear serial correlation.Figure 5 demonstrates a perfect linearserial correlation for the data. A compari-son of the two graphs provides a visualpicture of the differences between linearand nonlinear serial correlation.

FIGURE 5 HERE

An analyst attempting to formulate aMonte Carlo simulation model is going tobe overwhelmed by all of the lagged rela-tionships going back into time. Addingnonlinear relationships just makes theburden unbearable.

Nonstationary Distributions

This is another area of interest when mod-eling market returns. Monte Carlo simula-tion assumes that the market decides onone distribution with one mean and onestandard deviation and that the marketwill continue to follow that distributionuntil the end of time. To be honest, we dohear about mean reversion where every-one is conjecturing that the market will bereverting to its long-term mean sometime



N a w r o c k i

• Correlation Nasdaq and S&P lagged one day = 0.0863 (3.64)• Correlation S&P and Nasdaq lagged one day = 0.0474 (1.99)T-test in Parentheses


Nasdaq 100 Index

S&P 500 Index

Cross-Serial Correlation Between the Nasdaq 100 IndexAnd the S&P 500 Index Lagged One Day—Daily Data 1970 to 1994

F I G U R E 3

Ret

urn

Return Lagged One Day

Serial Correlation = 0.09Annual Return = 9.78%

1970–August 2000

Nonlinear Serial Correlation S&P 500

F I G U R E 4

–5.5 –4.5 –3.5 –2.5 –1.5 –0.5 0.5 1.5 2.5 3.5 4.5

1.5

1.0

0.5

0

–0.5

in the next ten years. This may be com-forting to an analyst; unfortunately, theeconomic conditions that generate finan-cial market returns are constantly chang-ing. Thus, the mean and standard devia-tion of the market returns are constantlychanging. Recent empirical work byWhitelaw [1994] and Perez-Quiros andTimmermann [2000] indicate that meansand standard deviations vary throughoutthe business cycle. There isn’t a stable dis-tribution to hang your hat on. Again, non-stationary distributions are very difficultto model for inclusion into a Monte Carlosimulation.

Fortunately, there is a simple alterna-tive to Monte Carlo simulation:exploratory simulation using historicaldata. The data represents the model orlaboratory that is used in exploratory sim-ulation and it contains all linear and non-linear correlations, all normal and nonnor-mal distributions and all stationary andnonstationary distributions. There is noneed to try to model them as with Monte

Carlo simulation. In addition, it is anunfortunate fact that most analysts are notgoing to model these relationships. Theywill use Monte Carlo simulation withlinear correlations of zero and normal dis-tributions as did the two papers thatappeared recently in this journal. Thissimplification can lead to incorrectanswers, especially with financial marketdata. Financial market data is quite com-plex with nonlinear relationships andstrangely shaped return distributions.

A Demonstration Case Study

Table 2 provides the results from ademonstration study using exploratorysimulation and Monte Carlo simulation.An investment fund is currently equallyinvested into stocks and timber. The man-ager wants to know whether to sell someor all of the timber and increase theamount invested in stocks. The data con-sists of semiannual returns from 1960 to

1998. The results are summarized inTable 2. It is easy to see that timber has alower return, a higher standard deviationand a higher downside risk (semidevia-tion) than the stock investment. So itseems to be a clear decision to investsolely in stocks, yet there is the matter ofthe low correlation between the twoinvestments. It is slightly negative but notsignificantly different from zero. How-ever, a zero correlation is still able to con-tribute considerable diversification to aportfolio.

TABLE 2 HERE

The next step is to study the perform-ance of the portfolio that has an equal allo-cation between the two investment alter-natives. Using Monte Carlo simulation,the following assumptions are made.

The distributions for stocks and timberare normally distributed. The cross corre-lation between the two investments iseffectively zero. All serial correlations arezero as are all cross-serial correlations.Finally, all relationships are linear.

The portfolio performance generatedby Monte Carlo simulation to model theportfolio is lackluster. It does havereduced risk compared with an all-stockportfolio but with a lower return. Thelower return results in poor risk-rewardperformance as evidenced in a lowerreward to semivariability (R/SV) ratio.(The reward to variability (R/V) ratio isalso lower. The R/V ratio uses the stan-dard deviation as the risk measure whilethe R/SV ratio uses the downside riskmeasure, semideviation.) The manager,given this analysis, is likely to shift fundsfrom timber to stocks. In fact, an optimalallocation (from an optimizer that maxi-mizes the Roy reward-to-variability ratio)would allocate 96 percent to stocks and 4percent to timber.

The alternative is exploratory simula-tion, which uses historical data to simulatethe performance of the portfolio over a rel-



N a w r o c k i

Ret

urn

Return Lagged One Day

Serial Correlation = 1.0

–5.5 –4.5 –3.5 –2.5 –1.5 –0.5 0.5 1.5 2.5 3.5 4.5

6.0

0

–2.0

4.0

2.0

–4.0

–6.0

Perfect Linear Serial Correlation

F I G U R E 5

evant historical period. The portfolioresulting from this analysis includes alldependent relationships, all nonlinear rela-tionships and does not assume a normaldistribution. Note that the exploratorysimulation portfolio with equal allocationto timber and stocks has a higher returnthan the Monte Carlo simulation. Thestandard deviation is lower, and althoughthe semideviation is higher, the R/SV ratiois higher than the R/SV ratio for stocks bythemselves. The timber-stock portfolioprovides a better risk-return performancethan the stocks-only portfolio.

How can the return be higher for theexploratory simulation portfolio as com-pared with the Monte Carlo simulationportfolio? Simply stated, the exploratorysimulation portfolio return is the actualeconomic return of the portfolio. TheMonte Carlo return is a calculated returnassuming normal distributions, serial inde-pendence and linear relationships. The

exploratory simulation technique makesnone of these assumptions.

The Monte Carlo simulation suffersfrom a number of problems, which aredemonstrated in Table 2 and Figure 6.

The first problem is that the distribu-tions for timber returns and stock returnsare not normal distributions. The stockdistribution has a higher kurtosis thanwould be expected if the distribution wasnormal. The timber distribution is signifi-cantly skewed and has a higher kurtosis.Second, there is a significant lagged rela-tionship between timber and the S&Plagged six months.

Finally, there is the problem of nonlin-ear relationships. Figure 6 demonstratesthat the cross-correlation relationship ofthe S&P and timber is nonlinear. Thegraph indicates that there is a negativerelationship between timber and stocks inthe –3 percent to +4 percent range. How-ever, the linear correlation coefficient



N a w r o c k i

1960 to 1998

Semi-Annual

Investment

Equity (S&P)

Timber

Simulation of 50-50 Portfolios

Std. Dev. Semidev. Skewness Kurtosis

Correlation T-Test

R/SV Ratio R/V RatioAnnual Return

12.26

5.06

11.6883

16.1417

6.2129

8.6058

0.5612

0.0019

0.2983

0.0035

0.1972

2.1890*

3.5348

5.9212

Monte Carlo

Exploratory

8.55

10.48

9.9830

9.0052

6.1201

4.3520

0.2806

0.6065

0.1721

0.2931

0.1517

0.7646*

3.0987

5.0010

Cross-Correlation S&P—Timber

Cross-Serial S&P—Timber Lagged 1 Period

Cross-Serial Timber—S&P Lagged 1 Period

*Significant at five percent confidence intervalR/SV Ratio = (Semiannual Portfolio Return – T-Bill Rate)/SemideviationR/V Ratio = (Semiannual Portfolio Return – T-Bill Rate)/Standard DeviationMonte Carlo—10,000 iterations using semiannual returnsT-Bill Rate = 5.00%

–0.1196

–0.0747

0.2158*

–1.0502

–0.6485

1.9137

Investment Results from Monte Carlo Simulation andExploratory Simulation of an Investment Portfolio

T A B L E 2

Tim

ber

Ret

urn

S&P Return

1960–1998

–10.0 –3.0 –0.5 0.5 4.0

12.0

6.0

4.0

10.0

8.0

2.0

0.0

Nonlinear CorrelationBetween S&P and Timber

F I G U R E 6

underestimates the strength of the rela-tionship. The computed correlation isclose to zero and is not statistically signifi-cant. All of these problems conspire toprovide the wrong decision—that is, sellthe timber investment.

FIGURE 6 HERE

Forecasting

A recently received e-mail about econo-mists included the four golden rules ofeconometric forecasting: (1) think bril-liantly, (2) be infinitely creative, (3) beoutstandingly lucky and (4) otherwise,stick to being a theorist.

As investment literature is fond ofpointing out for legal reasons, past per-formance is not a guarantee of futurereturns. This counsel applies equally toboth exploratory simulation and MonteCarlo simulation. Picking historic periodsfor an exploratory simulation that areequivalent to the current situation is prob-

lematic. However, even more so is pickinga distribution to use in a Monte Carlo sim-ulation. It should be noted that we are notforecasting per se when we useexploratory simulation. We are trying dif-ferent policies to find out which one willbest meet our needs. We are stress testingthe policy with data that includes all ofthe nonnormal, nonlinear and nonstation-ary interrelationships.

Historic data has suffered from a poorreputation because of straight-line extrap-olations of expected returns and variances.Exploratory simulation with historic datais not a straight-line extrapolation. In addi-tion, it includes the factors that drive stockmarket returns. Monte Carlo simulationhomogenizes away the factors that drivestock returns. Can you forecast stockreturns without bringing forward a fore-cast for any financial variable? MonteCarlo simulation lets you do this bysimply specifying the distribution forstock returns. Exploratory simulationrequires that you have some idea of whatis currently happening and what could

happen next in the financial markets.Then, you pick a comparable period inhistory and test different investmentstrategies throughout that period. Next,you select the strategy that worked best inthe period and you hope for the bestbecause you are dealing with uncertainty.

The goal in the equity-timber casestudy is simply to see which strategy per-formed best during the various economicconditions that existed in the past 40years. Did the portfolio run into liquidityproblems during these decades thatincluded recessions, high inflation, marketcrashes, credit crunches, wars and energyshortages? It is important that the selectedstrategy be able to deal with these scenar-ios, as it is doubtful that any of these havebeen banished from our future. It isimportant to study these periods as timberand equities perform differently duringrecessions and during periods of highinflation. As a result of studying theseperiods, the investor should give timber ahigher allocation in the portfolio. We arenot forecasting that the portfolio will



N a w r o c k i

Co

rrel

atio

ns

6207 6407 6607 6807 7007 7207 7407 7607 7807 8007 8207 8407 8607 8807 9007 9207 9407 9607 9807

Dates

Means

Whitelaw 1994 JF Graph with CRSP Value Weighted Index Updated to December 1999

F I G U R E 7

0.09

0.03

0.01

0.07

0.05

–0.01

–0.03

–0.05

–0.07

–0.09

Volatility Peaks Troughs

return the 10.48 percent annually obtainedin Table 2. Rather, it is a forecast that thisstrategy will provide the best results givendifferent economic conditions over theyears.

The probability results from MonteCarlo simulation may look impressive to aclient. However, if that number is derivedfrom assumptions that are not realistic,there is no value to the number. It doesprovide a good excuse: “Well, the MonteCarlo model did tell us there was a 15 per-cent chance of this happening.” The realquestion is whether it provides a goodpolicy decision. Recent research in theJournal of Finance by Whitelaw [1994]and Perez-Quiros and Timmermann[2000] indicates that the stock marketmean returns and volatility are differentbetween peak-to-trough and trough-to-peak periods in the business cycle. In thecurrent environment (January 2001),would an investment advisor be comfort-able making a recommendation usingMonte Carlo simulation based on somenormal distribution derived from the last75 years? The alternative would be to setup an exploratory simulation that looks atrecent history (the last 30–40 years) forprevious peak-to-trough periods in orderto see which asset mix provides the bestresults after the economy has reached apeak.

The Whitelaw [1994] study is particu-larly interesting. He uses a smoothingtechnique to filter out the noise thatoccurs in the stock market data. Theresulting means and standard deviationsare varying over time and are related tothe business cycle. Note that the greatestvariations occur during recessions. Thereis also the unsettling tendency for the riskto increase while at the same time returnsare decreasing. Rarely does increasedvolatility lead to increased returns. Portfo-lio theory tells us that increased risk leadsto increased returns, but this refers to anoptimized outcome from a subset ofmarket data while the market itself has

never been optimized. It can do whateverit pleases, and does.

FIGURE 7 HERE

When should a financial planner useMonte Carlo simulation? Whenever a vari-able in the problem cannot be estimated oris not available. Appropriate variablesmight include a person’s life span or irreg-ular cash flow needs for a retirement prob-lem. Rekenthaler [2000] discussesHopewell’s problem of in-retirement plan-ning with irregular cash-flow needs. Thisseems to be an appropriate application forMonte Carlo simulation, but only forthose variables where the data are notavailable. Monte Carlo simulation gives upinformation through its assumption setwhenever variables with readily availabledata are used.

Summary

Monte Carlo simulation is useful for thosecases where data and analytic modelssimply are not available. Otherwise, itrequires more work and does not result ina demonstrably better answer than otheranalytic techniques. The benefit/cost ratiojust is not there.

The problem with Monte Carlo simu-lation is the assumptions that have to bemade in the model in order to easilydeploy Monte Carlo simulation. Since fewplanners have formal training in opera-tions research, they will tend to makethese assumptions without understandingtheir implications. Other forms of simula-tion, exploratory and tactical, do not makethese assumptions and are easier todeploy. Monte Carlo simulation implieswe are operating under conditions of riskand we know the underlying distributions.However, the financial markets are reallyoperating under conditions of uncertaintywhere we don’t know the distribution.Under these conditions, the best policy is

one that adapts to the uncertain condi-tions. It is important to stress test policiesto see which have proved the most adapt-able under severe conditions. This is therole of exploratory simulation with his-toric data.

Finally, McCarthy [2000] notes thatfinancial planners have been slow toemploy Monte Carlo simulation. Propo-nents of Monte Carlo simulation have todemonstrate the additional benefits ofusing Monte Carlo simulation before itcan be recommended for wider use withinthe profession. Its benefits do not lie in thearea of analyzing aggregate marketreturns. However, it could prove useful inother areas of a financial planning practicewhere data are not readily available. Pro-ponents of Monte Carlo should explorethose areas for it to become a valuableaddition to financial planning.

References

1. S. P. Abeysekera and E. S. Rosen-bloom, “A Simulation Model forDeciding Between Lump-Sum andDollar-Cost Averaging,” Journal ofFinancial Planning, 2000, Vol. 13, No.6, pp. 86–96.

2. C. Chau and F. Nordhauser, “AddingDynamics to the Master Budget: Flexi-ble Budgeting with Micro-ComputerSimulation Procedures,” Journal ofAccounting and Computers, 1995,available at http://swcollege.com/acct/jac/jac11/jac11_article3.html#foot2.

3. H. Evensky, “Heading for Disaster.”Financial Advisor, April 2001, pp.64–69.

4. N. Gressis, J. Hayya and G.C. Philip-patos, “Multiperiod Portfolio Effi-



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by John D. Kingston, CFP

John Kingston, CFP, is principal of a registered

investment adviser and provides financial planning

support services to advisory professionals across the

country. He may be reached at

[email protected].

These days you can find a number ofWeb sites and financial planningsoftware programs that suggest to

clients that they should feel comfortablethat their objectives will be met becausetheir financial plans have been run throughthousands of random mathematical simula-tions using historical asset class returns.

Supporters of Monte Carlo simulationsay it helps investors choose the mostattractive course of action, providing infor-mation about a range of outcomes such asbest and worst case scenarios and the prob-ability of reaching a specified target. Theybelieve it’s a better alternative to using“one” blended rate-of-return assumption inthe financial planning process and thenprojecting that by 20 or more years.

If “one” blended rate of return is theonly option in financial planning, thenMonte Carlo may yet serve a purpose inhelping consumers understand the risk ofmissing their objective. If not, the profes-sion may find this exercise is little morethan hype developed by mathematiciansand promoted by financial planning soft-ware companies. Monte Carlo simulationsmight only be a tool to show clients howtechnical we are, giving them one morepiece of evidence why they should placetheir trust in us financial planners in thefirst place. In many cases, the amount and

predictability of a client’s retirementincome (such as Social Security and pen-sions) and lifestyle changes make the uncer-tainty of investment returns a less impor-tant issue in the overall financial plan.

A Common Trait Among BearMarkets

Monte Carlo simulations may provideinsight regarding investment returns overa limited period of time, but when used toanalyze an overall financial plan, there areseveral problems. First, the longer thetime period in question, the closer thedeviation of returns should converge onthe long-term average—that is, of course,if the analysis properly takes into accountthe historical distribution of returns anddoesn’t cluster all the poor (or good) yearstogether. A totally random approach failsto replicate the effects of a significant bearmarket. I’m not talking about the short-term greater-than-20-percent hiccups thattechnically qualify as a bear market, butthe prolonged ones that go into the secondyear posting even greater losses, whereinvestors become so fed up with repeatedbear market sucker rallies that they don’tever want to see a stock again. Bottoms ofsignificant bear markets occur when thefinancial journals can only see and reporton the worst of a negative investing envi-ronment, further encouraging investors asthey throw in the towel. The 1973–74bear market, for example, saw losses in theS&P 500 of almost 50 percent and theValueline index by more than 70 percent.It then took approximately seven years forthe market to recover to break even.

While many observers believe that bear

markets exhibit some degree of randomnessin their occurrence (and I’m not one ofthem), there is one thing all bear marketsshare. They all have a second significantdown year followed by a longer recoveryperiod of higher-than-normal returns. Onlytwo bear markets lasted longer than 24months (1929 and 1938), but the bulk oftheir losses were sustained in two years. Ata minimum, it’s this sharper than normalpost-bear market recovery period thatteaches us that bear markets are not clus-tered together. Significant bear marketspose the greatest financial risk to investorsand they cannot be adequately duplicatedby a random approach.

Financial planning is inherently morecomplex than an analysis of investmentreturns, while Monte Carlo surrounds theclient with a range of investment out-comes so broad that the client may easilylose sight of the real goal. If the client’sfinancial plan has a five-year horizon andhe or she is overweighted in large-capstocks, then knowing the range of returnsfor that asset class over any five yearsmight be of interest. The longer the timeperiod the greater the uncertainty thatother lifestyle factors will affect the plan,and the less useful Monte Carlo becomes.

What is it you want to test? If the clienthas significant exposure to certificates ofdeposit or other short-term fixed incomesecurities, then Monte Carlo should beadjusted to the inflation environmentlinked to those rate-of-return probabilities.There is a linkage between interest ratesand expected future inflation rates, andshort-term fixed income securities have ahigher correlation to current inflation ratesthan do longer-term fixed income invest-ments. If one variable offsets another vari-



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Monte Carlo Simulation—Challengingthe Sacred Cow

able (even partially), you shouldn’t countone while ignoring the other. Therefore,Monte Carlo doesn’t seem to provide sub-stantially improved rate-of-return informa-tion for long-term financial plans, and theshort-term is flawed by ignoring the corre-lation of inflation rates to different assetclasses in the computation.

What may be worse is that MonteCarlo masks the client’s worst case andmost likely scenarios with a range of out-comes so large that it distorts reality. Therandomness of Monte Carlo simulation,by its nature of clustering the good timesor bad times in its range of scenarios, maypotentially turn a planner’s client into aprofessional liability.

For a retired client living off his or herinvestments, a significant bear marketbeginning immediately is probably a worst-case scenario. The software used at ourfirm allows us to model a client’s financialplan as close to his or her projected circum-stances as possible, before looking at theimpact of different rates of return (includ-ing the ultimate bear market exampleabove), to see how they may affect theclient’s objectives. We can input both rateof return and duration for a bear market,the recovery period and post-recoveryperiod depending on the client’s specificinvestment allocation, and then show theclient a graphical comparison with morenormalized scenarios. Our software allowsfor multiple qualified and nonqualified planaccounts so that we can provide individual-ized treatment, showing continuing deferralof employment income if applicable, orusing different asset classes or investmentstyles that may logically suggest using dif-ferent rate-of-return assumptions. Distribu-tions are taxed appropriately (Roth or regu-lar) and can be accelerated in years thatcash flow is needed. Additions or outflowssuch as stock options, sale of property,divorce transactions, major expendituresand gifts can be modeled as taxable oruntaxable based on their character and cal-culated with other income each year.

Lifestyle expenditures are adjusted forinflation and can be changed in any year. Ifrefinancing is not chosen, the software willautomatically show sale of the primary res-idence when liquidity is exhausted, show-ing the client how much longer the assetswill last if he or she rents or purchases asmaller residence.

In the end, Monte Carlo simulationseems to clash with the continuing devel-opment of a holistic approach that allowschanges in the client’s investment alloca-tion over time, with the correspondingchanges of anticipated rate of return overthe same time periods. Modern softwarecan and should incorporate greater flexi-bility within the financial planning model,making random rate-of-return simulationseven less relevant.

While the profession quietly questionsMonte Carlo simulation, the benefits arebeing loudly proclaimed by the softwareindustry as the hottest new innovation infinancial planning in decades. Marketinghype touts this new information as onemore item of value in a client’s financialplan, while opponents say that when weextend client expectations out 20 or 30years, identifying factors such as lifestyleexpenditures, tax rates, inflation and invest-ment preference based on risk may bemore important. Chaos theory teaches ushow small errors in the early years of afinancial plan can make dramatic conse-quences when compounded over a longperiod of time, so let’s not make our realproblems any bigger than they are. Some“what if” scenarios are necessary, but let’sdo it right and do it often. The real answeris to make the plan as representative as pos-sible under the circumstances and toupdate it regularly. The benefit that MonteCarlo simulation promises to provide mightbe better achieved by using common sensein the financial planning process.

ciency Analysis Via the GeometricMean,” Financial Review, 1974, Vol. 9,No.1, pp. 46–63.

5. D. Hertz, “Risk Analysis in CapitalInvestment,” Harvard BusinessReview, January–February 1964, Vol.42.

6. G. Kautt and L. Hopewell, “Modelingthe Future,” Journal of Financial Plan-ning, 2000, Vol. 13, No. 10, pp.90–100.

7. W. G. Lewellen and M.S. Long, “Sim-ulation Versus Single-Value Estimatesin Capital Expenditure Analysis,”Decision Sciences, 1972, Vol.3,reprinted in Myers, 1976.

8. E. McCarthy, “Monte Carlo Simula-tion: Still Stuck in Low Gear,” Journalof Financial Planning, January 2000,Vol. 13, No. 1 pp. 54–60.

9. S. Myers, “Postscript: Using Simula-tion for Risk Analysis,” Modern Devel-opments in Financial Management, S.Myers, ed. (New York: Praeger Publi-cations, 1976), pp. 457–463.

10.G. Perez-Quiros and A. Timmermann,“Firm Size and Cyclical Variations inStock Returns,” Journal of Finance,2000, Vol. 55, No. 3, pp. 1,229–1,262.

11.G. C. Philippatos, Financial Manage-ment: Theory and Techniques (SanFrancisco: Holden Day, 1973).

12.W. Rees and C. Sutcliffe, “Mathemati-cal Modeling And Stochastic Simula-tion Of Accounting Alternatives,”Journal of Business Finance &Accounting, Vol. 20, No. 3, pp.351–358.

13.J. Rekenthaler, “The Limits of MonteCarlo: Despite What You’re Hearing,Simulations Don’t Improve Upon Con-ventional Math,” Advisor.Morn-ingstar.Com, July 24, 2000, http://advisor.morningstar.com/News_ Edi-torial/Print/1,7060,551,00.html.

14.R. Y. Rubinstein, Simulation and theMonte Carlo Method (New York: JohnWiley and Sons, 1981).

15.G. Santayana, “Reason in CommonSense,” Life of Reason, Chapter 12,1905–6.

16.R. Whitelaw, “Time Variations andCovariations in the Expectation andVolatility of Stock Market Return.”Journal of Finance, 1994, Vol. 49, No.2 pp 515 541



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