Monte Carlo Simulation Monte Carlo Simulation and Personal Financeand Personal Finance

Jacob FoleyJacob Foley

Background on myselfBackground on myself

I work at Stephens Financial Partners I work at Stephens Financial Partners as a Financial Advisoras a Financial Advisor

Monte Carlo simulations are the most Monte Carlo simulations are the most popular simulations used by advisorspopular simulations used by advisors

These simulations failed after the These simulations failed after the 2008 market collapse2008 market collapse

Where did it come from?Where did it come from?

John von Neumann and Stanislaw John von Neumann and Stanislaw Ulam Ulam

Los Alamos Scientific LaboratoryLos Alamos Scientific Laboratory

Studying radiation shieldingStudying radiation shielding

Why call it Monte Carlo?Why call it Monte Carlo?

Neuman and Ulam’s work had to be Neuman and Ulam’s work had to be kept a secret because it was part of kept a secret because it was part of the Manhattan Projectthe Manhattan Project

Von Neuman chose the name "Monte Von Neuman chose the name "Monte Carlo". Carlo".

What is it?What is it?

Class of computational algorithmsClass of computational algorithms Used to solve large systemsUsed to solve large systems Used when it is unfeasible or Used when it is unfeasible or

impossible to compute an exact impossible to compute an exact resultresult

Basic Principle of the Monte Carlo Basic Principle of the Monte Carlo Method.Method.

The Task: Calculate a number The Task: Calculate a number I I (one (one number only. Not an entire functional number only. Not an entire functional dependence)dependence)

Example: Calculate piExample: Calculate pi Numerically: look for an appropriate Numerically: look for an appropriate

convergent series and evaluate this convergent series and evaluate this approximatelyapproximately

Monte Carlo: look for a stochastic model: Monte Carlo: look for a stochastic model: probability space with random variableprobability space with random variable

What makes a method a Monte What makes a method a Monte Carlo Method?Carlo Method?

Define a domain of possible inputs. Define a domain of possible inputs. Generate inputs randomly from the Generate inputs randomly from the

domain using a certain specified domain using a certain specified probability distribution. probability distribution.

Perform a deterministic computation Perform a deterministic computation using the inputs. using the inputs.

Aggregate the results of the individual Aggregate the results of the individual computations into the final result computations into the final result

Random NumbersRandom Numbers

Uniform DistributionUniform Distribution The random variable X is uniformly The random variable X is uniformly

distributed on the interval [a, b]distributed on the interval [a, b]

How many of you have played How many of you have played battleship?battleship?

Dull Monte CarloDull Monte Carlo

““hit or miss”hit or miss” Take a sample point Take a sample point The point has two outcomesThe point has two outcomes

True (“hit”)True (“hit”) False (“miss”)False (“miss”)

Total number of hits and divide it by the Total number of hits and divide it by the total trialstotal trials

X

f(x)

I = ∫ f(x) dx

I: unknown areaknown area

x1, uniform

x2

uniform

misshit

Hit or Miss

Crude Monte CarloCrude Monte Carlo

Write the integral such that Write the integral such that I I becomes the mean value of a becomes the mean value of a random variable.random variable.

Purposes we generate Purposes we generate BB numbers numbers Uniformly distributed from (0,1)Uniformly distributed from (0,1) Then take their averageThen take their average

Take Numerical Analysis Take Numerical Analysis

Professor Robert LewisProfessor Robert Lewis

Math 413 and 414Math 413 and 414

Applications in the Real WorldApplications in the Real World

Physical sciencesPhysical sciences Design and visualsDesign and visuals TelecommunicationsTelecommunications GamesGames Finance and businessFinance and business

Monte Carlo in FinanceMonte Carlo in Finance

First Introduced in 1964First Introduced in 1964

““Risk Analysis in Capital Investment”Risk Analysis in Capital Investment” David B HertzDavid B Hertz Harvard Business Review Article Harvard Business Review Article

So how does Monte Carlo apply to So how does Monte Carlo apply to Finance?Finance?

Used to value and analyzeUsed to value and analyze InstrumentsInstruments OptionsOptions PortfoliosPortfolios InvestmentsInvestments

How does it predict values?How does it predict values?

For each SimulationFor each Simulation The behavior of the factors impacting the The behavior of the factors impacting the

component instrument is simulated over component instrument is simulated over timetime

The values of the instrument are calculatedThe values of the instrument are calculated The value is then observedThe value is then observed The various values are then combined in a The various values are then combined in a

histogram (i.e. the probability distribution)histogram (i.e. the probability distribution) The statistical characteristics are then The statistical characteristics are then

observedobserved

How is it used in financial planning?How is it used in financial planning?

Simulates the overall marketSimulates the overall market

Predicts the probability of reaching a Predicts the probability of reaching a target numbertarget number

Changes are made to reach the Changes are made to reach the target numbertarget number

An ExampleAn Example

http://www.flexibleretirementplanner.com/

What works with Monte Carlo?What works with Monte Carlo?

Forecasting EarningsForecasting Earnings

Modeling portfolio lossesModeling portfolio losses

Provides flexibilityProvides flexibility

What is wrong with Monte Carlo?What is wrong with Monte Carlo?

Assumes normal return distributionsAssumes normal return distributions We know from history that extreme We know from history that extreme

returns occur more frequently than returns occur more frequently than expectedexpected

Can’t predict every outcomeCan’t predict every outcome Most clients see the simulation run Most clients see the simulation run

through thousands of iterations and through thousands of iterations and believe that they have seen all possible believe that they have seen all possible outcomesoutcomes

What is wrong with Monte Carlo?What is wrong with Monte Carlo?

Does not measure bear markets wellDoes not measure bear markets well

Does not include the human factorDoes not include the human factor

What is wrong with Monte Carlo?What is wrong with Monte Carlo?

Does not recognize that portfolio Does not recognize that portfolio performance depends at least as performance depends at least as much on the sequence of the rate of much on the sequence of the rate of return that it does on the average of return that it does on the average of those returnsthose returns

What can we do better?What can we do better?

Let’s look at an exampleLet’s look at an example AssumptionsAssumptions

20 year period20 year period Individual that has just retired in 1988Individual that has just retired in 1988 Has $1,000,000 invested in DJIAHas $1,000,000 invested in DJIA Withdraws $50,000 each year that Withdraws $50,000 each year that

increases by 3% to compensate for increases by 3% to compensate for inflationinflation

1988 11.80% $1,118,000.00 $1,068,000.00 $50,000.00

1989 27.00% $1,356,360.00 $1,304,860.00 $51,500.00

1990 -4.30% $1,248,751.02 $1,195,706.02 $53,045.00

1991 20.30% $1,438,434.34 $1,383,797.99 $54,636.35

1992 4.20% $1,441,917.51 $1,385,642.07 $56,275.44

1993 13.70% $1,575,475.03 $1,517,511.33 $57,963.70

1994 2.10% $1,549,379.06 $1,489,676.45 $59,702.61

1995 33.50% $1,988,718.06 $1,927,224.37 $61,493.69

1996 26.00% $2,428,302.70 $2,364,964.20 $63,338.50

1997 22.60% $2,899,446.11 $2,834,207.45 $65,238.66

1998 16.10% $3,290,514.85 $3,223,319.03 $67,195.82

1999 25.20% $4,035,595.42 $3,966,383.73 $69,211.69

2000 -6.20% $3,720,467.94 $3,649,179.89 $71,288.04

2001 -7.10% $3,390,088.12 $3,316,661.44 $73,426.69

2002 -16.80% $2,759,462.31 $2,683,832.83 $75,629.49

2003 25.30% $3,362,842.53 $3,284,944.16 $77,898.37

2004 3.10% $3,386,777.43 $3,306,542.11 $80,235.32

2005 -0.60% $3,286,702.86 $3,204,060.48 $82,642.38

2006 16.30% $3,726,322.33 $3,641,200.68 $85,121.65

2007 6.80% $3,888,802.33 $3,801,127.02 $87,675.30

2008 -49.80% $1,908,165.77 $1,817,860.20 $90,305.56

1988 -49.80% $502,000.00 $452,000.00 $50,000.00

1989 6.80% $482,736.00 $431,236.00 $51,500.00

1990 16.30% $501,527.47 $448,482.47 $53,045.00

1991 -0.60% $445,791.57 $391,155.22 $54,636.35

1992 3.10% $403,281.04 $347,005.59 $56,275.44

1993 25.30% $434,798.01 $376,834.31 $57,963.70

1994 -16.80% $313,526.14 $253,823.53 $59,702.61

1995 -7.10% $235,802.06 $174,308.36 $61,493.69

1996 -6.20% $163,501.25 $100,162.74 $63,338.50

1997 25.20% $125,403.75 $60,165.09 $65,238.66

1998 16.10% $69,851.67 $2,655.85 $67,195.82

1999 22.60% $3,256.08 $65,955.62 $69,211.69

2000 26.00% $83,104.08 $154,392.12 $71,288.04

2001 33.50% $206,113.48 $279,540.17 $73,426.69

2002 2.10% $285,410.51 $361,040.00 $75,629.49

2003 13.70% $410,502.48 $488,400.85 $77,898.37

2004 4.20% $508,913.68 $589,149.00 $80,235.32

2005 20.30% $708,746.25 $791,388.63 $82,642.38

2006 -4.30% $757,358.92 $842,480.58 $85,121.65

2007 27.00% $1,069,950.33 $1,157,625.63 $87,675.30

2008 11.80% $1,294,225.46 $1,384,531.02 $90,305.56

1988 11.80% $1,118,000.00 $1,068,000.00 $50,000.00

1989 27.00% $1,356,360.00 $1,304,860.00 $51,500.00

1990 -4.30% $1,248,751.02 $1,195,706.02 $53,045.00

1991 20.30% $1,438,434.34 $1,438,434.34 $0.00

1992 4.20% $1,498,848.58 $1,423,219.09 $75,629.49

1993 13.70% $1,618,200.11 $1,540,301.74 $77,898.37

1994 2.10% $1,572,648.07 $1,492,412.75 $80,235.33

1995 33.50% $1,992,371.02 $1,909,728.63 $82,642.39

1996 26.00% $2,406,258.07 $2,321,136.42 $85,121.66

1997 22.60% $2,845,713.25 $2,758,037.94 $87,675.31

1998 16.10% $3,202,082.05 $3,111,776.48 $90,305.57

1999 25.20% $3,895,944.16 $3,802,929.42 $93,014.73

2000 -6.20% $3,567,147.80 $3,471,342.62 $95,805.18

2001 -7.10% $3,224,877.30 $3,224,877.30 $0.00

2002 -16.80% $2,683,097.91 $2,683,097.91 $0.00

2003 25.30% $3,361,921.68 $3,361,921.68 $0.00

2004 3.10% $3,466,141.25 $3,358,311.65 $107,829.60

2005 -0.60% $3,338,161.78 $3,227,097.30 $111,064.49

2006 16.30% $3,753,114.16 $3,753,114.16 $0.00

2007 6.80% $4,008,325.92 $3,890,497.62 $117,828.30

2008 -49.80% $1,953,029.80 $1,831,666.66 $121,363.15

1988 -49.80% $502,000.00 $452,000.00 $50,000.00

1989 6.80% $482,736.00 $482,736.00 $0.00

1990 16.30% $561,421.97 $508,376.97 $53,045.00

1991 -0.60% $505,326.71 $450,690.36 $54,636.35

1992 3.10% $464,661.76 $464,661.76 $0.00

1993 25.30% $582,221.18 $524,257.48 $57,963.70

1994 -16.80% $436,182.22 $376,479.61 $59,702.61

1995 -7.10% $349,749.56 $349,749.56 $0.00

1996 -6.20% $328,065.08 $328,065.08 $0.00

1997 25.20% $410,737.48 $410,737.48 $0.00

1998 16.10% $476,866.22 $386,851.49 $90,014.73

1999 22.60% $474,279.93 $381,564.75 $92,715.17

2000 26.00% $480,771.59 $385,274.96 $95,496.63

2001 33.50% $514,342.08 $415,980.55 $98,361.53

2002 2.10% $424,716.14 $349,086.65 $75,629.49

2003 13.70% $396,911.52 $319,013.15 $77,898.37

2004 4.20% $332,411.70 $252,176.37 $80,235.33

2005 20.30% $303,368.18 $220,725.79 $82,642.39

2006 -4.30% $211,234.58 $126,112.93 $85,121.66

2007 27.00% $160,163.42 $160,163.42 $0.00

2008 11.80% $179,062.70 $57,699.55 $121,363.15

Have multiple buckets of moneyHave multiple buckets of money

Don’t just have your money in the Don’t just have your money in the stock marketstock market

Have money growing outside of the Have money growing outside of the stock marketstock market

HomeworkHomework

Estimate Pi using Monte CarloEstimate Pi using Monte Carlo

Thank You!Thank You!

Any Questions?Any Questions?