Monte Carlo Simulation
and Personal Finance
Jacob Foley
Background on myself



I work at Stephens Financial Partners as a
Financial Advisor
Monte Carlo simulations are the most
popular simulations used by advisors
These simulations failed after the 2008
market collapse
Where did it come from?

John von Neumann and Stanislaw Ulam

Los Alamos Scientific Laboratory

Studying radiation shielding
Why call it Monte Carlo?


Neuman and Ulam’s work had to be kept a
secret because it was part of the
Manhattan Project
Von Neuman chose the name "Monte
Carlo".
What is it?



Class of computational algorithms
Used to solve large systems
Used when it is unfeasible or impossible to
compute an exact result
Basic Principle of the Monte Carlo
Method.


The Task: Calculate a number I (one
number only. Not an entire functional
dependence)
Example: Calculate pi


Numerically: look for an appropriate
convergent series and evaluate this
approximately
Monte Carlo: look for a stochastic model:
probability space with random variable
What makes a method a Monte
Carlo Method?




Define a domain of possible inputs.
Generate inputs randomly from the
domain using a certain specified
probability distribution.
Perform a deterministic computation using
the inputs.
Aggregate the results of the individual
computations into the final result
Random Numbers

Uniform Distribution

The random variable X is uniformly distributed
on the interval [a, b]
How many of you have played
battleship?
Dull Monte Carlo

“hit or miss”


Take a sample point
The point has two outcomes
True (“hit”)
 False (“miss”)


Total number of hits and divide it by the total
trials
Hit or Miss
f(x)
x2
uniform
I: unknown area
known area
miss
hit
I = ∫ f(x) dx
X
x1, uniform
Crude Monte Carlo

Write the integral such that I becomes the
mean value of a random variable.



Purposes we generate B numbers
Uniformly distributed from (0,1)
Then take their average
Take Numerical Analysis

Professor Robert Lewis

Math 413 and 414
Applications in the Real World





Physical sciences
Design and visuals
Telecommunications
Games
Finance and business
Monte Carlo in Finance

First Introduced in 1964

“Risk Analysis in Capital Investment”


David B Hertz
Harvard Business Review Article
So how does Monte Carlo apply to
Finance?

Used to value and analyze




Instruments
Options
Portfolios
Investments
How does it predict values?

For each Simulation





The behavior of the factors impacting the
component instrument is simulated over time
The values of the instrument are calculated
The value is then observed
The various values are then combined in a
histogram (i.e. the probability distribution)
The statistical characteristics are then
observed
How is it used in financial
planning?



Simulates the overall market
Predicts the probability of reaching a
target number
Changes are made to reach the target
number
An Example

http://www.flexibleretirementplanner.com
/
What works with Monte Carlo?

Forecasting Earnings

Modeling portfolio losses

Provides flexibility
What is wrong with Monte Carlo?

Assumes normal return distributions


We know from history that extreme returns
occur more frequently than expected
Can’t predict every outcome

Most clients see the simulation run through
thousands of iterations and believe that they
have seen all possible outcomes
What is wrong with Monte Carlo?

Does not measure bear markets well

Does not include the human factor
What is wrong with Monte Carlo?

Does not recognize that portfolio
performance depends at least as much on
the sequence of the rate of return that it
does on the average of those returns
What can we do better?


Let’s look at an example
Assumptions




20 year period
Individual that has just retired in 1988
Has $1,000,000 invested in DJIA
Withdraws $50,000 each year that increases
by 3% to compensate for inflation
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 money


Don’t just have your money in the stock
market
Have money growing outside of the stock
market
Homework

Estimate Pi using Monte Carlo
Thank You!
Any Questions?