Monte Carlo Simulation and Personal Finance Jacob Foley

Background on myself I work at Stephens Financial Partners as a Financial Advisor I work at Stephens Financial Partners as a Financial Advisor Monte Carlo simulations are the most popular simulations used by advisors Monte Carlo simulations are the most popular simulations used by advisors These simulations failed after the 2008 market collapse These simulations failed after the 2008 market collapse

Where did it come from? John von Neumann and Stanislaw Ulam John von Neumann and Stanislaw Ulam Los Alamos Scientific Laboratory Los Alamos Scientific Laboratory Studying radiation shielding 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 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". Von Neuman chose the name "Monte Carlo".

What is it? Class of computational algorithms Class of computational algorithms Used to solve large systems Used to solve large systems Used when it is unfeasible or impossible to compute an exact result 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) The Task: Calculate a number I (one number only. Not an entire functional dependence) Example: Calculate pi Example: Calculate pi Numerically: look for an appropriate convergent series and evaluate this approximately Numerically: look for an appropriate convergent series and evaluate this approximately Monte Carlo: look for a stochastic model: probability space with random variable 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. Define a domain of possible inputs. Generate inputs randomly from the domain using a certain specified probability distribution. Generate inputs randomly from the domain using a certain specified probability distribution. Perform a deterministic computation using the inputs. Perform a deterministic computation using the inputs. Aggregate the results of the individual computations into the final result Aggregate the results of the individual computations into the final result

Random Numbers Uniform Distribution Uniform Distribution The random variable X is uniformly distributed on the interval [a, b] 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” “hit or miss” Take a sample point Take a sample point The point has two outcomes The point has two outcomes True (“hit”) True (“hit”) False (“miss”) False (“miss”) Total number of hits and divide it by the total trials Total number of hits and divide it by the total trials

X f(x) I = ∫ f(x) dx I: unknown area known area x 1, uniform x 2 uniform miss hit Hit or Miss

Crude Monte Carlo Write the integral such that I becomes the mean value of a random variable. Write the integral such that I becomes the mean value of a random variable. Purposes we generate B numbers Purposes we generate B numbers Uniformly distributed from (0,1) Uniformly distributed from (0,1) Then take their average Then take their average

Take Numerical Analysis Professor Robert Lewis Professor Robert Lewis Math 413 and 414 Math 413 and 414

Applications in the Real World Physical sciences Physical sciences Design and visuals Design and visuals Telecommunications Telecommunications Games Games Finance and business Finance and business

Monte Carlo in Finance First Introduced in 1964 First Introduced in 1964 “Risk Analysis in Capital Investment” “Risk Analysis in Capital Investment” David B Hertz David B Hertz Harvard Business Review Article Harvard Business Review Article

So how does Monte Carlo apply to Finance? Used to value and analyze Used to value and analyze Instruments Instruments Options Options Portfolios Portfolios Investments Investments

How does it predict values? For each Simulation For each Simulation The behavior of the factors impacting the component instrument is simulated over time The behavior of the factors impacting the component instrument is simulated over time The values of the instrument are calculated The values of the instrument are calculated The value is then observed The value is then observed The various values are then combined in a histogram (i.e. the probability distribution) The various values are then combined in a histogram (i.e. the probability distribution) The statistical characteristics are then observed The statistical characteristics are then observed

How is it used in financial planning? Simulates the overall market Simulates the overall market Predicts the probability of reaching a target number Predicts the probability of reaching a target number Changes are made to reach the target number Changes are made to reach the target number

An Example / / / /

What works with Monte Carlo? Forecasting Earnings Forecasting Earnings Modeling portfolio losses Modeling portfolio losses Provides flexibility Provides flexibility

What is wrong with Monte Carlo? Assumes normal return distributions Assumes normal return distributions We know from history that extreme returns occur more frequently than expected We know from history that extreme returns occur more frequently than expected Can’t predict every outcome Can’t predict every outcome Most clients see the simulation run through thousands of iterations and believe that they have seen all possible outcomes 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 measure bear markets well Does not include the human factor 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 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 Let’s look at an example Assumptions Assumptions 20 year period 20 year period Individual that has just retired in 1988 Individual that has just retired in 1988 Has $1,000,000 invested in DJIA Has $1,000,000 invested in DJIA Withdraws $50,000 each year that increases by 3% to compensate for inflation Withdraws $50,000 each year that increases by 3% to compensate for inflation

% $1,118,000.00$1,068,000.00$50, % $1,356,360.00$1,304,860.00$51, % $1,248,751.02$1,195,706.02$53, % $1,438,434.34$1,383,797.99$54, % $1,441,917.51$1,385,642.07$56, % $1,575,475.03$1,517,511.33$57, % $1,549,379.06$1,489,676.45$59, % $1,988,718.06$1,927,224.37$61, % $2,428,302.70$2,364,964.20$63, % $2,899,446.11$2,834,207.45$65, % $3,290,514.85$3,223,319.03$67, % $4,035,595.42$3,966,383.73$69, % $3,720,467.94$3,649,179.89$71, % $3,390,088.12$3,316,661.44$73, % $2,759,462.31$2,683,832.83$75, % $3,362,842.53$3,284,944.16$77, % $3,386,777.43$3,306,542.11$80, % $3,286,702.86$3,204,060.48$82, % $3,726,322.33$3,641,200.68$85, % $3,888,802.33$3,801,127.02$87, % $1,908,165.77$1,817,860.20$90,305.56

%$502,000.00$452,000.00$50, %$482,736.00$431,236.00$51, %$501,527.47$448,482.47$53, %$445,791.57$391,155.22$54, %$403,281.04$347,005.59$56, %$434,798.01$376,834.31$57, %$313,526.14$253,823.53$59, %$235,802.06$174,308.36$61, %$163,501.25$100,162.74$63, %$125,403.75$60,165.09$65, %$69,851.67$2,655.85$67, %$3,256.08$65,955.62$69, %$83,104.08$154,392.12$71, %$206,113.48$279,540.17$73, %$285,410.51$361,040.00$75, %$410,502.48$488,400.85$77, %$508,913.68$589,149.00$80, %$708,746.25$791,388.63$82, %$757,358.92$842,480.58$85, %$1,069,950.33$1,157,625.63$87, %$1,294,225.46$1,384,531.02$90,305.56

% $1,118,000.00$1,068,000.00$50, % $1,356,360.00$1,304,860.00$51, % $1,248,751.02$1,195,706.02$53, % $1,438, $ % $1,498,848.58$1,423,219.09$75, % $1,618,200.11$1,540,301.74$77, % $1,572,648.07$1,492,412.75$80, % $1,992,371.02$1,909,728.63$82, % $2,406,258.07$2,321,136.42$85, % $2,845,713.25$2,758,037.94$87, % $3,202,082.05$3,111,776.48$90, % $3,895,944.16$3,802,929.42$93, % $3,567,147.80$3,471,342.62$95, % $3,224, $ % $2,683, $ % $3,361, $ % $3,466,141.25$3,358,311.65$107, % $3,338,161.78$3,227,097.30$111, % $3,753, $ % $4,008,325.92$3,890,497.62$117, % $1,953,029.80$1,831,666.66$121,363.15

%$502,000.00$452,000.00$50, %$482, $ %$561,421.97$508,376.97$53, %$505,326.71$450,690.36$54, %$464, $ %$582,221.18$524,257.48$57, %$436,182.22$376,479.61$59, %$349, $ %$328, $ %$410, $ %$476,866.22$386,851.49$90, %$474,279.93$381,564.75$92, %$480,771.59$385,274.96$95, %$514,342.08$415,980.55$98, %$424,716.14$349,086.65$75, %$396,911.52$319,013.15$77, %$332,411.70$252,176.37$80, %$303,368.18$220,725.79$82, %$211,234.58$126,112.93$85, %$160, $ %$179,062.70$57,699.55$121,363.15

Have multiple buckets of money Don’t just have your money in the stock market Don’t just have your money in the stock market Have money growing outside of the stock market Have money growing outside of the stock market

Homework Estimate Pi using Monte Carlo Estimate Pi using Monte Carlo

Thank You! Any Questions?