Monte Carlo Simulation
A powerful approach to assess uncertainty
July 08, 2013
Ravi Saraogi | IFMR Investment Adviser
Structure
 Approach to Uncertainty
 Introduction to Monte Carlo (MC) Simulations
 When to use and when not to use MC Simulations
 Impediments, Benefits and Pitfalls
 Stylized example
 Recent criticisms of MC Simulations
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Approach to Uncertainty
Non Probabilistic
• Point estimate – one value as the ‘best guess’ for the population parameter
• E.g. Sample mean is a point estimate for Population mean
Probabilistic
• Interval estimate – Range of values that is likely to contain the population parameter
• E.g. Sample mean lies within [a.b] with 95% confidence (i.e. a confidence interval)
Population
Parameter
Point Estimate or Interval Estimate
Sample
Statistic
3
Three types of probabilistic approach
 Scenario Analysis
• Best case, most likely, worst case
• Multiple scenarios
• Discrete outcomes
 Decision tree
• Discrete outcomes
• Sequential decisions
 Monte Carlo Simulation
• Combines both scenario analysis and decision trees
• Continuous outcomes
4
Manhattan Project - First electronic computation of Monte Carlo
simulation
•
John Von Neumann, Stanislaw Ulam, and
Nicholas Metropolis were part of the
Manhattan project during World War II
•
Solving the chain reaction in highly enriched
uranium was too complex with algebraic
equations
•
Hence, they used simulations- plugging many
different numbers into the equations and
calculating the result.
•
Systematically plugging in and trying numbers
would have taken too long
•
So they created a new approach -- plugging in
randomly chosen numbers into the equations
and calculating the results.
•
Analysing the results with statistics, they were
able to make very good predictions which were
instrumental in designing of an atomic bomb.
5
Birth of Modern Monte Carlo Simulation
“…….The question was what are the chances that a
Canfield solitaire laid out with 52 cards will come out
successfully? After spending a lot of time trying to
estimate them by pure combinatorial calculations, I
wondered whether a more practical method than
‘abstract thinking’ might not be to lay it out say one
hundred times and simply observe and count the
number of successful plays…..”
-Eckhardt, Roger (1987). Stan Ulam, John von Neumann, and the Monte Carlo
method, Los Alamos Science
Stan Ulam
Nicholas Metropolis, named it such as Stan’s Uncle would
borrow money by saying “I just have to go to Monte Carlo”
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What is Monte Carlo Simulation?
•
Uncertain inputs are represented by probability distributions instead of one most likely value
•
Random sampling is used to select input values from the defined probability distributions, and
the model is run over and over again (the idea is to capture all valid combinations of possible
inputs) to simulate all possible outcomes
Uncertainty in
model outputs
Uncertainty in
model inputs
(1) Input Distributional
Assumptions
(4) Output
distribution
X
(2)
(3)
Functional relationship (f) + Random sampling
Y
A
Z
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When is a simulation ‘Monte Carlo’?
•
The defining characteristic of Monte Carlo simulation is
sampling techniques that are entirely random in
principle
•
Monte Carlo sampling satisfies the purist's desire for an
unadulterated random sampling method.
•
Historical simulation is not Monte Carlo simulation
–
–
–
–
•
Past is not prologue – non stationary data
All historical path get equal probability
Tail risk may not be captured
New assets/market
To avoid clustering, stratified random sampling can be
used
– Latin Hypercube sampling
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Application of Monte Carlo Simulation
•
Can be solved mathematically – do not use simulation
– Probability of Heads/Tails in tossing of an unbiased coin
– The likelihood of getting Heads 7 times out of 10 coin tosses
•
Cannot be solved mathematically- use simulation
– Suppose your commute to work consists of the following:
• Drive 3 kms on a highway, with 90% probability you will be able to average
20 KMPH the whole way, but with a 10% probability that a free road will
result in average speed of 80 KMPH.
• Come to an intersection with a traffic light that is red for 10 minutes, then
green for 3 minutes.
• Travel 1 km, averaging 50 KMMPH with a standard deviation of 10 KMPH.
• Come to an intersection with a traffic light that is red for 1 minutes, then
green for 30 seconds.
• Travel 1 more km, averaging 30 KMMPH with a standard deviation of 5
KMPH.
• You want to know how much time to allow for the commute in order to have
a 75% probability of arriving at work on time.
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When to use Monte Carlo Simulation
•
Not possible to solve mathematically the functional relationship between input and
output
•
You can't work out what the distribution of the output variable or you don't want to do
integrals numerically, but you can take samples from input distribution.
•
Simulation should be used when there are complex interactions that requires input
from multiple variables
"Calculate when you can, simulate when you can't!”
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Steps of MC Simulations
 Select variables
o Unlike scenario analysis and decision trees, there is no constraint on how many
variables can be allowed to vary in a simulation
 Specify probability distributions of these variables
o This is the key and the most difficult step in the analysis.
o Three ways of defining probability distributions
o Historical data – Suitable only in the case of stationary data/no structural
shifts
o Cross sectional data
o Pick a statistical distribution that best captures the variability in the input and
estimate the parameters for that distribution.
 Sample randomly from the defined distributions and run the inputs past the
Model to get the output
 Analyze the distribution of the output variable
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Impediments to Simulation
 Estimating distributions of input variables
o It is far easier to estimate an expected growth rate of 8% in revenues for the next
5 years than it is to specify the distribution of expected growth rates – the type of
distribution, parameters of that distribution.
 Simulation ban be time and resource intensive
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Benefits and Pitfalls
Benefits
•
Simulation gives a way out when stuck against a complicated intractable
mathematical dilemma
•
Output is a distribution rather than a point estimate
– Investment with a higher expected return may have a fat tailed distribution
Pitfalls
•
Garbage in, garbage out: For simulations to have value, the distributions should be
based upon analysis and data, rather than guesswork
•
Benefits that decision-makers get by having a fuller picture of the uncertainty may be
more than offset by misuse of simulation
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Stylized Example
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Risk in a portfolio of assets
•
VaR for a single asset
– Assume normality
– Mean of Rs 120 lakhs and an annual standard deviation of Rs 10 lakhs
– With 95% confidence, you can say that the value of this asset will not drop below
Rs 100 lakhs (two standard deviations below from the mean) or rise above Rs
140 lakhs (two standard deviations above the mean) over the next year.
•
VaR for a portfolio of assets
– Assume normality
– Portfolio mean will be weighted mean of individual assets
– For getting variance of the portfolio, co-variances will have to be estimated
• In a portfolio of 10 assets, there will be 45 co-variances that need to be
estimated, in addition to the 10 individual asset variances
Portfolio with three assets-
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Evaluating risk in a portfolio of assets
P(d))
Bond 1
Structure
P(d))
Bond 2
P(d))
Bond 3
Costs
Interest Received
Investor Payouts
Investor Return
Repayments
Reinvestments
P(d))
Bond 10
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Inputs
•
•
•
•
•
•
•
Cumulative default probabilities will be used as cut offs in a uniform distribution for a binary
outcome – default or no default
– 5yr: 15%
3yr: 10%
2yr: 5%
1yr: 2%
Special case: Use correlated random numbers (r=0.33)
Point of default in a bond’s maturity is random
Recovery rate is 20%
Interest rate assumptions across the time of the investment
Semi annual interest payout: 1% (building a default cushion)
Surplus cash reinvested as per automated algorithm
Bonds give quarterly coupon and bullet principal repayment
IRR, Redemption Premium
Randomizer
Amount
Bond
(Rs Lakhs)
Bond 1
15
Bond 2
5
Bond 3
10
Bond 4
5
Bond 5
15
Bond 6
5
Bond 7
10
Bond 8
5
Bond 9
20
Bond 10
10
Pricing
10.00%
12.00%
8.00%
9.00%
9.00%
10.00%
11.00%
12.00%
12.00%
10.00%
Tenure
(years)
5
3
5
3
5
3
5
3
5
3
Default (Yes/No)
Timing of Default
x1
x2
x3
Cash flow model
x4
Interest Received
Repayments
Reinvestments
x5
x6
x7
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Simulation Methods
•
•
•
•
Set up cash flows and randomizer in Excel, recalculate (F9)
and store the value of output variable. Repeat for number of
trials.
Excel macros
Dedicated software/Add ins
Data table
Multi way data table in Excel
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Results
Without correlation
With correlation (r=0.33)
Value at Risk (VAR)We can say with 95% confidence that
after meeting all interest payments the
redemption premium will be at least Rs
2.3 lakhs
With correlation (r=0.33)
We can say with 95% confidence that
after meeting all interest payments the
loss on principal will not exceed Rs 11
lakhs
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Recent criticisms
These models were supposed to help quantify and
manage the risks of mortgage-backed securities,
credit-default swaps and other complex
instruments. But given the events of the past
couple of years, it appears that the models often
gave big institutions, as well as small investors, a
false sense of security.
Also controversial is that many
Monte Carlo simulations
assume that market returns
fall along a bell-curve-shaped
distribution.
Critics emphasize that the problem isn't Monte Carlo itself, but the assumptions that go into it.
Since no standard approach exists, one user might plug in a range of assumptions on interest
rates, inflation or volatility that is different from another user.
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Thank you
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