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Stochastic Models
Decisions
Walt Pohl
Universität Zürich
Department of Business Administration
February 28, 2013
Making Decisions
The ultimate purpose of models in business is to make
decisions.
Project Good Outcome Bad Outcome
A
50 million
0 million
B
15 million
15 million
Which is the right decision?
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
2 / 11
Mean
The simplest criterion, and the one most often used
outside of financial applications, is the use the mean.
Examples:
Expected waiting time for customers.
Expected cost.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
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Return on Stocks versus Bonds
Here are the stats for the monthly return on US stocks
and US government bonds since 1920.
Mean Std. Dev.
1%
5%
Stocks 0.009
0.054 -0.150 -0.078
Bonds 0.003
0.003
0
0
If you buy bonds, are you making a mistake? Maybe.
Or maybe you are just risk averse. Everything else being
equal, you prefer a certain outcome over an uncertain
one.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
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Mean-Variance Analysis
In the 50s, Markowitz suggested that we also consider
the variance. Stocks have much more variance than
bonds: a monthly standard deviation of 5% versus less
than 1%.
Stocks-versus-bonds face a risk-versus-return tradeoff.
Stocks have higher return, but more risk of dropping.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
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Limitations of the Variance
Variance does not account for “fat tails”: the chances of
rare but disastrous outcomes.
Consider these two distributions:
Normal with mean zero and variance 2.
t with 3 degrees of freedom.
They both have mean zero and variance 2.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
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Fat Tails
Here are the lower quantiles for t versus Normal.
1%
5%
Normal -2.33 -1.65
t
-4.54 -2.35
Under the t distribution, 1% of the time you will lose
twice as much than you would under the normal
distribution.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
7 / 11
Value at Risk
Value-at-risk (VAR) is a rule widely used by banks. You
consider how sensitive to extreme bad outcomes you are.
Bad outcome is how much you would lose an extreme
quantile such as the 1% or 5% quantile occurred, relative
to the mean. (This quantile is known as the confidence
level.)
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
8 / 11
Value at Risk, cont’d
For stocks and bonds, the historical VARs are, assuming
that you invest 1 US dollar at the beginning of the
month:
1%
5%
Stocks -0.159 -0.087
Bonds -0.003 -0.003
Note that this depends on the window: here monthly. A
bigger window would see bigger numbers.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
9 / 11
Expected Shortfall
VAR suffers from one obvious shortcoming.
Suppose that you have an investment that 99.9% of the
time earns you 1 dollar, and 0.1% of the time you lose 1
trillion dollars. Unless you pick a very small confidence
interval (0.1% or smaller), this has a VAR of zero.
Expected shortfall corrects for that. The Expected
shortfall is the expected value of the investment,
conditioning on the fact that we are below the VAR
cutoff.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
10 / 11
Statistical Decision Theory
This combines the statistical and decision-making into
one step. Widely used in machine learning.
You view chosing the parameters as a decision problem.
We will see more about this later.
Walt Pohl (UZH QBA)
Stochastic Models
February 28, 2013
11 / 11