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Stochastic Models
Statistics
Walt Pohl
Universität Zürich
Department of Business Administration
February 28, 2013
The Value of Statistics
Business people tend to underestimate the value of
statistics. Statistics allows us to learn from history.
Finance people tend to overestimate the value of
statistics. History rarely repeats itself exactly.
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Stochastic Models
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Statistical Methods
1
2
3
4
Method of moments
Maximum likelihood
Regression
many more...
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Method of moments
Simplest (and oldest) method of doing statistics. Pick
some moments (usually mean and variance), and match
with the data.
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Example: Lognormal Model of Stock
Prices
1
2
3
Changes in stock prices are unpredictable.
Modeled by random changes in the growth rate.
We assume these random changes can be modeled
by a lognormal distribution.
In other words, if St is the stock price, then the log
growth rate,
log St+1 /St
is normally distributed.
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Example: Lognormal Model of Stock
Prices
The model can be written as
log St+1 = µ + log St + t+1 ,
where t is normally distributed Normal(0, σ 2 ).
In financial circles, the variance or standard deviation is
known as the volatility.
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Stochastic Models
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Example: Lognormal Model of Stock
Prices, cont’d
This model has two parameters, the mean µ and
variance σ 2 , both of which can be computed from
moments. (We usually work with the standard deviation,
since the variances are usually tiny.)
Example: S&P 500 index.
Mean = 0.000
Standard Deviation = 0.015
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Example: Lognormal Model of Stock
Prices, cont’d
Is this a good model? No!
1
2
Distribution is “fatter tails” than the normal: more
extreme events than predicted by the normal
distribution.
Variances themselves vary over time.
We consider the second.
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Example: The EWMA Volatility Model
Forecasting is the art of predicting the future from today.
The easiest, and frequently most successful, method is to
predict that the future will be like the recent past.
The exponentially-weighted moving average (EWMA)
model for forecasting volatility is a simple example. We
forecast tomorrow’s volatility by a weighted sum of
today’s forecast and today’s observation
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Stochastic Models
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Example: The EWMA Volatility Model,
cont’d
Let σt2 be the variance of t for period t. Then the
EWMA forecast is
2
σt2 = λσt−1
+ (1 − λ)2t .
But how do we find λ?
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Stochastic Models
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Maximum Likelihood Estimation
Maximum likelihood estimation is a general-purpose
method for estimating parameters in models.
One limitation: you must completely specify your
probability distributions. (Here we must specify that t is
normally distributed).
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Stochastic Models
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From PDF to Likelihood
A probability density function (PDF) depends on
1
parameters
2
data
Once we plug in the data, we get a function of the
parameters alone – the likelihood.
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Stochastic Models
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Example: the Normal Distribution
Recall the PDF of the normal distribution, Normal(µ, σ):
√
1
2πσ 2
e
−(x−µ)2
2σ 2
If we have N independent draws, of a normal random
variable with unknown mean µ and variance σ.
N Y
−(xi −µ)2
1
√
e 2σ2
2
2πσ
i
We choose as our estimates the µ and σ to maximize
this expression. This is the maximum likelihood estimate
(MLE).
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Stochastic Models
February 28, 2013
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Example: the Normal Distribution, cont’d.
We can actually compute the MLE for the normal
distribution explicitly:
µMLE =
2
σMLE
=
N
X
i
N
X
xi /N
(xi − µMLE )2 /N
i
These are the sample mean and standard deviation.
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Stochastic Models
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Likelihood for EWMA
The likelihood for the EWMA is similar, but now σi
depends on the data. Let i = log Si+1 − log Si − α
(where α is the sample mean). Then, given lambda, we
know that
2
σi2 = λσi−1
+ (1 − λ)2i .
The (log) likelihood is then just
X i
1
2
− log(2πσi )
−
2σi2 2
i
We must resort to numerical methods to maximize this
expression.
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Stochastic Models
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Linear Regression
Linear regression is a technique for finding linear
relationships between variables,
Yt = α + βXt + Requires no distributional assumptions about Yt or Xt .
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Example: Capital Asset Pricing Model
Model to explain stock returns. Investors make money
on the stock market from:
1
Dividends, Dt – actual case payments to investors.
2
Capital gains, St − St−1 – price changes in the stock.
CAPM models total returns Rt
Rt =
Walt Pohl (UZH QBA)
St + Dt − St−1
.
St−1
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Example: Capital Asset Pricing Model,
cont’d
1
2
3
Investors always have an alternative investment
available: “risk-free” government bonds.
The important quantity to explain, then, is excess
returns, the return over the risk-free rate Rf .
The Capital Asset Pricing Model claims that excess
returns for a stock are (up to an error term), a linear
function of the excess returns on the total market.
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Stochastic Models
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Example: Capital Asset Pricing Model,
cont’d
Let RM be return on the market. Then the CAPM says
R − Rf = β (RM − Rf ) + CAPM is usually estimated using linear regression:
R − Rf = α + β (RM − Rf ) + .
Note the extra α. CAPM predicts α = 0.
For “market return” we use the return on a broad-based
index, such as the S&P 500 for US stocks. In the
example, I use the CRSP value-weighted index, which
includes all stock traded on the major US exchanges.
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Stochastic Models
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Failures of the CAPM
The CAPM is wildly popular, but doesn’t have a good
track record.
Well-known failures include:
Stocks for small firms do better than expected,
large firms do worse than expected.
Stocks for “value firms” – firms with lots of assets
for their size – do better than “growth” firms –
firms few assets for their size.
Here “size” means total market capitalization – the cost
of buying up all of the firms’s stock.
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Stochastic Models
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Example: Fama-French 3 Factor Model
The main academic alternative to CAPM is factor
models. Define several additional return differences, and
compute several betas using multivariate regression.
The Fama-French 3 factor model uses
Market factor – return on market minus risk-free
rate (just like CAPM).
SMB – return on small stocks minus return on big
stocks.
HML – return on value stocks minus return on
growth stocks.
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Stochastic Models
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Beyond Classical Econometrics
Traditional statistics depends on stringent assumptions.
Maximum likelihood requires specifying a
distribution.
Linear regression requires positing a linear
relationship.
There’s essentially no reason to believe in these
assumptions.
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Stochastic Models
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Big Data
There are newer techniques that go under the generic
name of “machine learning”. They require
Lots of data.
Computer power.
We will discuss these techniques more later.
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Stochastic Models
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