Stochastic Models - Statisticsffffffff-858f-2321-ffff...Walt Pohl (UZH QBA) Stochastic Models...

Stochastic ModelsStatistics

Walt Pohl

Universitat ZurichDepartment of Business Administration

February 28, 2013

The Value of Statistics

Business people tend to underestimate the value ofstatistics. Statistics allows us to learn from history.

Finance people tend to overestimate the value ofstatistics. History rarely repeats itself exactly.

Walt Pohl (UZH QBA) Stochastic Models February 28, 2013 2 / 23

Statistical Methods

1 Method of moments2 Maximum likelihood3 Regression4 many more...


Method of moments

Simplest (and oldest) method of doing statistics. Picksome moments (usually mean and variance), and matchwith the data.


Example: Lognormal Model of StockPrices

1 Changes in stock prices are unpredictable.2 Modeled by random changes in the growth rate.3 We assume these random changes can be modeled

by a lognormal distribution.

In other words, if St is the stock price, then the loggrowth rate,

log St+1/St

is normally distributed.


Example: Lognormal Model of StockPrices

The model can be written as

log St+1 = µ + log St + εt+1,

where εt is normally distributed Normal(0, σ2).

In financial circles, the variance or standard deviation isknown as the volatility.


Example: Lognormal Model of StockPrices, cont’d

This model has two parameters, the mean µ andvariance σ2, both of which can be computed frommoments. (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


Example: Lognormal Model of StockPrices, cont’d

Is this a good model? No!

1 Distribution is “fatter tails” than the normal: moreextreme events than predicted by the normaldistribution.

2 Variances themselves vary over time.

We consider the second.


Example: The EWMA Volatility Model

Forecasting is the art of predicting the future from today.The easiest, and frequently most successful, method is topredict that the future will be like the recent past.

The exponentially-weighted moving average (EWMA)model for forecasting volatility is a simple example. Weforecast tomorrow’s volatility by a weighted sum oftoday’s forecast and today’s observation


Example: The EWMA Volatility Model,cont’d

Let σ2t be the variance of εt for period t. Then theEWMA forecast is

σ2t = λσ2t−1 + (1− λ)ε2t .

But how do we find λ?


Maximum Likelihood Estimation

Maximum likelihood estimation is a general-purposemethod for estimating parameters in models.

One limitation: you must completely specify yourprobability distributions. (Here we must specify that εt isnormally distributed).


From PDF to Likelihood

A probability density function (PDF) depends on1 parameters2 data

Once we plug in the data, we get a function of theparameters alone – the likelihood.


Example: the Normal Distribution

Recall the PDF of the normal distribution, Normal(µ, σ):

1√2πσ2

e−(x−µ)2

2σ2

If we have N independent draws, of a normal randomvariable with unknown mean µ and variance σ.(

1√2πσ2

)N∏i

e−(xi−µ)2

2σ2

We choose as our estimates the µ and σ to maximizethis expression. This is the maximum likelihood estimate(MLE).


Example: the Normal Distribution, cont’d.

We can actually compute the MLE for the normaldistribution explicitly:

µMLE =N∑i

xi/N

σ2MLE =N∑i

(xi − µMLE )2/N

These are the sample mean and standard deviation.


Likelihood for EWMA

The likelihood for the EWMA is similar, but now σidepends on the data. Let εi = log Si+1 − log Si − α(where α is the sample mean). Then, given lambda, weknow that

σ2i = λσ2i−1 + (1− λ)ε2i .

The (log) likelihood is then just

−∑i

(εi

2σ2i− 1

2log(2πσ2i )

)We must resort to numerical methods to maximize thisexpression.


Linear Regression

Linear regression is a technique for finding linearrelationships between variables,

Yt = α + βXt + ε

Requires no distributional assumptions about Yt or Xt .


Example: Capital Asset Pricing Model

Model to explain stock returns. Investors make moneyon 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 =St + Dt − St−1

St−1.


Example: Capital Asset Pricing Model,cont’d

1 Investors always have an alternative investmentavailable: “risk-free” government bonds.

2 The important quantity to explain, then, is excessreturns, the return over the risk-free rate Rf .

3 The Capital Asset Pricing Model claims that excessreturns for a stock are (up to an error term), a linearfunction of the excess returns on the total market.


Example: Capital Asset Pricing Model,cont’d

Let RM be return on the market. Then the CAPM says

R − Rf = β (RM − Rf ) + ε

CAPM is usually estimated using linear regression:

R − Rf = α + β (RM − Rf ) + ε.

Note the extra α. CAPM predicts α = 0.For “market return” we use the return on a broad-basedindex, such as the S&P 500 for US stocks. In theexample, I use the CRSP value-weighted index, whichincludes all stock traded on the major US exchanges.


Failures of the CAPM

The CAPM is wildly popular, but doesn’t have a goodtrack 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 assetsfor their size – do better than “growth” firms –firms few assets for their size.

Here “size” means total market capitalization – the costof buying up all of the firms’s stock.


Example: Fama-French 3 Factor Model

The main academic alternative to CAPM is factormodels. Define several additional return differences, andcompute several betas using multivariate regression.

The Fama-French 3 factor model uses

Market factor – return on market minus risk-freerate (just like CAPM).

SMB – return on small stocks minus return on bigstocks.

HML – return on value stocks minus return ongrowth stocks.


Beyond Classical Econometrics

Traditional statistics depends on stringent assumptions.

Maximum likelihood requires specifying adistribution.

Linear regression requires positing a linearrelationship.

There’s essentially no reason to believe in theseassumptions.


Big Data

There are newer techniques that go under the genericname of “machine learning”. They require

Lots of data.

Computer power.

We will discuss these techniques more later.


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