Statistics and Data Analysis

1
Statistics and Data Analysis

• Professor William Greene
• Stern School of Business
• Department of IOMS
• Department of Economics

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Statistics and Data Analysis
Part 11 Random Walks
3
A Model for Stock Prices
18/46

• Preliminary
• Consider a sequence of T random outcomes,
  independent from one to the next, ?1, ?2,, ?T.
  (? is a standard symbol for change which will
  be appropriate for what we are doing here. And,
  well use t instead of i to signify something
  to do with time.)
• ?t comes from a normal distribution with mean µ
  and standard deviation s.

4
Application
21/46

• Suppose P is sales of a store. The accounting
  period starts with total sales 0
• On any given day, sales are random, normally
  distributed with mean µ and standard deviation s.
  For example, mean 100,000 with standard
  deviation 10,000
• Sales on any given day, day t, are denoted ?t
• ?1 sales on day 1,
• ?2 sales on day 2,
• Total sales after T days will be ?1 ?2 ?T
• Therefore, each ?t is the change in the total
  that occurs on day t.

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Using the Central Limit Theorem to Describe the
Total
19/46

• Let PT ?1 ?2 ?T be the total of the
  changes (variables) from times (observations) 1
  to T.
• The sequence is
• P1 ?1
• P2 ?1 ?2
• P3 ?1 ?2 ?3
• And so on
• PT ?1 ?2 ?3 ?T

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Summing
20/46

• If the individual ?s are each normally
  distributed with mean µ and standard deviation s,
  then
• P1 ?1 Normal µ, s
• P2 ?1 ?2 Normal 2µ, sv2
• P3 ?1 ?2 ?3 Normal 3µ, sv3
• And so on so that
• PT NTµ, svT

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Application
21/46

• Suppose P is accumulated sales of a store. The
  accounting period starts with total sales 0
• ?1 sales on day 1,
• ?2 sales on day 2
• Accumulated sales after day 2 ?1 ?2
• And so on

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This defines a Random Walk
22/46

• The sequence is
• P1 ?1
• P2 ?1 ?2
• P3 ?1 ?2 ?3
• And so on
• PT ?1 ?2 ?3 ?T

• It follows that
• P1 ?1
• P2 P1 ?2
• P3 P2 ?3
• And so on
• PT PT-1 ?T

9
A Model for Stock Prices
23/46

• Random Walk Model Todays price yesterdays
  price a change that is independent of all
  previous information. (Its a model, and a very
  controversial one at that.)
• Start at some known P0 so P1 P0 ?1 and so
  on.
• Assume µ 0 (no systematic drift in the stock
  price).

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Random Walk Simulations
24/46

• Pt Pt-1 ?t
• Example P0 10, ?t Normal with µ0, s0.02

11
Uncertainty
25/46

• Expected Price EPt P0TµWe have used µ 0
  (no systematic upward or downward drift).
• Standard deviation svT reflects uncertainty.
• Looking forward from now time t0, the
  uncertainty increases the farther out we look to
  the future.

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Using the Empirical Rule to Formulate an Expected
Range
26/46
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Application
27/46

• Using the random walk model, with P0 40, say µ
  0.01, s0.28, what is the probability that the
  stock will exceed 41 after 25 days?
• EP25 40 25(.01) 40.25. The standard
  deviation will be 0.28v251.40.

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Prediction Interval
28/46

• From the normal distribution,Pµt - 1.96st lt X lt
  µt 1.96st 95
• This range can provide a prediction interval,
  where µt P0 tµ and st svt.

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Random Walk Model
29/46

• Controversial many assumptions
• Normality is inessential we are summing, so
  after 25 periods or so, we can invoke the CLT.
• The assumption of period to period independence
  is at least debatable.
• The assumption of unchanging mean and variance is
  certainly debatable.
• The additive model allows negative prices.
  (Ouch!)
• The model when applied is usually based on logs
  and the lognormal model. To be continued

16
Lognormal Random Walk

• The lognormal model remedies some of the
  shortcomings of the linear (normal) model.
• Somewhat more realistic.
• Equally controversial.
• Description follows for those interested.

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Lognormal Variable
30/46
If the log of a variable has a normal
distribution, then the variable has a lognormal
distribution. Mean Expµs2/2 gt Median Expµ
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Lognormality Country Per Capita Gross Domestic
Product Data
31/46
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Lognormality Earnings in a Large Cross Section
32/46
20
Lognormal Variable Exhibits Skewness
33/46
The mean is to the right of the median.
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Lognormal Distribution for Price Changes
34/46

• Math preliminaries
• (Growth) If price is P0 at time 0 and the price
  grows by 100? from period 0 to period 1, then
  the price at period 1 is P0(1 ?). For example,
  P040 ? 0.04 (4 per period) P1 P0(1
  0.04).
• (Price ratio) If P1 P0(1 0.04) then P1/P0
  (1 0.04).
• (Math fact) For smallish ?, log(1 ?)
  ?Example, if ? 0.04, log(1 0.04) 0.39221.

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Collecting Math Facts
35/46
23
Building a Model
36/46
24
A Second Period
37/46
25
What Does It Imply?
38/46
26
Random Walk in Logs
39/46
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Lognormal Model for Prices
40/46
28
Lognormal Random Walk
41/46
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Application
42/46

• Suppose P0 40, µ0 and s0.02. What is the
  probabiity that P25, the price of the stock after
  25 days, will exceed 45?
• logP25 has mean log40 25µ log40 3.6889 and
  standard deviation sv25 5(.02).1. It will be
  at least approximately normally distributed.
• PP25 gt 45 PlogP25 gt log45 PlogP25 gt
  3.8066
• PlogP25 gt 3.8066 P(logP25-3.6889)/0.1 gt
  (3.8066-3.6889)/0.1)PZ gt 1.177 PZ lt
  -1.177 0.119598

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Prediction Interval
43/46

• We are 95 certain that logP25 is in the
  intervallogP0 µ25 - 1.96s25 to logP0 µ25
  1.96s25. Continue to assume µ0 so µ25
  25(0)0 and s0.02 so s25 0.02(v25)0.1 Then,
  the interval is 3.6889 -1.96(0.1) to 3.6889
  1.96(0.1)or 3.4929 to 3.8849.This means that
  we are 95 confident that P0 is in the
  rangee3.4929 32.88 and e3.8849 48.66

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Observations - 1
44/46

• The lognormal model (lognormal random walk)
  predicts that the price will always take the form
  PT P0eS?t
• This will always be positive, so this overcomes
  the problem of the first model we looked at.

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Observations - 2
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• The lognormal model has a quirk of its own. Note
  that when we formed the prediction interval for
  P25 based on P0 40, the interval is
  32.88,48.66 which has center at 40.77 gt 40,
  even though µ 0. It looks like free money.
• Why does this happen? A feature of the lognormal
  model is that EPT P0exp(µT ½sT2) which is
  greater than P0 even if µ 0.
• Philosophically, we can interpret this as the
  expected return to undertaking risk (compared to
  no risk a risk premium).
• On the other hand, this is a model. It has
  virtues and flaws. This is one of the flaws.

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Summary
46/46

• Normal distribution approximation to binomial
• Approximate with a normal with same mean and
  standard deviation
• Continuity correction
• Sums and central limit theorem
• Random walk model for stock prices
• Lognormal variables
• Alternative random walk model using logs