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Statistics Review 1

Marriott School of Management

Fall 2014

Rob Schonlau

Last updated Sept 10, 2014

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Statistics review part 1

As introduced earlier, we think of risk in terms of the likelihood

of observing return outcomes that are far different than the

expected outcome.

Financial theory states that there is a risk-return relationshipwhere people must be compensated for bearing additional risk.

Before we can discuss the formal financial risk-return models

we need to first review some of the related statistical concepts.

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Lecture 4 outline

Discuss statistical concepts that are useful for thinking about an

assets performance and risk.

Random variables

Expectations Probability density functions (PDFs) Variance and standard deviation

Review the normal distribution. Use properties of the normal

distribution to answer probability questions and to solve for the

value at risk.

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DefinitionsRandom Variable: A variable whose value is uncertain. I find it helpful

to think of a data generating process that generates all the possible

outcome values in each time period for the variable in question.

Example: IBM stock returns

Observation: The observed value from a single outcome of the random

variable. You can think about an observation as a single draw or a

single example observed out of a whole underlying population of

possible values.

Next years observed IBM annual return will be a singleobservation from the underlying possible set of IBM returns.

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Average weekly return: .0019

Standard deviation: .046

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Definitions continued

Probability density function (PDF): A function that describes the

probability of each outcome for a random variable.

Expectation or Expected Value: The mean or expected value of a

random variable is a single value that summarizes the value you would

observe on average if you could observe the outcome of the random

variable many times. If r is a random variable then the expectation

notation is E[r], or sometimes m.

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Discrete PDFExample of a discrete PDF

Only a finite number of outcomes (in this example there are twopossible outcomes or two states) The value of the function, p(s), tells us the exact probability of

observing a given state.

The sum of the probabilitiesacross all possible outcomes(states) must equal 1.

Discrete PDFs are not very realistic. Why do we use them?

Provides statistical intuition for more complex distributions Part of the CFA curriculum

%15for25.0

%10for75.0)(

r

rsp

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Continuous PDF

Example: Normal PDF

Infinite number of outcomes even within defined range The integral of the function between two points tells us the

probability of getting an outcome between those two points.

The integral of the function over the range of possible outcomesmust equal 1.

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Expectation

For a discrete probability function with Spossible outcomes (states)

where p(s) = probability of each of S possible statesr(s) = observed return if state s occurs

For example given the following probability function:

E[r]=0.75*(.10)+.25*(-.15)= 3.75%

S

ssrsprE

1)()(][

%15for25.0%10for75.0)(

rrsp

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Example of calculating expected return

Using the general PDF notation the information in this table can be

summarized as:

State of

Economy

Scenarios

(states)

Probability of

each state

Returns

Boom 1 .25 44%

Normal growth 2 .50 14%

Recession 3 .25 -16%

%16for25.0

%14for50.0

%44for25.0

)(

r

r

r

sp

10E[r]=.25(.44) + .50(.14) + .25(-.16) = .14

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Expectations using continuous PDFs

For a continuous probability function

The idea is the same as for a discrete PDF. We just integrate

across all possible values rather than sum over the discrete

values.

drsrsprE )()(][

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Expectation of a function of a random

variable

At times we are interested in the expectation of a function of a random

variable. For example, assume the following discrete PDF for random

variable r:

What is the E[r], E[r2], and E[3r+5]?

E[r] = .65*(.08) + .35*(-.10) = 0.017 E[r2] = .65*(.082) +. 35*(-.102) = 0.008 E[3r+5] = .65*(3*.08+5)+ .35*(3*(-.10)+5) = 5.051

%10for35.0

%8for65.0

)( r

r

sp

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PDFs, expectations, and stock returns

We never know the true future distribution (PDF) of returns for any

investment.

However, we can observe the actual returns over time of aninvestment and with those historical returns infer the nature of the

(unobservable) underlying process generating those outcomes.

For example, we dont know the true underlying PDF for future IBM

stock returns. But if we know that prior IBM returns have averaged10% a year with a standard deviation of 4% we can get an idea of

the distribution of IBMs future returns.

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Average weekly return: .0019

Standard deviation: .046

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Summary of Expectations

Given an assumed PDF we can find the expectation as follows

When we dont know anything about the PDF, but rather, observe a

sample generated by an underlying process, we can estimate theexpected value as a simple average. This works for both discrete and

continuous PDFs.

drsrsprE

srsprE S

s

)()(][

)()(][1

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Statistics rule #1

Rule 1: Let x and y be any two random variables. If z = ax + by,

where a and b are constants, and x and y are random variables,

then

Note that because

bE[y]aE[x]E[z]

aaE ][][][ xaEaxE

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Statistics rule #1: Example

Assume you own portfolio Z with 30% of your wealth in asset A and

70% in asset B. Assume you have gathered data on the returns to A,

and B and inferred the following PDF.

If there is an expansion: rZ= .3(10%) + .7(5%) = 6.5%. Expansions

occur with probability 0.80.

If there is a recession: rZ= .3(0%)+.7(3%) = 2.1%. Recessions occur

with probability 0.20.

What is the expected return for portfolio z?

)(recession3and%0for20.0)(expansion5and%10for80.0)(

%rr%rrsp

BA

BA

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The return on portfolio Z (rZ) is a random variable that is itself a

function of two other random variables (rAand rB). In either state of

the world (expansion or recession) rZcan be represented by the

formula rZ =.3(rA) + .7(rB).

Using expectations (statistics rule #1):

E[rZ] = .3 E(rA) + .7 E(rB)

First solve for E(rA) and E(rB) and then for E[rZ]

)(recession3and%0for20.0

)(expansion5and%10for80.0)(

%rr

%rrsf

BA

BA

5.62%0.7(0.046)0.3(0.08)]E[

046.0)03.20(.)05.80(.][

08.0)020(.)10.80(.][

Z

r

rE

rE

B

A

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Discrete PDF Example: Which of these

investments would you choose? Why?Investment #1 PDF:

E[r] = 3.75%

Investment #2 PDF:

E[r] = 3.75%

%15for25.0

%10for75.0)(

r

rsp

otherwise0

%75.3for1

)(

r

sp

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One possible measure of risk: Expected

deviation from mean.To estimate the risk involved with the two investments, lets

calculate the expected deviation from the mean.

The deviation from the mean for any observed return is r - E[r].

Hence the expected deviation from the mean is: E[r - E[r]]

#1: E[r-E[r]] = .75*(.10 - .0375) + .25*(-.15 - .0375) = 0

#2: E[r-E[r]] = 1*(.0375 - .0375) = 0

Not a very helpful measure!

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Another possible measure of risk: Variance

How about finding the expected squared deviation (variance) from

the mean?

Investment #1variance = .75*(.10-.0375)2 + .25*(-.15-.0375)2

= 0.0117

Investment #2

variance = 1*(.0375-.0375)2 = 0

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Variance

For a discrete probability function with Soutcomes

An alternative formula for variance:

2

1

2])[)()(()(

S

s

rEsrsprVar

222 ][][)( rErErVar

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Theoretical vs estimated

Again . . . we consider PDFs to be the underlying number

generating machines

In real life, we dont know the true properties of the underlying

(theoretical) PDF that generates the returns we observe. But the

returns we observe allow us to learn something about theproperties of the PDF that created them.

We have formulas for the theoretical expectation and the

variance. But the underlying PDF is not known so we have to use

estimates of the expectation and the variance.

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Estimation

To estimate the variance using sample observations, just take the

simple averageof the squared deviations from the estimated mean

with a slight correction for estimation error.

Example:

Sample of returns: 0.10, 0.05, 0, -.03

.)(1

1 22

ii rrn

03.4/)03.005.010.0( r

0033.

])03.03.()03.0()03.05.0()03.10.0[(*)]14/(1[ 22222

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Statistics rule #2

If z = ax + c, where a and c are constants, and x is a random

variable, then

If z = ax + by + c, where a and c are constants, and x and y are

random variables, then

xz

xz

a

a

222

xyyxyxz baba 222222

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Lecture 4 outline

Discuss statistical concepts that are useful for thinking about an

assets performance and risk.

Random variables

Expectations PDFs Variance and standard deviation

Review the normal distribution. Use properties of the normal

distribution to answer probability questions and to solve for the

value at risk.

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What are the benefits of assuming a normal

distribution?

Easy to use in math models.

The normal distribution has well known properties and is easily

accessible via Excel and other software packages.

Are returns distributed normally?

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Normal distribution

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Are returns normally distributed?

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3 Ann Taylor

This is a time series plot of the return process. Thex-axis is

time, and the y-axis is the value of the return at that time.

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How good is the normal assumption?

Ann Taylor

This is a histogram of returns. Thex-axis represents possible outcomes for

the return. We divide thex-axis into bins or intervals and count the number

of returns that fall into each interval. The y-axis tells us how many days had a

return within the corresponding interval.

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The mean variance framework

The variance on any investment measures the disparity between

actual and expected returns.

Expected Return

Low Variance Investment

High Variance Investment

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Normal distribution

Assume the PDF for your investment return is a normal

distribution.

If we know E[r] and [r] we can integrate under the normal

curve over any region using calculus (or Excel). That is we can

find theprobability the return will fall within any given range.

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Using the PDF distribution to gain

understanding of possible outcomes.Assume the PDF for your investment return is a normal

distribution with E[r]=10% and [r]=0.15.

What is the probability that r < - 20%?

What is the probability that r > 30%?

What is the probability that -20% < r

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According to your calculations, over the next year:

E[r] = 0.10

= 0.20

Find the losses you expect to incur with 5% probability.

5% VAR = 0.101.64*0.20 = -0.23 During any given year, you should expect to lose 23% or

more of your portfolio value with 5% probability.

Example application of the normal

distribution: 5% Value-at-Risk (VAR)

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