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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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