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Chapter 8: What is Risk?
Chapter 9: Portfolio Statistics
FIN 450
Dr. Anthony May
Working with stock prices in
excel
Computing
realized returns
Realized return on an asset during a
given period is defined as follows:
P1  P0  D1
R
P0
R = Realized Return
P0 = Price at beginning of period
P1 = Price at end of period
D1 = Total amount of dividends (or
coupon payment) paid during the
period.
Example: Computing Returns
Kellogg
stock Example
Working with stock prices
Stock
price data from most commercial
sources like Yahoo! Finance are already
adjusted for dividends (and stock splits).
This means that you can simply compute
the return as the percentage change in
the adjusted price – this will give you the
correct return
AP1  AP0
R
AP0
R = Realized Return
AP0 = Adjusted Price at beginning of period
AP1 = Adjusted Price at end of period
Getting Stock Price Data from
Yahoo! Finance
www.finance.yahoo.com
Enter
the ticker of your stock in the
“Quotes” box and hit Enter.
Click on Historical Prices
Choose frequency (daily, weekly,
monthly) and then choose date range
Click “Get Prices” button
Click “Download to Spreadsheet” at
bottom
Getting Stock Price Data from
Yahoo! Finance
Excel
file should open
Data is in reverse chronological order
(most recent date first) – but we
usually work with stock prices in
chronological order
Click on Data tab, Sort. Next to “Sort
By” choose date
“Adjusted Close” is the data that we
want – this is the stock price adjusted
for dividends and stock splits
Example: Getting Stock Price Data
from Yahoo! Finance
Get
adjusted stock price data for
Microsoft (ticker MSFT)
Get monthly prices between Jan. 1,
1990 and Jan. 3, 2012.
Compute monthly returns using the
adjusted prices
Compute the average monthly return,
using the Average function
What is risk?
Most
important and problematic concept
in finance
In finance we define risk as the degree
of uncertainty about what an asset’s
future returns (or cash flows) will be.
How is risk measured?
Measuring Investment Risk: Variability of
Stock Returns
 Variance
and Standard Deviation
 Variance of returns: The variance of an
asset’s realized returns over a given period of
time is a measure of how volatile or how
spread out those returns are around the
average.
Variance and Standard Deviation
(2) - the expected value of squared
deviations from the average
 Variance
N
 ( R  R)
2
t
Variance   
2
t 1
N 1
• The unit of the variance is percentage-squared, which
is difficult to interpret.
• Standard deviation () =square root of the variance.
Taking the square root gets the units back to
percentage terms.
• Standard deviation and variance are the most
common measures of a stock’s risk.
Variance and Standard
Deviation
In
Excel, we will use the functions
VAR and STDEV to compute variance
and standard deviation.
Example: Variance and Standard
Deviation
 Compare
the monthly returns of Wal-Mart and
Arena Pharmaceuticals.
 Which stock had a higher average monthly
return?
 But…which stock was more risky
Making a frequency distribution
(histogram)
 Chapter
8, page 270
 Frequency function counts number of data points in
specific ranges
 Frequency is an array function – works differently
than a normal function
Fist Mark the range for the function, then type
=Frequency(
Mark the data array (stock returns in this case) and
type a comma
Mark the bins array, then type )
DON’T HIT ENTER!!!!
Instead, hold down Ctrl and Shift and then hit
Enter
Making a frequency distribution
(histogram)
WHEN
GRAPHING A FREQUENCY
DISTRIBUTION, ALWAYS USE A
SCATTER CHART
Use one that connects the markers with a
line
If you want, you can use one without
markers (just lines).
DO
NOT USE A LINE GRAPH TO
MAKE A FREQUENCY
DISTRIBUTION
Example: Frequency Distribution
of Daily Returns
Compare
the distribution of Wal-Mart’s
daily returns to those of Arena
Pharmaceuticals by making a
frequency distribution and graphing it
Excel functions used
Average
Sqrt
Var
Stdev
Covariance.s
Correl
Frequency
Count
Ln
(advanced)
Additional Issues: Graphing Stock prices (or
Returns) with Dates on the X-Axis
Graph
on following slide: McDonald’s
adjusted stock price during 1998-2008
Dates are on the X-axis
If dates are on X-axis, USE A LINE
CHART
If dates are not on the x-axis, use a
scatter chart
McDonald's Stock Prices
31 Dec 1998 - 31 Dec 2008
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60
50
40
30
20
10
0
28-Dec-08
29-Dec-07
29-Dec-06
29-Dec-05
29-Dec-04
30-Dec-03
30-Dec-02
30-Dec-01
30-Dec-00
31-Dec-99
31-Dec-98
Portfolio Returns



Most investors hold a portfolio of multiple assets
Return on a portfolio over a given period is the
weighted average of returns on the individual assets
in the portfolio, where the weights reflect the amount
of money invested in each asset
Return on a portfolio of two stocks
Rp  wa  Ra  wb  Rb
Rp= Portfolio Return
Ra = Return on Stock A
Rb = Return on Stock B
wa = weight of Stock A in the portfolio
wb = weight of Stock B in the portfolio
Example: Portfolio Returns
 Average,
variance, and standard deviation of
portfolio returns computed the same way as
before
 In-Class Example: Compute the returns for a
portfolio with 50% invested in Exxon and 50%
invested in Kellogg
Additional statistical concepts
 Covariance
and Correlation
 Both are measures of how the returns of two
assets move together.
 The Covariance between two stocks, denoted i
and j, is defined as:
N
(R
i ,t
Covariance 
 Ri )( R j ,t  R j )
t 1
N 1
Additional statistical concepts:
Covariance and Correlation
 Covariance:
Do the returns on two stocks tend
to move in same direction? Meaning: When
one stock goes up (down), does the other tend
to go up (down) also?
If “on average, yes,” then Covariance > 0
If “on average, no, they tend to move in opposite
directions,” then Covariance < 0.
Sign of covariance is interpretable, but magnitude
of covariance is not directly interpretable without
additional info, i.e., generally we cannot use
covariance alone to determine how closely two
assets returns move together
Additional statistical concepts:
Covariance and Correlation
 Correlation:
Generally a more useful measure of
how two stocks “move” together.
 For two stock’s, denoted i and j, correlation is
defined as:
Correlationi,j 
Covariancei,j
σiσ j
 From
the formula above, you can see that another
way to express the Covariance is:
Covariancei,j  Correlationi,jσ iσ j
Additional statistical concepts:
Covariance and Correlation
 Correlation
always > -1 and < +1
 Correlation > 0: returns of two assets tend
to move in same direction
 Correlation ≤ 0: returns of two assets tend
to move in opposite direction
 Higher (more positive) the correlation, more
closely the two assets’ returns move
together
 Correlation +1 or -1 means perfect linear
relation between two assets
Covariance and Correlation in
Excel
Use
COVARIANCE.S function to
compute covariance
Use CORREL function for correlation
Example: Covariance and
Correlation of Exxon and Kellogg
Facts about covariance and
correlation
Symmetry:
Covariance between Exxon
and Kellogg = Covariance between
Kellogg and Exxon
Same for correlation
Diversification
•
Average return of a portfolio = weighted
average of average returns on all the stocks in
the portfolio.
• Standard deviation of the portfolio is not a
weighted average of individual stock standard
deviations.
• As long as the stocks in the portfolio are not
perfectly positively correlated (correlation=1),
the portfolio standard deviation will be less than
the weighted average of standard deviations of
individual stocks in the portfolio.
Diversification
•
•
•
Why? Idiosyncratic fluctuations of the
different assets partially cancel each
other out.
If correlations of the stocks are low
enough, you can even achieve a
portfolio standard deviation that is lower
than the standard deviation of any
individual stock in the portfolio
Diversification = elimination of some
risk.
Diversification
•
•
In general, the benefits of diversification
(reduction of volatility) increase as you
add more and more individual assets to
the portfolio, but only up to a point.
When investing in risky assets in the real
world, generally you cannot completely
eliminate all risk, i.e., get a portfolio
standard deviation of zero.
Diversification
See
Diversification examples in
Lecture Spreadsheets
Risk-Return Tradeoff



Make a data table that shows how the standard deviation of a portfolio
consisting of Exxon and Kellogg changes as you vary the weight of
Kellogg from 0 to 1 (use increments of 0.1).
Do the same for the portfolio average return.
Using your data table values, make a graph that plots the portfolio’s
average return on the Y-axis and standard deviation on the X-axis.
Make sure to use a SCATTER CHART. DO NOT USE A LINE CHART.
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
 If
a firm has one or more stock splits during a
given period of time, you cannot use the raw
(unadjusted) stock prices to directly compute
stock returns.
 You must first adjust the closing stock prices
(and dividends) for splits.
 Then you can compute the proper stock
returns using the split-adjusted prices (and
dividends).
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
 The
daily (unadjusted) stock prices and dividends
of a firm are given in the lecture spreadsheet.
 The firm had a 2 for 1 stock split on Feb. 18, 2003.
This means that the firm’s shares outstanding were
doubled on this date.
 Firm had a 3 for 1 stock split on Nov. 10, 2003.
Shares outstanding were tripled (each share was
converted into 3 shares).
 Adjust the stock prices and dividends for splits in
such a way that the adjusted prices and dividends
can be used to compute the daily stock returns
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
 What
to do?
 Make an “adjustment factor.” We will multiply the
raw stock prices and dividends by this factor to
get the split-adjusted prices and dividends
 Start with the last (latest) date and set the
adjustment factor to 1.
 Work your way up and change the adjustment
factor on every date preceding a split.
 The split factor will determine how the
adjustment factor changes.
 See next few slides for details
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Multiply
the stock prices and dividends
by the adjustment factor
Use the split-adjusted prices and
dividends to compute returns.
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 1
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 2
If
you have annual or monthly stock
prices (and dividends), the process is
the same.
The annual closing prices and dividends
for a firm are in the lecture spreadsheet.
The firm had a 2 for 1 stock split in May
1997.
The firm had a 3 for 2 stock split in April
1999.
Adjusting Raw Stock Prices (and
dividends) for Stock Splits: Example 2
Adjusting Raw Stock Prices (and dividends)
for Reverse Splits: Example 1
 Whereas
Stock Splits increase the number of
outstanding shares, Reverse Splits decrease
the number of outstanding shares.
 The annual stock returns and dividends of a
firm are in the lecture spreadsheet.
 In 1996, the firm had a 1 for 3 reverse stock
split.
Means that every three shares were converted
into one share
 In
1998, the firm had 2 for 1 stock split.
Each share was converted into two shares (same
as before).
Adjusting Raw Stock Prices (and
dividends) for Reverse Splits: Example 1
Chapter 10
Portfolio Returns and the
Efficient Frontier
FIN 450
Dr. Anthony May
This chapter
Formula
definitions for expected
return and standard deviation of
portfolio of two assets
Portfolio risk and return
Finding the minimum variance
portfolio and other efficient portfolios
Three-Asset Portfolios
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Excel in Chapter 10
Excel
Functions: Average, Var, Stdev,
Covariance.s,
Data Table and Graphing
Solver
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Refresher: Average, Variance, and St. Deviation
of Portfolio of 50% Kellogg and 50% Exxon

In previous lecture, we used the
AVERAGE, VAR and STDEV
functions to compute the average,
variance, and standard deviation of
historical returns for a portfolio
consisting of 50% Kellogg and 50%
Exxon.
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Expected Return on a Portfolio
of two assets
Often
we use the term “expected”
synonymously with average.
“Expected return” often means the
average return we expect for a stock in
the future.
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Expected Return of a Two-Asset
Portfolio
Expected
(or average) return of a
portfolio of two assets
Rportfolio= waRa + wbRb
wa = weight of asset a
wb = weight of asset b
Ra = expected return of Asset a
Rb = expected return of Asset b
Rportfolio = expected return of portfolio
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Variance of a Two-Asset Portfolio
VARportfolio= wa2VARa + wb2VARb + 2wawbCOVa,b
SDportfolio=√ VARportfolio
VARportfolio = Variance of portfolio
wa = weight (or proportion) of asset a
wb = weight of asset b
VARa = Variance of Asset a
VARb = Variance of Asset b
COVa,b = Covariance of Asset a with Asset b
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Variance of a Two-Asset Portfolio
COVa,b= (CORa,b)(SDa)(SDb),
formula for variance of a portfolio is
sometimes written as:
Since
VARportfolio= wa2VARa + wb2VARb + 2wawb(CORa,b)(SDa)(SDb),
SDa = Standard Deviation of Asset a
SDb = Standard Deviation of Asset b
CORa,b = Correlation of Asset a with Asset b
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Implementing the formulas
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Past (realized) Returns vs.
Expected Future Returns
 In
the previous slide, we applied the portfolio
formulas to returns realized in the past.
 Investors are often interested in estimating the future
expected return and risk of a portfolio.
You can use the portfolio average and variance formulas to
do this, as long as you have estimates of the future expected
returns, variances, and covariances of stocks in the portfolio.
A variety of methods are used in practice to obtain estimates
of future expected returns, variances, and covariances, e.g.,
asset pricing models (CAPM), statistical models, etc. We will
talk about the CAPM in a subsequent chapter.
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Expected Return and St. Deviation of Different
Portfolio Choices
Data Table
57
Some portfolios are obviously
suboptimal
 Portfolios on the bottom (downward sloping) portion of the curve are
suboptimal.
 For the same level of risk, you could achieve a higher portfolio return by
moving to a portfolio on the top (upward sloping) portion of the curve.
Expected return E(rp)
12%
11%
Portfolio Returns: Expected Return
and Standard Deviation
10%
9%
8%
7%
6%
5%
10%
11%
12%
13%
14%
15%
16%
17%
Standard deviation of return, p
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Some are more interesting!
 The upward sloping part of the curve is called the “efficient frontier.”
 Every portfolio on the efficient frontier is “efficient” because it
provides the highest portfolio return, given a specific level of risk.
Expected return E(rp)
12%
11%
Portfolio Returns: Expected Return
and Standard Deviation
10%
9%
8%
7%
6%
5%
10%
11%
12%
13%
14%
15%
16%
17%
Standard deviation of return, p
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The minimum variance portfolio
The Minimum Variance Portfolio
Expected return E(rp)
12%
11%
10%
9%
8%
7%
6%
5%
10%
11%
12%
13%
14%
15%
16%
17%
Standard deviation of return, p
Minimum variance portfolio can be located using:
 Trial and error
 A formula
 Excel’s Solver tool
60
The Efficient Frontier
Expected Return and Standard Deviation of
Portfolio Return--Showing Efficient Frontier
Expected portfolio return, E(rp)
12%
11%
10%
9%
8%
7%
6%
5%
10%
11%
12%
13%
14%
15%
16%
17%
Standard deviation of portfolio return, p
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The meaning of the efficient
frontier
The
efficient frontier: includes all
portfolio choices that maximize the
expected return subject to a given level
of risk (standard deviation).
All the portfolios represent a tradeoff
between expected return and risk
On the efficient frontier
Higher expected return is accompanied by
higher risk
62
Solver
We
will use Solver to find the
minimum-variance portfolio and
portfolios along the efficient frontier.
63
Install Solver






Click the File button (or Microsoft Office Button in
2007)
Click Options (Excel Options in 2007)
Click Add-Ins
In the Manage box, select Excel Add-ins. Click Go. In the
Add-Ins available box, select the Solver Add-in check
box, and then click OK.
If Solver Add-in is not listed in the Add-Ins available box,
click Browse to locate it.
If you get prompted that the Solver Add-in is not currently
installed on your computer, click Yes to install it.
After you load the Solver Add-in, the Solver command is
available on the Data tab, in the Analysis group
64
Solver: Example 1
Your
portfolio consists of K and XOM
stock.
Find the minimum variance portfolio.
65
Solver: Example 2
 Find
the weights of K and XOM that maximize
the portfolio’s expected return, subject to the
constraint that the portfolio’s standard deviation
is less than or equal to 13%.
$B$14 < = 0.13
66
Solver: Example 3
Stock A has
expected return of 7%
and a standard deviation of 12.25%.
Stock B has expected return of 9.2%
and a standard deviation of 17.89%.
The correlation between A and B is
0.55.
Find the minimum variance portfolio
that consists of Stocks A and B (i.e.,
find the weights on A and B that
minimize portfolio variance).
67
Solver: Example 3
68
Solver: Example 4
 You
have a portfolio with 40% invested in Stock
1 and 60% invested in Stock 2.
 Stock 1 has expected return of 10% and
variance of 0.015.
 Stock 2 has variance of 0.032.
 Your portfolio has an expected return of 11.8%
and standard deviation of 14.84%
 What is the Expected Return of Stock 2?
 What is the Correlation between Stock 1 and
Stock 2?
69
Solver: Example 4
See
next slide for Solver Parameters
70
Solver: Example 4
71
Expected Return of a Portfolio of
Three assets
Rportfolio= waRa + wbRb + wcRc
72
Variance and Standard Deviation of
a Three-Asset Portfolio
VARportfolio= wa2VARa + wb2VARb + wc2VARc +
2wawbCOVa,b + 2wawcCOVa,c + 2wbwcCOVb,c
SDportfolio=√ VARportfolio
73
Expected Return and Variance of
a Three Asset Portfolio
 Calculate
the expected return, variance, and standard
deviation of a portfolio that consists of 25% Stock A,
35% Stock B, and 40% Stock C.
74
Three-Asset Portfolio: Solver
Example 1
 Find
the weights that minimize the risk of a portfolio
consisting of Stocks A, B, and C.
75
Three-Asset Portfolio: Solver
Example 2
 Find
the weights that maximize the expected return of
the three-asset portfolio, subject to the constraint that
the portfolio’s standard deviation is less than or equal
to 45%.
76
Notes on Ch. 10 Practice
Exercises
 #12
The question asks you to compute the “return
statistics” of the second stock. This means you
need to compute the expected return, variance,
and standard deviation of the second stock
 A1, A2
Located in the Appendix of Ch. 10 (Page 343).
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