Chapter 11 Risk and Return in Capital Markets: Historical
Perspectives
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2 Chapter Outline 1. A First Look at Risk and Return 2.
Historical Risks and Returns of Stocks 3. The Historical Tradeoff
Between Risk and Return 4. Common Versus Independent Risk
5.Diversification in Stock Portfolios 6.Expected Return and Risk
7.Arithmetic average versus Geometric average
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3 Bull Market, Bear Markets, Risk and Return Source: Google
Finance
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4 Recent History: Yield Spreads 4
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5 Returns on Some Extreme Days
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6 The Market of 1987 2009, Bumps Along the Way Period% Decline
in S&P 500 Aug. 25, 1987 Oct. 19, 1987-33.2% Oct. 21, 1987 Oct.
26, 1987-11.9% Nov. 2, 1987 Dec. 4, 1987-12.4% Oct. 9, 1989 Jan.
30, 1990-10.2% July 16, 1990 Oct. 11, 1990-19.9% Feb. 18, 1997 Apr.
11, 1997-9.6% Jul. 17, 1998 Oct. 8, 1998-19.2% Jul. 19, 1999 Oct.
18, 1999-12.1% Aug. 28, 2000 Sep. 30, 2002-46.2% May. 16, 2008 Mar.
6, 2009-52.1%
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7 Historical Returns, Standard Deviations, and Frequency
Distributions: 19262009
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8 Geometric versus Arithmetic Averages, 1926-2006, 1926-2012
Source: Jordan, Miller, and Dolvin (2012) textbook
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9 11.1 A First Look at Risk and Return Consider how an
investment would have grown if it were invested in each of the
following from the end of 1929 until the beginning of 2010:
Standard & Poors 500 (S&P 500) Small Stocks World Portfolio
Corporate Bonds Treasury Bills
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10 Figure 11.1 Value of $100 Invested at the End of 1925 in
U.S. Large Stocks (S&P 500), Small Stocks, World Stocks,
Corporate Bonds, and Treasury Bills
Slide 11
11 Computing historical return and risk
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12 11.2 Historical Risks and Returns of Stocks Computing
Historical Returns Individual Investment Realized Returns The
realized return from your investment in the stock from t to t+1 is:
(Eq. 11.1)
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13 Example 11.1 Realized Return Microsoft paid a one-time
special dividend of $3.08 on November 15, 2004. Suppose you bought
Microsoft stock for $28.08 on November 1, 2004 and sold it
immediately after the dividend was paid for $27.39. What was your
realized return from holding the stock? Using Eq. 11.1, the return
from Nov 1, 2004 until Nov 15, 2004 is equal to This 8.51% can be
broken down into the dividend yield and the capital gain
yield:
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14 11.2 Historical Risks and Returns of Stocks Computing
Historical Returns Individual Investment Realized Returns For
quarterly returns (or any four compounding periods that make up an
entire year) the annual realized return, R annual, is found by
compounding: (Eq. 11.2)
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15 Example 11.2 Compounding Realized Returns Problem: Suppose
you purchased Microsoft stock (MSFT) on Nov 1, 2004 and held it for
one year, selling on Oct 31, 2005. What was your annual realized
return?
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16 Example 11.2 Compounding Realized Returns Solution: We need
to analyze the cash flows from holding MSFT stock for each quarter.
In order to get the cash flows, we must look up MSFT stock price
data at the purchase date and selling date, as well as at any
dividend dates. Next, compute the realized return between each set
of dates using Eq. 11.1. Then determine the annual realized return
similarly to Eq. 11.2 by compounding the returns for all of the
periods in the year. In Example 11.1, we already computed the
realized return for Nov 1, 2004 to Nov 15, 2004 as 8.51%. We
continue as in that example, using Eq. 11.1 for each period until
we have a series of realized returns. For example, from Nov 15,
2004 to Feb 15, 2005, the realized return is
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17 Example 11.2 Compounding Realized Returns (contd): We then
determine the one-year return by compounding. By repeating these
steps, we have successfully computed the realized annual returns
for an investor holding MSFT stock over this one-year period. From
this exercise we can see that returns are risky. MSFT fluctuated up
and down over the year and ended-up only slightly (2.75%) at the
end.
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18 Example: GM Stock
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19 Example: FORD Stock Problem: What were the realized annual
returns for Ford stock in 1999 and in 2008? DatePrice ($)Dividend
($)ReturnDatePrice ($)Dividend ($)Return 12/31/199858.69
12/31/20076.730 1/31/199961.440.265.13% 3/31/20085.720-15.01%
4/30/199963.940.264.49% 6/30/20084.810-15.91%
7/31/199948.50.26-23.74% 9/30/20085.208.11%
10/31/199954.880.2913.75% 12/21/20082.290-55.96%
12/31/199953.31-2.86%
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20 Example (contd) Solution Note that, since Ford did not pay
dividends during 2008, the return can also be computed as:
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21 Realized Return for the S&P 500, GM, and Treasury Bills,
19992008
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22 11.2 Historical Risks and Returns of Stocks Average Annual
Returns Example: S&P 500 Annual Returns (Eq. 11.3)
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23 Figure 11.2 The Distribution of Annual Returns for U.S.
Large Company Stocks (S&P 500), Small Stocks, Corporate Bonds,
and Treasury Bills, 19262012
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24 Figure 11.3 Average Annual Returns in the U.S. for Small
Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury
Bills, 19262012, 1926-2009
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25 11.2 Historical Risks and Returns of Stocks The Variance and
Volatility of Returns: Variance Standard Deviation (Eq. 11.4) (Eq.
11.5)
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26 Example 11.3 Computing Historical Volatility Using the data
from Table 11.1, what is the standard deviation of the S&P 500s
returns for the years 2005-2009? In the previous section we already
computed the average annual return of the S&P 500 during this
period as 3.1%, so we have all of the necessary inputs for the
variance calculation:
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27 Example 11.3 Computing Historical Volatility Alternatively,
we can break the calculation of this equation out as follows:
Summing the squared differences in the last row, we get 0.233.
Finally, dividing by (5-1=4) gives us 0.233/4 =0.058 The standard
deviation is therefore: Our best estimate of the expected return
for the S&P 500 is its average return, 3.1%, but it is risky,
with a standard deviation of 24.1%.
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28 Figure 11.4 Volatility (Standard Deviation) of U.S. Small
Stocks, Large Stocks (S&P 500), Corporate Bonds, and Treasury
Bills, 1926-2012, 19262009
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29 11.2 Historical Risks and Returns of Stocks The Normal
Distribution 95% Prediction Interval: About two-thirds of all
possible outcomes fall within one standard deviation above or below
the average (Eq. 11.6)
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30 Frequency Distribution of Returns on Common Stocks,
19262009
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31 Price Changes and Normal Distribution Coca Cola - Daily %
change 1987-2004 Proportion of Days Daily % Change
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32 The Normal Distribution and Large Company Stock Returns
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33 Example 11.4 Confidence Intervals Problem: In Example 11.3
we found the average return for the S&P 500 from 2005-2009 to
be 3.1% with a standard deviation of 24.1%. What is a 95%
confidence interval for 2010s return? Solution: Average (2 standard
deviation) = 3.1% (2 24.1%) to 3.1% + (2 24.1% ) = 45.1% to 51.3%.
Even though the average return from 2005 to 2009 was 3.1%, the
S&P 500 was volatile, so if we want to be 95% confident of
2010s return, the best we can say is that it will lie between 45.1%
and +51.3%.
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34 Example 11.4b Confidence Intervals Problem: The average
return for corporate bonds from 2005-2009 was 6.49% with a standard
deviation of 7.04%. What is a 95% confidence interval for 2010s
return? Solution: Even though the average return from 2005-2009 was
6.49%, corporate bonds were volatile, so if we want to be 95%
confident of 2010s return, the best we can say is that it will lie
between -7.59% and +20.57%.
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35 Table 11.2 Summary of Tools for Working with Historical
Returns
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36 Historical tradeoff between risk and return
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37 11.3 Historical Tradeoff between Risk and Return The Returns
of Large Portfolios Investments with higher volatility, as measured
by standard deviation, tend to have higher average returns Larger
stocks have lower volatility overall Even the largest stocks are
typically more volatile than a portfolio of large stocks The
standard deviation of an individual security doesnt explain the
size of its average return All individual stocks have lower returns
and/or higher risk than the portfolios in Figure 11.6
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38 Figure 11.6 The Historical Tradeoff Between Risk and Return
in Large Portfolios, 1926 2010
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39 Systematic risk, unsystematic risk, and diversification
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40 11.4 Common Versus Independent Risk Types of Risk Common
Risk Independent Risk An Example: Consider two types of home
insurance an insurance company might offer: theft insurance and
earthquake insurance
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41 Example 11.5 Diversification Problem: You are playing a very
simple gambling game with your friend: a $1 bet based on a coin
flip. That is, you each bet $1 and flip a coin: heads you win your
friends $1, tails you lose and your friend takes your dollar. How
is your risk different if you play this game 100 times in a row
versus just betting $100 (instead of $1) on a single coin flip?
Solution: Plan: The risk of losing one coin flip is independent of
the risk of losing the next one: each time you have a 50% chance of
losing, and one coin flip does not affect any other coin flip. We
can compute the expected outcome of any flip as a weighted average
by weighting your possible winnings (+$1) by 50% and your possible
losses (-$1) by 50%. We can then compute the probability of losing
all $100 under either scenario.
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42 Example 11.5 Diversification Execute: If you play the game
100 times, you should lose about 50 times and win 50 times, so your
expected outcome is 50 (+$1) + 50 (-$1) = $0. You should
break-even. Even if you dont win exactly half of the time, the
probability that you would lose all 100 coin flips (and thus lose
$100) is exceedingly small (in fact, it is 0.50100, which is far
less than even 0.0001%). If it happens, you should take a very
careful look at the coin! Evaluate: In each case, you put $100 at
risk, but by spreading-out that risk across 100 different bets, you
have diversified much of your risk away compared to placing a
single $100 bet. If instead, you make a single $100 bet on the
outcome of one coin flip, you have a 50% chance of winning $100 and
a 50% chance of losing $100, so your expected outcome will be the
same: break-even. However, there is a 50% chance you will lose
$100, so your risk is far greater than it would be for 100 one
dollar bets.
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43 Example 11.5a Diversification Problem: You are playing a
very simple gambling game with your friend: a $1 bet based on a
roll of two six-sided dice. That is, you each bet $1 and roll the
dice: if the outcome is even you win your friends $1, if the
outcome is odd you lose and your friend takes your dollar. How is
your risk different if you play this game 100 times in a row versus
just betting $100 (instead of $1) on a roll of the dice? Solution:
Plan: The risk of losing one dice roll is independent of the risk
of losing the next one: each time you have a 50% chance of losing,
and one roll of the dice does not affect any other roll. We can
compute the expected outcome of any roll as a weighted average by
weighting your possible winnings (+$1) by 50% and your possible
losses (- $1) by 50%. We can then compute the probability of losing
all $100 under either scenario.
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44 Example 11.5a Diversification Execute: If you play the game
100 times, you should lose about 50% of the time and win 50% of the
time, so your expected outcome is 50 (+$1) + 50 (-$1) = $0. You
should break-even. Even if you dont win exactly half of the time,
the probability that you would lose all 100 dice rolls (and thus
lose $100) is exceedingly small (in fact, it is 0.50100, which is
far less than even 0.0001%). If it happens, you should take a very
careful look at the dice! If instead, you make a single $100 bet on
the outcome of one roll of the dice, you have a 50% chance of
winning $100 and a 50% chance of losing $100, so your expected
outcome will be the same: break-even. However, there is a 50%
chance you will lose $100, so your risk is far greater than it
would be for 100 one dollar bets. Evaluate: In each case, you put
$100 at risk, but by spreading-out that risk across 100 different
bets, you have diversified much of your risk away compared to
placing a single $100 bet.
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45 Diversification using Historical Returns: ExxonMobil (XOM)
and Panera Bread (PNRA)
Slide 46
46 Why Diversification Works, Covariance: A measure of how the
returns for two securities moves together. Correlation Coefficient
Correlation is a standardized covariance scaled by the product of
two standard deviations, so that it can take a value between -1 and
+1. The correlation coefficient is denoted by Corr(R A, R B ) or
simply, A,B.
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47 Figure 11.8 The Effect of Diversification on Portfolio
Volatility
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48 11.5 Diversification in Stock Portfolios Unsystematic vs.
Systematic Risk Stock prices are impacted by two types of news:
1.Company or Industry-Specific News, or unsystematic risk,
diversifiable risk 2.Market-Wide News, or systematic risk, non-
diversifiable risk Lets re-define risk more precisely. Stand-alone
Risk (or total risk) = Market Risk + Diversifiable Risk
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49 Systematic Risk Risk factors that affect a large number of
assets such as market-wide news Also known as non-diversifiable
risk or market risk Includes such things as changes in GDP,
inflation, interest rates, etc.
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50 Unsystematic Risk Risk factors that affect a limited number
of assets such as firm-specific news Also known as unique risk,
diversifiable risk, idiosyncratic risk, and firm-specific risk
Includes such things as labor strikes, part shortages, etc.
Slide 51
51 Pop Quiz: Systematic Risk or Unsystematic Risk? The
government announces that inflation unexpectedly jumped by 2
percent last month. Systematic Risk One of Big Widgets major
suppliers goes bankruptcy. Unsystematic Risk The head of accounting
department of Big Widget announces that the companys current ratio
has been severely deteriorating. Unsystematic Risk Congress
approves changes to the tax code that will increase the top
marginal corporate tax rate. Systematic Risk
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52 Diversification and Risk
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53 11.5 Diversification in Stock Portfolios Diversifiable Risk
and the Risk Premium The risk premium for diversifiable risk is
zero Investors are not compensated for holding unsystematic risk
The Importance of Systematic Risk The risk premium of a security is
determined by its systematic risk and does not depend on its
diversifiable risk Table 11.4 Systematic Risk Versus Unsystematic
Risk
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54 11.5 Diversification in Stock Portfolios The Importance of
Systematic Risk While volatility or standard deviation might be a
reasonable measure of risk for a large portfolio, it is not
appropriate for an individual security. There is no relationship
between volatility and average returns for individual securities.
Consequently, to estimate a securitys expected return we need to
find a measure of a securitys systematic risk.
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55 An Intuitive Example Suppose type S firms are only affected
by the systematic risk. Holding a large portfolio of type S stocks
will not diversify the risk. Suppose type U firms are only affected
by the unsystematic risks. Holding a large portfolio of type U
firms will diversify the risk because these risks are
firm-specific. Of course, actual firms are not similar to type S or
U firms. Typical firms are affected by both systematic, market-wide
risks, as well as unsystematic risks. When firms carry both types
of risk, only the unsystematic risk will be eliminated by
diversification when we combine many firms into a portfolio The
volatility will therefore decline until only the systematic risk,
which affect all firms, remain.
Slide 56
56 Figure 11.7 Volatility of Portfolios of Type S and U
Stocks
Slide 57
57 Table 11.5 The Expected Return of Type S and Type U Firms,
Assuming the Risk-Free Rate Is 5%
Slide 58
58 Expected return and risk: Probability- weighted return
Slide 59
59 Common Measures of Risk and Return Probability Distributions
When an investment is risky, there are different returns it may
earn. Each possible return has some likelihood of occurring. This
information is summarized with a probability distribution, which
assigns a probability, P R, that each possible return, R, will
occur. Assume BFI stock currently trades for $100 per share. In one
year, there is a 25% chance the share price will be $140, a 50%
chance it will be $110, and a 25% chance it will be $80.
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60 Probability Distribution of Returns for BFI
Slide 61
61 Probability Distribution of Returns for BFI
Slide 62
62 Expected Return Expected (Mean) Return Calculated as a
weighted average of the possible returns, where the weights
correspond to the probabilities.
Slide 63
63 Variance and Standard Deviation Variance The expected
squared deviation from the mean Standard Deviation The square root
of the variance Both are measures of the risk of a probability
distribution
Slide 64
64 Variance and Standard Deviation (cont'd) For BFI, the
variance and standard deviation are: In finance, the standard
deviation of a return is also referred to as its volatility. The
standard deviation is easier to interpret because it is in the same
units as the returns themselves.
Slide 65
65 Another Example
Slide 66
66 Probability Distributions for BFI and AMC Returns
Slide 67
67 Alternative Example Problem TXU stock is has the following
probability distribution: What are its expected return and standard
deviation? ProbabilityReturn.258%.5510%.2012%
70 Arithmetic Avg. vs. Geometric Avg. - 50% + 100% 0% or 25%?
Geometric Avg. Return What was your avg. compound return per year?
Arithmetic Avg. Return What was your returns in an average (
typical) year?
Slide 71
71 Arithmetic Avg. vs. Geometric Avg.
Slide 72
72 Arithmetic Averages versus Geometric Averages The arithmetic
average return answers the question: What was your return in an
average year over a particular period? The geometric average return
answers the question: What was your average compound return per
year over a particular period? When should you use the arithmetic
average and when should you use the geometric average? First, we
need to learn how to calculate a geometric average.
Slide 73
73 Example: Calculating a Geometric Average Return
Slide 74
74 Arithmetic Averages versus Geometric Averages The arithmetic
average tells you what you earned in a typical year. The geometric
average tells you what you actually earned per year on average,
compounded annually. When we talk about average returns, we
generally are talking about arithmetic average returns. For the
purpose of forecasting future returns: The arithmetic average is
probably "too high" for long forecasts. The geometric average is
probably "too low" for short forecasts. 74
Slide 75
75 Geometric versus Arithmetic Averages, 1926-2006, 1926-2012
Source: Jordan, Miller, and Dolvin (2012) textbook
Slide 76
76 Chapter 11 Quiz 1.Historically, which types of investments
have had the highest average returns and which have been the most
volatile from year to year? Is there a relation? 2.For what purpose
do we use the average and standard deviation of historical stock
returns? 3.What is the relation between risk and return for large
portfolios? How are individual stocks different? 4.What is the
difference between common and independent risk? 5.Does systematic
or unsystematic risk require a risk premium? Why?
