Chapter 5: Risk and Return

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• Chapter 5 Risk and Return
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• Overview Risk vs. Return
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• A. risk the chance that unfavorable events will
  occur
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• 1. stand alone risk
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• 2. portfolio risk
•  

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B. return the benefit from an undertaking   C.
key intuition 1. for rational investors, no
investment will be undertaken unless the
_______________ is enough to compensate investors
for the ___________________they
face.           2. in equilibrium, no investment
will have an expected return that compensates
investors for ____________ than they would face
if they reduced the risk of that investment as
much as possible.
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• A review of statistics
• A. probability distribution a list of all
  possible events and the probability that each
  will occur
• 1. e.g., a discrete distribution
  outcome probability good 0.4 bad
  0.4 ugly 0.2
• (a) each event has a specific positive
  probability
•  
• 2. e.g., a continuous distribution the
  standard normal distribution f(x) (1/2?)0.5
  exp( - 0.5 x2)
•   
• (a) each event has a minute (zero!)
  probability, but we can
• calculate a probability that number will fall
  between two
• numbers. This calculation simply involves
  calculating the
• area under the curve between the two points.

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B. expected return or average return a measure
of the anticipated or historical payoff on an
investment     C. variance a measure of the
__________________ returns
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D. standard deviation another (better) measure
of the dispersion (or risk) of returns ?
_____________________     E. correlation
coefficient __________________________________
____________________ _________________________
1. can take values between (a) -1 (b) 0
(c) 1
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F. examples 1 100 initial investment outcom
e probability payoff return
good 0.4 200 100 bad 0.4 90 -10
ugly 0.2 40 -60 (a) expected return  
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Example Continued outcome probability p
ayoff return good 0.4 200 100
bad 0.4 90 -10 ugly 0.2 40 -60 (b)
variance     (c) standard deviation
64.68
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2. stock returns Month Return Month Return
Jan 2.1 Apr 0.4 Feb 0.9 May 1.3
Mar -1 Jun -0.1  (a) average return
  (b) variance         (c) standard
deviation 1.088
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• More terminology
• A. risk aversion the degree to which investors
  prefer to avoid risk.
• 1. An example Suppose that a risk averse
  investor must choose one of six possible
  investments. The investments have the following
  characteristics Stock expected
  return standard deviation A 6 12
  B 8 10 C 9 10 D 4 8
  E 10 16 F 11 14
• Which investments would the investor definitely
  not choose?

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B. risk premium the extra expected return that
must be given to risk averse investors to induce
them to invest. 1. Suppose you have 100,000 to
invest and you have two choices. Investment A
pays 0 with probability 0.5 and 300,000 with
probability 0.5. Investment B pays off with
certainty. At what payoff for Investment B would
you be indifferent between the two
choices? (a) Suppose you say
140,000.   (b) The expected return on
Investment B is 40. The expected return on
Investment A is 50. Why are the two
different?     (c) The 10 difference in
expected return is called the risk premium for
Investment A.
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C. expected portfolio return 1. example An
investor invests a total of 80,000 in four
assets. Investment Expected Return
A 10,000 10 B 25,000 12
C 30,000 8 D 15,000 9 What is the
expected return on the portfolio? (a) wA
wB wC
wD (b) Notice that wAwBwCwD
(c) expected return
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D. portfolio risk  1. example An investor
has 100 to invest and has two choices.
Investment A triples the amount invested with
probability 0.5 and pays zero with probability
0.5. Investment B also triples the amount
invested with probability 0.5 and pays zero with
probability 0.5. The investments are independent
(uncorrelated).
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Example continued (a) What is the expected
return on each investment? (i) expected return
0.5 (200) 0.5 (-100) 50 (b)Suppose the
investor chooses to invest the 100 ONLY in
Investment A. What is the standard deviation
of the investment? (i) variance 0.5 (200-50)2
0.5 (-100-50)2 22,500 (ii) standard
deviation 150
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• (c) Suppose instead that the investor chooses to
  invest 50 in Investment A and 50 in Investment
  B.

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(i) There are four possible outcomes
outcome probability total payoff return A pays
off, B does not 0.25 150 50 B pays off,
A does not 0.25 150 50 Both pay off
0.25 300 200 Neither pays off 0.25
0 -100 (ii) What is the expected return on
the portfolio?     (iii) What is the risk of
this investment?  (1) var.   (2) standard
deviation
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2. a simple example An investor has 100 to
invest and has two choices. Investment A triples
the amount invested with probability 0.5 and pays
zero with probability 0.5. Investment B also
triples the amount invested with probability 0.5
and pays zero with probability 0.5. The
investments are perfectly positively correlated.
Suppose we place half of our money in each
asset. (a) Notice that both investments have an
expected return of 50. (b) Since the
investments are perfectly correlated, there are
only two possibilities outcome probability payo
ff return Both pay off 0.5 300 200
Neither pays off 0.5 0 -100 (c)
variance 0.5 (200-50)2 0.5 (-100-50)2
22,500 (d) standard deviation 150 Recall
that the standard deviation of each investment
alone was 150, so we got no benefit (in terms of
diversification) from splitting our money.
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3. another simple example An investor has 100
to invest and has two choices. Investment A
triples the amount invested with probability 0.5
and pays zero with probability 0.5. Investment B
also triples the amount invested with probability
0.5 and pays zero with probability 0.5. The
investments are perfectly negatively
correlated. (a) Notice that both investments
have an expected return of 50. (b) Since the
investments are perfectly correlated, there are
only two possibilities outcome probability payo
ff return A pays off, B doesn't 0.5
150 50 B pays off, A doesn't 0.5
150 50 (c) variance (d) standard
deviation So, by dividing our money into two
perfectly negatively correlated assets, we are
able to completely eliminate our risk.
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4. A simple way to calculate portfolio risk.
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(a) recall the examples in the previous
section. (i) ?A 150, ?B150 (ii) if ?0 and
we put half our money in each, ?P2 .
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(iii) if ?1 and we put half our money in each,
?P2 .
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(iv) if ?-1 and we put half our money in each,
?P2 0.52 (150)2 0.52 (150)2 0.5 (0.5)
(-1) (150) 150 0.5 (0.5) (-1) (150) 150
0?P 0 .
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(b) example Asset w std. dev. correlation
coefficient A 0.5 35 ?A,B 0.6
B 0.2 40 ?B,C 0.2 C 0.3 20
?A,C 0.3 (i) variance 0.52 (35)2
0.22 (40)2 0.32 (20)2 (0.5) (0.2) (0.6) (35)
(40) (j ,k ) (0.5) (0.3) (0.3) (35)
(20) (j ,k ) (0.2) (0.5) (0.6) (40)
(35) (j ,k ) (0.2) (0.3) (0.2) (40)
(20) (j ,k ) (0.3) (0.5) (0.3) (20)
(35) (j ,k ) (0.3) (0.2) (0.2) (20)
(40) (j ,k ) 656.45 (ii) standard
deviation 25.62
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(c) example 2 using the variance covariance
matrix on the previous example (i) Simply fill
in a matrix with the correlation coefficient
multiplied by the product of the two standard
deviations. For instance, the element in Column
B, Row A is ?A,B?A?B 0.6 (40) (35)
840 A B C A 1225 840 210 B 840 1600
160 C 210 160 400 (ii) To calculate the
portfolio variance, simply multiply each element
of the matrix by the product of the two
weights. A B C A 306.25 84 31.5 B
84 64 9.6 C 31.5 9.6 36 (iii) then add up
the elements of the matrix to get 656.45 (iv)
the standard deviation is then 25.62
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• Diversifiable risk vs. Non-diversifiable risk
• unsystematic or irrelevant vs. market or
  systematic or relevant
• A. Recall our simple examples in which we split
  our money and were able to reduce our risk. If
  we had the opportunity to split our money, but
  chose not to, would the expected return on the
  asset be enough to compensate us for the risk we
  face?
• 1. The answers suggest that investors will only
  be compensated for the risk that they would face
  given that they reduce it as much as possible.

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2) The risk that remains after it has been
reduced as much as possible is called   _________
_____________ a) risk associated with
        3) The risk that has been taken away is
called _________________   a) risk associated
with only the firm,
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B) example Suppose that there are only 100
independent assets available for investment.
Each asset has a standard deviation of 150.
Suppose we split our money evenly amongst the
assets. That is, we put 1 of our money in each
asset. 1) portfolio variance       2)
standard deviation ? 3) non-diversifiable
risk   4) diversifiable risk
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C) example Suppose that there are only 10000
independent assets available for investment.
Each asset has a standard deviation of 150.
Suppose we split our money evenly amongst the
assets. That is, we put 0.01 of our money in
each asset. 1) portfolio variance
    2) standard deviation ?   3)
non-diversifiable risk   4) diversifiable risk

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C) Notice that as the number of independent
assets increase, the non-diversifiable risk
approach zero.   D) key point Risk that is
non-diversifiable must come from correlation
between assets.   E) Since investors can
costlessly eliminate a stocks diversifiable
or firm-specific risk, they do not deserve a
return for it. Their risk premium should be
based only on a stocks non-diversifiable (market)
risk. In other words, the market risk is the
only risk that is relevant to a stock return.
  To understand how risk and return are related
in stock markets, we need a theory or a model
that relates required return to the market risk
(relevant risk).   The CAPM model was the first
to do this.
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• The Capital Asset Pricing Model (CAPM)
• A) The CAPM will do 2 things for us
• 1)
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• 2)
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• B) market portfolio a portfolio consisting of
  investments in each and every possible asset.
  What assets are included?

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• C) We need to be able to measure the risk left
  after all possible
• diversification.
• 1) Risk of the market portfolio the
  stand-alone risk of the market portfolio (
  )
• is_______________________________________.
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• Because_________ is all relevant risk (market
  risk), you can envision that investors observe
  _______ and determine the return that they
  require, Ekm.
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• B) The required risk premium for the market
  portfolio is determined by
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• C) The market portfolio gives an initial (risk
  and risk-premium) reference point.

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2) CAPM defines a stocks relevant risk as
      Relevant risk it the standalone risk
times the correlation coefficient that exist
between the stocks returns and the market
portfolios returns.   3) beta scales the
relevant risk of a stock be the measure of the
systematic (non-diversifiable risk) of an
investment. (a) security beta
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(i) ? 1..investment has the same
non-diversifiable risk as the market   (ii) ?
0.investment has no non-diversifiable risk
(Treasury bills, for instance)   (iii) ? lt
0.investment is negatively correlated with the
market (gold perhaps). These can be very useful
for diversification   (iv) 0 lt ? lt 1.investment
has less non-diversifiable risk than the
market   (v) ? gt 1.investment has more
non-diversifiable risk than the market.
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(b) portfolio beta    
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D) The security market line we won't go through
the derivation of the equation, but the
definition of ? is such that the following
equation should hold
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1) examples kRF 4, kM 12 (a) ? 0 kI
4 0 (12 - 4) 4 (b) ? 1 kI 4
1 (12 - 4) 12 (c) ? 2 kI 4 2
(12 - 4) 20 (d) ? -0.25 kI 4
(-0.25) (12 - 4) 2 ! (e)higher risk
higher return
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2) more examples kRF 3, kM 10 (a) ki
10?1 (b) 15?1.71 (c)
20..?2.43
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2) example kRF 4, kM 12 (a)
portfolio (i) wi ?i 0.1 0 0.4 0.9 0.3 1.2 0.
2 2.4 (b) ?P         (c) kp
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E) risk premium 1) market risk premium kM -
kRF         2) asset (or portfolio) risk
premium ?i (kM - kRF)  
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