Lecture note 8

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Lecture note 8

• (Ch10)

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Outline

• Expected return, variance and covariance
• Return and risk of a portfolio
• Efficient set
• A portfolio with a risk-free asset
• The optimal portfolio
• The beta
• Capital asset pricing model

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Expected return, variance and covariance.

• Consider the returns on the stocks of two
  companies, stock A and stock B. Returns on stock
  A possibly take four values depending on the
  state of economy. Similarly, returns on stock B
  possibly takes four values. The possible values
  and probability distributions are given in the
  following slide.

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10.2 Expected Return, Variance, and Covariance

• Using this example, we will review the
  computation of expected return, variance,
  standard deviation, covariance and correlation.

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Expected return

• Let R1,..Rn be the possible values of returns,
  and Pr1,..,Prn be the probability for each state
  of economy. Then the expected return is given by
• Expected return may be interpreted as the best
  prediction of the return, or the average of the
  return.
• Exercise. Use the excel sheet Expected
  Return, Variance and Covariance to compute the
  expected return the stocks of Company A and B.

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Variance and standard deviation

• Variance of the return is given by
• Standard deviation is given by
• Both Variance and standard deviation are the
  measure of the risk.
• Exercise Compute the variances and standard
  deviations of the returns of stock A and stock B.

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Covariance

• Covariance shows how two random variables are
  related. More specifically, covariance shows the
  direction of the relationship. If Covariance is
  positive, two random variables are positively
  correlated. If it is negative, then the two
  random variables are negatively correlated. Let
  RA and RB be the returns for stock A and stock B.
  We use Cov(RA,RB) or sAB to denote the
  covariance.
• Let RA1 RA2, RAn be the possible values of
  Company As stock, and RB1, RB2, RBn be the
  possible values of company Bs stock. Then
  Covariance between the returns of stock A and
  stock B is defined as
• Exercise Use the excel sheet to compute the
  covariance between stock A and stock B returns

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Correlation

• Correlation also shows how two random variables
  are related to each other. More specifically, it
  shows the strength of linear relationship. If the
  correlation is 1, two random variables have
  perfect positive linear relationship. If it is
  1, the two random variables have perfect
  negative linear relationship. As it becomes close
  to zero, the linearity in the relationship
  weakens.
• Correlation between the returns of the stock A
  and stock B is written as?AB, and this is defined
  as
• Exercise, compute the correlation of stock A
  and stock B returns.

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Return and Risk of a Portfolio

• Often investors have a combination of different
  stocks. Such a combination of stocks are called a
  portfolio.
• One of the main reasons why we would like to hold
  a combination of different stocks is to reduce
  the risk-return tradeoff by exploiting the
  situation where correlations between the returns
  of different stocks is less than one. To see how
  it might reduce the tradeoff, let us first look
  at the expected return and standard deviation of
  the returns on a portfolio of two stocks.
  
• See next page

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Expected return of a portfolio

• Consider a portfolio of the stock A stock B. You
  have a fixed amount of money to invest. Let Xa be
  the proportion of the money you invest in stock
  A, and Xb be the proportion of money you invest
  in stock B. Let E(RA) and E(RB) be the expected
  returns stock A and stock B.
• Expected return of this portfolio is computed as
• Expected return of the portfolio Xa E(RA)
  Xb E(RB)
• Exercise
• Continue using the previous example. If you
  invest 60 of the money into stock A, what is the
  expected return of this portfolio.?

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Variance and standard deviation of a portfolio

• Let sA and sB be the standard deviation of the
  returns stock A and stock B. Let sAB be the
  covariance and?AB be correlation of the returns
  of stock A and B. Then the variance of the
  portfolio is given by
• Var(Portfolio)Xa2sA 2 2XaXbsAB
  Xb2sB2
• Xa2sA 2 2XaXb?AB
  sA sB Xb2sB2
• Exercise
• Continue using the same example. If you
  invest 60 of your money in stock A, what will be
  the variance and the standard deviation of the
  portfolio?

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The relationship between the return and SD of a
portfolio of two stocks

• To see how a portfolio may reduce the return-risk
  tradeoff, it is constructive to plot the expected
  return against the standard deviation of
  portfolio for different weights.
• Use the excel file return and SD of portfolio
  to plot the relationship between standard
  deviation and return of the portfolio of stock A
  and stock B for different weights.

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Relationship between standard deviation and
expected return of the portfolio of stock A and B.

Stock A
Stock B

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Relationship between standard deviation and
expected return of the portfolio of stock A and
B. (Contd)

• As can be see, as you change the weights of the
  portfolio, you will have portfolio with different
  risk (SD) and return combinations.
• Without constructing portfolio, only possible
  risk-return combinations are given by two end
  points. Thus, constructing portfolio gives you
  greater choice of risk-return combinations.
• Now, let us see how such a portfolio reduces the
  risk-return tradeoff. Such reduction in the
  risk-return tradeoff is called diversification
  effect.

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Diversification effect

• To see the diversification effect (reduction in
  the risk-return tradeoff), it is constructive to
  consider the hypothetical case where there is a
  perfect correlation between stock A and stock B,
  (i.e., correlation 1).
• See next page.

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Diversification Effect (contd)

• If there correlation between stock A and stock B
  return is 1, for any combination of the
  portfolio, the risk-return combination would be
  given by the straight line below. The comparison
  of the straight line and the curved line shows
  the diversification effect.
• For any given risk, a portfolio whose correlation
  is less than one (curved line) will give you
  higher return than a portfolio whose correlation
  is 1.
• Also, for any given return, a portfolio whose
  correlation is less than one (curved line) will
  give you lower risk than a portfolio whose
  returns are 1.

For any given risk (SD), a portfolio whose
correlation is less than one gives higher return.

Stock A
For any given return, a portfolio with
correlation is less than 1 risk (SD)

Stock B
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Diversification effect

• To re-iterate the point, we found that a
  portfolio of stocks whose correlation is less
  than one will have lower risk and higher return.
• To understand the intuition, consider a case
  where correlation between two stock is negative
  (thus less than 1). This means that if there is a
  negative shock to one stock, the other stock
  price is likely to rise, thus offsetting the
  risk.
• The same intuition holds even if the correlation
  is not negative. Consider stock A and B whose
  correlation is positive but less than one (say
  0.5). If there is a negative shock to stock A,
  the price of Stock B is likely to fall, but not
  as much as stock A. Thus, combining stock A and B
  still reduces the risk.
• Next page

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Diversification Effect (contd)

• By exploiting the correlation between the
  returns of stocks, portfolio reduces the
  risk-return trade off.
• This is seem by a flatter relationship between
  risk and return for the portfolio whose
  correlation is less than one.

Risk-return relationship is flatter for the
portfolio whose correlation is less than one.


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Efficient set for two asset

• Notice that the curve is bent backward.
  Obviously, an investor would not choose the
  portfolio from the backward bending part, because
  this is the part where return is decreasing while
  risk is increasing. Therefore, an investor will
  choose portfolio only from the upper part of the
  feasible set.
• Therefore, the upper part of the feasible set is
  called the efficient set (or efficient frontier).

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Efficient set for many risky assets

• Portfolio can contain more than two risky assets.
• When there more than two risky assets in the
  portfolio, the feasible set (feasible
  combinations of risk-return) are no longer a
  line. It will be an area. This is illustrated in
  the next slide.

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Feasible set for many risky assets
return
Feasible sets
Individual Assets
?P

• Feasible set is the area which is enveloped by
  the curved line.
• Although the feasible set is an area, there is a
  boundary given by the red curved line.

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Efficient set for many risky assets
Efficient set
return
Feasible sets
Individual Assets
?P

• An investor would choose a portfolio from the
  upper boundary of the feasible set since this
  curve gives the best risk-return tradeoff.
• Thus, for a portfolio with many risky assets, the
  upper boundary of the feasible set is called the
  efficient set

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Minimum variance portfolio
return
efficient set
minimum variance portfolio
Individual Assets
?P

• Also you can notice that a portfolio attains
  its minimum variance at the leftmost part of the
  efficient set. This portfolio is called minimum
  variance portfolio

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Feasible set for a portfolio composed of a
risk-free asset and a risky asset.

• We have so far considered a portfolio of risky
  assets. How does a feasible set look like if you
  construct a portfolio with one risk-free asset
  and one risky asset?
• Consider you invest in a portfolio of the stock
  of Merville and Risk-Free asset such as T-Bills.
  The expected returns and standard deviations are
  given by the table below.

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Feasible set for a portfolio of a risk-free asset
and a risky asset (contd)

• We would like to obtain the feasible set of the
  portfolio of Merville and the risk free asset.
• To do so, making the following assumption will
  become convenient later on
• Assumption The investor can borrow at the same
  rate equal to the risk-free-rate.
• To understand what this assumption does, consider
  you initially have 1000 to invest. Now consider
  that you borrow 200 at the risk free rate to
  make additional investment in the Merville stock.
  This means that you invested 120 of the initial
  amount. You can consider this as a special kind
  of portfolio a portfolio formed by borrowing
  money. How would you compute the expected return
  and standard deviation of this special portfolio?
• Computing the expected return of this portfolio
  is equivalent to computing the return from
  investing 1200 in an asset with 14 return and
  investing 200 in an asset with return equal to
  10. This becomes equivalent to
• Expected return (120)(return on
  Merville)(20)(Return on risk free asset)
• 
  (120)(0.14)(20)(0.1)14.8

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Feasible set for a portfolio of a risk-free asset
and a risky asset (contd)

• Exercise
• Using Excel sheet feasible set for a
  portfolio with risk-free asset, find the
  feasible set for the portfolio of Merville and
  the Risk-free asset.

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Feasible set for a portfolio of a risk-free asset
and a risky asset (contd)
Borrowing 20 of the initial amount



Merville
Risk free asset
Therefore, the feasible set is a straight line
with the intercept equal to the risk free rate,
and which goes through the point given by the
Merville.
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Optimal portfolio

• Now, we combine what we have discussed so far to
  find the optimal portfolio. The tools that we are
  going to use are (1) the efficient set for a
  portfolio of many risky assets, and (2) feasible
  set for a portfolio with one risk free and one
  risky asset.

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Optimal portfolio

• Let rf be the risk free rate. First, consider
  drawing a straight line with intercept equal to
  rf, and which is tangent to the efficient set.
• See next page.

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Optimal portfolio (contd)
Capital Market Line
return
Optimal risky portfolio
rf
?

• Draw a tangent line that goes through rf. The
  tangency point is the optimal risky portfolio.
  The reason why it is called optimal will become
  clear later. The tangent line is called the the
  capital market line.

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Optimal portfolio (contd)
Capital Market Line
return
Optimal risky portfolio
rf
?

• First, consider the capital market line. This is
  the feasible set of a portfolio with two assets
  one is risk-free asset and the other is the
  optimal risky portfolio. (We are treating the
  optimal risky portfolio as if it were an asset
  with a single stock.)
• An investor can achieve any risk-return
  combinations that are on the capital market line.

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Optimal portfolio (contd)
Capital market line always gets higher return for
any given risk.
Capital Market Line
This line is non-optimal because, for any given
risk, the capital market line always gets higher
return.
return
A
rf
?

• Now, let us consider why the tangent point is
  called the optimal risky portfolio. To
  understand why, let us choose any other
  portfolios from the feasible sets, such as point
  A.
• You can see that this line is not optimal because
  for any given risk, capital market line always
  gives you a higher return.
• Therefore, the tangent line is the optimal risky
  portfolio

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Optimal portfolio (contd)
Capital Market Line
return
Optimal risky portfolio
rf
?

• Finally, which point on the capital market line
  would an investor choose? This depends on the
  preference of the investor towards risk. If the
  investor is very risk averse, he/she would choose
  a point near the y-axix. If he/she is less risk
  averse, he/she may choose a point away from
  y-axix.

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Market portfolio.
return
CML
efficient frontier
A
rf
?P

Financial economist often imagine a world where
all investors have the same belief about the
expected returns, variance and covariance. This
assumption is called homogeneous assumption.
Under this assumption, all the investors will
choose a portfolio of risky asset given by Point
A. Point A is called Market Portfolio. Return
of the market portfolio is called Return on
Market.
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Market portfolio

• Financial economist uses the broad-based index as
  proxy for the market portfolio market return.
• Broad-Based Index
• An index designed to reflect the movement of
  the entire market. Examples include Dow Jones
  Industrial Average and the SP 500.

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The beta

• Researchers have shown that the best measure of
  the risk of a security in a large portfolio is
  the beta (b)of the security.
• Beta measures the responsiveness of a security to
  movements in the market portfolio.
• Next few slides explain what the beta is.

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Computing beta

• Consider the return on Jelco.inc and on the
  market portfolio.

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Computing beta (contd)

• The slope of this line is the beta for Jelco.Inc.
• The line is called the characteristic line.
• Question. Compute the beta for Jeleco.Inc

Return on Jelco
(0.15,0.2)
Slope
Return on market
(-0.05,-0.1)
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Interpretation of beta

• The beta of Jelco is 1.5. From the figure in the
  previous page, the interpretation of beta is
  intuitive.
• The beta of 1.5 means that the returns of Jelco
  is magnified 1.5 times over those of market If
  market does well, Jelco will do even better, but
  if market does poorly, Jelco will do even worse..
• If beta is greater, the fluctuation of the return
  will be greater, and vice versa. Thus, beta can
  be interpreted as a measure of a risk.

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Formal definition of the beta

• More formally, the beta of the security of
  company i is given by

• Where Ri is the return on company is stock, and
  RM is the market return.
• If you have historical data of market returns and
  the Ri, this definition of ß turns out to be the
  same as regression slop coefficient of the
  following regression equation RiaßRM

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Estimates of b for Selected Stocks
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Relationship between Risk and Expected Return
(Capital Asset Pricing Model)

• Expected Return on the Market

• Financial economists show that, under plausible
  assumptions, the expected return on individual
  security i is related to beta by the following
  equation.

Market Risk Premium
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Example

• The beta of stock A is 0.8. The risk-free rate is
  6 and the market risk premium is 8.5. Assume
  that the capital asset pricing model holds. What
  is the expected return on stock A?

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Beta of a portfolio

• Beta of a portfolio is the weighted average of
  the betas.
• Example Consider the following portfolio (10.38)
• The risk-free rate is 4. Expected return on
  market is 15. What is the beta of this
  portfolio? What is the expected return on the
  above portfolio?