Teori Pasaran Modal dan CAPM Reference: RWJ Chapter 10

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Teori Pasaran Modal dan CAPM
Reference RWJ Chapter 10
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Lecture Objectives

• Know how to calculate expected returns
• Understand portfolio returns, variance and
  standard deviation
• The Efficient Set for Two Assets
• The Efficient Set for Many Securities
• Diversification An Example
• Riskless Borrowing and Lending
• Market Equilibrium
• Relationship between Risk and Expected Return
  (CAPM)

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Expected Returns

• Expected returns are based on the probabilities
  of possible outcomes
• In this context, expected means average if the
  process is repeated many times
• The expected return does not even have to be a
  possible return

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Example Expected Returns

• Suppose you have predicted the following returns
  for stocks C and T in three possible states of
  nature. What are the expected returns?
• State Probability C T
• Boom 0.3 0.15 0.25
• Normal 0.5 0.10 0.20
• Recession ??? 0.02 0.01
• RC
• RT

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Variance and Standard Deviation

• Variance and standard deviation still measure the
  volatility of returns
• Using unequal probabilities for the entire range
  of possibilities
• Weighted average of squared deviations

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Example Variance and Standard Deviation

• Consider the previous example. What are the
  variance and standard deviation for each stock?
• Stock C
• Stock T

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Another Example ASSIGNMENT

• Consider the following information
• State Probability ABC, Inc.
• Boom .25 .15
• Normal .50 .08
• Slowdown .15 .04
• Recession .10 -.03
• What is the expected return?
• What is the variance?
• What is the standard deviation?

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Portfolios

• A portfolio is a collection of assets
• An assets risk and return is important in how it
  affects the risk and return of the portfolio
• The risk-return trade-off for a portfolio is
  measured by the portfolio expected return and
  standard deviation, just as with individual assets

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Example Portfolio Weights

• Suppose you have RM15,000 to invest and you have
  purchased securities in the following amounts.
  What are your portfolio weights in each security?
• RM2000 of A
• RM3000 of B
• RM4000 of C
• RM6000 of D

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Portfolio Expected Returns

• The expected return of a portfolio is the
  weighted average of the expected returns for each
  asset in the portfolio
• You can also find the expected return by finding
  the portfolio return in each possible state and
  computing the expected value as we did with
  individual securities

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Example Expected Portfolio Returns

• Consider the portfolio weights computed
  previously. If the individual stocks have the
  following expected returns, what is the expected
  return for the portfolio?
• A 19.65
• B 8.96
• C 9.67
• D 8.13

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Portfolio Variance

• Compute the portfolio return for each stateRP
  w1R1 w2R2 wmRm
• Compute the expected portfolio return using the
  same formula as for an individual asset
• Compute the portfolio variance and standard
  deviation using the same formulas as for an
  individual asset

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Example Portfolio Variance

• Consider the following information
• Invest 50 of your money in Asset A
• State Probability A B
• Boom .4 30 -5
• Bust .6 -10 25
• What is the expected return and standard
  deviation for each asset?
• What is the expected return and standard
  deviation for the portfolio?

Portfolio
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Expected (Ex Ante) Return, Variance

• Expected Return E(R) S (ps x Rs)
• Variance s2 S ps x Rs - E(R)2
• Standard Deviation s

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Return and Risk for Portfolios (2 Assets)

• Expected Return of a Portfolio
• E(Rp) XAE(R)A XB E(R)B
• Variance of a Portfolio
• sp2 XA2sA2 XB2sB2 2 XA XB sAB

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Another Example

• Consider the following information
• State Probability X Z
• Boom .25 15 10
• Normal .60 10 9
• Recession .15 5 10
• What is the expected return and standard
  deviation for a portfolio with an investment of
  RM6000 in asset X and RM4000 in asset Y?

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Calculating Expected Return and Standard Deviation
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WHAT IS COVARIANCE AND CORRELATION?

• Covariance sAB S ps x Rs,A - E(RA) x Rs,B
  - E(RB)
• Correlation Coefficient rAB sAB / (sA sB)
• Variance and Standard Deviation measure the
  variability of individual stock.
• Covariance and Correlation measure how two random
  variables are related

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PERFECT POSITIVE CORRELATION
Returns
A

0
B
-

• Corr(RA , RB ) 1

TIME
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PERFECT POSITIVE CORRELATION
RETURNS
A

0
B
-

• Corr(RA , RB ) -1

TIME
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ZERO CORRELATION
Returns
A

0
B
-

• Corr(RA , RB ) 0

TIME
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FROM PREVIOUS EXAMPLE
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Calculating Covariance and Correlation
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• Covariance
• sAB S ps x Rs,A - E(RA) x Rs,B - E(RB)
• Correlation Coefficient
• rAB sAB / (sA sB)

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Two-Security Portfolios with Various Correlations
return
100 stocks
? -1.0
? 1.0
? 0.2
100 bonds
?
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Portfolio Risk/Return Two Securities Correlation
Effects

• Relationship depends on correlation coefficient
• -1.0 lt r lt 1.0
• The smaller the correlation, the greater the risk
  reduction potential
• If r 1.0, no risk reduction is possible

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Portfolio Risk as a Function of the Number of
Stocks in the Portfolio
In a large portfolio the variance terms are
effectively diversified away, but the covariance
terms are not.
?
Diversifiable Risk Nonsystematic Risk Firm
Specific Risk Unique Risk
Portfolio risk
Nondiversifiable risk Systematic Risk Market
Risk
n
Thus diversification can eliminate some, but not
all of the risk of individual securities.
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The Efficient Set for Many Securities
return
Individual Assets
?P

• Consider a world with many risky assets we can
  still identify the opportunity set of risk-return
  combinations of various portfolios.

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The Efficient Set for Many Securities
return
minimum variance portfolio
Individual Assets
?P

• Given the opportunity set we can identify the
  minimum variance portfolio.

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The Efficient Set for Many Securities
return
efficient frontier
minimum variance portfolio
Individual Assets
?P

• The section of the opportunity set above the
  minimum variance portfolio is the efficient
  frontier.

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Optimal Risky Portfolio with a Risk-Free Asset
return
100 stocks
rf
100 bonds
?

• In addition to stocks and bonds, consider a world
  that also has risk-free securities like T-bills

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Riskless Borrowing and Lending
CML
return
100 stocks
Balanced fund
rf
100 bonds
?

• Now investors can allocate their money across the
  T-bills and a balanced mutual fund

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Riskless Borrowing and Lending
CML
return
efficient frontier
rf
?P

• With a risk-free asset available and the
  efficient frontier identified, we choose the
  capital allocation line with the steepest slope

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Market Equilibrium
efficient frontier
return
CML
M
rf
?P

• With the capital allocation line identified, all
  investors choose a point along the linesome
  combination of the risk-free asset and the market
  portfolio M. In a world with homogeneous
  expectations, M is the same for all investors.

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The Separation Property
CML
return
efficient frontier
M
rf
?P

• The Separation Property states that the market
  portfolio, M, is the same for all investorsthey
  can separate their risk aversion from their
  choice of the market portfolio.

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The Separation Property
CML
return
efficient frontier
M
rf
?P

• Investor risk aversion is revealed in their
  choice of where to stay along the capital
  allocation linenot in their choice of the line.

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Market Equilibrium
CML
return
100 stocks
Balanced fund
rf
100 bonds
?

• Just where the investor chooses along the Capital
  Asset Line depends on his risk tolerance. The big
  point though is that all investors have the same
  CML.

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Market Equilibrium
CML
return
100 stocks
Optimal Risky Porfolio
rf
100 bonds
?

• All investors have the same CML because they all
  have the same optimal risky portfolio given the
  risk-free rate.

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The Separation Property
CML
return
100 stocks
Optimal Risky Porfolio
rf
100 bonds
?

• The separation property implies that portfolio
  choice can be separated into two tasks (1)
  determine the optimal risky portfolio, and (2)
  selecting a point on the CML.

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Optimal Risky Portfolio with a Risk-Free Asset
CML1
CML0
return
100 stocks
Second Optimal Risky Portfolio
First Optimal Risky Portfolio
100 bonds
?

• By the way, the optimal risky portfolio depends
  on the risk-free rate as well as the risky assets.

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Definition of Risk When Investors Hold the Market
Portfolio

• 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.

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Estimating b with regression
Security Returns
Return on market
Ri a i biRm ei
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The Formula for Beta
Clearly, your estimate of beta will depend upon
your choice of a proxy for the market portfolio.
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Relationship between Risk and Expected Return
(CAPM)

• Expected Return on the Market

• Expected return on an individual security

Market Risk Premium
This applies to individual securities held within
well-diversified portfolios.
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Expected Return on an Individual Security

• This formula is called the Capital Asset Pricing
  Model (CAPM)

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Relationship Between Risk Expected Return
Expected return
b
1.0
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Relationship Between Risk Expected Return
Expected return
b
1.5