Portfolio%20Theory%20Capital%20Market%20Theory%20Capital%20Asset%20Pricing%20Model

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Portfolio Theory Capital Market TheoryCapital
Asset Pricing Model

• Do not put all your eggs in one basket

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Expected Returns

• Expected return average return on a risky asset
  expected in the future.

where Pi probability of each state of the
economy Ri expected return under each
state and Risk premium E ( R ) - Rf
Expected return Risk free rate
3
Standard Deviation

• Standard deviation is calculated as

where Pi probability of each state of the
economy Ri expected return under each
state E(R) expected return for the security
4

• Example Stock A gives average return 4 on the
  normal situation. During the crisis,Stock A
  plumped and generated negative return -2.
  However,Stock A expected to generate return 10
  on bullish economy. Given the possibility of
  crisis, normal, and bullish to be 30,50, and
  20 respectively. What is the expected return and
  standard deviation (total risk) of stock A?
• E (R) 0.30 x ( -2) 0.50 x (4) 0.20 x
  (10)
• E (R) 3.4
• SDA 0.3 x(-2 - 3.4)2 0.5 x(4 - 3.4)2
  0.2x(10 3.4)2 0.5
• SDA 4.2

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Calculation of Expected Return
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Calculation of Standard Deviation
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Portfolio

• Portfolio Group of assets such as stocks and
  bonds held by an investor.
• Efficient portfolio portfolio that maximize the
  expected return given a level of risk.
• Optimal portfolio the most preferred efficient
  portfolio that an investor selects.
• Risk averse investor with the same expected
  return but two different risks, he will prefer
  the lower risk.

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Portfolio

• Portfolio weights Percentage of a portfolio's
  total value invested in a particular asset.
• Portfolio expected returns the weighted average
  combination of the expected returns of the assets
  in the portfolio.
• Portfolio Varianceit is the combination of the
  weighted average of the individual security's
  variance and securities covariance factor.

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Portfolio Expected Returns
Two ways to calculate portfolio expected return
Xj weight of each stock in the
portfolio E(Rj) expected return of each stock
in the portfolio n total number of different
stocks in the portfolio E(Rps) expected return
of portfolio under state s Ps Probability of
state s m total number of all states in the
future
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Portfolio Risk

• Portfolio variance

cov (Ri, Rj) p1ri1 - E(Rj) rj1 E(Rj)
p2ri2 - E(Rj) rj2 E(Rj) .
pNriN - E(Rj) rjN E(Rj) Where rin and
rjn the nth possible rate of return for asset i
and j respectively pn the probability of
attaining the rate of return n for assets i and
j N the number of possible outcomes for the
rate of return
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Covariance and Correlation

• Relationship between covariance and correlation

Correlation represent only the direction between
two assets. It values between -1 to
1 Covariance represent both direction and
magnitude between two assets.
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Covariance and Correlation

• Calculating covariance and correlation

cov (Ri, Rj) 0.50(15-11)( 8- 8)
0.30(10-11)(11-8) 0.13( 5 -11)( 6- 8)
0.05( 0 -11)( 0- 8) 0.02(-5-11)(-4- 8)
8.90 corr (Ri, Rj) 8.9 /
(4.91)(3.05) 0.60
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Example Portfolio Return and Risk
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Feasible and Efficient Portfolios
Figure 3.2 Feasible and efficient sets of
portfolios for stocks C and D
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Feasible and Efficient Portfolios
Figure 3.2 Feasible and efficient sets of
portfolios for stocks C and D
1 is not include in Markowitz efficient set
Because it was dominated by 2,3, and 4 Space on
the left side of 2,3,4, and 5 are not attainable
from combinations of C and D. Space on the right
side of 2,3,4, and 5 are not include in Markowitz
efficient set. i.e. 6 has same return as 2 but 6
gives higher risk. In another word, 6 has same
risk as 4 but 6 gives lower return
5
4
3
2
1
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The minimum-variance frontier of risky assets
E(r)
Efficient frontier
Individual assets
Global minimum variance portfolio
Minimum variance frontier
St. Dev.
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Markowitz Efficient Frontier

• Efficient Frontier
• Set of portfolios with the maximum return for a
  given risk level.
• Data needed to find Efficient Frontier
• Expected returns on all assets
• Standard deviations on all assets
• Correlation coefficients between every pair of
  assets

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Feasible and Efficient Portfolios
U3
Moving from left to right on the frontier,
although risk increases, so does the expect
return. Which point is the best portfolio to
hold? The answer is optimal portfolio depend on
the investors preference or utility as the
trade-off between risk and return.
U2
U1
Optimal portfolio
U1,U2,U3 Indifference curves with U1 lt U2 lt U3
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Correlation Coefficient

• Correlation The tendency of the returns on two
  assets to move together. Corr(RA, RB) or ?A,B
• -1.0 ? ? ? 1.0
• Perfect positive correlation 1.0 gives no risk
  reduction
• Perfect negative correlation -1.0 gives complete
  risk reduction
• Correlation coefficient between -1.0 and 1.0
  gives some, but not all, risk reduction
• The smaller the correlation, the greater the risk
  reduction potential

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Portfolio Diversification
Diversification Spreading an investment across a
number of assets will eliminate some, but not
all, of the risk.
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Returns Distributions for Two Perfectly
Positively Correlated Stocks (? 1.0)
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Returns Distribution for Two Perfectly Negatively
Correlated Stocks (? -1.0)
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Risk Return with 2 Assets
E(r)
13
r -1
r 0
r .3
r -1
8
r 1
12
20
St. Dev
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Problem 3

• Use the following information to calculate the
  expected return and standard deviation of a
  portfolio that is 40 invested in Kuipers and 60
  invested in SuCo.
• Kuipers SuCo
• Expected return, E(R) 30 28
• Standard deviation , ? .65 .45
• Correlation .30
• E(RP) .4 x (.30) .6 x (.28) 28.8
• ?P .402 x (.65)2 .602 x (.45)2
• 2 (0.4)(0.6)(0.65 x .45 x .30
• (0.18262)1/2 42.73

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CAPM Capital Asset Pricing Model

• Assumptions
• 1.Two-Parameter Model Investors rely on E(R)
  and Risk in making decision
• 2.Rational and Risk Averse Investors are
  rational and risk averse.
• 3.One-Period Investment Horizon Investors all
  invest for the same period of time
• 4.Homogeneous Expectations Investors share all
  expectations about assets

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CAPM Capital Asset Pricing Model

• Assumptions
• 5.Existence of a risk-free asset and unlimited
  borrowing and lending at the risk-free rate
  Investors can borrow and lend any amount at the
  risk-free rate.
• 6.Capital markets are completely competitive and
  frictionless The sufficiently large number of
  buyerssellers. Also, no transaction costs.

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CML Capital Market Line

• From
• Var(Rp) (WRf)2 Var(Rf) (Wm)2 Var(Rm)
  2WRfWmcov(Rf,Rm)
• Since, risk-free asset has no variability, and
  therefore does not move at all with the return on
  the market portfolio
• Var(Rp) (Wm)2 Var(Rm)
• Wm SD(Rp) / SD(Rm)
• From
• E(Rp) WRfRf Wm E(Rm) which WRf Wm 1
• E(Rp) (1-Wm)Rf Wm E(Rm)
• E(Rp) Rf Wm (E(Rm) - Rf )
• Therefore, E(Rp) Rf (E(Rm) - Rf )/ SD(Rm)
  SD(Rp)

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CML Capital Market Line
Every combination of the risk-free asset and the
Markowitz efficient portfolio M is shown on the
capital market line (CML)
Rf
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CML Capital Market Line

• Because investors have the same optimal risky
  portfolio given the risk-free rate, all investors
  have the same CML. What if they have different
  levels of risk-free rate?

CML1
CML0
return
100 stocks
Second Optimal Risky Portfolio
First Optimal Risky Portfolio
100 bonds
?
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Announcements News

• Announcement expected part surprise

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Expected Return or Unexpected Return

• Total Return Expected Return Unexpected
  Return
• Total Return - Expected Return Unexpected
  Return

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Risk Systematic Unsystematic

• Risk The unanticipated part of the return, the
  portion resulting from surprises.
• Types of risk
• Systematic or market
• Unsystematic or unique or asset-specific

Risk that influences a large number of assets.
Also called market risk.
Risk that influences a single company or a small
group of companies. Also called unique or
asset-specific risk.
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Components of Risk

• R - E(R) Systematic portion Unsystematic
  portion
• R - E(R) U m ?

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Components of Risk
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Components of Risk

• Unsystematic Risk
• Unsystematic risk is essentially eliminated by
  diversification, so a portfolio with many assets
  has almost no unsystematic risk.
• Diversifiable risk / unique risk / asset-specific
  risk
• Systematic Risk
• Systematic risk affects all assets and can not be
  diversified away (even in a larger portfolio).
• Non diversifiable risk / market risk

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Systematic Risk (Beta)

• Systematic risk can not be eliminated by
  diversification.
• Since unsystematic risk can be eliminated at no
  cost, there is no reward for bearing it.
• Systematic Risk Principle
• Measuring Systematic Risk Beta or ?

The reward (expected return) for bearing risk
depends only on the systematic risk of an
investment.
Measure of the relative systematic risk of an
asset. Assets with betas larger (smaller) than 1
have more (less) systematic risk than market
average and will have greater (lower) expected
returns.
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Beta Coefficients
Beta Company Coefficient ( ?i) Exxon 0.65 AT
T 0.90 IBM 0.95 Wal-Mart 1.10 General
Motors 1.15 Microsoft 1.30 Harley-Davidson 1.6
5 America Online 2.40
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Portfolio Beta

• With a large number of assets in a portfolio,
  multiply each asset's beta by its portfolio
  weight, and then sum the results to get the
  portfolio's beta

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Portfolio Beta
Amount PortfolioStock Invested Weights Beta (1)
(2) (3) (4) (3) x (4) Haskell Mfg.
6,000 50 0.90 0.450 Cleaver, Inc. 4,000 33 1.10
0.367 Rutherford Co. 2,000 17 1.30 0.217 Portfoli
o 12,000 100 1.034
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Portfolio Beta

• High beta security is more sensitive to market
  movements
• Low beta security is relatively insensitive to
  market movements.
• Sensitivity depends on
• How closely is the securitys return correlated
  with the market
• How volatile the security is relative to the
  market

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Portfolio Beta

• Beta is computed as

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Portfolio Beta

• Beta is estimated as

Security Returns
Return on market
Ri a i biRm ei
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Beta Expected Return of a portfolio with one
risky and one risk free asset

• If you invest in WA in Asset A and the rest
  (1-WA) in Risk-Free Asset (with ß0), then
• ßP WA ßA (1 - WA) 0 WA ßA
• If you own 40 in A with an expected return of
  18, with the remainder in the Risk-Free Asset
  with a 8 return. The beta on Asset A is 1.4.
• E(RP) (0.40) (0.18) (1 - 0.40) (0.08) 12
• ßP WA ßA (1 - WA ) 0 (0.40)(1.4) 0.56

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Why do Betas Differ?

• Using daily, weekly, monthly, quarterly, or
  annual returns.
• Estimating betas over short periods (a few weeks)
  versus long periods (5-10 years).
• Choice of the market index (SP 500 index versus
  all risky assets).
• Some sources adjust betas for statistical and
  fundamental reasons (such as Value Line).

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SML Security Market Line

• E(Ri) Rf (E(Rm) - Rf )/ SD(Rm) SD(Ri)
• by assuming the unsystematic risk is zero and
  since,
• ,therefore,
• E(Ri) Rf cov (Ri, Rm) / var(Rm ) (E(Rm) -
  Rf )
• SML shows that it is not the variance or
  standard deviation of an asset that affects its
  return. Actually, Covariance of the assets
  return with the markets return affects its
  return.
• From equation above, positive covariance implies
  higher expected return than the risk-free asset.
  However, with positive covariance, it increases
  the risk of an asset in a portfolio, thats why
  investors will buy that asset only if they expect
  to earn a return higher than the risk-free asset.

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SML Security Market Line
Assume you wish to hold a portfolio consisting
of Asset A and a risk-less asset. Given the
following information Asset A has a beta of
1.2 and an expected return of 18. The risk-free
rate is 7. Use Asset A weights of 0, 25,
50, 75, and 100.
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SML Security Market Line
Proportion Proportion Invested
in Portfolio Invested in Risk-Free Expected Port
folio Asset A () Asset () Return () Beta
0 100 7.00 0.00 25 75 9.75 0.30
50 50 12.50 0.60 75 25 15.25 0.90
100 0 18.00 1.20
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SML Security Market Line
Graphical representation of the linear
relationship between systematic risk and expected
return in financial markets
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Reward-to-Risk Ratio

• Risk premium of X per unit of systematic risk
• Based on CAPM assumptions, the reward-to-risk
  ratio is the same for all securities. Hence,

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SML Security Market Line
Asset Expected Return E(Rj)
Slope of SML E(Rj) - Rf / ßj or E(RM) - Rf
SML
D
E(RA)
A
E(RM)
C
Rf
ßM 1.0
ßA
Asset Beta ßj
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• Question
• Asset A has an expected return of 12 and a beta
  of 1.40. Asset B has an expected return of 8 and
  a beta of 0.80. Are these assets valued correctly
  relative to each other if the risk-free rate is
  5?
• a. For A (.12 - .05) / 1.40 ________
• b. For B (.08 - .05) / 0.80 ________
• What would the risk-free rate have to be for
  these assets to be correctly valued?
• (.12 - Rf) / 1.40 (.08 - Rf) / 0.80
• Rf ________

0.050
0.0375
0.0267
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CAPMCapital Asset Pricing Model

• This result gives us the CAPM

The Capital Asset Pricing Model (CAPM) is an
equilibrium model of the relationship between
risk and return in a competitive capital
market. The CAPM shows the expected return for
an asset depends on 1.Pure time value of money
risk-free rate 2.Reward for bearing systematic
risk market risk premium 3.Amount of systematic
risk beta coefficient
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CAPMCapital Asset Pricing Model
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Risk Return Summary

• Total risk variance (standard deviation) of an
  assets return
• Total return expected return unexpected return
• Total unexpected return Systematic risk
  Unsystematic risk
• Systematic risk unanticipated events that affect
  almost all assets to some degree
• Unsystematic risk unanticipated events that
  affect single assets or small groups of assets

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Risk Return Summary

• Effect of diversification elimination of
  unsystematic risk via the combination of assets
  into a portfolio
• Systematic risk principle beta reward for
  bearing risk depends only on its level of
  systematic risk
• Reward-to-risk ratio ratio of an assets risk
  premium to its beta
• Capital asset pricing model expected return on
  an asset can be written as
• E(Rj) Rf E(RM) - Rf x ßj

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Arbitrage Pricing Theory

• Arbitrage - arises if an investor can construct a
  zero investment portfolio with a sure profit.
• Since no investment is required, an investor can
  create large positions to secure large levels of
  profit.
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear.

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APT Arbitrage Pricing Theory Model
Concept The return on a security is linearly
related to H factors. The APT does not specify
what these factors are, but it is assumed that
the relationship between security returns and the
factors is linear.

• E(Ri) Rf bi,f1E(Rf1) Rf bi,f2E(Rf2)
  Rf .
• bi,fnE(Rfn) Rf

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• Question
• You own a stock portfolio invested 30 in Stock
  Q, 20 in Stock R, 10 in Stock S, and 40 in
  Stock T. The betas for these for stocks are 1.2,
  0.6, 1.5, and 0.8, respectively. What is the
  portfolio beta?
• Solution
• ßP .3(1.2) .2(.6) .1(1.5) .4(.8) .95

Question A stock has a beta of 1.2, the
expected return on the market is 17, and the
risk-free rate is 8. What must the expected
return on this stock be? Solution ERj .08
(.17 - .08)(1.2) .188 18.8
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• Question
• Stock Y has a beta of 1.59 and an expected
  return of 25. Stock Z has a beta of .44 and an
  expected return of 12. If the risk-free rate is
  6 and the market risk premium is 11.3, are
  these stocks correctly priced?
• Solution
• Erj .06 .113ßj
• ErY .06 .113(1.59) .2397 lt .25
• so Y plots above the SML and is undervalued
• ErZ .06 .113(.44) .1097 lt .12
• so Z plots above the SML and is undervalued
• Question
• what would the risk-fee rate have to be for the
  two stocks to be correctly priced?
• Solution
• .25 - Rf / 1.59 .12 - Rf / 0.44 ,
  therefore, Rf .0703

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Risks associated with investing in Financial
Assets
Price risk An assets value drops. Default
risk The issuer of an asset cannot meet its
obligations. Inflation risk The rate of
inflation erodes the value of an
asset. Currency risk The rate of exchange erodes
the value of a foreign- denominated
asset.
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Risks associated with investing in Financial
Assets
Reinvestment risk The cash flow received must
be reinvested in a similar vehicle that offers
a lower return. Liquidity risk An asset
cannot easily be sold at a fair price. Call
risk The issuer of an asset exercises
its right to pay off the amount borrowed.