Chapter 8 Capital Market Theory

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Chapter 8Capital Market Theory

• J. D. Han
• Kings College, UWO

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1. Risk

• Credit Risk Default Risk
• Liquidity Risk
• Inflation Risk
• Market Risk Variability of the Rate of Return
  of an asset

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How to evaluate/measure Risks?

• Credit Risk
• Credit rating
• Liquidity Risk
• Usually proportional to the maturity length, but
  not always
• Inflation Risk
• Proportional to expected money creation rates
• Market Risk
• Volatility

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(Bond) Credit Rating

• Only 5 rating companies have the Nationally
  Recognized Statistical Rating Organization
  (NRSRO) designation, and are overseen by the SEC
  in their assignment of credit ratings
• Standard Poor's (SP), Moody's, Fitch, A. M.
  Best and Dominion Bond Rating Service.

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Scales of Credit Ratings
Below BBB, or Baa are Junk Bonds.
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Examples

• Dominion Bond Rating Services
• http//www.dbrs.com/intnlweb/jsp/search/listResul
  ts.faces
• Standard and Poors in Canada
• http//www2.standardandpoors.com/portal/site/sp/e
  n/ca/page.topic/ratings_corp/2,1,3,0,0,0,0,0,0,0,4
  ,0,0,0,0,0.html

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Risk requires Compensation for Investor, Risk
Premium
By John Hull at Rotman School of Business
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2. How to measure Market Risk of Individual Asset?

• 1. Variability Deviation from its own Average
  Rate of Return
• Mean Variance Approach
• 2. Co-movement with the Market Index Relative
  Variability of Rate of Return to the Market Index
• Capital Market Pricing Model

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3. Mean Variance MethodMarket Risk and Return
for a Single Asset

• How to characterize an asset?
• With Expected Returns, and Market Risk
• rA Distribution(E(rA), sA )

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1) Expected Return a Statistical Statement

• What will be the expected return for asset A
  rA for next year?
• There are many possible contingencies
• Assume that history will repeat in the future
• - Look back at the historical data of various ri
  that have hanged over time in different
  contigencies.
• - Get the mean value (weighted average for all
  possible states of affairs) as the expected rate
  of return.
• -

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• Mathematically,
• Mean Value, or rA bar
• Expected Value E(rA)
• S rA.i prA.i
• rA.1 prA.1 rA.2 prA... rA.m prA.m
• where
• rA.i annualized rate of returns of asset
  A in situation i
• prA.i probability of situation i taking place

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2) Market Risk by Standard Deviation

• Mean Variance Approach measure the risk by
  standard deviation
• How mcuh do the actual rates of return deviate
  from its own average value over time?

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• SD comes from variance
• s2A
• S (rA.i E rA)2 prA.i
• (rA.1 E rA)2 prA.1 (rA.2 E rA)2 prA...
• (rA.m E rA)2 prA.m

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Numerical Example How to calculate the
variance and the standard deviation?

• 1) Stock B Data of r over 3 years are 4, 6,
  and 8
• E (r ) (4 6 8)/3 6
• s 2 1/3(4- 6)2 1/3(6-6)2 1/3(8-6)2 8/3
• (8/3)1/2
• B (6, (8/3)1/2 )
• 2) Stock C Data r 3 times of 4, 5 times of 6,
  twice of 8

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Various Assets

• Expected Rate of returns of a Stock (ith
  companys stock) E (r s I)
• Expected Rate of returns of a Bond (ith
  institutions bond) E( r b i )
• Expected Rate of returns of a T-Bill E (r
  T-bill i) ) rf (risk free asset)
• Expected Rate of returns of the Market Portfolio
  E( rm)
• Expected Rate of returns of gold E(rg)
• Expected Rate of returns of Picasso Print
  rpicasso

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Risk and Returns
re
rstock i
rbond i
rPicasso
rT-bill i
s
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Stylized Fact

• The Higher the Standard Deviation, the Higher the
  Average Rate of Returns
• - The Higher the Market Risk, the Higher the
  Risk Premium an Asset should pay to the investor.
  
• Otherwise, no investor will hold this asset
• However, the Risk Premium does NOT rise in
  proportion to the Market Risk

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4. Portfolio Diversification Multiple Assets

• Mixing Two or More Assets for Investment
• Spreading Investment over two or more assets
• We will see
• First
• Combine Two (or more) Risky Assets
• Second
• Risky Assets and Risk-Free Asset

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1) Why Diversification?

• Expanded Opportunity Set More Options for
  different combination of returns and risk
• or
• Taking advantage of non-linear trade-off between
  returns and risk

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2) Return and Risk for Combining Two Risky Assets

• Asset A ( E(rA), sA)
• Asset B (E(rB), sB)
• Suppose we mix A and B at ratio of w1 to w2for a
  portfolio
• Resultant Portfolio Ps
• Expected Rate of Return?
• Market Risk?

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Return of Portfolio

• Return E(rp) w1 E (rA) w2 E(rB)
• Simple weighted average of two assets individual
  average rate of return

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Risk
rA B is the correlation coefficient of rA and
rB. sA B is the correlation coefficient of rA
and rB.
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• Depending on s A B,, there are 3 different cases

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• Case 1. rAB 1
• rA and rB are perfectly positively correlated
• Return E(rp ) w1 E(rA) w2 E(rB)
• RiskWeighted average of risk of two component
  assets

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In this case, the Investment Opportunity Set
looks like
E (Rp)
As Bs portion w2 rises,
E (Rp)
B
w2
sp
Portfolio 1 0.9 A 0.1B
A
sp
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• Case 2. rAB -1
• rA and rB are perfectly negative correlated
  Return E (rp) w1 E(rA) w2 E(rB)
• Riskweighted difference between risks of two
  assets

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In this case, the Investment Opportunity Set
looks like
As Bs portion w2 rises,
E (Rp)
E (Rp)
B
Portfolio X a A b B Perfect Hedge
sp
w2
Portfolio 1 0.9 A 0.1B
A
sp
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Perfect Hedge Portfolio P which has zero market
risk- At what ratio should A and B be mixed?

• Two equations and two unknowns
• sp I w1 sA - w2 sB I 0
• w1 w2 1
• Solve for w1 and w2

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Case 3. Generally 1lt rABlt 1 Imperfect
Correlation between A and Bs returns

• Return E (Rp ) w1 E( RA) w2 E( RB )
• Risk

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In this case, the Opportunity Set Looks
LikeNote that the expected value of the
portfolio is the linear function of the expected
rates of returns of the assets, and the standard
deviation is less than the weighted average
unless r AB 1.
E (Rp)
E (Rp)
B
w2
Portfolio 1 0.9 A 0.1B
sp
A
sp
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Prove sp lt w1 sA w2 sB in general

• Square the both sides.
• The above is, sp2 versus (w1 sA w2 sB)2
• First, left-hand side-
• Recall sp2 w12 sA2 w22 sB2 2 w1 w2 rAB sA
  sB
• Second,-right hand side-
• w12 sA2 w22 sB2 2 w1 w2 sA sB
• w12 sA2 w22 sB2 2 w1 w2 x 1x sA sB
• The comparison boils down to rAB versus 1.
• Recall rAB is equal to or less than 1.
• Thus, the left-hand side is equal to or less than
  the right-hand side.

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3) Efficient Frontier the upper part of
investment opportunity set is superior to the
lower part

Minimum Variance Portfolio
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What if there are more than 2 risky-assets?Gener
al Case of Mean Variance Approach

• Risk or SD is given by the square root of

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What if there are more than one set of risky
assets? Step 2. Get the Best Results of Combing
a pair of risky assets, and get their envelope
curve for Efficient Frontier

D
B
C
A
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Combining Market-Risk- Free
Lending/Borrowing, and Risky Asset

• Risk Free Asset (rf , 0)
• Correlation coefficient with any other asset 0
• Portfolio which mixes Risk free asset and Asset A
  at w1 to w2
• return w1 rf w2 E(rA)
• market risk w2 sA
• - This is on a straight line between Risk free
  asset and Asset A

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With Market-Risk-Free Borrowing/Lending, the
Efficient Frontier is a Straight Line
sM
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Application Question 1 Should a Canadian
investment include a H.K. stock?

• H.K. has currently depressed stock market
• H.K. stocks have lower rates of returns and a
  higher risk (a larger value of SD) compared to
  the Canadian Stocks.
• What would the possible benefit for a Canadian
  fund including a H.K. stock(with a lower return
  and a higher risk)?
• surely, more comparable investment options
• Maybe, a possibility of some new superior options
• Show this on a graph

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Application Question 2 How much of foreign
stocks a Canadian should include in his portfolio?
100 International Stock(MSCI World Index)
15.5
14.6
Minimum Risk Portfolio 76 of MSCI and 24 of TES
300
100 Canadian Equities(TSE 300)
10.9
Source About 75 Foreign Content Seems Ideal
for Equity Portfolio, Gordon Powers, Globe and
Mail, March 6, 1999
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Application Question 3 As you are mixing more
and more assets, the Mean-Variance Risk of the
portfolio falls
Total risk sp
Unique (Diversifiable) Risk
Market (Systematic) Risk
of assets
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Appliation Example XYZ Fund
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5. Choice of Optimum Portfolio for an Individual
Customer

• Tangent Point of
• Efficient Frontier of Portfolios Return and
  Risk
• Individual Customers Indifference Curve
  showing his Risk Preference (- Attitude towards
  Risk and Return)

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Risk Preference of Client may vary

• Risk-Averse vs Risk-Loving

Indifference Curves
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In case there is no risk-free asset, we can
choose the Optimum now.
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What will be the graph of choice like for the
case with Market-Risk-Free Lending/Borrowing and
Risky Assets?
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Answer Choice depending on Preference in case
where risk-free lending and borrowing is possible
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Tobins Separation Theorem

• The investment decision of which portfolio of
  risky assets to hold is separate from the
  financing decision of how to allocate investment
  between the risky assets and the risk-free
  assets.
• In other words, there is one optimal portfolio
  of risk assets for all investors.
• Of course, a risk-loving person will hold more of
  risk-free assets, and a risk-averse person will
  hold more of risky assets. However, for both, the
  best relative combination of different risky
  assets is the same
• - financial advisors should recommend the same
  proportion of risky assets in clients portfolio
• - In reality, this is not the case

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5. Capital Asset Pricing Model

• Improve on Mean-Variance Approach
• Risk Premium depends on Assets Systematic Risk
  only
• Systematic Risk is measured by b
• Co-movement of Return on an asset and the
  Market Portfolio (index).

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1) Why is b a superior measure of market risk
than Mean-Variance s?

• Asset B

• Asset A

RA and Rm over time
RB and Rm over time
sB1 bB -1 Extremely Desirable Asset for
Portfolio Diversification
sA1 bA1 Typical Asset
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Comparison of SD and b

• Beta of CAPM model
• -Measuring only the portion of fluctuations of
  the rate of returns which move along with the
  Market
• -Measuring only
• Systematic Risk

• Standard Deviation
• (lt- Mean-variance)
• -Measuring the entirety of fluctuations of the
  rate of returns over time
• -Measuring
• Systematic and
• Non-systematic risks

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Two Component of Market Risk

• Systematic Risk
• changes in price of an asset when the entire
  market (prices) moves.
• Market-wide Risk
• Foreseen Risk
• Non-diversifiable Risk
• risk premium for it.

• Non-systematic Risk
• unrelated to the entire market movement
• Firm-specific Risk
• Idiosyncratic Risk
• Unforeseen Risk
• Diversifiable Risk
• No risk premium for this

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Market Pays Risk Premium only on Systematic
RiskWhy?

• Anybody can remove unsystematic risk by portfolio
  diversification
• -gt positive deviation of one asset may offset
  negative deviation of another asset
• If the market pays risk premium on non-systematic
  risk, nobody would try hard to diversify his
  portfolio
• -gt risk premium on non-systematic risk would
  discourage due diligence for portfolio
  diversification

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• b measures the degree to which an asset's returns
  covaries with the returns on the overall market,
  or the relative market risk of an asset to the
  typical market to the market portfolio (market
  index) as a whole
• b 2 means that this asset has twice as much as
  variation in price as the market index as a
  whole.
• Thus this asset is twice as risky as the
  market portfolio.
• -b lt1 Defensive
• 1 Typical
• gt1 Aggressive

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Some Canadian Examples in the Stock Market

• Cetricom 2.92
• Clearnet 1.77
• Air Canada 1.66
• Noranda 1.57
• BCE 1.22
• Chapters 1.01
• Bank of Nova Scotia 1.03
• Bombardier 0.68
• Hudsons Bay 0.58
• Loblaw 0.35
• Source Compustat, Feb 2000

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2) Market Risk by b
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3) Risk Premium
Beta x Market Portfolios Risk Premium
4) Required Rate of Return on this Asset
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5) Security Market Line(SML)

• Risk Premium

ri - rf
Slope of SML ( rM rf )/ bM risk premium /
risk risk premium per unit of risk price of
(a unit of) systematic risk
rM - rf
bM 1 bi
0
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Intuitionthe slope of the CML indicates the
market price of risk

• Suppose that the Market Portfolio has 12 of
  expected returns and 30 of standard deviation.
  The risk free rate on a 30-day T-Bills is 6.
  What is the slope of the CML?
• -gtAnswer 20 (0.12-0.06)/0.30
• -gt The market demands 0.20 percent of additional
  return for each one percent increase in a
  portfolios risk measured by its s.

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Security Market Line (SML) Visual Presentation
of CAPM model
Required Yields or Expected Rates
E(Ri)
E(RM)
Rf
b
bM 1
bi
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Numerical Example

• Suppose that the correlation coefficient between
  Inert Technologies Ltd and the stock market index
  is 0.30. The rate of return on a 30-day T-Bill
  is 8. Overall, the rates of return on stocks
  are 9 higher than the rate of return on T-Bills.
  The standard deviation of the stock market index
  is 0.25, and the standard deviation of the
  returns to Inert Technologies Ltd is 0.35.
• What is the required rate of return on a Inert
  Technologies Ltd stock?
• Covariance rAB sA sB
• Thus the covariance 0.3 x 0.35 x 0.25 0.02625
• Beta covariance / variance of market portfolio
  0.02625/(0.25)2 0.42
• Required Rate 0.08 0.42 (0.09) 0.117

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6) Evidence Regarding the CAPM Ex-Post or Actual
Ri may differ from ex-ante or required Ri or E
(Ri )

• Note that e is random unexpected error, or
  unsystematic risk, idiosyncratic risk.
• e has an average value of 0 it is diversifiable
  risk
• The market does not pay any risk premium for this
  as it cannot be anticipated and it can be
  diversified.

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Undervalued?

• Suppose that X is observed ex-post as having the
  following rate of return and risk. What does this
  mean?

X
Security Market Line
bX