Chapter 8 Capital Market Theory

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Chapter 8Capital Market Theory

• J. D. Han
• Kings College, UWO

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1. Market Risk and Return

• How to characterize an asset?
• With Returns, 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?
• - look back at the historical data of ri that
  have hanged over time(variable).
• - get the mean value (weighted average for all
  possible states of affairs) as the expected rate
  of return.
• - Mathematically,
• E(rA) rA bar S rA.i prA.i rA.1 prA.1
  rA.2 prA... rA.m prA.m
• 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

• What is the market risk?
• -A measure that the actual rate of return may
  deviate from the expected rate of return
• Market risk is measured by standard deviation
• sA.
• 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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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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2. Portfolio Diversification

• 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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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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3. 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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• Portfolio (E(rP), sp) which has A and B assets
  at the ratio of w1 and w2 (w1 w2 1.0)
• Return E(rp) w1 E (rA) w2 E(rB)
• Risk
• rAB is the correlation coefficient of rA and rB.
• sAB is the correlation coefficient of rA and
  rB.
• sAB rAB sA sB

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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

• 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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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 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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What if there are more than 2 risky-assets?

• Diversification

Total risk sp
Unique (Diversifiable) Risk
Market (Systematic) Risk
of assets
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Example XYZ Fund
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Example
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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Consider Preference of Client

• 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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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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4. Combining Risk Free Asset 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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Introducing Borrowing and Lending
Diversification between Risk Free Asset and
Market Portfolio
sM
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Security Market Line(SML)

• Risk Premium

ri - rf
Slope of CML ( rm rf )/ bM risk premium /
risk risk premium per unit of risk price of
(a unit of) 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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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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Choice depending on Preference in case where
risk-free lending and borrowing is possible
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5. Capital Asset Pricing Model

• Risk Premium depends on Assets Systematic Risk
  only
• Systematic Risk means the Co-movement of Return
  on an asset and the Market Portfolio (index).

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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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• -Remarks
• b measures the degree to which an asset's
  returns covaries with the returns on the market
  relative measure of risk.
• -b lt1 Defensive
• 1 Typical
• gt1 Aggressive

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Why is b superior than s as a measure of
market risk?

• 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 beta

• 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- 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
• Market-wide Risk
• Foreseen Risk
• Non-diversifiable Risk

• Non-systematic Risk
• Firm-specific Risk
• Idiosyncratic Risk
• Unforeseen Risk
• Diversifiable Risk

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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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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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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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Evidence Regarding the CAPM Ex-Post or Actual
Ri may differ from ex-ante 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