Financial Market III: Risk Premium Theories 2- Market Risk

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Financial Market IIIRisk Premium Theories 2-
Market Risk

• J. D. Han
• Kings College, UWO

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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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1. Mean-Variance ApproachMarket Risk and Return
for a Single Asset

• How to characterize an asset over time?
• With Time-series data of the rates of return on
  it, get
• Expected Returns average/mean value of
  rates of return and Market Risk standard
  deviation
• rA Distribution(E(rA), sA )

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• Case of a Single Financial Asset
• risk is measured by standard deviation(SD) of
  a single financial asset.
• Case of Multiple Financial Asset in a Portfolio
• variance of the portfolio is non-linear
  combination of SDs of each individual asset and
  covariance among them.

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• Mean-Variance Approach of a Single Asset

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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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• Statistically,
• Suppose that there are n possible outcomes for
  rA.
• And each event/outcome has probability of pr1,
  pr2, ..prn.
• Mean Value, or rA bar
• Expected Value E(rA)
• S rA.i pri
• rA.1 pr1 rA.2 pr2... rA.n prn
• where
• rA.i annualized rate of returns of asset
  A in situation i
• pri 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 pri
• (rA.1 E rA)2 pr1 (rA.2 E rA)2 pr2..
• (rA.n E rA)2 prn

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

• Bond A Time series data of r over 3 years are
  4, 6, and 8 then
• 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
• Thus s (8/3)1/2
• B (6, (8/3)1/2 )
• Note that here time sequence does not matter.

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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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Stylized fact Risk and Returns
re
rstock i
rbond i
rPicasso
rT-bill i
s
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• 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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• Mean-Variance of Multiple Assets in a Portfolio
• - case without risk-free asset
• - case with risk-free asset with return rf
• free access at rf for deposits and loans

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Diversified Portfolio Multiple Assets

• Mixing Two or More Assets for Investment in the
  way to minimize the resultant SD of the portfolio
• We will see
• First
• Combine Two (or more) Risky Assets
• Second
• Risky Assets and Risk-Free Asset

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• First we will examine the combination of two
  risky assets, and then move onto
• The combination of multiple risky assets and the
  risk-free asset here comes Tobins Separation
  Theorem saying The best combination portfolio of
  risk assets is the same for everybody.

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1) Why Diversification?

• Suppose that we have two assets A and B, shown by
  two dots
• Diversification Mixing the two at different
  rates gives the lines of return-risk profile.
• We can see the advantage of diversification could
  be either
• i) Expanded Opportunity Set More Options for
  different combinations of returns and risk or
• ii) Taking advantage of some reduced risk or
  smaller SD than is given by the liner
  aggregation

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• Of course, the second one is better. Whether the
  second one is available depends on the
  covariance/correlation between Asset A(s rates
  of return) and Asset B(s rates of return) over
  time.
• Unless the two are perfectly correlated, the
  second one is available.
• Even if the two are perfectly correlated,
  diversification means different options of
  combinations of assets A and B.

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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 covariance coefficient of rA
and rB.
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• Recall
• rA B sA B / (sA . sB)
• sA B
• S (rA.i E rA) (rB.i E rB) pri
• (rA.1 E rA) (rB.1 E rB) pr1 (rA.2 E
  rA) (rB.2 E rB)pr2..
• (rA.n E rA) (rB.n E rB) prn

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Numerical Example

• Click here for a practice question

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• Depending on r A B,, there are 3 different
  impacts on the combined risk

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• Case 1. rAB 1 rA and rB are perfectly
  positively correlated
• Return E(rp ) w1 E(rA) w2 E(rB)
• Portfolio Risk weighted average of risks 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. 1lt rABlt 1 Imperfect Correlation
between A and Bs returns General Case

• Return E (Rp ) w1 E( RA) w2 E( RB )
• Risklt weighted average of two risks

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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 case of
rAB lt1

• Square sp and w1 sA w2 sB
• It is now, sp2 versus (w1 sA w2 sB)2
• Compare the size of the left and the right side.
• First, left-hand side is sp2
• Recall sp2 w12 sA2 w22 sB2 2 w1 w2 rAB sA
  sB
• Recall rAB is less than 1.
• 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.
  
• Thus, the left-hand side is equal to or less than
  the right-hand side.

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• This general case includes the one where
• the rates of returns on two assets are
  completely independent of each other
• Still the risk of the portfolio will be smaller
  than the risk of the less risky asset of the two
  components.
• The arched-out part of the lower part of the
  locus(curve) of portfolio has lower risk,and the
  upper arched-part is efficient.

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Suppose that the two assets are independent of
each other.If you start with less risky asset,
the risk falls as you include some risky asset
first, and, past H point, the risk starts
increasing. The arrow line shows the locus. The
blue arrow indicates the efficient portfolios,
and the red arrows are not efficient.
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• The principle of choice of assets for portfolio
• - The smaller the correlation between the
  component assets, the larger the benefits of
  reduced risk of the portfolio.
• We search for assets whose returns are
  hopefully less-positively-correlated and
  more-negatively-correlated.
• - The curve of return-risk will be arched to
  the left to the maximum.

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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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Application 4. Buying Art for portfolio
diversification

• An inferior single asset can be a great element,
  if taken in a small amount, in the portfolio.
• It lowers the rate of return of the portfolio,
  but it may lower the risk even more so.
• Click here for J. Pesandos paper

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Returns and Risks of the Art

• Investment on Art, especially, on Picassos
  prints.

r

rstock i
rbond i
rT-bill i
rArt Prints
s
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Remark

• The art prints have the lower rate of return at a
  given risk, compared with other financial assets.
  In other words, the art prints seem to be
  inferior For the same risk, the returns are
  lower.

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Would we include these prints in our portfolio?

• The answer
• Not as a single investment item.
• But, we may include them in the portfolio.
• Why? Lets explain.

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The Art Prints have a very desirable property in
terms of portfolio diversification a Negative
Correlation Coefficient with some Financial Assets
Prints Stocks Bonds T-Bills Inflation
Prints 1 0.3 -0.10 (-0.17) -0.21 (-0.27) 0.03 (0.08)
stocks 1 0.46 0.27 -0.31
bonds 1 0.73 -0.56
T-Bills 1 -0.73
Inflation 1
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• The prints could provide an attractive investment
  as their small amount of inclusion in a portfolio
  of traditional financial assets may reduce the
  mean return a little but it may reduce the entire
  risk by a substantially larger margin.

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Returns and Risks

• When T bills and prints are mixed at the ratio of
  94 to 6(), the portfolio has the minimum
  variance.

r

rT-bill i
s
rPicasso
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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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• Note that depending on his preference an investor
  can end up on any point on the efficient
  frontier it will be his optimal portfolio.
• However, regardless of preferences, the
  combination of the risk assets is the same for
  everybody, and it is called here market
  portfolio.

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• Tobins Separation Theorem
• Investment decision(of choosing the right
  combination of risky assets), and
• Financing decision(of depositing or borrowing
  from banks at the risk-free rate) are independent
  of each other.

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Tangent Portfoliomarket portfolio
Optimum-risk portfolio

• It is not overall-Optimum portfolio.
• It is the optimum portfolio only with risky
  assets.
• It has the highest
• Sharp Ratio E(rp ) rf
• sp

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Importance of the unlimited access to borrowing
and lending at the risk-free rate

• Without it, the choice of (overall) optimal
  portfolio would be on the Curved Line of the
  portfolio locus.
• The curved line is in general inferior to the
  capital market line.
• -gt This smooth combination of investment(securitie
  s business) and commercial banking would be
  important
• lt- The Financial Holding company by G-L-B act in
  the U.S. may be justifiable in this contribution
• In practice, a portfolio manager of a
  securities company can coordinate with a credit
  officer of a commercial bank within the same FHC
  for a clients loans and deposits at the risk
  free rate so that the client can finance his
  investment along the straight line of Efficient
  Frontier.

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Is there only one market portfolio?

• Because of different available set of assets for
  different financial companies, it varies.
• However, across the board, the return of the
  market portfolio is similar.

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• Practice Question of Making your own Portfolio
• Here is a detailed instruction.

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2. 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