CHAPTER 5 Risk and Rates of Return

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CHAPTER 5Risk and Rates of Return

• Stand-alone risk
• Portfolio risk
• Risk return CAPM / SML

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Return Risk

• Return is the annual income received plus any
  change in market price of an asset or investment.
  
• Risk is the variability of actual return from the
  expected return associated with a given asset.

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Rate of Return

• The rate of return on an investment for a period
  (which is usually a period of one year) is
  defined as follows,
• Annual Income (Ending price Beginning price)
• Beginning price

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Rate of Return

• Price at the beginning of the year Tk. 60.00
• Dividend paid toward the end to the year Tk.
  2.40
• Price at the end of the year Tk. 66.00
• 2.40 (66.00 60.00)
• 60.00
• 0.14 or 14.

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Rate of Return Current Yield Capital
gains/Losses

• Current Yield
• Capital gains/Losses Yield

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Current Yield Capital gains/Losses

• Annual Income Ending Beginning Price
• Beginning Price Beginning Price
• (Current Yield) (Capital Gain/Losses)

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Current Yield Capital gains/Losses

• 2.40 (66.00 60.00)
• 60.00 60.00
•  
• 4 10
• Current Yield Capital gains/Losses
•  

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Current Yield Capital gains/Losses

• If the price of a share on April 1 is TK. 25, the
  annual dividend received at the end of the year
  is TK. 1 and the year end price on March 31 is TK
  30.
• Find the Rate of Return
• Find the Current Yield
• Find the Capital gains/Losses Yield.

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• 1 (30.00 25.00)
• Rate of return 25.00
• 0.24 or 24.
• Current Capital gain/Losses
• 1 (30.00 25.00)
• 25 25.00
•  
• 4 20
• Current Yield Capital gains/Losses

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What is investment risk?

• Two types of investment risk
• Stand-alone risk
• Portfolio risk
• Investment risk is related to the probability of
  earning a low or negative actual return.
• The greater the chance of lower than expected or
  negative returns, the riskier the investment.

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Measurement of Risk Single Asset

• The risk associated with single asset is
  assessed from both,
• Behavioral point of view
• Sensitivity Analysis
• Probability Distribution
• Statistical point of view
• Standard Deviation
• Coefficient of Variation

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Measurement of Risk Single AssetBehavioral
Point of View

• This approach is to estimate the worst
  (pessimistic), the expected (most likely) and the
  best (optimistic) return associated with the
  asset.
• The level of outcome may be related to the
  economic conditions namely, recession, growth and
  Boom.
• The difference between pessimistic and optimistic
  outcome is the RANGE which is the measurement of
  RISK.
• The greater the RANGE, the more RISKY the Asset.

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Sensitivity Analysis

• Particular Asset X Asset Y
• Initial Outlay 50 50
• Annual Return ()
• Pessimistic 14 8
• Most Likely 16 16
• Optimistic 18 24
• RANGE 4 16
• (optimistic Pessimistic)

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Measurement of Risk Single AssetBehavioral
Point of View Probability Distribution

• The probability of an event represent the
  chance of its occurrence.
• Probability Distribution is model that relates
  probabilities to the associated outcome.

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Probability Distribution Asset X
Possible Outcome (1) Probability (2) Returns (3) Expected Returns (2)X(3)4
Pessimistic Most Likely Optimistic 0.20 0.60 0.20 1.00 14 16 18 2.8 9.6 3.6 16.00
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Probability Distribution Asset Y
Possible Outcome (1) Probability (2) Returns (3) Expected Returns (2)X(3)4
Pessimistic Most Likely Optimistic 0.20 0.60 0.20 1.00 8 16 24 1.6 9.6 4.8 16.00
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Measurement of Risk Single AssetStatistical
Point of ViewStandard Deviation

• Risk refers to the dispersion of returns around
  an expected value.
• The most common statistical measure of risk of an
  asset is the standard deviation from the
  mean/expected value of return.
• ? (R-R)2 X pr

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Standard Deviation
Asset X Asset X Asset X Asset X Asset X Asset X Asset X
i R R R - R (R R)2 Pr (R R)2x Pr
1 14 16 -2 4 .20 .80
2 16 16 0 0 .60 0
3 18 16 2 4 .20 .80 1.6
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Standard Deviation

• ? (R-R)2 X pr
• 1.6
• 1.26

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Standard Deviation
Asset Y Asset Y Asset Y Asset Y Asset Y Asset Y Asset Y
i R R R - R (R R)2 Pr (R R)2x Pr
1 8 16 -8 64 .20 12.80
2 16 16 0 0 .60 0
3 24 16 8 64 .20 12.80 25.6
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Standard Deviation

• ? (R-R)2 X pr
• 25.6
• 5.06
• The greater the Standard Deviation of Returns,
  the greater the risk.

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Measurement of Risk Single AssetStatistical
Point of ViewCoefficient of Variation

• it is the measure of relative dispersion used in
  comparing the risk of assets with differing
  expected returns.
• ?
• CV
• R

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Measurement of Risk Single AssetStatistical
Point of ViewCoefficient of Variation

• The coefficient of variation of assets X Y are
  respectively,
• Asset X ( 1.26 / 16) 0.079
• Asset Y (5.06 / 16 ) 0.316
• The larger the CV, the larger the relative risk
  of the asset.

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Risk Return of PORTFOLIO

• Portfolio means a combination of two or more
  Assets.
• Each portfolio has risk return characteristics
  of its own.
• Portfolio theory developed by Harry Markowitz,
  shows that portfolio risk, unlike portfolio
  return, is more than simple aggregation of the
  risks of individual assets. This depends on the
  interplay between the returns on assets
  comprising the portfolio.

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Portfolio Expected return

• E (rp) ?wi E(ri)
• E (rp) Expected return from portfolio
• Wi Proportion invested in asset i
• E(ri) Expected return for asset i
• n number of assets in portfolio

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Portfolio Expected return

• The expected return on two assets L and H are
  12 16 respectively. If the corresponding
  weights are 0.65 0.35. Calculate Portfolio
  Expected return
• E (rp) ?wi E(ri)
• 0.65 x 0.12 0.35 x 0.16
• 0.134
• 13.4.

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Portfolio RiskTwo Asset portfolio

• ?2p w21?21 w22?22 2 w1 w2 (?12)
• Alternatively,
• ?2p (w1?1)2 (w2?2 )2 2 w1 w2 (P 12 ?1 ?
  2)
• W1 fraction of total portfolio invested in
  Asset 1
• W2 fraction of total portfolio invested in
  Asset 2
• ?21 Variance of asset 1
• ?1 Standard deviation of Asset 1
• ?22 Variance of asset 2
• ?2 Standard deviation of Asset 2
• ?12 Covariance between returns of two assets (P
  12 ?1 ? 2)
• P 12 Coefficient of correlation between the
  returns of two asset.

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Portfolio RiskTwo Asset portfolio

• The expected return on two assets L and H are
  12 16 respectively. The standard deviations
  of assets L H are 16 and 20 respectively. If
  the coefficient of correlation between their
  returns is 0.6 and the two assets are combined in
  the ratio of 31.
• (1) Calculate expected rate of return
• (2) variance of Portfolio
• (3) Standard Deviation

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Portfolio Expected return

• E (rp) wL E(rL) wH E(rH)
• (0.75 x 0.12) (0.25 x 0.16)
• 94
• 13.

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The Variance of the Portfolio

• 2 ?2p (w1?1)2 (w2?2 )2 2 w1 w2 (P 12 ?1 ?
  2)
• (0.75 x 16)2 (0.25 x 20) 2 2 (0.75) (0.25)
  (0.06) (16 x 20)
• 144 25 (0.375)(192)
• 144 2572
• 241
• 3 ?p 241
• 15.52

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Portfolio Risk

• The above discussion shows that the portfolio
  risk depends on 3 factors
• 1 Variance or Standard deviation of each asset
  in portfolio.
• 2 Relative importance or weight of each asset
  in the portfolio
• 3 Interplay between returns on two assets
• Among these only weights can be controlled by the
  portfolio managers. Therefore his/her primary
  task is to decide the proportion of each security
  in the portfolio.

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Investor attitude towards risk

• Risk aversion assumes investors dislike risk
  and require higher rates of return to encourage
  them to hold riskier securities.
• Risk premium the difference between the return
  on a risky asset and less risky asset, which
  serves as compensation for investors to hold
  riskier securities.

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Breaking down sources of risk

• Stand-alone risk Market risk Firm-specific
  risk
• Market risk portion of a securitys stand-alone
  risk that cannot be eliminated through
  diversification. Measured by beta.
• Firm-specific risk portion of a securitys
  stand-alone risk that can be eliminated through
  proper diversification.

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Capital Asset Pricing Model (CAPM)

• Model based upon concept that a stocks required
  rate of return is equal to the risk-free rate of
  return plus a risk premium that reflects the
  riskiness of the stock after diversification.
• It is the logical major extension of the
  portfolio theory of Markowitz by william Sharpe
  (1964), John Linter ( 1965) Jan Mossin (1967).

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Capital Asset Pricing Model (CAPM)

• CAPM is a theory that explains how asset prices
  are formed in the market place.
• CAPM provides the framework for determining the
  equilibrium expected return for risky return. It
  uses the results of capital market theory to
  derive the relationship between expected return
  and systematic risk of individual
  assets/securities and portfolio.

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Capital Asset Pricing Model (CAPM)

• The CAPM has implication for
• Risk-Return relationship for an efficient
  Portfolio
• Risk-Return relationship for an individual asset
• Identification of over valued or under valued
  assets traded in in the market
• Pricing of assets not yet traded in the market
• Effect of leverage on cost of equity.

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Capital Asset Pricing Model (CAPM)

• Capital budgeting decision cost of capital
• Risk of the firm through diversification of the
  project portfolio.

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Capital Asset Pricing Model (CAPM) Assumption

• All investors are price takers. There number is
  so large that no single investor can affect
  prices
• Assets/securities are perfectly divisible
• All investors plan for one identical holding
  period
• Investors can lend or borrow at an identical
  risk-free rate.
• There is no transaction costs income Tax

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Capital Asset Pricing Model (CAPM)

• The elements of the model
• K K RF (KM - K RF) ß
• Where,
• K RF Risk Free Return
• KM required rate of return of market
• ß Beta (systematic risk of the asset)

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Beta

• It measure the risk of an individual asset
  relative to the market portfolio. Beta shows how
  the price of securities responds to market force.
  In practice, the more responsive the price of
  security is to changes in the market, the higher
  will be its beta. The beta for the overall market
  is equal to 1.00 Beta can be positive or
  negative. Investors will find beta helpful in
  assessing systematic risk and understanding the
  impact the market movement can have on the return
  expected from a share or stocks.

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Calculating betas

• The ABC Company is considering a new capital
  investment proposal. The projects risk structure
  is very similar to that of the companys existing
  business. Return for this companys stocks for
  the past ten years are given in the following
  table together with returns for a countrys stock
  market index. The Govt. Treasury Bill return
  (Risk Free Return) was around 5.6 per annum.
•  

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Calculating betas
Year Companys Stock Return Stock Market Index Return
1992 0.09 0.07
1993 0.10 0.09
1994 0.10 0.10
1995 0.11 0.12
1996 0.10 0.11
1997 0.11 0.10
1998 0.11 0.10
1999 0.10 0.09
2000 0.09 0.08
2001 0.07 0.07
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Calculating betas

• Required
• Calculate the
• 1 Beta
• 2 Required Return according to CAPM model

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Comments on beta

• If beta 1.0, the security is just as risky as
  the average stock.
• If beta gt 1.0, the security is riskier than
  average.
• If beta lt 1.0, the security is less risky than
  average.
• Most stocks have betas in the range of 0.5 to 1.5.

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Problem

• Assume a security with beta of 1.2 being
  considered at a time when the risk free rate is
  4 and the market return is expected to be 12.
  Substitute those data by using CAPM equation.
• Calculate Expected Return according to CAPM
  model

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Problem

• There are three assets- X, Y Z with beta value
  of 0.5, 1.0 1.5 respectively. The risk free
  rate is assumed to be 5 and the market return is
  expected to be 15.
• calculate the expected return

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Illustrating the calculation of beta
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The Security Market Line (SML)Calculating
required rates of return

• SML ki kRF (kM kRF) ßi
• Assume kRF 8 and kM 15.
• The market (or equity) risk premium is RPM kM
  kRF 15 8 7.

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Illustrating the Security Market Line
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Factors that change the SML

• What if investors raise inflation expectations by
  3, what would happen to the SML?

ki ()
SML2
D I 3
SML1
18 15 11 8
Risk, ßi
0 0.5 1.0 1.5
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Factors that change the SML

• What if investors risk aversion increased,
  causing the market risk premium to increase by
  3, what would happen to the SML?

ki ()
SML2
D RPM 3
SML1
18 15 11 8
Risk, ßi
0 0.5 1.0 1.5
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Verifying the CAPM empirically

• The CAPM has not been verified completely.
• Statistical tests have problems that make
  verification almost impossible.
• Some argue that there are additional risk
  factors, other than the market risk premium, that
  must be considered.

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More thoughts on the CAPM

• Investors seem to be concerned with both market
  risk and total risk. Therefore, the SML may not
  produce a correct estimate of ki.
• ki kRF (kM kRF) ßi ???
• CAPM/SML concepts are based upon expectations,
  but betas are calculated using historical data.
  A companys historical data may not reflect
  investors expectations about future riskiness.