Module 5

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Module 5

• Risk and Return

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Outline

• Define the return and risk
• Calculate the expected return and risk (standard
  deviation) of a single asset
• Calculate the expected return and risk (standard
  deviation) for a portfolio
• Understanding the concept of correlation between
  the returns on assets in a portfolio
• Explain the principle of diversification.
• Distinguish between systematic and non-systematic
  risk.
• Explain the capital asset pricing model (CAPM).

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Return on Investment

• Return is the total gain or loss of an investment
• Dollar return
• Rate of return
• Ex Income of an investment in security
  dividends or interest received, and increase or
  decrease in the market price of security

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Return Example
Pt 37.00 Pt1 40.33 Dt1 1.85
Per dollar invested we get 5 cents in dividends
and 9 cents in capital gainsa total of 14 cents
or a return of 14 per cent.
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Risk

• Risk is the degree of uncertainty associated with
  a future outcome. It is a chance of loss.
• Risk premium when investing in risky assets,
  reward is required by investors for bearing risk

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Sources of risk

• Firm-specific risk
• business risk the chance that the firm will not
  be able to cover its operating costs.
• Financial risk the chance that the firm will not
  be able to cover its financial obligations
• Shareholders specific risk
• interest rate
• liquidity
• Market the value of an investment will be
  affected by market factors that are independent
  of the investment (economic, political and
  social events)
• Unexpected event have huge impact on the value
  of the firm or a specific investment.

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Investors Attitude to Risk

• Risk-neutral investor one who neither likes nor
  dislikes risk
• Risk-averse investor one who dislikes risk
• Risk-seeking investor one who prefers risk

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Risk Attitudes (cont.)

• The standard assumption in finance theory is
  risk-aversion.
• This does not mean an investor will refuse to
  bear any risk at all. Rather an investor regards
  risk as something undesirable, but which may be
  worth tolerating if compensated with sufficient
  return.

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The Normal Distribution

• It is often assumed that an investments
  distribution of returns follows a normal
  distribution, so an investments distribution of
  returns can be fully described by its expected
  return and risk.

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Frequency of Returns on Ordinary Shares 19782002
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Monthly Returns on SP 500 (1928-1999)
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The Measurement of Risk

• The variance and standard deviation describe the
  dispersion (spread) of the potential outcomes
  around the expected value
• Greater dispersion generally means greater
  uncertainty and therefore higher risk

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The Measurement of Risk

• Measure variability of returns on the investment
• how much a particular return deviates from an
  average return (i.e. variance or standard
  deviation)
• The square root of Variance is standard deviation

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ExampleVariance
ABC Co. have experienced the following returns in
the last five years
Calculate the average return and the standard
deviation.
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ExampleVariance
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ExampleVariance
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Example

• Calculate the standard deviation of security
  returns on All Ordinaries Index (AOI) from 1996
  to 2000

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• Average return is 11.7
• The standard deviation of returns is 9.7
• Implication?

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Standard Deviation of Returns

• The importance of standard deviation of returns
  is that it assists in estimating the range of
  future possible outcomes.
• It follows that the higher the standard
  deviation of returns, the higher the range of
  possible outcomes, and hence the more risk that
  is associated with this stock-market investment.

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A 68 chance that future returns on the market
will lie between 2.0 and 21.4.
Source Frino et al. 2004, Introduction to
Corporate Finance.
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68.26 of the actual returns would be within /
1 standard deviations and 95.46 of the actual
returns would be within / 2 standard
deviations and 99.74 of the actual returns
would be within / 3 standard deviations.
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Return and Risk

• In the long term, the greater the risk, the
  greater the potential reward.

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The Historical Record
Conclusion Historically, the riskier the asset,
the greater the return.
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Measuring Risk and return using a probability
distribution

• There is uncertainty associated with returns from
  an investment.
• Probability is the chance that a given outcome
  will occur

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Probability Distributions

• A probability distribution is simply a listing of
  the probabilities and their associated outcomes
• Probability distributions are often presented
  graphically as in these examples

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Expected Return and Standard Deviation

• Expected returnthe weighted average of the
  distribution of
• possible returns in the future.

• Variance (or standard deviation) of returnsa
  measure of the dispersion of the distribution of
  possible returns.

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• Example Possible return and probabilities of
  occurrence of a companys share are as follows

The expected return is the mean of the
distribution 0.02 x 0.100.07 x0.250.12
x0.300.17 x0.250.22 x0.10 0.12
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ExampleCalculating Expected Return
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ExampleCalculating Variance
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Measure Risk Coefficient of variation

• The relationship between the risk and return of
  the investment is known as the coefficient of
  variation

The higher the CV the more is the risk per unit
of return
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Measure Risk Coefficient of variation

• The expected returns on two assets A and B are 6
  and 13 respectively.
• The standard deviations of returns are 8 and
  15 respectively
• Which asset is risky?

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Portfolios

• Investors usually invest in a number of assets (a
  portfolio) and will be concerned about the return
  and risk of their overall portfolio.
• The riskreturn trade-off for a portfolio is
  measured by the portfolios expected return and
  standard deviation, just as with individual
  assets.
• Markowitz(1952) developed portfolio theory as a
  normative approach to investment choice under
  uncertainty based on the following assumptions.

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Portfolio Theory

• Assumptions
• Investors perceive investment opportunities in
  terms of a probability distribution defined by
  expected return and risk.
• Investors expected utility is an increasing
  function of return and a decreasing function of
  risk (risk-aversion).
• Investors are rational

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Measuring Return for a Portfolio

• Portfolio Return (Rp) is a weighted average of
  all the expected returns of the assets held in
  the portfolio
• where wj the proportion of the portfolio
  invested in asset j
• n the number of securities in the
  portfolio.

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Portfolio Return Calculation

• Assume 60 per cent of the portfolio is invested
  in Security 1 and 40 per cent in Security 2. The
  expected returns of the securities are 0.08 and
  0.12 respectively. The Rp can be calculated as
  follows

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Example portfolio return

• Assume 50 per cent of portfolio in asset A and 50
  per cent in asset B.

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Measurement of Portfolio Risk

• E(Rp) (0.5 x E(RA)) (0.50 x E(RB)).
• But Var(Rp) ? (0.50 x Var(RA)) (0.50 x
  Var(RB)).
• use standard deviation formula

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Measurement of Portfolio Risk
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• From above example it is noted that the standard
  deviation of the expected return of the
  portfolio, 0.0245, is less than the weighted
  average of the standard deviations of A and B,
  0.173. Why?
• By combining assets in a portfolio, the risks
  faced by the investor can significantly change as
  the returns on assets invested do not increase or
  decrease at the same rate. In this case, the
  riskiness of one asset may tend to be canceled by
  that of another asset.

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Portfolio Risk Conclusion

• Portfolio risk depends on
• the proportion of funds invested in each asset
  held in the portfolio (w)
• the riskiness of the individual assets comprising
  the portfolio (?2)
• the relationship between each asset in the
  portfolio with respect to risk ( ??

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The formula of calculating portfolio risk
For a two-asset portfolio, the variance can be
calculated using the following formula
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Portfolio Risk Calculation

• Given the variances of security 1 and 2 are
  0.0016 and 0.0036 respectively and the ??1,2 is
  -0.5

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Relationship Measures

• Covariance
• Statistic describing the relationship between two
  variables.
• How to measure covariance?
• Correlation coefficient
• describes the goodness of fit about a linear
  relationship between two variables.

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Relationship Measures (cont.)

• The correlation is equal to the covariance
  divided by the product of the assets standard
  deviations.

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The Correlation Coefficient

• The correlation coefficient between returns of
  two assets can range from -1.00 to 1.00 and
  describes how the returns move together through
  time.

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• When the correlation coefficient (?12 ) is
• 1, the returns are said to be perfect
  positively correlated. This means that if the
  return on security 1 is high, then the return on
  security2 will also be high to precisely the same
  degree.
• -1, the returns are said to be perfect
  negatively correlated. This means that if the
  return on security 1 is high, then the return on
  security 2 will be paired with low returns on
  security.
• 0, which indicates the absence of a systematic
  relationship between the returns on the two
  securities.

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Example Two-Asset Case

• Two-asset share portfolio comprising
• 50.60 invested in BHP shares (company A)
• 49.40 invested in NAB shares (company B)
• BHP
• average monthly return (rA) 0.04
• standard deviation of monthly returns (?A)
  4.85
• NAB
• average monthly return (rB) 1.92
• standard deviation of monthly returns (?B)
  4.97

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Example Two-Asset Case

• The average monthly portfolio return on the BHP
  and NAP portfolio is
• This translates to an expected annual return for
  this portfolio of 11.64 (12 ? 0.97)

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Example Two-Asset Case

• The standard deviation of this portfolio depends
  on the degree and direction of correlation
  between the returns for BHP and NAB
• These two companies operate in different
  industries
• These industries are influenced by different
  company- and economy-specific factors
• Their returns are unlikely to be perfectly
  positively correlated

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Example Two-Asset Case

• Assuming the returns are perfect positively
  correlated (?AB 1), the standard deviation of
  the portfolio is
• The overall standard deviation of the portfolio
  is 4.91

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Example Two-Asset Case

• To show that there are no diversification
  benefits when the asset returns are perfectly
  positively correlated, we can calculate the
  weighted average standard deviation for the
  portfolio
• Thus, there is no risk reduction from forming a
  portfolio of perfectly correlated assets

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Example Two-Asset Case

• Assuming the returns are perfectly negatively
  correlated (?AB -1), the standard deviation of
  the portfolio is
• Here, risk has been completely eliminated, due to
  the negative correlation of asset returns

Source Bishop et al. (2004), Corporate Finance
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Reducing RiskThe Principle of Diversification

• Diversification can substantially reduce the
  variability of returns (i.e. risk) without an
  equivalent reduction in expected returns.
• This reduction in risk arises because worse than
  expected returns from one asset are offset by
  better than expected returns from another.
• However, there is a minimum level of risk that
  cannot be diversified away and that is the
  systematic portion.

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by combining assets with low or negative
correlation, we reduce the overall risk of the
portfolio
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Example of diversification

• The graph of monthly returns shows that BHP and
  CCL do not move in tandem
• i.e. they are not positively correlated, but
  are negatively correlated,
• which allows for offsetting of returns and
  reduction in risk
• Source Frino et al. 2004, introduction to
  corporate finance, 2nd edition, Pearson,
  Australia.

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Determinants of Portfolio Risk

• Portfolio risk/variance depends on
• the proportion of funds invested in each asset
  held in the portfolio (w)
• the riskiness of the individual assets comprising
  the portfolio (?2 )
• the relationship between each asset in the
  portfolio with respect to risk ( ??

• Portfolios that offer a high return and a low
  risk are considered
• to be efficient

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Gains from Diversification

• The gain from diversifying is closely related to
  the value of the correlation coefficient.
• The degree of risk reduction increases as the
  correlation between the rates of return on 2
  securities decreases.
• Combining two securities whose returns are
  perfectly positively correlated results only in
  risk averaging, and does not provide any risk
  reduction.

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Gains from Diversification (cont.)

• When the correlation coefficient is less than
  one, the third term in the portfolio variance
  equation is reduced, reducing portfolio risk.
• If the correlation coefficient is negative, risk
  is reduced even more.

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Gains from Diversification summary

• Heres the moral
• The lower the correlation, the more risk
  reduction (diversification) you will achieve.

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Diversification with Multiple Assets

• for every pairwise relationship in the portfolio
• With n assets there will be a n n matrix. The
  properties of the variancecovariance matrix are
• it will contain n2 terms, n are the variances of
  individual assets and the remaining (n2 n)
  terms are the covariances between the various
  pairs of assets in the portfolio.
• the two covariance terms for each pair of assets
  are identical
• it is symmetrical about the main diagonal which
  contains n variance terms

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Diversification with Multiple Assets (cont.)

• A portfolio becomes larger, the effect of the
  covariance terms on the risk of the portfolio
  will be greater than the effect of the variance
  terms.
• This is because the number of assets increases,
  the number of covariance terms increases much
  more rapidly than the number of variance terms.
• For a diversified portfolio, the variance of the
  individual assets contributes little to the risk
  of the portfolio.
• The portfolio risk depends largely on the
  covariances between the returns on the assets.

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Can all the risk of a well-diversified portfolio
be eliminated?
Unsystematic risk
Systematic risk
Portfolio Diversification
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Component of Total Risk Systematic and
Unsystematic Risk

• Total risk systematic unsystematic risk
• Systematic risk (market-related risk or
  non-diversifiable risk)
• that component of total risk that is due to
  economy-wide factors
• Unsystematic risk (unique risk, or diversifiable
  risk)
• that component of total risk that is unique to
  the firm and may be eliminated by diversification

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Systematic and Unsystematic Risk (cont.)

• Unsystematic risk is removed by holding a
  well-diversified portfolio.
• The returns on a well-diversified portfolio will
  vary due to the effects of market-wide or
  economy-wide factors.
• Systematic risk of a security or portfolio will
  depend on its sensitivity to the effects of
  these market-wide factors.

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• If you holds a portfolio of 50 stocks and is
  considering the addition of an extra stock to the
  portfolio. What will you concern? Its individual
  variance? Or just the covariance of this new
  stock with the portfolio?

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Risk of an Individual Asset in a Diversified
Portfolio

• Risk of an asset is largely determined by the
  covariance between the return on that asset and
  the return on the holders existing portfolio
  
• Well-diversified portfolios will be
  representative of the market as a whole, thus the
  risk of these portfolios will depend on the
  market risk of the securities included in the
  portfolio.

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Measuring Systematic (Market) Risk by Beta

• Relevant measure of risk is the covariance
  between the return on the asset and the return on
  the market Cov(Ri, Rm)
• The beta coefficient, a measure of a securitys
  systematic risk, describing the amount of risk
  contributed by the security to the market
  portfolio.
• Std Deviation Beta
• Security A 30 0.60
• Security B 10 1.20
• Security A has greater total risk but less
  systematic risk (more non-systematic risk) than
  Security B.

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Characteristic Line
Characteristic line
Beta slope of characteristic
line
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Beta

• What does beta tell us?
• - A beta of 1 implies the asset has the same
  systematic risk as the overall market.
• - A beta lt 1 implies the asset has less
  systematic risk than the overall market.
• Ex if the market moves by 10 in response to a
  market event, the stock will move by less than
  10
• - A beta gt 1 implies the asset has more
  systematic risk than the overall market.
• Ex if the market moves 10 the stock will move
  by more than 10

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Some benchmark betas

• ß 0.5 Stock is only half as volatile as the
  average relevant index
• ß 1.0 Stock has average risk
• ß 2.0 Stock is twice as volatile

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Beta Coefficients for Selected Companies
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Portfolio Beta

• portfolio beta (ß) is weighted average of
  individual asset betas
• where
• n number of assets in the portfolio
• wi proportion of the current market value
• of portfolio p constituted by the i th
  asset

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ExamplePortfolio Beta Calculations

Amount
Portfolio
Share
Invested
Weights
Beta

(1)
(2)
(3)
(4)
(3)
(4)
6 000
ABC Company
50
0.90
0.450
LMN Company
4 000
33
1.10
0.367
XYZ Company
2 000
17
1.30
0.217
Portfolio
12 000
100
1.034
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The Pricing of Risky Assets

• What determines the expected rate of return on an
  individual asset or particular investment?
• The two main factors for any investment are
• The perceived riskiness of the investment
• Investors need to be sufficiently compensated for
  taking on the risks associated with the
  investment
• The required returns on alternative investments
• An alternative way to look at this is that the
  required return is the sum of the RFR and a risk
  premium
• E(R) Risk Free Rate of Return Risk Premium

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Portfolio theory view of required rate of
return The Security Market Line

• Modern portfolio theory assumes that the required
  return is a function of the RFR, the market risk
  premium, and an index of systematic risk (i.e.
  BETA)
• This model is known as the Capital Asset Pricing
  Model (CAPM).
• It is also the equation for the Security Market
  Line (SML)

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Risk and Return Graphically
The Market Line
Rate of Return
RFR
Risk
b or s
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CAPM Security Market Line

• The security market line presents the
  relationship between the expected return of any
  security and its systematic risk(ß )
• SML depicts the required return for each level of
  ß
• SML is upward-sloping in (ß, Ri) space
• Slope E(RM) Rf market risk premium

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The Capital Asset Pricing Model Security Market
Line (SML)
Asset expectedreturn (E (Ri))
market risk premium
E (RM) Rf
E (RM)
Rf
Assetbeta (?i)
M 1.0
How to calculate the expected return on an asset
or a portfolio invested?
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The Capital Asset Pricing Model (CAPM)

• An equilibrium model of the relationship between
  systematic risk and expected return.
• What determines an assets expected return?
• The risk-free ratethe pure time value of money.
• The market risk premiumthe reward for bearing
  systematic risk.
• The beta coefficienta measure of the amount of
  systematic risk present in a particular asset.

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The Capital Asset Pricing Model (CAPM)
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• CAPM Example
• Suppose the Treasury bond rate is 4, the average
  return on the All Ords Index is 11, and XYZ has
  a beta of 1.2. According to the CAPM, what
  should be the required rate of return on XYZ
  shares?
• Rj Rf ßj( Rm Rf )
• Here
• Rf 4
• Rm 11
• ßj 1.2
• Rj 4 1.2 x ( 11 4 )
• 12.4
• According to the CAPM, XYZ shares should be
  priced to give a 12.4 return

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Implication of CAPM

• The capital market will only reward investors for
  bearing risk that cannot be eliminated by
  diversification.
• CAPM states the reward for bearing systematic
  risk is a higher expected return.

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Summary

• Portfolio theory tells us that diversification
  reduces risk.
• Diversification works best with negative or low
  positive correlations between assets and asset
  classes.
• Risk can be divided into two categories
• Systematic risk-cannot be diversified away.
• Unsystematic risk-can be diversified away.
• Systematic risk of an asset is measured by the
  assets Beta. Risk of asset is relative to market.

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Summary (cont.)

• CAPM provides the relationship between risk and
  expected return for risky assets.
• CAPM uses assets beta and assumes linear
  relationship between expected return and risk
  relative to market, measured by beta.