Chapter 13: Return, Risk, and the Security Market Line

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Chapter 13Return, Risk, and the Security Market
Line

• 2504 Essentials of Business Finance

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Risk and Return Simple Rule

• The required return depends on the risk of the
  investment.
• There is a reward for bearing risk.
• The greater the risk, the greater the potential
  reward.

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Returns

• Two components of return on investment
• Income component cash you receive while owning
  the investment, e.g., dividends.
• Capital gain/loss due to the changes in the
  value of the asset.

where P0 price of the stock at time 0 Pt
price of the stock at time t D0 dividend paid
on the stock during the time period 0 t.
The above is the nominal rate of return. The
real rate of return (nominal rate of return
1)/(inflation rate 1) -1.
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13.1 Expected Returns

• Suppose you have invested in Stock L and U.
• Stock L
• The return will be 70 (!!!) if the economy is
  good and
• -20 if the economy is bad.
• Stock U
• The return will be 10 if the economy is good
  and
• 30 if the economy is bad.
• There is a 50 of chance that the economy is good.

Risk Premium For Stock L, 25 - 10 15 For
Stock U, 20 - 10 10
Ojvalue of the jth outcome Pj associated
probability of occurrence
In general,
Expected Returns Given the above projections,
what do I expect to earn in the future? If I
assume everything is normally distributed, then I
can simply state that, on average, I expect to
earn an average of what I expect to occur.
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13.1 How about Variance?
In general,
Variances/Standard Deviations How risky is the
expected return? What is the likelihood of
actually earning the expected return? Measure the
squared deviation of each observation from the
expected return.
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13.2 Portfolios
Group of assets held by an investor

• You put a half of your money in Stock L and the
  other half in Stock U. The return on this
  portfolio is.
• If the economy is good,

Return on Stock L

• If the economy is bad,

Return on Stock U
Therefore
Portfolio Weight
Probability
In general
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13.2 Portfolios
Alternative way to calculate the expected return
on the portfolio
E(R) on Stock L
E(R) on Stock U
Portfolio Weight
In general,
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13.2 Portfolio Variance?
Variance on Stock L
Variance on Stock U
Portfolio Weight
WRONG!!! WHY?
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13.2 Portfolio Variance?
Right Answer
Return on p in this case
Return on p in this case
E(Rp)
E(Rp)
Probability of bad economy
Probability of good economy
In general
So, what kind of relationship exists between
portfolio variance and variance of each stock
comprising the portfolio?
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13.2 Portfolio Variance?
Alternative way to calculate portfolio variance
Covariance
Correlation Coefficient
Now, we know why the first answer is wrong it
ignores the effect of correlation on the
portfolio.
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13.2 Covariance Correlation Coefficient
In general
Covariance How do these 2 securities vary
around their expected values, relative to each
other? If A is above its expected value in a
given month, what can we say about B? Correlation
Coefficient How do these two securities vary
relative to each other? Contrary to the
covariance measure above, we want to find out
what happens to B if A increases or decreases. If
we run a simple Ordinary Least Squares regression
of A on B. That is The extent to which the
returns on two assets move together. -1.0 lt
CORR lt 1.0
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13.2 Correlation b/w Stock L and U
Stock L and U have a perfect negative
correlation, i.e., CORR -1.0. Note CORR is
NOT the slope of this curve, but the degree to
which the returns on stock L explain the return
on stock U (it is the square root of R2)
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13.2 Positive Correlation
CORRA,B gt 0, A and B are positively correlated,
i.e., when A increases, B increases.
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13.2 Negative Correlation
CORRA,B lt 0, A and B are negatively correlated,
i.e., when A increases, B decreases.
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13.2 No Correlation
CORRA,B 0, A and B are uncorrelated.
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13.2 Portfolio Variance and Correlation
Now, lets calculate the portfolio variance using
the formula
E(RL) 25, Variance 20.25. E(RU) 20,
Variance 1.00. E(Rp) 22.5, Variance
3.0325, thanks to the negative CORR. We can
reduce the risk of investments while keeping
relatively high return, by constructing a
portfolio Diversification Effect.
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13.2 Diversification

• There are benefits to diversification whenever
  the correlation between two stocks is less than
  perfect (p lt 1.0)

When CORR -1.0, it is possible to create a
zero-variance portfolio!
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13.4 Systematic Risk

• Risk affecting the economy/industry as a whole
• Also known as non-diversifiable risk or market
  risk
• Includes such things as changes in GDP,
  inflation, interest rates, etc.

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13.4 Unsystematic / Firm Specific Risk

• Risk affecting just one company
• Also known as unique risk and asset-specific risk
• Includes such things as labor strikes, shortages,
  etc.

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13.5 Portfolio Diversification
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13.5 Portfolio Diversification
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13.5 The Principle of Diversification

• Diversification can substantially reduce the
  variability of returns 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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13.5 Diversifiable (Unsystematic) Risk

• The risk that can be eliminated by combining
  assets into a portfolio
• Synonymous with unsystematic, unique or
  asset-specific risk
• If we hold only one asset, or assets in the same
  industry, then we are exposing ourselves to risk
  that we could diversify away
• The market will not compensate investors for
  assuming unnecessary risk

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13.5 Total Risk

• Total risk systematic risk unsystematic risk
• The standard deviation of returns is a measure of
  total risk
• For well diversified portfolios, unsystematic
  risk is very small
• Consequently, the total risk for a diversified
  portfolio is essentially equivalent to the
  systematic risk

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13.5 Diversification

• Portfolio diversification is the investment in
  several different asset classes or sectors
• Diversification is not just holding a lot of
  assets
• For example, if you own 50 internet stocks, you
  are not diversified
• However, if you own 50 stocks that span 20
  different industries, then you are diversified

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Review

• The total risk associated with an asset can be
  decomposed into two components systematic and
  unsystematic risk.
• Unsystematic can be essentially eliminated by
  diversification.
• Systematic risk cannot be eliminated by
  diversification.
• Systematic risk principle the reward for bearing
  risk depends only on the systematic risk of an
  investment.
• Because unsystematic risk can be eliminated at
  virtually no cost (by diversifying), there is no
  reward for bearing it. The market does not
  reward risks that are born unnecessarily.
• The expected return on an asset depends only on
  that assets systematic risk.
• Since we will only be compensated for the
  systematic risk, is there a way to determine how
  much a particular stock contributes to the
  overall systematic risk of a portfolio? In other
  words, since we use portfolios in order to
  diversify away the risks that are specific to the
  firm, what is left over and how does it impact on
  the portfolio?

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13.7 The Capital Asset Pricing Model

• Gives us (1) a method of measuring the
  systematic risk of an individual stock and (2) a
  way to predict the required return for a security
  given its systematic risk (ties us directly to
  chapter 14, where we need to determine what the
  cost of raising funds in the equity market will
  be).
• The risk associated with a well-diversified
  portfolio comes from the market, or systematic
  risk of the securities in the portfolio.
• Market risk specifically refers to the
  sensitivity of an individual securitys returns
  to market-wide movements. The beta (?) of a
  security measures its responsiveness to market
  movements, where the market is usually proxied by
  an aggregate stock index like the SP 500 or
  SP/TSX Composite.

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13.6 What is ??
Ri
Slope ?
Rm

• Plot individual stock (excess) returns against
  the (excess) returns on the overall market. This
  is called the characteristic line and the slope
  that it generates is ?.
• ? can also be calculated by considering that it
  measures the responsiveness of a securitys
  returns relative to the movement in returns of
  the benchmark or index. Therefore, we can obtain
  the covariance of an individual assets returns
  and the market, and compare this to the variance
  of the market. In other words

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13.6 Volatility High and Low Betas
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13.6 Examples of Beta Coefficients
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13.6 Systematic Risk and Beta

• Beta Coefficient (ß) how much systematic risk a
  particular asset has relative to an average
  asset.
• ? ß, ? systematic risk, ? expected return.
• You put half of your money in Bank of Nova Scotia
  and half in Nortel Networks. What would the beta
  of this portfolio be?
• ?p 0.50 ßBNS 0.50 ßNortel
• 0.50 1.19 0.50 3.81
• 2.50
• Note A portfolio beta can be calculated just
  like a portfolio expected return, i.e., we do not
  have to consider the correlation. WHY?

Portfolio weight
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The Security Market Line
R
Rm
Rf
?
? m 1

• Where ? measures the riskiness (systematic) of an
  individual security relative to the riskiness of
  the market as a whole. For every security within
  the market portfolio, there will be a covariance
  with the market portfolio. Therefore, there will
  be a beta for every security within the market as
  well. If we calculate all the betas for all the
  securities and plot them against their individual
  returns, we obtain the Security Market Line
  (SML). 
• Since beta measures riskiness relative to the
  market, ?m will measure the markets riskiness
  relative to itself, or 1 to 1. Therefore, the
  beta on the market is 1. The beta of the
  risk-free assets (such as T-bills) are 0.

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The Security Market Line
R
The equation for the line (CAPM)
Rm
Rf
?
? m 1

• In order to find out the required return of an
  individual security (the return that you feel you
  need given the market risk associated with that
  stock), simply solve for the return in the above
  graphical relationship. The equation for the line
  in the graphical relationship above is your
  typical Y MX B. Y is R above, the intercept,
  B, is the risk-free rate, X is the ? above, and
  the slope M is the rise over the run. Rise is (Rm
  Rf), and run is (?m 1).
• Once you have distributions of returns, you can
  calculate individual betas. Once you have
  individual betas, you can calculate the return
  investors require given the systematic risk they
  must assume. Remember investors are always able
  to diversify away all the firm-specific risk. As
  such, the market will not compensate them for it.

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13.7 Security Market Line

• An expected return of Asset A (E(RA)) is 20
• A Beta of Asset A (ßA) is 1.6
• The risk free rate (Rf) is 8.
• Consider a portfolio made up of Asset A and a
  risk-free asset (e.g., T-bill).
• If 25 of the portfolio is invested in Asset A,
  the expected return on the portfolio is
• E(Rp) 0.25E(RA) (1 - 0.25) Rf
• 0.250.20 (1 - 0.25) 0.08
• 0.11
• The beta on the portfolio (ßp) is
• ßp 0.25 ßA (1 - 0.25)0
• 0.251.6
• 0.40
• If 50 of the portfolio is invested in Asset A,
  the expected return on the portfolio is..if
  75?...if 150?

Portfolio weight
Why is the beta for risk free asset zero?
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13.7 Portfolio Weights, Beta, and Expected Return
Investor borrows at the risk-free rate
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13.7 Portfolio Expected Returns and Betas

• Slope risk premium on Asset A / Asset As beta
• Asset A offers a reward-to-risk ratio of 7.5
  (Asset A has a risk premium of 7.5 per unit of
  systematic risk).

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13.7 Reward-to-Risk Ratio

• Asset A E(RA) 20 (ßA) is 1.6
• Asset B E(RB) 16 (ßB) is 1.2
• Rf 8.
• Reward-to-risk ratio for Asset A
• (E(RA) Rf)/ ßA (20 - 8)/1.6 7.5
• Reward-to-risk ratio for Asset B
• (E(RB) Rf)/ ßB (16 - 8)/1.2 6.67
• Who prefers Asset B?
• The reward-to-risk ratio must be the same for all
  the assets in the market.

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13.7 Security Market Line
Describe the relationship between systematic risk
and expected return in financial markets.
Market Risk Premium

• Market portfolio a portfolio made up of all of
  the assets in the market.
• Market portfolio has a beta of one (since it is
  representative of all the assets in the market,
  it must have average systematic risk).

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13.7 The Capital Asset Pricing Model

• The expected return for a particular asset
  depends on
• Pure time value of money. As measured by the
  risk free rate, this is the reward for waiting
  for your money, without taking any risk.
• Reward for bearing systematic risk As measured
  by the market risk premium, this component is the
  reward the market offers for bearing an average
  amount of systematic risk.
• Amount of systematic risk As measured by beta,
  this is the amount of systematic risk present in
  a particular asset, relative to an average asset.
  

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See you!