CHAPTER 8 Risk and Rates of Return

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CHAPTER 8Risk and Rates of Return

• Outline
• Stand-alone return and risk
• Return
• Expected return
• Stand-alone risk
• Portfolio return and risk
• Portfolio return
• Portfolio risk
• Beta
• Link Risk return
• CAPM
• Security Market Line

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I-1 Return What is my reward of investing?
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Investment returns

• If 1,000 is invested and 1,100 is returned
  after one year, the rate of return for this
  investment is
• (1,100 - 1,000) / 1,000 10.
• The rate of return on an investment can be
  calculated as follows
• (Amount received Amount invested)
• Return ________________________
• 
  Amount invested

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Rates of Return stocks

HPR Holding Period Return P1 Ending price P0
Beginning price D1 Dividend during period
one Define return? Your gain per dollar
investment
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Rates of Return Example

• Ending Price 24
• Beginning Price 20
• Dividend 1
• HPR ( 24 - 20 1 )/ ( 20) 25

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I-2 Expected return describe the uncertainty
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Calculating expected return

• Two scenarios and the concept of expected return
• Extending to more than two scenarios

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Investment alternatives
Economy Prob. T-Bill HT Coll USR MP
Recession 0.1 5.5 -27.0 27.0 6.0 -17.0
weak 0.2 5.5 -7.0 13.0 -14.0 -3.0
normal 0.4 5.5 15.0 0.0 3.0 10.0
strong 0.2 5.5 30.0 -11.0 41.0 25.0
Boom 0.1 5.5 45.0 -21.0 26.0 38.0
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Calculating the expected return
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Summary of expected returns

• Expected return
• HT 12.4
• Market 10.5
• USR 9.8
• T-bill 5.5
• Coll. 1.0
• HT has the highest expected return, and appears
  to be the best investment alternative, but is it
  really? Have we failed to account for risk?

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I-3. Stand-alone risk
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Calculating standard deviation
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Standard deviation for each investment
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Comparing standard deviations
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Comments on standard deviation as a measure of
risk

• Standard deviation (si) measures total, or
  stand-alone, risk.
• The larger si is, the lower the probability that
  actual returns will be closer to expected
  returns.
• Larger si is associated with a wider probability
  distribution of returns.

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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 a risk free asset, which
  serves as compensation for investors to hold
  riskier securities.

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Comparing risk and return
Security Expected return, r Risk, s
T-bills 5.5 0.0
HT 12.4 20.0
Coll 1.0 13.2
USR 9.8 18.8
Market 10.5 15.2

Seem out of place.
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Selected Realized Returns, 1926 2001

• Average Standard
• Return Deviation
• Small-company stocks 17.3 33.2
• Large-company stocks 12.7 20.2
• L-T corporate bonds 6.1 8.6
• Source Based on Stocks, Bonds, Bills, and
  Inflation (Valuation Edition) 2002 Yearbook
  (Chicago Ibbotson Associates, 2002), 28.

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Coefficient of Variation (CV)

• A standardized measure of dispersion about the
  expected value, that shows the risk per unit of
  return.

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Risk rankings, by coefficient of variation

• CV
• T-bill 0.0
• HT 1.6
• Coll. 13.2
• USR 1.9
• Market 1.4

• Collections has the highest degree of risk per
  unit of return.
• HT, despite having the highest standard deviation
  of returns, has a relatively average CV.

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II Risk and return in a portfolio
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Portfolio constructionRisk and return

• Assume a two-stock portfolio is created with
  50,000 invested in both HT and Collections.

• Expected return of a portfolio is a weighted
  average of each of the component assets of the
  portfolio.
• Standard deviation is a little more tricky and
  requires that a new probability distribution for
  the portfolio returns be devised.

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II-1. Portfolio return
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Calculating portfolio expected return
Economy Prob. HT Coll Port.
Recession 0.1 -27.0 27.0
weak 0.2 -7.0 13.0
normal 0.4 15.0 0.0 7.5
strong 0.2 30.0 -11.0
Boom 0.1 45.0 -21.0
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Calculating portfolio expected return
Economy Prob. HT Coll Port.
Recession 0.1 -27.0 27.0 0.0
weak 0.2 -7.0 13.0 3.0
normal 0.4 15.0 0.0 7.5
strong 0.2 30.0 -11.0 9.5
Boom 0.1 45.0 -21.0 12.0
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An alternative method for determining portfolio
expected return
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II-2. Portfolio risk and beta
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Calculating portfolio standard deviation and CV
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Comments on portfolio risk measures

• sp 3.4 is much lower than the si of either
  stock (sHT 20.0 sColl. 13.2).
• sp 3.4 is lower than the weighted average of
  HT and Coll.s s (16.6).
• Therefore, the portfolio provides the average
  return of component stocks, but lower than the
  average risk.
• Why? Negative correlation between stocks.

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Returns distribution for two perfectly negatively
correlated stocks (? -1.0)
40
40
15
15
-10
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Returns distribution for two perfectly positively
correlated stocks (? 1.0)
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Creating a portfolioBeginning with one stock
and adding randomly selected stocks to portfolio

• sp decreases as stocks added, because they would
  not be perfectly correlated with the existing
  portfolio.
• Expected return of the portfolio would remain
  relatively constant.
• Eventually the diversification benefits of adding
  more stocks dissipates (after about 10 stocks),
  and for large stock portfolios, sp tends to
  converge to ? 20.

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Illustrating diversification effects of a stock
portfolio
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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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Beta

• Measures a stocks market risk, and shows a
  stocks volatility relative to the market.
• Indicates how risky a stock is if the stock is
  held in a well-diversified portfolio.
• Portfolio beta is a weighted average of its
  individual securities beta

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Calculating betas

• Run a regression of past returns of a security
  against past returns on the market.
• The slope of the regression line is defined as
  the beta coefficient for the security.

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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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III CAPM
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What risk do we care?

• Stand alone?
• Risk that can not be diversified?

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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.

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Capital Asset Pricing Model (CAPM)

• Model linking risk and required returns. CAPM
  suggests that a stocks required return equals
  the risk-free return plus a risk premium that
  reflects the stocks risk after diversification.
• ri rRF (rM rRF) bi
• Risk premium RP additional return to take
  additional risk
• The market (or equity) risk premium is (rM rRF)
  

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Calculating required rates of return

• rHT 5.5 (5.0)(1.32)
• 5.5 6.6 12.10
• rM 5.5 (5.0)(1.00) 10.50
• rUSR 5.5 (5.0)(0.88) 9.90
• rT-bill 5.5 (5.0)(0.00) 5.50
• rColl 5.5 (5.0)(-0.87) 1.15

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Applying CAPM

• Portfolio beta Beta of a portfolio is a weighted
  average of its individual securities betas.
• Computing other variables risk free rate, market
  return, market risk premium
• Computing the difference of return between two
  stocks.
• Computing price in the future when current price
  is given

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CAPM in a graph the Security Market Line
SML ri 5.5 (5.0) bi
ri ()
SML
.
HT
.
.
rM 10.5 rRF 5.5
.
USR
T-bills
.
Risk, bi
-1 0 1 2
Coll.
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Applying CAPM in real world(optional)

• Total Risk vs. Beta. An experiment
• The difference between commonly referred risk and
  beta (Are these high beta stocks really high
  beta)
• High risk( total risk), low beta stock can hedge
  your portfolio (reduce portfolio risk)

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Problems with CAPM (optional)

• Measurement error of beta
• Empirical relationship between beta and return is
  weak
• Size and Book-to-market factors
• Momentum

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Optional diversification in real world

• Stock Index ETF
• Style Value vs. Growth
• Style Small vs. Big
• Performance, Risk, Expense(0.1 is low, 0.5 is
  about average)
• Examples
• Vanguard Small Cap Value ETF  VBR
• Small growth VBK
• Large value VTV
• Large growth VUG

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diversification in real world

• Foreign ETFRBL
• Pros
• More diversification
• Low PE ratio
• cons
• Higher risk
• Higher expense 0.6 vs. 0.1
• Higher spread
• Poor prior performance