Outlines

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Part 2-3 ??

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Outlines

• Statistical calculations of risk and return
  measures
• Risk Aversion
• Systematic and firm-specific risk
• Efficient diversification
• The Capital Asset Pricing Model
• Market Efficiency

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Rates of Return Single Period

• HPR Holding Period Return
• P0 Beginning price
• P1 Ending price
• D1 Dividend during period one

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Rates of Return Single Period Example

• Ending Price 48
• Beginning Price 40
• Dividend 2
• HPR (48 - 40 2 )/ (40) 25

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Return for Holding Period Zero Coupon Bonds

• Zero-coupon bonds are bonds that are sold at a
  discount from par value.
• Given the price, P (T ), of a Treasury bond with
  100 par value and maturity of T years

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Example - Zero Coupon Bonds Rates of Return
Horizon, T Price, P(T) 100/P(T)-1 Risk-free Return for Given Horizon
Half-year 97.36 100/97.36-1 .0271 rf(.5) 2.71
1 year 95.52 100/95.52-1 .0580 rf(1) 5.80
25 years 23.30 100/23.30-1 3.2918 rf(25) 329.18
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Formula for EARs and APRs

• Effective annual rates, EARs
• Annual percentage rates, APRs

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Table - Annual Percentage Rates (APR) and
Effective Annual Rates (EAR)
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Continuous Compounding

• Continuous compounding, CC
• rCC is the annual percentage rate for the
  continuously compounded case
• e is approximately 2.71828

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Characteristics of Probability Distributions

• Mean
• most likely value
• Variance or standard deviation
• Skewness

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Mean Scenario or Subjective Returns

• Subjective returns
• ps probability of a state
• rs return if a state occurs

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Variance or Dispersion of Returns

• Subjective or Scenario
• Standard deviation variance1/2
• ps probability of a state
• rs return if a state occurs

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Deviations from Normality

• Skewness
• Kurtosis

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Figure - The Normal Distribution
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Figure - Normal and Skewed (mean 6 SD 17)
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Figure - Normal and Fat Tails Distributions (mean
.1 SD .2)
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Spreadsheet - Distribution of HPR on the Stock
Index Fund
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Mean and Variance of Historical Returns

• Arithmetic average or rates of return
• Variance
• Average return is arithmetic average

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Geometric Average Returns

• Geometric Average Returns
• TV Terminal Value of the Investment
• rG geometric average rate of return

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Spreadsheet - Time Series of HPR for the SP 500
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Example - Arithmetic Average and Geometric
Average
Year 1 2 3 4
Return 10 -5 20 15
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Measurement of Risk with Non-Normal Distributions

• Value at Risk, VaR
• Conditional Tail Expectation, CTE
• Lower Partial Standard Deviation, LPSD

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Figure - Histograms of Rates of Return for
1926-2005
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Table - Risk Measures for Non-Normal Distributions
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Investors View of Risk

• Risk Averse
• Reject investment portfolios that are fair games
  or worse
• Risk Neutral
• Judge risky prospects solely by their expected
  rates of return
• Risk Seeking
• Engage in fair games and gamble

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Fair Games and Expected Utility

• Assume a log utility function
• A simple prospect

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Fair Games and Expected Utility (cont.)
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Diversification and Portfolio Risk

• Sources of uncertainty
• Come from conditions in the general economy
• Market risk, systematic risk, nondiversifiable
  risk
• Firm-specific influences
• Unique risk, firm-specific risk, nonsystematic
  risk, diversifiable risk

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Diversification and Portfolio Risk Example
Normal Year for Sugar Normal Year for Sugar Abnormal Year
Bullish Stock Market Bearish Stock Market Sugar Crisis
.5 .3 .2
Best Candy 25 10 -25
SugarKane 1 -5 35
T-bill 5 5 5
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Diversification and Portfolio Risk Example (cont.)
Portfolio Expected Return Standard Deviation
All in Best 10.50 18.90
Half in T-bill 7.75 9.45
Half in Sugar 8.25 4.83
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Components of Risk

• Market or systematic risk
• Risk related to the macro economic factor or
  market index.
• Unsystematic or firm specific risk
• Risk not related to the macro factor or market
  index.
• Total risk Systematic Unsystematic

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Figure - Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
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Figure - Portfolio Diversification
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Two-Security Portfolio Return

• Consider two mutual fund, a bond portfolio,
  denoted D, and a stock fund, E

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Two-Security Portfolio Risk

• The variance of the portfolio, is not a weighted
  average of the individual asset variances
• The variance of the portfolio is a weighted sum
  of covariances

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Table - Computation of Portfolio Variance from
the Covariance Matrix
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Covariance and Correlation Coefficient

• The covariance can be computed from the
  correlation coefficient
• Therefore

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Example - Descriptive Statistics for Two Mutual
Funds
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Portfolio Risk and Return Example

• Apply this analysis to the data as presented in
  the previous slide

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Table - Expected Return and Standard Deviation
with Various Correlation Coefficients
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Figure - Portfolio Opportunity Set
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Figure - The Minimum-Variance Frontier of Risky
Assets
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Figure - Capital Allocation Lines with Various
Portfolios from the Efficient Set
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Capital Allocation and the Separation Property

• A portfolio manager will offer the same risky
  portfolio, P, to all clients regardless of their
  degree of risk aversion
• Separation property
• Determination of the optimal risky portfolio
• Allocation of the complete portfolio

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Capital Asset Pricing Model (CAPM)

• It is the equilibrium model that underlies all
  modern financial theory.
• Derived using principles of diversification with
  simplified assumptions.
• Markowitz, Sharpe, Lintner and Mossin are
  researchers credited with its development.

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Figure - The Efficient Frontier and the Capital
Market Line
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Slope and Market Risk Premium

• Market risk premium
• Market price of risk, Slope of the CML

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The Security Market Line

• Expected return beta relationship
• The securitys risk premium is directly
  proportional to both the beta and the risk
  premium of the market portfolio
• All securities must lie on the SML in market
  equilibrium

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Figure - The Security Market Line
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Sample Calculations for SML

• Suppose that the market return is expected to be
  14, and the T-bill rate is 6
• Stock A has a beta of 1.2
• If one believed the stock would provide an
  expected return of 17

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Efficient Market Hypothesis (EMH)

• Do security prices reflect information ?
• Why look at market efficiency?
• Implications for business and corporate finance
• Implications for investment

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Random Walk and the EMH

• Random Walk
• Stock prices are random
• Randomly evolving stock prices are the
  consequence of intelligent investors competing to
  discover relevant information
• Expected price is positive over time
• Positive trend and random about the trend

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Random Walk with Positive Trend
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Random Price Changes

• Why are price changes random?
• Prices react to information
• Flow of information is random
• Therefore, price changes are random

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Figure - Cumulative Abnormal Returns before
Takeover Attempts Target Companies
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EMH and Competition

• Stock prices fully and accurately reflect
  publicly available information.
• Once information becomes available, market
  participants analyze it.
• Competition assures prices reflect information.

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Forms of the EMH

• Weak form EMH
• Semi-strong form EMH
• Strong form EMH

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Information Sets