Chapter 2: Portfolio theory

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Chapter 2Portfolio theory

• Derives the principles by which investors choose
  securities
• Provide theory for market prices and implications
  for portfolio selection
• Simple portfolio goal - Want high return and low
  risk
• What is return and risk?
• Introduce tools to be used through the class

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Different concepts of risk and return

• Rates of Return
• Holding period
• Cash flow adjusted
• Statistical
• Time weighted
• Internal rate of return
• Risk
• Variance or standard deviation
• Coefficient of variation

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Holding period return

• Total return from an investment from all sources
  including price appreciation and income
• The holding period is not specifically defined

• NOTE We usually state returns net of principal
  invested (V0)

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When is holding period return not adequate?

• Bill, Hillary, and Chelsea each started with 1
  million on January 1
• Each was given additional 500,000 to invest but
  at different times
• Bill was given the extra cash at the end of the
  1st quarter, Hillary at the end of the 2nd
  quarter, and Chelsea at the end of the 3rd qtr
• If all end up with 2 million at the end of the
  year, who did the best?

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Cash Flows Adjusted (CFA) Rate of Return

• We must account for cash flows in and out of the
  portfolio

• Note the formula above assumes that period is a
  month and that date is in days (see example on p.
  46)
• How do we adjust if we have quarterly info?

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Statistical rate of return

• Use arithmetic total and arithmetic mean

• Annualized return - multiply the average return
  per period by the number of periods per year
• Statistical rate of return is best estimate of
  one-period return

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Time weighted rate of return

• Assumes the portfolio is reinvested every period
• Measure of average return over multiple periods
• Use geometric total and geometric averages

• Annualized return - Take the per period average
  return and raise it to the appropriate power

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A word of caution

• There is not an industry standard that must be
  used when publishing rates of return
• This is important when comparing different
  investments

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Internal rate of return (IRR)

• IRR is the interest rate that will equate the
  present value of cash flows with the initial
  market price
• To find IRR, write out the cash flows and use
  trial and error
• Use Goal Seek in Excel

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Risk - some preliminaries

• Risk is uncertainty of future outcomes or returns
• It is assumed that investors are risk averse
• Investors require compensation by way of higher
  expected returns to accept more risk
• Experiment

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Standard deviation

• Measure of spread between good outcomes and bad
  outcomes
• Higher standard deviation means higher
  probability of loss
• If mean is the same

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Standard deviation

• The standard deviation is the square root of the
  variance
• In expectations

• If using historical data

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Coefficient of variation (CV)

• Used as a standardized measure of risk to compare
  assets with two different returns
• It is the amount of risk per unit of return

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Portfolio Math

• Risk and return of combination of two securities
• If wi, wj are the percentage of the portfolio
  invested in securities i and j, respectively

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Example

• Consider two securities, A and B
• rA 8, rB 20, sA 10, sB 40
• Suppose we hold wA80 and wB1-wA20

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Efficient frontier

• A portfolio is efficient if increasing return
  must increase risk
• For a given risk level, it maximizes return
• For a given return level, it minimizes risk

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Question?

• Compare the following portfolios(A) 100
  invested in Microsoft stock (B) 50 in Microsoft
  stock and 50 in Illinois Power bonds
• Which has more risk? Give two reasons for your
  conclusion

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Example

• There are two firms, Dime-a-Beer Co. and Fancy
  Liquor Inc.

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Diversification

• Holding many stocks reduces risk because stock
  prices and returns are imperfectly correlated
• Dont put all your eggs in one basket
• As long as the assets are not perfectly
  positively correlated (rho 1), there are
  benefits to diversification

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Systematic vs. unsystematic risk

• Systematic risk reflects market wide forces
• No matter how many stocks you own, you cannot get
  rid of systematic risk
• Thus it is also known as market risk
• Unsystematic risk is firm specific and can be
  diversified away
• Also called residual or idiosyncratic risk

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Portfolio Diversification
nonsystematic risk
systematic risk
Lesson you need to have a portfolio with many
securities
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Estimating Systematic Risk of a Stock

• Suppose we assume that one economic factor is
  driving stock returns
• Use a proxy to represent the economy
• Return on the market or Rm
• How sensitive is a stock to the economy?
• How well does return on the market explain the
  return on our stock?

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Use the Tools, Data Analysis, Regression option
in Excel
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Factor models

• Statistical model to help distinguish systematic
  and firm specific risk
• When the market is proxied by an index (e.g., SP
  500), it is called an index model
• A regression that tries to explain how security
  returns correlate with market returns
• How well do market movements explain the
  securitys movements

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Explanation of variables

• Ri,t is security is return during period t
• Rm,t is the market return at time t
• Usually proxied by SP 500 or other index
• bi is security sensitivity to market return
• Result of regression is characteristic line
• R2 measures the fit of the regression
• NOTE Characteristic line is based on history

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More about beta

• Beta is a measure of systematic risk in an asset
  or portfolio
• What is beta of market? Of risk-free security?
• Is beta2 a risky portfolio?
• T-bills are usually used to proxy risk-free rate
• Beta of portfolio is weighted sum of individual
  asset betas

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Capital Asset Pricing Model

• CAPM provides a specific form for the risk-return
  trade-off
• Why is there a risk-return tradeoff?
• Risk premium is the excess return over the
  risk-free rate
• CAPM says investors are only compensated for
  taking systematic risk

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CAPM

• Individual securities risk premia must
  compensate investors in proportion to total
  portfolio (market) risk
• Expected return-risk relationship

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Security market line (SML)

• Graph the expected return-risk equation
• Slope is market risk premium
• y-intercept is rf
• Looks at individual securities
• Fairly priced assets plot on line

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Estimating beta

• Run regression between realized (excess) monthly
  return on market and (excess) monthly return on
  stock
• Include dividends
• Use T-bill as risk-free rate
• Market proxied by an index

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Arbitrage Pricing Theory

• CAPM specified risk-return trade-off based only
  on systematic or market risk
• APT suggests that there may be many factors that
  are important
• Unfortunately, APT does not specify the number of
  factors nor the exact factors
• If F1, F2, are the risk premia for various risk
  factors

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Summary Chapter 2

• Risk and return measures
• Risk reduction
• Diversification
• Return maximization / risk reduction
• Efficient frontier
• Risk-return tradeoff
• CAPM
• APT