Chapter 5 Concepts and Issues: Return, Risk and Risk Aversion

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Chapter 5 Concepts and Issues Return, Risk and
Risk Aversion

• Objective To introduce key concepts related to
  risk, return, and portfolio theory.
• Determinants of interest rates
• The historical record
• Risk and risk aversion
• Portfolio risk

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1. Determinants of interest rates

• real (r), nominal (R) rates and inflation (i)
  R ? r i
• Example r 3, i 6 ? R ? 9
• Fisher effect Exact relation

Example r (0.09-0.06)/1.062.83
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Equilibrium real rate of interest
Supply (household) demand (business) Monetary
and fiscal policies determine the
governments net supply and demand.
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Equilibrium nominal rate of interest

• Rr E(i)
• Fisher effect one to one increase between
  inflation and nominal interest rate.
• ? Nominal interest rate predicts expected
  inflation.
• Fisher effect is correct, if real interest is
  constant. However, real interest changes
  unpredictably.
• Nominal interest rate is the sum of real interest
  rate plus a noisy forecast of inflation.

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2. Risk and risk premium

HPR Holding Period Return P0 Beginning
price P1 Ending price D1 Dividend paid during
the period Return capital gains dividend
yield The probability distribution summarize the
uncertainties in future price and dividend .
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mean, variance and risk premium
mean (expected) return E(r)?sp(s)r(s) variance
?2 ?s p(s)r(s)-E(r)2 p(s) probability
that scenario s will occur r(s) return if
scenario s occurs risk premiumrisky return
risk free return (reward to bearing risk) How
much does an investor bear? It depends on risk
aversion.
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Example

• E(r) (.1)(-.05)(.2)(.05)...(.1)(.35) 0.15
• ?2 (.1)(-.05-.15)2(.2)(.05- .15)2
• 0.01199
• (0 .01199)1/2 0.1095

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Historical record
Annual returns in Canada (1957-2003)
Normal distribution mean-? ltretltmean? 68
-5.8, 27.1 mean-2?ltretltmean2? 95
-22.3, 43.6
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Annual returns in U.S. (1926-2002)
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5. Return distributions and value at risk

• The skewness of a distribution is a measure of
  the asymmetry of the distribution
• Normal distribution is a symmetric distribution.
• It is equal to the average cubed deviation from
  the mean
• Canadian stock returns are negatively skewed,
  implying that large losses are more probable than
  for a normal distribution

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Value at Risk (VaR)

• Value at risk (VaR) is another measure of risk
• It represents the potential loss from extreme
  negative returns
• The 5 percent VaR is equal to the 5 quantile of
  the cumulative distribution.
• Suppose the 5 VaR is -6.9. It means one can
  expect a loss equal to or greater than 6.9 with
  a 5 probability.

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6 7 Global view and the long run

• Over the entire 20th century the performance in
  Canadian financial markets has been very close to
  those of the US and other industrialized
  countries, in both risk and return
• There is some concern that the large observed
  average risk premium for stocks cannot persist in
  the long run
• US equity premium (Fama and French, 2000)
• 1872-1949 4.6
• 1950-1999 8.4

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8. Risk and risk aversion
A. risky payment
E(W) pW1 (1-p)W2 0.5m s2 pW1 - E(W)2
(1-p) W2 - E(W)2 s2 0.25 and s
0.5 B. riskfree, certain payment of 0.5m.
Risk averse prefer A Risk neutral
indifferent Risk loving prefer B
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Risk Aversion Utility

• There is always trade-off between risk and
  return. A utility function is one way to model
  the trade-off.
• Utility Function
• U E(r) 0.5 A s2
• A is a measure of risk aversion

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Dominance Principle
2 dominates 1 has a higher return 2
dominates 3 has a lower risk 4 dominates 3
has a higher return
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Indifference Curves

• Represent an investors willingness
• to trade-off return and risk

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9. Portfolio risk

• rule 1 mean return E(r)?sp(s)r(s)
• rule 2 variance ?2 ?s p(s)r(s)-E(r)2
• rule 3 portfolio return Erp w1 Er1 w2 Er2
• rule 4 ?p wrisky asset ?risky asset
• stdev of a portfolio with a risk-free asset
• rule 5 variance of a portfolio