Return and Risk: Analyzing the Historical Record

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Chapter 5
Return and Risk Analyzing the Historical Record
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Chapter Summary

• Objective To introduce key concepts and issues
  that are central to informed decision making
• Determinants of interest rates
• The historical record
• Risk and risk aversion
• Portfolio risk

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Factors Influencing Interest Rates

• Supply (of funds)
• Households
• Demand
• Businesses
• Governments Net Supply and/or Demand
• Central Bank Actions
• (Inflation)

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Equilibrium Level of Real Interest Rates
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Equilibrium Level of Nominal Interest Rates

• RrE(i), i.e., for rconst., E(i)f(R)

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Real (r) vs. Nominal Rates (R)

• Fisher effect Approximation
• R r i or r R - i
• Example r 3, i 6
• R 9 36 or r 3 9-6
• Fisher effect Exact

or
Numerically
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Return for Holding Period Zero Coupon Bonds
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Example 5.2 Annualized Rates of Return
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Formula for EARs and APRs
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Table 5.1 Annual Percentage Rates (APR) and
Effective Annual Rates (EAR)
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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

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Characteristics of Probability Distributions
(APP.5B)

• 1) Mean most likely value
• 2) Variance or standard deviation (uncertainty)
• 3) Skewness (asymmetry of the distribution),
  desired positive
• 4) Kurtosis
• If a distribution is approximately normal, the
  distribution is described by characteristics 1
  and 2

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Figure 5.4 The Normal Distribution
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Figure 5.5A Normal and Skewed (mean 6 SD
17)
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Figure 5.5B Normal and Fat Tails Distributions
(mean .1 SD .2)
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Measuring Mean Scenario or Subjective Returns
Subjective returns
s number of scenarios considered pi
probability that scenario i will occur ri
return if scenario i occurs
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Numerical example Scenario Distributions
E(r) (.1)(-.05)(.2)(.05)...(.1)(.35) E(r)
.15 15
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Measuring Variance or Dispersion of Returns

• Subjective or Scenario Distributions

Standard deviation variance1/2 s
Using Our Example s2(.1)(-.05-.15)2(.2)(.05-
.15)2 .01199 s .011991/2 .1095
10.95
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Geometric Average Returns
TV Terminal Value of the Investment
g geometric average rate of return
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Sharpe Ratio
Risk Premium
Sharpe Ratio for Portfolios
SD of Excess Return
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Summary Reminder

• Objective To introduce key concepts and issues
  that are central to informed decision making
• Determinants of interest rates
• The historical record
• Risk and risk aversion
• Portfolio risk

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Annual HPRsCanada, 1957-2001
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Annual HP Risk Premiums and Real Returns, Canada
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Annual HPRsUS, 1926-1999
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Annual HP Risk Premiums and Real Returns, US
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Figure 5.10 Standard Deviations of Real Equity
and Bond Returns Around the World, 1900-2000
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Summary Reminder

• Objective To introduce key concepts and issues
  that are central to informed decision making
• Determinants of interest rates
• The historical record
• Risk and risk aversion
• Portfolio risk

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Risk - Uncertain Outcomes
W 100
E(W) pW1 (1-p)W2 122 s2 pW1 - E(W)2
(1-p) W2 - E(W)2 s2 1,176 and s
34.29
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Risky Investments with Risk-Free Investment
100
Risk Premium 22-5 17
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Risk Aversion Utility

• Investors view of risk
• Risk Averse
• Risk Neutral
• Risk Seeking
• Utility
• Utility Function
• U E ( r ) .005 A s 2
• A measures the degree of risk aversion

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Risk Aversion and Value The Sample Investment

• U E ( r ) - .005 A s 2
• 22 - .005 A (34) 2
• Risk Aversion A Utility
• High 5 -6.90
• 3 4.66
• Low 1 16.22

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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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Utility and Indifference Curves

• Represent an investors willingness to trade-off
  return and risk
• Example (for an investor with A4)

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Indifference Curves
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Summary Reminder

• Objective To introduce key concepts and issues
  that are central to informed decision making
• Determinants of interest rates
• The historical record
• Risk and risk aversion
• Portfolio risk

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Portfolio MathematicsAssets Expected Return

• Rule 1 The return for an asset is the
  probability weighted average return in all
  scenarios.

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Portfolio MathematicsAssets Variance of Return

• Rule 2 The variance of an assets return is the
  expected value of the squared deviations from the
  expected return.

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Portfolio Mathematics Return on a Portfolio

• Rule 3 The rate of return on a portfolio is a
  weighted average of the rates of return of each
  asset comprising the portfolio, with the
  portfolio proportions as weights.
• rp w1r1 w2r2

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Portfolio MathematicsRisk with Risk-Free Asset

• Rule 4 When a risky asset is combined with a
  risk-free asset, the portfolio standard deviation
  equals the risky assets standard deviation
  multiplied by the portfolio proportion invested
  in the risky asset.

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Portfolio MathematicsRisk with two Risky Assets

• Rule 5 When two risky assets with variances s12
  and s22 respectively, are combined into a
  portfolio with portfolio weights w1 and w2,
  respectively, the portfolio variance is given by