Part 6: Risk and Return

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• Part 6 Risk and Return
• A Statistical review
• Topics Covered
• Expected Return and Standard Deviation
• Correlation
• Normal Distribution

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6.1 Meaning of Risk

• The simplest definition of risk
• More than one outcome can occur
• Investors are not indifferent as which outcome
  does occur
• Other things being equal, most investors dislike
  risk. Therefore, investors demand a higher
  expected return from a risky project or
  investment.
• Start with the risk free rate
• Add a risk premium
• The risk premium is determined by the Capital
  Asset Pricing Model

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6.2 Return and Risk Based on Historical Data

• With a series of (realized) asset returns over
  time, the mean return is
• Based on realized returns, calculate variance
  and standard deviation.

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6.3 An Example
ABC Co experienced the following returns in the
last five years
Calculate the average return and the standard
deviation.
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6.4 Example Continued
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6.5 Example (concluded)

• We cab use the Excel functions average() and
  stdev() to get the mean and standard deviation.
• We also use the term volatility, or risk to
  refer to the standard deviation.

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6.6 Expected Return

• We want to know expected return and risk, i.e.,
  those of before-the-fact (ex ante).
• Expected return and risk measures are based on
  probability distributions.
• A stock has following return distribution
• Expected return is

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6.7 Expected Risk

• Ex ante variance and standard deviation of return
  is
• General formula, based on probability
  distribution

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6.8 Expected and Realized Return

• It can be difficult to make realistic assumptions
  on probability distribution of stock return.
• Thus, we often use realized return and risk
  measures to approximate expected counterparts.
• Justification is Law of Large Numbers - if the
  number of realized returns from the same
  distribution is large (e.g., over a long period
  of time), then their statistical mean and
  variance are good approximations of their
  expected return and variance.

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6.9 Covariance/Correlation

• Covariance measures how closely two stocks move
  together.
• There are two ways to calculated covariance
  (a) Based on forecast
• (b) Or, based on historical observations,

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6.10 Coefficient of Correlation

• The coefficient of correlation is defined as,
• The coefficient of correlation is always between
  -1 and 1. Negative correlations indicate that
  the variables tend to move in opposite
  directions. Positive correlations indicate that
  the variables tend to move in the same direction.
  

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6.11 Correlations among Chinese, Australian and
U.S. Stock Markets

• Based on the total return data from 23/7/1992 to
  23/7/2002
• What does this table tell us?

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6.12 Further Examples of Correlation

• Company A runs a number of resorts and Company B
  manufactures umbrellas.
• The profits of the two firms should be negatively
  correlated because in the rainy season, B is
  doing better while A suffers. It is the opposite
  for the sunny season.
• Gold Price (US/ounce) and SP 500 (U.S. stock
  market index) are negatively correlated.
• The coefficient of correlation between the two is
  -0.40475 based on 25 years data (from 1981 to
  2006).

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6.13 Diversification on Portfolio Variance
Portfolio returns50 A and 50 B
Stock B returns
Stock A returns
0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03
0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04
-0.05
0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03
Note rA,B lt 0
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6.14 Normal Distribution

• We usually assume that the rate of return on an
  asset follows a normal distribution (Bell-shaped
  distribution curve).
• Two important parameters of a normal distribution
  are mean and standard deviation.
• The mean and standard deviation give us a
  risk-return profile of an asset.

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6.15 The Normal Distribution
Return

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6.16 An Example of Risk-Return Profile

• Suppose that the rate of return of ADC stock has
  a mean of 10 per year and a standard deviation
  of 20. What does this mean?
• If you invest 1000 in the ADCs stock. Then on
  average, your payoff will be 1100 a year later.
• The probability of your payoff between 900 and
  1300 (within one st.dev. from the mean) is
  68.3.
• What is the probability of at least breakeven on
  this investment? Using Excel to find it.
• 1 - normdist(0,10,20, true) 1 - 0.308
  0.692 69.2