Risk and Return Primer

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Risk and ReturnPrimer
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Expectations

• Expected value (µ) is weighted sum of possible
  outcomes
• E(X) µ p1X1 p2X2 . psXs
• E(X) Expected value of X
• Xi Outcome of X in state i
• pi Probability of state i
• s Number of possible states
• Probabilities have to sum to 1
• p1 p2 .. ps 1

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Horse Race

• There are three horse racing in the Finance
  Derby. Your horse is Love of NPV. If your horse
  has a 30 chance of coming in first, and a 40
  chance of coming in second. How much do you
  expect your horse to win?
• 1st pays 1,500
• 2nd pays 750
• 3rd pays 250

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What is risk?
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Measuring Risk

• There is no universally agreed-upon measure
• However, variance and standard deviation are both
  widely accepted measures total risk

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Statistics Review Variance

• Variance (s2) measures the dispersion of possible
  outcomes around µ
• Standard deviation (s) is the square root of
  variance
• Higher variance (std dev), implies a higher
  dispersion of possible outcomes
• More uncertainty

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Different Variances
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Variance Calculation

• Variance s2 Spi (Xi µ)2 Use this one
• Alternative formulas you may have seen
• s2 S(Xi µ)2 / N
• s2 S(Xi µ)2 / (N-1)
• Alternatives give very different answers with
  small samples
• Ex. s3
• s2 p1 (X1 µ)2 p2 (X2 µ)2 p3 (X3
  µ)2

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Risk Example

• Economy is Good with 20 probability DJIA will
  return 20
• Economy is Fair with 30 probability DJIA will
  return 5
• Economy is Bad with 50 probability DJIA will
  return -9

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Calculations

• Expected Return
• Variance
• Standard Deviation

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Historical Data

• In practice we do not know all of the possible
  states of the world, so we use historical data to
  form expectations
• Idea Look at what has happened in the past and
  we can calculate the mean and variance
• What is each states probability of occurring?

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Risk Example 2
1996 1997 1998 1999 2000
20 15 -5 5 10

• Sample Mean
• Sample Variance
• Standard Deviation

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Risk

• A risky asset is one in which the rate of return
  is uncertain.
• Risk is measured by ________________

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General Securities

• T-bills are a very safe investment
• No default risk, short maturity
• Risk free asset
• Stocks are much riskier
• Bonds riskiness is between T-bills and Stocks

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Why Do We Demand a Higher Return

• Investors seem to dislike risk (ex. insurance)
• Risk Averse
• If the expected return on T-Bills (risk-free), is
  10, and the expected return for Ford is 10,
  which would you buy?

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Return Breakdown

• A risky assets return has two components
• Risk free rate Risk premium
• Risk free rate The return one can earn from
  investing in T-Bills
• Risk Premium The return over and above the risk
  free rate
• Compensation for bearing risk

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Average Risk Premiums (1926-2005)

• Small company stocks
• 17.4 3.8 13.6
• Large company stocks
• 12.3 3.8 8.5
• Long-term corporate bonds
• 6.2 3.8 2.4
• The more risk the larger the risk premium

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The Risk-Return Tradeoff
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Quick Quiz

• Which of the investments discussed has had the
  highest average return and risk premium?
• Which of the investments discussed has had the
  highest standard deviation?
• Why is the normal distribution informative?
• What is the difference between arithmetic and
  geometric averages?

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Why we care?

• This is the very basics of investing
• General knowledge that finance people possess