RISK, CAPITAL BUDGETING, AND DIVERSIFICATION

1
CHAPTER 11

• RISK, CAPITAL BUDGETING, AND DIVERSIFICATION

2
RISK, CAPITAL BUDGETING, AND DIVERSIFICATION

• Review of Some Statistical Concepts
• Portfolio Diversification

3
STATISTICAL CONCEPTS

• Review of Some Statistical Concepts
• Measures of Location
• Measures of Dispersion
• Measures of Co-movement

4
STATISTICAL CONCEPTS

• Measures of Location
• Two common measures of location
• Expected Value (Mean)
• Median
• Expected Value is the weighted average of all
  possible outcomes
• Median is the middle value of the ranked outcomes

5
STATISTICAL CONCEPTS

• Expected value
• If we know the probability of each outcome, we
  have to use the formula to find the expected
  value
• E(X) ?piXi
• Here pi is the probability of outcome i, and Xi
  is the value of outcome i.
• We can use array function in Excel to find
  expected value under this situation.
• e.g., imagine that we know that there is a jar
  with 10 balls, three balls with .25 written on
  it, five with .15, one with .05, and one with
  -.05. What is the expected value of the number
  you will get from the ball?

i1
6
STATISTICAL CONCEPTS

• Expected value (estimation)
• If we do not know probability of each outcome, we
  need to make an educated guess of the expected
  value, if we already have some observation. This
  process is called estimating.
• e.g., now, lets assume that we have drawn one
  ball from the jar each year for five years, and
  have obtained five readings. We do not know the
  exact number of balls in the jar. We want to
  estimate what the average reading we may get in
  the future.
• Solution assign each outcome with equal
  probability, and find weighted average of all
  observed outcome.
• In Excel, we can use average function.

7
STATISTICAL CONCEPTS

• 2. Measures of Dispersion
• Variance and Standard Deviation
• Variance
• Var(X) ?X2 ? pi(Xi E(x))2
• Standard Deviation
• ?X (?X2 )1/2
• Note
• In this case, we know the probability of each
  outcome.
• (Xi E(x)) gives us the deviation of each
  outcome from the mean (Are these deviations
  positive or negative?)
• Positive deviations and negative deviation will
  cancel each other out, leaving a weighted average
  of zero.
• To correct for negative deviations, we square the
  deviations
• To get the unit back to original scale, we take
  the square root of the variance to obtain the
  standard deviation.
• Square root function in Excel is SQRT()

8
STATISTICAL CONCEPTS

• Variance and Standard Deviation
• e.g., continue with our jar example, we can use
  the array function to find variance and standard
  deviation.
• If we do not know the distribution, instead, we
  only have five years of annual drawing from the
  Jar, we have to estimate the variance and
  standard deviation by assigning each deviation
  with a probability of 1/(N-1).
• Note some people will use 1/N
• When we have enough observations, 1/(N-1) and 1/N
  are very close to each other.
• In excel
• we can use VAR (STDEV) to find variance and
  standard deviation. (probability assigned is
  1/(N-1)).
• we can use VARP (STDEVP) to find variance and
  standard deviation. (probability assigned is
  1/N).

9
STATISTICAL CONCEPTS

• 3. Measures of Co-movement
• Covariance and correlation coefficient
• Covariance
• ?XY ? pi(Xi E(X)) (Yi E(Y))
• Correlation coefficient
• Covariance and correlation coefficient show the
  degree at which two variables (stock prices) are
  moving together.

10
PORTFOLIO DIVERSIFICATION

• Total risk
• Systematic risk
• Unsystematic risk
• Diversification can reduce the unsystematic risk
  in an investment and hence the total risk.
• Via diversification, one can earn higher returns
  for the same level of risk, or reach lower risk
  for the same return.

11
PORTFOLIO DIVERSIFICATION

• The benefit of diversification comes from the
  correlation between different stocks.
• For example if correlation between two stocks is
  1, this is called perfect positive correlation.
  It means that prices of both stocks move in
  exactly the same direction.
• If the correlation is 1, prices of these two
  stocks are moving exactly in the opposite
  directions.
• Note that correlation coefficient is between 1
  and 1.

12
PORTFOLIO DIVERSIFICATION

• To benefit from portfolio diversification, one
  should invest in stocks that have the lowest
  correlation.
• Stocks with negative correlations are the best
  combination to reduce risk.
• If you invest in stocks that are perfectly
  positively correlated, there will be no benefit
  from diversification.

13
PORTFOLIO DIVERSIFICATION

• Expected Return of Portfolio, E(Rp)
• E(Rp) ? wiE(Ri)
• Here wi is the proportion invested in portfolio
  i, E(Ri) is the expected return of portfolio i,
  and N is the number of securities in the
  portfolio.

N
i1
14
PORTFOLIO DIVERSIFICATION

• Variance and Standard Deviation of a Portfolio

• Again, the Standard Deviation is the square root
  of the variance