Lecture 11: Tue, Oct 15

1
Lecture 11 Tue, Oct 15

• Todays material
• Portfolio selection
• Binomial Distribution

2
Portfolio Problems

• An investor has two possible assets, A1 and A2.
  The investor wants to form an optimal portfolio
  consisting of some fraction of each asset.
• Let R1 return on asset 1
• Let R2 return on asset 2
• Let w1fraction of your wealth invested in A1.
• Let w2fraction of your wealth invested in A2.
• Let Rp w1R1w2R2 return on portfolio.

3

• Question How to choose w1, w2 ?
• 1. Maximize expected return of portfolio, E(Rp)
• 2. Minimize variance of portfolio, V(Rp)
• 3. Maximize the Sharpe Ratio, E(Rp) / V(Rp).

4
Covariance and Correlation

• First, recall that
• Rearranging, we get

5
Interpreting Correlation

Complete Diversification
No Diversification
More Diversified
Less Diversified
6

• Assume we are given
• The mean and variance of the portfolio are given
  by

7
Example

• Suppose that you wish to invest 1 million.
  After careful consideration of investment
  opportunities, you reduce the number of choices
  to two. The means and standard deviations of the
  two investments are listed in the following
  table. The returns are highly correlated with
  Discuss whether you should put all your
  money in investment 1, investment 2, or a
  portfolio composed of an equal amount of
  investments 1 and 2.

8
(No Transcript)
9
Example (cont)

• Repeat the last question, assuming
• Half in each

10
Example (cont)

• Repeat the last question, assuming
• Half in each

11
Portfolio Problems with k assets

• Assume we have k assets, A1,, Ak, with returns
  R1,,Rk, respectively. The mean and variance of
  the return on the portfolio are given by

12
Exercise 7.68

• The semiannual returns for three stocks over an
  18-year-period are stored in file XR07-68.
  Calculate the mean and variance of the three
  stocks, and compare that to the mean and variance
  of a portfolio consisting of 1/3 in each.

Variance/Covariance
13
Exercise 7.68

• With w1w2w31/3, the mean and variance of the
  portfolio are

14
Some Simple Discrete Distributions

• The Binomial Distribution
• The binomial experiment is one where the result
  is only one of two possible outcomes.
• Typical cases where the binomial experiment
  applies
• A coin flipped results in heads or tails
• An election candidate wins or loses
• An employee is male or female
• A car uses 87octane gasoline, or another gasoline.

15
Binomial Experiment

• Conditions for a Binomial Experiment
• There are n trials (n is finite and fixed).
• Each trial can result in a success or a failure.
• The probability p of success is the same for all
  the trials.
• All the trials of the experiment are independent.
• Binomial Random Variable
• The binomial random variable counts the number of
  successes in n trials of the binomial experiment.
• By definition, this is a discrete random variable.

16
Calculating the Binomial Probability
In general, The binomial probability distribution
can be represented as
17
Calculating the Binomial Probability

• Example 7.9 7.10
• Pat Statsdud is registered in a statistics course
  and intends to rely on luck to pass the next
  quiz.
• The quiz consists on 10 multiple choice questions
  with 5 possible choices for each question, only
  one of which is the correct answer.
• Pat will guess the answer to each question
• Find the following probabilities
• Pat gets no answer correct
• Pat gets two answer correct?
• Pat fails the quiz

18
Calculating the Binomial Probability

• Solution
• Checking the conditions
• An answer can be either correct or incorrect.
• There is a fixed finite number of trials (n10)
• Each answer is independent of the others.
• The probability p of a correct answer (.20) does
  not change from question to question.

19
Calculating the Binomial Probability

• Solution Continued
• Determining the binomial probabilities
• Let X the number of correct answers

20
Calculating the Binomial Probability

• Solution Continued
• Determining the binomial probabilities
• Pat fails the test if the number of correct
  answers is less than 5, which means less than or
  equal to 4.

21
Mean and Variance of Binomial Variable

• E(X) m np
• V(X) s2 np(1-p)
• Example 7.11
• If all the students in Pats class intend to
  guess the answers to the quiz, what is the mean
  and the standard deviation of the quiz mark?
• Solution
• m np
• s np(1-p)1/2

22
Quick Quiz

• An analysis of the stock market produces
  the following information
• about the returns of two stocks.
• Stock 1 Stock 2
• Expected Returns 15
  18
• Standard Deviations 20
  32
• Assume that the returns are positively correlated
  with 0.80.
• Find the mean and standard deviation of the
  return on a portfolio
• consisting of an equal investment in each of the
  two stocks.

23
(No Transcript)
24
Problem 2

• Consider a binomial random variable X with n 5
  and p 0.40.
• a. Find the probability distribution of X.
• b.      Find P(X lt 3).
• c.       Find P(2 X4) .
• d.      Find the mean and the variance of X.

25
(No Transcript)