Only prepared from in(slide share) for teach. Not for business

1. Investment Management
Chapter 4.1-Optimal
Portfolio Choice
Lectured by : Mr. HENG Leangpheng (MBA)
Tel: 095 433 369 / 081 895 695
E-mail: leangpheng.heng@yahoo.com
1

2. Chapter Outline
1. The Expected Return of a Portfolio
2. The Volatility of a Two-Stock Portfolio
3. The Volatility of a Large Portfolio
4. Risk Versus Return: Choosing an
Efficient Portfolio
5. Risk-Free Saving and Borrowing
6. The Efficient Portfolio and Required
Returns
2

3. Learning Objectives
1. Given a portfolio of stocks, including the holdings in each stock
and the expected return in each stock, compute the following:
1. portfolio weight of each stock (equation 11.1)
2. expected return on the portfolio (equation 11.3)
3. covariance of each pair of stocks in the portfolio (equation 11.5)
4. correlation coefficient of each pair of stocks in the portfolio (equation
11.6)
5. variance of the portfolio (equation 11.8)
6. standard deviation of the portfolio
2. Compute the variance of an equally weighted portfolio, using
equation 11.12.
3. Describe the contribution of each security to the portfolio.
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4. 4. Use the definition of an efficient portfolio from Chapter 10 to
describe the efficient frontier.
5. Explain how an individual investor will choose from the set of
efficient portfolios.
6. Describe what is meant by a short sale, and illustrate how short
selling extends the set of possible portfolios.
7. Explain the effect of combining a risk-free asset with a portfolio of
risky assets, and compute the expected return and volatility for
that combination.
8. Illustrate why the risk-return combinations of the risk-free investment
and a risky portfolio lie on a straight line.
4 Learning Objectives

5. 9. Define the Sharpe ratio, and explain how it helps identify the
portfolio with the highest possible expected return for any level of
volatility, and how this information can be used to identify the
tangency (efficient) portfolio.
10. Calculate the beta of investment with a portfolio.
11. Use the beta of a security, expected return on a portfolio, and the
risk-free rate to decide whether buying shares of that security will
improve the performance of the portfolio.
12. Explain why the expected return must equal the required return.
13. Use the risk-free rate, the expected return on the efficient
(tangency) portfolio, and the beta of a security with the efficient
portfolio to calculate the risk premium for an investment.
5 Learning Objectives

6. 1.The Expected Return of a Portfolio
Portfolio Weights
• The fraction of the total investment in the
portfolio held in each individual investment in
the portfolio
oThe portfolio weights must add up to 1.00 or 100%.
6
i
Value of investment
Total value of portfolio

i
x

7. Then the return on the portfolio, Rp , is the
weighted average of the returns on the
investments in the portfolio, where the
weights correspond to portfolio weights.
7
1 1 2 2     P n n i ii
R x R x R x R x R
1.The Expected Return of a Portfolio

8. Textbook Example 4.18

9. 9 Textbook Example 4.1

10. The expected return of a portfolio is the
weighted average of the expected
returns of the investments within it.
10
           P i i i i i ii i i
E R E x R E x R x E R
1.The Expected Return of a Portfolio

11. Textbook Example 4.211

12. Alternative Example 4.2
Problem
• Assume your portfolio consists of $25,000 of
Intel stock and $35,000 of ATP Oil and Gas.
• Your expected return is 18% for Intel and 25%
for ATP Oil and Gas.
• What is the expected return for your
portfolio?
12

13. Solution
• Total Portfolio = $25,000 + 35,000= $60,000
• Portfolio Weights
o Intel: $25,000 ÷ $60,000 = .4167
o ATP: $35,000 ÷ $60,000 = .5833
• Expected Return
o E[R] = (.4167)(.18) + (.5833)(.25)
o E[R] = 0.075006 + 0.145825 = 0.220885 = 22.1%
13 Alternative Example 4.2

14. 2.The Volatility of a Two-Stock Portfolio
Combining Risks
• Table 11.1 Returns for Three Stocks, and Portfolios of Pairs of Stocks
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15. • While the three stocks in the previous table
have the same volatility and average return,
the pattern of their returns differs.
oFor example, when the airline stocks performed
well, the oil stock tended to do poorly, and when
the airlines did poorly, the oil stock tended to do
well.
15 2.The Volatility of a Two-Stock Portfolio

16. • Consider the portfolio which consists of equal
investments in West Air and Tex Oil. The
average return of the portfolio is equal to the
average return of the two stocks
• However, the volatility of 5.1% is much less
than the volatility of the two individual stocks.
16 2.The Volatility of a Two-Stock Portfolio

17. • By combining stocks into a portfolio, we
reduce risk through diversification.
• The amount of risk that is eliminated in a
portfolio depends on the degree to which the
stocks face common risks and their prices
move together.
17 2.The Volatility of a Two-Stock Portfolio

18. Determining Covariance and
Correlation
 To find the risk of a portfolio, one must
know the degree to which the stocks’ returns move
together.
 Covariance
• The expected product of the deviations of two returns
from their means
• Covariance between Returns Ri and Rj
18
( , ) [( [ ]) ( [ ])]  i j i i j jCov R R E R E R R E R

19. • Estimate of the Covariance from Historical Data
o If the covariance is positive, the two returns tend to
move together.
o If the covariance is negative, the two returns tend to
move in opposite directions.
19
, ,
1
( , ) ( ) ( )
1
  

i j i t i j t jt
Cov R R R R R R
T

20. Correlation
• A measure of the common risk shared by stocks that
does not depend on their volatility
oThe correlation between two stocks will always be
between –1 and +1.
20
( , )
( , )
( ) ( )
i j
i j
i j
Cov R R
Corr R R
SD R SD R


21. Figure 4.1 Correlation21

22. Textbook Example 4.322

23. Table 4.2 Computing the Covariance and
Correlation Between Pairs of Stocks
23

24. Table 4.3 Historical Annual Volatilities
and Correlations for Selected Stocks
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25. Textbook Example 4.425

26. Textbook Example 4.526

27. Alternative Example 4.5
Problem
• Using the data from Table 4.3, what is the
covariance between General Mills and
General Motors?
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28. Solution
28
General Mills Ford General Mills Ford General Mills Ford( , ) ( , ) ( ) ( )
(0.07)(0.18)(0.42) .005292
Cov R R Corr R R SD R SD R
 

29. Computing a Portfolio’s Variance
and Volatility
For a two security portfolio:
• The Variance of a Two-Stock Portfolio
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1 1 2 2 1 1 2 2
1 1 1 1 1 2 1 2 2 1 2 1 2 2 2 2
( ) ( , )
( , )
( , ) ( , ) ( , ) ( , )

  
   
P P PVar R Cov R R
Cov x R x R x R x R
x x Cov R R x x Cov R R x x Cov R R x x Cov R R
2 2
1 1 2 2 1 2 1 2( ) ( ) ( ) 2 ( , )  PVar R x Var R x Var R x x Cov R R

30. Textbook Example 4.630

31. 31

32. Alternative Example 4.6
Problem
• Continuing with Alternative Example 4.2:
oAssume the annual standard deviation of
returns is 43% for Intel and 68% for ATP Oil
and Gas.
• If the correlation between Intel and ATP is
.49, what is the standard deviation of your
portfolio?
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33. Solution
33
2 2
1 1 2 2 1 2 1 2D( ) ( ) ( ) 2 ( , )  PS R x Var R x Var R x x Cov R R
2 2 2 2
D( ) (.4167) (.43) (.5833) (.68) 2(.4167)(.5833)(.49)(.43)(.68)  PS R
D( ) (.1736)(.1849) (.3402)(.4624) 2(.4167)(.5833)(.49)(.43)(.68)  PS R
D( ) .0321 .1573 .0696 0.259 .5089 50.89%     PS R

34. 3.The Volatility of a Large Portfolio
The variance of a portfolio is equal to the
weighted average covariance of each stock
with the portfolio:
• which reduces to:
34
 ( ) ( , ) , ( , )   P P P i i P i i Pi i
Var R Cov R R Cov x R R x Cov R R
j( ) ( , ) ( , )
( , )
  

 
 
P i i P i i j ji i
i j i ji j
Var R x Cov R R x Cov R x R
x x Cov R R

35. Diversification with an Equally
Weighted Portfolio of Many Stocks
Equally Weighted Portfolio
• A portfolio in which the same amount is invested
in each stock
Variance of an Equally Weighted Portfolio
of n Stocks
35
1
( ) (Average Variance of the Individual Stocks)
1
1 (Average Covariance between the Stocks)

 
  
 
PVar R
n
n

36. Figure 4.2 Volatility of an Equally Weighted
Portfolio Versus the Number of Stocks
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37. Textbook Example 4.737

38. 38

39. Textbook Example 4.839

40. Diversification with General Portfolios
 For a portfolio with arbitrary weights, the standard deviation is
calculated as:
• Volatility of a Portfolio with Arbitrary Weights
 Unless all of the stocks in a portfolio have a perfect positive
correlation of +1 with one another, the risk of the portfolio will be
lower than the weighted average volatility of the individual stocks:
40
Security ’s contribution to the
volatility of the portfolio
Amount Total Fraction of ’
of held Risk of
( ) ( ) ( , )  
  

i
P i i i pi
i
i i
SD R x SD R Corr R R
s
risk that is
common to P
( ) ( ) ( , ) ( )  P i i i p i ii i
SD R x SD R Corr R R x SD R

41. 4.Risk Versus Return: Choosing an
Efficient Portfolio
Efficient Portfolios with Two Stocks
• Identifying Inefficient Portfolios
oIn an inefficient portfolio, it is possible to find
another portfolio that is better in terms of both
expected return and volatility.
• Identifying Efficient Portfolios
oRecall from Chapter 10, in an efficient portfolio
there is no way to reduce the volatility of the
portfolio without lowering its expected return.
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42. Efficient Portfolios with Two Stocks
• Consider a portfolio of Intel and Coca-Cola
Table 4.4 Expected Returns and Volatility for Different
Portfolios of Two Stocks
42