Modern Portfolio Theory proposes that an efficient portfolio maximizes expected return for a given level of risk or minimizes risk for a given level of return. The efficient frontier graphically shows the set of efficient portfolios. The optimal portfolio is the point selected on the efficient frontier based on an investor's preferences. Markowitz Portfolio Theory builds on diversification to reduce risk. It models portfolio risk as a function of the variances and covariances of individual securities rather than just summing individual risks.

1. Modern Portfolio Theory

2. EFFICIENT PORTFOLIO
• An efficient portfolio is a portfolio maximize the expected return with
a certain level of risk that is willing underwritten, or portfolio that
offers the lowest risk with a certain rate of return.
• Smallest portfolio risk for a given level of expected return
• Largest expected return for a given level of portfolio risk
• From the set of all possible portfolios
• Only locate and analyze the subset known as the efficient set
• Lowest risk for given level of return

3. • All other portfolios in attainable set are dominated by efficient set
• Global minimum variance portfolio
• Smallest risk of the efficient set of portfolios
• Efficient set
• Part of the efficient frontier with greater risk than the global minimum
variance portfolio

4. 7-4
x
B
A
C
y
Risk = 
E(R)
• Efficient frontier or
Efficient set (curved line
from A to B)
• Global minimum
variance portfolio
(represented by point A)

5. OPTIMUM PORTFOLIO
• The optimal portfolio is a portfolio chosen by investors from many the
choices that are in the collection efficient portfolio.
• The portfolio chosen by investors is portfolio according to preference
the investor is concerned with return and against willing risks bear.

6. OPTIMAL PORTFOLIO FORMATION: SINGLE INDEX
MODEL
• Calculate the mean return
= i + i + e
• Calculating abnormal returns (excess return or abnormal return).
• Estimate (beta) with a single index model for each security return
(Ri) to market return (Rm).
Ri = i + i Rm + 
• Calculating risk is not systematic
iR mR
 Fi RR 
  
2
1
2 1


t
t
mtiiitei RR
t


7. • Calculates the performance of abnormal returns relative to  (Ki):
After the Ki value is obtained, securities are sorted by Ki score from
highest to lowest
i
Fi RR



8. MARKOWITZ PORTFOLIO MODEL
• Portfolio theory with the Markowitz model based on three
assumptions, namely:
• Single investment period, for example 1 year.
• There are no transaction fees.
• Investor preferences are just based on expected returns and risks.
• Markowitz Diversification
• Non-random diversification
• Active measurement and management of portfolio risk
• Investigate relationships between portfolio securities before making a decision to invest
• Takes advantage of expected return and risk for individual securities and how security
returns move together

9. • Simplifying Markowitz Calculations
• Markowitz full-covariance model
• Requires a covariance between the returns of all securities in order to calculate portfolio
variance
• n(n-1)/2 set of covariances for n securities
• Markowitz suggests using an index to which all securities are related to
simplify

10. Portfolio Expected Return
• Weighted average of the individual security expected returns
• Each portfolio asset has a weight, w, which represents the percent of the total
portfolio value
• Calculating Expected Return
• Expected value
• The single most likely outcome from a particular probability distribution
• The weighted average of all possible return outcomes
• Referred to as an ex ante or expected return



n
1i
iip )R(Ew)R(E
i
m
1i
iprR)R(E 



11. Portfolio Risk
• Portfolio risk not simply the sum of individual security risks
• Emphasis on the risk of the entire portfolio and not on risk of
individual securities in the portfolio
• Individual stocks are risky only if they add risk to the total portfolio
• Measured by the variance or standard deviation of the portfolio’s
return
• Portfolio risk is not a weighted average of the risk of the individual securities
in the portfolio
2
i
2
p
n
1i i
w  



12. Risk Reduction in Portfolios
• Assume all risk sources for a portfolio of securities are independent
• The larger the number of securities the smaller the exposure to any
particular risk
• “Insurance principle”
• Only issue is how many securities to hold
• Random diversification
• Diversifying without looking at relevant investment characteristics
• Marginal risk reduction gets smaller and smaller as more securities are added
• A large number of securities is not required for significant risk reduction
• International diversification benefits

13. Portfolio Risk and Diversification
p %
35
20
0
Number of securities in portfolio
10 20 30 40 ...... 100+
Portfolio risk
Market Risk

14. Calculating Portfolio Risk
• Encompasses three factors
• Variance (risk) of each security
• Covariance between each pair of securities
• Portfolio weights for each security
• Goal: select weights to determine the minimum variance combination for a
given level of expected return
• Generalizations
• the smaller the positive correlation between securities, the better
• Covariance calculations grow quickly
• n(n-1) for n securities
• As the number of securities increases:
• The importance of covariance relationships increases
• The importance of each individual security’s risk decreases

15. Measuring Portfolio Risk
• Needed to calculate risk of a portfolio:
• Weighted individual security risks
• Calculated by a weighted variance using the proportion of funds in each security
• For security i: (wi × i)2
• Weighted comovements between returns
• Return covariances are weighted using the proportion of funds in each security
• For securities i, j: 2wiwj × ij