The document summarizes the Markowitz model for building optimal investment portfolios. It discusses key aspects of the model such as diversification to reduce risk, defining the efficient frontier of portfolios with maximum return for a given level of risk, and including both risky and risk-free assets as well as leverage to construct portfolios. The model provides a framework for investors to analyze risk and return tradeoffs across different portfolio combinations.

2. THE MARKOWITZ MODEL
We all agree that holding two stocks is less risky as
compared to one stock. But building the optimal portfolio
is very difficult. Markowitz provides an answer to it with
the help of risk and return relationship.

3. Simple diversification
 In case of simple diversification securities are selected at
random and no analytical procedure is used.
 The simple diversification reduces total risk. The reason
behind this is that the unsystematic price fluctuations are
not correlated with the market fluctuations.
 As the portfolio size increases the total risk starts
declining. It flattens out after a certain point. Beyond that
limit risk cannot be reduced.

4. `
 Problems of vast diversification
 Purchase of poor performers
 Information adequacy
 High research cost
 High transaction cost

5. Assumptions
For a given level of risk, investors prefers higher return to
lower return. Likewise for given level of return investors
prefer low risk as compared to high risk.

6. The concept
In developing the model, Markowitz has given up the
single stock portfolio and introduced diversification. The
single stock portfolio would be preferable if the investor is
perfectly certain that his expectation of higher return
would turn out to be real. But in this era of uncertainty
most of the investors would like to join Markowitz rather
than single stock. It can be shown with the help of
example.

7. Stock ABC Stock XYZ
Return % 11 or 17 20 or 8
Probability .5 each return .5 each return
Expected return 14 14
Variance 9 36
Standard deviation 3 6

8. ABC expected return: .5 x 11+ .5x17= 14
XYZ expected return: .5 x 20+ .5x8= 14
ABC variance = .5(11-14)² + .5(17-14) ²= 9
XYZ variance= .5(20-14) ² + .5(8-14) ²= 36
ABC standard deviation= 3
XYZ standard deviation= 6

9. Now ABC and XYZ have same expected return of 14 % but
XYZ stock is much more risky as compared to ABC
because the standard deviation is much more high.
Suppose the investor holds 2/3 of ABC and 1/3 of XYZ the
return can be calculated as follows
Rp=∑X₁ R₁
Rp= return form portfolio
X₁= proportion of total security invested in security 1.
R₁= expected return of security 1.

10. Let us calculate the expected
return for both possibilities.
possibility 1= 2/3 x 11 + 1/3 x 20 = 14
possibility 2= 2/3 x 17 + 1/3 x 8 = 14
In both the cases the investor stands to gain if the worst
occurs, than by holding either of security individually.
Holding two securities may reduce portfolio risk too. The
portfolio risk can be calculated with the help of following
formula.

11. __________________________
Ϭp= √ X₁²Ϭ₁² + X₂²Ϭ₂² + 2 X₁ X₂( r₁₂ Ϭ₁Ϭ₂)
Ϭp= std. deviation of portfolio
X₁= proportion of stock X₁
X₂= proportion of stock X₂
Ϭ₁= std. deviation of stock X₁
Ϭ₂= std. deviation of stock X₂
r₁₂= correlation coefficient of both stocks

12. r₁₂= covariance of X₁₂
Ϭ₁ Ϭ₂
Using the same example given in the return analysis , the
portfolio return can be estimated
Cov of X₁₂= 1/N ∑(R₁ - Ṝ₁) (R₂ - Ṝ₂)
= ½ [(11-14)(20-14) + (17-14)(8-14)]
= -18
Now r = -18/6x3= -1

13. In our example the correlation coefficient is -1.0.
That means there is perfect negative correlation between
the two and the return moves in opposite direction. If the
correlation is +1 it means securities will move in same
direction and if it is zero the return of both the securities
is independent. Thus the correlation between two
securities depend upon the covariance between the two
securities and the standard deviation of each security.

14. ___________________________
Ϭp= √ X₁²Ϭ₁² + X₂Ϭ₂² + 2 X₁ X₂( r₁₂ Ϭ₁Ϭ₂)
______________________________________
= √ (2/3)² x 9 + (1/3)² x 36 + 2x 2/3 x 1/3 (-1x3x6)
______________
= √ 4+4-8
= 0
The portfolio risk is nil here.

15. The change in portfolio proportions can change the
portfolio risk. Taking same example of ABC and XYZ stock,
the portfolio std. deviation is calculated for different
proportions.
Stock ABC Stock XYZ Portfolio std. deviation
100 0 3
66.66 33.34 0
50.00 50.00 1.5
0 100 6

16. Markowitz efficient frontier
The risk and return of all portfolios plotted in risk-return
space would be dominated by efficient portfolios.
Portfolio may be constructed from available securities. All
the possible combination of expected return risk
compose attainable set. The following example shows the
expected return and risk of different portfolios.

17. PORTFOLIO EXPECTED RETURN % RISK
A 17 13
B 15 8
C 10 3
D 7 2
E 7 4
F 7 8
G 10 12
H 9 8
J 6 7.5

18. The attainable set of portfolios are illustrated in fig.
Each of the portfolios along the line or within the line
ABCDEFGJ is possible. It is not possible for the investor to
have portfolio of this perimeter because no combination
of expected return and risk exists there.
But there are some attractive options. Portfolio B is more
attractive than portfolio F and H because it offers more
return on same level of risk. Likewise, C is more attractive
than portfolio G as for same return there is lower level of
risk.

19. EFFICIENT FRONTIER
0
2
4
6
8
10
12
14
16
18
0 2 4 6 8 10 12 14
Y-Values
Y-Values

20. UTILITY ANALYSIS
Utility is the satisfaction the investors enjoy from the
portfolio return. An ordinary investor is assumed to
receive greater utility from higher return and vice-versa.
In a fair gamble which costs Re 1, the outcomes are A and B
events. A event will yield Rs. 2. occurrence of B event is a
dead loss i.e. 0. The chance of occurrence of both the
events are 50-50. The expected value of investment is
(1/2 x 2 + 1/2x 0) =Re 1. The expected value of gamble is
exactly equal to cost. Hence it is a fair gamble.

21. Risk averter rejects a fair gamble because the disutility of
the loss is greater for him than the utility of equivalent
gain. Risk neutral investors means that he is indifferent to
whether a fair gamble is undertaken or not. The risk
seeking investor would select a fair gamble. The expected
utility of investing is higher than the expected utility of
not investing.

22. Leveraged portfolio
In the above model, the investor is assumed to have certain
amount of money to make investment for fixed period of time.
There is no borrowing and lending opportunities. When the
investors is not allowed to use the borrowed money, he is
denied the opportunity of having financial leverage.
Again the investor is assumed to be investing only on risky
assets. Riskless assets are not included in the portfolio. To
have a leveraged portfolio investor has to consider not only
risky assets but also risk free assets. Secondly, he should be
able to borrow and lend money at given rate of interest.

23. RISK FREE ASSET
The features of risk free assets are:
(i) Absence of default risk
(ii) Full payment of principal and interest amount

24. Inclusion of risk free asset
Now, the risk free asset is introduced and the investor can
invest part of his money on risk free asset and the
remaining amount on risky assets. It is also assumed that
investor would be able to borrow money at risk free rate
of interest. When risk free asset is included in portfolio,
the feasible efficient set of portfolios is altered.
But return from risk free asset is less as compared to risky
assets so investor will make a combination of risk free and
risky assets to maximize his return.