Prepared by Lucky Yona
Management Consultant
Introduction
 A portfolio is the collection of different
investments that make up an investor’s total
holding.
 It is a bundle or combination of individual assets
or shares
 A portfolio might be
 The investments in stocks and shares of an investor
 The investments in capital projects of a company
 Portfolio Theory provides a guideline for building
up of a portfolio of stocks and shares or a portfolio
of projects.
 It provides a normative approach to investors
decision to invest in assets or securities under risk.
 This implies that investors hold well –diversified
portfolios instead of investing their entire wealth
in a single asset or security
Major Concern of Investors
 If the investor holds a well diversified portfolio ,
then his concern should be the expected return
and risk of the portfolio rather than an individual
asset or securities.
 The assumptions underlying portfolio theory is
that the mean ( expected Value) and variance (
standard deviation) analysis is the foundation of
Portfolio Decisions.
Factors to be considered when an investor
choosing an investments
 Security
 Liquidity
 Return
 Spreading the risk
 Growth Prospects
Portfolio Expected Returns
 The expected return of a portfolio will be the weighted
average of the expected returns of the investment in
the portfolio , weighted by the proportion of total
funds invested in each.
Example 1- One case Asset
 Suppose the return from an investment has the
following probability distribution
 Return
Probability
8%
0.2
10%
0.2
12%
0.5
14%
0.1
Calculate the Expected Return and Standard
Deviation
11% and 1.84%
 The expected return of this investment is 11% and
Standard Deviation is 1.84%.
 The risk of investment might be high or low,
depending on the nature of investments
 Low risks investments usually give low returns
 High risks investments might give high returns, but
with more risk of disappointing results.
Two Case Asset- Expected Returns
 Example
 Consider the following two projects which have
different possible returns in different state of
conditions .
State of Economy
A
B
C
D
Returns (%)
x
y
20
16
10
12
25
20
8
16
Probabilities
0.4
0.3
0.2
0.1
What is the expected return of each of the project
Expected Return for Project x and Y
State
Rx
Ry
P
Where Rx = Possible Returns for X
E( Rx)
E(Ry)
RxP
RyP
Ry = Possible Returns for Y
P= Probabilities
A
20
25
0.4
8
10
B
16
20
0.3
5
6
C
12
8
0.2
2
2
D
10
12
0.1
1
1
16
19
Expected Return for X = 16%
Expected Return for Y = 19%
Expected Return of a Portfolio
Example.
Using the previous example, an investor decided to
invest 40% of his wealth in X and 60% in Y. What will
be the expected return of the portfolio?
Solution- Expected Return of the Portfolio
Method 1
 Step 1- Calculate the Expected Return of the individual
asset portfolio
 Attach the proportion of the investment
 Weight the individual return with the required
proportion
 Add the weighted Individual returns and obtain the
expected return of the portfolio
 From our Calculation
E(Rx) = 16% and E(Ry) = 19%
We have the proportions for X as 40% and for Y is 60%
Therefore
Expected Return of Portfolio
= 0.4( 16%) + 0.6 (19%) = 17.8%
Two Case Asset- Risk Measurement
 Measuring Risk of portfolio is the same as measuring
risk of individual project risk.
 We will use standard Deviation to calculate the risk of
portfolio
 Example
State of Economy
A
B
C
D
20
16
10
12
Returns (%)
x
y
25
20
8
16
Probabilities
0.4
0.3
0.2
0.1
What is the standard Deviation of each of these
projects?
Standard Deviation of Individual Project
Correlation Between Investments
 Correlation measures the degree to which the returns
on investments vary with each other.
 Portfolio theory states that individual investments
cannot be viewed simply in terms of their risk and
return.
 Their relationship between the return from one
investment and the return from other investments is
just as important.
 The relationship between investments can be one
of the three types
1. Positive Correlation
 When there is positive correlation between investments
, if one investments does well ( or badly) it is likely that
the other will perform like wise . Thus if you buy shares
in one company making umbrellas and in another which
sells raincoats you would expect both companies to do
badly in dry weather.
 Negative Correlation
 If one investment does well the other will do badly, and
vice versa. Thus if you hold shares in one company
making umbrellas and in another which sells ice cream,
the weather will affect the companies differently.
 No correlation
 The performance of one investment will be
independent of how the other performs. If you hold
shares in a mining company and in leisure company, it is
likely that there would be no relationship between the
profits and returns from each.
 This relationship between the returns from different
investments is measured by the correlation coefficient
.
 A figure Close to + 1indicates high positive correlation,
and a figure close to -1 indicates high negative
correlation.
 A figure of 0 indicates no correlation.
 If investments show high negative correlation, then by
combining them in a portfolio overall risk would be
reduced.
 Risk will also be reduced by combining in a portfolio
investments which have no significant correlation.
Calculation of Correlation
State of Economy
A
B
C
D
Returns (%)
x
y
20
16
10
12
25
20
8
16
Probabilities
0.4
0.3
0.2
0.1
What is the standard Deviation of each of these
projects? What is the correlation of Coefficient?
Calculation
x
y
xy
X²
y²
20
25
500
400
625
16
20
320
256
400
10
8
80
100
64
12
16
192
144
256
58
69
1092
900
1345
Information
 Totals for Calculations are
n=4
X = 58
Y = 69
XY = 1092
X² = 900
Y² = 1345
Coefficient of Correlation
 r=
n ∑xy – ∑x∑ y
√ n∑ x²-(∑x)² √n∑ y²-(∑y)²
4(1092)- (58)(69)
√ 4(900)- 58² x √4(1345)-69²
= 0.957
Coefficient of Colleration
Consider the following investments
,Investment A and Investment B.
Investment A
Outcome
Returns
Worst Outcome 10% 0.3
Most Likely
15% 0.4
Best
20% 0.3
Investment B
Outcome
Returns
Worst Outcome 9%
Most Likely
16%
Best
21 %
Probability
0.3
0.4
0.3
1.What is the expected return of each investment?
2. What is the expected return of the total portfolio if the
proportion of investment is 30% for investment A and 70%
for investment B?
3. What is the correlation Coefficient of the investments?
4. Calculate the standard Deviation of individual asset
5.Standard Deviation of the Portfolio.
Expected Returns
Investment – Asset A = 15%
Asset B = 15.4%
.
Expected Return of the Portfolio
 Ep = W1(ERX) + W2(ERy)
Where
E = Expected Return of the portfolio
p
W1= Proportion of investment in investment 1
W2= Proportion of investment in investment 1
ERX = Expected Return of Investment 1
ERY = Expected Return of Investment 2
 Ep = W1(ERX) + W2(ERy)
= 0.3 (15%) + 0.7( 15.4%)
= 15.28%
Correlation Coefficient
 r=
n ∑xy – ∑x∑ y
√ n∑ x²-(∑x)² √n∑ y²-(∑y)²
Information Gathered
n= 3
∑X = 45
∑Y = 46
∑XY = 750
∑X² = 725
∑Y² = 778
Putting the above information in the formula you
Obtain r = 0.996
Calculating the likely variation of investment
and Portfolio.
 In this case we have to calculate the standard deviation
of individual investments and there after the standard
deviation of the whole portfolio.
 Standard Deviation for Investment 1= 3.87
 Standard Deviation for Investment 2= 4.67
Standard Deviation of the Portfolio
δ (Rp) =
√ W² δ ² + W² δ ² +2WaWb r δ δ
a
a
b
b
a
b
Where
Wa= Proportion of investment in investment a
Wb= Proportion of investment in investment b
δa - Standard Deviation of Investment a
δB - Standard Deviation of Investment b
r =Correlation of the Portfolios
Solution
Standard Deviation of Portfolio δ (Rp) = 4.43
Discussion -1


An investor has invested his money in two securities, A and B. The expected returns under different
outcomes for each of the securities is expected to have the following information.
Investment A
Investment B




Outcomes
Worst Outcome
Most likely
Best

Assume that the investor need to invest 40% of his fund in security A and 60% in Security B.






Required
1. Calculate the Expected Return of each asset
2. Calculate the expected return of the whole investment portfolio
3. Calculate the standard deviation of investments
4. Calculate the Coefficient of Correlation of the investment.
4. Calculate the Standard Deviation of the Portfolio.
Returns
31%
26 %
35%
Probability
0.3
0.4
0.3
Returns
20%
25%
31%
Probability
0.3
0.4
0.3
Discussion-2
 Portfolio Theory application is the most needed
theory now by the African Business organizations.
Assess the Relevance of this theory in the Context of
African Business Environment and discuss the possible
challenges in its application.