Portfolio Managment

Portfolio
Theory
The benefits of
diversification
Expected Return
The expected return on an investment is the expected value of

the probability distribution of possible returns it can provide to
investors.
The return on the investment is an unknown variable that has

different values associated with different probabilities.
Expected return is calculated by multiplying potential outcomes

(returns) by the chances of each outcome occurring, and then
calculating the sum of those results (as shown below).
Expected Return for a Single
Investment
Let us take an investment A, which has a 20% probability of

giving a 15% return on investment, a 50% probability of
generating a 10% return, and a 30% probability of resulting in a
5% loss.
This is an example of calculating a discrete probability

distribution for potential returns.
Expected Return for a Single
Investment
Expected Return of A = 0.2(15%) + 0.5(10%) + 0.3(-5%)
(That is, a 20%, or .2, probability times a 15%, or .15, return;

plus a 50%, or .5, probability times a 10%, or .1, return; plus a
30%, or .3, probability of a return of negative 5%, or -.5)
= 3% + 5% – 1.5%
= 6.5%
Therefore, the probable long-term average return for
Investment A is 6.5%.
Expected Return for Portfolio
let’s assume the portfolio is comprised of investments in three

assets – X, Y, and Z. $2,000 is invested in X, $5,000 invested in
Y, and $3,000 is invested in Z. Assume that the expected
returns for X, Y, and Z have been calculated and found to be
15%, 10%, and 20%, respectively. Based on the respective
investments in each component asset, the portfolio’s expected
return can be calculated as follows:
Expected Return for Portfolio
Expected Return of Portfolio = 0.2(15%) + 0.5(10%) +

0.3(20%)
= 3% + 5% + 6%
= 14%
Thus, the expected return of the portfolio is 14%.

Measurement of the Co-
movement of security returns
Covariance :
Covariance is a statistical tool that is used to determine the

relationship between the movement of two asset prices.
When two stocks tend to move together, they are seen as

having a positive covariance; when they move inversely,
the covariance is negative.
Covariance is a significant tool in modern portfolio theory used

to ascertain what securities to put in a portfolio.
Risk and volatility can be reduced in a portfolio by

Measurement of the Co-
movement of security returns
Covariance :
Positive which highlight that the returns on two securities try to

move in one direction at the same time. This means that when
return of one security increases, the other try to copy the same.
The positive covariance results in a positive correlation
coefficient.
Negative which point out that the returns on two securities try
to move in opposite direction. This means that when return of
one security increases, the other try to decrease. The negative
covariance result into negative correlation coefficient.
Zero which point out that the returns on two securities are
Stock returns from the Excelsior Corporation and the
Adirondack Corporation from the years 2008 to
2012, as shown here:
Year Excelsior Adirondack
Corp. Corp.
Annual Annual
Return Return
(percent) (percent)
(X) (Y)
2008 1 3
2009 –2 2
2010 3 4
2011 0 6
What 2012
are the covariance
3 and 0
correlation between the stock
returns?
Step 1: The sample mean of X & Y is
This table shows the remaining calculations for the sample covariance:
The denominator equals the sample size minus one, which is 5 – 1
= 4. (Both samples have five elements, n = 5.) Therefore, the
sample covariance equals
To calculate the sample correlation coefficient, divide the sample covariance
by the product of the sample standard deviation of X and the sample
standard deviation of Y:
To calculate the sample correlation coefficient, divide the sample covariance
by the product of the sample standard deviation of X and the sample
standard deviation of Y:
Portfolio Standard
Deviation
Portfolio of 3 Assets
» Consider the portfolio of three securities.
» The correlation coefficients are shown in the correlation matrix

below.
Calculation Example 2
(Cont…)
» Putting these values into the formula;
Calculation Example 2
(Cont…)
» Putting these values into the formula;
Efficient Frontier
» Different combinations of securities produce different levels of

return.
» The efficient frontier represents the best of these securities

combinations -- those that produce the maximum expected
return for a given level of risk.
» The efficient frontier is the basis for modern portfolio theory.
» According to Markowitz, for every point on the efficient frontier,

there is at least one portfolio that can be constructed from all
available investments that has the expected risk and return
corresponding to that point.
Efficient Frontier
Efficient Frontier
» Imagine a 50/50 allocation between just two securities. Assuming

that the year-to-year performance of these two securities is not
perfectly in sync -- that is, assuming that the great years and the
lousy years for Security 1 don't correspond perfectly to the great
years and lousy years for Security 2.
» The standard deviation of the 50/50 allocation will be less than

the average of the standard deviations of the two securities
separately.
Efficient Frontier
Efficient Frontier
» The relationship securities have with each other is an important

part of the efficient frontier. Some securities' prices move in the
same direction under similar circumstances, while others move in
opposite directions.
» The more out of sync the securities in the portfolio are (that is,
the lower their covariance), the smaller the risk (standard
deviation) of the portfolio that combines them.
» The efficient frontier is curved because there is a diminishing

marginal return to risk. Each unit of risk added to a portfolio
gains a smaller and smaller amount of return.
Example
You are considering two assets, Asset A and Asset B, for investment.
The following table shows your expectation of their expected return
andAsset
standard deviation:
E(R) Standard Deviation
A 12% 6%
B 8% 4%
Correlation coefficient between

returns on both assets is expected to
be 0.6
Exam
ple
We can invest all our money in A or B or in some combination of A

and B. The following table shows our portfolio’s expected return and
standard deviation at different mix of A and B.
Portfolio
Portfolio
Asset A Weight Asset B Weight Standard
Expected Return
Deviation
100% 0% 6.00% 12.00%
80% 20% 4.87% 11.20%
60% 40% 3.94% 10.40%
40% 60% 3.39% 9.60%
20% 80% 3.42% 8.80%
0% 100% 4.00% 8.00%
Exam
ple
The results above can be plotted to get the following graph. It is

called efficient frontier.
Any
query
Thankyo could
be
u asked
at
email
mentio
ned at

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Portfolio

The expected return on an investment is the expected value of

The return on the investment is an unknown variable that has

Expected return is calculated by multiplying potential outcomes

Let us take an investment A, which has a 20% probability of

This is an example of calculating a discrete probability

Expected Return of A = 0.2(15%) + 0.5(10%) + 0.3(-5%)

(That is, a 20%, or .2, probability times a 15%, or .15, return;

let’s assume the portfolio is comprised of investments in three

Expected Return of Portfolio = 0.2(15%) + 0.5(10%) +

Thus, the expected return of the portfolio is 14%.

Covariance is a statistical tool that is used to determine the

When two stocks tend to move together, they are seen as

Covariance is a significant tool in modern portfolio theory used

Risk and volatility can be reduced in a portfolio by

Positive which highlight that the returns on two securities try to

» The correlation coefficients are shown in the correlation matrix

» Different combinations of securities produce different levels of

» The efficient frontier represents the best of these securities

» The efficient frontier is the basis for modern portfolio theory.

» According to Markowitz, for every point on the efficient frontier,

» Imagine a 50/50 allocation between just two securities. Assuming

» The standard deviation of the 50/50 allocation will be less than

» The relationship securities have with each other is an important

» The efficient frontier is curved because there is a diminishing

Correlation coefficient between

We can invest all our money in A or B or in some combination of A

The results above can be plotted to get the following graph. It is

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