### Chapter 2 Diversification and Risky Asset Allocation

```Chapter 2
Diversification and
Risky Asset
Allocation



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Expected Return and Variances
Portfolios
Diversification and Portfolio Risk
Correlation and Diversification
Markowitz Efficient Frontier
Asset Allocation and Security
Selection Decisions of Portfolio
Formation
2-1
Diversification

Intuitively, we all know that if you hold many
investments
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Through time, some will increase in value
Through time, some will decrease in value
It is unlikely that their values will all change in
the same way
Diversification has a profound effect on
portfolio return and portfolio risk.
But, exactly how does diversification work?
2- 2
Diversification and Asset Allocation



Our goal in this chapter is to examine the role of
diversification and asset allocation in investing.
In the early 1950s, professor Harry Markowitz was the first to
examine the role and impact of diversification.
Based on his work, we will see how diversification works, and
we can be sure that we have “efficiently diversified
portfolios.”
 An efficiently diversified portfolio is one that has the
highest expected return, given its risk.
 You must be aware of the difference between historic
return and expected return!
2012 McGraw-Hill Ryerson
2- 3
Expected Return


Expected return is the “weighted average” return on a risky
asset, from today to some future date. The formula is:
To calculate an expected return, you must first:
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Decide on the number of possible economic scenarios that might occur,
Estimate how well the security will perform in each scenario, and
Assign a probability to each scenario
(BTW, finance professors call these economic scenarios, “states.”)
The next slide shows how the expected return formula is used
when there are two states.

Note that the “states” are equally likely to occur in this example. BUT!
They do not have to be. They can have different probabilities of
occurring.
n
expected
return
i

 p
s
 return
i, s

s 1
2- 4
Expected Return II




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Suppose there are two stocks: NetCap and JMart
We are looking at a period of one year.
Investors agree that the expected return:
for NetCap is 25 percent and for JMart is 20 percent
Why would anyone want to hold JMart shares when
NetCap is expected to have a higher return?
Jmart is expected to return 20 percent
But the realized return on Jmart could be significantly
higher or lower than 20 percent
Similarly, the realized return on NetCap could be
significantly higher or lower than 25 percent.
2- 5
Expected Return
2- 6

Recall:
expected


i
 expected
return
i
 riskfree
rate
Suppose riskfree investments have an 8% return.
So, in the previous slide, the risk premium expected on Jmart
is 12%, and 17% for Netcap.
This expected risk premium is the difference between the
expected return on the risky asset in question and the certain
return on a risk-free investment.
2- 7
Calculating the Variance of Expected Returns

The variance of expected returns is calculated as the sum of the
squared deviations of each return from the expected return,
multiplied by the probability of the state. Here is the formula:
  p  return
n
Variance
i
s
i, s
 expected
return

2
i

s 1

This formula is not as difficult as it appears. It says:
̶
add up the squared deviations of each return from its expected
return
̶
after it has been multiplied by the probability of observing a
particular economic state (denoted by “s”).
2- 8
Standard Deviation

The standard deviation is simply the square root of the
variance.
Standard

Deviation
 
Variance
The following slide contains an example that shows how
to use these formulas in a spreadsheet.
2- 9
Expected Returns and Variances
Equal State Probabilities
Calculating Expected Returns:
Netcap:
(1)
(2)
(3)
(4)
Return if
State of
Probability of
State
Product:
Economy
State of Economy Occurs (2) x (3)
Recession
0.50
-0.20
-0.10
Boom
0.50
0.70
0.35
Sum:
1.00
E(Ret):
0.25
Jmart:
(5)
Return if
State
Occurs
0.30
0.10
E(Ret):
(6)
Product:
(2) x (5)
0.15
0.05
0.20
Calculating Variance of Expected Returns:
Netcap:
(1)
(2)
(3)
(4)
(5)
(6)
Return if
State of
Probability of
State Expected Difference: Squared:
Economy
State of Economy Occurs Return:
(3) - (4)
(5) x (5)
Recession
0.50
-0.20
0.25
-0.45
0.2025
Boom
0.50
0.70
0.25
0.45
0.2025
Sum:
1.00
Sum is Variance:
Standard Deviation:
(7)
Note that
the second
is only for
Netcap.
What would
you get for
Jmart?
Product:
(2) x (6)
0.10125
0.10125
0.20250
0.45
2- 10
Expected Returns and Variances,
Netcap and Jmart
2- 11
Portfolios



Portfolios are groups of assets, such as stocks
and bonds, that are held by an investor.
One convenient way to describe a portfolio is
by listing the proportion of the total value of the
portfolio that is invested into each asset.
These proportions are called portfolio weights.


Portfolio weights are sometimes expressed in
percentages.
However, in calculations, make sure you use
proportions (i.e., decimals).
2- 12
Portfolios: Expected Returns

The expected return on a portfolio is the weighted average, of
the expected returns on the assets in that portfolio. The
formula, for “n” assets, is:
E R P
n
   w i  E R i 
i1
In the formula:
E(RP) =
expected portfolio return
wi = portfolio weight in portfolio asset i
E(Ri) =
expected return for portfolio asset i
2- 13
Example: Calculating Portfolio Expected Returns
Note that the portfolio weight in Jmart = 1 – portfolio
weight in Netcap.
Calculating Expected Portfolio Returns:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Netcap
Netcap
Jmart
Jmart
Portfolio
Return if Portfolio Contribution Return if Portfolio Contribution
Return
State of
Prob.
State
Weight
Product:
State Weight
Product:
Sum: Product:
Economy of State Occurs in Netcap:
(3) x (4)
Occurs in Jmart: (6) x (7)
(5) + (8) (2) x (9)
Recession 0.50
-0.20
0.50
-0.10
0.30
0.50
0.15
0.05
0.025
Boom
0.50
0.70
0.50
0.35
0.10
0.50
0.05
0.40
0.200
Sum:
1.00
Sum is Expected Portfolio Return:
0.225
2- 14
Variance of Portfolio Expected Returns


Note: Unlike returns, portfolio variance is generally
not a simple weighted average of the variances of the
assets in the portfolio.
If there are “n” states, the formula is:
VAR R P  
 p  E R   E R  
n
2
s
p, s
P
s 1


In the formula,
VAR(RP) = variance of portfolio expected return
ps = probability of state of economy, s
E(Rp,s) = expected portfolio return in state s
E(Rp) = portfolio expected return
Note that the formula is like the formula for the variance of the
expected return of a single asset.
2- 15
Calculating Variance of Portfolio Returns

It is possible to construct a portfolio of risky assets
with zero portfolio variance! What? How? (Open the
spreadsheet and set the weight in Netcap to 2/11ths.)
Calculating Variance of Expected Portfolio Returns:
(1)
(2)
(3)
(4)
(5)
(6)
Return if
State of
Prob.
State Expected Difference: Squared:
Economy of State Occurs: Return:
(3) - (4)
(5) x (5)
Recession
0.50
0.05
0.225
-0.18
0.0306
Boom
0.50
0.40
0.225
0.18
0.030625
Sum: 1.00
Sum is Variance:
Standard Deviation:
Limited
(7)
Product:
(2) x (6)
0.01531
0.01531
0.03063
0.175
2- 16
Diversification and Risk, I.
Limited
2- 17
Diversification and Risk, II.
2- 18
Why Diversification Works, I.
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Correlation: The tendency of the returns on two assets to
move together. Imperfect correlation is the key reason why
diversification reduces portfolio risk as measured by the
portfolio standard deviation.
Positively correlated assets tend to move up and down
together.
Negatively correlated assets tend to move in opposite
directions.
Imperfect correlation, positive or negative, is why
diversification reduces portfolio risk.
2- 19
Why Diversification Works, II.


The correlation coefficient is denoted by Corr(RA, RB)
or simply, A,B.
The correlation coefficient measures correlation and
ranges from:
From:
-1 (perfect negative correlation)
Through:
To:
0 (uncorrelated)
+1 (perfect positive correlation)
Limited
2- 20
Why Diversification Works, III.
2- 21
Why Diversification Works
McGraw-Hill Ryerson
Why Diversification Works
McGraw-Hill Ryerson
Calculating Portfolio Risk

For a portfolio of two assets, A and B, the variance of
the return on the portfolio is:

2
p
  A
2
A
  B  B  2 A  B COV ( A , B )

2
p
  A
2
A
  B  B  2 A  B  A  B CORR ( R A R B )
2
2
2
2
2
2
Where: wA = portfolio weight of asset A
wB = portfolio weight of asset B
such that wA + wB = 1.
(Important: Recall Correlation Definition!)
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Correlation and Diversification

Suppose that as a very conservative, risk-averse
investor, you decide to invest all of your money in
a bond mutual fund. Very conservative, indeed?
Uh, is this decision a wise one?
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Correlation and Diversification
Note: A correlation of 0.10 is assumed in calculations
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Correlation and Diversification
2- 27
Correlation and Diversification
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The various combinations of risk and return available
all fall on a smooth curve.
This curve is called an investment opportunity set,
because it shows the possible combinations of risk
and return available from portfolios of these two
assets.
A portfolio that offers the highest return for its level
of risk is said to be an efficient portfolio.
The undesirable portfolios are said to be dominated
or inefficient.
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Correlation and Risk &Return Trade off
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Expected Standard
Inputs
Return Deviation
Risky Asset 1
14.0%
20.0%
Risky Asset 2
8.0%
15.0%
Correlation
30.0%
Expected Return
Efficient Set--Two Asset Portfolio
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
0%
5%
10%
15%
20%
25%
30%
Standard Deviation
2- 30
Risk and Return with Multiple Assets
2- 31
The Markowitz Efficient Frontier

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The Markowitz Efficient frontier is the set of portfolios
with the maximum return for a given risk AND the
minimum risk for a given a return.
For the plot, the upper left-hand boundary is the
Markowitz efficient frontier.
Efficient Frontier
All the other possible combinations are inefficient.
That is, investors would not hold these portfolios
because they could get either


more return for a given level of risk, or
less risk for a given level of return.
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Portfolio Formation Decisions

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Investors can form their using the following financial
assets :
1- Domestic Stocks, 2- Domestic Bonds (government,
municipal, provincial bonds, corporate bonds) 3Derivative Instruments (options, futures, swaps), 4Money market instruments (T-Bills, certificate of
deposits, repurchase agreements, etc), 5- Foreign
Stocks and Bonds.
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Asset Allocation and Security Selection
Investors face with two problems when they form
portfolios of multiple securities from different asset
classes.
1. Asset Allocation Decision
Asset allocation problem involves a decision of what
percentage of investor’s portfolio should be allocated
among different asset classes (stocks, bonds, derivatives,
money market instruments, foreign securities).
2. Security Selection Decision
Security selection is deciding which securities to pick in
each class and what percentage of funds to allocate to
these securities (for example choosing different stocks and
their percentages within the asset class).
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Portfolio Decisions
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Suppose you have \$100,000 for investment. First you have to
decide which assets you want to hold. For example, you may
decide to invest \$20,000 in money market instruments, \$30,000
in bonds and \$50,000 in domestic stocks. Then, your portfolio
consists of 20% of money market instruments, 30% of bonds
and 50% in stocks. This is your asset allocation decision.
Then you may decide within the stock part of your portfolio, to
allocate \$50,000 equally among the following five stocks:
Nortel, Research In Motion, Royal Bank, Bombardier, and
Molson. That decision, your security selection decision,
indicates that you will buy \$10,000 worth of stocks of each
company.
2- 35
Useful Internet Sites
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www.411stocks.com (to find expected returns)
www.investopedia.com (for more on risk measures)
www.teachmefinance.com (also contains more on risk
measure)
www.morningstar.com (measure diversification using “instant
x-ray”)
www.moneychimp.com (review modern portfolio theory)
www.efficentfrontier.com (check out the online journal)
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