Chapter
11
Diversification and
Risky Asset Allocation
Learning Objectives
To get the most out of this chapter,
spread your study time across:
1. How to calculate expected returns and variances
for a security.
2. How to calculate expected returns and variances
for a portfolio.
3. The importance of portfolio diversification.
4. The efficient frontier and importance of asset
allocation.
2
A Simple Example of Diversification using Historical Returns:
ExxonMobil (XOM) and Panera Bread (PNRA)
3
Expected Returns, I.
• Expected return is the “weighted average” return on a risky asset,
from today to some future date. The formula is:
expected return i   p s  return i,s 
n
s 1
• To calculate an expected return, you must first:
–
–
–
–
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 equally likely--they can have different
probabilities of occurring.
4
Calculating Expected Returns
5
Expected Return, II.
• Suppose:
– There are two stocks:
• Starcents
• Jpod
– We are looking at a period of one year.
• Investors agree that the expected return:
– for Starcents is 25 percent
– for Jpod is 20 percent
• Why would anyone want to hold Jpod shares when
Starcents is expected to have a higher return?
6
Expected Return, III.
• The answer depends on risk
• Starcents is expected to return 25 percent
• But the realized return on Starcents could be significantly
higher or lower than 25 percent
• Similarly, the realized return on Jpod could be
significantly higher or lower than 20 percent.
7
Expected Risk Premium
• Recall:
Expected Risk Premium  Expected Return  Riskfree Rate
• Suppose riskfree investments have an 8% return. If so,
– The expected risk premium on Jpod is 12%
– The expected risk premium on Starcents is 17%
• This expected risk premium is simply the difference between
– The expected return on the risky asset in question and
– The certain return on a risk-free investment
8
Calculating the Variance of
Expected Returns
• The variance of expected returns is calculated using this formula:
n

Variance  σ   p s  return s  expected return 
2
s 1
2

• This formula is not as difficult as it appears.
• This formula says is to 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”).
• The standard deviation is simply the square root of the variance.
Standard Deviation  σ  Variance
9
Example:
Calculating
Expected
Returns and
Variances:
Equal State
Probabilities
Calculating Expected Returns:
Starcents:
Jpod:
(1)
(2)
(3)
(4)
(5)
(6)
Return if
Return if
State of
Probability of
State
Product:
State
Product:
Economy
State of Economy
Occurs (2) x (3)
Occurs
(2) x (5)
Recession
0.50
-0.20
-0.10
0.30
0.15
Boom
0.50
0.70
0.35
0.10
0.05
Sum:
1.00
E(Ret):
0.25
E(Ret):
0.20
Calculating Variance of Expected Returns:
Starcents:
(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 = the Variance:
(1)
(2)
State of
Probability of
Economy
State of Economy
Recession
0.50
Boom
0.50
Sum:
1.00
Standard Deviation:
Jpod:
(5)
(6)
(3)
(4)
Return if
State Expected Difference: Squared:
Occurs Return:
(3) - (4)
(5) x (5)
0.30
0.20
0.10
0.0100
0.10
0.20
-0.10
0.0100
Sum is Variance:
Standard Deviation:
(7)
Product:
(2) x (6)
0.10125
0.10125
0.20250
0.45
(7)
Product:
(2) x (6)
0.00500
0.00500
0.01000
0.10
10
Expected Returns and Variances,
Starcents and Jpod
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).
12
Portfolios: Expected Returns
• The expected return on a portfolio is a linear
combination, or weighted average, of the expected
returns on the assets in that portfolio.
• The formula, for “n” assets, is:
n
ERP    w i  ERi 
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
13
Example: Calculating Portfolio Expected Returns
Note that the portfolio weight in Jpod = 1 – portfolio weight in Starcents.
Calculating Expected Portfolio Returns:
(1)
(2)
State of
Economy
Recession
Boom
Prob.
of State
0.50
0.50
Sum:
1.00
(3)
(4)
Starcents
Return if
Portfolio
State
Weight
Occurs in Starcents:
-0.20
0.40
0.70
0.40
(5)
Starcents
Contribution
Product:
(3) x (4)
-0.08
0.28
(6)
(7)
Jpod
Return if Portfolio
State
Weight
Occurs in Jpod:
0.30
0.60
0.10
0.60
(8)
Jpod
Contribution
Product:
(6) x (7)
0.18
0.06
(9)
Sum:
(5) + (8)
0.10
0.34
Sum is Expected Portfolio Return:
(10)
Portfolio
Return
Product:
(2) x (9)
0.050
0.170
0.220
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RP    p s  ERp,s   ERP 
n
s 1
2

• 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.
15
Variance of Portfolio
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
0.10
0.220
-0.12
0.0144
Boom
0.50
0.34
0.220
0.12
0.0144
Sum:
1.00
Sum is Variance:
Product:
(2) x (6)
0.00720
0.00720
0.01440
Standard Deviation:
0.120
(7)
16
It is possible to construct a portfolio of risky assets
with zero portfolio variance!
• What? How? (Open this spreadsheet, scroll up, and set
the weight in Starcents to 2/11ths.
Calculating Expected Portfolio Returns:
(1)
(2)
State of
Economy
Recession
Boom
Sum:
(1)
State of
Economy
Recession
Boom
Sum:
(3)
Starcents
Return if
Prob.
State
of State
Occurs
0.50
-0.20
0.50
0.70
1.00
(4)
Portfolio
Weight
in Starcents
0.18
0.18
(5)
Starcents
Contribution
Product:
(3) x (4)
-0.04
0.13
(6)
Jpod
Return if
State
Occurs
0.30
0.10
(8)
(9)
Jpod
Portfolio Contribution
Weight
Product:
Sum:
in Jpod
(6) x (7) (5) + (8)
0.82
0.25
0.21
0.82
0.08
0.21
Sum is Expected Portfolio Return:
Calculating Variance of Expected Portfolio Returns:
(2)
(3)
(4)
(5)
(6)
Return if
Prob.
State
Expected
Difference: Squared:
of State Occurs:
Return:
(3) - (4)
(5) x (5)
0.50
0.209
0.209
0.00
0.0000
0.50
0.209
0.209
0.00
0
1.00
(7)
Sum is Variance:
(10)
Portfolio
Return
Product:
(2) x (9)
0.105
0.105
0.209
(7)
Product:
(2) x (6)
0.00000
0.00000
0.00000
17
Calculating return and risk of portfolio:
Three Stocks
• Example: Suppose you had $1 million to invest
on stocks. You bought stock A for $500,000,
stock B for $250,000, and stock C for $250,000,
respectively. Your brokerage firm sent the
following projections on these stocks. What are
the portfolio’s expected returns and standard
deviations?
Returns
State of
Economy Probability Stock A
Boom
0.4
10%
Bust
0.6
8%
Stock B
15%
4%
Stock C
20%
0%
18
Calculating returns of portfolio
• Step One: Calculate the weighted average of returns for
each of given economy status
– Expected Return of portfolio for “Boom” Economy
= (500K/1,000K)10% + (250K/1,000K)15% + (250K/1,000K)20%
= 13.75%
– Expected Return of portfolio for “Bust” Economy
= 5%
• Step Two: Compute the expected return of portfolio as
we did for the single stock
– Expected Return of Portfolio = 13.75% (.4) + 5% (.6) = 8.50%
19
Calculating Risk: Portfolio Case
State of
Economy Probability
Boom
0.4
Bust
0.6
1
Rate of
Return
13.75%
5.00%
Expected
Product
Return
0.055
8.50%
0.03
8.50%
8.50%
Return Deviation
from Expected
Return
0.0525
-0.035
Squared
Deviation
0.002756
0.001225
Product
0.001103
0.000735
0.001838
4.29%
20
Diversification
• Intuitively, we all know that if you hold many investments
• 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?
21
Diversification and Asset Allocation
• Remember that 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.
22
Diversification and Risk, I.
23
Portfolio Standard Deviations
• In the beginning of this semester, we learned that the
standard deviation of the annual return on a portfolio of
large company common stocks was about 20 percent per
year.
• Spreading an investment across many assets will
eliminate some of the risk. (Diversifiable risk)
• However, there is a minimum level of risk that cannot be
eliminated by simple diversifying. (Non-diversifiable risk)
24
Diversification and Risk, II.
25
Systematic and Unsystematic Risk
• Systematic risk is risk that influences a large number of
assets. Also called market risk.
• Unsystematic risk is risk that influences a single
company or a small group of companies. Also called
unique risk or firm-specific risk.
Total risk = Systematic risk + Unsystematic risk
26
The Fallacy of Time Diversification, I.
• Young people are often told that they should hold a large
percent of their portfolio in stocks.
• The advice could be correct, but often the typical
argument used to support this advice is incorrect.
– The Typical Argument: Even though stocks are more
volatile, over time, the volatility “cancels out.”
– Sounds logical, but the typical argument is incorrect.
• This argument is the fallacy of time diversification fallacy
• How can such a plausible argument be incorrect?
27
The Fallacy of Time Diversification, II.
•
How can such logical-sounding advice be bad?
– You might remember from your statistics class that we can add
variances.
– This fact means that an annual variance grows each year by multiplying
the annual variance by the number of years.
•
Standard deviations cannot be added together: An annual standard deviation
grows each year by the square root of the number of years.
•
As we showed earlier in the chapter, a randomly selected portfolio of largecap stocks has an annual standard deviation of about 20%.
– If we held this portfolio for 16 years, the standard deviation would be
about 80 percent, which is 20 percent multiplied by the square root of 16.
– Bottom line: Volatility increases over time—volatility does not
“cancel out” over time.
•
Investing in equity has a greater chance of having an extremely large
value AND increases the probability of ending with a really low value.
28
The Very Definition of Risk—a Wider Range of
Possible Outcomes from Holding Equity
29
So, Should Younger Investors Put a
High Percent of Their Money into Equity?
• The answer is probably still yes, but for logically sound reasons that
differ from the reasoning underlying the fallacy of time diversification.
• If you are young and your portfolio suffers a steep decline in a
particular year, what could you do?
•
You could make up for this loss by changing your work habits (e.g.,
your type of job, hours, second job).
• People approaching retirement have little future earning power, so a
major loss in their portfolio will have a much greater impact on their
wealth.
• Thus, the portfolios of young people should contain relatively more
equity (i.e., risk).
30
Why Diversification Works,
• 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.
31
Why Diversification Works,
• Covariance:
– A measure of how the returns for two securities moves together.
n


COV ( AB)   ( RAi  R A )( RBi  R B ) Pi
i 1
• Correlation Coefficient
– Correlation is a standardized covariance scaled by the product of
two standard deviations, so that it can take a value between -1
and +1. The correlation coefficient is denoted by Corr(RA, RB) or
simply, A,B.
 A, B 
COV ( A, B )
 A B
32
33
Why Diversification Works,
34
Combining Negatively Correlated Assets to
Diversify Risk
35
Investing on Metals and Business Cycle
36
Why Diversification Works
37
Why Diversification Works,
38
Calculating Portfolio Risk
• For a portfolio of two assets, A and B, the variance of the return on
the portfolio is:
σ p2  x 2A σ 2A  xB2 σB2  2x A xBCOV(A,B)
σ p2  x 2A σ 2A  xB2 σB2  2x A xBσ A σ BCORR(R ARB )
Where: xA = portfolio weight of asset A
xB = portfolio weight of asset B
such that xA + xB = 1.
(Important: Recall Correlation Definition!)
39
The Importance of Asset Allocation, Part 1.
• 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?
40
Correlation and Diversification
41
Asset Allocation
• With 100 percent in the stock fund, the expected return is
12% and our standard deviation is 15%.
• As we being to move out of the stock fund and into the
bond fund, we are not surprised to see both expected
return and the standard deviation declines.
• However, what might be surprising to you is the fact hat
the standard deviation falls only so far and then begins to
rise again.
• In other words, beyond a point, adding more of the lower
risk bond fund actually increases your risk!
42
Correlation and Diversification
43
Correlation and Diversification
• 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.
44
More on Correlation & the Risk-Return Trade-Off
(The Next Slide is an Excel Example)
45
Example: Correlation and the
Risk-Return Trade-Off, Two Risky Assets
Inputs
Return
Deviation
Risky Asset 1
14.0%
20.0%
Risky Asset 2
8.0%
15.0%
Correlation
Risk-Return Profile
-0.20
80
Percentage
in Risky
Standard Expected
Asset 1
Deviation Return
-60.0%
28.9%
4.4%
-50.0%
26.4%
5.0%
-40.0%
23.9%
5.6%
-30.0%
21.5%
6.2%
-20.0%
19.2%
6.8%
Exp Ret
Portfolio
18.0%
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
20.0%
SD
40.0%
46
The Markowitz Efficient Frontier
• The Markowitz Efficient frontier is the set of portfolios with
the maximum return for a given risk AND the minimum
risk given a return.
• For the plot, the upper left-hand boundary is the
Markowitz 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.
47
The Efficient Frontier with Two Stocks
48
The Efficient Frontier:
More General Case with Numerous Assets
49
The Importance of Asset Allocation, Part 2.
• We can illustrate the importance of asset allocation with 3 assets.
• How? Suppose we invest in three mutual funds:
• One that contains Foreign Stocks, F
• One that contains U.S. Stocks, S
• One that contains U.S. Bonds, B
Expected Return
Standard Deviation
Foreign Stocks, F
18%
35%
U.S. Stocks, S
12
22
U.S. Bonds, B
8
14
• Figure 11.6 shows the results of calculating various expected
returns and portfolio standard deviations with these three assets.
50
Risk and Return with Multiple Assets, I.
51
Risk and Return with Multiple Assets, II.
• Figure 11.6 used these
formulas for portfolio return
and variance:
• But, we made a simplifying
assumption. We assumed
that the assets are all
uncorrelated.
• If so, the portfolio variance
becomes:
rp  x FRF  x SR S  x BRB
σ p2  x F2σ F2  x S2 σ S2  x B2 σ B2
 2x F x Sσ Fσ SCORR(R FR S )
 2x F x Bσ Fσ BCORR(R FRB )
 2x S x Bσ Sσ BCORR(R SRB )
σ p2  x F2σ F2  x S2 σ S2  x B2 σ B2
52
Useful Internet Sites
•
•
•
•
•
•
www.411stocks.com (to find expected earnings)
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 reading list)
53
Chapter Review, I.
•
Expected Returns and Variances
– Expected returns
– Calculating the variance
•
Portfolios
– Portfolio weights
– Portfolio expected returns
– Portfolio variance
•
Diversification and Portfolio Risk
– The effect of diversification: Another lesson from market history
– The principle of diversification
•
Correlation and Diversification
– Why diversification works
– Calculating portfolio risk
– More on correlation and the risk-return trade-off
•
The Markowitz Efficient Frontier
– Risk and return with multiple assets
54
• There is some combination of stock and bonds which
produces a lower risk portfolio than that which is
obtainable from an all-bond portfolio.
A) True
B) False
55
• Which one of the following statements is correct?
A) One hundred stocks are required to eliminate the bulk of
the diversifiable risk from a portfolio.
B) Standard deviation measures diversifiable risk.
C) The number of stocks needed to highly diversify a
portfolio is constant over time.
D) A fully diversified portfolio still contains nondiversifiable
risk.
56
• You own a portfolio consisting of two stocks. The stocks
have a zero correlation. If stock A increases its return by
10 percent over the course of one year, the return on
Stock B will:
A) decrease by 10 percent.
B) decrease by up to 10 percent.
C) change in an unpredictable manner.
D) remain constant.
E) increase by 10 percent.
57
• Tracy just inherited a portfolio valued at $50,000 from her
grandmother. The portfolio consists of two stocks. The
first stock is worth $28,000 and has a standard deviation
of 5.4 percent. The second stock has a standard
deviation of 11.8 percent. What is the portfolio standard
deviation if the correlation of these two stocks is .62?
A) 6.28 percent
B) 6.39 percent
C) 6.81 percent
D) 7.45 percent
E) 7.54 percent
58
• All efficient portfolios which consist of the same two
assets plot at or above the:
A) maximum obtainable rate of return.
B) maximum variance portfolio.
C) minimum variance portfolio.
D) average portfolio rate of return.
E) average portfolio standard deviation.
59
• Adding foreign securities to a portfolio containing
domestic stocks and bonds tends to shift the Markowitz
efficient frontier:
A) downward and to the left.
B) downward and to the right.
C) upward and to the left.
D) upward and to the right.
E) upward only.
60
• You are analyzing the possible portfolios which can be
created from combining bonds and stocks. If you select
the one portfolio that has the lowest standard deviation,
you have identified the _____ portfolio.
A) one and only efficient
B) minimum return
C) maximum return
D) minimum variance
E) inefficient
61