### Portfolio Theory II

```FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Portfolio Theory II
1
Learning objectives
1.
2.
3.
4.
5.
6.
Explain the benefit of diversification and identify the
source of this benefit.
Understand the concepts of covariance and correlation.
Compute covariance and correlation between two risky
assets.
Compute the expected return and standard deviation of
a portfolio of two risky assets.
Define the minimum variance portfolio and compute its
weights in the 2-risky asset case.
Define the investment opportunity set and the efficient
frontier.
2
Learning objectives
7.
8.
9.
10.
11.
12.
Compute the expected return and standard deviation of
a portfolio consisting of a risky asset/portfolio and a
risk-free asset.
Define the Capital Allocation Line.
Define and compute the reward-to-variability (Sharpe)
ratio.
Define the tangency portfolio and compute its weights
in a 2-risky asset case.
Describe the process of efficient diversification with
many risky asset and a risk-free asset.
Understand the idea of the separation property.
3
Portfolio Theory II: Concept Map
Diversification
Separation
Property
Portfolio
Portfolio
Theory
II
Efficient
frontier
Covariance,
Correlation
4
Why diversify?
To reduce risk.
This is the basic benefit of
diversification.
5
How does diversification work?
By exploiting the way different assets relate to
each other.
 We use a statistical measure called covariance
to characterize the tendency of two assets to
“move” with each other.



Covariance: A measure of the extent to which the
returns of two assets tend to vary with each other.
An equivalent and easier to interpret measure is
the correlation coefficient.
6
Simple example

Consider two assets: a stock and a bond. There
are 3 possible scenarios over the coming year:
HPR (%)
Scenario
Probability
Stock
Bond
Recession
Normal
0.3
0.4
-11
13
16
6
Boom
Expected return
S.D.
0.3
27
10
14.92
-4
6
7.75
7
Portfolio of 60% in stock & 40% in bond

What is the expected return and S.D. of this
portfolio?
Scenario
Prob. HPR
%
Prob.
X HPR
Deviation
from E( r )
Squared
deviation
Prob. X
squared
deviation
Recession
0.3
-0.2
-0.06
-8.60
73.96
22.188
Normal
0.4
10.2
4.08
1.80
3.24
1.296
Boom
0.3
14.6
4.38
6.20
38.44
11.532
E( r )
8.40
Variance
35.016
S.D.
5.92
8
Diversification lowers risk!




Compare the standard deviations of the stock, bond and
portfolio:
Stock
Bond
Portfolio
14.92%
7.75%
5.92%
The portfolio is actually less risky than either of the two
assets. Diversification lowers investment risk.
Why? Stock and bond returns tend to move in opposition
to each other.
How do we measure this tendency? -- Covariance
9
Covariance
Deviation from
mean return
Scenario
Covariance
Prob.
Stock
Bond
Product of
Dev
Prob. x
Product of Dev
Recession
0.3
-21
10
-210
-63
Normal
0.4
3
0
0
0
Boom
0.3
17
-10
-170
-51
Covariance
-114
HPR (%)
Scenario
Recession
Normal
Boom
Expected return
S.D.
Probability
0.3
0.4
Stock
-11
13
Bond
16
6
0.3
27
-4
10
6
14.92
7.75
10
Covariance formula

Suppose there are S possible scenarios
(i = 1,2,…,S), and the probability of scenario i is
p(i), then covariance between assets A and B is
S
Cov
(rA(r,Sr,BrB)) 
p
(
i
)[
r
(
i
)

r
][
r
(
i
)

r
]




Cov
p
(
i
)
r
(
i
)

r
r
(
i
)

r
A
A
B
B

S
S
B
B
S
ii11
Asset A’s return in
scenario i



Asset B’s return in
scenario i
11
Apply covariance formula

Go back to the example, but now suppose the returns on
the stock are -14%, 13% and +30%. Compute the
stock’s expected return, variance, and covariance with
bond.
HPR (%)
Scenario
Recession
Normal
Boom
Expected return
S.D.
Probability
0.3
0.4
Stock
-14
13
Bond
16
6
0.3
30
?
?
-4
6
7.75
12
Correlation between
assets A & B, rAB , is
r AB 

Cov(rA, rB )
 A B
Given this formula, we can write covariance as
Cov(rA, rB )  r AB A B
Going back to our example, the correlation between stock
and bond = -114/(14.92 x 7.75) = -0.99
13
Interpreting the
correlation coefficient

Correlations range from -1 to 1.
Correlation
Meaning
-1, Perfect negative correlation Strongest tendency for two
assets to vary inversely/
move in opposite direction.
+1, Perfect positive correlation Strongest tendency for two
assets to move in tandem.
0, Uncorrelated
Returns on the two assets
are unrelated to each other.
14
Three rules of two-risky asset
portfolio
Suppose you have a portfolio consisting of two risky
assets, A and B. You invest the proportion, wA, of the
portfolio in A and the proportion, wB , in B.
You know the following:
 The return on asset A is rA ; the return on asset B is rB.
 The expected return on asset A is E(rA) ; the expected
return on asset B is E(rB).
 The standard deviation of A is A; the standard deviation
of B is B.
 The correlation coefficient between the returns of A and
B is rAB.
15
Rule 1: Portfolio rate of return
A
portfolio’s rate of return (rP) is a
weighted average of the returns on the
component assets, with the investment
proportions as weights,
rP = wA rA + wB rB
16
Rule 2: Portfolio expected return
A
portfolio’s expected return is a weighted
average of the expected returns on the
component assets with the same portfolio
proportions as weights,
E(rP) = wA E(rA) + wB E(rB)
17
Rule 3: Portfolio variance
A
portfolio’s variance (P2) is
P2 = (wAA)2 + (wBB)2 + 2(wAA) (wBB)rAB

Since cov(rA,rB) = ABrAB , we can also write
portfolio variance as
P2 = (wAA)2 + (wBB)2 + 2wAwBcov(rA,rB)

Taking the square root of P2 gives us the
portfolio standard deviation.
18
Applying these rules
Suppose you have two assets (A,B) with the
following details:
E(rA)=6%, E(rB)=10%, A=12%, B=25%, rAB=0
wA = 0.5

 Verify
that:
portfolio expected return is 8%
 portfolio SD is 13.87%

19
Discussion
 By
investing in two assets, the portfolio
volatility is smaller than the average of
volatilities of the two assets.
 Why
not use SD=(12+25)/2=18.5%?
20
Minimum variance portfolio

Weight on asset A is:
 B  r AB A B
WA  2
2
 A   B  2 r AB A B
2
Weight on asset B is = 1 – wA.
 Using this formula, verify that the min variance
portfolio has wA=81.27%, wB=18.73%.

21
Investment opportunity set

The set of all attainable combinations of risk and
return offered by portfolios formed using the
available assets in differing proportions.
B
Z
A
22
23
Mean variance criterion & the
efficient frontier
 Mean-variance
criterion: choosing
assets/portfolios based on the expected
return and variance (or SD) of portfolios.
 Using this criterion, we prefer portfolio A to
portfolio B (or A dominates B) if:
E(rA)  E(rB) and A  B

Efficient frontier: portfolios on the upward sloping
portion of the investment opportunity set.
24
Practice 2
 Chapter
5: CFA problems: 4,7,8,9,11.
 Chapter 6: CFA problems: 1, 3(a) only.
25
Homework 2
1
2
1. If required risk premium= 2 A , where A=4, and risk free rate = 1%, what are the
annual required rates of return for the following investments? What will be the value of
these investments after 2 years, if these investments have achieved the required return
rates?
Investment
standard deviation
value today
1
0.4
\$1,000
2
0.5
\$1,000
3
0.6
\$1,000
4
0.2
\$1,000
2. You are considering investing in 2 assets. A asset has an expected return of 15% and
standard deviation of 32%. B asset has an expected return of 9% and standard deviation
of 23%. The correlation between A and B is 0.15.
a. What is the covariance between A and B?
b. What is the weight of A and B in the minimum variance portfolio?
c. What is the expected return and variance of the minimum variance portfolio?
26
Extreme cases
Suppose rAB= 1,
P2 = (wAA + wBB)2
P = wAA + wBB

No gains from diversification only in this case!

Whenever r < 1, there are gains from diversification.

Suppose rAB= -1,
P2 = (wAA - wBB)2
P = ABS[ wAA – wBB ]
27
Investment opportunity set with
different correlations
B
A
28
Problem



Jane Marple has an \$800,000 fully diversified portfolio. She
subsequently inherits Rafael Aerospace Inc. stock worth \$200,000.
Her financial advisor provided her with the following information for
the coming year:
Expected
annual HPR (%)
Std Dev of annual
return (%)
Original portfolio
6.7
9.3
Rafael Aerospace
8.2
15.6
The correlation between the original portfolio and Rafael Aerospace
is 0.3.
Jane will keep the new stock. (a) Calculate the expected return and
standard deviation of the new portfolio. (b) Calculate the covariance
between the original portfolio and Rafael Aerospace.
29
Follow-up problems
1.
Jane decides to keep Rafael Aerospace but
wants to rebalance the new portfolio so that risk
is reduced to the minimum. What are the
weights, the expected return and std dev of the
minimum variance portfolio?
2.
Continuing with the previous problem, if Jane
keeps the original composition, what is the
portfolio expected return and SD if the
correlation is (a) 1, (b) -1.
30
Asset allocation involving
risky assets and a risk-free asset.
 What’s

a risk-free asset?
An asset that provides a sure (for certain)
nominal rate of return. It is a default free
asset. Examples:
 U.S.
Treasury securities (Treasury bills, Treasury
notes, Treasury bonds),
 Portfolios invested in these securities, e.g.,
T-bill money market fund.
 In practice, money market instruments like bank
certificate of deposits (CDs) and commercial paper
are treated as effectively risk-free.
31
Properties of risk-free asset
 There
is no uncertainty in return.
Variance of risk-free asset is 0
 Likewise, standard deviation is 0

 The
risk-free asset does not vary with a
risky asset/ portfolio. So,
Cov(risk-free asset, risky asset) = 0
 Corr(risk-free asset, risky asset) = 0

32
3 rules when there is one risk-free
asset & one risky asset (1)
P: risky asset/ portfolio (wP)
+
Complete portfolio, C
Risk-free asset (wf)
P
Return
rP
Expected return E(rP)
P
S.D.

Risk-free
rf
rf
0
What is the correlation between P and the risk-free asset?
33
3 rules when there is one risk-free
asset & one risky asset (2)
1) Portfolio return, rC = wP rP + wf rf
2) Portfolio expected return, E(rC) = wP E(rP) + wf rf
Portfolio risk premium, E(rC) - rf = wP [ E(rP) – rf ]
3) Portfolio variance, C2 = (wPP)2
Portfolio standard deviation, C = wPP
34
Capital Allocation Line (CAL)




If we plot the expected return and standard deviation
combinations of a complete portfolio, we get the Capital
Allocation Line (CAL).
CAL: The plot of risk-return combinations available by
varying portfolio allocation between a risk-free asset and
a risky asset/portfolio.
CAL is just the investment opportunity set when we have
a risk-free asset and a risky asset/portfolio!
Capital Market Line (CML): the CAL when the market
index portfolio is used as the risky portfolio.
35
Capital Allocation Line (CAL)
36
Apply these rules
Suppose the risky portfolio P has an expected
return of 15% and a standard deviation of 22%.
The risk-free asset promises a return of 7%.
 Compute the expected return, risk premium and
standard deviation of the complete portfolio if





100% of the portfolio is invested in P
100% of the portfolio is invested in the risk-free asset
Equal weight is placed on P and the risk-free asset.
Draw the CAL.
37
Results
Expected return
Portfolio SD
100% in P
(1 x 15) + (0 x 7) = 15
1 x [15 – 7] = 8
1 x 22 = 22
100% in risk-free
(0 x 15) + (1 x 7) = 7
0 x [15 – 7] = 0
0 x 22 = 0
Equal weight
(0.5 x 15) + (0.5 x 7) = 11
0.5 x [15 – 7] = 4
0.5 x 22 = 11
38
Capital Allocation Line (CAL)
Slope of CAL = increase in expected return per unit of additional
standard deviation (SD), i.e., the extra return per extra risk.
The slope is also called the Sharpe ratio, reward-to-variability
ratio.
39
Sharpe ratio, S

Sharpe ratio is the slope, so use “rise” over “run”.


For our example, Sharpe ratio= (15 – 7)/(22 – 0)
= 8/22
= 0.36
In general, Sharpe ratio of portfolio
S = portfolio risk premium / portfolio std dev

Sharpe ratio is the same for all portfolios that plot on
the CAL.
40
Problems involving risk-free
and risky assets
Assume you manage a risky portfolio with an expected rate of return
of 17% and a standard deviation of 27%. The T-bill rate is 7%.
a) Your client invests 70% of his portfolio in your fund and 30% in a Tbill money market fund. What is the expected return and standard
b) Suppose your risky portfolio includes the following investments in the
given proportions:
Stock A
27%

Stock B
33%
Stock C
40%
What are the investment proportions of your client’s overall portfolio,
including the position in the T-bills?
c) What is the Sharpe ratio of your risky portfolio and your client’s
overall portfolio?
d) Draw the CAL and point the positions of your and your client’s
portfolios.
41
Optimal risky portfolio
with a risk-free asset
When we have risky assets and a risk-free
asset, we can identify one single risky portfolio
that gives us the highest possible Sharpe
(reward-to-variability) ratio.
 This risky portfolio is the ‘optimal’ or ‘tangency’
portfolio.
 The CAL formed using the optimal risky portfolio
has the steepest slope.

42
Go back to the 2 risky assets: A and B but assume
correlation is 0.2. Now add a risk-free asset with 5% return
B
CAL2
CAL1
2
A
1
Recall: E(rA)=6%, E(rB)=10%, A=12%, B=25%, rAB=0.2.
43
Optimal risky portfolio weights
[ E (rA )  rf ] B  [ E (rB )  rf ] A B r AB
2
WA 
[ E (rA )  rf ] B  [ E (rB )  rf ] A  [ E (rA )  rf  E (rB )  rf ] A B r AB
2
2
WB  1  WA
Using the formulas, the weights in the optimal portfolio (O):
wA = 32.99%,
wB = 67.01%
Expected return, SD, Sharpe ratio:
E(rO) = 8.68%
O = 17.97%
SharpeO = (8.68 – 5)/17.97 = 0.20
44
Risk aversion and portfolio choice
Preferred complete portfolio: 55% in
Portfolio O, 45% in risk-free asset.
45
Consider the following


A pension fund manager is considering 3 mutual funds.
The first is a stock fund, the second is a corporate bond
fund, and the third is a T-bill money market fund that
yields a sure rate of 5.5%. The expected return and
sigma of the risky funds are:
Expected return
Std Dev
Stock fund (S)
14%
30%
Bond fund (B)
8
20
The correlation between the risky fund returns is 0.25.
46
1)Compute the expected return and standard deviation of
the minimum variance portfolio.
2)Compute the expected return and standard deviation of
the tangency portfolio.
3)What is the Sharpe ratio of the tangency portfolio?
4)Suppose you want to form a complete portfolio on the
CAL. The portfolio must yield an expected return of 12%.

What is the standard deviation of the portfolio?

What is the proportion invested in the T-bill fund and each
of the two risky funds?
47
Efficient diversification with many risky
assets & a risk-free asset (1)
Efficient diversification entails 3 separate steps:
1)
Form efficient frontier of risky assets

Efficient frontier: the collection of portfolios
that maximizes expected return at each level
of portfolio risk/ standard deviation.


Northwestern-most portfolios
Inputs: expected return and SD of every risky
asset, plus correlation coefficients between each
pair of assets.
48
Efficient frontier of risky assets
Inefficiently
diversified.
N
W
Dominated by portfolios on
efficient frontier.
E
S
49
Efficient diversification with many risky
assets & a risk-free asset (2)
2.
Use the risk-free rate and efficient frontier to
find the tangency portfolio or optimal risky
portfolio

3.
Recall that the tangency portfolio gives us the CAL
with the highest Sharpe ratio
Investor chooses the preferred complete
portfolio based on his/her risk aversion.

Each investors will use tangency portfolio in forming
his/her complete portfolio
50
Separation Property
The act of choosing the appropriate portfolio can
be separated into two independent tasks.
1. Find the optimal risky portfolio.


2.
This consists of steps 1 and 2 and is a purely
technical problem.
Given the input data, the best risky portfolio is the
same for all clients regardless of risk aversion.
Investor constructs complete portfolio using
risk-free asset and optimal risky portfolio.

Depends on the investor’s risk/personal preference.
Here the client is the decision maker.
51
Summary
Efficient diversification reduces risk.
 Benefit of diversification depends on how assets
co-vary with each other.
 Efficient frontier is the collection of portfolios
offering the highest expected return for each
level of risk.
 Introducing a risk-free asset allows us to identify
the tangency portfolio and the CAL.
 Tangency portfolio has the highest Sharpe ratio
and therefore is most desirable.
 Implications of the separation property.

52
Practice 3
 Chapter
5: 13,14,18,19.
 Chapter 6: 8,9,10,11,12,19.
53
Homework 3
1.
Assume you manage a risky portfolio with an expected rate of return of
20% and a standard deviation of 30%. The T-bill rate is 5%. Your client
invests 60% of his portfolio in your fund and 40% in a T-bill money market
fund. What is the expected return and standard deviation of your client’s
portfolio? What is the Sharpe ratio of your client’s portfolio?
2.
A asset has an expected return of 20% and standard deviation of 30%. B
asset has an expected return of 10% and standard deviation of 23%. C
asset is risk-free with a rate of 5%. The correlation between A and B is
0.15. a) What is the expected return and standard deviation of the optimal
risky portfolio? b) Suppose your complete portfolio must yield an expected
return of 15% and be efficient. What is the standard deviation of your
portfolio? c) What is the proportion of your portfolio invested in A and B,
respectively?
54
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