Diversification

Week 5
Investments FNCE30001

Dr Bryan Lim
Finance
 Warning

This material has been reproduced and communicated to you by

or on behalf of the University of Melbourne pursuant to Part VB
of the Copyright Act 1968 (the Act).

The material in this communication may be subject to copyright

under the Act.

Any further copying or communication of this material by you

may be the subject of copyright protection under the Act.

Do not remove this notice

Can we make money
trading stocks with
each other?

3
 Before Trading

Alison 1 share of MSFT

Bonnie $300 cash

Calvin $500 cash

Total 1 share of MSFT and $800 cash

First, Alison sells her share to Bonnie for $250

 After one trade

Alison $250 cash

Bonnie 1 share of MSFT and $50 cash

Calvin $500 cash

Total 1 share of MSFT and $800 cash

Next, Bonnie sells the share to Calvin for $150

 After two trades

Alison $250 cash

Bonnie $200 cash

Calvin 1 share of MSFT and $350 cash

Total 1 share of MSFT and $800 cash

 So why does trading exist?
Imagine a simple, tropical economy with just two stores
• You sell umbrellas, and I sell sunscreen

This is risky for both of us

• Rainy days: you make money
• Sunny days: I make money

Trading shares of our company with each other à hedge our risk
 Trading financial assets in
secondary markets…
DOES NOT increase the overall wealth in the economy

DOES allow us to consume a more reliable stream of income

 Last week
Last week, we constructed a portfolio of the form:

Asset Weight
Risky Asset (P) 25%
Risk-Free Asset 75%

But we did not specify how P was constructed

9
 How do we optimally
construct the portfolio
of risky assets?
“Modern Portfolio Theory”
“Markowitz Model”

10
 Lecture Overview
Diversification
Portfolios of Two Risky Assets
The Optimal Risky Portfolio
The Optimal Complete Portfolio
Portfolios of More than Two Risky Assets

11
 Reading
7.1 Diversification and Portfolio Risk
7.2 Portfolios of Two Risky Assets
7.3 Asset Allocation with Stocks, Bonds, and Bills
7.4 The Markowitz Portfolio Optimization Model

12
Diversification

13
 Diversification
Diversification: holding multiple assets to reduce the risk of the resulting
portfolio

You open a store near a tropical beach selling only sunscreen lotion and
umbrellas
• When it’s sunny, you sell lotion
• When it rains, you sell umbrellas

• No matter what the weather is, you make money

• à You are well diversified

14
 Two Main Types of Risk
1. Market risk (systematic risk; undiversifiable risk): risk that is attributable
to economy-wide risk sources

2. Diversifiable risk (nonsystematic risk; idiosyncratic risk): risk that is specific

to a firm but can be eliminated by diversification

15
Total Risk = Diversifiable Risk +
Market Risk

16
# of stocks in Portfolio vs Portfolio
Risk (s.d.)

17
Portfolios
of Two
Risky
Assets

18
 Probability Review: Covariance

State Probability rRIO rBHP rTRS

Good 20% 12% 13% 1%
Normal 35% 5% 8% 3%
Bad 45% -3% -4% 4%

Covariance between RIO and BHP:

!!"#,%&' = ∑$%&'())(%!"# ) − , %!"#

̃ )(%%&' ) − , %%&'
̃ )

• prob(s) = probability of state s

• r(s) is the return in state s

19
 Solving for covariance
First:
, %!"#
̃ = .028, , %%&'
̃ = .036, , %(!)
̃ = .0305

Then:
!!"#,%&' = 0.2 0.092×0.094
+0.35 0.022×0.044
+0.45 −0.058×−0.076
= 0.004052

!!"#,(!) = −0.00063, !%&',(!) = −0.00072

20
 Covariance vs. Correlation
Recall from probability theory: for two random variables "! and #!

$%* ", ! #!
! #! = )!,# =
$%&& ",
+! +#
Range of values: −1 ≤ ρX,Y ≤ +1

In the preceding example:

0.004052
)$%&,'() = = 0.984
0.058×0.071
−0.00063
)$%&,*$+ = = −0.987
0.058×0.011
−0.00072
)'(),*$+ = = 0.984
0.071×0.011
21
 Portfolios of Two Risky Assets
Consider a portfolio P:
• wD in a debt (bond) fund
• wE = 1 − wD in a equity (stock) fund

rP = wD rD + wE rE

From probability theory:

• Expected Return
, %'̃ = ;* , %*̃ + ;+ , %+̃
• Variance
!', = ;*, !*, + ;+, !+, + 2;* ;+ <&=(%*̃ , %+̃ )

22
 Minimising Portfolio Variance
All else equal, the lower the covariance is, the lower the portfolio variance is:

!', = ;*, !*, + ;+, !+, + 2;* ;+ <&=(%*̃ , %+̃ )

Even if the covariance term is positive (correlation coefficient is less than 1):

All else equal, investors should prefer to hold assets with low or negative
correlation with each other in their portfolios

23
 How low can portfolio variance
be?
If we had two perfectly negatively correlated assets in our portfolios
(impossible, in practice)…
à then the portfolio standard deviation could be reduced to zero by choosing
appropriate weights

On average, assets are positively correlated with each other (˜.3)

24
 Portfolio Variance: Example
Consider two risky portfolios called Debt and Equity:

Debt Equity
![#$] 8% 13%
S.D. 12% 20%
Correlation 0.3

What are the characteristics of a portfolio that is 40% D and 60% E?

25
 Portfolio Characteristics
when ρ< 1
E %'̃ = 0.4 ⋅ 8% + 0.6 ⋅ 13% = 11%

σ-,. = 0.3 ∗ 0.12 ∗ 0.20 = 0.0072

σ,/ = 0.40 , .12 , + 0.60 , 0.2 , + 2 0.40 0.60 0.0072

= 0.02
σ/ = 0.02 = 0.142

26
 What if...?
What if we erroneously calculated σP as a weighted average of the s.d.s of D
and E?
σweighted = (0.40)0.12 + (0.60)0.2 = 0.168

Key point:
ρ < 1 → σactual < σweighted
(portfolio standard deviation < weighted average of the individual securities’
standard deviations)

σweighted − σactual: measure of the diversification benefit

27
 wd $# !P]
&E[r σP
0% 13.0% 20.0%
• We can calculate E %'̃ 10% 12.5% 18.4%
and σp for all possible 20% 12.0% 16.9%
weights wD on D 30% 11.5% 15.5%
40% 11.0% 14.2%
50% 10.5% 13.1%
• This table lists the feasible
(E %'̃ ,σp) combinations 60% 10.0% 12.3%
from portfolios of D and E 70% 9.5% 11.7%
80% 9.0% 11.5%
90% 8.5% 11.6%
100% 8.0% 12.0%

28
 Feasible Portfolios of D and E
(Opportunity Set of Risky Assets)
Feasible Portfolios of D and E

E [r ]
E

29
 Correlation and the Opportunity
Set of Risky Assets
Feasible Portfolios of D and E

E [r ]
E

30
Optimal
Risky
Portfolio

31
 Efficient Portfolios
An efficient portfolio has the maximum possible expected return for a given
level of risk

Our goal: combine risky assets to generate a set of efficient portfolios

32
 Efficient Portfolios
Correlation and the Opportunity Set of Risky Assets

E [˜
r]
E

33
 Investment Opportunities
Having sketched out the curve (“opportunity set”) between D and E, we know
that
1. The investor can invest in any point A on the curve
2. The investor can invest in any point on a CAL through A.
3. If all investors are risk averse, they prefer portfolios with highest expected
return for a given level of standard deviation. (Recall the discussion of
“quadrants” from last week.)

34
 Capital Allocation Lines and the
Opportunity Set of Risky Assets
Correlation and the Opportunity Set of Risky Assets

E [˜
r]
E

35
 Choose P to maximise the Sharpe
Ratio
Since risk-averse investors want to be as far “northwest” as possible:
• Identify the portfolio P composed of D and E that yields the highest slope of
the resulting CAL
• Equivalent to finding the portfolio P with the highest Sharpe Ratio:

E %'̃ − %0
max
σ/
where

E rG' = w-E rG* + 1 − w- E rG+

σ/ = , σ, + w , σ, + 2w w cov r
w- G-, rG.
- . . - .

36
 Solving for Optimal wD
• Rewrite the problem in terms of wD
• Take the derivative with respect to wD and set it equal to zero
• Then solve for wD*

∗ E rG- − r2 σ,. − E rG. − r2 σ-.

w- =
E rG- − r2 σ,. + E rG. − r2 σ,- − E rG- − r2 + E rG. − r2 σ-.
w.∗ = 1 − w- ∗

These are the weights of D & E in our optimal risky portfolio.

37
 Asset Weights
Suppose the risk-free rate is 5%
Plugging in our numbers for D and E, we have
.08 − .05 .04 − .13 − .05 .0072
w"∗ =
(.08 − .05)(.04) + (.13 − .05)(.0144) − (.08 − .05 + .13 − .05)(.0072)
= .40
w$∗ = 1 − .40 = .60
E r% = 0.40 0.08 + 0.60 0.13 = 0.11
σ% = 0.40& 0.12& + 0.60& 0.2& + 2 0.40 0.60 0.0072 '/&

= 0.142

0.11 − 0.05
S% = = 0.42
0.142
38
 Optimal Risky Portfolio (“Tangency
Portfolio”) with 2 Risky Assets
Correlation and the Opportunity Set of Risky Assets

E [˜
r]
E

39
Optimal
Complete
Portfolio

40
 Optimal Risky Portfolio à Optimal
Complete Portfolio
Last week: given a risky portfolio P and the risk-free asset
à determine optimal complete portfolio

Now: We find the optimal risky portfolio T

à Use individual investor’s risk aversion (risk preferences) to find the
optimal complete portfolio

41
 Optimal Complete Portfolio
Correlation and the Opportunity Set of Risky Assets

E [˜
r]
E

42
 Mean-Variance Preferences
Recall:

1 ,
N = , %̃ − O!
2

à Optimal proportion y of the portfolio in a portfolio P of risky assets given the

investor’s risk aversion level A:

,[%'̃ ] − %0
P∗ =
O!',

43
 Example
If A = 4, 9[&*̃ ] = 0.11, and σT = 0.142, then

9[&*̃ ] − &- 0.11 − 0.05

=∗ = = = 0.7439
?+*. 4 1.042.

The investor should put

• 74.39% of her funds into the tangency portfolio T
• 25.61% into the risk-free asset

Remember, portfolio T consists of a debt asset D and an equity asset E. So

y∗wD = (0.7439)(0.40) = 0.2976 of total funds in D
y∗wE = (0.7439)(0.60) = 0.4463 of total funds in E

44
 Constructing the Optimal
Complete Portfolio
1. Find the return characteristics of all securities
2. Establish the optimal risky portfolio
a. Calculate the tangency portfolio
b. Calculate the expected return and standard deviation of the tangency
portfolio using the weights obtained in the previous step.
3. Allocate funds between the tangency portfolio and the risk-free asset
a. Calculate the proportion of the complete portfolio allocated to the
tangency portfolio and to the risk-free asset.
b. Calculate the share of the complete portfolio invested in each risky
asset.

45
 Portfolio Separation Theorem
(Tobin, 1958)
Acc. to this theorem, the portfolio construction problem can be divided into two
independent steps:
1. Determination of the optimal risky portfolio
– A portfolio manager should offer the same risky portfolio to all clients
regardless of their degree of risk aversion
2. Determination of the complete portfolio that also includes the risk-free
asset
– The degree of risk aversion becomes important only in choosing the desired
point along the CAL

46
Optimal Risky
Portfolios
With More
Than 2 Risky
Assets

47
More than 2 risky assets

48
Different Beliefs à Different
Tangent Portfolios

49
s of a portfolio with n assets
MathsMaths of with
of a portfolio a portfolio
n assets with n assets
n
X
E [rP ] = wi · E 6[ri ]
X n
EE[rn P%/]̃ n =
= S ww3 i⋅·EE [r
%3̃ i ]
i=1
X X 345 i=1
2
P = wi Xwnj Xi,j
n
2 j=1
i=1P = wi wj i,j
Xn i=1Xn
j=1X n
= wi2 i2 +
Xn wX
inwjX
ni,j
2 2
i=1 = wi j6=
i=1 i i+ wi wj i,j
i=1 i=1 j6=i

For example,
r example, the variance
the variance of a 3-asset
of a 3-asset portfolio
portfolio is: by
is given
For example, the variance of a 3-asset portfolio is given by
2
P3 = wA22 A2 + w 2 22 2
+
= Bw B + wC w 22 22
+ wC2 2
P3 A A B CB C
+2wA wB +2w
A,B +
A w2w
B A wC+ A,C
A,B 2wA+wC2w B w+C 2w
A,C B wC
B,C B,C

FNCE30001
E30001 Investments (UoM) Investments (UoM) 3: Optimal Risky3:Portfolio
Optimal Risky Portfolio Semester
Semester 1, 2022 1, 2022
4

50
 Practical Considerations of the
Markowitz model
Without any restrictions on the security selection problem
à the tangency portfolio T is the optimal risky portfolio

But with restrictions or constraints (e.g., taxes or transaction costs)

à Some investors might optimally select a different risky portfolio P
which is nearby T

51
 Conceptual issues with the
Markowitz model
Markowitz: assume returns followed a normal (Gaussian) distribution
• Plenty of evidence suggests stock returns do not follow a normal
distribution

Fama (1970): Markowitz framework works for any stable distribution (i.e., a
linear combination of two independent random variables with this distribution
has the same distribution)
• Plenty of evidence suggests stock returns do not follow a stable
distribution, either

52
 Big Question: How do we estimate
correlations?
For an economy with n assets, Markowitz model requires
• n estimates of expected returns
• n estimates of variance
• n * (n-1) / 2 estimates of covariance

If n = 1,000 à ~500,000 estimates needed!

Predicting even one random variable is hard

53
 Legacy of the Markowitz model
Even if it’s not 100% accurate, Markowitz model remains hugely influential in
how we understand risk
Groundbreaking idea: (some) risk can be diversified without affecting expected
returns
Laid the groundwork for subsequent models of asset pricing:
• CAPM
• single-index model
• multi-factor models

54
 Key Questions
What is the purpose of diversification?
How does correlation affect the opportunity set of risky assets?
What is the tangency portfolio?
What is the difference between the optimal risky portfolio and the optimal
complete portfolio?

55
 COMMONWEALTH OF AUSTRALIA

Copyright Regulations 1969

Warning

This material has been reproduced and communicated to you by

or on behalf of the University of Melbourne pursuant to Part VB
of the Copyright Act 1968 (the Act).

The material in this communication may be subject to copyright

under the Act. Any further copying or communication of this
material by you may be the subject of copyright protection
under the Act.

Do not remove this notice

Investments
FNCE30001
Problem Set – Week 5

1. An investor considers various allocations of a risky portfolio and risk-free treasury notes to
construct their complete portfolio. How would the Sharpe ratio of the complete portfolio be
affected by the allocation choice?
sharpe ratio is unaffected .

thenumerator and denominator

Why ? because
changes to the weight of the risky asset in the portfolio change
of the
portfolio sharpe ratio
by equal amount.

1
2*. MAIN George Stephenson’s current portfolio of $2.0 million is invested as follows:
Summary of Stephenson’s current portfolio
Value Per cent of Expected Annual
total annual return standard
deviation
Short-term bonds $200 000 10% 4.6% 1.6%
Domestic large-cap 600 000 30% 12.4% 19.5%
equities
Domestic small-cap 1200 000 16.0% 29.9%
equities
Total portfolio $2 000 000 100% 13.8% 23.1%

Stephenson expects to receive an additional $2.0 million soon and plans to invest the entire
amount in an index fund that best complements the current portfolio. Stephanie Wright, CFA,
is evaluating the four index funds shown in the following table for their ability to produce
portfolio that will meet two criteria relative to the current portfolio: (1) maintain or enhance
expected return and (2) maintain or reduce volatility.

Each fund is invested in an asset class that is not substantially represented in the current
portfolio.

Index fund characteristics

Index fund Expected annual Expected annual Correlation of
return % standard returns with current
deviation % portfolio
Fund A 15 25 +0.80
Fund B
- 11 Lower 22 +0.60
Fund C
- 16 25 +0.90 high correlation
-
increase vol

Fund D Chosen 14 22 +0.65 correlation

Lower

State which fund Wright should recommend to Stephenson. Justify your choice by describing
how your chosen fund best meets both of Stephenson’s criteria. No calculations are required.

2
3. Consider the following assets in which to potentially invest:

E[r] ![r] "

Fund ABC 10% 18%
0.3
Fund DEF 9% 25%
Government Bond 4% 0% --

a) Without doing any calculations, what which risky asset, ABC or DEF, would you
expect to hold more of in the optimal risky portfolio? Why?
ABC as it has higher expected return and lower standard deviation

b) Draw the efficient frontier (use 20% increments for the weights) of risky assets
between ABC and DEF
SDrp
WABC WDEF Erp
0% 100 % 1x 9 %: 9 % ↓ 101 10 181" 111310 25)" + 210/((((0
. + .
. 340 . 18x0 .
25) = 25%

21 4 %
(10 21 10 812 10 =
849 %: 9 2% 1812 + 10 25)" + 210 2/10 8710 3x0 , 18x0 25)
.

2x10 % +
. .

80 %
0
20%
.
.
0. . .
. . . .

40 % 60 % . 4x10% +
0 0 . 6x9 %: 9 4 % .

(10 41" (0 181"

.
. + 10 . 61210 .
25)" + 210 4/10 6) 10
.
. . 3x0 . 18x0 25)
.
= 18 5 %
.

60 % 40 % 0 . 6x10 % + 0 4x9 %:. 9 .

6%
(10 61 (0 181" 10.41210 25)" + 210 6/10 47(0
.
. + .
. .
. 3x0 . 18x0 25)
. = 16 8 %
.

20 % . 8x10 % +
0
0 .
2x9 %: 9 8%. ↓ 10 . 8 1"10 181"
. + 10 .
21 210 .
25)" + 210 8) 10 27 (0
. . . 3x0 . 18x0 25)
.
= 16 6 %
.

80 %

10 %
↓
) 1110 181" 101310 25)" + 211(10(10
. + .
· 340 . 18x0 25)
.
: 18 %

100% 0 % 1x 10 % :

3
c) What is the weight of ABC in the optimal risky portfolio?

2 - 10 09 04) (0 3 x0 18X0 25)

10 . 1 - 0. 04) (0 25) .
.
-
0. . .
.

* = 0 79 or 79 %
W ABC 04) 10 1812
.

= 10 1.
-
0 . 04/10 . 2532 + 10 . 09 = 0 .
.
-
10 . 1 - 0 ,04 + 0 09,
-0 .
04/ (0 .
3x0 18x0
. .
25)

* *
ABC 1 21 21 %
W DEF : 1 - W = -
0 79 .
= 0 . or

d*) For an investor with a risk-aversion parameter A equal to 3, what is weight of the risk-
free asset in the optimal complete portfolio?
* Ecrp]- of
y =

A 2p

x10 % + 21 % x 9 % =
9 79 %
Erp] 79 %
.

2532 + 210 79)10 21) (0 3

x0 18x0 21) = 0 275
1892
. .
. .

+ 10 21x0
.

(0 79x0
.

22
.
= . .

04
0 .
0979 = 0 .

Y*
= 0 .
7 or 70 % >
- risky assets
3(0 -
0275)

*
) 30 % risk free asset
(1-y
>
-
=

e*) For an investor with A<3, do you expect she will hold more, less, or the same amount
of the risk-free asset in her optimal complete portfolio compared to the investor in (c)?
portfolio compared
of risk free to AC3 investors.
Less ,
because investor with A=3 has higher proportion

because Less risk averse investors are more willing to hold

risky portfolio
Why ?

4
 Elrp) (0 7 x 0. . 17) + 10 3 . x0 . 07) = 0 14
. or 14 % per year

27 0 189 18 9 %
Up 7 x 0 =
: 0 . .
. or .

. >X0 27
0 .
= 0 .
189 or 18 9 % .

.
0 7 x0 33 .
= 0 . 231 or 23 1 .
%

0 .
7X0 , 4 = 0 . 28 or 28 %

0. 17-0 07 .

Sharpe ratio 3704

: = 0 .

0 . 2
 ECrp7 :
15%

0 . 15 = 0 07W + 0 17
. . (1 w]-

0 15 .
= -0 12 + 0 17
. .

-
0 .02 = -0 . 1 w

W= 0 2 or 20 % (risk free
.

(1 -w) = 0 .
80080% (risky

Up = y x27 %

20% = +x27 %

y = 0 . 7407 0r74 .
87 %

90%

10%

Earpy =
(0 . 9 x0 . 67) + 10 1x0
.
. 42) = 0 645
.
%

Cov = 0 4 . x 2 . 37 % x 2 95 %: .
0 . 00028 or 0 . 028 %
op =
/(0 . 9x0 .
023732 + 10 /x0. . 0295)" + (2x0 . 9x0 1x0
. .
00028)

= 0 0227
.
or 2 27 .
%

Ezrpy = (0 9 x0 67 % )
. . + 10 . 1x0 42 % )
.
= 0 . 645 %

Cor = 0 the variance of a random variable with

,

a constant is 0 Government securities is a constant

,

Op =
/(0 92 . x2 374+
. 10 1x0) +
. (2x0 9x0 /X0/
. .

= 2 . 13 %

No ,
it
depends on what the correlation between

returns of ,
ABC XY2 and the
,
original portfolio are

Why ? because even if the expected returns and

standard deviation of both security are

equal ,
the covariance may be different. If the covariance is

different , selecting randomly may not result in a lower standard

deviation. Security that lowers standard deviation is

preferred.

If returns are not symmetric , standard deviation does not

reflect differences in positive and negative price movement.

than the for

why? risk of loss is more important opportunity
return . Using variance or standard deviation does not

distinguish between positive or

negative price movement

Thus limiting her goal

ECrA] :
10 3 .
x 0 .
32) + 10 5 . x 0 . 1) + 10 2x
.
-
0 . 03)

= 0 . 14

E[rB] :
10 3 .
x 0 .
08) + 10 5 . x 0 . 05) + 10 2x0
.
· 07
= 0 . 063

Covariance (FA , B) : 0 3 (0 32
. .
- 0 .
14)(0 08 .
-
0 .
063) +

0 .
5 (0 .
1 -
0 . 14) (0 05 .
-
0 .
063) +

0 . 2( - 0 03 .
- 0 . 14)10 . 07 -
0 . 063)

= 0 . 00094

Correlation (PaB) =
0 .
00094
= 0 . 545
= 0. 1277X 0 . 0135
 Solutions

(*Solutions to questions with an asterisk will be covered in lecture.)

1. The Sharpe ratio of the portfolio will be unaffected. Changes to the weight of the risky asset
in the portfolio change the denominator and numerator of the portfolio Sharpe ratio by equal
amounts.

3. a) ABC. Notice that ABC has higher expected return and lower standard deviation, so we
expect that it will have higher weight in the optimal risky portfolio.

b) Use formulae for E(r_P) and SD(r_P) to find:

W_ABC W_DEF E[r_P] SD(r_P)

0% 100% 9.0% 25.0%
20% 80% 9.2% 21.4%
40% 60% 9.4% 18.5%
60% 40% 9.6% 16.8%
80% 20% 9.8% 16.6%
100% 0% 10.0% 18.0%

E[r_P]

0.1

0.08

0.06

0.04

0.02

0
0 0.05 0.1 0.15 0.2 0.25 0.3

c) The optimal risky portfolio is given by

Plugging in our numbers, we have

5
The optimal risky portfolio has approximately 79% invested in ABC and the remaining
21% invested in DEF.

Notice that on the graph you drew for (b), if you draw a straight line from the risk free rate of
4% tangent to the curve, the point of intersection will be very close to the portfolio with 80%
in ABC.

6
 Online Quiz Questions

B1. Assume that you manage a risky portfolio with an expected rate of return of 17% and a
standard deviation of 27%. The T-note rate is 7%. Your client chooses to invest 70% of a
portfolio in your fund and 30% in a T-note cash fund.

B1a. What is the expected return and standard deviation of your client’s portfolio? (Questions
1&2)

B1b. Suppose your risky portfolio includes the following investments in the given
proportions:
Share A 27%
Share B 33%
Share C 40%

What are the investment proportions of your client’s overall portfolio, including the position
in T-notes? (Question 3)

B1c. What is the reward-to-volatility ratio (S) of your risky portfolio and your client’s overall
portfolio? (Question 4)

B1d. Suppose the client decides to invest in your risky portfolio a proportion (y) of his total
investment budget so that his overall portfolio will have an expected rate of return of 15%.
What is the proportion y? (Question 5)

B1e. Suppose the client prefers to invest in your portfolio a proportion (y) that maximises the
expected return on the overall portfolio subject to the constraint that the overall standard
deviation will not exceed 20%. What is the proportion y? (Question 6)

a. (Question 1) E(rP) = (0.3 × 7%) + (0.7 × 17%) = 14% per year

(Question 2) sP = 0.7 × 27% = 18.9% per year

b. (Question 3)
Security Investment proportions
T-notes 30.0%
Share A 0.7 ´ 27% = 18.9%
Share B 0.7 ´ 33% = 23.1%
Share C 0.7 ´ 40% = 28.0%

17 - 7
c. (Question 4)Your reward-to-variability ratio = S = = 0.3704
27
14 - 7
Client's reward-to-variability ratio = = 0.3704
18.9

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 d. (Question 5) Mean of portfolio = (1 – y)rf + y rP = rf + (rP – rf )y = 7 + 10y
If the expected rate of return for the portfolio is 15%, then, solving for y:
15 - 7
15 = 7 + 10y Þ y = = 0.8
10
Therefore, in order to achieve an expected rate of return of 15%, the client must invest
80% of total funds in the risky portfolio and 20% in T-notes.

e. (Question 6) Portfolio standard deviation = sP = y × 27%

If the client wants a standard deviation of 20%, then:
y = (20%/27%) = 0.7407 = 74.07% in the risky portfolio

B2. Anne Grace has a $900 000 diversified portfolio. She subsequently inherits ABC
Company common stock worth $100 000. Her financial adviser provided her with the
following estimates:

Risk and return characteristics

Expected monthly returns % Standard deviation of
monthly returns %
Original portfolio 0.67 2.37
ABC Company 1.25 2.95

The correlation coefficient of the ABC stock returns with the original returns is 0.40.

a. The inheritance changes Grace’s overall portfolio and she is deciding whether to keep the
ABC stock. Assuming Grace keeps the ABC stock, calculate the:

i. expected return of her new portfolio which includes the ABC stock. (Question 7)
ii. covariance of ABC stock returns with the original portfolio returns. (Question 8)
iii. standard deviation of her new portfolio which includes the ABC stock. (Question
9)

b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government
securities yielding 0.42% monthly. Assuming Grace sells the ABC stock and replaces it
with the government securities, calculate the:
i. expected return of her new portfolio which includes the government securities.
(Question 10)
ii. covariance of the government security returns with the original portfolio returns.
(Question 11)
iii. standard deviation of her new portfolio which includes the government securities.
(Question 12)

c. Based on conversations with her husband, Grace is considering selling the $100 000 of
ABC stock and acquiring $100 000 of XYZ Company common stock instead. XYZ stock
has the same expected return and standard deviation as ABC stock. Her husband
comments, ‘It doesn’t matter whether you keep all of the ABC stock or replace it with
$100 000 of XYZ stock.’ State whether her husband’s comment is correct or incorrect.
Justify your response.

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d. In a recent discussion with her financial advisor, Grace commented, ‘If I just don't lose
money in my portfolio, I will be satisfied.’ She went on to say, ‘I am more afraid of
losing money than I am concerned about achieving high returns.’ Describe one weakness
of using standard deviation of returns as a risk measure for Grace.
Answer:
a. Subscript OP refers to the original portfolio, ABC to the new share, and NP to the
new portfolio.
(Question 7)
i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9 ´ 0.67) + (0.1 ´ 1.25) =
0.728%
(Question 8)
ii. Cov = # ´ sOP ´ sABC = 0.40 ´ .0237 ´ .0295 = .00028

Be careful! The question asked for “List your answer in decimal form, rounded to 4 decimal
places (i.e. 0.0329, not 3.29% or 329%2). The margin for error is 0.0002.”

This means your calculations and answers should use percent in the form of “. 0.0329, not
3.29%”. However, if you were consistent with your answer (i.e. you kept everything in
percent and you got the answer 2.7966) then you received full credit for this question.

(Question 9)
iii. sNP = [wOP2 sOP2 + wABC2 sABC2 + 2 wOP wABC (CovOP , ABC)]1/2
= [(0.92 ´ .02372) + (0.12 ´ .02952) + (2 ´ 0.9 ´ 0.1 ´ .00028)]1/2
= .0227 or 2.27%
b. Subscript OP refers to the original portfolio, GS to government securities, and NP to
the new portfolio.

(Question 10)
i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9 ´ 0.67) + (0.1 ´ 0.42) = 0.645%

(Question 11)
ii. Cov = 0: the covariance of a random variable with a constant is zero.

(Question 12)
iii. sNP = [wOP2 sOP2 + wGS2 sGS2 + 2 wOP wGS (CovOP , GS)]1/2
= [(0.92 ´ 2.372) + (0.12 ´ 0) + (2 ´ 0.9 ´ 0.1 ´ 0)]1/2
= 2.133% @ 2.13%

c. (Question 13) The comment is not correct. Although the respective standard deviations and
expected returns for the two securities under consideration are equal, the covariances
between each security and the original portfolio are unknown, making it impossible to draw
the conclusion stated. For instance, if the covariances are different, selecting one security
over the other may result in a lower standard deviation for the portfolio as a whole. In such
a case, that security would be the preferred investment, assuming all other factors are equal.

NOTE: The husband is correct if we assume a CAPM world and that the portfolio is (as
in the original question) “fully diversified”. Assuming the stocks are correctly priced, we
can tell from the expected returns that the CAPM Beta of the ABC and XYZ, must be the

9
 same; therefore, their correlation with the market must be the same. So, Grace should be
indifferent (except transaction costs).

Best answer: “No. It depends on what the correlation between returns of ABC, XYZ,
and the original portfolio are.”

Second best answer: “No. The husband's comment is ignoring losses due to transaction
costs of selling ABC to buy XYZ.” The much more important issue is that of the
comovement. Transaction costs are generally fairly small for stock purchases and sales.

The “Yes” answer is wrong because of the statement “Since the standard deviations are
the same, we are indifferent.”

d. (Question 14) Grace clearly expressed the sentiment that the risk of loss was more
important to her than the opportunity for return. Using variance (or standard deviation) as
a measure of risk in her case has a serious limitation because standard deviation does not
distinguish between positive and negative price movements.

“If returns are not symmetric, standard deviation does not reflect differences in
positive and negative price movements.” This is the best answer as it captures the spirit
of ‘I am more afraid of losing money than I am concerned about achieving high returns.’

“Standard deviation does not measure the total risk of the portfolio.” Is wrong.

“Standard deviation does not measure the risk of the individual asset.” While not
completely wrong, because covariance risk with the existing portfolio is usually more
important, it is not completely correct either, because if an individual asset were your sole
investment, then standard deviation would be the correct measure of risk.

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B3. (Question 15) Consider the following table, which gives a security analyst’s expected
return on two stocks for in different states of the economy (SHOW YOUR WORK):

State Probability Aggressive Stock (rA) Defensive Stock (rD)

Boom .3 32% 8%
Normal .5 10% 5%
Bust .2 -3% 7%
E[r]
s 12.77% 1.35%
What is the correlation between the returns on the Aggressive and Defensive stocks?
Answer:
First, we need the expected returns.
$

$ [&̃! ] = * +(-)&!,#
#%&
$[&̃! ] = .3 × .32 + .5 × .1 + .2 × (−.03)
$ [&̃! ] = .096 + .05 − .009 = .14

$ [&̃' ] = .3 × .08 + .5 × .05 + .2 × (. 07)

$ [&̃' ] = .024 + .025 + .014 = .063

Next we need the covariance.

$

=!' = * +(-)>&!,# − $ [&̃! ]?>&',# − $ [&̃' ]?

#%&
=!' = .3(. 32 − .14)(. 08 − .063) + .5(. 10 − .14)(. 05 − .063)
+ .2(−.03 − .14)(. 07 − .063)
=!' = 0.000918 + 0.00026 − 0.000238 = 0.00094

Finally, we can calculate correlation.

=!' . 00094
#!' = = = .545
=! =' . 1277 ∗ .0135

11