change, what are the new portfolio weights?

Capital
2. Consider aAsset Pricing
portfolio of Model
250 shares of Homework
firm A worth Problems
$30/share and 1500 shares of firm B
worth $20/share. You expect a return of 4% for stock A and a return of 9% for stock B.

Portfolio
(a) What is weights and
the total value expected
of the return
portfolio, what are the portfolio weights and what is
the expected return?
Homework 6
1. Consider a portfolio
(b) Suppose of share
firm A’s 300 shares
price of firm
falls to A
$24worth $10/share
andFall
firm and price
B’s share 50 shares of firm
goes up B
to $22.
FIN2020 2022
worthWhat
$40/share.
is the You expect
newDue
value a return
of of 8% forWhat
the 12pm
portfolio? stock return
A and adid
return of 13%
itBeijing
earn? for the
After stockprice
B.
at Noon Tuesday Nov. 8, 2022 Time
change, what are the new portfolio weights?
(a) What is the total value of the portfolio, what are the portfolio weights and what is
the expected return?
Problem
Portfolio1 firm
(b) Suppose volatility
A’s share price goes up to $12 and firm B’s share price falls to $36.
What is the new value of the portfolio? What return did it earn? After the price
3. For the following
change, whatproblem
are the please refer to weights?
new portfolio Table 1 (Table 11.3, p. 336 in Corporate Finance
by Berk and DeMarzo).
2. Consider
Table 1: aHistorical
portfolio ofAnnual
250 shares of firm A and
Volatilities worthCorrelations
$30/share andfor 1500 shares of
Selected firm B(based
Stocks
(a)
worth What is
$20/share.
on monthly the covariance
You expect
returns, between the returns for Alaskan Air and General
a return of 4% for stock A and a return of 9% for stock B.
1996-2008). Mills?
(b) What is the volatility of a portfolio with
(a) What is the total value of the portfolio, what areSouthwest
Alaskan the portfolioFord
weightsGeneral
and whatGeneral
is
i. equal amounts invested in these two stocks?
the expected return? Microsoft Dell
ii. 20% invested in Alaskan Air and 80%Air investedAirlines
in General Motor
Mills? Motors Mills
(b) Volatility
Suppose (StDev)
firm A’s share price
37% Airfalls
50% to $24 and firm B’s31% share price goes up to $22. 18%
iii. 80% invested in Alaskan and 20%38% invested in General 42% Mills? 41%
What is thewith:
Correlation new value of the portfolio? What return did it earn? After the price
change,
4. Suppose what
theMicrosoftare the
historical new1.00
portfolio
volatility 0.62weights?
(standard 0.25
deviation) of the0.23
return 0.26 0.23stock is0.10
of a mid-cap
50% and the correlationDell 0.62 the1.00
between returns of0.19 0.21 is 30%.
mid-cap stocks 0.31 0.28 0.07
Alaska Air 0.25 0.19 1.00 0.30 0.16 0.13 0.11
Portfolio
(a) What is volatility
the
Southwest Airlines average variance AvgV
0.23 0.21 ar of a mid-cap
0.30 stock?
1.00 0.25 0.22 0.20
(b) WhatFord is theMotor 0.26 0.31
average covariance AvgCov of 0.16
a mid-cap 0.25
stock? 1.00 0.62 0.07
3. For
(c)the following
General
Consider a problem
portfolioplease
Motors of n0.23refer0.28
mid-capto Table
stocks. 10.13
(Table
What 11.3, p.estimate
is an0.22 336 in0.62
Corporate Financeof0.02
1.00
of the volatility
by Berksuch and DeMarzo).
General Millswhen n 0.10
a portfolio = 10? 0.07
n = 20? n0.11 = 40? What0.20 0.07 volatility?
is the limiting 0.02 1.00

(a) What is the covariance between the returns for Alaskan Air and General Mills?
1
(b) What is the volatility of a portfolio with
5. Consider a portfolio of two stocks. Data shown in Table 2. Let x denote the weight on
i. equal
Stock A amounts
and 1 xinvested in these
denote the twoonstocks?
weight Stock B. Correlation coefficient equals ⇢AB .
ii. 20% invested in Alaskan Air and 80% invested in General Mills?
iii.(a) Write
80% downina Alaskan
invested mathematical expression
Air and for theinportfolio’s
20% invested mean return and volatility
General Mills?
(standard deviation) as a function of x.
4. Suppose(b) theWhat
historical
is thevolatility (standard
portfolio’s deviation)
mean return of the return
and volatility whenofx a= mid-cap stock
0.4 if ⇢AB = 0?is ⇢AB =
50% and the+1? correlation
⇢ = 1?between the returns of mid-cap stocks is 30%.
AB
(c) is
(a) What Suppose ⇢AB =
the average 1? Are
variance there
AvgV ar portfolio weights
of a mid-cap that will result in a portfolio with
stock?
no volatility? If so, what are the weights?
(b) What is the average covariance AvgCov of a mid-cap stock?
(c) Consider a portfolio of n mid-cap stocks. What is an estimate of the volatility of
such a portfolio when n = 10? n = 20?Table
n = 40?
2: What is the limiting volatility?

Stock Expected
1 Return Volatility
Stock A 15% 40%
Stock B 7% 30%

Minimum variance portfolio

6. Consider the data shown in Table 2. The risk-free rate is rf = 3%.
1
(a) What is the minimum variance portfolio when ⇢AB = 0? What is its expected return
and volatility?
 Alaska
+1? ⇢ABAir
= 1? 0.25 0.19 1.00 0.30 0.16 0.13 0.11
5. Consider
Southwest a portfolio0.23
Airlines of two0.21stocks. Data
0.30 shown in Table 0.25
1.00 2. Let x denote
0.22 the 0.20
weight on
Stock Expected Return ⇢Volatility
(c) Stock
Suppose = 1? Are there portfolio weights that will result in a portfolio with
Ford15% A and 1 x denote
Motor
AB
0.26 0.31the weight0.16
on Stock B.0.25
Correlation
1.00 coefficient
0.62 equals0.07
⇢AB .
tock A no volatility? If40%
so, what are the weights?
tock B General Motors 0.23 0.28 0.13 0.22 0.62 1.00 0.02
(a) 7%
Write down30% a mathematical expression for the portfolio’s mean return and volatility
General Mills 0.10 0.07 0.11 0.20 0.07 0.02 1.00
(standard deviation) as a function of x.
Table 2:
Problem 2 is the portfolio’s mean return and volatility when x = 0.4 if ⇢AB = 0? ⇢AB =
(b) What
+1? ⇢AB = 1? Stock Expected Return Volatility
5. Consider
nce portfolio (c) Suppose a portfolio of two stocks. Data shown in Table 2. Let x denote the weight on
= 1?
⇢ABStock A Are there portfolio
15% weights40% that will result in a portfolio with
Stock A and 1 x denote the weight on Stock B. Correlation coefficient equals ⇢AB .
no volatility?Stock
If so,Bwhat are the weights?
7% 30%
wn in Table 2. The risk-free rate is rf = 3%.
(a) Write down a mathematical expression for the portfolio’s mean return and volatility
mum variance portfolio (standard
when ⇢AB deviation)
= 0? What as aisfunction of x.return
its expected
Table 2:
(b) What is the portfolio’s mean return and volatility when x = 0.4 if ⇢AB = 0? ⇢AB =
Minimum +1? ⇢AB variance
= 1? portfolio
Stock Expected Return Volatility
(c)2 Suppose ⇢AB = 1? Stock A portfolio weights
Are there 15% that will 40%result in a portfolio with
6. Consider the data shown Stock
in Table B 2. The risk-free
no volatility? If so, what are the weights? 7%
rate is r f =30%3%.

(a) What is the minimum variance portfolio when ⇢AB = 0? What is its expected return
and volatility? Table 2:
Minimum variance portfolio
Stock Expected 2 Return Volatility
(b) What is the minimum variance portfolio when ⇢AB = 0.4? What is its expected
Stock A 15% 40%
return and
6. Consider thevolatility?
data Stock
shownBin Table 2. The risk-free rate is rf = 3%.
7% 30%
(c) What is the minimum variance portfolio when ⇢AB = 0.4? What is its expected
(a) What
return and isvolatility?
the minimum variance portfolio when ⇢AB = 0? What is its expected return
and volatility?
7. Consider two stocks, A and B, such that A = 0.30, B = 0.80, R̄A = 0.10, R̄B = 0.06
Minimum
and rf = 0.02. variance portfolio 2

(a) Whatthe
6. Consider is the minimum
data shown invariance
Table 2.portfolio when rate
The risk-free = r0f and
⇢AB is what is its volatility?
= 3%.
(b) What is the minimum variance portfolio when ⇢AB = 0.6 and what is its volatility?
(a) What is the minimum variance portfolio when ⇢AB = 0? What is its expected return
(c) What is the minimum variance portfolio when ⇢AB = 0.6 and what is its volatility?
and volatility?
8. Consider three risky assets whose covariance matrix ⌃ is
0 2 1
0.09 0.045 0.01
B C
⌃ = @ 0.045 0.25 0.06 A, (1)
0.01 0.06 0.04

and whose expected returns are R̄1 = 0.08, R̄2 = 0.10, R̄3 = 0.16. The risk-free rate is
rf = 0.03. The inverse of the covariance matrix is
0 1
12.2137 2.2901 0.3817
B C
⌃ 1
= @ 2.2901 6.6794 9.4466 A. (2)
0.3817 9.4466 39.0744

What is the minimum variance portfolio and what is its volatility?

9. Consider three risky assets whose covariance matrix ⌃ is

0 1
2 1 0
B C
⌃ = @ 1 2 1 A. (3)
0 1 2

The expected returns are R̄1 = 0.11, R̄2 = 0.09, R̄3 = 0.05. The risk-free rate is rf = 0.02.
Solve for the minimum variance portfolio using the2 first-order optimality conditions, i.e.,
without computing the inverse of the covariance matrix. What is the minimum variance?
(Suggestion: By symmetry x⇤1 = x⇤3 . )
 ECON 337901 - Financial Economics Peter Ireland
For Extra
Boston College, Department Practice - Not Collected or Graded
of Economics Spring 2018

For Extra Practice - Not Collected or Graded

1. Portfolio Allocation with Mean-Variance Utility, Part I

Consider
Problem
1. 3an investor
Portfolio with with
Allocation preferences over the mean
Mean-Variance and variance
Utility, Part I of the returns on his or
her portfolio that are described by the utility function
Consider an investor with preferences over the mean ✓ and ◆ variance of the returns on his or
her portfolio that are described U by(µthe utility function
2 A 2
P , P ) = µP P,
✓ ◆ 2
2 A
where a higher value of A corresponds U (µP , toP )a = µP aversion P2to, risk. Suppose that this investor
larger 2
is able to form a portfolio from a risk-free asset with return rf and a risky asset with expected
return aequal
where highertovalue
µr and
of Aa corresponds
standard deviation of its
to a larger returntoequal
aversion risk. to r . In that
Suppose this this
context, the
investor
risky
is ableasset may
to form simply be
a portfolio thea only
from risky
risk-free asset
asset withavailable
return rto f the
and a investor
risky or
asset it
withcan be the
expected
tangency
return portfolio
equal to µr that
and arepresents
standard the optimalofcombination
deviation its return equal of manyto individual risky assets.
r . In this context, the
Eitherasset
risky way, may
our analysis
simply befromtheclass
onlyshowed that the
risky asset relationship
available to thebetween
investorµor P and for the
it canP be the
portfolio that
tangency combines
portfolio the risk-free
that represents theasset
optimalwithcombination
the risky asset is linear:
of many individual risky assets.
Either way, our analysis from class showed that ✓ the relationship
◆ between µP and P for the
portfolio that combines the risk-free asset µrthe rrisky
with f asset is linear:
µP = r f + P.
✓ r ◆
µr r f
By substituting this constraint into µP the
= rfinvestor’s
+ .
utility Pfunction, his or her problem can be
r
solved as one of choosing P to maximize
By substituting this constraint into✓the investor’s ◆ utility ✓ ◆ function, his or her problem can be
solved as one of choosing P torfmaximize µr r f A 2
+ P P.
✓ r ◆ ✓ ◆ 2
µr r f A 2
Use the first-order condition for rf +this problem toPsolve 2for the P . optimal choice of P .
r

2. Portfolio
Use Allocation
the first-order with
condition Mean-Variance
for this Utility,
problem to solve for thePart II choice of
optimal P.

OurPortfolio
2. analysis from class also
Allocation showed
with that the portfolio
Mean-Variance formed
Utility, Partfrom
II the risk-free and risky
asset that has a return with standard deviation P allocates the fraction
Our analysis from class also showed that the portfolio formed from the risk-free and risky
P
w=
asset that has a return with standard deviation P allocates the fraction
r
P
of wealth to the risk asset and the remaining w = fraction 1 w to the risk-free asset. Use this
r
expression, together with your solution for the optimal choice of P from question 3, above,
of
to wealth
find thetoinvestors’
the risk asset and choice
optimal the remaining
of w. How fraction
does 1the woptimal
to the risk-free
share of asset. Useasset
the risky this
expression, together with your solution for the optimal choice
depend on the investor’s risk aversion, as measured by the parameter of P from question 3, above,
A from the utility
to find theHow
function? investors’ optimal
does the optimalchoice
shareofof w.
the How
risky does
assetthe optimal
depend on µshare
r r f ,of the
the risky
risk asset
premium
depend on the
defined as theexpected
investor’sreturn
risk aversion, as measured
on the risky asset minus bythe
therisk-free
parameter A from the utility
return?
function? How does the optimal share of the risky asset depend on µr rf , the risk premium
defined as the expected return on the risky asset 1 minus the risk-free return?

3
① ⑨ Covariance between the return for Alaskan air & General Mills = 0,1 . 0,18 noise =
0,007524

⑥ ① equal am Mount totality : 50% Alaskan air & General Mills

SD = ÉÉÉÉoo7Ñ = 0,219=21,9 %

④ 20% invested in alaskan air & 80% in General Mills

SD = VÉ,7 = 0,1701=17,01 %

80% invested in alaskan air & 20% in General Mills

D= ÉPÉE) = 0,31=31 %

② ⑨ ✗ in stock A ≥

%¥?→ ◦+%÷=!%- ,
=
=
% .
36%

expected return = 0,36^15%-1 111-01367 7- % -

= 9.88%

6 =
fÉÉ =
24%

⑥ f AB =
0,4 →
GAB =
0,4 .
0,3 0,4 . =
0,048

%!¥%-a
✗ insta" A
%¥÷%→
:
= 27 . # %
=
,
,

Expected return =
012727 •
15%-1 (1-0,2727) •

7% = 9118%

6 = ÉBÉ77-012727T = 28103%

⑥ LAB =
-014 → GAB =
0,4 .

0,3 . 1-0,41=-0,048

expected return : 013988 .

15%-1 (1-0,3988) 7% = 10.19%

6 =
TÉ+ÉÉ = 18.7%

③
( %jI ) Gp (E)
'
Mat rf -
GP
6 P

Foc : %fˢ - A6P* = ◦

→ Gp
*
= Mf%ˢ

Mnf÷=Ma÷¥
optimal choice of w
%→w*%÷* = →
Sinew =
① The optimal choice of risky assets Is Inversely Proportional with investor risk aversion

If A large
•
*
is then w will decrease
^
If A- is
*
small then W Will Increase

② The optimal choke of risky assets is direct H Proportional with the risk Premium
•

if Mr -
it is large then W will increase

•
if Mr rf -
Is small then W☆ will decrease