15.401 Recitation
6: Portfolio Choice
Learning Objectives

R i off Concepts
Review
C
t
o Portfolio basics
o Efficient frontier
o Capital market line

Examples
o XYZ
o Diversification
o Sharpe ratio
o Efficient frontier
2010 / Yichuan Liu
2
Review: portfolio basics

A portfolio is a collection of N assets (A1, A2, …, AN)
with weights (w1, w2, …, wN) that satisfy
N
o
w
i
1
i1

Each asset Ai has the following characteristics:
o Return: ~ri (random variable)
o Mean return: ri
2

o Variance and std.
std de
dev. of return
ret rn: i ,  i
o Covariance with Aj:  ij
2010 / Yichuan Liu
3
Review: portfolio basics

The return of a portfolio is
N
~
rp   wi ~
ri
i 1

The mean/expected return of a portfolio is
E rp   rp   wi ri
N
i 1

The variance of a portfolio is
N
N
 p2   wi w j ij ;  p   p2
i1 j1

Note:  ii   i2 ;  ij   ij i j
2010 / Yichuan Liu
4
Example 1: XYZ
E(r)

X
15%
Y
10%
Z
20%
Variance Covariance
Variance‐Covariance
X
Y
Z
0.090
0.125
0.144
0.040
‐0.036
0.625
Whatt is
Wh
i th
the expected
t d return
t
and
d variance
i
off a
portfolio of …
a. (X, Y) with weights (0.4, 0.6)?
b. (X, Y, Z) with weights (0.2, 0.5, 0.3)?
c. (X, Y, Z) with weights (1/3, 1/3, 1/3)?
2010 / Yichuan Liu
5
Example 1: XYZ

Answer:
 
E r   14%; 
E r   15%; 
a. E r  12%;  2  0.08880;   29.80%
p
p
p
b.
c.
2010 / Yichuan Liu
p
2
p
 0.10133;  p  31.83%
p
2
p
 0.13567;  p  36.83%
6
Example 1: XYZ
 What is the minimum possible variance of a portfolio

with only Y and Z?
E(r)
2010 / Yichuan Liu
X
15%
Y
10%
Z
20%
Variance‐Covariance
X
Y
Z
0 090
0.090
0 125
0.125
0 144
0.144
0.040
‐0.036
0.625
7
Example 1: XYZ
 Answer:

Let (w, 1–w ) be the weights for (Y, Z), then

arg min w  0.04
04  2w
2 w1
1 w 0.036
0 036   1
1 w  0.625
0 625
2
2

w
 First‐order condition:
2w  0.04  21 2w 0.036   21 w  0.625  0
w*  0.8969
 The minimum variance portfolio is
0.8969,0.1
0.8969,0.1031
031
2010 / Yichuan Liu
8
Example 2: diversification

Supp
ppose that your portfolio consists of N eq
quallyy
weighted identical assets in the market, each of
which has the following properties:
o Mean = 15%
o Std dev = 20%
o Covariance with any other asset = 0.01

What is the expected return and std dev of return of
your portfolio if…
o N = 2?
o N = 5?
o N = 10?
o N = ∞?
2010 / Yichuan Liu
9
Example 2: diversification

Answer:
o Expected return
1


E r    00.15
15  0.15
15
N
p
i1
N
o Variance
 0.2 2 
0.2 2 N
0.01
0.01
 rp    2   2  N  2   N N  1 2
N
i1 N
i1 j i N
 N 
N
0.04 
1
0.03

 1 0.01  0.01
N
N
 N
2010 / Yichuan Liu
10
Example 2: diversification

Answer:
o N = 2:
E rp   15%;  p2  0.0250;  p  15.81%
o N = 5:
E rp   15%;  p2  0.0160
0160;;  p  12.65%
12 65%
o N = 10:
E rp   15%
15%;  p2  0
0.0130;
0130  p  11.40%
11 40%
o N = ∞:
E rp   15%;  p2  0.0100;  p  10.00%
2010 / Yichuan Liu
11
Example 2: diversification
20%
Standard Devviation of Return
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
0
2010 / Yichuan Liu
5
10
15
N
20
25
30
12
Review: diversification
20%
Standard Devviation of R
Return
18%
Idiosyncratic risk can be diversified away;
investors are not compensated for such risk.
risk
16%
14%
12%
10%
8%
Systematic risk cannot be diversified away;
investors are compensated
p
with higher
g
expected returns.
6%
4%
2%
0%
0
2010 / Yichuan Liu
5
10
15
N
20
25
30
13
Review: efficient frontier
 Given two assets, we can form portfolios with

weights (w, 1–w). As we vary w, we can plot the path
of the mean return and standard deviation of
return of the resulting portfolio.
 The shape of the path depends on the correlation
b t
between
the
th two assets.
t
 When the correlation is low, a large portion of asset
return variation comes from idiosyncratic risk that
can be diversified away.
2010 / Yichuan Liu
14
Review: efficient frontier
ρ=1
perfectly correlated
no risk reduction potential
 ‐1 < ρ < 1
imperfectly correlated
some risk reduction potential
 ρ = ‐1
1
perfectly negatively correlated
most risk reduction potential

14
E
13
Expected Return (%)
12
p = -1
11
p=1
10
p = .30
p=0
9
D
8
7
6
5
0
2
4
6
8
10
12
14
16
18
20
Standard Deviation (%)
Image by MIT OpenCourseWare.
2010 / Yichuan Liu
15
Review: efficient frontier

We can repeat the previous exercise for N assets:
E(r)
Efficient Frontier
Global Minimum
Variance Portfolio
Individual Assets
Minimum-Variance Frontier
σ
2010 / Yichuan Liu
Image by MIT OpenCourseWare.
16
Review: efficient frontier
 The efficient frontier can be described by a function
function
σ*(rp), which minimizes the portfolio std dev given
an expected return:
* rp   min
wi 
N
N
 w w 
i 1 j 1
i
j
ijj
N
 wi  1
s.t.  iN1
 wi ri  rp
 i 1
 Analytical solution for σ
σ*((rp) is possible but difficult
to derive.
2010 / Yichuan Liu
17
Review: capital market line

Efficient frontier + risk
risk‐free
free asset = CML
Expected Return
Market Portfolio
Risk-Free
Asset
Efficient Frontier
Risk
Image by MIT OpenCourseWare.
2010 / Yichuan Liu
18
Example 3: Sharpe ratio
 The Sharp
pe ratio measures the reward‐risk tradeoff of an
asset or a portfolio. It is defined as
S
S
r  rf

 The higher Sharpe ratio, the more desirable an asset / a
portfolio is. Suppose
pp
rf = 5%.
5 What is the portfolio of (A,, B))
with the highest Sharpe ratio?
E(r)
2010 / Yichuan Liu
A
15%
B
10%
COV‐VAR
A
B
0.090
0.015
0.040
4
19
Example 3: Sharpe ratio

Answer:
max S p  max
w

w
wrA  1 wrB  rf
w   2 w11 w AB  11 w  B2
2
2
A
2
Method 1: grid search
1. S
Sett up a grid
id f
for w, e.g., w = 0, 0.1, 0.2, …, 1.0
The finer the grid, the more accurate the result
2. Calculate the Sharpe ratio for each w
3. Find the maximum Sharpe ratio.
2010 / Yichuan Liu
20
Example 3: Sharpe ratio

Method 1: grid search
w
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2010 / Yichuan Liu
1–w
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
rp – rf
0.0500
0.0550
0.0600
0.0650
0.0700
0.0750
0.0800
0.0850
0.0900
0.0950
0.1000
σp
0.2000
0.1897
8
0.1844
0.1844
0.1897
0.2000
0.2145
0.2324
0.2530
0.2757
0.3000
Sp
0.2500
0.2899
8
0.3254
0.3525
0.3689
0.3750
0.3730
0.3658
0.3558
0.3446
0.3333
21
Example 3: Sharpe ratio
0.40
0.38
0.36
Maximum:
w* = 0.52
E(r) = 12.60%
σ = 20.26%
Max S = 0.3752
0 3752
0 34
0.34
0.32
0.30
0.28
0.26
0.24
0.22
0.20
0
2010 / Yichuan Liu
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
22
Example 3: Sharpe ratio

Method 2: Excel Solver
A
B
C
D
E
E(r)
Asset A
Asset B
=B3
=B4
0.15
0.09
0.015
01
0.1
0 015
0.015
0 04
0.04
6
rp – rf
σp
S
7
=f
=g
=C7/D7
1
2
3
Asset A
4
Asset B
=1 –B3
B3
5
Solver
Set Target Cellll:
$E$7
Equal To:
Max
By Changing Cell:
$B$3
$B$
f: SUMPRODUCT(B3:B4, C3:C4) – 0.05
g: SQRT(B3*D2*D3+B3*E2*E3+B4*D2*D4+B4*E2*E4)
2010 / Yichuan Liu
23
Example 3: Sharpe ratio

Method 2: Excel Solver
A
B
1
C
D
E
E(r)
()
Asset A
Asset B
0.52
0.48
2
3
Asset A
0.52
0.15
0.09
0.015
4
A tB
Asset
0.48
8
0.1
0.015
0.04
6
rp – rf
σp
S
7
0.076
5
2010 / Yichuan Liu
0.202583 0.375154
25
Example 3: Sharpe ratio

Method 3: analytical solution
o Full derivation:
S

w

rA  rB  p2 
1
2
  2w
1 2
p
2

 12
2
A

 21 2w AB  21 w B2 rp  rf 
 
1
2
2
p
rA  rB w2 A2  2w1 w AB  1 w2  B2  w A2  1 2w AB  1 w B2 wrA  1 wrB  rf 
 p2
0

 r w 
 
 w

 r
 2w1 w  1 w 
 1 2w  1 w wr  r   r
 r  r w  1 w   w  1 2w  1 w r  r 
 r  r      r  r  r  r        2   r  r w
 r  r   r  r   r  r      r  r    w
r  r   r  r 

r  r      r  r    
0  rA  rB  w2 A2  2w1 w AB  1 w  B2  w A2  1 2w AB  1 w B2 wrA  1 wrB  rf 
2
A
A
A
A
w*
B
2
A
B
AB
B
2
B
f
2
B
A
A
 0.52
2010 / Yichuan Liu
2
f
2
B
2
B
AB
B
f
22
B
2
AB
f
2
B
AB
2
A
B
B
A
f
f
AB
2
A
2
A
A
B
2
B
f
2
B
AB
2
B
AB
2
B
AB
f
AB
B
2
B
2
A
AB
B
f
B
B
B
 rf 
f
AB
2
A
A
2
B
B
f
AB
AB
26
Example 3: Sharpe ratio

Method 3: analytical solution
o Result only:
The general solution for the 2‐asset Sharpe ratio
maximization
i i ti problem
bl
i
is
w*

r  r   r  r 

r  r       r  r  
A
A
2010 / Yichuan Liu
f
f
2
B
2
B
AB
B
B
f
f
AB
2
A
  AB

27
Example 4: efficient frontier
 Given the risky assets A and B in the previous

question, what is the efficient frontier?
E(r)
A
15%
B
10%
COV VAR
COV‐VAR
A
B
0.090
0.015
0.040
 Given 5% risk

risk‐free
free rate,
rate what is the capital market
line?
2010 / Yichuan Liu
28
Example 4: efficient frontier

Table from the previous question:
w
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2010 / Yichuan Liu
1–w
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
rp
0.1000
0.1050
0.1100
0.1150
0.1200
0.1250
0.1300
0.1350
0.1400
0.1450
0.1500
σp
0.2000
0.18
897
0.1844
0.1844
0.1897
0.2000
0.2145
0.2324
0.2530
0.2757
0.3000
29
Example 4: efficient frontier

Scatter plot of (rp, σp) pairs:
0.16
Efficient frontier
0.14
0.12
Inefficient portion
of the frontier
0.10
0.08
0.06
0.04
2010 / Yichuan Liu
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
30
Example 4: efficient frontier

Capital market line:
0.16
Tangency portfolio:
w = 0.52
E(r) = 12.60%
σ = 20.26%
Max S = 0.3752
0.14
0.12
0.10
0.08
CML is the line passing through (0, 0.05) and
tangent to the efficient frontier.
0.06
0.04
2010 / Yichuan Liu
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
31
Example 4: efficient frontier

The moral of the story:
o The CML is tangent to the efficient frontier at the
tangency portfolio.
o The
Th tangency
t
portfolio
tf li is
i th
the portfolio
tf li off risky
i k assets
t th
thatt
maximizes the Sharpe ratio.
o The slope of the CML is the maximum Sharpe ratio.
o Rational investors always hold a combination of the
tangency portfolio and the risk‐free asset. The
proportion depends on investors’
investors risk preferences.
preferences.
2010 / Yichuan Liu
32
Sneak Peak: CAPM
The tangency portfolio is the market portfolio
portfolio..
An asset’s systematic risk is measured by beta, which
is defined as the correlation of its return and the
market return, normalized by the variance of market
return :
 im
i  2
m
 Since investors are only
y comp
pensated for sy
ystematic
risk, asset return is an increasing function of beta:
E ~
ri   rf   i ~
ri  rf 


2010 / Yichuan Liu
33
MIT OpenCourseWare
http://ocw.mit.edu
15.401 Finance Theory I
Fall 2008
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