Fin 2008 & Econ 3060 Investment
Chapter 7.
Optimal Risky Portfolios
Bokyung Park
Department of Finance
National Taiwan University
Spring, 2023
NTU. Dept of Fin. Investment
1
Last Class
Capital Allocation Line (CAL)
E(rc) = yE (rp ) + (1 − y)rf
Capital Allocation Line
(CAL)
P
E(rp)
•
E (rc ) = rf +
E (rp ) − rf
p
c
F
rf
σp
σc = y p
slope
= Sharpe
Ratio
2
Recall
Personal decision: based on
investor’s risk aversion
Capital
Allocation
Safe Asset
Last Class
Risky Asset(s)
1
Technical decision: based on
assets’/securities' covariance
structure with other securities
“Portfolio
Optimization”
N
Today
Asset
Allocation/
Security
Selection
3
Overview
▪ Asset allocation with risky assets
□ Agenda 1. Expected return, variance, and co-movement of Risky Portfolio
□ Agenda 2. Portfolios of Two risky assets
□ Agenda 3. Portfolios of Three risky assets
□ Agenda 4. Portfolios of N risky assets
• Markowitz Portfolio Optimization model
NTU. Dept of Fin. Investment
4
Agenda 1
Power of Diversification: A Simple Example
Measures of Co-movement
Expected Return & Variance of Risky Portfolios
5
Power of Diversification
Example: Ice Cream Co. vs. Straw Sandal Co.
Pr
rain
1/2
sunshine 1/2
mean
s.d .
Ice Cream Co Straw Sandal Co
-2%
4%
-1%
2%
1%
3%
0.5%
1.5%
Portfolio :1/2
in each stock
-1.5%
3%
0.75%
2.25%
cannot reduce risk at all by combining 2 stocks
Case of Perfect Positive Correlation (=1)
6
Power of Diversification
Example: Umbrella Co. vs. Straw Sandal Co.
Pr
rain
1/2
sunshine 1/2
mean
s.d .
2%
-1%
-1%
2%
Portfolio :1/2
in each stock
0.5%
0.5%
0.5%
1.5%
0.5%
1.5%
0.5%
0
Umbrella Co Straw Sandal Co
a complete elimination of risk by combining 2 stocks
Case of Perfect Negative Correlation ( = -1)
7
Power of Diversification
Definition of Diversification
▪ investment strategy designed to reduce risk by spreading the
portfolio across many securities / stocks.
always useful unless stocks perfectly move together (corr =1)
NTU. Dept of Fin. Investment
8
Measures of Co-Movement
Covariance ▪ expected value of the product of the deviations
from mean for two variables


Covariance (ri , rj ) = Cov (ri , ri ) = E [ri − E (ri )][ rj − E (rj )] =  ij
n
=  p( s)  [ri ( s ) − E (ri )][rj ( s ) − E (rj )]
s =1
Correlation ▪ covariance divided by s.d. of the two variables
(coefficient) ▪ always lies between -1 and 1
Correlatio n (ri , ri ) = Corr (ri , r j ) =
Cov(ri , r j )
V (ri )  V (r j )
=
 ij
 i j
=
9
Measures of Co-Movement
Calculating Covariance / Correlation
Example: Perfect Positive Correlation
Pr
rain
1/2
sunshine 1/2
mean
s.d .
Ice Cream Co Straw Sandal Co
(rIC − rIC )(rSS − rss )
-2%
4%
-1%
2%
(-3)(-1.5) = 4.5
(3) (1.5) = 4.5
1%
3%
0.5%
1.5%
Cov = 4.5
4.5
=1
Corr =
(3)(1.5)
10
Measures of Co-Movement
Calculating Covariance / Correlation
Example: Perfect Negative Correlation
Pr
rain
1/2
sunshine 1/2
mean
s.d .
Umbrella Co Straw Sandal Co
(rIC − rIC )(rSS − rss )
4%
-2%
-1%
2%
(3) (-1.5) = -4.5
(-3)(1.5) = -4.5
1%
3%
0.5%
1.5%
Cov = -4.5
− 4. 5
Corr =
= −1
(3)(1.5)
11
Measures of Co-Movement
Calculating Covariance / Correlation
Example: No Correlation
Pr
rain
hot sunshine
1/3
1/3
cold sunshine 1/3
mean
s.d .
Ice Cream Co Straw Sandal Co (rIC − rIC )(rSS − rss )
-2%
4%
-1%
0.33%
2.625%
-1%
2%
7.25%
2.75%
3.41%
(-7/3)(-3.75) = 26.25/3
(11/3)(-0.75) = -8.25/3
(-4/3)(4.5) = -18/3
Cov = 0
Corr =
0
=0
(2.625 )(3.41)
12
Expected Return & Variance of a Risky Portfolio
A Portfolio of 2 Risky Assets
stock 1
stock 2
Return
r1
r2
mean
E (r1 )
E (r2 )
variance Var (r1 ) Var (r2 )
=  12
=  22
a “portfolio” of stock 1 and 2
rp= w1r1 + w2 r2
E(rp)= ?
V(rp)= ?
13
Expected Return & Variance of a Risky Portfolio
Additional Properties of Expected Returns
If a and b are constants, then;
n
E (ar1 + br2 ) =  p(s)[ar (s) + br (s)]
s =1
1
2
n
n
s =1
s =1
= a  p( s )r1 ( s ) + b p( s )r2 ( s )
= aE (r1 ) + bE (r2 )
14
Expected Return & Variance of a Risky Portfolio
Additional Properties of Variance
If a and b are constants, then;
V (ar1 + br2 ) = E[( ar1 + br2 ) − (aE (r1 ) + bE (r2 ) )]2
= E[a(r1 − E (r1 ) ) + b(r2 − E (r2 ) )]
2
= E[a (r1 − E (r1 ) ) + b (r2 − E (r2 ) ) + 2ab(r1 − E (r1 ) )(r2 − E (r2 ) )]
2
2
2
2
= a E (r1 − E (r1 )) + b E (r2 − E (r2 )) + 2abE[(r1 − E (r1 ) )(r2 − E (r2 ))]
= a 2V (r1 ) + b 2V (r2 ) + 2abCov (r1 , r2 )
2
2
2
2
15
Expected Return & Variance of a Risky Portfolio
Alternative Ways of Calculating Variance and Covariance
Variance (r ) = E[[ r − E (r )]2 ] = E[r 2 − 2rE (r ) + (E (r ) )2 ]
= E ( r 2 ) − 2 E ( r ) E ( r ) + (E ( r ) )
2
= E ( r ) − (E ( r ) )
2
2


Covariance(ri , rj ) = E [ri − E (ri )][rj − E (rj )]


= E ri rj − E (ri )rj − E (rj )ri + E (ri ) E (rj )]
= E (ri rj ) − E (ri ) E (rj ) − E (rj ) E (ri ) + E (ri ) E (rj )]
= E (ri rj ) − E (ri ) E (rj )
16
Expected Return & Variance of a Risky Portfolio
A Portfolio of 2 Risky Assets
stock 1
stock 2
Return
r1
r2
mean
E (r1 )
E (r2 )
variance Var (r1 ) Var (r2 )
=  12
=  22
a “portfolio” of stock 1 and 2
rp= w1r1 + w2 r2
E(rp)= w1 E (r1 ) + w2 E (r2 )
V(rp)= w12 12 + w22 22 + 2w1w2 12
= w12 12 + w22 22 + 2w1w2  1 2
17
Expected Return & Variance of a Risky Portfolio
A Portfolio of 2 Risky Assets
Variance of a Portfolio of 2 Risky Assets: In Detail
2 2
2 2
w

+
w
V(rp)= 1 1
2  2 + 2 w1 w2  1 2
 ( w1  1 + w2 2 ) 2= w12 12 + w22 22 + 2w1w2 1 2
s.d.(rp) ≤ ( w1  1 + w2 2 )
s.d. of a portfolio is smaller
than weighted average of
individual stock’s s.d.
=> Power of Diversification!!
18
Expected Return & Variance of a Risky Portfolio
A Portfolio of 2 Risky Assets
Example: Calculating E(rp)& V(rp) of a portfolio using MS Excel
Scenario Probability
1
0.14
2
0.36
3
0.30
4
0.20
rates of return
r1
r2
-0.10
-0.35
0.00
0.20
0.10
0.45
0.32
-0.19
Form a portfolio that
invests 0.4 in stock 1
and 0.6 in stock 2
What is the expected return and variance of this portfolio?
2 methods
(1) Formula Approach
(2) Scenario Approach
19
Expected Return & Variance of a Risky Portfolio
A Portfolio of 2 Risky Assets
Example: Calculating E(rp)& V(rp) of a portfolio using MS Excel
w1=
w2=
0.4
0.6
Scenario Probability
1
2
3
4
0.14
0.36
0.30
0.20
Mean
SD
Covariance
Correlation
What happens when you
change these weights?
rates of retu rn
r1
r2
-0.10
0.00
0.10
0.32
-0.35
0.20
0.45
-0.19
0.08
0.1359
0.12
0.2918
-0.0034
-0.0847
Portfolio retu rn
w 1*r1+w 2*r2
-0.2500
0.1200
0.3100
0.0140
0.104
0.1788
0.104
0.1788
↑
↑
Formula Approach
Scenario Approach
utilizing mean, sd,
& covariances only
utilizing full probability
distribution
20
Agenda 2
Effect of Changes in Weights on
Expected Return
Variance
of risky portfolios
Derivation of Portfolio Opportunity Set
Construction of Optimal Risky Portfolio
Construction of Optimal Complete Portfolio
21
Effect of Changes in Weights
What Happens when you change the weights?
Example: continued
rates of return
r1
r2
Mean
SD
0.08
0.1359
Covariance
Correlation
0.12
0.2918
-0.0034
-0.0847
1.00
0.99
Portfolio
E(rp)
s.d.(rp)
0.1200 0.2918
0.1196 0.2887
0.40
0.60
0.1040 0.1788
0.99
0.01
0.0804 0.1344
1.00
0.00
0.0800 0.1359
w1
0.00
0.01
w2
22
Effect of Changes in Weights
Relationship between w1 and E(rp)
23
Effect of Changes in Weights
Relationship between w1 and s.d.(rp)
24
Effect of Changes in Weights
Relationship between w1 and E(rp)
Short Sales Allowed
When short-sale is allowed
some weights can be
negative and some can be
greater than one. The
securities shorted have
negative weights and the
securities purchased with the
proceeds of short-sale have
weights greater than one.
→ 𝑤1 < 0, 𝑤2 > 1
but the sum of all weights still
adds up to one.
෍ 𝑤𝑖 = 1
25
Effect of Changes in Weights
Relationship between w1 and s.d.(rp)
Short Sales Allowed
26
Effect of Changes in Weights
What happens when ρ changes?
Effect of changes in ρ on [w1 - E(rp)] and [w1 - s.d.(rp)] relationship
[w1 - E(rp)] relationship: Not affected!!
[w1 – s.d.(rp)] relationship: ? Let’s Check with Excel
As ρ gets closer to +1 :
s.d.(rp) becomes more linear
As ρ gets closer to -1 :
minimum s.d.(rp) becomes closer to zero
27
Derivation of Portfolio Opportunity Set
Combine [w1 - E(rp)] & [w1 - s.d.(rp)] plot => Create [s.d.(rp) - E(rp)] plot
Investment Opportunity Set is the set
of available portfolio risk-return
combinations
E(rp)=10.4%
•
stock 1
stock 2
A risky portfolio of 40% in
stock 1, 60% in stock 2
σp=17.88%
28
Derivation of Portfolio Opportunity Set
Short Sales Allowed
Combine [w1 - E(rp)] & [w1 - s.d.(rp)] plot => Create [s.d.(rp) - E(rp)] plot
What if short sales are allowed?
• stock 2
•stock 1
29
Derivation of Portfolio Opportunity Set
What happens when ρ changes?
Effect of changes in ρ on [s.d.(rp) - E(rp)] relationship
Let’s Check with Excel
As ρ gets closer to +1 : No benefit from diversification
As ρ gets closer to -1 :
Possible to eliminate risk completely
30
Expected Return & Variance of a Risky Portfolio
▪ Minimum-variance portfolio
□ A standard deviation smaller than that of either of the
individual components assets
V (ar1 + br2 ) = 𝑤12 𝜎12 + 𝑤22 𝜎22 + 2𝑤1 𝑤2 𝐶𝑂𝑉 𝑟1 , 𝑟2
= w12 12 + w22 22 + 2w1w2  1 2
□ −1 < 𝜌 < 1
𝑤𝑀𝑖𝑛
□ 𝜌 = −1
𝜎22 − 𝐶𝑂𝑉 𝑟1 , 𝑟2
𝑟1 = 2
𝜎1 + 𝜎22 − 2𝐶𝑂𝑉 𝑟1 , 𝑟2
𝜎2
(𝑤1 𝜎1 − 𝑤2 𝜎2 )2 = 0 𝑤𝑀𝑖𝑛 𝑟1 = 𝜎 + 𝜎
1
2
As ρ gets closer to +1 : No benefit from diversification
As ρ gets closer to -1 : Possible to eliminate risk completely
NTU. Dept of Fin. Investment
32
Construction of Optimal Risky Portfolio
Recall the Previous 2 Risky Assets Example
Mean
SD
Covariance
Correlation
rates of return
r1
r2
0.08
0.12
0.1359
0.2918
-0.0034
-0.0847
From above information, we
derived portfolio opportunity set
stock 2
stock 1
Q: Out of all available portfolios, which one is the optimal?
Let’s combine these risky portfolios with risk-free asset
33
Construction of Optimal Risky Portfolio
Recall from last class: Portfolio of Risk-free Asset and One Risky Asset
▪ P: risky asset (or a portfolio of risky assets)
▪ rp: (uncertain) rate of return of P
=> E(rp): expected return, σp: standard deviation of rp Previous class, P was arbitrarily
given, so E(rp) and s.d.(rp) was fixed
▪ F: risk-free asset
▪ rf: (certain) rate of return of F
▪ C: portfolio of P and F
▪ rc: rate of return of C => rc = yrp + (1 − y )r f
E(rc) = rf + y[ E (rp ) − rf ]
 c = y p
Today, we choose P from available
portfolios that provides best result
E (rc ) = r f +
Optimal Risky
Portfolio
[ E (rp ) − r f ]
p
c
34
Construction of Optimal Risky Portfolio
A Portfolio of 2 Risky Assets
a risky asset P
Key Criteria in mixing
possible P’s with F:
+ Risk-free Asset
rf = 7% risk-free asset F
Optimal Risky Portfolio
What is w1 at this point?
choose w1 (and w2) such that;
Sharpe Ratio =
E (rp ) − rf
will be maximized
CAL 3
CAL 2
•
•
s.d .(rp )
F
rf
CAL 1
•
•
35
Construction of Optimal Risky Portfolio
The Optimal CAL
▪ The optimal CAL is the one with the highest slope (or highest Sharpe
ratio).
□ The optimal CAL is the tangency line between the risk-free rate and the risky
investment opportunity curve
□ The risky portfolio that the optimal CAL is tangent with is called the tangency
portfolio.
𝑬(𝒓 )−𝒓
𝑬(𝒓 )−𝒓
𝐌𝐚𝐱 𝑺𝒍𝒐𝒑𝒆𝒑 =
𝑾𝒊
𝒑
𝒇
𝝈𝒑 −𝟎
=
𝒑
𝝈𝒑
𝒇
(Sharpe Ratio of the portfolio)
s.t 𝑤𝑎 + 𝑤𝑏 =1
𝑤𝑏 =
𝐸 𝑟𝑏 − 𝑟𝑓 𝜎𝑎2 − 𝐸 𝑟𝑎 − 𝑟𝑓 𝐶𝑂𝑉(𝑟𝑎 , 𝑟𝑏 )
𝐸 𝑟𝑏 − 𝑟𝑓 𝜎𝑎2 + 𝐸 𝑟𝑎 − 𝑟𝑓 𝜎𝑏2 − 𝐸 𝑟𝑏 − 𝑟𝑓 + 𝐸 𝑟𝑎 − 𝑟𝑓 𝐶𝑂𝑉(𝑟𝑎 , 𝑟𝑏 )
NTU. Dept of Fin. Investment
36
Construction of Optimal Risky Portfolio
A Portfolio of 2 Risky Assets
+ Risk-free (7%)
Optimal Risky Portfolio P (using a formula)
w1 = 52%
w2 = 48%
E(rp) = 9.92%
s.d.(rp) = 15.14%
Sharpe Ratio =
E (rp ) − rf
9.92 − 7
=
15.14
s.d .(rp )
Q: What about asset allocation between
optimal risky portfolio & risk-free asset?
= 19.3%
need information about
investor’s risk aversion
37
Construction of Optimal Complete Portfolio
A Portfolio of 2 Risky Assets
+ Risk-free (7%)
CAL
+ Risk Aversion
(ex. A = 4)
y =
*
Optimal Risky Portfolio
E ( rp ) − r f
A p2
0.0992 − 0.07
=
4 * 0.1514 2
indifference
curve
= 31.85%
Optimal Complete Portfolio
What is y* (proportion invested in risky asset) at this point?
NTU. Dept of Fin. Investment
38
Construction of Optimal Complete Portfolio
Optimal Complete Portfolio (2 risky assets): Summary
Assumptions
r1
Mean
SD
Covariance
0.08
0.1359
-0.0034
r2
0.12
0.2918
rf = 7%
A=4
w1 = 52% w2 = 48%
E(rp) = 9.92%
s.d.(rp) = 15.14%
y * = 31.85%
Final Asset Allocation
Proportion invested in stock 1 = 52%*31.85% = 16.56%
Proportion invested in stock 2 = 48%*31.85% = 15.29%
Proportion invested in risk-free asset = 1-31.85% = 68.15%
E(rc) = 31.85%*9.92% + 68.15%*7% = 7.93%
your advice to your client!!!
s.d.(rC) = 31.85%*15.14% = 4.82%
39
Agenda 3
Variance of a Portfolio: 3 Risky Assets
Derivation of Portfolio Opportunity Set
Construction of Optimal Risky Portfolio
Construction of Optimal Complete Portfolio
42
Variance of a Risky Portfolio
A Portfolio of 3 Risky Assets
Recall variance of a portfolio of 2 risky assets
2 2
2 2
V(rp)= w1  1 + w2 2 + 2w1w2  1 2
= w  + w  + 2w1w2 12
2 2
1 1
2
2
2
2
= w1 w1  11 + w2 w2 22 + w1w2 12 + w1w2 12
w1
w2
w1
w1*w1*Cov(r1,r1)
w2*w1*Cov(r2,r1)
w2
w1*w2*Cov(r1,r2)
w2*w2*Cov(r2,r2)
43
Variance of a Risky Portfolio
A Portfolio of 3 Risky Assets
Variance of a portfolio of 3 risky assets
w1
w2
w3
w1
w2
w3
w1*w1*Cov(r1,r1)
w2*w1*Cov(r2,r1)
w3*w1*Cov(r3,r1)
w1*w2*Cov(r1,r2)
w2*w2*Cov(r2,r2)
w3*w2*Cov(r3,r2)
w1*w3*Cov(r1,r3)
w2*w3*Cov(r2,r3)
w3*w3*Cov(r3,r3)
V(rp)=
w12 12 + w22 22 + w32 32
+ 2w1w2 12 + 2 w2 w3 23 + 2w1w3 13
44
Variance of a Risky Portfolio
A Portfolio of 3 Risky Assets
Recall Previous 2 Risky Assets Example
Scenario Probability
1
2
3
4
0.14
0.36
0.30
0.20
Mean
SD
between
Covariance
Correlation
rates of retu rn
r1
r2
=> Let’s add a 3rd stock
r3
-0.10
0.00
0.10
0.32
-0.35
0.20
0.45
-0.19
0.14
-0.10
0.17
0.05
0.08
0.1359
0.12
0.2918
0.0446
0.1163
Form a portfolio of 3 stocks
1-2
2-3
1-3
-0.0034
-0.0847
0.0016
0.0483
0.0028
0.1753
45
Variance of a Risky Portfolio
A Portfolio of 3 Risky Assets
Example: continued
rates of return
r1
r2
Mean
SD
between
Covariance
Correlation
w1
0.3
r3
0.08
0.1359
0.12
0.2918
0.0446
0.1163
1-2
-0.0034
-0.0847
2-3
0.0016
0.0483
1-3
0.0028
0.1753
w2
Portfolio
w3
E(rp) s.d.(rp)
0.2 0.0929 0.1521
0.5
What happens when you
change these weights?
46
Derivation of Portfolio Opportunity Set
What Happens when you change the weights?
Example: continued
rates of return
r1
r2
Mean
SD
between
Covariance
Correlation
w1
0.08
0.1359
0.12
0.2918
0.0446
0.1163
1-2
-0.0034
-0.0847
2-3
0.0016
0.0483
1-3
0.0028
0.1753
w2
0.0
0.0
0.9
1.0
r3
0.0
0.1
0.1
0.0
Portfolio
w3
E(rp) s.d.(rp)
1.0 0.0446
0.1163 0.1163
0.0000
0.9 0.1338 0.0932
0.0
0.0
0.0840 0.1233
0.0800 0.1359
47
Derivation of Portfolio Opportunity Set
Portfolios of 3 Risky Assets
stock 2
stock 1
stock 3
48
Derivation of Portfolio Opportunity Set
Portfolios of 3 Risky Assets
stock 2
stock 1
Previous 2 stocks case:
Portfolio opportunity set
with stocks 1 and 2 only
stock 3
49
Derivation of Portfolio Opportunity Set
Portfolios of 3 Risky Assets
→ If we add more assets, the curve extends more
to the Northwest corner of the graph (diversification)
50
Derivation of Portfolio Opportunity Set
Portfolios of 3 Risky Assets
“Efficient Frontier”
global minimumvariance portfolio
•
stock 2
Efficient Frontier –
graph representing the
set of portfolios that
maximizes expected
return at each level of
portfolio risk.
stock 1
stock 3
minimum - variance frontier
The efficient frontier
shows us the best
possible returns we
can get for any risk
(standard deviation)
51
Derivation of Portfolio Opportunity Set
Q: How do we obtain weights of Efficient Portfolios?
=> Let’s use Excel Solver
[Efficient Weights based on Excel results]
w1
w2
w3
E(rp)
s.d.(rp)
0.00
0.25
0.50
0.68
0.56
0.43
0.31
0.15
1.00
0.75
0.50
0.28
0.21
0.13
0.06
0.00
0.00
0.00
0.00
0.04
0.23
0.43
0.63
0.85
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.2918
0.2186
0.1556
0.1194
0.1014
0.0915
0.0925
0.1042
52
Derivation of Portfolio Opportunity Set
Efficient Portfolios with 3 Risky Assets
53
Construction of Optimal Risky Portfolio
Mean
SD
between
Covariance
Correlation
rates of return
r1
r2
0.08
0.12
0.1359 0.2918
r3
0.0446
0.1163
1-2
-0.0034
-0.0847
1-3
0.0028
0.1753
2-3
0.0016
0.0483
From above information, we derived
efficient frontier
Q: Out of all available portfolios, which one is the optimal?
Let’s combine these risky portfolios with risk-free asset
NTU. Dept of Fin. Investment
54
Construction of Optimal Risky Portfolio
+ Risk-free Asset
rf = 2%
a risky asset P
risk-free asset F
A Portfolio of 3 Risky Assets
Key Criteria in mixing possible P’s
with F:
CAL 2 Optimal Risky Portfolio
What is w1 at this point?
choose w1 (and w2) such that;
Sharpe Ratio =
E ( rp ) − r f
•
•
s.d .(rp )
will be maximized
F
rf
CAL 1
•
NTU. Dept of Fin. Investment
55
Construction of Optimal Risky Portfolio
A Portfolio of 3 Risky Assets
+ Risk-free (2%)
[Optimal Risky Portfolio based on Excel Solver]
Optimal Risky Portfolio
56
Construction of Optimal Risky Portfolio
A Portfolio of 3 Risky Assets
+ Risk-free (2%)
Optimal Risky Portfolio (based on Excel Solver)
w1 = 60%
w2 = 23%
w3 = 17%
E(rp) = 8.3%
s.d.(rp) = 10.61%
Sharpe Ratio =
E (rp ) − rf
s.d .(rp )
8.3 − 2
=
10.61
What about asset allocation between
optimal risky portfolio & risk-free asset?
= 59.4%
need information about
investor’s risk aversion
57
Construction of Optimal Complete Portfolio
A Portfolio of 3 Risky Assets
+ Risk-free (2%)
+ Risk Aversion
(ex. A = 6)
indifference
curve
y =
*
What is y* (proportion invested in
risky asset) at this point?
Optimal Complete Portfolio
E ( rp ) − r f
A p2
0.083 − 0.02
=
6 * 0.10612
= 93.27%
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58
Construction of Optimal Complete Portfolio
Optimal Complete Portfolio (3 risky assets): Summary
Assumptions
Mean
SD
between
Covariance
Correlation
rates of return
r1
r2
0.08
0.12
0.1359 0.2918
r3
0.0446
0.1163
1-2
-0.0034
-0.0847
1-3
0.0028
0.1753
2-3
0.0016
0.0483
rf = 2%
A=6
w1 = 60%
w2 = 23%
w3 = 17%
E(rp) = 8.3%
s.d.(rp) = 10.61%
y * = 93.27%
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59
Construction of Optimal Complete Portfolio
Optimal Complete Portfolio (3 risky assets): Summary
Final Asset Allocation
Proportion invested in stock 1 = 60%*93.27% = 55.96%
Proportion invested in stock 2 = 23%*93.27% = 21.45%
Proportion invested in stock 3 = 17%*93.27% = 15.86%
Proportion invested in risk-free asset = 1-93.27% = 6.73%
E(rc) = 93.27%*8.3% + 6.73%*2% = 7.88%
s.d.(rC) = 93.27%*10.61% = 9.896%
NTU. Dept of Fin. Investment
your advice to
your client!!!
60
Construction of Optimal Complete Portfolio
General Steps to Obtain Optimal Complete Portfolio
1. Specify the return characteristics of all securities (E(ri), s.d.(ri),Cov(ri,rj))
2. Establish the risky portfolio
- Calculate the optimal risky portfolio, P (i.e. the weights, wi’s)
- Calculate the properties of P (E(rp), s.d.(rp), Sharpe Ratio) using wi’s
3. Allocate funds between the (optimal) risky portfolio and risk-free asset
- Determine the weights between P and risk-free asset
- Calculate final weights allocated to each security in P
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61
Agenda 4
N risky assets
Expected Return & Variance of Risky Portfolios
Diversification and Portfolio Risk
62
Expected Return & Variance of a Risky Portfolio
Multiple Risky Assets
stock 1…stock N
Return
mean
r1
E (r1 )
rN
E (rN )
variance Var (r1 ) Var (rN )
a “portfolio” of N stocks
N
rp =  wi ri
i =1
portfolio weight
on stock i
N
w = 1
i =1
i
N
N


E (rp ) = E  wi ri =  wi E (ri )
i =1
 i =1
N

Var (rp ) = Var  wi ri  =
i =1

?
63
Expected Return & Variance of a Risky Portfolio
Multiple Risky Assets
Recall variance of a portfolio of 3 risky assets
w1
w2
w3
w1
w2
w2
w1*w1*Cov(r1,r1)
w2*w1*Cov(r2,r1)
w3*w1*Cov(r3,r1)
w1*w2*Cov(r1,r2)
w2*w2*Cov(r2,r2)
w3*w2*Cov(r3,r2)
w1*w3*Cov(r1,r3)
w2*w3*Cov(r2,r3)
w3*w3*Cov(r3,r3)
V(rp)= w12 12 + w22 22 + w32 32 + 2w1w2 12 + 2w2 w3 23 + 2w1w3 13
64
Expected Return & Variance of a Risky Portfolio
Multiple Risky Assets
Variance of a portfolio of N risky assets
w1
|
wN
w1
………………..…..
wN
w1*w1*Cov(r1,r1)
|
wN*w1*Cov(rN,r1)
………………..…..
w1*wN*Cov(r1,rN)
|
wN*wN*Cov(rN,rN)
………………..…..
N
V(rp)=
N
N
 w Var (r ) +  w w Cov(r , r )
i =1
2
i
i
i j
i
j
i
j
65
Expected Return & Variance of a Risky Portfolio
Multiple Risky Assets
The shaded boxes contain (weighted)
variance terms;
1
2
3
the remainder contain (weighted)
covariance terms.
4
weights
5
6
To calculate the portfolio’s
variance, add up the boxes
N
1
2
3
4
5
6
N
weights
66
Expected Return & Variance of a Risky Portfolio
Multiple Risky Assets
stock 1…stock N
Return
r1
rN
a “portfolio” of N stocks
N
rp =  wi ri
i =1
mean
E (r1 )
E (rN )
variance Var (r1 ) Var (rN )
portfolio weight on stock i
N
w = 1
i =1
i
N
 N
E (rp ) = E  wi ri  =  wi E (ri )
 i =1
 i =1
N

Var (rp ) = Var  wi ri 
 i =1

N
N
=  wi w j Cov(ri , rj )
i =1 j =1
N
N
i =1
i j
N
=  wi2Var (ri ) +  wi w j Cov(ri , rj )
67
Diversification and Portfolio Risk
What happens to portfolio variance when N goes to infinity?
N
N
N
 x Var (r ) +  x x Cov(r , r )
i =1
2
i
i
i j
i
j
i
2
N N
1
 1  1 
Var
(
r
)
+


  Cov(ri , r j )


i
 N  N 
i =1  N 
i j
N
assume xi = 1/N, then
j
1
= 
N
2 N
 1  1  N N
Var (ri ) +    Cov(ri , rj )

 N  N  i  j
i =1
N
N
Var (r )
1
= 
N
i
i =1
N
 1  1 
+   ( N 2 − N )
 N  N 
N
 Cov(r , r )
i
i j
j
N2 − N
1
1

=  average variance + 1 − average covariance
N
 N
68
Diversification and Portfolio Risk
What happens to portfolio variance when N goes to infinity?
N
N
i =1
i j
N
2
x
 i Var (ri ) +  xi x jCov(ri , rj )
1
1

average
variance
+
1
−


average covariance
N
 N
assume xi = 1/N, then = 
further assume all securities have common standard deviation, σ, and all security pairs have a
common correlation coefficient, ρ. Then;
1 2
1 2 
=   + 1 −  
N
 N
69
Diversification and Portfolio Risk
What happens to portfolio variance when N goes to infinity? (σ = 0.5)
∞
0.00
31.62
Source: BKM, Chap 7
70
Diversification and Portfolio Risk
What happens to portfolio variance when N goes to infinity?
Case 1: Average Covariance = 0
Case 2: Average Covariance > 0
average
covariance
71
Diversification and Portfolio Risk
What happens to portfolio variance when N goes to infinity?
Source: Statman, 1987, Journal of Financial and Quantitative Analysis
72
Derivation of Efficient Frontier
Consider stocks A, B, and C
We can form portfolios from A and B or
from B and C (blue curves)
We can also make portfolios out of the
portfolios we created (black curve)
This provides us even greater
diversification benefits and shifts the
possibilities to higher returns and lower
standard deviations.
Derivation of Efficient Frontier
E(Rp)
The composite of all stock sets constitutes the efficient frontier
Global Minimum
Variance Portfolio
Each half egg shell represents the possible weighted
combinations for two stocks (portfolios).
σp
Derivation of Efficient Frontier
Mean-Variance Optimization
N
=  wi E (ri )
i =1
N
N
=  wi w j Cov(ri , rj )
i =1 j =1
75
Markowitz’s Key Result / Implications
What did Markowitz’s dissertation committee think about his paper ?
…His work on portfolio theory, of which this article was the beginning, won him a
share of the Nobel prize in economics in 1990. Apparently, the importance of his work
was not widely-recognized at first. Milton Friedman, a member of Markowitz’s
doctoral dissertation committee and who also became a Nobel laureate, questioned
whether the work met the requirements for an economics Ph.D. See Bernstein
(Bernstein 1992).
Bernstein, Peter L., 1992, Capital Ideas: The Improbable Origins of Modern Wall Street (Free
Press). Requoted from G. Pennacchi, 2002, Asset Pricing
76
Markowitz’s Key Result / Implications
▪ The phrase “don’t put all your eggs in one basket” existed long before
modern finance theory.
□ It was not until 1952, however, that Harry Markowitz published a formal model
of portfolio selection embodying diversification principles
□ His model is precisely step one of portfolio management: the identification of
the efficient set of portfolios, or the efficient frontier of risky assets.
• The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are interested
only in that portfolio with the highest expected return.
• Alternatively, the frontier is the set of portfolios that minimizes the variance for any target expected
return.
77
Derivation of Efficient Frontier
Optimum Portfolio Selection (without risk-free asset)
E(Rp)
indifference
curves
Global Minimum
Variance Portfolio
σp
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78
Derivation of Optimal Risky Portfolio
Introduce Risk Free Asset (J. Tobin, 1958)
risky
assets
risk free
asset
Return
rp
rf
mean
E (rp )
E (rf )
= rf
variance Var (rp ) Var (rf )
=  p2
=  2f = 0
definition of
risk free asset
a “combination” of risky assets and risk free asset
rc = yrp + (1 − y )rf

E (rc ) = E yrp + (1 − y)rf


= yE (rp ) + (1 − y)rf
Var (rc ) = Var yrp + (1 − y)rf

= y 2 p2
E (rc ) = rf +
E (rp ) − rf
NTU. Dept of Fin. Investment
p
 c => CAL
79
Derivation of Optimal Risky Portfolio
Optimal Risky Portfolio
CAL3
E(Rc) p3 = Optimal Risky Portfolio
Risk-free rate
(ex. T-Bill)
CAL2
CAL1
E(rp3)
slope = Sharpe Ratio
E (rc ) = rf +
rf
σp1σp2 σp3
E (rp ) − rf
p
c
σc
80
Derivation of Optimal Risky Portfolio
Optimum Portfolio Selection (with risk-free asset)
CAL3
E(Rc)
indifference
curves
E(rp3)
p3 = Optimal Risky Portfolio
rf
σp3
σc
81
Portfolio Selection: A Summary
Step 1: Construction of Efficient Frontier
Specify the return characteristics of all securities (E(ri),
s.d.(ri),Cov(ri,rj)) : inputs to mean variance optimization
* number of estimates required for N securities
N estimates of expected returns, N estimates of variances
(N2 – N)/2 estimates of covariances
Total: 0.5N(N+3) number of estimates
ex. If N = 50, then total estimates = 1,325
 Obtain efficient frontier through mean-variance optimization
82
Portfolio Selection: A Summary
Step 2: Construction of Optimal Risky Portfolio
Introduce risk-free asset
Mix (i.e. form a portfolio) with risky portfolios along the frontier
 Obtain CAL with the highest slope (Sharpe Ratio)
 Obtain optimal risky portfolio P
P will be the same for all clients/investors!!!! Separation Property
84
Portfolio Selection: A Summary
Step 3: Construction of Optimal Complete Portfolio
Determine the weights between P and risk-free asset
=> Depends on individual’s preferences towards risk (risk aversion)
85
Portfolio Selection: A Summary
Step 3: Construction of Optimal Complete Portfolio
Example: Risk Tolerance Questionnaire
86
Portfolio Selection: A Summary
Separation Property
▪ Portfolio choice problem may be separated into two independent tasks
- Product Decision (steps 1& 2): determination of the optimal risky portfolio
(will generally be the same for all investors)
= > technical issue
- Consumption Decision (step 3) : allocation between risky portfolio and
risk-free asset (can be different across investors)
= > personal issue
87
Quiz
σP = Absolute value [wAσA − wBσB]
0 = 5 × wA − [10  (1 – wA)]  wA = 0.6667
The expected rate of return for this risk-free portfolio is:
E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667%
Therefore, the risk-free rate is: 11.667%
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89
Exercise Questions (BKM)
▪ Problem set #7-12
▪ For midterm exam,
□ Agenda 1,2, and 4 in these slides
□ However, you should understand key concepts in Agenda 3.
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