OPTIMAL RISKY PORTFOLIOS

Top-down process
◦ Capital allocation between risky portfolio and
risk-free assets
◦ Asset allocation across broad asset classes
(bond/ stock)
◦ Security selection of individual assets within
each asset class


Risky portfolio (E/D )
Risky portfolio and risk-free asset (E/D/Rf)
6.1 Diversification and Portfolio risk



Consider a portfolio composed of only one
stock, what would be the sources of risk?
What if add one stock?
Insurance principle
Market risk
◦ Risk that remains even after extensive
diversification, attributable to market wide risk
sources
◦ Systematic or non-diversifiable

Firm-specific risk
◦ Risk that can be eliminated by diversification
◦ Diversifiable or nonsystematic
6.2 Portfolio of Two Risk Assets

Study the efficient diversification
◦ Construct risky portfolios to provide the lowest
possible risk for any given level of expected return

Consider a portfolio comprised of two mutual funds,
a bond portfolio D, and a stock fund E
rp

rP
 Portfolio Return
wr
D
D
 wE r E
wD  Bond Weight
rD
 Bond Return
wE  Stock Weight
rE
 Stock Return

Rate of return on the portfolio:
rP  wD rD  wE rE

Expected rate of return on the portfolio:
E (rP )  wD E (rD )  wE E (rE )

Variance of the rate of return on the portfolio:
  wD  D  wE  E  2wD wE Cov  rD , rE 
2
P
2
2
2
2
rp

rP

wr
D
D
 wE r E
Portfolio Return
wD  Bond Weight
rD
 Bond Return
wE  Equity Weight
rE
 Equity Return
E (rp )  wD E (rD )  wE E (rE )
  w   w   2wD wE Cov(rD , rE )
2
P
2
D

2
D

2
E
2
D
2
E
2
E
= Variance of Security D
= Variance of Security E
= Covariance of returns for
Cov(rD , rE )
Security D and Security E
S
Cov(rE , rD )   p(i )  rE (i )  rE   rD (i )  rD 
i 1

Covariance:
S
Cov(rE , rD )   p (i )  rE (i )  rE   rD (i )  rD 
i 1
S
Cov(rE , rE )   p(i )  rE (i )  rE   rE (i )  rE    E2
i 1

Another way to express variance of the
portfolio:
 P2  wD wDCov(rD , rD )  wE wE Cov(rE , rE )  2wD wE Cov(rD , rE )

Variance is reduced
  w  
 w1
  2wD wE Cov(rD , rE )
2
P


2
D
2
D
2
E
2
E
Correlation coefficient
 ED 
Portfolio variance
Cov(rE , rD )
 E D
 w D D  w E E  2w D w E D E 
p

  Portfolio Variance
  Portfolio Standard Deviation
2
2
2
p
2
p
2
2
2
D,E
Range of values for  D,E
-1.0 <  < 1.0
If  = 1.0, the securities would be
perfectly positively correlated
If  = - 1.0, the securities would be
perfectly negatively correlated

When Perfect positive correlation

Portfolio variance:
 ED  1

2
p
 w D D  w E E  2w D w E D E
2
2
  wD D  wE E 
2
2
2
 P  wD D  wE E

Portfolio SD is no bigger than the weighted average
of the individual security SD

When perfect positive
correlation


p
 wD D  wE E
When perfect negative
correlation

p

wD D  w E E




Hedge asset: negative correlation with the other assets
in the portfolio
Expected return is unaffected by correlation, standard
deviation is less than the weighted average of the
component standard deviation
portfolio of less than perfectly correlated assets always
offer better risk-return opportunities than the
individual component securities on their own
The lower the correlation, the greater the gain in
efficiency.

Perfect hedged position
◦ When
  1
To solve

p
 wD D  w E E  0
wD  wE  1
E
D
wD 
, wE 
E D
E D
E (rP )  8wD  13wE
  12 w  20 w  2*12*20*0.3* wE wD
2
P
2
2
D
2
2
E
 144wD2  400wE2  144wE wD
Experiment with different proportions
Minimum
variance
portfolio


The Minimum-Variance Portfolio
In the Example   0.3 ,what is the minimum level
Min
wD
2
p
wD  wE  1
wmin D
 E2  Cov(rD , rE )
 2
 0.82
2
 D   E  2Cov(rD , rE )
wmin E  0.18
 min  11.45%
E (rP )  8.9%

Diversification effect: Minimum-variance portfolio has a
standard deviation smaller than that of either of the
individual component assets
◦ Pass through the two undiversified portfolios of WD=1,WE=1


Standard deviation is smaller than that of
either of the individual component assets
Figure 7.3 and 7.4 combined
demonstrate the relationship between
portfolio risk
PORTFOLIO
OPPORTUNITY
SET:
For any pair of
wD and wE
EXPECTED RETURN
EXPECTED RETURN
20
15
10
5
0
EXPECTED…
WEIGHT IN E
weight in E
1.
5
1.
3
1.
1
0.
9
0.
7
0.
5
0.
3
0.
1
-0
.5
-0
.3
-0
.1
portfolio std deviation
standard deviation
40
35
30
25
20
15
-1
0
0.3
1
10
5
0

Portfolio Opportunity set:
◦ All combination of portfolio expected return and std
deviation that can be constructed from the two
available assets

The best portfolio will depend on risk aversion

The relationship depends on the
correlation coefficient
◦ -1.0 <  < +1.0
◦ The smaller the correlation, the greater the
risk reduction potential
◦ If  = +1.0, no risk reduction is possible
◦ When  = -1.0 , perfect hedging
opportunity and maximum advantage from
diversification
6.3 Asset Allocation with Stocks,
Bonds and Bills

Asset allocation across the three key asset classes:
◦ Stocks, bonds, risk-free assets


Risk-free T-bills yielding 5%
Two possible CAL, comparing their reward-tovariability ratio
◦ CAL A
◦ CAL B
Si 
E (ri )  rf
i
B: Wd=0.7,wE=0.3
SB 
E (rB )  rf
B

9.5  5
 0.38
11.7
A: minimum
variance portfolio
wD=0.82,wE=0.18
SA 
E (rA )  rf
A

8.9  5
 0.34
11.45
Tangency portfolio, yield the CAL with
highest feasible reward-to-volatility ratio,
the optimal risky portfolio to mix with T-bills


Maximize the slope of the CAL for any
possible portfolio, p
The objective function is the slope:
SP 
E (rP )  rf
P

Maximize the slope of the CAL
◦ The tangency portfolio is the
optimal portfolio to mix with T-bills
Max S P 
wi
w
i
E (rP )  rf
P
1
wD
 E (rD )  rf   E2   E (rE )  rf  Cov(rD , rE )

 E (rE )  rf   D2   E (rD )  rf   E2   E (rD )  rf  E (rE )  rf  Cov(rD , rE )
wE  1  wD
The solution for the optimal risky portfolio
(see xls : optimal risky portfolio)

wD  0.4, wE  0.6
E (rP )  11%
 P  14.2

The optimal complete portfolio
◦ Specify the return characteristics of all
securities (expected return, variance,
covariance)
◦ The opportunity set of risky assets
◦ The optimal risky portfolio: The CAL
tangent with the opportunity set
◦ Use the investor’s degree of risk aversion

Investor with risk aversion A=4

y 


E  rP   rf
A p2
0.11  0.05

 0.7439
2
4*0.142
Percentage in
bonds=0.7439*0.4=0.2976
Percentage in
stocks=0.7439*0.6=0.4463
Indifference Curve
Opportunity Set of Risky Assets



The optimal combinations result in lowest
level of risk for a given return
The optimal trade-off is described as the
efficient frontier
These portfolios are dominant
6.4 The Markowitz Portfolio
Selection Model

Generalize the portfolio construction
problem to many risky assets and a risk-free
asset
◦ To determine the risk-return opportunities
available (Minimum-variance frontier) from the set
of risky assets
◦ Involve the risk-free asset (CAL tangent to the
efficient frontier) and get the optimal risky
portfolio P
◦ Choose the appropriate mix between P and T-bill

Minimum-variance frontier
◦ A graph of the lowest possible variance that can
be attained for a given portfolio expected return

Efficient frontier of risky assets
◦ The part of the frontier that lies above
search
for the
CAL
with the
highest
rewardtovariabili
ty ratio

Now the individual chooses the
appropriate mix between the optimal
risky portfolio P and T-bills as in Figure
7.8
n
 
2
P
i 1
n
 w w Cov(r , r )
j 1
i
j
i
j

Efficient frontier of risky assets
◦ Harry Markowitz, 1952
◦ Principle: for any risk level, we are interested
only in that portfolio with the highest
expected return, the frontier is the set of
portfolios that minimizes the variance for any
target expected return

The separation property
◦ The portfolio choice problem may be separated
into two independent tasks
 Determination of the optimal risky
portfolio is purely technical
 Allocation of the complete portfolio to Tbills versus the risky portfolio depends on
personal preference


Remember:
n
 
2
P
i 1
1 n 1 2 n
   i  
n i 1 n
j 1
j i
j 1
i
j
i
j
n
1
Cov(ri , rj )

2
i 1 n
If define the average variance and average
covariance of the securities as:
1 n 2
   i
n i 1
n
1
Cov 

n(n  1) j 1
2

 w w Cov(r , r )
For an equally weighted portfolio
2
p

n
j i
n
 Cov(r , r )
i 1
i
j
variance of an equally weighted portfolio as:
1 2
n 1
   
Cov
n
n
2
P

variance of an equally weighted portfolio as:
1 2
n 1
   
Cov
n
n
2
P


If average covariance is zero (uncorrelated),
when all risk is firm-specific, portfolio variance
approaches zero (power of diversification)
when n gets larger
Assume all securities same, discuss   0,  0,  0
1 2 n 1
2
   

n
n
2
P
6.5 Risk Pooling, Risk Sharing
and Risk in the Long Run

Property value is $100,000, payouts on the 1-year policy
as following
Loss: payout = $100,000
p = .001
No Loss: payout = 0
1 − p = .999
◦ Risk free rate=5%, up-front charge=$120,Compute the
risk premium and SD
 Expected return=
 100000*(1+5%)+120*(1+5%)-0.001*100000
 Risk premium=0.26%
 SD=3160.7/100000=3.16%
Risk premium
  0.001*999002  0.999*1002  3160.7

Sell 10000 of policies, uncorrelated, same E(r)
and SD
1
◦ the variance of the portfolio:

  
2
P
2
n
SD of the 10000 policies=

3.16%
p 

 0.0316%
n
10000

selling more policies causes risk to fall

When we combine n uncorrelated insurance policies each with
an expected profit of $ , both expected total profit and SD
grow in direct proportion to n:

E (n )  nE ( )
Var (n )  n Var ( )  n 
2
2
2
SD(n )  n

Ratio of mean and SD not change when n increases, riskreturn trade-off not improve with additional policies



What does explain the insurance business?
◦ Risk sharing or the distribution of a fixed amount of risk among
many investors
Risk sharing
◦ Example: insure a fraction of the project risk, fixed amount of
equity capital
◦ Underwriter diversifies its risk by allocating its investment budget
across many projects that are not perfectly correlated
◦ Limit exposure to any single source of risk by sharing the risk
with other underwriters
◦ Each one diversifies a largely fixed portfolio across many projects
Risk pooling: pooling many sources of risk in a portfolio of given
size
6.6 A Spreadsheet Model
6.7 Optimal Portfolios with Restrictions on
the Risk-Free Asset

Unique optimal risky portfolio
◦ When all investors can borrow and lend at the
risk-free rate
◦ Maximize the reward-to-variability ratio

Without a risk-free asset
◦ No tangency portfolio
◦ Superimpose a personal set of indifference
curves on the efficient frontier
Expected
return
More risk-tolerant
investor
B
●
●
Q
P
S
●
More risk-averse
investor
Std deviation

When risk-free investment available,
but cannot borrow
◦ CAL exist but limited to FP
◦ A: net lenders at rate of
r
f
◦ B: borrowing at risk-free, risk-tolerant
investors
◦ Q: with restriction on borrowing, B turned
to Q
Expected
return
B●
CAL
●
●
P
Q
A
●
rf
F
Std deviation


Investors wish to borrow to invest in a
risky portfolio at a rate higher than riskfree rate
Borrowing rate greater than the lending
rate
◦ CAL1
 FP1, efficient portfolio set for risk-averse
investors
 P1 as the optimal risky portfolio
 A as the complete portfolio
Expected
return
●
rBf
●
P2
CAL1
CAL2
B
Efficient Frontier
P1
●
rf
A
F
Std deviation
◦ CAL2
 Right of P2, efficient portfolio set for risk-tolerant
r
investors, borrow at the higher rate Bf to invest
 Left of P2 unavailable, because lending only at
 P2 as the optimal risky portfolio
 B as the complete portfolio
rf
CAL1
B CAL2
Expected
return
●
rBf
●
P1
P2
●
Efficient Frontier
●
rf
A
F
Std deviation

Investors in the middle range
◦ Choose the risky portfolio from range P1P2,
CAL1
CAL2
Expected
return
●
rBf
●
P1
C
●
P2
Efficient Frontier
rf
Std deviation

Rule 1: Expected return of an asset
E (r )   p ( s )r ( s )

Rule 2: Variance ofs an asset’s return
2
   p( s )  r ( s )  E (r ) 
2
s

Rule 3: Rate of Return on a Portfolio
E (rP )   wi ri
i

Rule 4: portfolio: a risky asset + a risk-free
asset
 P  wrisky risky



To quantify the hedging or
diversification potential of an asset,
use covariance and correlation
Portfolio risk depends on the
correlation between the returns of
the assets in the portfolio
Covariance and the correlation
coefficient provide a measure of the
returns on two assets to vary

Covariance:
S
Cov(rS , rB )   p (i )  rS (i )  rS   rB (i )  rB 
i 1

Correlation
Coefficient:
 SB 
Cov(rS , rB )
 S B

Rule 5: when two risky assets with
2
variances  2and  12 respectively, are
combined into a portfolio with portfolio
weights w1 and w2 , the portfolio variance
2
is given by  P
 P2  (w11 )2  (w2 2 )2  2w1w2 cov  r1, r2 

Correlation coefficient
12 



Range of values for
Cov (r1 , r2 )
 1 2
12
-1.0 < 12 < 1.0
If 12 = 1.0, the securities would be perfectly
positively correlated
If 12 = - 1.0, the securities would be perfectly
negatively correlated

Humanex portfolio:
• 50% T-bill , rate of return 5%
• 50% Best Candy stock
•Scenario analysis of best candy stock
Normal year
probability
Abnormal year
bullish
bearish
crisis
0.5
0.3
0.2
)  0.5*0.25  0.3*0.1  0.2*(0.25)  10.5%
Rate of returnE(r 25%
10%
-25%
best
•Expected rate of return of best candy stock

Variance
 best 2
 0.5*  0.25  0.105   0.3*  0.10  0.105   0.2*  0.25  0.105 
2
 357.25

Standard deviation
 best  18.9%
2
2

The portfolio’s
expected rate of
return
E (rP )   wi ri  0.5*0.105  0.5*0.05  7.75%
i

The portfolio’s
standard deviation
 P  wrisky risky  0.5*0.189  9.45%

Humanex portfolio:
• 50% Best Candy stock
• 50% Sugarcane’s stock
•Scenario analysis of Sugarcane’s stock
Normal year
probability
E (r )  6%
Abnormal year
bullish
bearish
crisis
0.5
0.3
0.2
-5%
35%
best
Rate of return 1%
•Expected rate of return of best candy stock
 best  14.73%
•Scenario analysis of the portfolio
Normal year
probability
Abnormal year
bullish
bearish
crisis
0.5
0.3
0.2
2.5%
5%
Rate of return 13%
•Expected rate of return of best candy stock
E (rp )  8.25%
 p  4.83%

Summarize
Portfolio
All in best candy
Expected
return
10.5%
Standard
deviation
18.9%
Half in T-bills
7.75%
9.45%
Half in
Sugarcane
8.25%
4.83%

Covariance of the return of best candy and
sugarcane stock
rsug  0.06 rbest  0.105
Cov(rbest , rsug )   p(i )  rbest (i )  rbest   rsug (i )  rsug 
i
 0.5*(0.25  0.105)  0.01  0.06   0.3*(0.1  0.105)  0.05  0.06 
0.2*(0.25  0.105)  0.35  0.06 
 2.405%
best , sug 
Cov(rbest , rsug )
 best sug
2.405%

 0.86
18.9%*14.73%

Portfolio variance
 P2  ( w1 1 ) 2  ( w2 2 ) 2  2 w1w2 cov  r1 , r2   0.233%
 P  4.83%