Optimal portfolios and
index model

Suppose your portfolio has only 1 stock, how many sources of
risk can affect your portfolio?
◦ Uncertainty at the market level
◦ Uncertainty at the firm level
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
Market risk
◦ Systematic or Nondiversifiable
Firm-specific risk
◦ Diversifiable or nonsystematic
If your portfolio is not diversified, the total risk of portfolio
will have both market risk and specific risk
If it is diversified, the total risk has only market risk



Why the std (total risk) decreases when more stocks are added to
the portfolio?
The std of a portfolio depends on both standard deviation of each
stock in the portfolio and the correlation between them
Example: return distribution of stock and bond, and a portfolio
consists of 60% stock and 40% bond
state
Recession
Normal
Boom
Prob.
0.3
0.4
0.3
stock (%)
-11
13
27
Bond (%)
16
6
-4
Portfolio

What is the E(rs) and σs?

What is the E(rb) and σb?

What is the E(rp) and σp?

Bond
Stock
Portfolio
E(r)
6
10
8.4
σ
7.75
14.92
5.92




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When combining the stocks into the portfolio, you get the average
return but the std is less than the average of the std of the 2 stocks in the
portfolio
Why?
The risk of a portfolio also depends on the correlation between 2 stocks
How to measure the correlation between the 2 stocks
Covariance and correlation
n
Cov(rs , rb )   pi rs (i )  E (rs ) rb (i )  E (rb ) 
i 1
Corr (rs , rb ) 
Cov(rs , rb )
 s b




Prob
0.3
0.4
0.3
rs
-11
13
27
E(rs)
10
10
10
rb
16
6
-4
E(rb)
6
6
6
P(rs- E(rs))(rb- E(rb))
-63
0
-51
Cov (rs, rb) = -114
The covariance tells the direction of the relationship between the 2 assets,
but it does not tell the whether the relationship is weak or strong
Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99


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 way returns two
assets vary
rp

rP
 Portfolio Return
wr
D
D
 wE r E
wD  Bond Weight
rD
 Bond Return
wE  Equity Weight
rE
 Equity Return
E (rp )  wD E (rD )  wE E (rE )
rp  wB rB  ws rs
E (rp )  wB E (rB )  ws E (rs )
  w   w   2w w   
  Portfolio Variance
  Portfolio Standard Deviation
2
2
2
2
2
p
B
B
S
S
2
p
2
p
B
S
S
B
B,S

Another way to express variance of the
portfolio:
 P2  wD wDCov(rD , rD )  wE wE Cov(rE , rE )  2wD wE Cov(rD , rE )
Cov(rD,rE) = DEDE
D,E = Correlation coefficient of
returns
D = Standard deviation of
returns for Security D
E = Standard deviation of
returns for Security E
Range of values for 1,2
+ 1.0 >
 > -1.0
If  = 1.0, the securities would be perfectly
positively correlated
If  = - 1.0, the securities would be
perfectly negatively correlated
E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )
2p = w1212 + w2222 + w3232
+ 2w1w2
Cov(r1,r2)
+ 2w1w3 Cov(r1,r3)
+ 2w2w3 Cov(r2,r3)
 E2  CovrD , rE 
wmin ( D)  2
 D   E2  2Cov(rD , rE )
wmin ( E )  1  wmin ( D)
example
 D , E  0.30
wmin ( D)  0.82
wmin ( E )  1  0.82  0.18
 p ,min  11.45%
E ( R p )  8 .9 %


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




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


Maximize the slope of the CAL for any
possible portfolio, p
The objective function is the slope:
SP 
E (rP )  rf
P
The solution of the optimal portfolio is as follows
E ( RD ) E2  E RE CovRD , RE 
wD 
E RD  E2  E RE  D2  E RD   E RE CovRD , RE 
wE  1  wD
E RD   E rD  r f
E RE   E rE  r f
(8  5)400  (13  5)72
 0.40
(8  5)400  (13  5)44  (8  5  13  5)72
wE  1  wD  0.60
wD 
E (rp )  (0.4  8)  (0.6  13)  11%
 p  0.4 144   0.6  400   2  0.4  0.6  72   14.2%
2
2
1
2
27
An investor with risk-aversion coefficient A
= 4 would take a position in a portfolio P
y
E rp   rf
A p2
.11  .05

 0.7439
2
4  .142
The investor will invest 74.39% of
wealth in portfolio P, 25.61% in Tbill. Portfolio P consists of 40% in
bonds and 60% in stock, therefore,
the percentage of wealth in stock
=0.7349*0.6=44.63%, in bond =
0.7349*0.4=29.76%

Security Selection
◦ First step is to determine the riskreturn opportunities available
◦ All portfolios that lie on the
minimum-variance frontier from the
global minimum-variance portfolio
and upward provide the best riskreturn combinations

We now search for the CAL with the
highest reward-to-variability ratio

Now the individual chooses the
appropriate mix between the optimal
risky portfolio P and T-bills as in
Figure 7.8
n
n
 
2
P
i 1
 w w Cov(r , r )
j 1
i
j
i
j

The separation property tells us that
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 T-bills versus the risky portfolio
depends on personal preference


Remember:
n
 
2
P
i 1
n
 w w Cov(r , r )
j 1
i
j
i
j
If we define the average variance and average
covariance of the securities as:
1 n 2
   i
n i 1
2
n
1
Cov 

n(n  1) j 1
j i

n
 Cov(r , r )
i 1
i
j
We can then express portfolio variance as:
1 2
n 1
   
Cov
n
n
2
P


The efficient frontier was introduced by Markowitz (1952) and
later earned him a Nobel prize in 1990.
However, the approach involved too many inputs, calculations
◦ If a portfolio includes only 2 stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?
◦ If a portfolio includes only 3 stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?
◦ If a portfolio includes only n stocks, to calculate the variance of the
portfolio, how many variance and covariance you need?
 n variances
 n(n-1)/2 covariances
Ri   i   i Rm  ei
Ri  ri  rf
Rm  rm  rf
Ri : risk premium of stock i
Rm : risk premium of market
 i : intercept
 i : responsive ness of stock i to the market
 i Rm : component of return due to uncertaint y at the market level
ei : component of return due to uncertaint y at the firm level

Risk and covariance:
◦ Total risk = Systematic risk + Firm-specific risk:
 i2  i2 M2   2 (ei )
◦ Covariance = product of betas x market index risk:
Cov(ri , rj )  i  j M2
◦ Correlation = product of correlations with the
market index
i  j M2 i M2  j M2
Corr (ri , rj ) 

 Corr (ri , rM ) xCorr (rj , rM )
 i j
 i M  j M

Portfolio’s variance:
      (eP )
2
P

2
P
2
M
2
Variance of the equally weighted portfolio of
firm-specific components:
2
1 2
1 2
2
 (eP )      (ei )   (e)
n
i 1  n 
n

When n gets large,  2 (eP ) becomes negligible
 i2   i2 m2   ei2
 i2 : Total risk
 i2 m2 : systematic risk component
 ei2 : specific risk
When we diversify, all the specific risk will go away, the only
risk left is systematic risk component
 p2  12 m2  ..........  n2 m2
Now, all we need is to estimate beta1, beta2, ...., beta n,
and the variance of the market. No need to calculate n
variance, n(n-1)/2 covariances as before
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Run a linear regression according to the index model, the
slope is the beta
For simplicity, we assume beta is the measure for market risk
Beta = 0
Beta = 1
Beta > 1
Beta < 1
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Reduces the number of inputs for
diversification
Easier for security analysts to specialize