Combining Individual Securities
Into Portfolios
(Chapter 4)





Individual Security Return and Risk
Portfolio Expected Rate of Return
Portfolio Variance
Combination Lines
Combination Line Between a
Risky Asset and a Risk-Free Asset
Individual Security Return and Risk

Expected Rate of Return
n
E(rA )   h i rA,i
i 1
where:
– E(rA) = Expected rate of return on security (A)
– rA,i = i(th) possible return on security (A)
– hi = probability of getting the i(th) return

Variance and Standard Deviation
n
σ 2 (rA )   h i [rA,i  E(rA )]2
i 1
σ(rA )  σ 2 (rA )
Portfolio Expected Rate of Return

A weighted average of the expected returns on the
portfolio’s component securities.
m
E(rp )   x jE(r j )
j1
where:
– E(rp) = Expected rate of return on portfolio (p)
– m = Number of securities in portfolio (p)
– xj = Weight of security (j)
 Note: The contribution of each security to portfolio
expected return depends on:
– 1. the security’s expected return
– 2. the security’s weight
Portfolio Variance

To compute the variance of a portfolio, you need:
– (1) the covariances of every pair of securities in
the portfolio, and
– (2) the weight of each security.

Example: (Three Security Portfolio)
Cov(r1 , r1 ) Cov(r1 , r2 ) Cov(r1 , r3 ) 


Cov(r
,
r
)
Cov(r
,
r
)
Cov(r
,
r
)
2
1
2
2
2
3 

Cov(r3 , r1 ) Cov(r3 , r2 ) Cov(r3 , r3 )
m
m
σ 2 (rp )   x j x k Cov(r j , rk )
j1 k 1

Take each of the covariances in the matrix
and multiply it by the weight of the security
identified on the row (security j) and then
again by the weight of the security identified
on the column (security k). Then, add up all of
the products.
σ 2 (rp )  x1x1Cov(r1 , r1 )  x1x 2Cov(r1 , r2 )  x1 x 3Cov(r1 , r3 )
 x 2 x1Cov(r2 , r1 )  x 2 x 2Cov(r2 , r2 )  x 2 x 3Cov(r2 , r3 )
 x 3 x1Cov(r3 , r1 )  x 3 x 2Cov(r3 , r2 )  x 3 x 3Cov(r3 , r3 )
The Covariance Between a Security
and Itself is Simply Its Own Variance
n
Cov(r , r )   h [r  E(r )][r  E(r )]
1 1
i 1, i
1 1, i
1
i 1
n
=  h [r  E(r )]2
i 1, i
1
i 1
= σ 2 (r )
1
Example : (Three Security Portfolio)
σ 2 (r1 ) Cov(r1 , r2 ) Cov(r1 , r3 ) 


2
Cov(r2 , r1 ) σ (r2 ) Cov(r2 , r3 )


2
Cov(r3 , r1 ) Cov(r3 , r2 ) σ (r3 )
m
σ 2 (rp ) 
m
 x x C ov(r , r )
j k
j k
j1 k 1
m
=

m
x 2jσ 2 (rj ) 
j1
on th ediagon al
m
 x x C ov(r , r )
j k
j k
j1 k 1
jk
off th ediagon al
σ 2 (rp )  x12σ 2 (r1 )  x1x 2C ov(r1 , r2 )  x1x 3C ov(r1 , r3 )
 x 2 x1C ov(r2 , r1 )  x 22σ 2 (r2 )  x 2 x 3C ov(r2 , r3 )
 x 3 x1C ov(r3 , r1 )  x 3x 2C ov(r3 , r2 )  x 23σ 2 (r3 )
Each Element Above the Diagonal is
Paired With an Identical Element
Below the Diagonal
[e . g ., C ov(r1 , r2 )  C ov(r2 , r1 )]
m
σ 2 (rp ) 
m
 x x C ov(r , r )
j k
j k
j1 k 1
m
=

m
x 2jσ 2 (rj ) 
j1
 x x C ov(r , r )
j k
m

j k
j1 k 1
jk
on th edi agon al
=
m
m
off th edi agon al
m
  x x C ov(r , r )
x 2jσ 2 (rj )  2
j1
on th edi agon al
j k
j k
j1 k  j1
above th edi agon al
Exam ple: (Thre eS e cu rityPortfolio)
σ 2 (r ) C ov(r , r ) C ov(r , r )
1
1 2
1 3 


σ 2 (r2 ) C ov(r2 , r3 ) 


2

σ (r3 ) 


σ 2 (rp )  x12σ 2 (r1 )  x 22σ 2 (r2 )  x 23σ 2 (r3 )
+ 2 x1x 2C ov(r1 , r2 )
+ 2 x1x 3C ov(r1 , r3 )
+ 2 x 2 x 3C ov(r2 , r3 )
Using the Correlation Coefficient
Instead of Covariance

Recall:
ρ j,k 

C ov(rj , rk )
σ(rj ) σ(rk )
As a Result:
Cov(rj , rk )  ρ j,k σ(rj ) σ(rk )

Therefore:
m
σ 2 (rp ) =

j1
m
=

j1
m
m
  x x C ov(r , r )
x 2jσ 2 (rj )  2
j k
j k
j1 k  j1
m
m
 x x ρ
x 2jσ 2 (rj )  2
j k j,k σ(rj ) σ(rk )
j1 k  j1
Exam ple: (Th re eS e cu ri tyPortfoli o)
σ 2 (r ) ρ σ(r ) σ(r ) ρ σ(r ) σ(r )
1
1,2 1
2
1,3 1
3 



2
σ
(r
)
ρ
σ(r
)
σ(r
)
2
2,3
2
3 



2
σ (r3 )


σ 2 (rp )  x12σ 2 (r1 )  x 22σ 2 (r2 )  x 23σ 2 (r3 )
+ 2 x1x 2ρ1,2σ(r1 ) σ(r2 )
+ 2 x1x 3ρ1,3σ(r1 ) σ(r3 )
+ 2 x 2 x 3ρ 2,3σ(r2 ) σ(r3 )
COMBINATION LINES

A curve that shows what happens to the risk and
expected return of a portfolio of two stocks as the
portfolio weights are varied.

Example 1 (Perfect Negative Correlation)
hi
rA,i
rB,i
.25
.25
.25
.25
2%
4%
6%
8%
20%
14%
8%
2%
n
E(rA ) 
h r
i A,i  .25(2) .25(4) .25(6) .25(8) 5%
i 1
n
E(rB ) 
h r
i B,i  .25(20) .25(14) .25(8) .25(2) 11%
i 1
n
σ 2 (rA ) 

h i [rA,i  E(rA )]2
i 1
= .25(2 5)2  .25(4 5)2  .25(6 5)2  .25(8 5)2  5
σ(rA )  5  2.236%
n
σ 2 (rB ) 

h i [rB,i  E(rB )]2
i 1
= .25(20 11)2  .25(14 11)2  .25(8 11)2  .25(2 11)2  45
σ(rB )  45  6.708%
n
C ov(rA , rB ) 
 h [r
i
A,i  E(r A )][rB,i  E(rB )]
i 1
= .25(2 5)(20 11)  .25(4 5)(14 11)
+ .25(6 5)(8 11)  .25(8 5)(2 11)
= - 15
ρ A,B 
C ov(rA , rB )
 15

 - 1.00
σ(rA ) σ(rB ) (2.236)(6.
708)
Now, for various weights, portfolio risk and return
can be calculated:
E(rp )  x A E(rA )  xBE(rB )
2 2
σ 2 (rp )  x 2Aσ 2 (rA )  xB
σ (rB )  2 x A xBρ A,Bσ(rA ) σ(rB )
σ(rp )  σ 2 (rp )


Students are encouraged to prove that the following
portfolio standard deviations and expected rates of
return are indeed correct for the weights given.
Note: With perfect negative correlation, we can create
a riskless portfolio by taking positive positions in both
stocks.
xA
xB
E(rp)
(rp)
-.3
0
.5
.75
1.0
1.5
1.3
1.0
.5
.25
0
-.5
12.8%
11.0%
8.0%
6.5%
5.0%
2.0%
9.4%
6.7%
2.2%
0%
2.2%
6.7%
COMBINATION LINE
(Perfect Negative Correlation)
Expected Rate of Return (%)
14
12
xA = .5, xB = .5
10
xA = -.3, xB = 1.3
All (B)
8
xA = .75, xB = .25
6
xA = 1.5, xB = -.5
4
All (A)
2
0
0
2
4
6
8
Standard Deviation of Returns (%)
10

Example 2 (Perfect Positive Correlation)
hi
rA,i
rB,i
.25
.25
.25
.25
2%
4%
6%
8%
2%
8%
14%
20%
E(rA) = 5% E(rB) = 11%
(rA) = 2.236% (rB) = 6.708%
Cov(rA,rB) = +15 A,B = +1.00

Note: When the stocks’ standard deviations are not
equal and the stocks are perfectly positively
correlated, we can always create a riskless portfolio
by selling one of the two stocks short.
xA
xB
E(rp)
(rp)
-.3
0
.5
1.0
1.5
1.75
1.3
1.0
.5
0
-.5
-.75
12.8%
11.0%
8.0%
5.0%
2.0%
.5%
8.0%
6.7%
4.5%
2.2%
0%
1.1%
COMBINATION LINE
(Perfect Positive Correlation)
Expected Rate of Return (%)
14
xA = -.3, xB = 1.3
12
xA = .5, xB = .5
10
All (B)
8
All (A)
6
4
xA = 1.5, xB = -.5
2
xA = 1.75, xB = -.75
0
0
2
4
6
8
Standard Deviation of Returns (%)
10

Example 3 (Zero Correlation)
hi
rA,i
rB,i
.25
.25
.25
.25
2%
4%
6%
8%
8%
20%
2%
14%
E(rA) = 5% E(rB) = 11%
(rA) = 2.236% (rB) = 6.708%
Cov(rA,rB) = 0 A,B = 0
xA
_____
-.3
0
.5
.9
1.0
1.5
xB
E(rp)
_____ _____
1.3
12.8%
1.0
11.0%
.5
8.0%
.1
5.6%
0
5.0%
-.5
2.0%
(rp)
_____
8.7%
6.7%
3.5%
2.1%
2.2%
4.7%
COMBINATION LINE
(Zero Correlation)
Expected Rate of Return (%)
14
xA = -.3, xB = 1.3
12
xA = .5, xB = .5
10
All (B)
8
xA = .9, xB = .1
6
4
All (A)
2
xA = 1.5, xB = -.5
0
0
2
4
6
8
Standard Deviation of Returns (%)
10
PATHS OF COMBINATION LINES

E(rp) is influenced by E(rj) and xj
 (rp) is influenced by (rj), j,k, and xj
 j,k determines the path between two
securities
 Moving along the path occurs by varying the
weights.
COMBINATION LINES
Expected Rate of Return (%)
14
12
 = -1.00
10
 = +1.00
8
6
Stock (B)
Stock (A)
4
=0
2
0
0
2
4
6
8
Standard Deviation of Returns (%)
10
Combination Line Between a Risky Stock
(or Portfolio) and a Risk-Free Bond


Example:
– Risky Stock (A): E(rA) = 10%, (rA) = 20%
– Risk-Free Bond (B): E(rB) = 6%, (rB) = 0%
Note on Risk:
2 2
σ 2 (rp )  x 2Aσ 2 (rA )  xB
σ (rB )  2 x A xBρ A,Bσ(rA ) σ(rB )
Since σ(rB )  0
σ 2 (rp )  x 2Aσ 2 (rA )
σ(rp )  x Aσ(rA )
Combination Line Between a Risky
Stock (or Portfolio) and a Risk-Free
Bond (Continued)
xA
xB
E(rp) = xAE(rA) + xBE(rB)
(rp) = xA(rA)
0
.5
1.0
1.5
1.0
.5
0
-.5
0(10) + 1.0(6) = 6%
.5(10) + .5(6) = 8%
1.0(10) + 0(6) = 10%
1.5(10) - .5(6) = 12%
0(20) = 0%
.5(20) = 10%
1.0(20) = 20%
1.5(20) = 30%
Combination Line When one of the
Assets is Risk-Free
Expected Rate of Return (%)
14
Lending
Borrowing
12
10
xA = 1.5, xB = -.5
8
xA = 1.0, xB = 0
xA = .5, xB = .5
6
4
xA = 0, xB = 1.0
2
0
0
10
20
Standard Deviation of Returns (%)
30
Combination Line Between a Risky Stock (or
Portfolio) and a Risk-Free Bond (Continued)

Note: When one of the two investments is
risk-free, the combination line is always a
straight line.
 Lending: When you buy a bond, you are
lending money to the issuer.
 Borrowing: Here, we assume that
investors can borrow money at the riskfree rate, and add to their investment in
the risky asset.