Combining Individual Securities Into Portfolios (Chapter 4) Individual Security Return and Risk...
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Combining Individual SecuritiesInto Portfolios
(Chapter 4)
Combining Individual SecuritiesInto 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
Individual Security Return and Risk
Expected Rate of Return
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
1iiA,iA rh)E(r
)(rσ)σ(r
)]E(r[rh)(rσ
A2
A
n
1i
2AiA,iA
2
Portfolio Expected Rate of Return
Portfolio Expected Rate of Return
A weighted average of the expected returns on the portfolio’s component securities.
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
m
1jjjp )E(rx)E(r
Portfolio VariancePortfolio 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)
m
1j
m
1kkjkjp
2
332313
322212
312111
)r,Cov(rxx)(rσ
)r,Cov(r )r,Cov(r )r,Cov(r
)r,Cov(r )r,Cov(r )r,Cov(r
)r,Cov(r )r,Cov(r )r,Cov(r
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.
)r,Cov(rxx)r,Cov(rxx)r,Cov(rxx
)r,Cov(rxx)r,Cov(rxx)r,Cov(rxx
)r,Cov(rxx)r,Cov(rxx)r,Cov(rxx)(rσ
333323231313
323222221212
313121211111p2
The Covariance Between a Security and Itself is Simply Its
Own Variance
The Covariance Between a Security and Itself is Simply Its
Own Variance
)1
(r2σ =
)]1
E(ri1,
[ri
h =
)]1
E(ri1,
[r)]1
E(ri1,
[ri
h)1
r,1
Cov(r
n
1i
2
n
1i
)(rσ )r,Cov(r )r,Cov(r
)r,Cov(r )(rσ )r,Cov(r
)r,Cov(r )r,Cov(r )(rσ
Portfolio) Security (Three :Example
32
2313
3222
12
312112
diagonal the off kj diagonal the on
)r,Cov(rxx )(rσx =
)r,Cov(rxx)(rσ
m
1j
m
1k
kjkj
m
1j
j22
j
m
1j
m
1k
kjkjp2
)(rσx)r,Cov(rxx)r,Cov(rxx
)r,Cov(rxx)(rσx)r,Cov(rxx
)r,Cov(rxx)r,Cov(rxx)(rσx)(rσ
322
323231313
3232222
21212
31312121122
1p2
Each Element Above the Diagonal is Paired With an
Identical Element Below the Diagonal
Each Element Above the Diagonal is Paired With an
Identical Element Below the Diagonal
diagonal the above diagonal the on
)r,Cov(rxx2 )(rσx =
diagonal the off kj diagonal the on
)r,Cov(rxx )(rσx =
)r,Cov(rxx)(rσ
)]r,Cov(r)r,Cov(r .,g.[e
m
1j
m
1jk
kjkj
m
1j
j22
j
m
1j
m
1k
kjkj
m
1j
j22
j
m
1j
m
1k
kjkjp2
1221
)r,Cov(rxx2+
)r,Cov(rxx2+
)r,Cov(rxx2+
)(rσx)(rσx)(rσx)(rσ
)(rσ
)r,Cov(r )(rσ
)r,Cov(r )r,Cov(r )(rσ
Portfolio) Security (Three :Example
3232
3131
2121
322
3222
2122
1p2
32
3222
312112
Using the Correlation Coefficient Instead of Covariance
Using the Correlation Coefficient Instead of Covariance
Recall:
As a Result:
Therefore:
)σ(r)σ(r
)r,Cov(rρ
kj
kjkj,
)σ(r)σ(rρ)r,Cov(r kjkj,kj
m
1j
m
1jk
kjkj,kj
m
1j
j22
j
m
1j
m
1jk
kjkj
m
1j
j22
jp2
)σ(r)σ(rρxx2 )(rσx=
)r,Cov(rxx2 )(rσx = )(rσ
)σ(r)σ(rρxx2+
)σ(r)σ(rρxx2+
)σ(r)σ(rρxx2+
)(rσx)(rσx)(rσx)(rσ
)(rσ
)σ(r)σ(rρ )(rσ
)σ(r)σ(rρ )σ(r)σ(rρ )(rσ
Portfolio) Security (Three :Example
322,332
311,331
211,221
322
3222
2122
1p2
32
322,322
311,3211,212
COMBINATION LINESCOMBINATION 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
.25 .25 .25 .25
rA,i
2% 4% 6% 8%
rB,i
20% 14% 8% 2%
11%.25(2).25(8).25(14).25(20)rh)E(r
5%.25(8).25(6).25(4).25(2)rh)E(r
n
1i
iB,iB
n
1i
iA,iA
6.708%45)σ(r
4511).25(211).25(811).25(1411).25(20=
)]E(r[rh)(rσ
2.236%5)σ(r
55).25(85).25(65).25(45).25(2=
)]E(r[rh)(rσ
B
2222
n
1i
2BiB,iB
2
A
2222
n
1i
2AiA,iA
2
Now, for various weights, portfolio risk and return can be calculated:
1.00- 708)(2.236)(6.
15
)σ(r)σ(r
)r,Cov(rρ
15- =
11)5)(2.25(811)5)(8.25(6 +
11)5)(14.25(411)5)(20.25(2 =
)]E(r[r)]E(r[rh)r,Cov(r
BA
BABA,
BiB,
n
1i
AiA,iBA
)(rσ)σ(r
)σ(r)σ(rρxx2)(rσx)(rσx)(rσ
)E(rx)E(rx)E(r
p2
p
BABA,BAB22
BA22
Ap2
BBAAp
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
-.3 0 .5
.75 1.0 1.5
xB
1.3 1.0 .5 .25 0
-.5
E(rp)
12.8% 11.0% 8.0% 6.5% 5.0% 2.0%
(rp)
9.4% 6.7% 2.2% 0%
2.2% 6.7%
COMBINATION LINE(Perfect Negative Correlation)
COMBINATION LINE(Perfect Negative Correlation)
0
2
4
6
8
10
12
14
0 2 4 6 8 10
Expected Rate of Return (%)
Standard Deviation of Returns (%)
All (B)
All (A)
xA = .75, xB = .25
xA = .5, xB = .5 xA = -.3, xB = 1.3
xA = 1.5, xB = -.5
Example 2 (Perfect Positive Correlation)
E(rA) = 5% E(rB) = 11%
(rA) = 2.236% (rB) = 6.708%
Cov(rA,rB) = +15 A,B = +1.00
hi
.25
.25
.25
.25
rA,i
2% 4% 6% 8%
rB,i
2% 8%
14% 20%
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
-.3 0 .5 1.0 1.5
1.75
xB
1.3 1.0 .5 0
-.5 -.75
E(rp)
12.8% 11.0% 8.0% 5.0% 2.0% .5%
(rp)
8.0% 6.7% 4.5% 2.2% 0%
1.1%
COMBINATION LINE(Perfect Positive Correlation)
COMBINATION LINE(Perfect Positive Correlation)
0
2
4
6
8
10
12
14
0 2 4 6 8 10
Expected Rate of Return (%)
Standard Deviation of Returns (%)
All (B)
All (A)
xA = -.3, xB = 1.3
xA = .5, xB = .5
xA = 1.5, xB = -.5
xA = 1.75, xB = -.75
Example 3 (Zero Correlation)
hi
.25
.25
.25
.25
rA,i
2% 4% 6% 8%
rB,i
8% 20% 2%
14%
E(rE(rAA) = 5% E(r) = 5% E(rBB) = 11% ) = 11%
(r(rAA) = 2.236% ) = 2.236% (r(rBB) = 6.708% ) = 6.708%
Cov(rCov(rAA,r,rBB) = 0 ) = 0 A,BA,B = 0 = 0
xA
_____ -.3 0 .5 .9
1.0 1.5
xB
_____ 1.3 1.0 .5 .1 0
-.5
E(rp) _____ 12.8% 11.0% 8.0% 5.6% 5.0% 2.0%
(rp) _____ 8.7% 6.7% 3.5% 2.1% 2.2% 4.7%
COMBINATION LINE(Zero Correlation)
COMBINATION LINE(Zero Correlation)
0
2
4
6
8
10
12
14
0 2 4 6 8 10
Expected Rate of Return (%)
Standard Deviation of Returns (%)
All (B)
All (A)
xA = .5, xB = .5
xA = .9, xB = .1
xA = 1.5, xB = -.5
xA = -.3, xB = 1.3
PATHS OF COMBINATION LINESPATHS 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 LINESCOMBINATION LINES
0
2
4
6
8
10
12
14
0 2 4 6 8 10
Stock (B)
Stock (A)
= -1.00
= +1.00
= 0
Expected Rate of Return (%)
Standard Deviation of Returns (%)
Combination Line Between a Risky Stock (or Portfolio) and a Risk-
Free Bond
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:
)σ(rx)σ(r
)(rσx)(rσ
0)σ(r Since
)σ(r)σ(rρxx2)(rσx)(rσx)(rσ
AAp
A22
Ap2
B
BABA,BAB22
BA22
Ap2
Combination Line Between a Risky Stock (or Portfolio) and a
Risk-Free Bond (Continued)
Combination Line Between a Risky Stock (or Portfolio) and a
Risk-Free Bond (Continued)
xA
0 .5 1.0 1.5
xB
1.0 .5 0
-.5
E(rp) = xAE(rA) + xBE(rB)
0(10) + 1.0(6) = 6% .5(10) + .5(6) = 8%
1.0(10) + 0(6) = 10% 1.5(10) - .5(6) = 12%
(rp) = xA(rA)
0(20) = 0%
.5(20) = 10% 1.0(20) = 20% 1.5(20) = 30%
Combination Line When one of the Assets is Risk-Free
Combination Line When one of the Assets is Risk-Free
0
2
4
6
8
10
12
14
0 10 20 30
Expected Rate of Return (%)
Standard Deviation of Returns (%)
xA = 1.5, xB = -.5
xA = 1.0, xB = 0xA = .5, xB = .5
xA = 0, xB = 1.0
LendingBorrowing
Combination Line Between a Risky Stock (or Portfolio) and a Risk-Free
Bond (Continued)
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 risk-free rate, and add to their investment in the risky asset.