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BASIC
PORTFOLIO
ANALYSIS
Fall 2002
Mean and Standard Deviation of Individual Securities Define: (1) ijR jth return on stock i
(2) iR expected return stock on i
(3) iσ standard deviation of return stock i
(4) M number of periods (5) N number of assets
M
ijRM
1jiR=Σ=
2
iR
ijRE
2
Mi
Rij
RM
1j2i
−=−
=Σ=σ
Note some use M-1.
Example: MONTH Return Dec 5% Nov -2% Oct 3% Sept 2% Aug -1% July -1% 6%
%16%6
6ij
R
iR ==Σ=
42)11(42)11(
12)12(42)13(
92)12(162)15(
=−−=−−
=−=−
=−−=−
326i
66/382i 3/2
=
==
σ
σ
MEAN AND VARIANCE OF PORTFOLIOS
Two General Rules:
1. 2R1R2
RE1
REj2
Rj1
RE +=+=+
2. 1RCj1
CRE =
Two Asset Case (both risky) Define:
iX as the proportion in security i.
(1) Return on portfolio
ijRiXj2
R2
Xj1
R1
XpjR Σ=+=
(2) Mean return on portfolio
+=j2
R2
Xj1
R1
XEPR
+=j2
R2
XEj1
R1
XE
iRiX2R2X1R1X Σ=+=
(3) Variance = 2
pRpRE2p
−=σ
2)
2R
2X
1R
1X()
j2R
2X
j1R
1X(E2p
+−+=σ
2)
2R
j2R(
2X)
1R
j1R(
1XE
−+−=
−−+−+−= )2Rj2R)(1Rj1R(2X1X22)2Rj2R(22X2)1Rj1R(2
1XE
−−+−+−= )2Rj2R)(1Rj1R(E2X1X22)2Rj2R(E22X2)1Rj1R(E2
1X
122X1X22
222X2
121X σσσ ++=
−−= )2
Rj2
R)(1
Rj1
R(E12σ
Note:
(1) Measures joint movement
(2) Unrestricted to sign
Example (assume equally likely) 6A Return Condition A B C Rainfall D Good 12 7 8 Heavy 8 Average 10 9 6 Average 6 Poor 8 11 4 Light 4 r 10 9 6 6 σ 8/3 8/3 8/3 8/3 Useful jiijij σσρσ =
1ij1 +≤≤− ρ
Calculating ABσ
(12 - 10) (7 - 9) = -4 (10 - 10) (9 - 9) = 0 (8 - 10) (11 - 9) = -4
38 AB −=σ
38
38
AB38 ρ=−
1AB −=ρ
Calculating ACσ 6C
(12 - 10) (8 - 6) = 4 (10 - 10) (6 - 6) = 0 (8 - 10) ( 4 - 6) = 4 8
38
AB=σ
38
38
AB38 ρ=
1AB=ρ
Calculating ADσ 6D
(12 - 10) (8 - 6) = +4 (12 - 10) (6 - 6) = 0 (12 - 10) (4 - 6) = -4 (10 - 10) (8 - 6) = 0 (10 - 10) (6 - 6) = 0 (10 - 10) (4 - 6) = 0 (8 - 10) (8 - 6) = -4 (8 - 10) (6 - 6) = 0 (8 - 10) (4 - 6) = +4
0AD
0AD
=
=
ρ
σ
Three Security Case
1. Return on portfolio ijRiXj3R3Xj2R2Xj1R1XPjR Σ=++=
2. Mean return on portfolio )j3R3Xj2R2Xj1R1X(EPR ++=
3R3X2R2X1R1XpR ++=
3. Variance of return
2)3R3X2R2X1R1X()
j3R3X
j2R2X
j1R1X(E2
P
++−++=σ
2)3R
j3R(3X)2R
j2R(2X)1R
j1R(1XE2
P
−+−+−=σ
Terms Variance
2)1Rj1R(E21X −
2)2Rj2R(E2
2X −
2)3Rj3R(E2
3X −
Terms Covariance
−− )2
Rj2
R)(1
Rj1
R(E2X1X2
−− )3
Rj3
R)(1
Rj1
R(E3X1X2
−−)3
Rj3
R)(2
Rj2
R(E3X2X2
General Formulas:
Mean Return on Portfolio: iRiXpR Σ=
Variance of Return on Portfolio
ikkXiX N
ik1k
N
1i2i 2
iX N
1i2p σσσ
≠=Σ
=Σ+
=Σ=
The Effect of Diversification
Assume random selection and equal amount in each security.
N1
iX =
ik
2
N1N
ik1k
N
1i 2
i
2
N1N
1i2P σσσ
≠=Σ
=Σ+
=Σ=
−≠=Σ
=Σ−+
=Σ=
ik1N1
N1N
ik1k
N
1iN1N
N
2i
N
1iN1 σ
σ
ikN1N2
iN1 σσ
−+=
ikN112
N1 σσ
−+=
ikik2iN
1 σσσ +−=
Efficient Set Theorem
(1). Holding PR constant minimize Pσ
(2). Holding Pσ constant maximize PR
Plotting Efficient Frontier
(two risky assets) R σ Proportion A 14 4 AX
B 8 2 )AX1(BX −=
Perfectly Positively Correlated Expected Return:
BR)AX1(ARAXpR −+=
)BRAR(AXBR −+=
BAAB)AX1(AX22B
2)AX1(2A
2AX2p σσρσσσ −+−+=
IF
1+=ρ
−+−+= 2B
2)A
X1(BA
)A
X1(A
X22A
2A
X2p σσσσσ
2
B)
AX1(
AAX
−+= σσ
or
B)AX1(AAXp σσσ −+=
)BA(AXBp σσσσ −+=
or
BA
BpAX
σσ
σσ
−
−=
Substituting into expected return equation:
−−
−+=
BR
AR
BA
BpBRpR
σσ
σσ
−
−+
−
−−=
BA
BR
AR
pBA
BR
AR
BBR
σσσ
σσσ
This is, of course, a straight line. With the example:
−−+
−−−=
24814
p2481428pR σ
p32pR σ+=
Perfect Negative Correlation
If 1−=ρ
−+−−= 2B
2)A
X1(BAA
X1A
X22A
2A
X2p σσσσσ
This can come from either
−−=B
)A
X1(AA
Xp σσσ
or
−+−=B
)A
X1(AA
Xp σσσ
and
)BA(AXBp σσσσ ++−=
)BA(AXB σσσ +−+=
BA
Bpor BA
BpAX
σσ
σσ
σσ
σσ
+
+−
+
+=
Substituting into expected return:
)BRAR(BA
BpBRpR −
+
++=
σσ
σσ
or
)BRAR(BA
BpBRpR −
+
+−+=
σσ
σσ
+
−±
+
−+=
BA
BR
AR
pBA
BR
AR
BBRpR
σσσ
σσσ
+−±
+−+=
24814
p2481428 σ
p10 σ±=
with other ρ 's not a straight line
In standard definition proceeds full usable 1X 2X pR
+2 -1 20 +3 -2 26 +4 -3 32
Efficient Frontier with Riskless Asset
)FRAR(XFRARXFR)X1(cR −+=+−=
where X is fraction in risky portfolio A
−++−=FAAF
)X1(X22A
2X2F
2)X1(2c σσρσσσ
A
cX2A
2X2c σσ
σσ =⇒=
)FRAR(A
cFRcR −+=
σσ
cA
FR
AR
FRcR σσ
−+=
(1). Separation Theorem: Investors optimum choice of a risky portfolio is separate from his or her preferences. (2). Two Fund Theorem: An investor is not hurt by restriction to a choice of two funds. (3). Unambiguous objective function.