BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B...
Transcript of BASIC PORTFOLIO ANALYSISkeechung/Lecture Notes and...p B B Rp R s s s s − − + − − = − A B...
BASIC
PORTFOLIO
ANALYSIS
Fall 2000
Mean and Standard Deviation of Individual Securities
Define:
(1)ij
R jth return on stock i
(2)i
R expected return stock on i
(3)i
σ standard deviation of return stock i
(4) M number of periods
(5) N number of assets
Mij
RM
1jiR
=Σ=
2
iR
ijRE
2
M
iR
ijRM
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(
=−−
=−−
=−
=−
=−−=−
326
i
66/382i
3/2
=
==
σ
σ
MEAN AND VARIANCE OF PORTFOLIOS
Two General Rules:
1. 2
R1
R2
RE1
REj2
Rj1
RE +=+=+
2. 1
RCj1
CRE =
Two Asset Case (both risky)
Define:
iX as the proportion in security i.
(1) Return on portfolio
ijR
iX
j2R
2X
j1R
1X
pjR Σ=+=
(2) Mean return on portfolio
+=j2
R2
Xj1
R1
XEPR
+=j2
R2
XEj1
R1
XE
iR
iX
2R
2X
1R
1X Σ=+=
(3) Variance = 2
pRpRE2p
−=σ
2)
2R
2X
1R
1X()
j2R
2X
j1R
1X(E2
p
+−+=σ
2)
2R
j2R(
2X)
1R
j1R(
1XE
−+−=
−−+−+−= )2
Rj2
R)(1
Rj1
R(2
X1
X22)2
Rj2
R(22
X2)1
Rj1
R(21
XE
−−+−+−= )2
Rj2
R)(1
Rj1
R(E2X1X22)2Rj2R(E22X2)1Rj1R(E2
1X
122X
1X22
222
X21
21
X σσσ ++=
−−= )2
Rj2
R)(1
Rj1
R(E12
σ
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
jiijijσσρσ =
1ij
1 +≤≤− ρ
Calculating AB
σ
(12 - 10) (7 - 9) = -4
(10 - 10) (9 - 9) = 0
(8 - 10) (11 - 9) = -4
38
AB−=σ
38
38
AB38 ρ=−
1AB
−=ρ
CalculatingAC
σ 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 AD
σ 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
ijR
iX
j3R
3X
j2R
2X
j1R
1X
PjR Σ=++=
2. Mean return on portfolio
)j3
R3
Xj2
R2
Xj1
R1
X(EPR ++=
3R
3X
2R
2X
1R
1XpR ++=
3. Variance of return
2)
3R
3X
2R
2X
1R
1X()
j3R
3X
j2R
2X
j1R
1X(E2
P
++−++=σ
2)
3R
j3R(
3X)
2R
j2R(
2X)
1R
j1R(
1XE2
P
−+−+−=σ
Terms Variance
2)1
Rj1
R(E21
X −
2)2
Rj2
R(E22
X −
2)3
Rj3
R(E23
X −
Terms Covariance
−− )2
Rj2
R)(1
Rj1
R(E2
X1
X2
−− )3
Rj3
R)(1
Rj1
R(E3
X1
X2
−−)3
Rj3
R)(2
Rj2
R(E3
X2
X2
General Formulas:
Mean Return on Portfolio:
iR
iXpR Σ=
Variance of Return on Portfolio
ikkX
iX
N
ik1k
N
1i2i
2i
X 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 4A
X
B 8 2 )A
X1(B
X −=
Perfectly Positively Correlated
Expected Return:
BR)
AX1(
AR
AXpR −+=
)B
RA
R(A
XB
R −+=
BAAB)
AX1(
AX22
B2)
AX1(2
A2A
X2p σσρσσσ −+−+=
IF
1+=ρ
−+−+= 2
B2)
AX1(
BA)
AX1(
AX22
A2A
X2p σσσσσ
2
B)
AX1(
AAX
−+= σσ
or
B)
AX1(
AAXp σσσ −+=
)BA
(A
XBp σσσσ −+=
or
BA
BpA
Xσσ
σσ
−
−=
Substituting into expected return equation:
−−
−+=
BR
AR
BA
BpB
RpRσσ
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
(A
XBp σσσσ ++−=
)BA
(A
XB
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
)F
RA
R(XF
RA
RXF
R)X1(cR −+=+−=
where X is fraction in risky portfolio A
−++−=
FAAF)X1(X22
A2X2
F2)X1(2
c σσρσσσ
A
cX2A
2X2c σ
σσσ =⇒=
)F
RA
R(
A
cF
RcR −+=σ
σ
cA
FR
AR
FRcR σσ
−+=
(1). Separation Theorem:
Investors optimum choice of a risky portfolio is separatefrom his or her preferences.
(2). Two Fund Theorem:
An investor is not hurt by restriction to a choice oftwo funds.
(3). Unambiguous objective function.