PART 4
M EAN -VARIANCE P ORTFOLIO T HEORY
93
Primary objectives:
Ï
Markowitz model: Computing minimum variance (risk)
portfolios
Ï
Random Variables, Variance, co-variance, mean
Ï
Quadratic programming: Modelling portfolio optimization as
quadratic program
Ï
Simple parameter estimation
Ï
Modelling: Transaction costs, diversifiaction
Ï
Robustness issues: Naive approaches
94
Further reading
INVESTMENT SCIENCE, by David G. Luenberger, Oxford University
Press
Today chapter 6
95
Assets
Asset is investment instrument that can be bought and sold
frequently
Return
Ï
Ï
Ï
X0 : amount invested, X1 : amount received
X1
Total return: R =
X0
X1 − X0
Rate of return: r =
X0
96
Short sales
Sell asset without owning it.
Ï
Borrow asset from someone who owns it (such as a brokerage
firm)
Ï
sell borrowed asset to someone else receiving X0
Ï
Repay your loan by purchasing asset for X1
Ï
Return asset to lender
Ï
Short selling is profitable if the asset price declines
97
Portfolios
Portfolio return
Ï
n assets are available with return Ri and X0 is to be invested
Ï
weight wi : fraction of asset i in portfolio
Pn
i=1 wi = 1
Ï
Ï
Return of Portfolio:
R =
=
Ï
Using formula R = 1 + r and
for the rate of return
Pn
i=1 wi X0 Ri
n
X
X0
wi R i
i=1
Pn
i=1 wi
= 1, one has r =
Pn
i=1 wi ri
98
Basic notions of probability
Expected value, variance
Ï
x a random variable over a finite probability space, expected
P
value E(x) or x of x is i xi pi , where pi is the probability of x to
attain value xi
Ï
Linearity of expectation: x, y random variables, α, β ∈ R, then
E(α x + β y) = αE(x) + βE(y)
Ï
Ï
Variance of x: var(x) = E[(x − x)2 ] = E(x2 ) − x̄2
p
Standard deviation: σ(x) = var(x)
Example: Rolling a dice
σ2 (x)
= E(x2 ) − 3.52
= 1/6(1 + 4 + 9 + 16 + 25 + 36) − 3.52
= 2.92
99
Covariance
Ï
Ï
Ï
Ï
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Covariance: cov(x, y) = E[(x − x)(y − y)] = E(x · y) − x · y
Correlation: ρ(x, y) = cov(x, y)/(σ(x) · σ(y)); notice |ρ(x, y)| É 1
Uncorrelated: ρ(x, y) = 0
Positively correlated: ρ(x, y) > 0
Negatively correlated: ρ(x, y) < 0
Example of samples of positive, negative and uncorrelated
variables
y
x
100
Variance of sum
var(
n
X
i=1
xi )
= E[
n
¡X
i=1
¢2
xi − xi ]
X
= E[ (xi − xi )(xj − xj )]
i,j
n
X
=
i,j=1
=
n
X
E(xi − xi )(xj − xj ))
cov(xi , xj ).
i,j=1
101
Random returns
Ï
Rates of return are considered to be random variables
Ï
n assets with random rates of return ri i = 1, . . . , n and expected
value r i respectively
Ï
Expected rate of return of portfolio
r=
Ï
n
X
r i wi
i=1
Variance of portfolio return
var(r) =
n
X
i,j=1
wi wj var(ri , rj ) = wT Qw
where Q is symmetric positive semi-definite matrix
102
The optimization problem
Task
Compute weights such that variance (risk) of portfolio is minimal
among those having at least a certain expected return.
103
Quadratic programming
Convex quadratic program
Q ∈ Rn×n positive semidefinite, c ∈ Rn A ∈ Rm×n and b ∈ Rm , then
min xT Qx + cT x
Ax = b
x Ê 0,
is (convex) quadratic program.
Efficient algorithms
Convex quadratic programs can be solved efficiently. We will treat
an approximation algorithm in greater detail later.
104
Mean-variance portfolio optimization as a quadratic program
Pn
i=1 wi r i
P
n
i=1 wi
w
Aw
Ê
=
Ê
É
min wT Qw
r
1
0 if no shortsales allowed
b additional constraints
105
Parameter estimation
Observed rates of return
Ï
Ï
Ï
Ï
Ï
Ï
Time series: r1 , . . . , rT
P
Mean value: r = T1 Tt=1 rt
P
Variance: var(r) = T1 Tt=1 (rt − r)2
Second time series: s1 , . . . , sT , mean: s, variance var(s)
P
Covariance: cov(r, s) = T1 Tt=1 (rt − r)(st − s)
Correlation: ρ(r, s) = cov(r, s)/(σ(r)σ(s))
106
Example
Ï
S&P 500 index for returns on stocks, 10 year treasury bond
index for returns on bonds, cash invested in money market a 1
day federal fund rate
Year
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
Stocks
20.2553
25.6860
23.4297
28.7463
33.4484
37.5813
33.7839
41.8725
46.4795
42.5448
44.2212
50.5451
60.1461
51.3114
37.7306
51.7772
64.1659
59.5739
63.4884
75.3032
99.7795
94.8671
Bonds
262.935
268.730
284.090
289.162
299.894
302.695
318.197
309.103
316.051
298.249
354.671
394.532
403.942
417.252
433.927
457.885
529.141
531.144
524.435
531.040
517.860
538.769
MM
100.00
102.33
105.33
108.89
113.08
117.97
124.34
129.94
137.77
150.12
157.48
164.00
172.74
189.93
206.13
216.85
226.93
241.82
266.07
302.74
359.96
404.48
Year
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Stocks
115.308
141.316
150.181
197.829
234.755
247.080
288.116
379.409
367.636
479.633
516.178
568.202
575.705
792.042
973.897
1298.82
1670.01
2021.40
1837.36
1618.98
1261.18
1622.94
Bonds
777.332
787.357
907.712
1200.63
1469.45
1424.91
1522.40
1804.63
1944.25
2320.64
2490.97
2816.40
2610.12
3287.27
3291.58
3687.33
4220.24
3903.32
4575.33
4827.26
5558.40
5588.19
MM
440.68
482.42
522.84
566.08
605.20
646.17
702.77
762.16
817.87
854.10
879.04
905.06
954.39
1007.84
1061.15
1119.51
1171.91
1234.02
1313.00
1336.89
1353.47
1366.73
107
Annual rates of return
Year
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
Stocks
26.81
-8.78
22.69
16.36
12.36
-10.10
23.94
11.00
-8.47
3.94
14.30
18.99
-14.69
-26.47
37.23
23.93
-7.16
6.57
18.61
32.50
-4.92
21.55
Bonds
2.20
5.72
1.79
3.71
0.93
5.12
-2.86
2.25
-5.63
18.92
11.24
2.39
3.29
4.00
5.52
15.56
0.38
-1.26
-1.26
-2.48
4.04
44.28
MM
2.33
2.93
3.38
3.85
4.32
5.40
4.51
6.02
8.97
4.90
4.14
5.33
9.95
8.53
5.20
4.65
6.56
10.03
13.78
18.90
12.37
8.95
Year
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
Stocks
22.56
6.27
31.17
18.67
5.25
16.61
31.69
-3.10
30.46
7.62
10.08
1.32
37.58
22.96
33.36
28.58
21.04
-9.10
-11.89
22.10
28.68
Bonds
1.29
15.29
32.27
22.39
-3.03
6.84
18.54
7.74
19.36
7.34
13.06
-7.32
25.94
0.13
12.02
14.45
-7.51
17.22
5.51
15.15
0.54
MM
9.47
8.38
8.27
6.91
6.77
8.76
8.45
7.31
4.43
2.92
2.96
5.45
5.60
5.29
5.50
4.68
5.30
6.40
1.82
1.24
0.98
108
Rates of return
rit = (Ii,t − Ii,t−1)/Ii,t−1 rates of return of asset i = 1, , 2, 3
Arithmetic mean
r i = 1/T ·
PT
t=1 rit
gives
Arithmetic mean r i
Stocks
12.06 %
Bonds
7.85%
MM
6.32 %
109
Geometric mean
Suppose I invest X0 at time 0. Rates of return are r1 = .25 and
r2 = −.25 on the first and second day. What is X2 ?
X2 = (1 − .25)(1 + .25)X0 = (1 − .0625)X0
Arithmetic mean of return is 0.
Geometric mean: (1 + r)2 = (1 − .25)(1 + .25) and thus
p
r = (1 − .25)(1 + .25) − 1.
Geometric mean
qQ
r i = T Ti=1 (1 + rit ) − 1.
Is constant yearly rate of return to be applied to obtain IiT starting
from Ii0 .
110
Example with geometric mean
Geometric mean r i
Stocks
10.73%
Bonds
7.37%
MM
6.27%
Covariance matrix
cov(Ri , Rj ) =
1
T
PT
t=1 (rit
− r it )(rjt − r jt ) :
Covariance
Stocks
Bonds
MM
Stocks
0.02778
0.00387
0.00021
Bonds
0.00387
0.01112
-0.00020
MM
0.00021
-0.00020
0.00115
111
Example cont.
Volatility (standard deviation)
σ(Ri ) =
p
cov(Ri , Ri )
Volatility
16.67 %
10.55 %
3.40 %
112
Setting up the quadratic program
min .02778 · wS2 + 2 · .00387wS wB + 2 · .00021wS wM
2
+.01112wB2 − 2 · .00020wB wM + 0.00115wM
such that
.1073wS + .0737wB + .0627wM Ê R
wS + wB + wM = 1
ws Ê 0, wB Ê 0, wM Ê 0
113
Diversification
Markowitz portfolio tends to have unreasonably large quantities of
individual assets and if short-sales are allowed, unusually large
quantities of short sales.
Example: Limit certain assets to m% means additional constraints
wi É m, for assets i.
Limiting the overall amount of short sales:
Split each wi into wi = wi+ − wi −
n
X
i=1
wi− É m′ ,
These go in additional constraint section of Markowitz model
114
Transaction costs
Re-optimizing a portfolio
w0 current portfolio and w the desired portfolio. Let yi be the
fraction of asset i sold and zi be the fraction of asset i bought.
wi − wi0 É yi , yi Ê 0
wi0 − wi É zi , zi Ê 0
Following constraint limits fraction of transaction to h:
n
X
(yi + zi ) É h
i=1
115
Robustness
Markowitz model is sensitive
Small change of mean and variance in input results often in large
changes of portfolio.
One way to deal with this is to re-sample returns from historical
data to generate alternative, but similar r and var estimates. Run
optimizer on the generated inputs and take mean.
116
Primary objectives:
Ï
Markowitz model: Computing minimum variance (risk)
portfolios
Ï
Random Variables, Variance, co-variance, mean
Ï
Quadratic programming: Modelling portfolio optimization as
quadratic program
Ï
Simple parameter estimation
Ï
Modelling: Transaction costs, diversifiaction
Ï
Robustness issues: Naive approaches
117
Primary objectives:
Ï
Markowitz model: Computing minimum variance (risk)
portfolios ✔
Ï
Random Variables, Variance, co-variance, mean ✔
Ï
Quadratic programming: Modelling portfolio optimization as
quadratic program ✔
Ï
Simple parameter estimation ✔
Ï
Modelling: Transaction costs, diversifiaction ✔
Ï
Robustness issues: Naive approaches ✔
118