Managing Portfolios of Products and Securities
S INS
MASSACHUSEr
OF TECHNOLOGY
Martin Quinteros
E
OCT 14 2008
Submitted to the Sloan School of Management
in partial fulfillment of the requirements for the degree of
LIBRARIES
Master of Science in Operations Research
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2008
@ Massachusetts Institute of Technology 2008. All rights reserved.
A
Author
................................................
Sloan chool of Management
August 14, 2008
Certified by ................
. .. . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gabriel Bitran
Professor, Sloan School of Management
Thesis Supervisor
Accepted by .................
Dimitris Bertsimas
Codirector, Operations Research Center
ARCHIVES
Managing Portfolios of Products and Securities
by
Martin Quinteros
Submitted to the Sloan School of Management
on August 14, 2008, in partial fulfillment of the
requirements for the degree of
Master of Science in Operations Research
Abstract
In this thesis we study modifications of the classical Mean-Variance Portfolio Optimization
model. Our objective is to identify an optimal subset of assets from all available assets
to maximize the expected return while incurring the minimum risk. In addition, we test
several approaches to measuring the effect of the variance of the portfolio on the optimal
asset allocation. We have developed a mixed integer formulation to solve the well known
Markowitz portfolio model.
Our model captures and solves the certain practical drawbacks that a real investor would
face with the Markowitz approach. For example, by selecting a limited number of assets
our procedure tends to prevent small allocations of assets. In addition, we find that in most
cases, the maximum drawdown increases as a function of the upper bound on the variance
of the portfolio and that this result is consistent with intuition, since portfolio risk increases
as the chance that a drawdown event occurs also increases. However, we have observed that
altering the composition of the portfolio can mitigate the risk of a drawdown event.
Thesis Supervisor: Gabriel Bitran
Title: Professor, Sloan School of Management
Acknowledgments
I would like to thank my family, Marta, Domingo, Pablo, Jorge, Ernesto, and my grandmother Porota for their deep love and support throughout my whole life. Although I spent
two years far away from you, I was always thinking of you.
I would like extend my deepest gratitude to Gabriel Bitran for being much more than an
adviser. He is a superb mentor and a close friend. Gabriel was a constant source of support,
especially through difficult times at MIT. In addition, Gabriel opened my eyes to new fields
beyond Operations Management. Working on my thesis project, I became strongly interested
in financial theory and applications and added a new facet to my interests and skills.
I would like to acknowledge Arnoldo Hax for his sincere guidance and support throughout
my tenure at MIT. Among many things, Arnoldo showed me the value of the management
and strategy aspect of business through many conversations and by attending his valuable
seminars. It is with his influence that I will now start working in the consulting industry.
I would like to acknowledge Andres Weintraub for providing me with the quantitative
skills to perform my research, and to instill in me a passion for Operations Research. It
is with his support, encouragement, and guidance through good times and bad that I have
come to MIT to pursue graduate study.
I would like to acknowledge the faculty at the MIT Sloan School of Management, in
particular professors Stephen Graves, David Gamarnik and Vivek Farias.
I would like to thank Alex Rikun who became my best friend in the United States,
and a great roommate. Thanks a lot for your partnership in many situations and our long
philosophical conversations at night. I will remember forever our time together and I hope
to see you in Chile very soon.
I would like to acknowledge Jonathan Kluberg for being a close friend, an excellent
roommate, and a partner in many motorcycle trips. It was great working with you through
hard problem sets. Upon moving back to Chile, I am sure to find my way to a few more
Shabbat dinners.
I would like to thank Karen Kotkow for being such a great friend at MIT. You were
always there for support and encouragement. In addition, this thesis would be impossible
to write without all of your help with the English language.
Thanks to the ORC and Sloan OM students for the rich and friendly environment. In
particular I would like to thank Thibault, Dan, Garreth, Shioulin, and Tor. It was fun to
work next to you and share many situations with you.
A special thanks to the golden ORC staff team: Paulette, Laura, and Andrew. You were
always willing to help me with any problems I had and speed up any administrative issues.
Finally, I would like to thank the Sloan Staff: Michelle Cole, Laura Taranto, Domingo
Altarejos, Conor Murphy, and Anna Piccolo. You were always nice, friendly, and helpful.
Contents
1 Introduction
1.1
1.2
1.3
Classical Mean-Variance Markowitz Portfolio Optimization Problem (MVO)
1.1.1
Version 1 ....................
1.1.2
Version 2
........
.......
1.1.3
Version 3
........
...
... .
13
... .
14
... .
14
Comments on the Parameters of MVO . . . . . . . .
. . . .
14
1.2.1
Arithmetic Rate of Return: fi .........
. . . .
15
1.2.2
Geometric Rate of Return pi . .........
. . . .
15
1.2.3
Covariances and the Power of Diversification .
. . . .
15
Contribution of This Thesis . . . . . . . . . . . . . .
. . . .
17
.....
.........
2 The Efficient Frontier and the Stability of Portfolios
19
2.1
1st Database: DB-1 ................
2.2
Overall Efficient Frontier period 1998-2007 . ...........
19
2.3
Efficient Frontier periods 1998-2000, 2001-2003 and 2004-2007 .
21
2.4
The Rolling Efficient Frontier Period 1998-2007
2.5
Selecting a Target Return
.........
19
. ........
..
.....................
25
3 Portfolio Optimization Models
3.1
2nd Database: DB-2 ...............................
3.2
Going Beyond the Classical MVO Model ...................
3.3
Selecting an ideal subset of k assets ...................
23
.
....
3.4 Minimum Holding per Asset ............................
3.5
The 7 M odels .. .. . . . .. ....
3.6
Building Different Scenarios ...........................
3.7
29
.. .. . . .. . .. ...
. . . . . .. .
..
29
3.6.1
Model 1 . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . ..
30
3.6.2
Model 2 .....................
31
3.6.3
Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ..
32
3.6.4
Model 4 .....................
...
33
3.6.5
Model 5 ...............................
..
34
3.6.6
Model 6 ...............................
..
35
3.6.7
Model 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
...........
...
...........
Analysis of Results for the Different Models ..................
.
4 Value of the Portfolio
4.1
4.2
29
Definitions ...........
36
37
39
.........................
.
39
4.1.1
Return of the Optimal Portfolio in Time 7 . ..............
39
4.1.2
Value of Optimal Portfolio in Time t . .................
39
Tables of V(t) for the 7 Models .........................
40
4.2.1
Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2.2
Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.2.3
Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.2.4
Model 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2.5
Model 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.6
Model 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
4.2.7
Model 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5 Maximum Drawdown Analysis
57
5.1 Definitions .......................................
5.1.1
Drawdown in Time ............................
5.1.2
Maximum Drawdown in Time ...........
8
.
57
57
. . .
....
. .
57
5.2
Computation of MDDs for the 7 Models
5.2.1
Results and Comments for Models
.........
. ..................
..
..
. . . . .
58
58
6 Conclusions
61
A Asset Allocation for Model 1
63
B Asset Allocation for Model 2
67
C Asset Allocation for Model 3
71
D Asset Allocation for Model 4
75
E Asset Allocation for Model 5
79
F Asset Allocation for Model 6
83
G Asset Allocation for Model 7
87
Chapter 1
Introduction
In the context of Financial Engineering one of the most interesting topics is Portfolio Optimization. The Mean-Variance Portfolio Optimization problem tries to solve the problem of
how to optimally allocate wealth among a given set of assets. In simple words, an investor
has n assets available to invest in and, knowing their expected returns and risks measured
in terms of volatility, he seeks to build the portfolio of assets that will allow him to get
the maximum expected return with the minimum risk. With these data, it is interesting
to characterize the points that satisfy the previous condition, also known as the Efficient
Prontier.
The notion of the Efficient Frontier was introduced by Harry Markowitz in 1952 (see
[2]), and it basically describes a curve in the return-volatility space such that every point
on the curve represents a particular portfolio that achieves a maximum return for a given
level of risk or, alternatively, a minimum risk for a given level of return. Clearly a rational
investor will select a portfolio on the efficient frontier. However, an important aspect of this
optimization problem is that its parameters may change over time. For example, a particular
asset can have a annual return of 10% in a particular year and decrease to 5% or 3% in the
following year, which implies that stability in the returns is a key factor to consider.
Mean-Variance-Efficient Frontier
I
2.5
I
I
I
2
-
-
C
a:
g 1.5
0
1
-
0.5
-
i
0
1
i
2
i
3
i
4
i
5
i
6
i
7
8
Risk (Standard Deviation)
1.1
Classical Mean-Variance Markowitz Portfolio Optimization Problem (MVO)
The theory of Mean-Variance Optimization (MVO) provides a mechanism for the selection
of portfolios of securities, or portfolios of assets, in such a way that balances expected returns
with risk. Let us consider assets s 1 , S2 ,... ,sn (n > 2) with random returns. Let ri and oi
denote the expected return (monthly or annual) and the standard deviation of the return of
asset si. For i 4 j, aj denotes the covariance between assets si and sj. Additionally, let
r = (ri, r2 , ... , rn)T and E = (oij) represent the expected return and the n x n symmetric
covariance matrix with aij = a i , respectively. So, if we define wi to represent the proportion
of money invested in security si, one can express the expected return and variance of the
resulting portfolio w = (wl,, ... ,wn) as follows:
E(rp) = riWi +.-.- + rnw = rTw
and
Zi Zj uljwiwj = wTw
This is also expressed as pij =
, which corresponds
aiaj '
to the correlations between assets si
and sj, therefore Pii = 1. It is important to note that since the variance is always nonnegative,
it follows that wtEw > 0 Vw, that is, E is positive semidefinite. One of the constraints that
we consider is:
n
Wi = 1,
i=1
which tells us that the investor is investing all his money. We say that a feasible portfolio
w is efficient if it has the maximum expected return among all portfolios with the same
variance, or, alternatively, if it has the minimum variance among all portfolios that have at
least a certain expected return. When we assume that E is positive definite, the variance
is a strictly convex function of the portfolio variables and there exists a unique portfolio in
the set of admissible portfolios that has the minimum variance. The MVO problem can be
formulated in three different but equivalent ways, as we detail in the following sections.
1.1.1
Version 1
The MVO problem can be solved by finding the minimum variance portfolio of the securities
1 to n that achieves at least a target value expected return b:
min
w2
st.
w TECW
rT
>R
Aw = b
Cw > d
The first constraint indicates that the expected return is no less than the target return
value R. Therefore, solving this optimization problem for different values of R in the range
Rmin
and R,,max yields all the efficient portfolios. The objective function corresponds to half
of the total variance of the portfolio, with the constant 1 added for convenience.
1.1.2
Version 2
Another formulation of the MVO problem involves finding the maximum expected return
for the portfolio of securities 1 to n that yields at most a given upper bound for the total
variance of the portfolio a 2:
max rT
2
WTW < a
st.
Aw = b
Cw > d
1.1.3
Version 3
Version 3 is similar to Version 2 but in this case the objective function is a risk-adjusted
return function where the constant A serves as a risk aversion constant.
ma rTW - AWT
w
2
st.
Aw = b
Cw > d
1.2
Comments on the Parameters of MVO
Assuming that we know the historical returns for assets sl, S2,... , sn in the time period
t = 1,2,..., T, named ri, there are usually three parameters to be calculated:
1.2.1
Arithmetic Rate of Return: ri
The arithmetic rate of return corresponds to the simple average of returns over time. It does
not take into account the effect of compounding on the rate of returns. It is computed in
the following way:
1
T
t
t=1
1.2.2
Geometric Rate of Return pi
The geometric rate of return is also called the effective rate of return. It takes into consideration the effect of compounding, so it is a more realistic model. The basic equation is the
following:
T
(1+ r)
(1+pi)T =
t=1
Therefore the effective return can be expressed as
1
=+
Ali
1.2.3
Covariances and the Power of Diversification
The covariance between two assets measures how similar the returns of the assets are during
a given period of time. The mathematical expression is as follows:
T
ij = Cov(ri, rj) =
(
t=1
As mentioned earlier, the variance of a portfolio in which n assets are allocated according
to each weight, w is
i=1 j=1
i=1
i=1 j=,j
> i
Therefore, it is interesting to analyze what determines the size of up. For this purpose,
let us consider a naive diversification strategy in which an equally weighted portfolio is
for each security. In this case, the previous equation can
constructed, that is, setting wi =
be rewritten as
i=1 j=l,j
i=1
i
or equivalently,
2
+ nn x n(n - 1Oj)
n n
nxx-1
X n=
We notice that there are n variance terms and n(n - 1) covariance terms in the previous
equation. If we define the average variance and average covariance of the assets as
n
Coy-
i=1
E
n(n- 1)
ij
i=1 j=1
we can write the variance of the portfolio as
2
12
n
n- 1
n
From the last equation we can analyze the effect of diversification. When the average covariance among security returns is zero, at it is when all risk is firm-specific, portfolio variance
can approach zero. The second term on the right-hand side of the previous equation will
be zero in this scenario, while the first term approaches zero as n becomes larger. Hence
when security returns are uncorrelated, the power of diversification to reduce portfolio risk
is unlimited. (see [4] chapter 7)
However, the most important situation is the one in which economy-wide risk factors impart
positive correlation among stock returns. In this case, as the portfolio becomes more highly
diversified portfolio variance remains positive. Although firm-specific risk, represented by
the first term in the equation, is still diversified away, the second term approaches to Cov
as n becomes greater. Thus the irreducible risk of a diversified portfolio depends on the
covariances of the returns of the component securities, which in turn is a function of the
importance of systematic factors in the economy.
1.3
Contribution of This Thesis
The main disadvantage of the classical Markowitz Portfolio Optimization Model is that the
optimal allocation of individual assets within the portfolio results in positions that can be
quite small, thus limiting its practical use. The approach that was employed here allows
one to select optimal subsets of assets, each with its corresponding allocation. We analyze
different upper bounds for the number of assets in the subset. For example, if there are 50
assets that are available, and a given investor wants to hold only 10 assets in his portfolio
our approach allows him to define the 10 assets with their corresponding allocations that
would yield the optimal return. Our approach is based solely on quantitative analysis, and
therefore, does not take into consideration other aspects of financial theory which may involve
behavioral analytic methods, or other considerations. An interesting extension of this work
might be to consider these factors as well.
Chapter 2
The Efficient Frontier and the
Stability of Portfolios
2.1
1st Database: DB-1
The first database that we have used, DB-1, consists of the historical monthly returns of 835
assets from the period January 1998 to December 2007. Hence, these data can be viewed as
a 835 by 120 matrix.
In order to be able to analyze the data in a more detailed way, we determined the efficient
frontier corresponding to the entire time period 1998-2007, and for three distinct intervals
from 1998-2000, 2001-2003 and 2004-2007. In addition we determined how the efficient
frontier changes over the entire time period (the Rolling Efficient Frontier).
2.2
Overall Efficient Frontier period 1998-2007
With the data in DB-1 we have used MATLAB to compute the covariance matrix and expected returns of the 835 assets during this period of time. These parameters were then used
to solve the actual optimization models corresponding to Version 1 of the MVO discussed in
Mean-Variance-Efficient Frontier
E
r-
aM
w
W
Risk (Standard Deviation)
Figure 2-1: Overall Efficient Frontier 1998-2007
Chapter 1.
The specific formulation of the quadratic optimization model was the following:
n
min,
2
n
ZW? + S S
i=1 j
i=1
Wj ij
i
Wi = 1
st.
i=1
n
Swiri =
p
i=1
i ER
From this model we have obtained the overall efficient frontier of optimal portfolios
represented in Figure 2-1.
It is important to remember that each point (ap, rp) on the curve represents a particular
portfolio for which the given target monthly return rp achieves a minimum volatility ap.
In the following table we have detailed 20 portfolios on the efficient frontier displaying the
monthly returns and the corresponding volatilities of each portfolio.
2.3
Portfolio
rp(%)
ap(%)
1
0.617
0.028
2
0.788
0.109
3
0.961
0.203
4
1.132
0.297
5
1.303
0.395
6
1.475
0.508
7
1.647
0.629
8
1.818
0.772
9
1.991
0.945
10
2.162
0.145
11
2.333
1.371
12
2.505
1.644
13
2.677
1.951
14
2.848
2.279
15
3.021
2.631
16
3.192
3.011
17
3.364
3.493
18
3.535
4.073
19
3.707
4.716
20
3.878
5.405
Efficient Frontier periods 1998-2000, 2001-2003 and
2004-2007
For these three periods of time, all of which cover 36 months, we have obtained the efficient
frontiers represented in Figures 2-2, 2-3 and 2-4:
Mean-Variance-Efficient Frontier
Risk (Standard Deviation)
Figure 2-2: Efficient Frontier 1998-2000
Mean-Variance-Efficient Frontier
Risk (Standard Deviation)
Figure 2-3: Efficient Frontier 2001-2003
Mean-Variance-Efficient Frontier
ra
1.5
g
0
!
'
•
·
•
··
·
0.5
1
·
·I
·
2
2.5
3
1.5
Risk (Standard Deviation)
·
3.5
·
4
4.5
Figure 2-4: Efficient Frontier 2004-2007
We see that the efficient frontiers of each period differ significantly from each other.
For example, if we want to achieve a portfolio monthly return of 2% we face fairly different
volatilities during each period: 0.9% in 1998-2000, 1.6% in 2001-2003 and 3.1% in 2004-2007.
Some examples of these differences are illustrated in the following table:
Target Return a 1998-2 000
2.4
a2001- 2003
o02004-2007
1%
0.2
0.5
0.8
2%
0.9
1.6
3.1
3%
2.5
3.2
6.1
The Rolling Efficient Frontier Period 1998-2007
Because the efficient frontier shifts over time, a portfolio that was once on the efficient frontier
may be not be on the efficient frontier in subsequent time periods. In addition, it is not clear
which portfolio on the efficient frontier to select. One solution is to study the time evolution
of efficient frontiers and to identify a sequence of portfolios that remain relatively stable
from one efficient frontier to the next. By plotting the frontier over time we can attempt to
Rolng EffcientFroniers
14
12
10
6
4
-----
·
2
-06
StdDeof RMns
Figure 2-5: Rolling Efficient Frontier 1998-2007
identify this region of stability. Figure 2 shows these efficient frontiers plotted as a function
of time. We have calculated the efficient frontiers of 40 portfolios during a fixed interval of
time that varies by 1 month intervals. To arrive at these data we have used the covariance
matrix and expected returns to calculate the efficient frontier, EF-1, corresponding to a
defined time period of 50 months from month 1 (Jan 1998) to month 50 (Feb 2002). In the
same fashion, analyzing from month 2 (Feb 1998) to month 51 (Mar 2002) we get EF-2, and
so on. The efficient frontier corresponding to the final interval, EF-61, is calculated from
data corresponding to month 71 (Oct 2003) and month 120 (Dec 2007).
2.5
Selecting a Target Return
The efficient frontier changes over time. This is particularly evident from the previous table
where we can see the variations in volatility for a given return. One exercise that is interesting
to perform is to fix a target monthly return, for example 1.0%, and analyze how the optimal
portfolio changes along the 61 efficient frontiers. Analysis of a subset of these data illustrates
that the optimal asset allocation changes significantly from one efficient frontier to another,
as shown in Figure 2-6. For example, Asset ID 10831 is allocated at 6.95% on EF-27, at
9.52% on EF-28, at 9.93% on EF-29, at 7.88% on EF-30, and then does not appear on the
efficient frontiers corresponding to EF-31 onward.
Asset ID
2
105
10103
10350
10481
10586
10636
10732
10812
10831
10849
10955
11102
11162
11240
11304
11408
11575
11715
11829
12014
12139
12273
12457
12598
12730
12887
13013
13256
13729
13929
14091
14370
14687
14973
15275
15435
15711
16063
16576
17335
17664
18017
18312
18656
18868
19315
19600
19836
20184
Eff Fro 27 Eff Fro 28 Eff Fro 29 Eff Fro 30 Eff Fro 31 Eff Fro 32 Eff Fro 33 Eff Fro 34
0
0
0
0
0
0
0
0
0.0609
0.064
0.12
0.0999
0.1023
0.0818
0.0508
0.0667
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0091
0
0
0.0657
0.0094
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0788
0.0993
0.0952
0.0695
0
0
0
0
0
0
0
0
0.0503
0.0293
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0161
0.0204
0.0339
0.0041
0
0.0264
0.0505
0.0505
0
0
0
0
0
0
0
0.1525
0
0
0
0
0
0
0.0684
0.0196
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.1423
0.1412
0.1516
0.2163
0.2083
0.167
0.1186
0.1459
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.1698
0.1561
0.1133
0.1308
0.0892
0.0859
0.0812
0.0552
0.3262
0.3745
0.384
0.3089
0.2922
0.3483
0.3085
0.2888
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.0104
0
0
0
0
0.057
0.0581
0.0404
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.014
0
0
0
0
0
0
0
0.0604
0.0631
0.0863
0.0732
0.0624
0.0858
0.0799
0.0645
0.1029
0.0934
0.0705
0.1475
0.1669
0.1055
0.0811
0.0775
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Figure 2-6: Rolling Efficient Frontier 1998-2007
Chapter 3
Portfolio Optimization Models
3.1
2nd Database: DB-2
The second database we have used, DB-2, consists of the historical returns of 39 assets over a
51 month period. More precisely, for every asset we have the monthly returns from January
2004 to March 2008. Hence, the data can be viewed as a 39 by 51 matrix. These data have
been provided by a portfolio manager who has been making his portfolio allocations on the
basis of experience.
The next graph (Figure 3.1) describes the correlation matrix for the 39 assets. We have
observed that many assets are negatively correlated, which is a desirable feature with respect
to the diversification of the portfolio. The red diagonal is equal to 1, since it represents the
correlation of an asset with itself.
3.2
Going Beyond the Classical MVO Model
In our models we seek to address inefficiencies in the classical MVO model. We address the
issue of selecting an ideal subset of assets, and holding a minimum weight of each chosen
asset. In addition we determine the effect of introducing several constraints into these models.
Figure 3-1: Graph of the correlation matrix for DB-2
3.3
Selecting an ideal subset of k assets
A portfolio manager may want to restrict the number of assets allowed in a portfolio. This
may be done to lower transaction costs, or for example, when he is attempting to construct
a portfolio tracking a benchmark using a limited set of assets. Therefore, selecting a subset
of k assets out of the n that are available would involve a complex enumeration strategy of
(n) combinations. To simplify this process we need to introduce a integer decision variable
xi which will take the value 1 if asset i is included in the subset and 0 otherwise. In terms
of the mathematical formulation, we need to add the constraint:
n
i=1
xi<_k,
and link the variable xi with the associated weight wi.
3.4
Minimum Holding per Asset
The solution of the classical MVO problem often results in a few large positions and many
small positions. In practice, due to transaction costs, small holdings are undesirable. In
order to eliminate small holdings threshold constraints of the following form can be used:
-i >- Ei I
where Ei is the minimum weight accepted for an asset i.
3.5
The 7 Models
The 7 models used in this study cover different combinations of the characteristics mentioned
above, and were suggested by the portfolio manager.
3.6
Building Different Scenarios
For each of the 7 models presented here we have built 13 different scenarios, named S-l,
S-2,..,S-13. In each scenario the value of delta (6), which is an upper bound for the monthly
standard deviation of the returns of the portfolio, is different. We derive 6 from the annual
standard deviation (6 A) of all returns of the portfolio using the following formula:
6A
Typically, the value of 6 A is 3.6%, yielding a 6 of 1.039%. Therefore, our calculations
for 6 employ values centered around 6 A = 3.6%, spanning the range from 3.2% to 4.4%. In
summary, we have solved 7 x 13, or 91, different optimization models.
The following sections describe in detail each model with its corresponding mathematical
formulation. In addition, at the end of the chapter a summary of the results is presented. A
full detail of the results, including the individual asset allocations of each model is presented
in the Appendices A, B, C, D, E, F and G.
3.6.1
Model 1
Model 1 corresponds to the original model. An investor is trying to maximize the expected
return of the portfolio, subject to an upper bound in the variance. In addition, we have
added several constraints to the model. A minimum monthly return of 1.5% for each month
is required. The value of k, the number of assets allowed in the portfolio, has been set to
15. We have linked xi to its weight in the portfolio. Finally, no more than 10% of our
wealth can be allocated into any given asset. The following model takes these constraints
into consideration:
n
max rp(b) = Z wr
i=1
n
Zwis=
st.
n
n
E EZWi,% •<62
2w
Vt
Zwiri -1.5%
i=i=1
Zxi <k
wi < 10% Vi
wi 2 -10%
Vi
x E {0, 1} Vi
3.6.2
Model 2
Model 2 has been derived from Model 1, with the exception that we are not allowing negative
weights. In order to illustrate the effects of eliminating some assets from the portfolio we
have excluded assets 9, 13, 18, 19, 22 and 32 and determined their effects on the overall
return.
max rp(6) =
iri
i=l
n
st.
= 1
i=1
n
n
+2Z2wiwji <
i=1
i=1
j
62
i
i=1
n
i=1
wi < xi Vi
wi < 10% Vi
X 9 =0, X 13 = 0, X 18 =0, X 19 = 0, X 22 =0, X 32
wi > 0 Vi
xi E {0, 1} Vi
0
3.6.3
Model 3
Model 3 has been derived from Model 2, except that all the weights of the assets are nonnegative. In addition, to determine the effect of a non-negative weight under these conditions
we have removed this constraint for asset 47. In other words, asset 47 is the only one that
can be shorted.
n
max rp(6) = Zwiri
i=1
n
st.
w =1
i=1
n
n
Zw%±ZT +
i=1
wiw,%j
<62
i=1 j54i
n
wr >_-1.5% Vt
i=1
Zxi
<k
i=1
wi <xi Vi
wi < 10% Vi
9 = 0,
13 =
0,
18
= 0, x 19 = 0,
wi > 0 Vi
47
xi E {0,1} Vi
22
= 0, x 32 = 0
3.6.4
Model 4
Model 4 has been derived from Model 3, however, there is no upper bound to the number of
assets that can be held in the portfolio. This has been done to determine how the number
of assets that are held affects the returns. Additional constraints include setting the upper
bound for the weights at 15%, setting the lower bound for the weight of asset 47 at -10%,
and in addition to the assets that were excluded in Model 3 we are also excluding asset 50
and 52.
n
Zwiri
max r (6)
i=1
n
1i
st.
i=1
n
n
j2±•ZZ
+
i=1
-Wi"
j
<•62
i=1 j~i
wir > -1.5%
Vt
i=1
wji < xi Vi
wi < 15% Vi
Z9
0, X13
0, X18
-
0, X19 = 0, 122 = 0, X32
wi > 0 Vi $ 47
w 47 > -0.1
xi E {0, 1} Vi
-
0, X50 = 0, X52 = 0
3.6.5
Model 5
Model 5 has been derived from Model 4, but the only difference is that the upper bound for
the weights is 10%.
n
max rp(6) = Zwiri
i=1
n
st.
wi
i=1
n
n
ZW2
U +-o
ii
wir
•2 62
1iaij
•
> -1.5%
Vt
i=1
Wi < xi Vi
wi < 10% Vi
X 9 = 0, X13 = 0, X18 = 0, 119 = 0, 122 = 0, X32 = 0,
wi > 0 Vi 7 47
w47 2 -0.1
xi E {0,1} Vi
X50 = 0, X52 = 0
3.6.6
Model 6
Model 6 has been derived from Model 5, however, in this model a lower bound for the
weights has been set at 1% for all the assets except asset 47, which has no constraints. This
constraint has been added so that we avoid holding assets with positions that are less than
1% of the total portfolio weight, since holding allocations of this size is not practical.
i=l
=
i
st.
i=1
n
n
L22
i=1
62
2
i=1 jji
n
wr > -1.5%
Vt
i=1
wi < xi Vi
0.01 * xi < wi V
Vi
47
wi < 10% Vi
X9
=
0, X13 = 0, X18 = 0, X19 = 0, X22
=
0, X32 = 0, X50
wi > 0 Vi f47
x E{O,1} Vi
=
0, X 52
=
0
3.6.7
Model 7
Model 7 has been derived from Model 6, except that the weight of asset 34 has been set at
23% and the weight for asset for 35 has been set at 10%, for the purposes of understanding
how fixed positions of assets affect returns.
n
max rp()=
wiri
i=l
n
st.
wi
i=1
Z
2
•2 62
i=l
i=1 j
i=1
i
wrt >_ -1.5% Vt
wi < Xi Vi
0.01 * i < wi Vi
wi•
47
10% Vi = 34
W34 = 23%
was = 10%
x 9 =- 0,
13 =
0, X 18 = 0, X 1 9 = 0, X2 2
wi 2 0 Vi
-
0,
47
xi E {0, 1} Vi
X32
0, X 50 = 0, X 52
0
3.7
Analysis of Results for the Different Models
As shown in Figure 3-2 the monthly return of the optimal portfolio increases with 6, which is
intuitive. In addition, given that our models are designed to maximize the expected returns
as an objective function, the return achieved through Model 1 is higher than that of any
other model. This is because Model 1 has the least constraints, or equivalently, it has the
largest number of feasible solutions.
Figure 3-3 shows a summary of the results of the different models. In particular, we can
see the monthly return, the number of assets used, and the maximum drawdown (MDD;
which will be explained in detail in Chapter 5). It is interesting to see that, even though
the number of assets for Models 6 and 7 is not formally constrained by the formulation of
the problem (in principle all 31 assets can be selected). Only 17 and 14 assets were selected
respectively.
From the practical point of view we observed that all the assets are selected when there
is no upper bound on the weights. In such instances, some of the assets might appear in the
optimal allocation with very small weight.
Monthly Return of Optimal Portfolio as a function of Delta
2.500
2.000
Model 1
Model 2
- Model 3
SModel 4
- Model 5
Model 6
-- Model 7
C
1.500
E
1
1.000
0.500
0.000
0.953
0.981 1.010
1.039 1.068 1.097
1.126 1.155 1.184 1.212 1.241
1.270
Delta
Figure 3-2: MDD for models 1,2,3,4,5,6 & 7
Annual StD(%)
Delta (monthly %)
Model 1
Model 2
Model 3
Model 4
Model 6
Model 6
Model 7
%Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
% Monthly Return
# Assets Used
MDD(%)
S-1
3.2
0.924
1.737
15
0.58
1.423
15
0.21
1.590
15
1.04
1.595
31
0.25
1.557
31
0.49
1.557
17
0.49
1.386
14
0.44
S-2
3.3
0.953
1.761
15
0.73
1.437
15
0.25
1.613
15
1.1
1.611
31
0.3
1.572
31
0.52
1.572
17
0.53
1.415
15
0.4
S-3
3.4
0.981
1.783
15
0.81
1.451
15
0.29
1.636
15
1.15
1.626
31
0.36
1.586
31
0.56
1.586
17
0.56
1.439
13
0.43
S-4
3.5
1.010
1.803
15
0.88
1.465
15
0.33
1.658
3.6
1.039
1.821
15
0.97
1.479
15
0.35
1.680
15 15
1.2 1.26
1.641 1.656
31
31
0.41 0.47
1.601 1.615
31
31
0.59 0.63
1.601 1.615
16
16
0.6 0.65
1.461 1.482
14
14
0.55
0.48
S-7
3.8
3.7
1.068 1.097
1.842 1.863
15
15
1.15 1.21
1.493 1.507
15
15
0.4 0.45
1.702 1.723
15
15
1.32 1.38
1.671 1.686
31
31
0.53 0.57
1.628 1.642
31
31
0.67 0.71
1.628 1.642
16
16
0.7 0.74
1.501 1.521
14
15
0.63 0.68
S-8
3.9
1.126
1.885
15
0.18
1.521
15
0.51
1.745
15
1.44
1.700
31
0.58
1.655
31
0.74
1.655
16
0.8
1.540
14
0.76
S-9
4.0
1.155
1.908
15
0.4
1.534
15
0.57
1.765
15
1.5
1.713
31
0.63
1.669
31
0.77
1.668
16
0.85
1.556
14
0.86
S-10 S-11
4.1
4.2
1.184 1.212
1.927 1.944
15
15
0.54 0.59
1.548 1.561
15
15
0.59
0.6
1.788 1.807
15
15
1.56
1.62
1.727 1.741
31
31
0.67 0.71
1.682 1.695
31
31
0.81 0.86
1.682 1.695
17
18
0.77
0.81
1.576 1.593
14
10
0., 0.98
S-12
4.3
1.241
1.961
15
0.64
1.574
15
0.62
1.826
15
1.73
1.754
31
0.75
1.708
31
0.9
1.708
16
0.85
1.610
14
1.01
Figure 3-3: Summary of results for models 1,2,3,4,5,6 & 7
S-13
4.4
1.270
1.977
15
0.68
1.587
15
0.67
1.646
15
1.83
1.767
31
0.79
1.721
31
0.95
1.721
16
0.92
1.626
14
1.05
Chapter 4
Value of the Portfolio
4.1
Definitions
4.1.1
Return of the Optimal Portfolio in Time 7
Result of the 91 different optimization models yield values for w, the weight of each asset
held in the portfolio.
We are interested in understanding how the return varies over time given the optimal
allocation of each asset during the respective time period. For a given period of time, r, the
return achieved for the optimal portfolio is expressed as
Rr(-J)
=
w
,r
Vr = 1,...T
i=l
4.1.2
Value of Optimal Portfolio in Time t
After computing the return of the optimal portfolio over time we can calculate the value of
each respective portfolio over time using the optimized allocations, or w.
In other words, an investor who allocates $1 at r = 1 will receive a compounded rate of
return during subsequent periods. Therefore, at r = T the value of his portfolio will be
t
V(t) =
I(1 + Rk)
Vt = 1,...,T
k=1
therefore, using the formula for R", we arrive at the following equation, which takes the
effect of compounding into consideration:
V(t)= 1-I
k=1
1+ EWiri
Vt = 1,...,T
i=1
For example, in Figure 4-2 corresponding to Model 1, we obtain 13 different values over
time for the portfolio under conditions of the 13 different scenarios, as shown by the 13 curves
on the graph. The x-axis indicates time, in months, from the period Jan04 to Mar08. The
y-axis indicates the value of the optimized portfolio in USD for each scenario. An investor
who invests $1 in Mar 04' would have received from $2.430 to $2.778 at the end of the period,
depending on the scenario that was employed.
4.2
Tables of V(t) for the 7 Models
In an attempt to validate this approach we have compared our results to that of a standard benchmark, the S&P500. The results demonstrate that this approach significantly
outperformed the market during these periods.
r
I
V1
V
i2
I
1053
v3
v44
..
V7 I
v
S4 I 1I S-5 I -· "~lr S-7
Snw
r
i ,z
.O I1 i0
I1V
026. I I.lR I 10482
I
8-1
1.UZO
1 i.046
1.056
I
1.062
_ . ..I
1.iU/
17
9i 1
1.086
1.068
.0..
1.0I9I
.
1.093
1.115
'
1.133
Vio
V11
1.13
1.147
1.176
V 12
I.M1u
'
1.047
1.057
1 .07
1.078
1.097
1.091
1.113
1 048
1
1.096
II
1".
1.133
1.148
1.176
1.211
1.1
. 08
"
0
10.05.
I 1078
047
1
1 050
i.U69
1.081
1 .(B1
051
1.000 I
1.100
058
1
601
1 099
|
1.004
1
1.095
1.120
S1.0971
1.122
1.143 1.142
1.156
1.156
1.184 1 1.185
1.215 1 .216
087
. .
1.086
1.110
1.135
1.152
1.181
1.212 1
1.152
1. . .
1.m09
1.085
1.109
1.237 1 239
-I
vis
V16 I
:1
2
V22
V25
V26
1.458
I•1as
I
1.609
vI :
I
--
..
.
·-
I .... ..
---.
1.777
If~I~
v37
f-
55oo
v 41
.
i
I.....
...
..
·. ·-::
-
:·--- -
-..
:--:
:'-::
I77 756 177
tnssr
V32
1.510
11740 117Z5 I1.748
--.-.
I ....
-
·--·
--.~R , ..--, ··-· ·, -----
'
1
216
'
217
1
218
1
1.'•9 ] 1.312
1
1
----
1
031
1
1.328
1.466
1.574 11.566 11.580
1_671
1.
----
1.752
.___
.....
1.156
1.188
. -.. I--
_
11 241
1.244 1.245
284 111.242
286 1288
11.318
.290 ]
.315
1A1
1.416
E 1.41
1.573 1
1.606
1137
1.137
312
219
----
315
1.331 1 335
3
8
1.338
13623
1.3662
.346
11.430
.426
1.362 I; 1.422
--6
.36
1.7
1.331
1.469
I1136
1153 1 1154 1 155
1.184
1185 1.186
· ...
1.335
1.387~
V21
**
CI~
060 11059
1.61.621070 11.069
068
I
1.090
11.084 1.1.00o8
8883 11.008683 I
08U
11.109
I
....
...
I ....
1 108 1.108 1.107
9
=.327
-- *~'·---'
=**
1137
1.136
1.152
1.183
1.214
1.264
1
I1.071
v=*
1.Uo
1.0(0
1 .070U
..
1068
1-080
.
...
9
c~·
101050
11+050
1.0
1o028
11028 1,028 1.028
v
,.V8V
1
1.48
049
i
I .0•7
r·~
e-o
<
we
a-a
1.U0i2
1806 1 786 8
I
. ..
.
-=1:
-1
1.426
1.430
. .. .
1.473 1.478 1.
1.526
1
1528
1532
1 545
1 .590
1 .658
1.552
1.599
1.667
1 .556
1 .604
1 .674
5
1.795
1774
1.5
1 .532
1 51
7
1.776
1.785
1.1 .560
1.609
1.79
.71
.781
.792
1.
1.786
1.7
1.
1.779 1.789 1795
1.81
1.4
1.835
.842
1.8
1.859
1.849
1.870
1854
1 .876
1.893
1.906
1
1.947
1.951
1.935
t
1.422
1.866
1.651
910
1.916
1.956
195
2.079t
2.043
2.017
2.030
29
2.110
2.120
.2.037
2T
2.
2.142
2.151
2.1
2T5
2.7577
9
2
2ES
6857
.0
2.118B
I•
v4s
-T.j34
2•
2
v so
.458
2.488
2.515
1 2"541 1 2.586 1 2.613 1 2.643
2.675
2.700
)
A.
I--
Value of Portfolio at each time period: Model 1
g
Cc
C
8
ta
C
.5
0
0L
Ca
E
0.
00
203
0"0
Cu
0
10
20
4
30
40
50
Months (1= Jan 04, 51= Mar 08)
Figure 4-2: Graph of V(t) for model 1
60
proxy for the market during this period of time. As illustrated in Figure 4-2, under the
conditions of every scenario we have tested, Model 1 significantly outperformed the market
index. Our results are even more impressive given that the overall market was declining
during this period of time. These data provide objective evidence that demonstrates the value
of our optimization models, and validates our general strategy of dynamic asset allocation.
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Chapter 5
Maximum Drawdown Analysis
5.1
Definitions
One of the most critical concerns of an investor is the possibility of a decrease in the value
of an asset he is holding, otherwise called drawdown. As a method of testing the robustness
of these optimization models we have considered the effects of drawdown on our returns.
5.1.1
Drawdown in Time
The drawdown is the percentage decrease in total returns from the start to the end of a
period of time. If the total equity time series is increasing over the time of an entire period,
drawdown is 0. Otherwise, it is a negative number, although often the absolute value is
considered.
DD(t) = V(t) - V(tmin)
V(t)
5.1.2
Maximum Drawdown in Time
The maximum drawdown is a proxy for downside risk that computes the largest drawdown
over all intervals of time that can be determined within a specific interval of time.
MDD (%)for Models 1,2,3,4,5,6 & 7
2
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Model 1
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delta
Figure 5-1: MDD for models 1,2,3,6 & 7
MDD = max DD(t)
t=1,...,T
5.2
5.2.1
Computation of MDDs for the 7 Models
Results and Comments for Models
As shown in Figure 5-1 the MDD increases with 6 for almost all the models. However, Model
1 exhibits a significant decline in MDD when we move from scenario 7 to 8. To understand
these results we can analyze the composition of assets and the optimal allocations of weights
of those assets in scenario 7 and 8 (see Appendix B). The primary difference between these
two scenarios is that under scenario 7 asset 44 was not selected, but asset 46 was selected,
and under scenario 8 asset 44 was selected and asset 46 was not selected. The difference in
monthly returns for these two assets are more than 2-fold; 1.81% for asset 44 and 0.81% for
asset 46 (see Appendix B for a table of effective monthly returns for all the assets).
As shown in Figure 5-1 Model 2, 3, 4, and 5 all show a consistent upward trend as we
move through each respective scenario.
Model 6 demonstrates the same phenomena as that of Model 1, albeit on a much smaller
scale. In this case, we observe a small decrease moving from scenario 9 to 10. The primary
difference between scenario 9 and 10 is that scenario 10 contains all the assets that are in
scenario 9, but in addition includes asset 7, which has an effective monthly return of 2.67%,
and asset 42, which has an effectively monthly return of 0.99%. It is important to note that
this is possible, because in Model 6 we have not constrained the number of assets that can
be held in the portfolio.
In Model 7 we observe a slight downward trend in MDD, moving from scenario 1 to 2.
From scenario 2 to 13 there is a consistent upward trend. The difference between scenario
1 and 2 is that scenario 2 contains all the assets of scenario 1 with the addition of asset 44,
which has an effective monthly return of 1.81%.
Chapter 6
Conclusions
The present results have been compared with that of a financial manager who was selecting
assets from the second database on the basis of his experience.
it is possible to outperform a sophisticated manager.
Our results show that
Most of the constraints that have
been introduced were that of the portfolio manager, reflecting what he has been trying to
accomplish on his own.
The results in Chapter 4 illustrate the trade off between variance and return of the
portfolio. They also provide some insight into the different types of constraints with regard
to this trade-off.
The contribution of Chapter 5, the maximum drawdown analysis, illustrates something
that is not intuitive, which is the possibility that the maximum drawdown can fall.
It is interesting to note that our models have significantly outperformed the S&P500
during this period when the index was used as a benchmark.
A next step in this line of research is to perform the portfolio optimization on a prospective
basis, since what we have done here is a retrospective, or posterior analysis.
Appendix A
Asset Allocation for Model 1
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annualSt[
delta
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0
1
0
1
1
0
1
1
0
1
1
0
0
0
0
0
1
1
1
0
0
0
0
1
0
0
0
0
0
0
1
0
1
0
1
0
1
0
0
Scenario 13
4.3 annualStC
1.241 delta
1.961 Retp
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xi
0
0
0.07528
0
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0.1
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0.1
0.078883
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1.977
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0.06406
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0
0.1
0
0
0
0
0
0
0.1
0
0.05063
0
-0.1
0
-0.1
0
0
Figure A-3: Asset Allocation for Model 1
Appendix B
Asset Allocation for Model 2
1$1·1·I11
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Scenario 5
annualStD
delta
Retp
xl
0
0
1
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
1
0
Scenario 6
3.6 annualStD
1.039
delta
1.479 Retp
wi
xi
0
0
0.058566
0.1
0.1
0.0060356
0
0.067402
0.051791
0.06728
0
0
0
0
0
0
0.1
0
0.1
0
0
0.057157
0
0
0
0
0.014414
0
0
0
0.1
0
0.031323
0.049307
0
0
0
0.096723
0
0
0
1
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
1
0
Scenario 7
3.7 annualStD
1.068
delta
1.493 Retp
wI
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0
0
0.044761
0.1
0.1
0.006315
0
0.069785
0.054139
0.07071
0
0
0
0
0
0
0.1
0
0.1
0
0
0.058947
0
0
0
0
0.013199
0
0
0
0.1
0
0.032386
0.049761
0
0
0
0.099997
0
0
0
1
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
1
0
Scenario 9
Scenario 8
3.8 annualStD
1.097
delta
1.507 Retp
wl
xl
0
0
0.031953
0.1
0.1
0.0065449
0
0.072241
0.05635
0.074913
0
0
0
0
0
0
0.1
0
0.1
0
0
0.061304
0
0
0
0
0.012865
0
0
0
0.1
0
0.033354
0.050477
0
0
0
0.1
0
0
0
1
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
1
0
3.9 annualStD
1.126
delta
1.521 Ret_p
wI
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0
0
0.01922
0.1
0.1
0.0067725
0
0.074684
0.05855
0.079091
0
0
0
0
0
0
0.1
0
0.1
0
0
0.063647
0
0
0
0
0.012532
0
0
0
0.1
0
0.034314
0.05119
0
0
0
0.1
0
Figure B-2: Asset Allocation for Model 2
69
0
1
0
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
0
1
0
4
1.155
1.534
wi
0
0.0045126
0
0.1
0.1
0.0067383
0
0.07751
0.061284
0.083177
0
0
0
0
0
0
0.1
0
0.1
0
0
0.066314
0
0
0
0
0.013054
0
0
0
0.1
0
0.035286
0.052123
0
0
0
0.1
0
Scenario 10
annualStD
delta
RetJp
xi
0
Scenario 11
4.1 annualStD
1.184
delta
1.548 Retp
wI
xl
0
0
ScenarIo 12
4.2 annualStD
1.212
delta
1.561 Retp
wl
xi
0
0
Scenario 13
4.3 annualStD
1.241
delta
1.574 Retp
wi
xi
0
0
4.4
1.270
1.587
wi
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
1
1
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
1
0
0
0
1
0
1
1
0
0
1
1
0
0.1
0.1
0.0075682
0
0.078882
0.062119
0.08807
0
0
0
0
0
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0.1
0
0.1
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0
0.06783
0
0
0
0
0.0075155
0
0
0
0.1
0
0.036264
0.051751
0
0
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1
1
1
0
1
1
1
0
0
0
0
0
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0
1
0
0
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0
0
0
0
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0
0
0
1
0
1
1
0
0
1
1
0
0.1
0.1
0.0087005
0
0.080114
0.061918
0.093764
0
0
0
0
0
0
0.1
0
0.1
0
0
0.067809
0
0
0
0
1.575E-07
0
0
0
0.1
0
0.037386
0.050309
0
0
6.22E-09
0.099999
0
1
1
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0
1
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0
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0.048558
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0.093135
0
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0.089625
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0.08606
0.059425
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0.068148
0
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4.247E-09
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0
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0
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0
0.041215
0.047137
0
0
1.757E-09
0.096722
0
Figure B-3: Asset Allocation for Model 2
70
Appendix C
Asset Allocation for Model
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Asset Allocation for Model 4
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