Fuzzy Portfolio Optimization Using Carlsson-Fullér-Majlender’s Trapezoidal Possibility

Model

YuxiangOu

Abstract Within the framework of Carlsson-Fullér-Majlender’s Trapezoidal Possibility Model, we

apply Lagrange Multiplier Method and Karush-Kuhn-Tucker Conditions to derive the

optimal solution to fuzzy portfolio selection problem.

Keywords: Porfolio selection; Trapezoidal fuzzy variables; Lagrange method; KKT

conditions

Introduction The portfolio selection problem concerns how to form an optimal portfolio, that is, how to

decide on the weights of every asset to generate the highest level of investor’s utility. Modern

portfolio analysis was pioneered by [7] Markowitz in 1952. As is known to us, people are

risk-averse return-seekers, which means we all want to maximize the return and minimize the

risk of the investment. There is a balance we need to figure out. The most essential point

Markowitz made in the article is that we can use the expected rate of return (mean) to model

the return and the variance of the rate to represent the risk. Markowitz then derived the

optimal choice with the belief that the investors have complete information, which, in most

instances, is not true. To account for uncertainty, it is better to utilize fuzzy set theory by [9]

Zadeh (1965) and [10] Bellman and Zadeh (1970). Many studies on the portfolio selection

problem using various fuzzy formulations emerge. In terms of membership function of the

fuzzy variable, researchers have proposed linear function, tangent type function, interval

linear function, exponential function, inverse tangent function, etc. to address the problem.

2

This paper is typically based on [1] Carlsson et al (2002), which presumes that the

membership function of fuzzy return is in a trapezoidal form. The model is basically the same

and some expressions might be identical. The novelty of this paper is that we reorganize

the argument and provide some proofs that are omitted in the original paper. Also, we use

real stock market data to give a numerical illustration instead of artificially assigning some

values to the model. That is why we get a different result from the original paper.

Furthermore, we develop some critical thinking of the Carlsson-Fullér-Majlender’s model

and state our concerns.

The rest of the paper goes as follows. In Section 2, we mention some preliminaries on the

related issues. In Section 3, we describe the optimization problem in different ways, apply

Lagrange method to deal with it and employ Karush-Kuhn-Tucker conditions to confirm the

minimizer. We list the generalized algorithm in Section 4 and apply it to realistic situations in

Section 5. The paper will cover some personal suggestions in Section 6 and conclude with

Section 7.

Preliminaries

2.1 Utility theory of portfolio investment

A utility function is viewed as a means of ranking portfolios. Higher utility values are

assigned to portfolios with more attractive risk-return profiles. Based on this rule, we can

design a function as follows:

𝑈 𝑃 = 𝐸 𝑟& − 0.005×𝐴×𝜎.(𝑟&)

where A is an index of the investor’s risk aversion (𝐴 ≈ 2.46 for an average investor in the

USA), 𝑟& is the rate of return on the portfolio and 𝐸 𝑟& and 𝜎.(𝑟&) represent its mean value

and variance, respectively. The scaling factor of 0.005 allows us to express the expected

return and variance as percentages rather than decimals.

Note that the sign of 𝐸 𝑟& is positive while that of 𝜎.(𝑟&) is negative, this utility function is

consistent with reality. Moreover, this utility function prevents us from dealing with

complicated multiobjective optimization problems.

3

2.2 Probability or possibility approach

Investors make decisions on portfolio selection according to their knowledge and anticipation

of capital market, budget constraints and available options. Due to limited or incomplete

information one can gather from the market, there exists uncertainty among the decision-

making process that we need to address.

Probability theory is the standard approach to this issue, with the belief that uncertainty is

equated with randomness. Nevertheless, this is not exactly true. Subjective judgement makes

a huge difference in decision-making but it seems difficult to incorporate it into the

probability theory. The assignment of the probabilities would also be problematic when we

demand a higher precision and more decimal places.

Alternatively, in this paper we will assume that the rates of return on assets are modeled by

possibility distributions. That is, the rate of return on the 𝑖th asset will be represented by a

fuzzy number 𝑟8, and 𝑟8 𝑡 , 𝑡𝜖ℛ will be interpreted as the degree of possibility of the

statement that “𝑡 will be the rate of return on the 𝑖th asset”, which is also named as

membership function. In our method, we will consider only trapezoidal possibility

distributions.

2.3 Trapezoidal fuzzy variable

2.3.1 Membership function

The definition of trapezoidal fuzzy variable is based on the membership function.

Definition. A fuzzy number A is called trapezoidal with tolerance interval [𝑎, 𝑏], left width 𝛼

and right width 𝛽 if its membership function has the following form:

𝐴 𝑡 =

1 −𝑎 − 𝑡𝛼 𝑖𝑓𝑎 − 𝛼 ≤ 𝑡 ≤ 𝑎,

1𝑖𝑓𝑎 ≤ 𝑡 ≤ 𝑏,

1 − 𝑡 − 𝑏𝛽 𝑖𝑓𝑎 ≤ 𝑡 ≤ 𝑏 + 𝛽,

0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

and we denote A by 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽).

This membership function can be visualized as

4

2.3.2 𝜸-level set

A 𝛾-level set of a fuzzy variable is composed of all the possibilities with the grade of

membership higher than 𝛾. Then we can modify Fig. 1 to get a closer look at the issue.

Proposition 1. Let 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽) be a trapezoidal fuzzy variable and [𝐴]O = [𝑎P 𝛾 , 𝑎.(𝛾)]

be the corresponding 𝛾-level set, then [𝐴]O = 𝑎P 𝛾 , 𝑎. 𝛾 = 𝑎 − 1 − 𝛾 𝛼, 𝑏 +

1 − 𝛾 𝛽 ,∀𝛾𝜖[0,1].

Proof. It can be easily to check that this proposition holds for 𝛾𝜖{0,1}. Let’s focus on

situations where 0 < γ < 1. From Fig. 2, we observe that 𝛾-level line intersects A’s

membership function at two points, i.e. 𝑎P 𝛾 and 𝑎. 𝛾 . Therefore, we need to derive these

two points here.

For 𝑎P 𝛾 , let 1 − VWXY= 𝛾. We can get 𝑎P 𝛾 = 𝑡 = 𝑎 − (1 − 𝛾)𝛼;

For 𝑎. 𝛾 , let 1 − XWZ[= 𝛾. We can get 𝑎. 𝛾 = 𝑡 = 𝑏 + (1 − 𝛾)𝛽.

Thus, [𝐴]O = 𝑎P 𝛾 , 𝑎. 𝛾 = 𝑎 − 1 − 𝛾 𝛼, 𝑏 + 1 − 𝛾 𝛽 . ¢

5

2.3.3 Possibilistic mean

The crisp possibilistic mean value of fuzzy variable A with [𝐴]O = [𝑎P 𝛾 , 𝑎.(𝛾)] is defined

as

𝐸 𝐴 = 𝛾 𝑎P 𝛾 + 𝑎.(𝛾) 𝑑𝛾P

] (1)

Proposition 2. Let A be a trapezoidal fuzzy variable denoted as 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽), then.

𝐸 𝐴 =𝑎 + 𝑏2 +

𝛽 − 𝛼6 (2)

Proof. According to the definition, we can calculate possibilistic mean of trapezoidal fuzzy

variable as follows:

𝐸 𝐴 = 𝛾 𝑎P 𝛾 + 𝑎. 𝛾 𝑑𝛾P

]

= 𝛾 𝑎 − 1 − 𝛾 𝛼 + 𝑏 + 1 − 𝛾 𝛽 𝑑𝛾P

]

= 𝛾 𝛼 − 𝛽 𝛾 + 𝑎 + 𝑏 + 𝛽 − 𝛼 𝑑𝛾P

]

= 𝛼 − 𝛽 𝛾.𝑑𝛾P

]+ 𝑎 + 𝑏 + 𝛽 − 𝛼 𝛾𝑑𝛾

P

]

=𝛼 − 𝛽3 +

𝑎 + 𝑏 + 𝛽 − 𝛼2

=𝑎 + 𝑏2 +

𝛽 − 𝛼2 −

𝛽 − 𝛼3

=𝑎 + 𝑏2 +

𝛽 − 𝛼6

¢

2.3.4 Possibilistic variance

The crisp possibilistic mean value of fuzzy variable A with [𝐴]O = [𝑎P 𝛾 , 𝑎.(𝛾)] is defined

as

𝜎. 𝐴 =12 𝛾 𝑎. 𝛾 − 𝑎P(𝛾) .𝑑𝛾

P

] (3)

Proposition 3. Let A be a trapezoidal fuzzy variable denoted as 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽), then.

6

𝜎. 𝐴 = (𝑏 − 𝑎2 +

𝛼 + 𝛽6 ). +

𝛼 + 𝛽 .

72 (4)

Proof. Similarly, we calculate the possibilistic variance:

𝜎. 𝐴 =12 𝛾 𝑎. 𝛾 − 𝑎P 𝛾

.𝑑𝛾P

]

=12 𝛾 𝑏 + 1 − 𝛾 𝛽 − 𝑎 + 1 − 𝛾 𝛼 .𝑑𝛾

P

]

=12 𝛾 𝑏 − 𝑎 + 𝛼 + 𝛽 − 𝛼 + 𝛽 𝛾 .𝑑𝛾

P

]

=12 𝛾 𝑏 − 𝑎 + 𝛼 + 𝛽 . − 2 𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽 𝛾 + 𝛼 + 𝛽 .𝛾. 𝑑𝛾

P

]

=12 𝑏 − 𝑎 + 𝛼 + 𝛽 .𝛾𝑑𝛾

P

]− 𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽 𝛾.𝑑𝛾

P

]

+12 𝛼 + 𝛽 .𝛾`𝑑𝛾

P

]

=𝑏 − 𝑎 + 𝛼 + 𝛽 .

4 −𝑏 − 𝑎 + 𝛼 + 𝛽 𝛼 + 𝛽

3 +𝛼 + 𝛽 .

8

=𝑏 − 𝑎 .

4 +𝑏 − 𝑎 𝛼 + 𝛽

2 +𝛼 + 𝛽 .

4 −𝑏 − 𝑎 𝛼 + 𝛽

3 −𝛼 + 𝛽 .

3 +𝛼 + 𝛽 .

8

=𝑏 − 𝑎 .

4 +𝑏 − 𝑎 𝛼 + 𝛽

6 +𝛼 + 𝛽 .

24

= (𝑏 − 𝑎2 +

𝛼 + 𝛽6 ). +

𝛼 + 𝛽 .

72

¢

Analysis of portfolio selection problem

3.1 Basic formulation of the optimization problem

Recalling that the utility function of portfolio investment in our model, which is introduced in

Section 2.1, is 𝑈 𝑃 = 𝐸 𝑟& − 0.005×𝐴×𝜎.(𝑟&).

Assume that

n: the number of available securities;

𝑥8: the proportion invested in security (or asset) 𝑖, 𝑖 = 1,2, … , 𝑛;

7

𝑟8: the rate of return on security 𝑖;

𝑟&: the rate of return on the portfolio.

Then we know that 𝑟& = 𝑟8𝑥8e8fP and 𝑥8e

8fP = 1. As we do not consider short-selling and

long-buying, we also have 0 ≤ 𝑥8 ≤ 1.

Accordingly, our portfolio selection problem is equivalent to the following mathematical

programming problem:

maxjk

𝑈 𝑃 = 𝐸 𝑟8𝑥8e

8fP− 0.005×𝐴×𝜎. 𝑟8𝑥8

e

8fP

s. t.{ 𝑥8e

8fP= 1, 𝑥8 ≥ 0, 𝑖 = 1,2, … , 𝑛}

(5)

Where 𝑟8 = 𝑎8, 𝑏8, 𝛼8, 𝛽8 , 𝑖 = 1,2, … , 𝑛 are fuzzy variables of trapezoidal form.

3.2 Translations of the optimization problem

Note that in Section 2.3.3 and 2.3.4 we have derived that the possibilistic mean and variance

of a trapezoidal fuzzy variable 𝐴 = (𝑎, 𝑏, 𝛼, 𝛽) are 𝐸 𝐴 = VnZ.+ [WY

o and 𝜎. 𝐴 =

(ZWV.+ Yn[

o). + Yn[ p

q., respectively.

Then for trapezoidal fuzzy number 𝑟8 = 𝑎8, 𝑏8, 𝛼8, 𝛽8 , 𝑖 = 1,2, … , 𝑛, we have

𝐸 𝑟8 = VknZk.

+ [kWYko

= P.[𝑎8 + 𝑏8 +

P`(𝛽8 − 𝛼8)], thus,

𝐸 𝑟8𝑥8e8fP = 𝑥8𝐸(𝑟8)e

8fP = P.[𝑎8 + 𝑏8 +

P`(𝛽8 − 𝛼8)]𝑥8e

8fP (6)

And since

𝜎. 𝑟8 = (ZkWVk.

+ Ykn[ko). + Ykn[k p

q.= (P

.𝑏8 − 𝑎8 +

P`𝛼8 + 𝛽8 ). + Ykn[k p

q., when we

ignore the covariance between rate of returns on different securities, we have

𝜎.( 𝑟8𝑥8e8fP ) = ( P

.𝑏8 − 𝑎8 +

P`𝛼8 + 𝛽8e

8fP 𝑥8). +Pq.[ 𝛼8 + 𝛽8 𝑥8e

8fP ]. (7)

If we introduce the notations as:

𝑢8 =P.[𝑎8 + 𝑏8 +

P`(𝛽8 − 𝛼8)],

𝑣8 =].]]tu.

𝑏8 − 𝑎8 +P`𝛼8 + 𝛽8 ,

𝑤8 =].]]tuq.

(𝛼8 + 𝛽8), then

𝐸 𝑟8𝑥8e8fP = P

.[𝑎8 + 𝑏8 +

P`(𝛽8 − 𝛼8)]𝑥8e

8fP = 𝑢8𝑥8e8fP ,

8

𝜎. 𝑟8𝑥8e

8fP=

12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8

e

8fP𝑥8

.

+172 𝛼8 + 𝛽8 𝑥8

e

8fP

.

=1

0.005𝐴𝑣8

e

8fP𝑥8

.

+172

720.005𝐴

𝑤8𝑥8e

8fP

.

=1

0.005𝐴 𝑣8e

8fP𝑥8

.+

10.005𝐴 𝑤8

e

8fP𝑥8

.

thus,

𝑈 𝑃 = 𝐸 𝑟8𝑥8e

8fP− 0.005×𝐴×𝜎. 𝑟8𝑥8

e

8fP

= 𝑢8𝑥8e

8fP− 0.005𝐴×

10.005𝐴 𝑣8

e

8fP𝑥8

.+

10.005𝐴 𝑤8

e

8fP𝑥8

.

= 𝑢8𝑥8e

8fP− 𝑣8

e

8fP𝑥8

.− 𝑤8

e

8fP𝑥8

.

and the optimization problem becomes

maxjk

𝑈 𝑃 = 𝑢8𝑥8e

8fP− 𝑣8

e

8fP𝑥8

.− 𝑤8

e

8fP𝑥8

.

s. t.{ 𝑥8e

8fP= 1, 𝑥8 ≥ 0, 𝑖 = 1,2, … , 𝑛}

(8)

Here the 𝑖th asset is represented by a triplet (𝑣8, 𝑤8, 𝑢8), where 𝑢8 denotes its possibilistic

expected value, and 𝑣8. + 𝑤8. denotes its possibilistic variance multiplied by the constant

0.005×𝐴.

The convex hull of { 𝑣8, 𝑤8, 𝑢8 : 𝑖 = 1,2, … , 𝑛}, denoted by 𝑇, and defined by

𝑇 = 𝑐𝑜𝑛𝑣 𝑣8, 𝑤8, 𝑢8 : 𝑖 = 1,2, … , 𝑛

= { 𝑣8e

8fP𝑥8, 𝑤8

e

8fP𝑥8, 𝑢8𝑥8

e

8fP: 𝑥8

e

8fP= 1, 𝑥8 ≥ 0, 𝑖 = 1,2, … , 𝑛}

is a convex polyhedron in ℛ`. We can move to any point in the polytope by varying the value

of 𝑥8. In other words, let 𝑣] = 𝑣8e8fP 𝑥8, 𝑤] = 𝑤8e

8fP 𝑥8, and 𝑢] = 𝑢8e8fP 𝑥8, we need to

find the point within the polytope generating the highest value of 𝑢] − 𝑣]. − 𝑤].. Then

problem (8) turns into the following three-dimensional non-linear programming problem:

maxyz,{z,|z

𝑈 𝑃 = 𝑢] − 𝑣]. − 𝑤].

s. t. 𝑣], 𝑤], 𝑢] 𝜖𝑇 (9)

Or, equivalently,

minyz,{z,|z

𝑈 𝑃 = 𝑣]. + 𝑤]. − 𝑢]

s. t. 𝑣], 𝑤], 𝑢] 𝜖𝑇 (10)

9

Note that 𝑇 is a compact and convex subset of ℛ`, and the implicit function

𝑔� 𝑣], 𝑤] = 𝑣]. + 𝑤]. − 𝑐

is strictly convex for any 𝑐𝜖ℛ. This means that any optimal solution to (10) must be on the

boundary of 𝑇. As 𝑇 is a polyhedron of ℛ` and the optimal solution must be on the boundary

of 𝑇, then any optimal solution can be obtained as a convex combination of at most 3 extreme

points of 𝑇. [1] Carlsson, Fullér and Majlender (2002) presented an algorithm for finding

such an optimal solution. In the algorithm, one should calculate: (i) the (exact) solutions to all

conceivable 3-asset problems with non-collinear assets, (ii) the (exact) solutions to all

conceivable 2-asset problems with distinguishable assets, and (iii) the utility value of each

asset. Then one can compare the utility values of all feasible solutions and portfolios with the

highest utility value will be chosen as optimal solutions to the portfolio selection problem.

3.3 Optimal solutions

3.3.1 3-asset problems

Consider three noncollinear assets 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3,

Proposition 4. For any noncollinear assets 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3, ∄ 𝛼P, 𝛼. 𝜖ℛ., 𝛼P, 𝛼. ≠ 0,

such that

𝛼P𝑣P𝑤P𝑢P

+ 𝛼.𝑣.𝑤.𝑢.

− 𝛼P + 𝛼.𝑣`𝑤`𝑢`

= 0.

Proof. Suppose there exists 𝛼P, 𝛼. 𝜖ℛ., 𝛼P, 𝛼. ≠ 0, such that

𝛼P𝑣P𝑤P𝑢P

+ 𝛼.𝑣.𝑤.𝑢.

− 𝛼P + 𝛼.𝑣`𝑤`𝑢`

= 0, then

𝛼P𝑣P − 𝑣`𝑤P − 𝑤`𝑢P − 𝑢`

+ 𝛼.𝑣. − 𝑣`𝑤. − 𝑤`𝑢. − 𝑢`

= 0, that is, we have

𝑣P − 𝑣`𝑤P − 𝑤`𝑢P − 𝑢`

= −YpY�

𝑣. − 𝑣`𝑤. − 𝑤`𝑢. − 𝑢`

if 𝛼P ≠ 0;

or 𝑣. − 𝑣`𝑤. − 𝑤`𝑢. − 𝑢`

= −Y�Yp

𝑣P − 𝑣`𝑤P − 𝑤`𝑢P − 𝑢`

if 𝛼. ≠ 0.

We find collinearity in both cases, which contradicts our noncollinear assumptions.

¢

10

Then the 3-asset optimal portfolio selection problem with not-necessarily non-negative

weights is

minj�,jp,j�

𝑈 𝑃 = (𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥`). + (𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥`).

− (𝑢P𝑥P + 𝑢.𝑥. + 𝑢`𝑥`)

s. t.𝑥P + 𝑥. + 𝑥` = 1

(11)

Let

𝐿 𝑥, 𝜆 = 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` . + 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` .

− 𝑢P𝑥P + 𝑢.𝑥. + 𝑢`𝑥` + 𝜆(𝑥P + 𝑥. + 𝑥` − 1) (12)

be the Lagrange function of the constrained optimization problem (11). Then the Kuhn-

Tucker necessity conditions are

2𝑣P 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤P 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢P + 𝜆 = 02𝑣. 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤. 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢. + 𝜆 = 02𝑣` 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤` 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢` + 𝜆 = 0

𝑥P + 𝑥. + 𝑥` − 1 = 0

(13)

Proposition 5. The Karush-Kuhn-Tucker necessity conditions listed as (13) can be

transformed into the following linear equality system:

𝑞P. + 𝑟P. 𝑞P𝑞. + 𝑟P𝑟.𝑞P𝑞. + 𝑟P𝑟. 𝑞.. + 𝑟..

𝑥P𝑥. =

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

𝑤ℎ𝑒𝑟𝑒𝑞P = 𝑣P − 𝑣`, 𝑞. = 𝑣. − 𝑣`, 𝑟P = 𝑤P − 𝑤`𝑎𝑛𝑑𝑟. = 𝑤. − 𝑤`

(14)

Proof. From the third equation in (13), we have

𝜆 = 𝑢` − 2𝑣` 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` − 2𝑤` 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` ,

from the fourth equation, we have 𝑥` = 1 − 𝑥P − 𝑥..

Substituting 𝜆 and 𝑥` with these two expressions,

the first equation in (13) becomes

2𝑣P 𝑣P𝑥P + 𝑣.𝑥. + 𝑣` 1 − 𝑥P − 𝑥. + 2𝑤P 𝑤P𝑥P + 𝑤.𝑥. + 𝑤` 1 − 𝑥P − 𝑥. − 𝑢P + 𝑢`− 2𝑣` 𝑣P𝑥P + 𝑣.𝑥. + 𝑣` 1 − 𝑥P − 𝑥.− 2𝑤` 𝑤P𝑥P + 𝑤.𝑥. + 𝑤` 1 − 𝑥P − 𝑥. = 0

that is,

2 𝑣P − 𝑣` (𝑣P − 𝑣`)𝑥P + (𝑣.−𝑣`)𝑥. + 𝑣` + 2(𝑤P− 𝑤`) (𝑤P − 𝑤`)𝑥P + (𝑤.−𝑤`)𝑥. + 𝑤` − 𝑢P + 𝑢` = 0

recalling that 𝑞P = 𝑣P − 𝑣`, 𝑞. = 𝑣. − 𝑣`, 𝑟P = 𝑤P − 𝑤`𝑎𝑛𝑑𝑟. = 𝑤. − 𝑤`, we have

2𝑞P(𝑞P𝑥P + 𝑞.𝑥. + 𝑣`) + 2𝑟P(𝑟P𝑥P + 𝑟.𝑥. + 𝑤`) − 𝑢P + 𝑢` = 0

11

i.e. 2[ 𝑞P. + 𝑟P. 𝑥P + 𝑞P𝑞. + 𝑟P𝑟. 𝑥. + (𝑞P𝑣` + 𝑟P𝑤`)] − 𝑢P + 𝑢` = 0

simplifying and rearranging,

𝑞P. + 𝑟P. 𝑥P + 𝑞P𝑞. + 𝑟P𝑟. 𝑥. =12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`

Similarly, for the second equation, we substitute 𝜆 and 𝑥` and then simplify and rearrange it

to

𝑞P𝑞. + 𝑟P𝑟. 𝑥P + 𝑞.. + 𝑟.. 𝑥. =12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

therefore, the equation system (13) is equivalent to

𝑞P. + 𝑟P. 𝑥P + 𝑞P𝑞. + 𝑟P𝑟. 𝑥. =12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`

𝑞P𝑞. + 𝑟P𝑟. 𝑥P + 𝑞.. + 𝑟.. 𝑥. =12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

which can be expressed in matrix form as

𝑞P. + 𝑟P. 𝑞P𝑞. + 𝑟P𝑟.𝑞P𝑞. + 𝑟P𝑟. 𝑞.. + 𝑟..

𝑥P𝑥. =

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

¢

Now we try to figure out the solution to equation (14). Before that, it would be helpful to

consider the uniqueness of the solution.

Proposition 6. If 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3 are not collinear, then equation (14) has a unique

solution.

Proof. Suppose that the solution to equation (14) is not unique, i.e.

𝑑𝑒𝑡 𝑞P. + 𝑟P. 𝑞P𝑞. + 𝑟P𝑟.𝑞P𝑞. + 𝑟P𝑟. 𝑞.. + 𝑟..

= 0

that is,

𝑑𝑒𝑡 𝑞P. + 𝑟P. 𝑞P𝑞. + 𝑟P𝑟.𝑞P𝑞. + 𝑟P𝑟. 𝑞.. + 𝑟..

= 𝑞P. + 𝑟P. 𝑞.. + 𝑟.. − 𝑞P𝑞. + 𝑟P𝑟. .

= 𝑞P.𝑞.. + 𝑞P.𝑟.. + 𝑟P.𝑞.. + 𝑟P.𝑟..

− 𝑞P.𝑞.. + 2𝑞P𝑞.𝑟P𝑟. + 𝑟P.𝑟..

= 𝑞P.𝑟.. + 𝑟P.𝑞.. − 2𝑞P𝑞.𝑟P𝑟.

= 𝑞P𝑟. − 𝑟P𝑞. .

= 𝑑𝑒𝑡𝑞P 𝑟P𝑞. 𝑟.

.

= 0

12

i.e. 𝑑𝑒𝑡𝑞P 𝑟P𝑞. 𝑟. = 0.

Thus, the row of 𝑞P 𝑟P𝑞. 𝑟. are not linearly independent: ∃(𝛼P, 𝛼.) ≠ 0 such that

𝛼P 𝑞P, 𝑟P + 𝛼. 𝑞., 𝑟. = 0 ⇔

𝛼P 𝑣P − 𝑣`, 𝑤P − 𝑤` + 𝛼. 𝑣. − 𝑣`, 𝑤. − 𝑤` = 0 (15)

Suppose 𝛼P ≠ 0, then 𝑞P = −YpY�𝑞. and 𝑟P = −Yp

Y�𝑟., (14) turns into

(−𝛼.𝛼P𝑞.). + (−

𝛼.𝛼P𝑟.). −

𝛼.𝛼P𝑞.. −

𝛼.𝛼P𝑟..

−𝛼.𝛼P𝑞.. −

𝛼.𝛼P𝑟.. 𝑞.. + 𝑟..

𝑥P𝑥. =

12 𝑢P − 𝑢` +

𝛼.𝛼P𝑞.𝑣` +

𝛼.𝛼P𝑟.𝑤`

12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

that is,

(𝑞.. + 𝑟..)

𝛼..

𝛼P.−𝛼.𝛼P

−𝛼.𝛼P

1

𝑥P𝑥. =

12 𝑢P − 𝑢` +

𝛼.𝛼P𝑞.𝑣` +

𝛼.𝛼P𝑟.𝑤`

12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

i.e.

(𝑞.. + 𝑟..)𝛼.. −𝛼P𝛼.

−𝛼P𝛼. 𝛼P.𝑥P𝑥. = 𝛼P

12𝛼P 𝑢P − 𝑢` + 𝛼.(𝑞.𝑣` + 𝑟.𝑤`)12𝛼P 𝑢. − 𝑢` − 𝛼P(𝑞.𝑣` + 𝑟.𝑤`)

Multiplying both sides by [𝛼P, 𝛼.] we get

0 = 𝛼P 𝛼P 𝛼.

12𝛼P 𝑢P − 𝑢` + 𝛼. 𝑞.𝑣` + 𝑟.𝑤`12𝛼P 𝑢. − 𝑢` − 𝛼P 𝑞.𝑣` + 𝑟.𝑤`

= 𝛼P.12𝛼P 𝑢P − 𝑢` + 𝛼. 𝑞.𝑣` + 𝑟.𝑤` + 𝛼P𝛼.

12𝛼P 𝑢. − 𝑢` − 𝛼P 𝑞.𝑣` + 𝑟.𝑤`

=12𝛼P

.[𝛼P 𝑢P − 𝑢` + 𝛼. 𝑢. − 𝑢` ]

Note that we suppose 𝛼P ≠ 0, then we get 𝛼P 𝑢P − 𝑢` + 𝛼. 𝑢. − 𝑢` = 0. Combine this

with equation (15) we have

𝛼P𝑣P𝑤P𝑢P

+ 𝛼.𝑣.𝑤.𝑢.

− 𝛼P + 𝛼.𝑣`𝑤`𝑢`

= 0

i.e. 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3, which contradicts our noncollinearity assumption.

If 𝛼P = 0 then 𝛼. ≠ 0, and from equation (15) we know that 𝑞. = 𝑟. = 0. Then equation

(14) becomes

13

𝑞P. + 𝑟P. 00 0

𝑥P𝑥. =

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`

12 𝑢. − 𝑢`

Multiplying both sides by [0,1], we obtain P.𝑢. − 𝑢` = 0, i.e. 𝑢. − 𝑢` = 0.

Note that 𝑞. = 𝑟. = 0 which implies 𝑣. − 𝑣` = 𝑤. − 𝑤` = 0, thus

𝑣. − 𝑣` = 𝑤. − 𝑤` = 𝑢. − 𝑢` = 0. This means that 𝑣8, 𝑤8, 𝑢8 , i = 1,2,3 are collinear,

which contradicts our noncollinearity assumption. ¢

Now we turn to the search for this unique solution.

Using the general matrix inverse formula:

𝑡P 𝑡.𝑡` 𝑡�

WP=

1𝑡P𝑡� − 𝑡.𝑡`

𝑡� −𝑡.−𝑡` 𝑡P

we find the optimal solution to (14) is

𝑥P∗𝑥.∗

=1

𝑞P. + 𝑟P. 𝑞.. + 𝑟.. − 𝑞P𝑞. + 𝑟P𝑟. .𝑞.. + 𝑟.. − 𝑞P𝑞. + 𝑟P𝑟.

−(𝑞P𝑞. + 𝑟P𝑟.) 𝑞P. + 𝑟P.

×

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

=1

𝑞P𝑟. − 𝑟P𝑞. .𝑞.. + 𝑟.. − 𝑞P𝑞. + 𝑟P𝑟.

−(𝑞P𝑞. + 𝑟P𝑟.) 𝑞P. + 𝑟P.

×

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

(16)

thus, we know that 𝑥∗ = 𝑥P∗, 𝑥.∗, 𝑥`∗ = (𝑥P∗, 𝑥.∗, 1 − 𝑥P∗ − 𝑥.∗) is a candidate for a constrained

minimizer. To ensure that our portfolio selection function given by equation (11) minimizes

at 𝑥 = 𝑥∗, it is necessary to check the Karush-Kuhn-Tucker sufficiency condition.

Proposition 7. 𝑥∗ = 𝑥P∗, 𝑥.∗, 𝑥`∗ = (𝑥P∗, 𝑥.∗, 1 − 𝑥P∗ − 𝑥.∗) satisfies the Kuhn-Tucker

sufficiency condition and constitutes a minimal solution to problem (11) if 𝑥P∗ ≥ 0, 𝑥.∗ ≥ 0

and 𝑥`∗ ≥ 0.

Proof. Recalling that

14

𝐿 𝑥, 𝜆 = 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` . + 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` .

− 𝑢P𝑥P + 𝑢.𝑥. + 𝑢`𝑥` + 𝜆(𝑥P + 𝑥. + 𝑥` − 1) (12)

We need to show that 𝐿′′(𝑥, 𝜆) is a positive definite matrix at 𝑥 = 𝑥∗ in the subset defined by

{𝑦 = 𝑦P, 𝑦., 𝑦` 𝜖ℛ`:𝑦P + 𝑦. + 𝑦` = 0}.

Since

∇j𝐿 𝑥, 𝜆 =2𝑣P 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤P 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢P + 𝜆2𝑣. 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤. 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢P + 𝜆2𝑣` 𝑣P𝑥P + 𝑣.𝑥. + 𝑣`𝑥` + 2𝑤` 𝑤P𝑥P + 𝑤.𝑥. + 𝑤`𝑥` − 𝑢P + 𝜆

then

∇j.𝐿 𝑥, 𝜆 = 2𝑣P. + 𝑤P. 𝑣P𝑣. + 𝑤P𝑤. 𝑣P𝑣` + 𝑤P𝑤`

𝑣P𝑣. + 𝑤P𝑤. 𝑣.. + 𝑤.. 𝑣.𝑣` + 𝑤.𝑤`𝑣P𝑣` + 𝑤P𝑤` 𝑣.𝑣` + 𝑤.𝑤` 𝑣`. + 𝑤`.

= 2(𝑣P𝑣.𝑣`

𝑣P𝑣.𝑣`

�

+𝑤P𝑤.𝑤`

𝑤P𝑤.𝑤`

�

)

hence, the following inequality

𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 =𝑦P𝑦.𝑦`

�

×2𝑣P𝑣.𝑣`

𝑣P𝑣.𝑣`

�

+𝑤P𝑤.𝑤`

𝑤P𝑤.𝑤`

�

×𝑦P𝑦.𝑦`

= 2

×𝑦P𝑦.𝑦`

�

×𝑣P𝑣.𝑣`

×𝑣P𝑣.𝑣`

�

×𝑦P𝑦.𝑦`

+𝑦P𝑦.𝑦`

�

×𝑤P𝑤.𝑤`

×𝑤P𝑤.𝑤`

�

×𝑦P𝑦.𝑦`

= 2× 𝑣P𝑦P + 𝑣.𝑦. + 𝑣`𝑦` . + 𝑤P𝑦P + 𝑤.𝑦. + 𝑤`𝑦` . ≥ 0

(17)

holds for any 𝑦 = 𝑦P, 𝑦., 𝑦` 𝜖ℛ`. That is, ∇j.𝐿 𝑥, 𝜆 is a positive semidefinite matrix.

If 𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 = 0, then from (17) we have

𝑣P𝑦P + 𝑣.𝑦. + 𝑣`𝑦` = 0, 𝑤P𝑦P + 𝑤.𝑦. + 𝑤`𝑦` = 0;

Suppose for some 𝑦 = 𝑦P, 𝑦., 𝑦` ≠ 0, 𝑦P + 𝑦. + 𝑦` = 0, these two equalities satisfy,

that is, 𝑣P 𝑣. 𝑣`𝑤P 𝑤. 𝑤`1 1 1

𝑦P𝑦.𝑦`

=000

has nonzero solutions, which implies

𝑑𝑒𝑡𝑣P 𝑣. 𝑣`𝑤P 𝑤. 𝑤`1 1 1

= 𝑑𝑒𝑡𝑣P − 𝑣` 𝑣. − 𝑣` 𝑣`𝑤P − 𝑤` 𝑤. − 𝑤` 𝑤`

0 0 1

= 𝑑𝑒𝑡𝑣P − 𝑣` 𝑣. − 𝑣`𝑤P − 𝑤` 𝑤. − 𝑤` = 𝑑𝑒𝑡

𝑞P 𝑞.𝑟P 𝑟.

15

= 𝑑𝑒𝑡𝑞P 𝑟P𝑞. 𝑟. = 0

As our proof of Proposition 6, this contradicts our noncollinearity assumption of

𝑣8, 𝑤8, 𝑢8 , i = 1,2,3. So 𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 ≠ 0.

Then from inequality (17), we know that 𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 > 0, i.e. 𝐿′′(𝑥, 𝜆) is a positive

definite matrix at 𝑥 = 𝑥∗, thus 𝑥 = 𝑥∗ is a minimizer of the utility function in problem (11).

¢

3.3.2 2-asset problems

Now consider a 2-asset problem with two assets, denoted as 𝑣P, 𝑤P, 𝑢P and 𝑣., 𝑤., 𝑢. ,

such that 𝑣P, 𝑤P, 𝑢P ≠ 𝑣., 𝑤., 𝑢. . The optimization problem turns into

minj�,jp,

𝑈 𝑃 = (𝑣P𝑥P + 𝑣.𝑥.). + (𝑤P𝑥P + 𝑤.𝑥.). − (𝑢P𝑥P + 𝑢.𝑥.)

s. t.𝑥P + 𝑥. = 1 (18)

The Lagrange function of this constrained problem is

𝐿 𝑥, 𝜆 = 𝑣P𝑥P + 𝑣.𝑥. . + 𝑤P𝑥P + 𝑤.𝑥. . − 𝑢P𝑥P + 𝑢.𝑥. + 𝜆(𝑥P + 𝑥. − 1) (19)

The Karush-Kuhn-Tucker necessity conditions are

2𝑣P 𝑣P𝑥P + 𝑣.𝑥. + 2𝑤P 𝑤P𝑥P + 𝑤.𝑥. − 𝑢P + 𝜆 = 02𝑣. 𝑣P𝑥P + 𝑣.𝑥. + 2𝑤. 𝑤P𝑥P + 𝑤.𝑥. − 𝑢. + 𝜆 = 0

𝑥P + 𝑥. − 1 = 0 (20)

Subtract the second equation from the first one in (20), we get

2 𝑣P − 𝑣. 𝑣P𝑥P + 𝑣.𝑥. + 2 𝑤P − 𝑤. 𝑤P𝑥P + 𝑤.𝑥. − (𝑢P − 𝑢.) = 0

and we substitute 𝑥. using the third equation:

2 𝑣P − 𝑣. [𝑣P𝑥P + 𝑣.(1 − 𝑥P)] + 2 𝑤P − 𝑤. [𝑤P𝑥P + 𝑤.(1 − 𝑥P)] − (𝑢P − 𝑢.) = 0

2 𝑣P − 𝑣. .𝑥P + 𝑤P − 𝑤. .𝑥P + 2 𝑣P − 𝑣. 𝑣. + 2 𝑤P − 𝑤. 𝑤. = (𝑢P − 𝑢.)

i.e.

𝑣P − 𝑣. . + 𝑤P − 𝑤. . 𝑥P =12 𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤. (21)

If 𝑣P − 𝑣. . + 𝑤P − 𝑤. . ≠ 0 then we find the solution 𝑥∗ = 𝑥P∗, 𝑥.∗ = (𝑥P∗, 1 − 𝑥P∗)

where

𝑥P∗ =1

𝑣P − 𝑣. . + 𝑤P − 𝑤. . ×[12 𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤.] (22)

Otherwise, if 𝑣P = 𝑣. and 𝑤P = 𝑤. then from equation (21) we find 𝑢P = 𝑢., which

contradicts the assumption that the two assets are not identical. Therefore, we can always

16

have a candidate solution to the constrained minimizer problem. The only question is whether

this candidate solution minimizes our selection function or not.

Similarly, we take a look at 𝐿′′(𝑥, 𝜆).

Since

𝐿 𝑥, 𝜆 = 𝑣P𝑥P + 𝑣.𝑥. . + 𝑤P𝑥P + 𝑤.𝑥. . − 𝑢P𝑥P + 𝑢.𝑥. + 𝜆(𝑥P + 𝑥. − 1) (19)

then

∇j𝐿 𝑥, 𝜆 = 2𝑣P 𝑣P𝑥P + 𝑣.𝑥. + 2𝑤P 𝑤P𝑥P + 𝑤.𝑥. − 𝑢P + 𝜆2𝑣. 𝑣P𝑥P + 𝑣.𝑥. + 2𝑤. 𝑤P𝑥P + 𝑤.𝑥. − 𝑢. + 𝜆

so

∇j.𝐿 𝑥, 𝜆 = 2 𝑣P. + 𝑤P. 𝑣P𝑣. + 𝑤P𝑤.𝑣P𝑣. + 𝑤P𝑤. 𝑣.. + 𝑤..

= 2(𝑣P𝑣.

𝑣P𝑣.

�+

𝑤P𝑤.

𝑤P𝑤.

�)

hence,

𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 =𝑦P𝑦.

�×2

𝑣P𝑣.

𝑣P𝑣.

�+

𝑤P𝑤.

𝑤P𝑤.

�×𝑦P𝑦.

= 2× 𝑣P𝑦P + 𝑣.𝑦. . + 𝑤P𝑦P + 𝑤.𝑦. . ≥ 0

holds for any 𝑦 = 𝑦P, 𝑦. 𝜖ℛ..

If 𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 = 0 then 𝑣P𝑦P + 𝑣.𝑦. = 0 and 𝑤P𝑦P + 𝑤.𝑦. = 0.

For any 𝑦 = 𝑦P, 𝑦. 𝜖ℛ. such that 𝑦P, 𝑦. ≠ 0 and 𝑦P + 𝑦. = 0, then 𝑦. = −𝑦P ≠ 0.

From 𝑣P𝑦P + 𝑣.𝑦. = 0 we have 𝑣P𝑦P − 𝑣.𝑦P = 𝑣P − 𝑣. 𝑦P = 0.

Note that 𝑦P ≠ 0, thus 𝑣P − 𝑣. = 0. Also, we can derive that 𝑤P − 𝑤. = 0.

As our proof of proposition 6, we find that the two assets are identical, which contradicts the

assumption. So 𝑦�∇j.𝐿 𝑥, 𝜆 𝑦 > 0, i.e. 𝐿′′(𝑥, 𝜆) is a positive definite matrix at 𝑥 = 𝑥∗, and

𝑥 = 𝑥∗ is a minimizer of the utility function in problem (18).

Generalized algorithm for n-asset problem For n-asset selection problem, we can break it down into 3-asset or 2-asset problems as what

we have discussed and provide a generalized algorithm for it. This algorithm will terminate in

𝑜(𝑛`) steps.

Step 1: Let 𝑐 ≔ +∞ and 𝑥� ≔ [0,… ,0].

Step 2: Choose three points from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1,… , 𝑛} which have not been

considered yet. If there are no such points then go to Step 9, otherwise denote these three

points by 𝑣�, 𝑤�, 𝑢� , 𝑣�, 𝑤�, 𝑢� and 𝑣�, 𝑤�, 𝑢� . Let 𝑣P, 𝑤P, 𝑢P ≔ 𝑣�, 𝑤�, 𝑢� ,

𝑣., 𝑤., 𝑢. ≔ 𝑣�, 𝑤�, 𝑢� and 𝑣`, 𝑤`, 𝑢` ≔ 𝑣�, 𝑤�, 𝑢� .

17

Step 3: If 𝑑𝑒𝑡𝑞P 𝑟P𝑞. 𝑟. = 𝑑𝑒𝑡

𝑣P − 𝑣` 𝑤P − 𝑤`𝑣. − 𝑣` 𝑤. − 𝑤` = 0

then go to Step 2, otherwise go to Step 4.

Step 4: Compute the first two components, [𝑥P∗, 𝑥.∗], of the optimal solution to (11) using

equation (16).

Step 5: If [𝑥P∗, 𝑥.∗, 1 − 𝑥P∗ − 𝑥.∗] > 0 then go to Step 6, otherwise go to Step 2.

Step 6: If 𝑈 𝑥P∗, 𝑥.∗, 1 − 𝑥P∗ − 𝑥.∗ < 𝑐 then go to Step 7, otherwise go to Step 2.

Step 7: Let 𝑐 = 𝑈 𝑥P∗, 𝑥.∗, 1 − 𝑥P∗ − 𝑥.∗ , and let

𝑥� = [0, … ,0, 𝑥P∗���

, 0, … ,0, 𝑥.∗���

, 0, … ,0, 𝑥`∗���

, 0, … ,0]

Step 8: Go to Step 2.

Step 9: Choose two points from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1,… , 𝑛} which have not been

considered yet. If there are no such points then go to Step 16, otherwise denote these two

points by 𝑣�, 𝑤�, 𝑢� and 𝑣�, 𝑤�, 𝑢� . Let 𝑣P, 𝑤P, 𝑢P ≔ 𝑣�, 𝑤�, 𝑢� and 𝑣., 𝑤., 𝑢. ≔

𝑣�, 𝑤�, 𝑢� .

Step 10: If 𝑣P − 𝑣. . + 𝑤P − 𝑤. . = 0 then go to Step 9, otherwise go to Step 11.

Step 11: Compute the first component, 𝑥P∗, of the optimal solution to (18) using equation (22).

Step 12: If 𝑥P∗, 𝑥.∗ = 𝑥P∗, 1 − 𝑥P∗ > 0 then go to Step 13, otherwise go to Step 9.

Step 13: If 𝑈 𝑥P∗, 1 − 𝑥P∗ < 𝑐 then go to Step 14, otherwise go to Step 9.

Step 14: Let 𝑐 = 𝑈 𝑥P∗, 1 − 𝑥P∗ , and let

𝑥� = [0, … ,0, 𝑥P∗���

, 0, … ,0, 𝑥.∗���

, 0, … ,0]

Step 15: Go to Step 9.

Step 16: Choose a point from the bag { 𝑣8, 𝑤8, 𝑢8 , i = 1,… , 𝑛} which has not been considered

yet. If there is no such point then go to Step 20, otherwise denote this point by 𝑣�, 𝑤�, 𝑢� .

Step 17: If 𝑈 𝑣�, 𝑤�, 𝑢� = 𝑣�. + 𝑤�. − 𝑢� < 𝑐 then go to Step 18, otherwise go to Step 16.

Step 18: Let 𝑐 = 𝑈 𝑣�, 𝑤�, 𝑢� = 𝑣�. + 𝑤�. − 𝑢�, and let

𝑥� = [0, … ,0, 1���

, 0, … ,0]

Step 19: Go to Step 16.

Step 20: 𝑥� is an optimal solution and – 𝑐 is the optimal value of the original portfolio

selection problem (8).

18

Numerical illustration We now use real-life data to demonstrate the proposed algorithm.

For simplicity, we consider a 3-asset problem. In order to alleviate the impact of correlation

between distinct assets, we look for companies from uncorrelated or less correlated industrial

sectors. Hence, we choose Facebook Inc. (FB), Exxon Mobil Corporation (XOM), and The

Coca-Cola Company (KO). Since Facebook held its initial public offering (IPO) on May 18,

2012, we pick monthly quotes of these three stocks from May, 2012 to April, 2016. All the

data are collected from http://finance.yahoo.com.

We first compute monthly rate of returns using the stock quotes by the following equation:

𝑟8X% = 100×𝑃8XnP − 𝑃8X

𝑃8X%

where 𝑟8X is the percentage of return on asset 𝑖. Note that in the utility function (see Section

2.1) we add up a scaling factor of 0.005 to avoid decimals, we now need to use percentages

rather than decimals of returns on the asset. There are 48 monthly stock quotes and thus we

can obtain 47 monthly percentage returns on each asset.

As 𝑟8 are assumed to be trapezoidal fuzzy variables with possibilistic distributions, we need

to figure out the exact trapezoidal forms. Normally, the researcher can use the Delphi Method

[4] to decide the trapezoidal form. In our illustration, we use the frequency statistic method

(see [3] Gupta et al, 2008) to estimate the trapezoidal fuzzy return rates.

19

The percentage returns on Facebook Inc. (FB) can be graphed as:

From Fig. 3 we observe that most of the historical data fall into the intervals [−12.0, −4.0],

[−4.0, 4.0], [4.0, 12.0] and [12.0, 20.0]. We take the mid-points of the intervals

[−12.0, −4.0] and [12.0, 20.0] as the left and the right end points of the tolerance interval,

respectively. Thus, the tolerance interval of the fuzzy percentage returns is [−8.0, 16.0]. By

going through all the historical data, we find the minimum possible value -30.2 and the

maximum possible value 47.9 and view them as the limits of uncertain percentage returns in

the future, respectively. Therefore, the left spread is 22.2 and the right spread is 31.9, and the

trapezoidal percentage returns on FB is 𝑟P = [−8.0, 16.0, 22.2, 31.9].

Likewise, we can obtain the trapezoidal returns on XOM, which is 𝑟. = [−4.6, 3.8, 4.3, 7.5],

and KO, which is 𝑟 = [−4.5, 4.5, 3.9, 3.9].

Assume that 𝐴 = 2.46, we can calculate

𝑣P, 𝑤P, 𝑢P = (2.331, 0.707, 5.617),

𝑣., 𝑤., 𝑢. = (0.684, 0.154, 0.133),

𝑣`, 𝑤`, 𝑢` = (0.643, 0.102, 0.000).

First consider the 3-asset problem with 𝑣P, 𝑤P, 𝑢P , 𝑣., 𝑤., 𝑢. and 𝑣`, 𝑤`, 𝑢` .

Since 𝑑𝑒𝑡𝑞P 𝑟P𝑞. 𝑟. = 𝑑𝑒𝑡

𝑣P − 𝑣` 𝑤P − 𝑤`𝑣. − 𝑣` 𝑤. − 𝑤` = 𝑑𝑒𝑡 1.688 0.605

0.041 0.052 = 0.063 ≠ 0,

20

we get

𝑥P∗𝑥.∗

=1

𝑞P𝑟. − 𝑟P𝑞. .𝑞.. + 𝑟.. − 𝑞P𝑞. + 𝑟P𝑟.

−(𝑞P𝑞. + 𝑟P𝑟.) 𝑞P. + 𝑟P.×

12 𝑢P − 𝑢` − 𝑞P𝑣` − 𝑟P𝑤`12 𝑢. − 𝑢` − 𝑞.𝑣` − 𝑟.𝑤`

=1

0.063.0.004 −0.100−0.100 3.214

1.6610.035

= 0.792−13.5072 .

Notice that 𝑥.∗ < 0, which is not feasible, then we found no qualified 3-asset candidate for an

optimal solution to (10).

Now we turn to all conceivable 2-asset problems:

○1 For the combination of FB and XOM,

since 𝑣P − 𝑣. . + 𝑤P − 𝑤. . = 3.018 ≠ 0, we get

𝑥P∗ =1

𝑣P − 𝑣. . + 𝑤P − 𝑤. . ×12 𝑢P − 𝑢. − 𝑣P − 𝑣. 𝑣. − 𝑤P − 𝑤. 𝑤.

=1

3.018×1.530 = 0.507

Thus, [0.507, 0.493, 0] is a qualified candidate for an optimal solution to (10), where

𝑈 0.507, 0.493, 0 = −0.417.

○2 For the combination of FB and KO,

since 𝑣P − 𝑣` . + 𝑤P − 𝑤` . = 3.214 ≠ 0, we get

𝑥P∗ =1

𝑣P − 𝑣` . + 𝑤P − 𝑤` . ×12 𝑢P − 𝑢` − 𝑣P − 𝑣` 𝑣` − 𝑤P − 𝑤` 𝑤`

=1

3.214×1.661 = 0.517

Thus, [0.517, 0, 0.483] is a qualified candidate for an optimal solution to (10), where

𝑈 0.517, 0, 0.483 = −0.435.

○3 For the combination of XOM and KO,

since 𝑣. − 𝑣` . + 𝑤. − 𝑤` . = 0.004 ≠ 0, we get

𝑥P∗ =1

𝑣. − 𝑣` . + 𝑤. − 𝑤` . ×12 𝑢. − 𝑢` − 𝑣. − 𝑣` 𝑣` − 𝑤. − 𝑤` 𝑤`

21

=1

0.004×0.035 = 8.75 > 1

Thus, this cannot be a qualified candidate for an optimal solution to (10).

Finally, we compute the utility values of all the 1-asset options:

𝑈 1, 0, 0 = 𝑣P. + 𝑤P. − 𝑢P = 0.316;

𝑈 0, 1, 0 = 𝑣.. + 𝑤.. − 𝑢. = 0.359;

𝑈 0, 0, 1 = 𝑣`. + 𝑤`. − 𝑢` = 0.424.

Comparing the function values of all feasible solutions we find that the optimal portfolio

would be 𝑥∗ = [0.517, 0, 0.483], i.e. the combination of Facebook (51.7%) and Coca-Cola

(48.3%).

Remarks on Carlsson-Fullér-Majlender’s model

6.1 Assumption of covariance

To calculate possibilistic variance of the linear combination of fuzzy variables, we shall use

the following theorem ([6] Sánta, 2012):

𝑉𝑎𝑟 𝜆] + 𝜆8𝐴8e

8fP= 𝜆8.𝑉𝑎𝑟 𝐴8

e

8fP+ 2 𝜆8𝜆� 𝐶𝑜𝑣 𝐴8, 𝐴�

e

8fP

𝑤ℎ𝑒𝑟𝑒𝐴8𝑎𝑟𝑒𝑓𝑢𝑧𝑧𝑦𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠𝑎𝑛𝑑𝜆8𝑎𝑟𝑒𝑟𝑒𝑎𝑙𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑖 = 1,… , 𝑛. (23)

A simple two-variable theorem has also been presented in [5] (Carlsson, 2001).

From these definitions, we learn that we need to take the covariance terms into account when

calculating the variance of fuzzy number combinations.

Note that in Carlsson-Fullér-Majlender’s model when we derive possibilistic variance of the

whole portfolio, we actually ignore the intercorrelation between different assets. That is, we

assume the covariance to be zero. The subsequent discussion is based on this hypothesis. In

that case, Carlsson-Fullér-Majlender’s model only applies to cases where the optional assets

are uncorrelated or significantly less correlated. In fact, however, it is unreal to find assets

that are totally uncorrelated, so this model is not as applicable and effective as we might

expect. We need to pick out the assets from differentiating industrial sectors carefully to

comply with the zero-covariance assumption.

22

6.2 Possibilistic variance of portfolio

Even if we rule out the covariance terms, there might still be some confusion in equation (7).

Note that 𝑥8 is a real number and 𝑟8 is a fuzzy number. Using the formula in (23), we can

derive the portfolio variance as

𝜎. 𝑟8𝑥8e

8fP= 𝑥8.𝜎. 𝑟8

e

8fP

= 𝑥8.e

8fP

𝑏8 − 𝑎82 +

𝛼8 + 𝛽86

.

+𝛼8 + 𝛽8 .

72 fromequation 4

= 𝑥8.e

8fP

12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8

.

+𝛼8 + 𝛽8 .

72

= 𝑥8.12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8

.e

8fP+

𝛼8 + 𝛽8 .𝑥8.

72e

8fP

=12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8 𝑥8

.e

8fP+172 𝛼8 + 𝛽8 𝑥8 .

e

8fP

≠12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8

e

8fP𝑥8

.

+172 𝛼8 + 𝛽8 𝑥8

e

8fP

.

which is given by equation (7).

If equation (7) is not true, neither is the rest of the discussion. The whole Carlsson-Fullér-

Majlender’s model, therefore, does not seem convincing to me. Considering that this is a

well-known model in fuzzy optimization, I am not sure if I have taken this the wrong way or

not.

If equation (7) is corrected to

𝜎. 𝑟8𝑥8e

8fP=

12 𝑏8 − 𝑎8 +

13 𝛼8 + 𝛽8 𝑥8

.e

8fP+172 𝛼8 + 𝛽8 𝑥8 .

e

8fP (24)

then the optimization problem turns into

maxjk

𝑈 𝑃 = 𝑢8𝑥8e

8fP− 𝑣8𝑥8 .

e

8fP− 𝑤8𝑥8 .

e

8fP

s. t.{ 𝑥8e

8fP= 1, 𝑥8 ≥ 0, 𝑖 = 1,2, … , 𝑛}

(25)

rather than problem (8), which is

maxjk

𝑈 𝑃 = 𝑢8𝑥8e

8fP− 𝑣8

e

8fP𝑥8

.− 𝑤8

e

8fP𝑥8

.

s. t.{ 𝑥8e

8fP= 1, 𝑥8 ≥ 0, 𝑖 = 1,2, … , 𝑛}

(8)

and thus the solutions will change correspondingly.

23

6.3 Feasibility of the solution

As we disregard short-selling and long-buying, the feasible set of the solution should be

{𝑥8𝜖ℛ:𝑥8𝜖 0,1 , 𝑖 = 1,… , 𝑛}. However, we do not include this condition into the constraints

of our optimization problem. The Carlsson-Fullér-Majlender’s model, in fact, computes the

not-necessarily feasible weights, so we need to check the feasibility every time we obtain a

candidate of the solution. This may cause some incovenience.

Conclusions In this paper, we introduce the Carlsson-Fullér-Majlender’s trapezoidal possibility model to

address fuzzy portfolio selection problem. We devise a utility function based on portfolio

selection theory formulated by [7] (Markowitz, 1952). Using some properties of trapezoidal

fuzzy variable as well as optimization theory, we translate the optimization problem into a

non-linear prgramming problem, in which we can employ the Lagrange Multiplier Method

and Karush-Kuhn-Tucker (KKT) Conditions to calculate the optimal solutions. We provide a

generalized algorithm for the problem and then use some real data for illustration. We end the

paper with some personal thinking of the model, including its limitations or even some faults.

24

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selecting portfolios with highest utility score." Fuzzy sets and systems 131.1 (2002):

13-21.

[2] Gupta, Pankaj, et al. Fuzzy Portfolio Optimization. Springer-Verlag, Berlin, 2014.

[3] Gupta, Pankaj, Mukesh Kumar Mehlawat, and Anand Saxena. "Asset portfolio

optimization using fuzzy mathematical programming." Information Sciences 178.6

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[4] Linstone, Harold A., and Murray Turoff, eds. The Delphi method: Techniques and

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[5] Carlsson, Christer, and Robert Fullér. "On possibilistic mean value and variance of

fuzzy numbers." Fuzzy sets and systems 122.2 (2001): 315-326.

[6] Sánta, Katalin. "Portfolio Optimization with Fuzzy Constraints." 2012.

[7] Markowitz, Harry. "Portfolio selection." The journal of finance 7.1 (1952): 77-91.

[8] Markowitz, Harry M. Portfolio selection: efficient diversification of investments. Vol.

16. Yale university press, 1968.

[9] Zadeh, Lotfi A. "Fuzzy sets." Information and control 8.3 (1965): 338-353.

[10] Bellman, Richard E., and Lotfi Asker Zadeh. "Decision-making in a fuzzy

environment." Management science 17.4 (1970): B-141.