Proceedings

of assets resides in traditional titles. The second resides in projects. The main
difference between the two types of investments is that decisional variables for
projects are binary, while the ones for securities are continuous. There is an available
budget for each project, and the budgets can be quantified by investors or experts. We
assume that the total cost of a project is the same as the budget needed to start the
project. Also, the project investments cannot be reallocated in any moment of time,
unlike title investments.
We consider the structure of securities from the mixed assets as having n risky titles
S i , i = 1,
2 ,..., n , that offer random earning rates, as well as m projects, Pj , j = 1,
2,...,
m . We
assume that the investor starts with a portfolio that includes only securities titles, and
then decides to rebuild the new mixed securities portfolio, including titles and
projects.
To formalize the problem we introduce the following notations:
~
r i : random variable, which represents rate of return on securities without transactional
costs S i , i = 1,
2 ,..., n , ;
r : expected net return for securities without transactional costs S i , i = 1,
2,...,
n,
;
i
~
j
R : random variable representing random net earnings of a project j 1,
2,...,
m
R : project expected net return after cost deduction , j = 1,
2,...,
m ;
j
M: total value of securities owned by an investor;
X : total value of investment on risky securities S i , i = 1,
2,...,
n,
i
i
x : rate of total investment in risky assets S i , i = 1,
2,...,
n,
0
i
~ 336 ~
P j
P j
, = ;
X : total value of risky securities investment from the existing portfolio , i 1,
2,...,
n,
0
i
S i = ;
S i , i = 1,
2,...,
n ;
x : rate of total value of risky securities investment in existing portfolio ,
k : rate of transactional costs for risky securities , i 1,
2,...,
n,
i
j
S i = and non risky asset n+
1
z : binary variable which indicates if the project is a selected asset or not , j = 1,
2,...,
m
z ⎧1,
if Pj project is selected for investments
j = ⎨
⎩0,
if not
⎛ ~ ~ ~ ~ ~ ~ ⎞
We assume that the vector of random variables ⎜ r 1,
r 2 ,..., r n,
R1
, R 2 ,..., R m ⎟ is distributed
⎝
⎠
on a given space { ( r1t ,..., rnt
, R1t
,..., R mt ) , t = 1,
2,...,
T } with known probabilities:
⎧⎛
~ ~ ~ ~ ~ ~ ⎞
⎫
pt = Pr ⎨⎜
r1
, r 2 ,..., r n,
R1
, R 2 ,..., R m ⎟ = ( r1t
,..., rnt
, R1t
,..., Rmt
) ⎬,
t = 1,
2,...,
T
⎩⎝
⎠
⎭
Then, expected net return i
is given by the relation: ri
= ∑
=
return j
r of risky securities
T
, i 1,
2,...,
n,
ptrit
, i = 1,
2,...,
n,
where it
t 1
T
R of Pj , j = 1,
2,...,
m project, is given by: R j = ∑ pt
R jt , j = 1,
2,....,
m , where jt
t=
1
be generated through predicted data.
P j
S ;
S i = without transactional costs
r can be predicted. Expected net
R can
Having a securities mixed portfolio ( x 1 , x2
,..., xn
, z1,
z 2 ,..., z m ) , the expected return of the
portfolio without transactional costs can be expressed by: