as steel, copper, and aluminum, and
exchange rates such as EUR/US$ and
GBP/US$ from the ECOWIN database.
Then, by using time series models, we
fitted the best model for each factor
(product prices and exchange rates).
Next, we put these time series models
into the proposed model as pijt.
We use AIC (Akaike Information
Criterion) measure to evaluate the adequacy of several proposed time series
models by choosing the one with the
Next, to consider correlation among
risk factors in the past derived from
historical data, and to expand the correlation to future price (simulated by
time series models), we generated a
covariance matrix. This matrix is
very important for determining the
co-movement of market factors.
In the previous steps, we determined
all the necessary parameters and put
them into the model. Then, we nominated the level of confidence for managerial preference regarding optimization of
CFaR. For this study, CFaR is optimized
at 5% (the model’s objective is to maximize 5% “left tail” of cash flow) and, as
mentioned in the proposed model section, this parameter in determining CFaR
is a risk-aversion parameter. Changing
its level can change the decision maker’s preference between risk and return.
In Table 2, project information is
shown. For example, project 1 can be
involved in the portfolio from period 1
until period 4. Also, if it is involved in
the portfolio during period 2, the portfolio will incur an expenditure to invest
in it, which is estimated by the normal
distribution (mean is 260000 and 2
is 26000). The Opt-Quest optimizer in
@Risk software is performed to determine all the variables that are binary
and to optimize CFaR. After running the
model, the best time for investing and
discarding each project is determined.
The result is shown in Table 3.
For example, as you can see in Table 3,
y111 1 means that project 1 in indus-
try 1 must be invested at period 1 or
the probability distribution of the initial
product price and production volume
(units are in dollars).
pijt is the product selling price,
estimated by time series models, and
its unit is dollars per unit of sales.
It should be noted that for projects
invested abroad in which income is
in other currencies (other than dollars)—for example, EUR or GBP—the
product price must be multiplied in
the exchange rate to convert the unit
to dollars. Therefore, pijt in this case is
replaced with (pijt 3 Et). To illustrate,
if a project invested abroad produces
zinc, to calculate the return in dollars,
we should consider the price of zinc, the
exchange rate of the related currency,
and the production volume.
vijt is a sample from the probability
distribution function of production volume or the quantity of product that will
be produced during the period. Its unit
is the unit of sales.
Computational Results and
Case Example A
As a test, the proposed model is applied
to maximize CFaR in a simulated case
(because of the inaccessibility of real
data). The case was developed to include
different projects with different start
times, returns, and risks in various industries. The planning horizon should be
specified. It depends on the accessibility
of the data. If a shorter period is chosen,
prediction accuracy will increase, and
vice versa. Here, we assume that the
planning horizon is composed of five
periods. According to the research methodology for each project, four parameters must be determined, including
required capital investment (Iijt), project
cost (Cijt), production volume (vijt), and
project selling price (Sijt) in each period.
These parameters are estimated by expert
judgment, and probability distribution
(normal) is applied to estimate them.
For considering price fluctuations in
the prediction horizon and forecasting
the future trend of prices, a time series
analysis is used. We collected the yearly
historical prices of different metals, such
in the portfolio or it is selected and has
not been discarded yet), and 0 other-
wise. 0ijt is another binary variable; it is
equal to 1 if xijt yijt, and 0 otherwise.
(It should be noted that all the binary
variables are unit-less.)
CFt is the cash flow at period t (at first, CF0
total available budget); aj is the minimum fund (capital) that must be invested
in industry j; bj is the maximum fund that
can be invested in industry j; and K is the
maximum number of allowed projects.
and are predetermined confidence levels. is considered a risk-aversion parameter; to clarify, if 50%,
the investor completely disregards risks
and the objective is to maximize returns;
also, by decreasing to 0%, the investor
becomes increasingly risk-averse until
he or she only wants to minimize risks.
In addition, is another confidence-level parameter for an uncertainty constraint that is determined by the decision
maker. It ensures that the constraint will
be satisfied at the level (in deterministic modeling, under a specific situation,
a constraint is either satisfied or not satisfied; in probabilistic modeling, under a
specific situation, because of probabilistic parameters in the model, a constraint
will be satisfied with different possibility
levels—for example, a constraint is satisfied in a 90% level through simulation
and the other constraint is satisfied in
an 80% level).
Iijt is a sample from the probability distribution function of the initial
required capital (for selecting the project in the portfolio; units are in dollars).
Cijt is a sample from the probability
distribution function of the project cost
(capital expenses during each period;
units are in dollars). Sijt is a sample from
the probability distribution function of
the project selling price (for discarding
project from the portfolio; units are in
dollars). r is a probability distribution
function of the discount rate. And ijt
is the project return, which depends on