cornerstones of Markowitz’s theory of
the optimal portfolio. From a financial
perspective, a covariance matrix is used
to establish a link between the risk factors
originated from the market variables. In
other words, the covariance matrix considers price fluctuations and their correlations. The matrix is based on historical
data series of market price movements,
collected from the ECOWIN database.
This matrix provides us not only
with the variability of individual market factors, but also the CO-movement
(the correlated or similar movement)
of the market factors. Thus, it would
contribute to our understanding of the
price process of the portfolio and consequently the model’s ability to provide
accurate and reliable volatility forecasts.
It should be noted that the covariance
matrix is considered and generated only
for the market factors and not for projects. Therefore, the number of components of this matrix is equal to the
number of historical data points that
have been obtained from the market.
This matrix catches the variance and
correlation in data (Alexander, 2008).
Variance and covariance are often displayed together in a variance-covariance
matrix. The variances appear along the
diagonal and covariance appears in the
off-diagonal elements, as shown below.
var( 1) cov( 1, 2) . . . cov( 1, k)
cov( 2, 1) var( 2) . . . cov( 2, k)
... ... ... ...
cov(k, 1) cov(k, 2) . . . var(k)
Alexander, C. (2008). Moving average
models for volatility and correlation,
and covariance matrices. Handbook of
finance. New York, NY: John Wiley & Sons.
Andrén, N., Jankensgård, H., &
Oxelheim, L. (2005). Exposure-based
cash-flow-at-risk: An alternative to VaR
for industrial companies. Journal of
Applied Corporate Finance, 17( 3), 76–86.
model is a generalized autoregressive
conditional (GARCH) model. Applications of GARCH models are widespread
in situations where the volatility of return
is mainly considered. GARCH models,
especially in financial applications, have
become important tools for the analysis
of time series data. They are particularly
useful when the goal of the study is to
analyze and forecast volatility.
A GARCH (p, q) model is more suitable for capturing the dynamics of a time
series conditional variance. Generally, a
GARCH (p, q) model is expressed as:
N (0,t ) (A. 1)
tut , ut N ( 0, 1 ) (A. 2)
j t2j (A. 3)
Wherep 0, q 0, 0, i 0 (i
1, 2, . . . p), j 0 (j 1, 2, . . . q), p is
the order of GARCH terms and q is the
order of the terms 2. An autoregressive integrated moving average (ARIMA)
model is used widely for forecasting nonstationary time series, expressed as follows (Tan et al., 2010):
0(B)(12B)dXt (B)t (A. 4)
Where Xt is a nonstationary time
series at time t, t is white noise (with
constant variance and zero mean), and
d is the order of differencing. B is a backward shift operator defined by BXt
Xt21, 0(B) 1201B202B22. . . 0pBp, and
(B) is the moving average operator
defined as (B) 1201B202B22. . . 0qBq
(Tan et al., 2010) AIC measure is used to
evaluate the adequacy of the model by
choosing the one that minimizes AIC
from several possible models.
A covariance matrix (variance-covariance
matrix) is a matrix whose element in the
ith row and jth column is the covariance
between the ith row and jth elements. The
covariance matrix has a long history in
financial analytics, and it is one of the
considered by covariance matrix, and
time series models are used to fore-
cast the volatility of prices. Moreover,
a new risk-aversion parameter (a) is
applied that enables the decision maker
to determine the desirable level of risk.
Analysis of the results indicates that the
best outcomes occur when projects are
selected from industries that have the
least correlation (all else being equal).
In other words, this study suggests that
decision makers should invest in indus-
tries with the lowest levels of correlation
to decrease overall risk.
The result of this study should be evaluated keeping in mind the limitations of
modeling and the fact that the choice of
methods and the assumptions will affect
the outcomes. For example, in this study
we proposed that some data should
be estimated by expert judgment and
we applied a probabilistic distribution
(such as the normal distribution); such
estimations, however, may decrease the
Masoud Mohammad Sharifi would specifically like to emphasize the ongoing
and undeniable support of Saeide Bah-rani during the completion of this study,
and to whom he dedicates this article.
Financial markets are affected by unexpected changes both in the money supply
and in the fluctuations of money. Therefore, the prediction of price volatility
seems to be an essential part of analyzing these markets. Management can
use a history analysis (like a time series
analysis) to make current decisions and
plans based on long-term forecasting. A
time series is a set of observations, each
recorded at time span t. Autoregressive-moving-average (ARMA) models prepare
a parsimonious description of a (weakly)
stationary stochastic process. If an
is assumed for the error variance, the