Modeling and forecast of the monthly, quarterly and half-yearly usd Libor Rates (página 2)

IV.
RESULTS

IV.1 Descriptive Statistics

The main descriptive statistics pertaining to the three
time series processed (pure random process) for the monthly,
quarterly and half-yearly LIBOR usd interest rates since January
the 2nd, 1987 to March the 24th, 2006 (19
years) are shown in Table 1, where can be seen the time series
Normal distributions, averages and standard deviations
validities.

Table 1. Main descriptive statistics obtained for the
three time series are as follow:

IV.2 Juncture analysis Long Run Trend

With the purpose of knowing better the time series
performance in the long run, monthly, quarterly and half-yearly
LIBOR usd interest rates, proceed to prepare the Juncture
Analysis, since 31/1/88 to 31/3/09, resulting the long run trend
LIBOR seasonally adjusted variation rates (Annex D) of the last
18 years, showing a growth trend in that juncture, as can be seen
in Table 2, appreciating that the best alternative is to
negotiate the monthly period, because achieve the least
percentage growth in the last 18 years (Annex D).

Table 2. Real percentage growth till 24/3/06

Reasonably it should be pointed out that the LIBOR
juncture variation rates expected values since 30/4/06 to 31/3/09
for the monthly period do not show significant acceleration,
recommending, if proceed, apply for a loan in this period,
because the forecasted expected interest rates will be low (Annex
D).

IV.3 Forecasts

Applying the process describe in point III of this
report and the statistical validity of the mathematical models
describe before, proceed to obtain the LIBOR usd interest rates
for the monthly, quarterly and half-yearly periods forecast with
a 95% confidence interval, whose expected values are shown in
Annex E in the figures 4, 5 and 6, where can be appreciated the
growth trend in the long run, over which oscillate the forecasted
values, growing in the period since 30/4/06 to
31/3/09.

IV.4 Forecast average errors

The average errors of the forecasted monthly, quarterly
and half-yearly LIBOR interest rates, corresponding to the last
day of the month since 30/4/06 to 31/3/09 are low (less than 10%)
as shown in Table 3.

Table 3. Forecast average errors.

The fitting and checking of the three ARIMA (0,1,9)
(0,1,1)12 multiplicative seasonal models are
good.

IV.5 Identified periodical behavior
patterns

In this kind of work is important to analyze the
variation causes, starting with the identified periodical
behavior patterns that are not included in the objective and
scope of the present work, directed to obtain a practical tool
that will permit to evaluate during the negotiation process the
credit that the bank offers and its financial impact, in some way
this theme should be included to orientate future
works.

Taking into account this aspect, in Table 4 are related
the periodical behavior patterns identified in the time series
under study and their forecasts, through the fitting and checking
inspection procedures and the growth, stability and decline
stages which are corresponded between them.

Table 4. Periodical behavior patterns for the analyzed
LIBOR patterns

5. CONCLUSION AND
   RECOMMENDATION

The LIBOR interest rates were sufficient to be
processed through mathematical and statistical techniques that
permitted to conceive and design a mathematical model well
based to forecast with the least average monthly error, the
last day of the month for the monthly, quarterly and
half-yearly periods from April, 2006 to March, 2009. The
Juncture Analysis was also applied.

Considering the financial impact of this procedure to
forecast the LIBOR interest rates, it is recommended to use the
values shown in Annex D during the bank credit negotiating
process to evaluate each offer and widen the forecast horizon
through the periodical real interest rates feedback.

VI.
BIBLIOGRAPHY

Boletín Panorama del Mercado, Banco Financiero
Internacional, 12/ago/05 – 24-mar-06; Cuba.

Web site,
LIBOR rate

Web site, http://www.megabolsa.com

Web site, http://www.finanzas.com

Time Series Analysis, Forecasting and Control, Box
& Jenkins, Holden Day, California, 1970.

Business Forecasting, John E. Hanke, Arthur G. Reitsch
and Dean W. Wichem, Prentice Hall, 2001.

Juncture Analysis Methodology, Statistical National
Office (ONE),
Cuba, 1996.

VII.
ANNEXES

Annex A. Terms and definitions
glossary

Juncture analysis: Reflect in a synthetic way the
principal characteristics of the economic situation in a given
moment, for the international, national, regional, branches,
enterprise groupings as a whole.

Cycle: Oscillating movement in the short-run (3
to 5 years), medium-run (5 to 15 years) and long-run (15 to 30
Years or more) of a time series.

Trend-cycle: Difference between the long-run and
medium-run trend curves of the time series, when they cross each
other form maximum and minimum cycles.

Seasonal adjustment: Adjust a time series to its
seasonal variation to show its trend in the long-run.

Cycle duration: Number of months that exist
between the observation in which we find the analyzed turning
point and the corresponding next one of different
symbol.

Seasonality: Oscillating movement in the yearly
period of a time series. It is determined, essentially, by
climate and institutional factors and do not respond to any type
of economic variable.

Autocorrelation function: Correlation that exists
between the observations of the same time series. It is used to
determine the seasonality of a time series and other
uses.

Irregularity: Random components, errors that can
not be explained in a time series. Correspond to movements in the
short-run. These irregularities in the time series could be
generated by economic factors, they have a transitory
characteristic and they are not repeated in the short-run. They
are not predictable.

Points of return: They are points which pass from
a phase of acceleration to other of desacceleration.

Time series: Observations or values taken at the
same time interval. It is also known as stochastic process
(probabilistic).

Trend: Oscillating performance of a long-run time
series. Its movement in the short-run has other characteristics.
It is composed mainly by economic factors. Include the economic
cycles. It is predictable. In technical analysis is the growth,
stability and decline trends of a time series in the long-run,
seasonal or cycle adjusted. It is obtain from a time series data,
applying the smoothing or least square methods.

Annex B. Time series graphs

Annex C. Mathematical model ARIMA (0,1,9)
(0,1,1)12 multiplicative and
seasonal

(1- B) (1- B12) Log Z t =
1 (B) 2 (B2)
3 (B3) 4
(B4) 5 (B5)
6 (B6) 7
(B7) 8 (B8)
9 (B9) 12
(B12) At

(1- B) (1- B12) Log Z t = (1-
1 B – 2 B2 –
3 B3 – 4
B4 – 5 B5 –
6 B6 – 7
B7 – 8 B8 –
9 B9) 12
(B12) At

(1- B – B12 + B13) Log Z
t = (1- 1 B – 2
B2 – 3 B3 –
4 B4 – 5
B5 – 6 B6 –
7 B7 –
8B8 – 9
B9) 12 (B12)
At

Note: The rest of the model is not expanded due
to its length. This is a mathematical model especially conceive
and design to forecast the LIBOR interest rates with a
satisfactory fitting.

where:

B = Lag operator such that Bm Z t
= Z t-m

(1- B) = Difference operator.

 (B) = 1 – 1 B –
2 B2 – . . . – q
Bq and the  are stationary moving averages
parameters.

 (BS) = 1 – S
BS – . . . – QS BQS and
the S are seasonal moving averages
parameters.

D, DS, S = Are not negative integers. (Stationary,
seasonal and frequency differences)

Z t = Original or transformed time series
values. These observations are taken at the same time
intervals.

At = Random perturbations which are supposed
independently

distributed as N (0,
2a).

Annex D. Juncture Analysis

Variation rate expression:

T= 100
((Zt + Zt-1 + . . . + Zt-k
∕ Zt-p + Zt-p-1 + . . . +
Zt-p-k ) – 1)

where:

k = Lag

p = Previous period

Seasonally adjusted expression:

Z t – Z t-12 = (1 –
B12) Z t

ANNEX E FORCASTED LIBOR DATA FOR
MONTHLY, QUARTERLY AND HALF-YEARLY PERIODS

Table E.1 Forcasted LIBOR usd monthly
rates

Table E.2 Forecasted LIBOR usd quarterly
rates

Table E.3 Forecasted LIBOR usd half-yearly
rates

Authors:

Msc. Jesús Mesa Oramas

Msc. Luis Pérez Suárez