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