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


Partes: 1, 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:

Indicators

Time series

LIBOR:

Monthly

Quarterly

Half-yearly

Sample size

231

231

231

Average

5.44467

5.12066

5.22076

Std. Deviation

2.23943

2.24937

2.25085

Std. Skewness

-0.415772

-0.298224

-0.356824

Std. Kurtosis

-1.79795

-2.03952

-1.87941

Variation Coefficient

41.1307

43.9274

43.1134

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

LIBOR

Percentage

Growth

Monthly

25.47

Quarterly

37.70

Half-yearly

32.84

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.

Indicators

Time series

LIBOR:

Monthly

Quarterly

Half-yearly

Monthly average errors

4.49%

1.97%

2.81%

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

  1. 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

DATE

FORECAST

Confidence

Interval

95%

Minimum

Maximum

30-abr-06

5.4631

4.79438

6.2251

31-may-06

5.5960

4.57875

6.83931

30-jun-06

5.7034

4.38489

7.41846

31-jul-06

5.7476

4.17009

7.92185

31-ago-06

5.7053

3.93677

8.26829

30-sep-06

5.5407

3.63345

8.44902

31-oct-06

5.5370

3.4545

8.87486

30-nov- 06

5.7059

3.35073

9.71644

31-dic-06

5.5237

3.0551

9.98685

31-ene-07

5.4828

2.83261

10.6125

28-feb-07

5.5209

2.68141

11.3672

31-mar-07

5.6663

2.59985

12.3497

30-abr-07

5.7895

5.10538

6.56517

31-may-07

5.8377

4.81078

7.08372

30-jun-07

5.8358

4.52913

7.51935

31-jul-07

5.7757

4.23975

7.86818

31-ago-07

5.7171

3.99946

8.17235

30-sep-07

5.5509

3.69853

8.33105

31-oct-07

5.5371

3.51773

8.71567

30-nov-07

5.7379

3.44036

9.56961

31-dic-07

5.6312

3.18868

9.94449

31-ene-08

5.5828

2.96169

10.5235

29-feb-08

5.6362

2.81844

11.2712

31-mar-08

5.7980

2.74568

12.2434

30-abr-08

5.9286

5.25277

6.69145

31-may-08

5.9657

4.9481

7.19247

30-jun-08

5.9711

4.67555

7.6256

31-jul-08

5.9251

4.39475

7.98841

31-ago-08

5.9143

4.18935

8.34956

30-sep-08

5.7819

3.90636

8.55783

31-oct-08

5.7858

3.73389

8.96531

30-nov-08

5.9657

3.63878

9.78072

31-dic-08

5.8706

3.38984

10.1668

31-ene-09

5.8095

3.14876

10.7186

28-feb-09

5.8779

3.00826

11.4851

31-mar-09

6.0362

2.93032

12.434

Table E.2 Forecasted LIBOR usd quarterly rates

DATE

FORECAST

Confidence

Interval

95%

Minimum

Maximum

30-abr-06

5.1541

4.67533

5.68192

31-may-06

5.3834

4.60207

6.29738

30-jun-06

5.5611

4.50103

6.87076

31-jul-06

5.5950

4.29472

7.28904

31-ago-06

5.6999

4.14627

7.83592

30-sep-06

5.7050

3.92319

8.29605

31-oct-06

5.6694

3.69497

8.69881

30-nov-06

5.6442

3.45242

9.22728

31-dic-06

5.5347

3.16389

9.68201

31-ene-07

5.4404

2.90631

10.1839

28-feb-07

5.5019

2.765

10.9479

31-mar-07

5.5810

2.65174

11.7459

30-abr-07

6.1516

5.40731

6.99823

31-may-07

6.4898

5.4258

7.76237

30-jun-07

6.5911

5.2356

8.29746

31-jul-07

6.2820

4.74062

8.32455

31-ago-07

6.5867

4.71155

9.20809

30-sep-07

6.6739

4.52011

9.85403

31-oct-07

6.7375

4.32339

10.4995

30-nov-07

6.7941

4.09269

11.2787

31-dic-07

6.6584

3.74438

11.8401

31-ene-08

6.5426

3.4498

12.4081

29-feb-08

6.6218

3.29323

13.3146

31-mar-08

6.7189

3.16603

14.2589

30-abr-08

6.8360

6.03474

7.74355

31-may-08

6.9748

5.87139

8.28568

30-jun-08

7.0596

5.65808

8.8083

31-jul-08

6.8299

5.20915

8.95489

31-ago-08

7.1196

5.15344

9.83592

30-sep-08

7.1756

4.92155

10.4619

31-oct-08

7.1762

4.66307

11.0436

30-nov-08

7.2424

4.41699

11.8751

31-dic-08

7.0829

4.03215

12.4418

31-ene-09

6.9545

3.71262

13.0272

28-feb-09

7.0404

3.54568

13.9794

31-mar-09

7.1403

3.40802

14.96

Table E.3 Forecasted LIBOR usd half-yearly rates

DATE

FORECAST

Confidence

Interval

95%

Minimum

Maximum

30-abr-06

5.3516

4.81845

5.94375

31-may-06

5.58498

4.71142

6.62053

30-jun-06

5.78897

4.60691

7.27433

31-jul-06

5.81033

4.37776

7.7117

31-ago-06

5.85511

4.18393

8.1938

30-sep-06

5.76506

3.90121

8.51937

31-oct-06

5.69484

3.66455

8.84998

30-nov-06

5.71773

3.45942

9.45026

31-dic-06

5.61981

3.18803

9.90653

31-ene-07

5.54858

2.94018

10.471

28-feb-07

5.61286

2.79674

11.2646

31-mar-07

5.71811

2.6926

12.1432

30-abr-07

5.7906

5.23234

6.40843

31-may-07

5.86836

4.9797

6.91559

30-jun-07

5.94161

4.76513

7.40854

31-jul-07

5.9112

4.4963

7.77136

31-ago-07

5.91151

4.27169

8.18084

30-sep-07

5.8257

3.99434

8.4967

31-oct-07

5.7736

3.77042

8.84104

30-nov-07

5.8599

3.6061

9.52231

31-dic-07

5.78599

3.34625

10.0045

31-ene-08

5.70798

3.09116

10.54

29-feb-08

5.7822

2.95098

11.3297

31-mar-08

5.89675

2.84981

12.2014

30-abr-08

5.97834

5.42213

6.59161

31-may-08

6.05415

5.16786

7.09245

30-jun-08

6.13465

4.96007

7.58737

31-jul-08

6.12457

4.70326

7.9754

31-ago-08

6.16014

4.50263

8.42782

30-sep-08

6.10167

4.23852

8.78381

31-oct-08

6.06302

4.01648

9.15232

30-nov-08

6.15512

3.84757

9.8466

31-dic-08

6.07262

3.57464

10.3162

31-ene-09

5.97985

3.30173

10.8303

28-feb-09

6.06748

3.16207

11.6425

31-mar-09

6.1844

3.05649

12.5133

 

Authors:

Msc. Jesús Mesa Oramas

Msc. Luis Pérez Suárez

 

Partes: 1, 2
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