Threshold Cointegration and Credit Dynamics

1. Threshold Cointegration and Credit Dynamics
Nov 2003, Ames, Iowa
Pin Chung

2. Introduction
Kao (2000)
–high-yield bond debacle (Junk bonds).
–large derivative losses (LTCM, 1998).
–a global credit/liquidity crisis (Argentina, 1999 to
2001).
Credit Risk Pricing Models
–1959 to 1992: 5 papers.
–1993 to 1999: more than 10 papers.
–1998 & 1999: more than 30 papers.
2

3. Introduction (continued)
Wilson & Jones (1990), Chang & Huang (1990), Adrangi &
Ghazanfari (1996/1997):
-seasonality, e.g., the January effect and the weekday effect;
-defy the market efficiency hypothesis.
Collin-Dufresne, Goldstein & Martin (1999):
-hedge funds are sensitive to changes in the credit spread;
-a common factor to explain the variation;
-gain from studying individual bonds is limited.
Pedrosa & Roll (1998):
-credit spread risks are non-diversifiable;
-credit spreads of indices are affected by some common factors.
3

4. The Corporate Debt Pricing Models
Segmentation Model:
Fisher (1959), Silvers (1973), Boardman & McEnally (1981).
Market Yield Premium Model:
Fons (1987), Altman & Bencivenga (1995).
Yield Spread Model:
Fridson & Jonsson (1995), Garman & Fridson (1996).
Yield Premium and Yield Spread model:
Barnhill, Joutz, & Maxwell (2000).
4

5. Neal, Rolph and Morris (2000, NRM)
• Johansen’ (1988, 1991) cointegration approach.
s
• U.S. government rates are cointegrated with corporate
rates.
• Time horizon dictates the dynamic relationships
between credit spreads and Treasury rates.
• Asymmetric Results:
– Short-run: Treasury rate ↑ è credit spread to narrow.
– Long-run: Treasury rate ↑ è credit spreads to widen.
– Corporate bonds more sensitive to interest rate movements.
– Time varying correlation between credit spreads and
interest rates. 5

6. Discontinuous Nonlinear Asymmetric
Adjustment Process
Main idea:
Asymmetric è specification error.
Why use threshold cointegration methods:
Capture the asymmetric behaviors.
We have conducted the tests:
Q1: Are interest rates cointegrated?
Q2: If rates are cointegrated, then:
1. Test same speeds of adjustment in a two-regime environment.
2. In a three-regime environment:
• a random walk process when rates are inside the band;
• a mean-reverting process to the equilibrium band, possibly with
different adjustment speeds.
6

7. OUR GOAL
•Offer non-linear, discontinuous, asymmetric
alternatives to the traditional linear,
continuous and symmetric approach.
•Examine credit spread dynamics for different
maturities and different investment grades.
•Identify the equilibrium adjustment process.
•Perform out-of-sample forecast performance
evaluations: symmetric vs. asymmetric.
7

8. Economic implications
• to study real economic activities.
• to price portfolios.
• to find the value of a firm.
• to price credit derivatives, improve credit management
quality.
• to apply to risk management and hedging activities.
• to price different financial instruments.
• to improve the adequacy of reserves held by banks and
insurance companies.
• to improve the profitability of trading, the accuracy of
current asset pricing and option pricing models. 8

9. Empirical studies: Threshold Cointegration
• Ghosh (1993), Brenner and Kroner (1995):
-futures and spot prices.
• Martens, Kofman and Vorst (1998):
-index-futures trading strategies.
• Balke and Wohar (1998):
-covered interest parity.
• Goodwin and Holt (1999):
-price linkages of U.S. beef market.
9

10. Models in this paper:
Asymmetric Threshold Cointegration Models:
-Lo and Zivot (2001)
-Hansen and Seo (2002)
-Enders and Siklos (2001)
Symmetric Cointegration Model:
-Engle and Granger (1987)
-Neal, Rolph and Morris (2000)
10

11. DATA
• Monthly data, 1/1960 to 12/1997, 456 observations.
– 10-year constant maturity Treasury note (Tsy).
– Ibbotson Bond Index for 20-year Treasury bond (Ibb).
– Moody’ Aaa Bond Index (Aaa).
s
– Moody’ Baa Bond Index (Baa).
s
• Table 4.1: Tsy, Ibb, Aaa, Baa.
• Table 4.2: high autocorrelations è (near) unit root
process.
11

12. %
0
3
6
9
12
15
18
21
Jan-60
Jan-63
Jan-66
Jan-69
Jan-72
Jan-75
Jan-78
Jan-81
Jan-84
Tsy-Ibb-Aaa-Baa
Jan-87
DATA (continued)
Jan-90
Jan-93
Jan-96
Ibb
Tsy
Baa
Aaa
12

13. DATA (continued)
• Four yield spreads:
– (Aaa-Tsy), (Aaa-Ibb), (Baa-Tsy), (Baa-Ibb).
• 5 unit root tests (Tables 4.3 to 4.6):
– Dickey-Fuller, TAR, M-TAR, C-TAR, and M-C TAR.
• Dickey-Fuller test:
– reject the null of a unit root process at 5% ( for all pairs).
• Symmetric Adjustment Speeds???:
– Aaa-Tsy: M-TAR, C-TAR @ 10%; M-C TAR @ 5%.
– Aaa-Ibb: M-C TAR @ 5%.
– Baa-Tsy: C-TAR @ 10%; M-TAR and M-C TAR @ 5%.
– Baa-Ibb: M-C TAR @ 21%.
13

14. Results from Lo-Zivot Model
•Tables 5.1.1 5.1.2: TVECM(3) with lag =
1, 2 against the null of Linear VECM.
•The null of no threshold effects:
–cannot be rejected for (Aaa, Ibb) and (Baa, Ibb)
pairs for either lag length.
•The null of no threshold effects:
–can be rejected for (Aaa, Tsy) and (Baa, Tsy)
pairs with lag = 2.
14

15. Results from Lo-Zivot Model (continued)
• Lag = 1 in levels:
– All long rate eqs (Aaa or Baa) in the 3rd regime have the
expected signs (negative) for their error-correction terms;
– not so noteworthy for the short rate (Tsy or Ibb) equations.
• Lag = 2 in levels:
– all long rate eqs (Aaa or Baa) in the 3rd regime have the
expected signs (negative) for their error-correction terms;
– for the short rate (Tsy or Ibb) equations only one
coefficient violates the negative sign expectation.
15

16. Results from Hansen-Seo Model
•Four pairs of interest rates.
•Lag = 1, 2 in levels.
•β = 1 or β is estimated from the model.
•(Aaa, Tsy): strong TC relationship.
•(Baa, Tsy): TC effect if β is estimated.
•(Aaa, Ibb) and (Baa, Ibb): less TC effect.
16

17. Results from Hansen-Seo Model (continued)
• If β is estimated from the model:
– six of eight β estimates are greater than unity, only two are
less than unity (Aaa, Ibb pair).
– in contrast to conventional assumption.
• 1st (2nd) regime is “typical”or “extreme”regime:
– depends on interest rate pairs, lag length and how we specify
the β value.
• Sensitivity Tests:
– stable estimation results.
– 100 vs. 300 grid points è AIC criterion improves.
– 1000 vs. 5000 replications è moderate changes of p-values.
17

18. Results from Enders-Siklos model
• Relax the restriction: [1,-1].
• Tables 5.3.1 to 5.3.4 report 5 cointegration tests:
– Linear: Engle-Granger.
– Nonlinear: TAR, M-TAR, C-TAR, M-C TAR.
• Engle-Granger test:
– reject the null of unit root process.
– cointegration relationship with symmetric adjustment.
• Asymmetric adjustment @ 5% significance level:
– (Aaa, Ibb): M-C TAR.
– (Aaa, Tsy): C-TAR, M-C TAR.
– (Baa, Ibb): C-TAR.
– (Baa, Tsy): M-C TAR. 18

19. Results from Enders-Siklos model (continued)
•M-C TAR provides stronger evidence of
asymmetric behavior than the M-TAR.
•Similar observation for C-TAR vs. TAR.
•Expect smaller AIC and BIC under M-C TAR:
–(Aaa, Tsy) and (Aaa, Ibb): Yes.
–(Baa, Tsy) and (Baa, Ibb): TAR.
19

20. Results from Enders-Siklos model (continued)
Error-Correction model: (Aaa, Tsy)
• Enders-Siklos M-C TVECM:
1. +1 unit deviation: Tsy 0.28%, Aaa 3.91%.
2. -1 unit deviation: Tsy 0.99%, Aaa 14.31%.
3. Stronger adjustment when Aaa rates wander
away under the negative change environment.
• Engle-Granger Linear VECM:
1. Symmetric adjustment.
2. Tsy 0.46%, Aaa 6.60%.
20

21. Forecasting Performance Evaluation
• One-step-ahead to six-step-ahead forecasts.
• (Aaa, Tsy) and (Baa, Tsy): 7 competing models
– (2 Lo-Zivot, 2 Hansen-Seo, Enders-Siklos, Engle-Granger, NRM)
• (Aaa, Ibb) and (Baa, Ibb): 6 competing models
– (2 Lo-Zivot, 2 Hansen-Seo, Enders-Siklos, Engle-Granger)
• Forecast period: 01/1998 to 12/2002 (60 months)
• Six accuracy measures:
– ME, EV, RMSE, RMSPE, MAE, MAPE.
21

22. Forecasting Performance Evaluation (continued)
(Aaa, Tsy):
• 7 under-estimate Aaa: 1.23 to 5.40 bp (me > 0).
• 6 over-estimate Tsy: 2.43 to 23.84 bp (me < 0).
– Lo-Zivot (lag=2) under-estimate Tsy 40.40 bp.
• 1-step-ahead:
– Enders-Siklos M-C TAR (rmse, rmpse, mae, mape).
– Under-estimate Aaa 4.63 bp.
– Over-estimate Tsy 2.51 bp.
• 6-step-ahead:
– NRM (ev, rmse), Engle-Granger (rmpse, mae), Hansen-
Seo (β=1) (mape).
22

23. Forecasting Performance Evaluation (continued)
(Baa, Tsy):
• 7 under-estimate Baa: 2.66 to 5.44 bp.
• 5 over-estimate Tsy: 1.83 to 4.59 bp.
– 2 L-Z under-estimate Tsy 1.08 to 10.59 bp.
• 1-step-ahead:
– Hansen-Seo (β=1) (rmse, rmpse, mae, mape).
– Under-estimate Baa 3.60 bp.
– Over-estimate Tsy 1.83 bp.
• 6-step-ahead:
– Hansen-Seo (β=1) (ev, rmse, rmpse, mae, mape)
– Under-estimate Baa 15.69 bp.
– Over-estimate Tsy 21.38 bp. 23

24. Forecasting Performance Evaluation (continued)
(Aaa, Ibb):
• 6 under-estimate Aaa: 1.77 to 14.21 bp.
• 4 over-estimate Ibb: 3.82 to 7.06 bp.
– 2 L-Z under-estimate Ibb 22.48 to 9.16 bp.
• 1-step-ahead:
– Hansen-Seo (β estimated) (ev, rmse, rmpse, mae, mape).
– Under-estimate Aaa 3.69 bp.
– Over-estimate Ibb 3.82 bp.
• 6-step-ahead:
– Hansen-Seo (β estimated) (ev, rmse, rmpse, mae, mape).
– Under-estimate Aaa 7.54 bp.
– Over-estimate Ibb 20.13 bp. 24

25. Forecasting Performance Evaluation (continued)
(Baa, Ibb):
• 5 under-estimate Baa: 1.06 to 5.07 bp.
– Lo-Zivot (lag=1) over-estimate Baa 1.30 bp.
• 4 over-estimate Ibb: 0.72 to 4.89 bp.
– 2 L-Z under-estimate Ibb 3.07 to 0.31 bp.
• 1-step-ahead:
– Lo-Zivot (lag=1) (ev, rmse, rmpse, mae, mape).
– Over-estimate Baa 1.30 bp.
– Under-estimate Ibb 3.07 bp.
• 6-step-ahead:
– Hansen-Seo (β=1) (ev, rmse, rmpse, mae, mape)
– Under-estimate Baa 12.95 bp.
25
– Over-estimate Ibb 6.22 bp.

26. Forecasting Performance Evaluation (continued)
Lo-Zivot with non-unity cointegrating vector:
• (Aaa, Tsy): [1,-1.028](NRW), [1,-1.039](Hansen-Seo)
• (Baa, Tsy): [1,-1.178](NRW), [1,-1.108](Hansen-Seo)
• (Aaa, Ibb): [1,-0.981] (Hansen-Seo)
• (Baa, Ibb): [1,-1.385] (Hansen-Seo)
• 1-step-ahead forecast:
– same leading models
• 6-step-ahead forecast:
– (Baa, Tsy) [1,-1.178] (NRW) (rmse, rmpse, mae, mape)
26

27. Aaa-Actual
1-step-ahead Aaa
Enders-Siklos lag=2 Aaa vs Tsy T sy-Actual
1-step-ahead T sy
10
8
6
%
4
2
0
Jan-98 Jan-99 Jan-00 Jan-01 Jan-02
27

28. Baa-Actual
1-step-ahead Baa
Hansen-Seo Fxd1 Baa vs Tsy Tsy-Actual
1-step-ahead T sy
10
8
6
%
4
2
0
Jan-98 Jan-99 Jan-00 Jan-01 Jan-02
28

29. Aaa-Actual
1-step-ahead Aaa
Hansen-Seo CI1 Aaa vs Ibb Ibb-Actual
1-step-ahead Ibb
10
8
6
%
4
2
0
Jan-98 Jan-99 Jan-00 Jan-01 Jan-02
29

30. Baa-Actual
1-step-ahead Baa
Lo-Zivot lag=1 Baa vs Ibb Ibb-Actual
1-step-ahead Ibb
10
8
6
%
4
2
0
Jan-98 Jan-99 Jan-00 Jan-01 Jan-02
30

31. Forecasting Performance Evaluation (continued)
• 1-step-ahead forecast:
– under-estimate long rate (Aaa, Baa).
– over-estimate short rate (Tsy, Ibb).
– threshold cointegration models perform better than linear
cointegration models.
• None of the threshold cointegration models dictates
the overall performance.
• Some gains by incorporating the non-zero
cointegrating vector into the Lo-Zivot specification.
31

32. Aaa
Forecast 2003: Aaa vs. Tsy (E-S) Tsy
Aaa-For
10 Tsy-For
9
8
7
6
5
%
4
3
2
1
-
J-03 F-03 M-03 A-03 M-03 J-03 J-03 A-03 S-03 O-03
32

33. Baa
Forecast 2003: Baa vs. Tsy (H-S) Tsy
Baa-For
10 Tsy-For
9
8
7
6
5
%
4
3
2
1
-
J-03 F-03 M-03 A-03 M-03 J-03 J-03 A-03 S-03 O-03
33

34. Aaa
Forecast 2003: Aaa vs. Ibb (H-S) Ibb
Aaa-For
10 Ibb-For
9
8
7
6
5
%
4
3
2
1
-
J-03 F-03 M-03 A-03 M-03 J-03 J-03 A-03 S-03 O-03
34

35. Baa
Forecast 2003: Baa vs. Ibb (L-Z) Ibb
Baa-For
10 Ibb-For
9
8
7
6
5
%
4
3
2
1
-
J-03 F-03 M-03 A-03 M-03 J-03 J-03 A-03 S-03 O-03
35

36. Conclusions
•Major findings:
–exist long run equilibrium relationships.
–all interest rates pairs follow the threshold
cointegration behavior.
–spreads are stationary.
–the speeds of adjustment are asymmetric.
–the threshold estimates are asymmetric in a
three-regime environment.
36

37. Conclusions (continued)
•Major findings (continued):
–1% ↑ in Treasury rates (Tsy or Ibb)
è more than 1% ↑ in corporate bond indices.
–the Baa bond index is more sensitive.
–above findings are coherent with NRW(2000)
but inconsistent with the view that increased
credit risk will make corporate bonds less
interest rate sensitive.
37

38. Conclusions (continued)
•Major findings (continued):
–no one particular threshold cointegration model
dictates the overall forecasting accuracy.
–for different interest rates pairs, different
threshold cointegration model offers a better fit.
–linear cointegration models perform relatively
less accurate than the threshold cointegration
models.
38

39. Conclusions (continued)
•Future work (theoretical):
–allow for two threshold variables in the Enders-
Siklos model (a three-regime setting).
–allow for estimating cointegrating vector, delay
variable and threshold variables simultaneously.
–allow for at least three variables in the threshold
cointegration model (i.e., multiple cointegrating
vectors.)
–develop a distribution theory for the parameter
estimates.
39

40. Conclusions (continued)
•Future work (empirical):
–extend the model to have multiple corporate
bond indices.
–include some other macroeconomic variables in
the setting to control for economic evolution.
–incorporate some other variables like, liquidity
risk, default risk, the expected loss in the event
of default to model the yield on risky debts.
40

41. ACKNOWLEDGEMENTS
•Dr. Barry Falk
•Dr. Harvey Lapan
•Dr. John Schroeter
•Dr. Dermot Hayes
•Dr. Rick Dark
41

42. Appendix
42