New Directions in Term Structure Modeling, SAFE, 2010

Do Interest Rate Options Contain Information about Excess Returns? by Caio Almeida, Jeremy Graveline and Scott Joslin Discussion by Anna Cie´slak University of Lugano Institute of Finance

June 29, 2010

c SAFE ( 2010 Anna Cie´slak)

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This paper... ⊲ This paper Factors

... is an important attempt to answer this question by estimating 3-factor ATSMs on swaps and interest rate caps jointly!

Predictability Conclusions

Thought-provoking results...  For premia, purely Gaussian models and stochastic vol (SV) models estimated on options deliver similar results, but... SV models estimated w/o options lag behind.  For options, different models give similar, relatively large, pricing errors.  For conditional yield volatilities, the best fit comes from a model estimated w/o options.

How to interpret and reconcile these findings?

c SAFE ( 2010 Anna Cie´slak)

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This paper

⊲ Factors Caps vs swaps IV vs pc’s Joint estimation OLS Pricing errors Intuition JK fit Predictability Conclusions

Factors in yields and caps Something in common?

c SAFE ( 2010 Anna Cie´slak)

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Caps versus swap zeros a. Caps implied vol

 In the last 2 decades, IVs have been high in recessions...

1y 2y 3y

60

% p.a.

50 40 30 20 10 0 1995

c SAFE ( 2010 Anna Cie´slak)

1997

1999

2001

2003

2005

2007

2009

4

Caps versus swap zeros a. Caps implied vol 1y 2y 3y

60 50 % p.a.

 In the last 2 decades, IVs have been high in recessions...  i.e. ... when Fed eases financial conditions to support real economy / fight unemployment.

40 30 20 10 0 1995

1997

1999

2001

2003

2005

2007

2009

b. Swap zero yields 8

6m 1y 2y 3y

% p.a.

6 4 2 0 1995

c SAFE ( 2010 Anna Cie´slak)

1997

1999

2001

2003

2005

2007

2009

4

Caps versus swap zeros R2 from regression of cap IVs on yields

 In the last 2 decades, IVs have been high in recessions...  i.e. ... when Fed eases financial conditions to support real economy / fight unemployment.

0.8 0.7

⇒ Let’s regress cap IVs on yields, one-by-one: IVtn

= β0 +

β1 ytm

+ ut

n = {1, 2, 3, 4, 5, 7, 10y}, m = {6m, 1y, ..., 10y}.

R2

0.6 0.5 0.4 0.3 0.2 10 10

7 5 cap maturity

c SAFE ( 2010 Anna Cie´slak)

8 4

3

6 2

4 1

2

yield maturity

0

4

Caps versus swap zeros R2 from regression of cap IVs on yields

 In the last 2 decades, IVs have been high in recessions...  i.e. ... when Fed eases financial conditions to support real economy / fight unemployment.

0.8 0.7

⇒ Let’s regress cap IVs on yields, one-by-one: IVtn

= β0 +

β1 ytm

+ ut

n = {1, 2, 3, 4, 5, 7, 10y}, m = {6m, 1y, ..., 10y}.

R2

0.6 0.5 0.4 0.3 0.2 10 10

7 5 cap maturity

8 4

3

6 2

4 1

2

yield maturity

0

Short-term yields explain up to 80% of variation in cap IVs across all cap maturities. The explanatory power of the yield curve for IVs declines with maturity of yields.

c SAFE ( 2010 Anna Cie´slak)

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Explaining variation in cap IVs with yield PCs Initially surprising result... that a 3-factor model can fit both yields and IVs T-stats and adj. R2 in a regression of cap IVs on yield PCs IVtn = β0 + β1′ P Ct + ut Cap maturity, n

1y

2y

3y

4y

5y

7y

10y

15.2 5.7 7.4

15.2 5.0 6.5

15.9 4.3 5.0

16.7 3.7 3.6

A. tstat (NW) pc1 pc2 pc3

13.2 12.0 6.4

14.7 9.1 8.8

15.0 6.6 8.3

B. Adj. R2 pc1 ∆ pc2 ∆ pc3

61% 18% 7%

66% 13% 10%

68% 11% 10%

69% 10% 9%

69% 9% 8%

71% 7% 6%

71% 6% 4%

Total

87%

89%

89%

87%

86%

84%

81%

...becomes clear from the perspective of this table: ⇒ Factors in yields (esp level) fit a majority of variation in cap IVs!

c SAFE ( 2010 Anna Cie´slak)

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Do we need the joint estimation? This paper Factors Caps vs swaps IV vs pc’s Joint estimation OLS Pricing errors Intuition JK fit

⊲

Predictability

For simplicity, let’s consider an OLS model for yields without imposing noarbitrage restrictions: ytm = am + b′m P Ct + eyt (1) The coefficients am and bm give the best linear fit in the sense of minimizing the mean cross-sectional squared error. Also, let’s neglect non-linearities in caps, and fit:

Conclusions

IVtn = cn + d′n P Ct + ect .

(2)

Q: How does this simple exercise compare to the performance of the complex ATSMs estimated using both swap zeros and caps?

c SAFE ( 2010 Anna Cie´slak)

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Caps IVs explained by pure yield curve factors Implied vol 3y cap

80

80

60

60 % p.a.

% p.a.

Implied vol 1y cap

40 20 0 95

40 20

97

99

01

03

05

07

0 95

09

97

99

50

50

40

40

30

30

20 10 0 95

03

05

07

09

07

09

Implied vol 10y cap

% p.a.

% p.a.

Implied vol 5y cap

01

20 10

97

99

01

03

05

07

09

0 95

97

99

01

03

05

Note: Observed, fitted ATM cap IVs, and 95% confidence bands. The IVs are fitted with the 3 first PC obtained from swap zero coupon curve with maturities 6m to 10y. All data is from Datastream.

c SAFE ( 2010 Anna Cie´slak)

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Caps IVs explained by pure yield curve factors Implied vol 3y cap

80

80

60

60 % p.a.

% p.a.

Implied vol 1y cap

40 20 0 95

40 20

97

99

01

03

05

07

0 95

09

97

99

50

50

40

40

30

30

20 10 0 95

03

05

07

09

Implied vol 10y cap

% p.a.

% p.a.

Implied vol 5y cap

01

observed fitted 95% CI

20 10

97

99

01

03

05

07

09

0 95

97

99

01

03

05

07

09

Note: Observed, fitted ATM cap IVs, and 95% confidence bands. The IVs are fitted with the 3 first PC obtained from swap zero coupon curve with maturities 6m to 10y. All data is from Datastream.

c SAFE ( 2010 Anna Cie´slak)

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Model performance... not so different Pricing errors in 3 factor models: Regressions vs ATSM τ

6m

1y

2y

3y

4y

5y

7y

10y

2.0 6.6

0.9 5.3

2.5 —

12.4% 9.0%

10.7% 8.3%

9.2% 8.9%

A. Yields (RMSE in bps) OLS Ao 2 (3) AGJ

2.4 6.8

3.9 9.3

2.0 —

1.5 4.5

2.1 6.2

B. Caps (RMSRE in %) OLS Ao 2 (3) AGJ

— —

22.7% 35.5%

17.6% 14.4%

15.2% 10.9%

13.6% 9.6%

Note: I compare results obtained from OLS regressions of yields and cap IVs on three yield PCs with those of the Ao 2 (3) model reported by Almeida, Graveline and Joslin (AGJ). AGJ estimation uses both yields and caps. Sample used for the regressions is weekly, 1995:01–2008:06. AGJ results are based on weekly data, 1995:01–2006:02.

My intuition...  OLS gives a similar fit to cap IVs as Ao2 (3) although it does not exploit any information in derivatives. Neither does great on this front...  Ao2 (3) trades off a bit of the fit of yields to improve on IVs.  This is how much a 3-factor model can do.

c SAFE ( 2010 Anna Cie´slak)

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More intuition This paper Factors Caps vs swaps IV vs pc’s Joint estimation OLS Pricing errors Intuition JK fit

Let’s consider A1 (3) model with state Xt = (X1t , X2t , Vt ) where Vt is the vol ′ state, and ytm = Am + Bm Xt . Conditional vol of a m-maturity yield becomes: m vart (yt+h )

⊲

=

′ ′ (Bm ⊗ Bm ) × vec [vart (Xt+h )]

(3)

=

b0 + b1 V t

(4)

Predictability Conclusions

But... In estimation w/o caps, Vt often becomes the level factor [misspecification, Feller condition].

Q: Does inclusion of caps change this rotation, i.e.

c SAFE ( 2010 Anna Cie´slak)

Vt is allocated correctly?

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Level in yields versus volatility, A1 (3) Volatility fit of the essentially affine A1 (3) model, Treasury curve 1970–2003 x 10-3

x 10-3 12 10 2-year

3-month

15 10 5

8 6 4

1970

1975

1980

1985

1990

1995

1970

2000

x 10-3

1975

1980

1985

1990

1995

2000

1980

1985

1990

1995

2000

1980

1985

1990

1995

2000

x 10-3 7 6 5-year

6-month

15 10 5

5 4 3

1970

1975

1980

1985

1990

1995

2000

1970

x 10-3 6 10-year

1-year

15 10 5 1970

1975

x 10-3

5 4 3

1975

1980

1985

1990

1995

2000

2 1970

1975

Source: Jacobs & Karoui, JFE, p.300. Note: Solid line shows the EGARCH(1,1) volatility estimates with AR(1) mean equation, dotted line depicts the conditional volatility implied by the essentially affine model A1 (3).

c SAFE ( 2010 Anna Cie´slak)

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Level in yields versus volatility, A1 (3) Volatility fit of the essentially affine A1 (3) model, swap curve 1991–2005 x 10-3

6

2.5

4

2

2-year

3-month

x 10-3

2 0 1991

1996

2001

1.5 1 1991

2005

-3

1996

2001

2005

1996

2001

2005

1996

2001

2005

-3

x 10

x 10

5

2.5

5-year

6-month

4 3 2

2 1.5

1 0 1991

1996

2001

1 1991

2005

x 10-3

3

2.5

2

2

10-year

1-year

x 10-3

1 0 1991

1996

2001

2005

1.5 1 1991

Source: Jacobs & Karoui, JFE, p.300. Note: Solid line shows the EGARCH(1,1) volatility estimates with AR(1) mean equation, dotted line depicts the conditional volatility implied by the essentially affine model A1 (3).

c SAFE ( 2010 Anna Cie´slak)

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This paper Factors

⊲ Predictability Predictability Conclusions

Predictability of excess returns Looking for benchmarks

c SAFE ( 2010 Anna Cie´slak)

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Predictability of excess bond returns rxnt+12m = α0 + α′ Factorst + εnt+12m Factor

rx2

rx3

rx4

rx5

rx6

rx7

rx8

rx9

rx10

A. Predictive R2 0. CP, fwd 10 1. Treasury premia, X1 , X2

43% 41%

47% 48%

49% 54%

50% 59%

51% 62%

51% 64%

50% 66%

49% 67%

47% 68%

2. Treasury vol, V11 3. IV cap 1y, 7y

17% 32%

16% 32%

14% 31%

12% 30%

11% 28%

9% 26%

8% 24%

7% 22%

5% 20%

57% 16%

62% 14%

66% 12%

68% 9%

70% 8%

71% 6%

71% 5%

71% 4%

72% 3%

0.8 8.7 0.7 4.5 -4.5

1.1 9.1 0.3 4.2 -4.2

1.5 9.3 0.0 3.9 -4.0

1.8 9.5 -0.3 3.6 -3.7

2.1 9.6 -0.5 3.2 -3.4

4. X, V, Cap (1.+2.+3.) Improvement 4.-1.

B. t-stat for regression 4 X1 X2 V11 IV cap 1y IV cap 7y

-0.5 5.4 2.1 3.2 -4.6

-0.4 6.4 1.9 4.1 -5.0

-0.1 7.4 1.6 4.6 -5.0

0.4 8.2 1.2 4.7 -4.8

Note: We report predictive regressions of 1-year holding period bond returns on swap zeros, sample 1995:01–2008:06. Panel A. provides adj. R2 obtained from various regressions: 0. Cochrane-Piazzesi factor, 1. Term premia factors from Treasuries (Cieslak&Povala, 2010), 2. Treasury long-maturity volatility (Cieslak&Povala, 2009), 3. Cap IVs with maturities 1y and 7y. Panel B. reports t-stats when all regressors are used jointly, in regression 4.

c SAFE ( 2010 Anna Cie´slak)

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Predictability of excess bond returns rxnt+12m = α0 + α′ Factorst + εnt+12m Factor

rx2

rx3

rx4

rx5

rx6

rx7

rx8

rx9

rx10

A. Predictive R2 0. CP, fwd 10 1. Treasury premia, X1 , X2

43% 41%

47% 48%

49% 54%

50% 59%

51% 62%

51% 64%

50% 66%

49% 67%

47% 68%

2. Treasury vol, V11 3. IV cap 1y, 7y

17% 32%

16% 32%

14% 31%

12% 30%

11% 28%

9% 26%

8% 24%

7% 22%

5% 20%

57% 16%

62% 14%

66% 12%

68% 9%

70% 8%

71% 6%

71% 5%

71% 4%

72% 3%

4. X, V, Cap (1.+2.+3.) Improvement 4. over 1.

 Benchmark... Vast predictability (> 60%) comes from factors within the yield curve, i.e. spanned, no derivatives are needed [Cieslak & Povala, 2010].  Extra... Vol factors (unspanned) and cap IVs add most predictability at short bond maturities. ⇒ But... your estimated models seem to do the opposite, i.e. improve on the long end [Table 6 and 7]. ⇒ ... and imply a much lower degree of predictability than reported here [after accounting for the holding period horizon]. c SAFE ( 2010 Anna Cie´slak)

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This paper Factors Predictability

⊲ Conclusions Sum up

Conclusions

c SAFE ( 2010 Anna Cie´slak)

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Putting things together This paper Factors Predictability Conclusions Sum up

⊲

My simple diagnostics could suggest that:  The estimated models have difficulties in identifying volatility states correctly, but options help to find the right direction.  There is little statistically significant difference for explaining term premia within a 3-factor model with and w/o options data.  Providing orthogonal information on vols (e.g. from straddles) would help identify the volatility states and the marginal contribution of derivatives for premia. I would like to see more on:  ... the model implied states, esp those driving vols in A1 (3), A2 (3) estimation with and w/o options.  ... evidence that derivatives are indeed key to improvement of these models on the premia front. Overall... An interesting project that can enhance our understanding of both ATSMs, and vols and premia in bonds.

c SAFE ( 2010 Anna Cie´slak)

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