Yield curve factors dynamics: A note Anna Cieslak∗ First version: November 2, 2009; last update: February 2010
We study the relation between level and slope of the yield curve over time. Based on statistical and economic grounds, we conclude that the changing correlation of these factors is not spurious. A simulation exercise discards the possibility that real data lies just an epsilon away from a Gaussian world. We reconcile the changing correlation of factors with the apparent stability of their loadings (eigenvectors). Tiny—by the eyeball metric—perturbations of eigenvectors can easily translate into significant correlation of initially orthogonal factors.
∗
Anna Cieslak is at the University of Lugano:
[email protected], University of Lugano, Institute of Finance, Via Buffi 13a,
6900 Lugano, Switzerland. This note was written when Cieslak was visiting the University of Chicago Booth School of Business, generously made possible by the Swiss National Science Foundation.
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1. Level, slope and (maybe) curvature do comove If a standard PCA gives an incomplete description of yields, this fact should be reflected through the time-varying comovement of factors it implies. We use loadings from the unconditional PCA of yields to construct the level, slope and curvature tick-by-tick. With high-frequency observations at hand, we are able estimate the realized correlation of factors. To fend off concerns that splining could distort our findings, we select only actively traded maturities (two, three, five and ten years). Here, the construction of the tick-by-tick zero curve is most robust. The result is plotted in Figure 1 together with the 90% confidence band.1 Factor correlations move around persistently over time. Importantly, this conclusion survives under several definitions of factors: While different specifications do influence the average level of correlations, be it simple portfolios, immunized portfolios or unconditional PCA loadings, the persistence remains intact. Judging by the confidence bands, we tend to give credit to the time varying comovement between level and slope. Their correlation oscillates between ±50%, and is generally lower during periods of monetary easings than it is during tightening cycles. Figure 2 superimposes ρt (lvl, slo) against Fed cycles, and shows that reversals in correlation often coincide with changes of the Fed regime. Using high-frequency data is like taking a microscopic view of the world. Whether this perspective allows to see more over and above a bird’s eye view is a legitimate question. Let us then describe what we see. In Figures 3 and 4, we look at yields, yield factors, and their correlations across four interesting periods in our sample. Figure 3 compares two widely-discussed tightening episodes: 1994:02–1995:01 and 2004:06–2005:06. Figure 4, instead, looks at recent easing environments: 2002:04–2003:09, i.e. the final stage of the easy money in the aftermath of the 2001 recession, and the 2007:01–2007:12 runway for the recent crisis and the zero-rate policy. ρt captures fine alterations in the level-slope link that cannot be gauged by looking at yields and factors alone. The different dynamics of ρt in 1994/95 versus 2004/05 could reflect the changing strength of monetary transmission in those periods. In particular, meandering around zero through 2004/05, ρt appears to align with the confession of Mr. Greenspan that the Fed lost control over the long-end of the curve in that period.2 Also, the 2002/03 and 2007 events are characterized by a markedly different 1
To compute the confidence bands around realized correlation estimates, we use the result derived in BarndorffNielsen and Shephard (2004). The asymptotic distribution of the realized correlation estimator is based on N → ∞ where N is the number of observations within a fixed interval (t, t + h). We define h = 1 week which translates into N = 288 for ten-minute sampling. Choosing weekly interval improves the precision of the realized correlation estimate. For expressions see Barndorff-Nielsen and Shephard (2004), equations (29), (35) and (40).
2
See his March 11, 2009 Wall Street Journal article: “The Fed Did Not Cause the Housing Bubble.”
2
a. Level & slope ρ(lvl,slo)
0.5 0 −0.5 1992
1994
1996
1998 2000 2002 b. Level & curvature
2004
2006
2008
1994
1996
1998 2000 2002 c. Slope & curvature
2004
2006
2008
1994
1996
1998
2004
2006
2008
ρ(lvl,cur)
0.5 0 −0.5 1992
ρ(slo,cur)
0.5 0 −0.5 1992
2000
2002
Fig. 1. Correlation of yield curve factors The figure plots realized correlations between yield curve factors constructed from the high-frequency zero curve. Level, slope and curvature factor loadings are obtained from the PCA decomposition of the unconditional covariance matrix of yields with maturities: two, three, five and ten years. Dashed lines mark the 90% confidence intervals (Barndorff-Nielsen and Shephard, 2004).
correlation dynamics. The exit from a recession in 2002/03 marks the most pronounced correlation rally, while the entry into the recession in 2007 is associated with its most abrupt decline in our sample period.
2. Assume level and slope do comove. Then? ρt (lvl, slo) is a complex object that aggregates different scenarios for the yield curve into one number. That this variable happens to move over time should have a meaning.3 An increasing
3
As inspiration for our intuition, the Cochrane-Piazzesi factor works as a clever aggregator of several variables into a single number that—by separating wheat from chaff—is able to predict premiums. Yet, its exact identity remains unknown.
3
hike
0
−0.5
nochg
Fed regime
ρ(lvl,slo)
0.5
cut 92
94
96
98
00
02
04
06
08
Fig. 2. Correlation of level and slope across easing and tightening cycles The figure plots daily realized correlations between level and slope constructed from the high-frequency zero curve, and smoothed over three months. The grey line shows Fed regimes: hike, no change or cut. The cycles are regarded as easing or tightening if at least three subsequent moves in the Fed funds target were in the respective direction. We use daily data to account for the exact timing of changes of the Fed target.
B. Jun 04 — Jun 05
A. Feb 94 — Jan 95 a1. Yields 2Y 3Y 5Y 10Y
6 4 2 01/94
05
08
b1. Yields
8
11
6 4 2 06/04
01/95
09
a2. Level & slope
08
11
16 14 12 10 8 6 06/04
0 01/95
1
09
a3. ρt (lvl, slo) 0.5
0
0
05
08
12
03/05
0 06
03/05
06
b3. ρt (lvl, slo)
0.5
−0.5 01/94
06 2
lvl
slo
1
05
03/05
b2. Level & slope 2
lvl
16 14 12 10 8 6 01/94
12
slo
8
11
−0.5 06/04
01/95
09
12
Fig. 3. Yields, factors, and realized correlations: 1994/1995 and 2004/2005 Left-hand panels (A) cover the period from February 1994 to January 1995. Right-hand panels (B) cover the period from June 2004 to June 2005. We plot yields with maturities two, three, five and ten years (panels a1, b1), the dynamics of level and slope (panels a2, b2), and the realized correlation smoothed over one-month window (panels a3, b3). Factor loadings are obtained from the unconditional covariance matrix, and kept constant across panels.
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C. Apr 02 — Sep 03
D. Jan 07 — Dec 07
c1. Yields
6
2Y 3Y 4 5Y 10Y 2
4 2 07
10
01/03
04
07
09
01/07
04
c2. Level & slope
07
10
12
d2. Level & slope 3
8
2
8
2
6 4 04/02
lvl
10 slo
3
lvl
10
6
1 07
10
01/03
04
07
4 01/07
09
1 04
c3. ρt (lvl, slo) 0.5
0
0
07
10
01/03
07
10
12
10
12
d3. ρt (lvl, slo)
0.5
−0.5 04/02
slo
04/02
d1. Yields
6
04
07
09
−0.5 01/07
04
07
Fig. 4. Yields, factors, and realized correlations: 2002/2003 and 2007 Left-hand panels (C) cover the period from April 2002 to September 2003. Right-hand panels (D) cover the period from January 2007 to December 2007. We plot yields with maturities two, three, five and ten years (panels c1, d1), the dynamics of level and slope (panels c2, d2), and the realized correlation smoothed over one-month window (panels c3, d3). Factor loadings are obtained from the unconditional covariance matrix, and kept constant across panels.
correlation involves at least two scenarios: (i) average yield curve increases and becomes steeper (long end moves up faster) or (ii) average yield curve declines and becomes flatter (long end moves down faster). Conversely, a decreasing correlation means that (iii) average curve increases and becomes flatter (short end moves up faster), or (iv) average curve declines and becomes steeper (short end moves down faster). Accordingly, persistent changes in correlations reflect the relative speed of adjustments along the curve. Having said that, we have more questions than answers as to what it really means. Are the movements we observe interesting after all? Is it the magnitude or the direction of ρt that matters, or maybe both? With the origins of this variable in mind: should we pay attention to its distance from zero or is it just the dynamics that play a role? Different methods of construction can (obviously) change the unconditional level of ρt but do not affect the dynamics—this would suggest that the latter matters more.
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Allowing some speculation, we would expect ρt (lvl, slo) to be related to the conditional (at time t) configuration of forces in the economy, e.g. constellations of the kind: strength of growth—danger of inflation. As such, it could be informative about the premia. Let us ignore for a second the confidence bands on ρt , and run the Cochrane-Piazzesi forecasting regressions (see Table 1). This is not a horse-race but rather a first check as to whether ρt is worthy of attention. The regressions reveal a systematic pattern.
ρt explains a significant portion (up to 20%)
of variation in premia across maturities, and is most significant at the short and intermediate maturities up to five years. These results show very little sensitivity to how we smooth the data, and whether the smoothing window is one or two months. The presence of the Cochrane-Piazzesi factor does not affect ρt loadings in a significant manner. Regression coefficients have consistently negative signs, but the effect of ρt withers at longer maturities. Figure 5 superimposes the time series of excess returns rxτt,t+1Y against ρt for different maturities (both variables are standardized). Returns tend to be high when the correlation factor drifts low. We are moderately excited about these results. Indeed, the level-slope interactions tend to carry economic content that is important for premia and at the same time is not spanned by the Cochrane-Piazzesi factor. We take it as evidence that ρt is not spurious. However, since ρt is most relevant for bonds with shorter durations, this moderates our excitement: We tend to think that a significant amount of premia is located at longer segments of the curve. As such, ρt captures only a particular part of their dynamics. More still, good t-stats and R2 ’s do not succumb our need for deeper understanding. While we have a “feeling” for the regression results, to avoid speculation, we refrain from their closer interpretation at this stage.
3. If the world were Gaussian how much would realized second moments wander around? While we do believe in the high-frequency theory of Jacod (1994) and Barndorff-Nielsen and Shephard (2004), and test against the role of microstructure noise in our data, the Gaussian world gives a valid reference point for assessing the robustness of our findings. We fit a standard three-factor Gaussian model to yields with maturities six months and two, three, five, seven and ten years using ML and the Kalman filter. With parameter estimates at hand, we simulate 20 years of 10-minute yields, and compute daily, weekly and 4-week rolling
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Table 1. Regressions of bond excess returns on ρt (lvl, slo) Panel a reports the regression coefficients, t-statistics and adjusted R2 ’s from regressions of one-year excess holding period bond returns on the level-slope correlation factor ρ(lvl,slo). In panel b, the regression is augmented with the Cochrane-Piazzesi (CP) factor. We construct the monthly correlation factor as the four-week average of the realized correlation between the level and slope. Level and slope are computed using unconditional PCA loadings. The CP factor is obtained from the GSW data set using 9 forward rates (using FB data does not alter the upshot). rx denotes the average excess return across all maturities. Data is monthly, and covers the 1992:01–2007:12 period. T-statistics use Newey-West adjustment with 12 lags. All variables have been standardized.
rx2
rx3
rx4
rx5
rx6
rx7
rx8
rx9
rx10
rx
a. Level-slope interaction factor, ρt (lvl, slo) ρ(lvl,slo)
-0.45 (-3.43)
-0.42 (-3.19)
-0.39 (-2.92)
-0.36 (-2.66)
-0.33 (-2.42)
-0.31 (-2.18)
-0.28 (-1.96)
-0.25 (-1.76)
-0.23 (-1.57)
-0.31 (-2.22)
Adj. R2
0.20
0.17
0.15
0.13
0.11
0.09
0.07
0.06
0.05
0.09
b. ... plus Cochrane-Piazzesi factor ρ(lvl,slo) CP Adj. R2
-0.43 (-3.19) 0.20 ( 1.35)
-0.39 (-3.02) 0.28 ( 1.92)
-0.36 (-2.80) 0.34 ( 2.40)
-0.32 (-2.57) 0.39 ( 2.84)
-0.29 (-2.32) 0.44 ( 3.27)
-0.26 (-2.08) 0.47 ( 3.69)
-0.23 (-1.85) 0.50 ( 4.08)
-0.20 (-1.62) 0.53 ( 4.44)
-0.18 (-1.41) 0.55 ( 4.75)
-0.27 (-2.11) 0.47 ( 3.63)
0.23
0.25
0.26
0.28
0.29
0.31
0.32
0.34
0.35
0.31
realized volatilities (RV’s). Figure 6 compares our empirical estimates with the Gaussian ones. Not surprisingly, Gaussian RV stays—but for some noise—constant at the unconditional level. On average, it is 10 times less volatile than the empirical proxy (mean standard deviation across maturities of 3.7 bps against 36 bps observed in the data). In previous sections we have cheered at the persistence of ρt as a signal for its economic content. Could the persistence be just an artefact of aggregation and smoothing? To answer this question, we replicate the empirical ρt dynamics of Figure 2 within a Gaussian experiment, i.e. we compute daily realized correlations and smooth them over the window of 66 days. Figure 7 plots the series generated in simulation. It is comforting that the correlations move in a narrow band of ±5% around zero, that is 10-times tighter than the one obtained empirically.
4. Eigenvectors seem not to move too much But what does it mean “much”? To answer this question, we launch a simple simulated example and then discuss real-data evidence. We conclude that the “eyeball metric” can be misleading in answering this question.
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a. 2Y bond rx ρ(lvl,slo)
2 0 −2 1992
1994
1996
1998
2000 2002 b. 3Y bond
2004
2006
2008
1994
1996
1998 2000 2002 c. 10Y bond
2004
2006
2008
1994
1996
1998
2004
2006
2008
2 0 −2 1992
2 0 −2 1992
2000
2002
Fig. 5. Excess bond returns and the level-slope correlation factor We plot one-year holding period excess bond returns rxt,t+1Y against the level-correlation factor measured at time t, ρt (lvl, slo). Both variables are standardized. The time axis represents time t.
Let us consider again the three-factor Gaussian model, and extract principal components from yields with maturities two, five and ten years: pct = Q′ yt . yt is a 3 × 1 vector of yields observed at time t, and Q is a 3 × 3 matrix of orthogonal eigenvectors (factor loadings), QQ′ = I. Now, ˜ by slightly perturbing Q, so that the interpretation of loadings does not change, obtain matrix Q ˜Q ˜ ′ = I is preserved. Compute perturbed factors: pc ˜ ′ yt . What does the and the property Q ˜t = Q ˜ imply for the correlation of pc small change in Q ˜ t ’s? For illustration, we consider the following eigenvector matrices:
−0.6557 −0.6752
Q = −0.5710 −0.4940
0.3379
0.1507 −0.8070 0.7220 0.4844
−0.5766 −0.7230
˜= Q −0.6101 −0.5434
8
0.3806
0.0712 −0.7891 . 0.6872 0.4822
(1)
250
a1. RV in Gaussian world
200
200
150
150
100
100
50 0
b1. RV in real world
250
2Y 10Y
2Y 10Y
50 200
400 600 Time
800
1000
92 94 96 98 00 02 04 06 08
a2. Density of Gaussian RV 0.12
b2. Density of real RV 0.02
2Y 10Y
0.1
2Y 10Y
0.015
0.08 0.01
0.06 0.04
0.005 0.02 0 0
50
100 150 200 RV (bps p.a.)
0 0
250
50
100 150 200 RV (bps p.a.)
250
Fig. 6. Realized volatility in the true and Gaussian worlds The figure compares empirical realized volatilities (RV) to the ones obtained from a Gaussian model. Panels a1 and b1 show the dynamics of weekly RV smoothed over 4 weeks. Panel a2 and b2 plot the kernel density of the respective weekly RVs. Numbers have been annualized and are expressed in basis points.
Figure 8, panel a, plots the columns of both matrices. A glance at the figure suggests that the interpretation of factors does not change: Without prior knowledge, had one asked us to pick the true Q, we would need to flip a coin to decide. However, panels b and c show that an innocuous shift in Q can have significant consequences for factor correlation. A regression of the simulated level on the slope makes this point clear. Indeed, with an R2 of 27% and a significant β coefficient, ˜ t and slo ˜ t are independent. Appendix A discusses a toy argument for it is hard to argue that lvl why this can be the case. The following empirical exercise confirms intuition from the Gaussian model. Let Q denote loadings from the unconditional PCA of weekly yields (1992:01–2007:12). To proxy for the empirically ˜ i ’s by decomposing the conditional (realized) valid perturbations of Q, we construct an array of Q
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66-day rolling ρt (lvl, slo) in the Gaussian model 0.5 0.25 0.1 0 −0.1 −0.25 −0.5 0
500
1000
1500
2000 2500 3000 Time (days)
3500
4000
4500
5000
Fig. 7. Correlation of level and slope in the Gaussian world We plot the realized correlation of the level and slope factors implied by the three-factor Gaussian model. We follow the same steps used to construct real-world correlation in Figure 2. The daily model-implied correlations are smoothed over the window of 66 days.
˜ iQ ˜ ′ = I for all i. We discard all Q ˜ i ’s that do covariance matrix of yields week by week. Clearly, Q i not preserve the level and slope interpretation of factors. This leaves us with 564 observations of ˜ i. Q To understand the influence that changes in eigenvectors have on the dependence between factors, we define the following distance measure: δij
=
r
′ ˜ i (:, j) ˜ i (:, j) , Q(:, j) − Q Q(:, j) − Q
j = {lvl, slo} and i = 1, . . . , 564.
(2)
As a reference, the Gaussian example above implies δlvl = 0.101 and δslo = 0.099. ˜ ′ yt ), their correlation, and regression For each i, we compute level and slope factors (pc ˜ i,t = Q i statistics. These statistics allow us to translate a vague notion of distance δ into a comprehendible set of numbers. Figure 9 scatter-plots correlations ρ(lvl, slo) as a function of δlvl , δslo . The correlations are in the range of −0.9 and 0.9. The general pattern of regression β’s and t-statistics follows the one of correlations. Already a small δ of about 0.1 can lead to a (statistically significant) correlation of ±0.5. This result favors the use of factor correlations rather than factor loadings to gauge the limitations of the unconditional PCA analysis.
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a. Eigenvectors 1
0.5
Q ˜ Q
0
−0.5
−1 2
3
4
5
6 Maturity
b. Orthogonal factors, Q′ yt 3
1
0
0
−2 −3 0
200
R2 =0.00 β =0.00 t-stat = 0.00 400 600 800 Time (weeks)
9
10
˜ ′ yt c. Perturbed factors, Q lvl slo
2
1
−1
8
3
lvl slo
2
7
−1
R2 =0.27 β =0.52 t-stat = 8.72
−2 −3 0
1000
200
400 600 Time (weeks)
800
1000
Fig. 8. Perturbations of factor loadings: Simulated example from the Gaussian three-factor model The figure illustrates the impact of perturbing eigenvectors on the correlation between level and slope. In panel a, ˜ is a perturbed matrix. Panels b Q denotes the original eigenvector matrix that gives perfectly orthogonal factors. Q ˜ respectively. For ease of comparison, both level and slope and c plot the dynamics of factors induced by Q and Q, factors have been standardized. In each panel, we report R2 , β and t-statistics from a regression of level on slope. t-statistics use the Newey-West adjustment with 12 lags.
5. Outlook We are convinced that yield curve states have nontrivial stochastic volatility and covary over time in reflection of economic conditions. Now, even though the unconditional PCA obscures some effects, it is certainly a comprehensive description of the term structure. Level, slope and curvature do explain almost the total variation in the curve—it is not our goal to contest that. We learn, however, that the unconditional PCA is just one particular point fixed on the space of time varying rotations of yields. Likewise, we are far from claiming that Gaussian models are wrong, and should be discarded as a matter of principle. Quite the opposite. While for our goals of modeling volatility it would be
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0.87 0.68
0.8 0.6
0.48
ρt (lvl,slo)
0.4 0.2
0.29
0 0.10
−0.2 −0.4
−0.10
−0.6 −0.8
−0.29
0.6
−0.49 0.4
distance δ slo
−0.68 0.2 0
0
0.3 0.2 distance δ lvl
0.1
0.4 −0.87
˜ Fig. 9. Empirical correlation of level and slope when Q becomes Q The figure scatter-plots the correlation between level and slope as a function of factor perturbation. Factor ˜ j , j = {lvl, slo} perturbation is measured as the Euclidean norm of the difference between the perturbed eigenvector Q j and the eigenvector which gives perfectly orthogonal level and slope factors Q , j = {lvl, slo}, see equation (2). The color-bar on the right-hand side shows the magnitude of the correlation from most negative to most positive. The graph is based on 564 data points.
hard to defend a Gaussian approach, Gaussian models (plus some nonlinear twists on the drift) can be very helpful in explaining the first moments. Finally, we subscribe to the view that other factors—beyond three standard PCs—reside in the curve. These can have little effect on today’s cross section but may matter for the dynamics of yields, bond volatilities or future excess returns. ρt could be one example. Understanding these factors, especially their link to the pricing of yields and to the term premia, is an important task. Which portion of movements in some of those “unspanned” factors can in fact be extracted from yields remains an open question.
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References Barndorff-Nielsen, O. E., Shephard, N., 2004. Econometric analysis of realized covariation: High-frequency based covariance, regression and correlation in financial economics. Econometrica 72, 885–925. Jacod, J., 1994. Limit of random measures associated with the increments of a Brownian semimartingale. Unpublished working paper. Universite Pierre et Marie Curie.
A. Appendix Level, slope and curvature are one of the most pervasive features of interest rate data. Their interpretation is unaffected whether one uses the covariance matrix of yields or the covariance matrix of yield changes. Both approaches are applied almost interchangeably in the literature. Yet, the Q matrix identified from yields does not orthogonalize yield changes and vice versa. While the reason for this is straightforward, we venture a short discussion with risk of becoming trivial. Consider a standard n-factor Gaussian model, and let xt+1 = µ + Φxt + Σεt+1
(3)
be an n-vector of factors underlying the term structure. Collect n yields with different maturities in a vector yt . Yields are affine in the state xt yt = A + Bxt ,
(4)
for a constant n-vector A and a constant n × n matrix B. For both yields and yield changes the conditional covariance matrix is given as Covt (yt+1 ) = Covt (∆yt+1 ) = BΣΣ′ B ′ .
(5)
Note, however, that this equivalence does not hold unconditionally, i.e. Cov(yt ) 6= Cov(∆yt ). Indeed, the unconditional covariance matrix of xt depends not only on Σ but also on the mean reversion parameters, Φ for yields and Φ − I for yield changes, respectively. (We spare the simple algebra and the expressions.) Thus, in general, Cov(yt ) and Cov(∆yt+1 ) are not simultaneously diagonalized by the same orthogonal eigenvector matrix, Q. As illustration of this point, the ˜ matrix in our Gaussian example (Figure 8) has been obtained from yield differences. perturbed Q
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