Bond Return Predictability. The Cochrane and Piazzesi model (CP-factor)

Table of contents

List of figures

List of tables

List of abbreviations

1 Introduction

2 Bond return predictability in financial literature
2.1 Introduction
2.2 The expectation hypothesis
2.3 The Fama and Bliss model
2.4 The Campbell and Shiller model

3 The Cochrane and Piazzesi model
3.1 Short overview
3.2 Data and statistical issues
3.3 Methodology
3.4 Results
3.5 What kind of risk is captured by the CP-factor?
3.6 Critical appraisal
3.7 Conclusion

4 Testing the CP-factor on different data sets
4.1 The extended UFB data set
4.2 The Gurkaynak data set
4.3 CP-factor on Canada
4.4 CP-factor on Germany
4.5 CP-factor on other countries (Datastream data set)
4.6 Summary of bond return predictability

5 The CP-factor and FX predictability
5.1 Theoretical background concerning FX rates and FX forward contracts
5.2 UIP: Empirical findings
5.3 UIP and the CP-factor differential
5.4 Excess returns in FX rates and the CP-differential

6 Conclusion

Appendix

List of figures

Figure 1: Plot of the regression coefficients of the restricted and unrestricted models

Figure 2: First three Principal Components in CP

Figure 3: Expected Return Factor

Figure 4: The CP-factor and NBER recession periods

Figure 5: Yield curves with different interpolation methods

Figure 6: UFB, NSB and SFB regression coefficients

Figure 7: Results for UFB and SFB when one maturity is dropped

Figure 8: Unrestricted and restricted models for extended time period

Figure 9: Unrestricted and restricted models for the Gurkaynak data set

Figure 10: Multicollinearity test for the Gurkaynak data set

Figure 11: Unrestricted and restricted models for the Canadian data set

Figure 12: Multicollinearity test for the Canadian data set

List of tables

Table 1: Fama and Bliss Model

Table 2: Campbell and Shiller Results 1

Table 3: Campbell and Shiller Results 2

Table 4: Estimates of the Single-Factor model

Table 5: Excess return forecast with principal components

Table 6: UFB vs. SFB

Table 7: values for the unrestricted model using different data sets

Table 8: Contribution of the principal components (PCs) to the predicting power of the first five PCs

Table 9: Coefficients of regression of excess return on five principal components of the yield curve

Table 10: Standard deviations of the principal components in basis points

Table 11: Estimates of the single factor model for the extended time period (until 2013)

Table 12: Fama-Bliss model 2003 vs. 2013

Table 13: Estimates of the principal components model for the UFB data until 2013

Table 14: Individual bond regression PC’s vs. restricted model

Table 15: Estimates of the single-factor model on the Gurkaynak data set

Table 16: Fama-Bliss model on the Gurkaynak data set

Table 17: Estimates of the PC’s model for Gurkaynak data

Table 18: Individual bond regression PC’s vs. restricted model on Gurkaynak data

Table 19: Estimates of the single-factor model for the Canadian data set

Table 20: Fama-Bliss model on the Canadian data set

Table 21: Estimates of the PC’s model for the Canadian data

Table 22: Individual bond regression PC’s vs. restricted model on Canadian data

Table 23: Estimates of the single-factor model for the German data set

Table 24: Fama-Bliss model on the German data set

Table 25: Estimates of the PC’s model for the German data

Table 26: Individual bond regression PC’s vs. restricted model on German data

Table 27: OLS estimates of the CP factor on Datastream data

Table 28: Newey-West 18 Lag standard errors and chi square values for CP model

Table 29: Fama-Bliss model on the Datastream data set

Table 30: OLS estimates with constant for the PC-model on the Datastream data set

Table 31: OLS estimates without constant for the PC-model on the Datastream data set

Table 32: Newey-West 18 Lag standard errors and chi square values for PC model with constant

Table 33: Newey-West 18 Lag standard errors and chi square values for PC model without constant

Table 34: Individual bond regression PC’s vs. restricted model on Datastream data

Table 35: Fama (1984), uncovered interest parity

Table 36: CP-differential regression on percentage changes in FX rates

Table 37: Uncovered interest parity regression

Table 38: Uncovered interest parity with CP-differential as risk premium

Table 39: CP-differential on excess returns

Table 40: Multiple linear regression with excess returns

List of abbreviations

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1 Introduction

The present work deals with the headline topic of bond return predictability and to some extent with foreign exchange predictability. I therefore start with an overview of past research concerning bond return predictability in section two. I note that in the early days of financial research, the so-called expectation hypothesis was formulated (which states that return predictability is not possible at all). This hypothesis held for many decades until Fama and Bliss (1987) and Campbell and Shiller (1991) showed that the hypothesis is wrong, and thus that bond returns are indeed predictable. Fama and Bliss (1987), for example, found that the forward-spot spread predicts the lagged excess return (thus a time varying risk premium) of U.S. government bonds with a value up to 18%. I show major results of both papers in greater detail. Many researchers were active in this field of studies in the following years, and then in 2005, Cochrane and Piazzesi (2005) found a seemingly revolutionary bond return predictability factor (the so-called CP-factor), which predicts lagged bond excess returns with a value up to 44% – thus more than twice better than the results of Fama and Bliss (1987). The uniqueness of the CP-factor is that it is a linear combination of forward rates, and unlike the Fama and Bliss model (which only predicts excess returns on specific maturities), it is able to predict excess returns of all maturities. When this return predictability factor is plotted, a tent-shaped pattern emerges. I present several further results by Cochrane and Piazzesi (2005) in a separate section (section three). Although the CP-factor seems very revolutionary, researchers such as Dai, Singleton and Yang (2004) have shown some concerns about the data that Cochrane and Piazzesi (2005) used. These three researchers show that the CP-factor is vulnerable to how the data is constructed. I present their results in section three, as well as brief insights into what time varying risk premium might be captured by the CP-factor. Additionally, the CP-model has only been tested on a few data sets exclusively within the U.S. The main goals of the present work are to test the CP-factor empirically on several different data sets, both within and outside the U.S. I test the CP-factor against both the Fama and Bliss (1987) model and the first three principal components of the yield curve in section four, to assess whether the CP-factor will produce superior results and is indeed a revolutionary finding in bond return predictability. I test the models on four long-term data sets (two within and two outside the U.S.) comprised of more than 25 years of observational data, as well as on 11 shorter data sets (all outside the U.S.) that include data up to 15 years. Besides the focus on values and statistical significance, I assess whether the tent-shaped pattern persists through all data sets. My second goal, which is presented in section five, is then to assess whether the CP-factor will not only be able to predict bond excess returns, but also to predict changes as well as excess returns in foreign exchange rates. It will be interesting to see how much information about future exchange rates is actually captured by the CP-factor. I therefore present some theoretical aspects of the link between interest rates in different countries and the corresponding exchange rates. At the same time I present the uncovered interest rate parity and the related forward premium puzzle, which states that paradoxically currencies of high interest rate levels tend to appreciate against currencies of low interest rate levels. This fact, which has been shown by many researchers including Fama (1984), Froot and Thaler (1990), and Chinn (2006) is to some extent puzzling because it seems irrational and provides arbitrage opportunities. I therefore try to use the CP-factor to solve that puzzle.

To summarize, my first and main goal is to test the CP-factor on different data sets of different countries against alternative models to evaluate whether the CP-factor outperforms the alternatives models in terms of bond returns predictability. My second goal is to use the CP-factor in order to predict changes and excess returns in foreign exchange rates, and with this in hand to show whether the CP-factor is able to contribute to solving the forward premium puzzle.

2 Bond return predictability in financial literature

2.1 Introduction

Bond return predictability has always been an active field of study. The question whether bond returns are predictable is of interest in relation to various economic and financial issues. Predictability provides the basis for investment and saving decisions for firms, investors and households. It also could provide valuable insights for central banks: if they understand how their actions affect market expectations and price dynamics, they can use that information to optimize their monetary policy. A third benefit of predictability could relate to derivative pricing and hedging strategies. A large number of researchers in financial academic literature have been active in this field of study, examining the predictability of bond returns from both theoretical and empirical points of view. A huge transition in views concerning predictability has taken place, starting with the opinion that bond excess returns are not predictable at all (which up to now had been rejected by many researchers).

2.2 The expectation hypothesis

The expectation hypothesis describes numerous statements about the relationship between bond yields, forward rates and returns. Its theoretical foundation was first developed by Macaulay (1938),^[1] who stated inter alia that “(the) forecasting of short term interest rates by long term interest is, in general, so bad that the student may well begin to wonder whether, in fact, there really is any attempt to forecast.”^[2] The literature distinguishes between the pure expectation hypothesis and the expectation hypothesis. The idea of the pure expectation hypothesis is that in efficient markets, long-term rates are equal to the sum of current and expected future short-term rates.^[3] This form of the expectation hypothesis suggests that there is no excess return for investing in a longer-term bond; investing in a bond that matures in one year yields the same expected one-year excess return as investing in a bond that matures in n years and is sold after one year. However, several theories were developed in the first half of the twentieth century to explain that rational investors actually can ask for a premium for long over short-term bonds. The most commonly known of these theories is probably the liquidity preference theory. In short, this theory, which was developed by Keynes (1936), says that investors have a preference for liquidity and therefore will demand a premium when liquidity risk rises. This apparently is the case for longer over shorter maturity bonds. Incorporating those theories, the expectation hypothesis states that in efficient markets, long-term rates are equal to the sum of current and expected future short-term rates plus a constant premium. To formulate the pure expectation hypothesis in mathematical terms we can write:

(1)

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where describes the forward rate and the yield. Numbers in parentheses stay for maturities and the index describes the time. The formula thus says that the forward rate should be equal to the expected value of the future spot rate:

(2)

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The expected holding period return should be the same on bonds of any maturity:

(3)

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The long-term yield should be equal to the average of the expected future short rates.^[4] In order to adjust for the expectation hypothesis, one just needs to add a term-specific but constant premium to the equations above. Neither the pure expectation hypothesis nor the expectation hypothesis allow for time-varying risk premiums, due to the assumption of rational investors and efficient markets. The common view until the 1970s was, with changing yields on specific maturities, investors would anticipate the changes and offset excess returns through their investment behaviour. As in the efficient market hypothesis, expected future prices are assumed to be martingales, meaning that loosely speaking, the best prediction for future prices are current prices (because all information is incorporated in current prices), predictability of returns should not be possible. Early researchers who tested the expectation hypothesis included, Sutch (1970), Sargent (1972, 1979), Shiller (1973, 1979, 1981) and Modigliani and Shiller (1973).^[5] Probably the most popular rejections of the expectation hypothesis were put forward by Fama and Bliss (1987), Campbell and Shiller (1991) and Cochrane and Piazzesi (2005) (CP). These researchers have shown that bond returns are indeed predictable to a certain degree and that risk premiums are time-varying, so they are neither zero nor constant. The puzzle of having rational investors and time-varying risk premiums at the same time can be explained by the consumption-based capital asset pricing model (CCAPM), which was developed by Lucas (1978). Based on the situation of the economy and its outlook for the future, it might be reasonable for rational investors to have time-varying risk preferences. Thus, bonds with different maturities might have different risks incorporated and therefore risk premiums that vary over time. Since there was evidence against the expectation hypothesis, many researchers focused on return predictability, including Campbell and Ammer (1993), Fama (1976, 1984a, 1984b, 1990), Fama and French (1989), Hansen and Hodrick (1980), Keim and Stambaugh (1986) and Stambaugh (1988).^[6] As the papers from Fama and Bliss (1987), Campbell and Shiller (1991) and CP are probably the most relevant for bond return predictability in the academic literature, I have expounded on each in their own sections. In the latest research, the CP-factor, invented from Cochrane and Piazzesi (2005), is part of parsimonious factor models that are used for forecast ability. Koijen, Lustig and van Nieuwerburgh (2012), for instance, use the CP-factor as one of three factors in order to explain cross-sectional excess returns of stocks and bonds. On the other hand, Gargano, Pettenuzzo and Timmermann (2014) use the CP-factor as one of three factors in their model in order to predict both in- and out-of-sample bond returns.

2.3 The Fama and Bliss model

Fama and Bliss (1987) have shown that risk premiums are not constant, but rather vary over time; they have therefore shown evidence against the expectation hypothesis.^[7] They extract forward rates from U.S. government coupon bond prices by an iterative method, also called bootstrapping. The discount rate function is then “extended each step by computing the forward rate necessary to price successively longer maturity bonds given the discount rate function fitted to the previously included issues.”^[8] The implied discount rates show kinks at the maturities of the coupon bonds, which probably lead to the method often being referred to in the academic literature as the unsmoothed Fama-Bliss method.^[9] The data set, which is available at the Centre for Research in Security Prices (CRSP), comprises zero coupon bond prices of one- to five-year maturities on a monthly basis. In order to show that the expectation hypothesis is false, Fama and Bliss (1987) did a simple regression where the left-hand variable is the lagged excess return and the right-hand variable is the corresponding forward-spot spread. The forward-spot spread is defined as the forward rate of maturity minus the yield of the bond that has a maturity of one year. As their data set consists of bond prices, Fama and Bliss took the natural logarithm of the prices ( and calculated the variables of the regression as follows:

(4)

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(5)

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(6)

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where denotes the log yield, the log forward rate and the log one-year excess return. The standard regression would then have the form:

(7)

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Rewriting formula (7) leads to the Fama and Bliss (1987) model:

(8)

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If the pure expectation hypothesis holds, and are supposed to be zero, as there should be no excess returns. However, the expectation hypothesis allows to be non-zero, but definitely needs to be zero as there should not be any predictability. If in the regression will be greater than zero, it would imply that the expected term premium varies through time.^[10] Table 1 shows the results for the original time period (1964-1985), as well as for the reproduction of CP (1964-2003).

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Table 1: Fama and Bliss Model

The table reveals the results for the two regression periods, where is the constant and is the slope of the regression.^[11] In order to overcome the problem of overlapping data (which I will describe in more detail in a later section), CP use a Hansen-Hodrick correction, which is just one possibility for correcting standard errors that result from an overlapped data set. therefore describes the corrected standard errors of . The and values are also reported, where the number in parentheses depicts the degrees of freedom. The critical values for significance levels of 10%, 5% and 1% are respectively 2.71, 3.84, and 6.63. As can clearly be seen, neither nor are constant or zero and coefficients for are all statistically significant. The results indicate that there is indeed predictability of excess bond returns, as the maximum is 14% in the period until 1985, while it is 18% for the period until 2003. According to CP, when performing a significance test all coefficients but the one for a maturity of five years are statistically significant at a 1% level for the original time period. For the time period until 2003, significance of all maturities have increased, so that the previously non-significant coefficient for the maturity of five years has become significant at a 5% level. This shows that the Fama and Bliss model holds for even a much longer time period than initially examined and therefore is not just an anomaly of earlier times.^[12] In the next section I describe the Campbell and Shiller (1991) model.

2.4 The Campbell and Shiller model

The Campbell and Shiller model focuses on the behavior through time of the yield spread. These researchers’ analysis starts with the statement of the expectation hypothesis that is given by equation 3, which I have already introduced. They modify the equation slightly to:

(9)

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where is the constant risk premium the expectation hypothesis allows for. The statement is that the expected -period return on a series of one-period bonds is supposed to be equal to the -period return on a -period bond, so the -period yield should be the average of expected one-period yields plus a constant, period-dependent premium. This implies that the yield spread is a constant risk premium and additionally that it is an optimal forecast of changes in future interest rates. They constructed the model in such a way that the coefficients of their regression (which regresses the appropriate changes onto the spread) are supposed to be one if the expectation hypothesis holds.^[13] Combining equation 9 for a -period (long period) and a -period (short period) bond will after rearrangement result in:

(10)

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which says that the yield spread is supposed to predict -period -period expected change in the yield. We can also rearrange equation 9 to get:

(11)

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which says that the cumulative expected change in the short yield over the life of the longer period bond is supposed to be predicted by the yield spread. The regressions for both statements will then have the form:

(12)

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(13)

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As stated previously, the coefficients and should be one if the expectation hypothesis holds.^[14] Campbell and Shiller (1991) used the McCulloch data, which consists of monthly notes and bond prices of maturities of 1, 2, 3, 4, 6, and 9 months and 1, 2, 3, 4, 5 and 10 years, measured over the period January 1952 to February 1987. The results are shown in the following two-dimensional tables for all possible pairs of maturities.^[15] Table 2 shows the regression coefficients (upper figure) and the respective Hansen-Hodrick corrected standard errors (lower figure in parentheses) for different pairs of maturities, of the regression stated in formula 12. According to Campbell and Shiller (1991), most coefficients are highly statistically significant and unequal to one, which rejects the expectation hypothesis. The regression coefficients even give the wrong directions of forecast for the change in yield of the -period bond over the life of the -period bond for almost any two maturities. For large and small , they used the approximation of (which is indicated by an after the numbers when used).^[16]

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Table 2: Campbell and Shiller Results 1

Source: Campbell, Shiller, 1991, p. 502

Table 3 shows the regression coefficients (upper figure) and the respective Hansen-Hodrick corrected standard errors (lower figure in parentheses) for different pairs of maturities, of the regression stated in formula 13. For the standard error with * a Newey-West (1987) correction was used, as the Hansen-Hodrick correction yields to negative standard errors.^[17] The results show that the regression coefficients for smaller than 36 months are substantially below one and statistically significant, which also rejects the expectation hypothesis. However, for longer maturities of , this regression cannot reject the expectation hypothesis as the coefficients are almost all very close to one. Overall, results from tables 2 and 3 are called the Campbell and Shiller paradox, as the first regression almost always gives a forecast in the wrong direction whereas the second regression gives a forecast in the right direction.^[18] In the next section I describe the CP approach that delivers the best bond prediction model for the data they have used.

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Table 3: Campbell and Shiller Results 2

Source: Campbell, Shiller, 1991, p. 504

3 The Cochrane and Piazzesi model

3.1 Short overview

In contrast to Fama and Bliss (1987), who predicted bond excess returns on specific maturities with a R[2] up to 18%, CP constructed a single factor (also referred to as the CP-factor) that predicts bond excess returns of all maturities with R[2] values up to 44%. As the expectation hypothesis states inter alia that long-term yields should be equal to the average of the expected future short rates and thus excess return predictability is not feasible, the results of CP have strengthened the evidence against the expectation hypothesis. Their model regresses the lagged excess return against a linear combination of forward rates.^[19]

3.2 Data and statistical issues

Cochrane and Piazzesi use the Fama-Bliss data from 1964 to 2003, which is available from the CRSP, to run their analysis. The data is available on a monthly basis apparently produced in the same manner as in Fama and Bliss (1987) (which I have already presented in a previous section), and contains one- through five-year zero coupon U.S. government bond prices. There are several issues when working with empirical data, especially for time series of prices. One of the biggest problems encountered is so-called overlapping data, which appears when the analyzed time period is larger than the frequency of the data used. For instance, CP calculate one year excess returns by using a data set of monthly observations. When rolling from one month to another, the overlapping of observations creates a moving average error term that makes hypothesis tests biased and ordinary least squares (OLS) parameter estimates inefficient.^[20] In order to provide asymptotically valid hypothesis tests, several heteroskedasticity and autocorrelation consistent (HAC) consistent estimators have been invented.^[21] The methods used by CP in order to test results are called Hansen and Hodrick (1980) 12 Lags, Newey and West (1987) 18 Lags and simplified Hansen and Hodrick. Furthermore, they also reduced the data in a way that none of it actually overlaps. The Hansen and Hodrick (1980) method is the standard approach for handling forecasting regressions with overlapping data; however, it does not always produce a positive definite covariance matrix.^[22] In order to make sure that the definite covariance matrix is positive in any sample, the Newey and West (1987) method with 18 lags is used. This weighted estimator creates a greater chance of correcting for the moving average error that results from overlapping data.^[23] The simplified Hansen and Hodrick method contains the idea that error correlation is due only to overlapping data and thus conditional heteroskedasticity will be ignored.^[24] Another issue in empirical studies might be the so-called small-sample bias. It is always a concern when dealing with relatively small samples and the dynamics of the data imply high serial correlations. It has been pointed out by Mankiw and Shapiro (1986) and Stambaugh (1986) that the existence of small-sample bias in tests of predictability leads to a rejection of the null hypothesis of nonpredictability too often.^[25] Bekaert, Hodrick and Marshall (1997) have shown their concerns about the small-sample bias regarding the rejection of the expectation hypothesis by Campbell and Shiller (1991), but concluded that their results display only little bias and thus the evidence against the expectation hypothesis persists.^[26] In order to overcome the problem of small-sample bias, a commonly used approach is to run a vector autoregression (VAR) to bootstrap the residuals from an augmented sample population. Cochrane and Piazzesi use 50,000 bootstrapped samples from, among other things, a 12 lag yield VAR to calculate the “Small T” standard errors.

3.3 Methodology

The following section describes the methodology that CP used to derive their results.^[27] As their data set contains prices, they first take the log of all prices so that we get:

(14)

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As in all previous formulas, numbers in parentheses display the maturity of the bonds and lower case denotes logs. The yield is then given by:

(15)

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As the data consists of one- through five-year zero coupon bonds, log forward rates can by calculated by:

(16)

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Log holding period returns are then calculated by the difference of the log price of a bond with maturity in time and the log price of a bond with maturity in time :

(17)

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The difference of log holding period returns and the yield of the bond with maturity of one year are defined as excess returns and can be expressed by:

(18)

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When using vectors across maturities, the same letter without index are used, e.g.:

(19)

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For right-hand variables, an intercept is included in the vectors, e.g.:

(20)

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(21)

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Averages across maturity are denoted by overbars, e.g.:

(22)

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The first regression CP run contains the lagged excess return as the dependent variable and a linear combination of all forward rates as the right-hand variable. The regression, which focuses on the one-year return horizon, can be expressed by:

(23)

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This regression is called the unrestricted model. As CP constructed a single factor, we first need to estimate a coefficient by regressing the average excess return on all forward rates:

(24)

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(25)

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In the second step, we can estimate a loading factor for each bond by regressing the excess return of each maturity on:

(26)

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As the single-factor model is restricted in , it is therefore also called the restricted model.

3.4 Results

In this section I describe the main findings of CP that show why their paper has been given such great attention in financial literature. As already mentioned, the analysis incorporates the time period from 1964 to 2003. Table 4 summarizes some of CP’s results.

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Table 4: Estimates of the Single-Factor model

Source: Cochrane, Piazzesi, 2005, p. 143

The upper part of the table shows the results of the estimated gammas of equation 25. It stands out that, ignoring the intercept, the coefficients are almost symmetric around . With a of 0.35 it is about double of the highest in the Fama and Bliss (1987) model, underscoring that , the CP-factor, captures a lot great deal of information about bond risk premiums. To recapitulate, the highest of the Fama-Bliss model for the same time period was just about 0.18. The middle portion of the table shows the standard errors in parentheses, as well as the critical values for a hypothesis test that tries to reject the null that all parameters are jointly zero. Although critical values decrease with chosen correction methods, all results are highly statistically significant as the 10%, 5% and 1% critical values for a are 9.2, 11.1 and 15.1 respectively. Keeping the possibility of small-sample biases in mind, we find the standard errors for the 12 Lag VAR. They are also significant at the 1% level, although they are approximately one third larger than the Newey and West 18 Lags standard errors. As an aside, CP did conduct further analyses on the small sample bias that are beyond of the scope of this work; nonetheless, it is noteworthy to mention that all of their further results were also significant at the 1% level. The 95% confidence interval for is reported in square brackets. The bottom part of the table depicts the results for the individual bond regressions of both the restricted and unrestricted models. The regression coefficients of the restricted models increase with maturity. Newey and West standard errors, critical values and “Small T” errors confirm high statistical significance. The values of the restricted and unrestricted models are practically the same, suggesting that the unrestricted model is not superior to the single-factor model (which predicts the excess return of all, not just specific maturities). It might be questioned why the maximum in table 4 is 37%, while I note in the introduction that CP found values up to 44%. In order to exclude measurement errors, CP regressed the average excess return across maturities on monthly lagged forward rates and found that those results have values up to 44%. I will not go into further detail on this as these analyses are not part of my further investigation.

Plotting the results of the restricted and unrestricted models reveals an interesting pattern in the coefficients. As already indicated, the regression coefficients are almost symmetric around , resulting in a tent-shaped pattern. Figure 1 depicts the estimated from the unrestricted regressions, as well as the estimates of from the single-factor model. The x-axis shows the maturity of the forward rates of the independent variables of the regression and the legend exhibits the maturity of the bonds of the dependent variable. The figure clearly shows that the results of both the unrestricted and restricted models are pretty much the same, confirming that the single-factor model almost captures the parameter of the unrestricted model.^[28]

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Figure 1: Plot of the regression coefficients of the restricted and unrestricted models

Source: Cochrane, Piazzesi, 2005, p.140

Another interesting finding by CP is that their single return-forecasting factor, the CP-factor, is unrelated to movements of the standard term structure model. Most term structure models imply that changes in the yield curve are almost completely described by level, slope and curvature. The foundation is based on so-called principal component analysis (PCA), which decomposes the variance of the data into components that are sorted in order of importance. As the PCA is a pure statistical method, the ordered components need to be interpreted. Plotting the first three principal components of the yield curve (the so-called yield factors) reveals why they are called “level”, “slope” and “curvature”.

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Figure 2: First three Principal Components in CP

Source: Cochrane, Piazzesi, 2005, p. 141

Figure 2 represents the first three principal components of the yields calculated from the data set. It shows how the yields for different maturities change when the respective factor changes. For example, a change in the level factor increases the yields of all five maturities about 0.5%; as it affects all maturities more or less equally and therefore has a flat structure, it is called the level factor. The first three principal components in the data set used by CP explain 99.97% (98.57% level, 1.37% level and 0.03% curvature) of the variance of the yield curve. Therefore it is natural to question how we could relate the return forecasting factor to the principal components of the yield curve. In order to make the forecasting factor comparable, we have to run an OLS of the average excess return on the yields instead of forward rates:

(27)

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As forward rates and yields span the same linear space, we can write .^[29] Figure 3 plots the results for .

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Figure 3: Expected Return Factor

Source: Cochrane, Piazzesi, 2005, p. 141

As can be clearly seen, the return-forecasting factor is unrelated to the first three principal components of the yield curve. While the curvature-factor is curved at the short end, the return-forecasting factor is curved at the long end of the yield curve. Whether the return-forecasting factor indeed has superior return predictive ability to the first three principal components of the yield curve can be assessed by regressing the average excess return on the principal components and comparing the results to the Cochrane and Piazzesi model.

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Table 5: Excess return forecast with principal components

Source: Cochrane, Piazzesi, 2005, p. 147

Table 5 depicts the results for the regression of lagged average excess returns on different combinations of the principal components. If only regressing on the slope, the is 0.22. When including the level to the regression, the rises to 0.24. The peaks at 0.26, when the first three principal components are included in the regression. The slope seems to be the most important of the three factors, as it predicts the vast majority of excess returns. It should also be noted that the vast majority of changes in the yields are explained by the level factor, whereas the slope explains changes of expected returns.

As the for the CP-factor regression (0.35) was above the maximum of the principal component regressions and all values are well above the 5% critical values, we can conclude for this data set that the CP-factor is indeed the better model to predict bond excess returns. In order to find out what drives the CP-factor, we can perform a PCA analysis for . It turns out that the first five principal components explain 9.1%, 58.7%, 7.6%, 24.3% and 0.3% of the variation of the CP-factor. It also clearly stands out here that the slope explains the vast of the majority of changes, but in contrast to the principal components of the yield curve, the fourth principal component explains the second highest number of variations. This is an important discovery, as it seems that the fourth principal component of the CP-factor (which has a big impact on the four- to five-year yield spread) does not carry much information about changes in the yield curve, although it does carry a great deal of information about future excess returns.^[30] This might also explains why the CP-factor is a better predictor then the first three principal components of the yield curve, as well as that commonly used factors in term structure models (namely level, slope and curvature of the yield curve) are not sufficient to predict excess returns.

3.5 What kind of risk is captured by the CP-factor?

As we have seen in the previous section, the CP-factor seems to be a revolutionary factor in predicting risk premiums; however, as of today nobody really knows why the CP-factor actually works so well. A natural question is therefore what exact time-varying risk premium could actually be captured by the CP-factor? Cochrane and Piazzesi realized that the CP-factor is “high in troughs and low at peaks”^[31] of economic activities. Koijen, Lustig and van Nieuwerburgh then analyzed the CP-factor in detail and found that it clearly forecasts future economic activity.^[32] They regressed the CP-factor on the -month lagged Chicago Fed National Activity Index (CFNAI), which is a proxy for economic activity. They found that the CP-factor predicts economic activity 12-24 months in advance.^[33]

illustration not visible in this excerpt

Figure 4: The CP-factor and NBER recession periods

Based on: Koijen, Lustig, van Nieuwerburgh, Appendix A

In figure 4, the CP-factor is plotted from March 1953 to November 2013. The bars in the graph indicate American recession periods according to the National Bureau of Economic Research (NBER). It stands out that the CP-factor is often low just before and very high at the end or just after a recession period, underscoring its predicative power for business cycles.

3.6 Critical appraisal

In this section I critically appraise the findings of CP solely based on the work of Dai, Singleton and Yang (2004) (DSY) and the reply from Cochrane and Piazzesi (which can be found on John Cochrane’s homepage)^[34].

Dai, Singleton and Yang try to assess whether bond return predictability using the CP-factor also works when applying different interpolation mechanisms to the data. As there are not enough real traded assets of government bonds across different maturities and most of the bonds that are traded are not zero bonds, it has always been a challenge to develop the “real” (zero) yield curve. In order to overcome this problem, interpolation mechanisms (several of which exist) must be used. As these methods have an impact on what the yield curve looks like, the interpolation method can influence the results of empirical studies. As already explained in previous sections, the Fama and Bliss (1987) approach (which is also called the unsmoothed Fama-Bliss (UFB) method) for example uses bootstrapping to create the yield curve. Dai, Singleton and Yang test the CP-factor on three further data sets that are based on the same set of underlying coupon bond prices but use different interpolation and smoothing mechanisms, namely the Fisher-Waggoner (FW), the Nelson-Siegel-Bliss (NSB) and the smoothed Fama-Bliss (SFB) methods. The FW method uses a cubic spline to approximate the forward rate function.^[35] The idea behind a cubic spline is that in comparison to a linear approximation, we construct a piecewise third degree polynomial between each knot point that is twice differentiable at each knot point.^[36] This means that the piecewise interpolation function passes through all knot points and that the slope and curvature of the function must match from both sides of each knot point, so that it creates a smooth curve. The drawback of a cubic spline is that the approximated curve can oscillate between each knot point and thus the interpolation might end up with unrealistic results. The FW method is a modified version of the Fisher, Nychka and Zervos (1995) model and uses a so-called variable roughness penalty that lowers the oscillating effect and fits the smoothed yield curve to more realistic results.^[37] The NSB approach is an extension of the parametric Nelson and Siegel (1987) method that was described by Bliss (1997). The idea behind the Nelson-Siegel method is that forward rates can be described by a parametric model with an exponential function of time to maturity:

(28)

Abbildung in dieser Leseprobe nicht enthalten

where is the time to maturity and , , , and are coefficients. Because the spot rate function is defined as the average of the forward rates:

(29)

Abbildung in dieser Leseprobe nicht enthalten

The yield curve can be described by:

(30)

Abbildung in dieser Leseprobe nicht enthalten^[38]

The basis of the interpolation is observed knot points of the term structure. As can be seen in equation 28, the Nelson-Siegel method has four parameters that are estimated in order to fit the observed knot points. There are several ways to make the estimations, although minimizing the sum of squared errors is probably the most popular choice. In comparison to the Nelson-Siegel method, the NSB method uses five instead of four parameters. The drawback of a parametric model is that the fitted curve does not necessarily have to pass through the observed knot points. The high correlation between the factors of the Nelson-Siegel method might additionally raise potential multicollinearity issues.^[39] The last smoothing mechanism DSY use is called the SFB method, which basically utilizes the NSB method to smooth the unsmooth Fama-Bliss data.

illustration not visible in this excerpt

Figure 5: Yield curves with different interpolation methods

Source: Dai, Singleton, Yang, 2004, p. 5

[...]

---

^[1] cf Cont, 2010, p. 622

^[2] Macaulay, 1938, p. 33

^[3] cf Bekaert, Hodrick, 2002, p. 3

^[4] cf Cochrane, 1999, p. 46

^[5] cf Cont, 2010, p. 226

^[6] cf Cont, 2010, p. 226

^[7] cf Fama, Bliss, 1987, p. 689 f

^[8] Bliss, 1997, p. 10

^[9] cf Dai, Singleton, Yang, 2004, p. 3

^[10] cf Fama, Bliss, 1987, p. 683

^[11] (***), (**), (*) denote significance at levels of 1%, 5% and 10% respectively

^[12] cf Fama, Bliss, (1987), p. 683

^[13] cf Campbell, Shiller, 1991, p. 4-5

^[14] cf Bulkley, Harris, Nawosah, 2008, p. 6

^[15] cf Campbell, Shiller, 1991, p. 2

^[16] cf Campbell, Shiller, 1991, p. 501

^[17] cf Campbell, Shiller, 1991, p. 504

^[18] cf Campbell, Shiller, 1991, p. 505

^[19] cf Cochrane, Piazzesi, 2005, p. 138

^[20] cf Harri, Brorsen, 2009, p. 79

^[21] cf Harri, Brorsen, 2009, p. 79

^[22] cf Cochrane, Piazzesi, 2005, p. 143

^[23] cf Cochrane, Piazzesi, 2005, p. 143

^[24] cf Cochrane, Piazzesi, 2005, p. 144

^[25] cf Nelson, Kim, 1993, p. 642

^[26] cf Beaert, Hodrick, Marshall, 1997, p. 343

^[27] The following cf Cochrane, Piazzesi, 2005, p. 140-142

^[28] cf Cochrane, Piazzesi, 2005, p. 142

^[29] cf Cochrane, Piazzesi, 2005, p. 146

^[30] cf Cochrane, Piazzesi, 2005, p. 147

^[31] Cochrane, Piazzesi (2005), p. 154

^[32] cf Koijen, Lustig, van Nieuwerburgh (2012), p. 1

^[33] cf Koijen, Lustig, van Nieuwerburgh (2012), p. 10

^[34] http://faculty.chicagobooth.edu/john.cochrane/research/papers/dsy.pdf

^[35] cf Dai, Singleton, Yang, 2004, p. 3

^[36] cf McCulloch, 1971, p. 23 f.

^[37] cf Waggoner, 1997, p. 9-11

^[38] cf Nelson, Siegel, 1987, p. 475

^[39] cf Diebold, Li, 2006, p. 343