## Abstract

This article investigates the out-of-sample predictability of bond excess returns. We assess the economic value of the forecasting ability of empirical models based on long-term forward interest rates in a dynamic asset allocation strategy. The results show that the information content of forward rates does not generate systematic economic value to investors. Indeed, these models do not outperform the no-predictability benchmark. Furthermore, their relative performance deteriorates over time.

## 1. Introduction

The predictability of bond excess returns has occupied the attention of financial economists for many years. In past decades, several studies have reported evidence that empirical models based on forward rates or forward spreads are able to generate accurate forecasts of bond excess returns. Because forward rates represent the rate on a commitment to buy a one-period bond at a future date, it is natural to hypothesize that they incorporate information that is useful for predicting bond excess returns. In support of this conjecture, Fama and Bliss (1987; henceforth FB) find that the forward-spot spread has predictive power for the change in the spot rate and excess returns and that the forecasting power increases as the forecast horizon lengthens. (Cochrane and Piazzesi, 2005; henceforth CP) extend FB's original work by proposing a framework in which bond excess returns are forecast by the full term structure of forward rates. They find that their specification is able to capture more than 30% of the variation of bond excess returns over the period January 1964–December 2003. Cochrane and Piazzesi (2008) confirm these results using a larger set of maturities.1^{,}2

We contribute to the existing literature on the predictive ability of forward rates for bond excess returns in two ways: First, because a model's in-sample predictive performance tends to correlate poorly with its ability to generate satisfactory out-of-sample forecasts (Inoue and Kilian, 2004, 2006), we evaluate the forecasting ability of predictive models based on forward rates in a genuine out-of-sample forecasting exercise. Second, given that statistical significance does not mechanically imply economic significance (Leitch and Tanner, 1991; Della Corte, Sarno, and Thornton 2008; Della Corte, Sarno, and Tsiakas 2009), we assess the economic value of the predictive power of forward rates by investigating the utility gains accrued to investors who exploit the predictability of bond excess returns relative to a no-predictability alternative associated with the validity of the expectations hypothesis.

In the spirit of Fleming et al. (2001), Marquering and Verbeek (2004), Della Corte, Sarno, and Thornton (2008), and Della Corte, Sarno, and Tsiakas (2009), we quantify how much a risk-averse investor is willing to pay to switch from a dynamic portfolio strategy based on a model with no predictable bond excess returns to a model that uses either forward spreads (FB) or the term structure of forward rates (CP), with and without dynamic volatility specifications. We consider two volatility specifications: a constant variance consistent with a standard linear regression and a rolling sample volatility model (Foster and Nelson, 1996; Fleming et al., 2001, 2003). The latter is computationally efficient and is flexible enough to capture the features of bond excess returns data. In order to take into account the problems arising from potential misspecification and parameter changes in models of conditional mean excess returns, the parameters are estimated over time using all past observations available up to the time of the forecast (recursive scheme) and a selected window of past observations (rolling scheme). In addition, we also allow for parameter uncertainty when constructing optimal portfolios. Specifically, we impose an informative prior to define the distribution of the parameter estimates used to carry out the asset allocation problem, as in Kandel and Stambaugh (1996) and Connor (1997).3

We find that none of the predictive models based on forward rates are able to add significant economic value to investors relative to the no-predictability benchmark. However, the extent of the underperformance varies across specifications, especially when parameter uncertainty is taken into account. Also, predictive regressions with conditional volatility show no significant improvement relative to the constant volatility alternative. Finally, we find that the relative performance of the predictive models deteriorates over time. In particular, as suggested by John Cochrane on his website, the predictive models seem to be especially unsatisfactory during the 2007–2009 financial crisis.

Various studies have attempted to validate, with mixed success, the early empirical findings reported in FB and CP. Rudebush et al. (2007) show that the empirical estimates of the term premia implied by CP are less correlated with other available measures and are more volatile. Similarly, using a reverse regression methodology, Wei and Wright (2010) find that ex ante risk premia on long-term bonds are both large and volatile because the underlying parameters appear to be imprecisely estimated.

Other studies have investigated the source of information embedded in forward rates and their genuine predictive power. Radwanski (2010) shows that the one-year-ahead expected inflation extracted from the cross-section of forward rates, together with a level factor capturing the average level of forward rates, is able to attain results similar to CP. Almeida et al. (2011) find that term-structure affine models that include interest rate option prices in the estimation are able to generate bond risk premia that better predict excess returns for long-term rates. The $$R^{2}$$ estimates that they obtain are similar in magnitude to those reported in earlier studies. Cieslak and Povala (2011) decompose long-term yields into a persistent inflation component and maturity-related cycles. They show that the CP predictive regressions are special cases of a more general return-forecasting regression where the CP factors are constrained linear combinations of cycles. Using this framework, they obtain in-sample $$R^{2}$$ that are twice those reported by CP. Duffee (2011) criticizes the notion that term structure models ought to rely on bond yields (and linear combinations of them, such as forward rates) to serve as the factors in theoretical and empirical models. He shows that almost half of the variation in bond excess returns can be associated with a (hidden) filtered factor that is not related to the cross-section of bond yields. Furthermore, when the filtered factor is added to the CP regressions, the term structure of forward rates is no longer statistically significant at conventional levels.

The statistical properties of bond yields also give a reason to be skeptical of the forecasting power of predictive regressions based on forward rates. First, the empirical frameworks proposed in this literature implicitly embed long-horizon returns. It is well known that OLS estimations of regressions of bond excess returns on the term structure of yields suffer from small-sample bias and size distortions that exaggerate the degree of predictability. Hence, the estimates of $$R^{2}$$ reported in the existing literature are inadequate measures of the true in-sample predictability (Kirby 1997; Valkanov 2003; Campbell and Yogo 2006; Boudoukh et al. 2008; Wei and Wright 2010, and the references therein).

Second, bond yields are highly serially correlated and correlated across maturities. If both regressors and regressands exhibit a high serial correlation, the predictive regressions based on forward rates may suffer from a spurious regression problem (Ferson et al. 2003a, 2003b, and the references therein)4. Consequently, evidence of in-sample predictability need not be a useful indicator of out-of-sample predictive performance. Moreover, the parameters of these empirical models may vary over time (Fama, 2006; Wei and Wright, 2010), and this in turn affects the performance of the empirical models when used out of sample.

Our article is closely related to those of Duffee (2010) and Barillas (2010), who explore the predictability of bond excess returns from a similar perspective. Duffee (2010) investigates the conditional maximal Sharpe ratios implied by fully flexible term structure models and finds that in-sample model overfitting leads to astronomically high Sharpe ratios. Barillas (2010) investigates the optimal bond portfolio choice of an investor in a model that captures the failure of the expectations hypothesis of interest rates. In an in-sample exercise, the author finds that investors conditioning on bond prices and macroeconomic variables would be willing to give up a sizable portion of their wealth in order to live in a world where the risk premia state variable is observable. Our analysis differs from these studies in three important respects. These studies investigate the in-sample predictability of predictive models, while we assess the economic value from using these models out of sample. In addition, we investigate the impact of parameter uncertainty on bond excess return predictions, and we also explicitly incorporate estimates of the conditional variances of bond returns into the portfolio allocation problem.5

The remainder of the article is as follows: Section 2 introduces the empirical framework used to model the conditional mean and volatility of bond excess returns. Section 3 discusses the framework for assessing the economic value of bond excess returns predictability for a risk-averse investor with a dynamic portfolio strategy. Section 4 reports the main empirical results, and Section 5 explores the performance of the predictive models over time during the past thirty years. Section 6 discusses the results of various robustness checks, and Section 7 concludes the article.

## 2. The Predictive Power of Forward Rates

In line with the existing literature, we define the log-yield of an $$n$$-year bond as

Using monthly data for bond yields with maturities ranging between one and five years, FB estimate the excess return equations6

CP propose a modified version of the FB excess returns equation. Specifically, they estimate a general regression where bond excess returns are predicted by the full term structure of forward rates and the one-period bond yield—that is,

Note that Equations (4) and (5) can be written more generally as

When β = **0**, bond excess returns are not predictable and are equal to a constant $$c$$. This case is consistent with the expectations hypothesis of the term structure of interest rates, which is frequently used as a benchmark against which other empirical bond excess return models are compared. We label this model as EH.

As reported in various studies, and documented later in this article, there is considerable evidence indicating that the volatility of bond yields and bond excess returns is time varying and predictable (Gray 1996; Bekaert et al. 1997; Bekaert and Hodrick 2001). Hence, in addition to Equation (6), we model the dynamics of the conditional variance-covariance matrix of bond excess returns with a simple linear regression model and with a rolling sample variance estimator (Foster and Nelson, 1996; Fleming et al., 2001, 2003). More specifically, the linear regression model assumes that the conditional covariance matrix of the residuals $${\Sigma _{t + 12|t}} = {E_t}\left[ {{\epsilon _t + 12}{{\epsilon '}_{t + 12}}} \right]$$ with $${\epsilon _{t + 12}} = \left[ {\epsilon _{t + 12}^{(2)}\quad \ldots \quad \epsilon _{t + 12}^{(5)}} \right]$$ is constant over time—that is, $${\widehat\Sigma _{t + 12|t}} = \widehat\Sigma $$ . The rolling sample variance estimator is of the general form

_{$$t- l$$}is a symmetric 4 × 4 matrix of weights and ⊙ denotes element-by-element multiplication. The logic behind this approach is that if $${\Sigma }_{t+12}$$ is time varying, then its dynamics are reflected in the sample path of past excess returns. Hence, if a suitable set of weights is applied to squares and cross-products of excess return innovations, it is possible to construct a time-series estimate of $${\Sigma }_{t+12}$$ (Foster and Nelson, 1996). In our empirical application, we follow Fleming et al. (2003) and select the optimal weight for a one-sided rolling estimator $$Ω_{t- l} = α \exp(- {\alpha }l)\bf{11}′$$, where

**1**denotes a $$4 × 1$$ vector of ones and α is the decay rate that governs the relative importance assigned to past excess return innovations. As in Fleming et al. (2001, p. 334), we impose that the decay parameter is unique across all cross products of excess return innovations in order to ensure the positivity of the matrix $${\Sigma }_{t+12}$$. We use this estimation method to compute the conditional covariance matrix $${\Sigma }_{t+12}$$ since it is not heavily parametrized and it is less difficult to estimate than multivariate ARCH and GARCH models8 (see, inter alia, Bawens et al. 2006 and the references therein). In fact, for certain choices of $$Ω_{t- l}$$ the rolling sample estimator resembles the $${\Sigma }_{t+12}$$ process implied by a multivariate GARCH model (Fleming et al., 2003, p. 479).

## 3. Assessing Bond Excess Returns Predictions

### 3.1 The asset allocation framework

This section explores the economic significance of the predictive information embedded in forward rates and forward spreads relative to the no-predictability alternative. A classic portfolio choice problem is used (Della Corte, Sarno, and Thornton 2008; Della Corte, Sarno, and Tsiakas 2009). Specifically, we consider an investor who optimally invests in a portfolio comprising $$K + 1$$ bonds similar in all respects but with different maturities: a risk-free one-period bond and $$K$$ risky $$n$$-period bonds. The investor constructs a monthly dynamically rebalanced portfolio that minimizes the conditional portfolio variance subject to achieving a given target of expected return.

Let the conditional expectation and the conditional variance-covariance matrix of the $$K \times 1$$ vector of bond excess returns, $${\bf{rx}}_{t+12}$$, be equal to $${\mu }_{t+12|t} = E_{t}(\bf{rx}_{t+12})$$ and $$Σ_{t+12|t} = E_{t}$$$$[({\bf{rx}}_{t+12} - {\mu }_{t+12|t})(\bf{rx}_{t+12} - {\mu }_{t+12|t})^\prime]$$, respectively. At the end of each period, the investor solves the following problem:

**w**$$_{t}{\prime }{\iota }$$, where ι is a $$K \times 1$$ vector of ones.

In the empirical analysis, we winsorize the weights for each of the $$n$$-period bonds to −1 ≤ $$w_{t}^{(n)} {\le } 2$$ to prevent extreme investments (Goyal and Welch, 2008; Ferreira and Santa Clara, 2011). These constraints essentially allow for the full proceeds of short sales (see, inter alia, Vayanos and Weill 2008 and the references therein).

### 3.2 Modeling bond excess returns and their volatility: The role of parameter uncertainty

In order to construct the optimal portfolio weights, **w**$$_{t}$$, estimates of conditional expected bond excess returns **$${\mu }$$**_{$$t+12|t$$} and conditional variance-covariance matrices $$Σ_{t+12|t}$$ are required. Three different conditional mean strategies are considered: the benchmark model of no predictability (EH), the FB model, and the CP model. The three models are estimated using both constant volatility (CVOL henceforth) and time-varying volatility (TVOL henceforth) to compute the volatility forecasts.

Given the statistical problems noted in the previous section, it seems likely that there could be uncertainty about the parameter estimates or even the overall parametrization of the data-generating process.9 This complicates the asset allocation process because the investor does not know over which parameter set to minimize the portfolio variance's function. In the spirit of the literature on portfolio choice under parameter uncertainty, we follow Kandel and Stambaugh (1996) and Connor (1997), who recommend imposing an informative prior to define the distribution of the parameter estimates used to carry out the asset allocation problem.10 More specifically, we advocate a weak-form efficiency of the bond markets consistent with the expectations hypothesis—that is, estimates of $$\hat \beta $$ are assumed equal to zero. Kandel and Stambaugh (1996) demonstrate that, in a Bayesian regression setup, this prior yields a posterior of the parameter estimates that is the product of the OLS estimates and a shrinking factor that is a function of the precision of the parameter estimates. The smaller the precision of the parameter estimates, the stronger the shrinkage toward their prior mean of zero.

The empirical investigation in this article takes into account Kandel and Stambaugh (1996) findings by implementing the procedure introduced by Connor (1997). That is, each of the $$j$$ parameter estimates in Equation (6) are computed as

### 3.3 The economic value of excess returns predictability

The economic value of the predictability of forward rates is assessed by assuming quadratic utility, as in West et al. (1993), Fleming et al. (2001), and Della Corte, Sarno, and Thornton (2008), and Della Corte, Sarno, and Tsiakas (2009); the average realized utility, $$\overline U( \cdot )$$ , for an investor with initial wealth $$W_{0}$$ is given by

As in Fleming et al. (2001), the measure of the economic value of alternative predictive models is obtained by equating average utilities of selected pairs of portfolios. For example, assume that holding a portfolio constructed using the optimal weights based on the EH strategy with constant volatility $$(EH_{CVOL})$$ yields the same average utility as holding the portfolio implied by the CP strategy with constant volatility $$(CP_{CVOL})$$. The latter portfolio is subject to management expenses, Φ, expressed as a fraction of wealth invested in the portfolio. If the investor is indifferent between these two strategies, then Φ can be interpreted as the maximum performance fee the investor would be willing to pay to switch from the $$EH_{CVOL}$$ to the $$CP_{CVOL}$$ strategy. In general, this criterion measures how much a risk-averse investor is willing to pay for conditioning on the information in the forward rates since the benchmark used implies no predictability in either the conditional mean or the conditional variance. The performance fee is the value of Φ that satisfies

In the context of mean-variance analysis, several other measures of performance are routinely employed. A measure frequently used is the Sharpe ratio (SR), which is calculated as the ratio of the average portfolio excess returns to its standard deviation—that is, $$\frac{1}{{T - 12 + 1}}\sum\limits_{t = 0}^{T - 12} {\left( {r_{p,t + 12}^\mathcal {F} - y_{t + 12}^{(1)}} \right)/{\sigma ^\mathcal {F}}} $$ for any predictive model, where $$r_{p,t + 12}^\mathcal {F} = R_{p,t + 12}^\mathcal {F} - 1$$ and $${\sigma ^\mathcal {F}}$$ denotes the standard deviation of $$\left( {r_{p,t + 12}^\mathcal {F} - y_{t + 12}^{(1)}} \right)$$ . The statistical significance of the difference of the SR from two competing models is tested by using the bootstrap procedure introduced by Ledoit and Wolf (2008). This procedure has been shown to be robust to portfolio returns that are nonnormal and serially correlated. In particular, we construct a studentized time-series bootstrap confidence interval for the difference of the SR, using a variant of the circular block bootstrap (Politis and Romano, 1992), and test whether zero is contained in the interval.13

While Sharpe ratios are commonly used, they exhibit some drawbacks. Specifically, they do not take into account the effect of nonnormality (Jondeau and Rockinger, 2008), they tend to underestimate the performance of dynamic strategies (Marquering and Verbeek, 2004; Han, 2006 and the references therein), and they can be manipulated in various ways (Goetzmann et al., 2007). In order to take into account these concerns, we follow Goetzmann et al. (2007), who suggest a set of conditions under which a manipulation-proof measure exists. This performance measure can be interpreted as a portfolio's premium return after adjusting for risk. We build on Goetzmann et al. (2007) and calculate a risk-adjusted abnormal return of the predictive models relative to the $$EH_{CVOL}$$ strategy as follows:

^{$$EH_{CVOL}$$}denote the value traded in the different bonds associated with the predictive models $${\mathcal F} = FB,CP$$ and the benchmark, respectively. In comparing any predictive model $${\mathcal F}$$ with $$EH_{CVOL}$$, an investor who pays transaction costs lower than $${\tau }^{BE}$$ will always prefer model $${\mathcal F}$$ to the benchmark. Break-even transaction costs are computed only when they can be meaningfully interpreted; that is, when the performance fees in Equation (13) are positive.

## 4. Empirical Results

### 4.1 Data and preliminary results

The data set used in this study, consistent with early studies on the predictability of bond excess returns, comprises monthly one- to five-year zero-coupon bond prices from June 1952 through December 2010.

Log-bond excess returns are computed from bond prices as described in Section 2. The summary statistics of the resulting time series are reported in Table 1, Panels A and B. Bond excess returns are found to be close to zero on average (ranging between 0.4% and 1% per annum), but all are statistically significant at the 5% statistical level. Panel A also reports the autocorrelation coefficients of order 1 and 12 for the individual time series that show that bond excess returns are highly serially correlated.

Panel A: Bond excess returns | ||||
---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |

Mean $$rx_{t}^{(n)}$$ | 0.004*** | 0.007*** | 0.009*** | 0.010*** |

Std Dev $$rx_{t}^{(n)}$$ | 0.017*** | 0.031*** | 0.043*** | 0.052*** |

$$Corr\left( {rx_t^{(n)},rx_{t - 1}^{(n)}} \right)$$ | 0.929*** | 0.933*** | 0.933*** | 0.923*** |

$$Corr\left( {rx_t^{(n)},rx_{t - 12}^{(n)}} \right)$$ | 0.189*** | 0.135*** | 0.105*** | 0.071** |

Panel A: Bond excess returns | ||||
---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |

Mean $$rx_{t}^{(n)}$$ | 0.004*** | 0.007*** | 0.009*** | 0.010*** |

Std Dev $$rx_{t}^{(n)}$$ | 0.017*** | 0.031*** | 0.043*** | 0.052*** |

$$Corr\left( {rx_t^{(n)},rx_{t - 1}^{(n)}} \right)$$ | 0.929*** | 0.933*** | 0.933*** | 0.923*** |

$$Corr\left( {rx_t^{(n)},rx_{t - 12}^{(n)}} \right)$$ | 0.189*** | 0.135*** | 0.105*** | 0.071** |

Panel B: Absolute bond excess returns | ||||
---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |

Mean |$$rx_{t}^{(n)}|$$ | 0.013*** | 0.024*** | 0.033*** | 0.040*** |

Std Dev |$$rx_{t}^{(n)}|$$ | 0.011*** | 0.020*** | 0.028*** | 0.034*** |

$$Corr\left( {\left| {rx_t^{(n)}} \right|,\left| {rx_{t - 1}^{(n)}} \right|} \right)$$ | 0.860*** | 0.867*** | 0.865*** | 0.847*** |

$$Corr\left( {\left| {rx_t^{(n)}} \right|,\left| {rx_{t - 12}^{(n)}} \right|} \right)$$ | 0.130*** | 0.178*** | 0.206*** | 0.214*** |

Panel B: Absolute bond excess returns | ||||
---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |

Mean |$$rx_{t}^{(n)}|$$ | 0.013*** | 0.024*** | 0.033*** | 0.040*** |

Std Dev |$$rx_{t}^{(n)}|$$ | 0.011*** | 0.020*** | 0.028*** | 0.034*** |

$$Corr\left( {\left| {rx_t^{(n)}} \right|,\left| {rx_{t - 1}^{(n)}} \right|} \right)$$ | 0.860*** | 0.867*** | 0.865*** | 0.847*** |

$$Corr\left( {\left| {rx_t^{(n)}} \right|,\left| {rx_{t - 12}^{(n)}} \right|} \right)$$ | 0.130*** | 0.178*** | 0.206*** | 0.214*** |

The table reports the descriptive statistics for bond excess returns (Panel A) and absolute bond excess returns (Panel B) computed over the different maturities, $$n$$. The data sample ranges from June 1952 to December 2010 for a sample size of 703 monthly observations. *, **, and *** denote statistical significance at 10%, 5%, and 1%, and statistical significance is evaluated using autocorrelation and heteroscedasticity–consistent standard errors (Newey and West 1987). *Mean* and *Std Dev* are reported in decimals per annum (i.e., 0.01 = 1 annual percentage point).

Panel B of Table 1 reports the same summary statistics for absolute bond excess returns used as a proxy for bond excess return volatilities. Absolute bond excess returns exhibit average values that are higher the longer the term to maturity, and they are all statistically significant at the conventional 5% significance level. Furthermore, in line with previous studies and the results reported in Panel A, absolute bond excess returns are also highly serially correlated.

In addition to reporting the autocorrelation coefficients, we also compute the correlation between all pairs of bond excess returns. Figures 1 and 2 show the average cross-correlations of bond excess returns and absolute excess returns over different ten-year subperiods of the sample. The correlation coefficients are high for both excess returns and absolute excess returns (larger than 0.8 in all cases), and there is some evidence of time variation over the sample period. In particular, the correlation coefficients across maturities increase between the 1960s and the 1980s and then generally decline over the past two decades. This pattern is also exhibited by the correlation coefficients between absolute bond excess returns. The finding of time-varying correlation among excess returns innovations is also corroborated by the Lagrange Multiplier test developed by Tse (2000) that rejects the null of constant conditional correlations with a $$p$$-value of virtually zero.14 In related contexts, there is evidence that shocks generated by negative news may have greater impact on subsequent volatilities than positive shocks of the same magnitude (Engle and Ng, 1993; de Goeij and Marquering, 2006). In order to investigate this issue, we have estimated the following threshold GARCH(1,1) model (Glosten et al., 1993; Zakoïan, 1994) for each of the four bond excess returns time series:

The preliminary exploration of the data is completed by estimating the parameters of the three candidate models over two sample periods: the full sample period 1952–2010 and the CP's sample period, 1964–2003. The in-sample estimates are reported in Table 2, Panels A–C. Estimates of FB and CP models computed over the sample period 1964–2003 are similar to those reported in Cochrane and Piazzesi (2005, 2006). A comparison of the FB model over the two sample periods shows that the estimates of $${\bar R^2}$$ are somewhat smaller over the full sample period; however, the estimates of the parameters changed little. Nevertheless, the parameters estimated over the full sample period generally lie outside the confidence interval of the ones estimated over the smaller sample period for all $$n$$.

Panel A: No predictability | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | 0.004*** | (0.001) | 0.007*** | (0.001) | 0.009*** | (0.001) | 0.010*** | (0.002) |

$$v$$ | 0.017 | 0.031 | 0.043 | 0.052 | ||||

$$logL$$ | 1384.22 | 1418.81 | 1193.13 | 1051.23 | ||||

1964–2003 | ||||||||

β_{0} | 0.005*** | (0.001) | 0.008*** | (0.001) | 0.010*** | (0.002) | 0.010*** | (0.002) |

$$v$$ | 0.019 | 0.034 | 0.048 | 0.059 | ||||

$$logL$$ | 1219.57 | 929.34 | 772.90 | 675.05 |

Panel A: No predictability | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | 0.004*** | (0.001) | 0.007*** | (0.001) | 0.009*** | (0.001) | 0.010*** | (0.002) |

$$v$$ | 0.017 | 0.031 | 0.043 | 0.052 | ||||

$$logL$$ | 1384.22 | 1418.81 | 1193.13 | 1051.23 | ||||

1964–2003 | ||||||||

β_{0} | 0.005*** | (0.001) | 0.008*** | (0.001) | 0.010*** | (0.002) | 0.010*** | (0.002) |

$$v$$ | 0.019 | 0.034 | 0.048 | 0.059 | ||||

$$logL$$ | 1219.57 | 929.34 | 772.90 | 675.05 |

Panel B: Fama and Bliss (1987) | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | 0.001* | (0.001) | 0.001 | (0.001) | −0.001 | (0.001) | 0.001 | (0.002) |

β_{1} | 0.759*** | (0.089) | 1.001*** | (0.113) | 1.272*** | (0.127) | 0.995*** | (0.147) |

$$v$$ | 0.016 | 0.029 | 0.040 | 0.051 | ||||

$$logL$$ | 1868.30 | 1455.97 | 1239.60 | 1073.43 | ||||

$${\bar R^2}$$ | 0.09 | 0.10 | 0.12 | 0.06 | ||||

1964–2003 | ||||||||

β_{0} | 0.001 | (0.001) | −0.001 | (0.001) | −0.004 | (0.001) | −0.008 | (0.003) |

β_{1} | 0.975*** | (0.104) | 1.301*** | (0.133) | 1.526*** | (0.152) | 1.181*** | (0.183) |

$$v$$ | 0.017 | 0.031 | 0.044 | 0.057 | ||||

$$logL$$ | 1259.57 | 973.05 | 818.62 | 694.95 | ||||

$${\bar R^2}$$ | 0.15 | 0.16 | 0.17 | 0.07 |

Panel B: Fama and Bliss (1987) | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | 0.001* | (0.001) | 0.001 | (0.001) | −0.001 | (0.001) | 0.001 | (0.002) |

β_{1} | 0.759*** | (0.089) | 1.001*** | (0.113) | 1.272*** | (0.127) | 0.995*** | (0.147) |

$$v$$ | 0.016 | 0.029 | 0.040 | 0.051 | ||||

$$logL$$ | 1868.30 | 1455.97 | 1239.60 | 1073.43 | ||||

$${\bar R^2}$$ | 0.09 | 0.10 | 0.12 | 0.06 | ||||

1964–2003 | ||||||||

β_{0} | 0.001 | (0.001) | −0.001 | (0.001) | −0.004 | (0.001) | −0.008 | (0.003) |

β_{1} | 0.975*** | (0.104) | 1.301*** | (0.133) | 1.526*** | (0.152) | 1.181*** | (0.183) |

$$v$$ | 0.017 | 0.031 | 0.044 | 0.057 | ||||

$$logL$$ | 1259.57 | 973.05 | 818.62 | 694.95 | ||||

$${\bar R^2}$$ | 0.15 | 0.16 | 0.17 | 0.07 |

Panel C: Cochrane and Piazzesi (2005) | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | −0.007*** | (0.001) | −0.012*** | (0.003) | −0.018*** | (0.004) | −0.024 | (0.005) |

β_{1} | −0.649*** | (0.105) | −1.118*** | (0.192) | −1.633*** | (0.261) | −2.035*** | (0.325) |

β_{2} | 0.378* | (0.416) | 0.049 | (0.371) | 0.130 | (0.504) | 0.255 | (0.628) |

β_{3} | 0.416*** | (0.168) | 1.553*** | (0.305) | 1.543*** | (0.415) | 1.550*** | (0.517) |

β_{4} | 0.328*** | (0.127) | 0.514** | (0.231) | 1.458*** | (0.314) | 1.491*** | (0.391) |

β_{5} | −0.329*** | (0.099) | −0.787*** | (0.181) | −1.218*** | (0.246) | −0.928*** | (0.306) |

$$v$$ | 0.015 | 0.028 | 0.038 | 0.047 | ||||

logL 1900.05 | 1486.62 | 1274.48 | 1123.40 | |||||

$${\bar R^2}$$ | 0.16 | 0.17 | 0.20 | 0.18 | ||||

1964–2003 | ||||||||

β_{0} | −0.015*** | (0.002) | −0.026*** | (0.004) | −0.037*** | (0.006) | −0.047*** | (0.008) |

β_{1} | −0.938*** | (0.124) | −1.711*** | (0.223) | −2.481*** | (0.302) | −3.101*** | (0.378) |

β_{2} | 0.470* | (0.254) | 0.326 | (0.458) | 0.598 | (0.620) | 0.899 | (0.775) |

β_{3} | 1.173*** | (0.213) | 3.010*** | (0.385) | 3.532*** | (0.520) | 4.023*** | (0.651) |

β_{4} | 0.334** | (0.156) | 0.461 | (0.282) | 1.385*** | (0.381) | 1.374*** | (0.477) |

β_{5} | −0.819*** | (0.129) | −1.745*** | (0.233) | −2.583*** | (0.316) | −2.649*** | (0.395) |

$$v$$ | 0.015 | 0.028 | 0.038 | 0.048 | ||||

$$logL$$ | 1310.81 | 1028.14 | 883.28 | 776.01 | ||||

$${\bar R^2}$$ | 0.30 | 0.33 | 0.36 | 0.33 |

Panel C: Cochrane and Piazzesi (2005) | ||||||||
---|---|---|---|---|---|---|---|---|

$$n = 2$$ | $$n = 3$$ | $$n = 4$$ | $$n = 5$$ | |||||

1952–2010 | ||||||||

β_{0} | −0.007*** | (0.001) | −0.012*** | (0.003) | −0.018*** | (0.004) | −0.024 | (0.005) |

β_{1} | −0.649*** | (0.105) | −1.118*** | (0.192) | −1.633*** | (0.261) | −2.035*** | (0.325) |

β_{2} | 0.378* | (0.416) | 0.049 | (0.371) | 0.130 | (0.504) | 0.255 | (0.628) |

β_{3} | 0.416*** | (0.168) | 1.553*** | (0.305) | 1.543*** | (0.415) | 1.550*** | (0.517) |

β_{4} | 0.328*** | (0.127) | 0.514** | (0.231) | 1.458*** | (0.314) | 1.491*** | (0.391) |

β_{5} | −0.329*** | (0.099) | −0.787*** | (0.181) | −1.218*** | (0.246) | −0.928*** | (0.306) |

$$v$$ | 0.015 | 0.028 | 0.038 | 0.047 | ||||

logL 1900.05 | 1486.62 | 1274.48 | 1123.40 | |||||

$${\bar R^2}$$ | 0.16 | 0.17 | 0.20 | 0.18 | ||||

1964–2003 | ||||||||

β_{0} | −0.015*** | (0.002) | −0.026*** | (0.004) | −0.037*** | (0.006) | −0.047*** | (0.008) |

β_{1} | −0.938*** | (0.124) | −1.711*** | (0.223) | −2.481*** | (0.302) | −3.101*** | (0.378) |

β_{2} | 0.470* | (0.254) | 0.326 | (0.458) | 0.598 | (0.620) | 0.899 | (0.775) |

β_{3} | 1.173*** | (0.213) | 3.010*** | (0.385) | 3.532*** | (0.520) | 4.023*** | (0.651) |

β_{4} | 0.334** | (0.156) | 0.461 | (0.282) | 1.385*** | (0.381) | 1.374*** | (0.477) |

β_{5} | −0.819*** | (0.129) | −1.745*** | (0.233) | −2.583*** | (0.316) | −2.649*** | (0.395) |

$$v$$ | 0.015 | 0.028 | 0.038 | 0.048 | ||||

$$logL$$ | 1310.81 | 1028.14 | 883.28 | 776.01 | ||||

$${\bar R^2}$$ | 0.30 | 0.33 | 0.36 | 0.33 |

The table reports the estimates of the no–predictability benchmark consistent with the expectations hypothesis (Panel A) and the Fama and Bliss (1987) and Cochrane and Piazzesi (2005) predictive regressions (Panels B and C). All equations are estimated assuming a constant conditional variance of excess returns innovations. Equations in Panels A, B, and C are estimated over two sample periods: June 1952–December 2010 and June 1964–December 2003. $$n$$ denotes the maturity of forward rates and bond excess returns expressed in years. $$logL$$ denotes the log–likelihood value of the regressions, and $$v$$ is the estimated constant conditional volatility of excess returns innovations. Values in parentheses are asymptotic standard errors computed using least–square estimators. *, **, and *** denote statistical significance at the 10%, 5%, and 1% levels, respectively. $${\bar R^2}$$ denotes the in–sample adjusted coefficient of determination. See also notes to Table 1.

The comparison of CP's model for the two samples shows a marked reduction in the estimates of $${\bar R^2}$$ : the estimates over the full sample are at least 45% lower relative to CP's sample period. While the tent shape of the parameter estimates noted by CP is evident in both samples, the estimates of the parameters are considerably different. Specifically, with the exception of estimates of β_{4}, the estimates are much smaller in absolute value over the full sample period, and in most cases the parameters estimated over the full sample period lie outside the confidence interval of the ones estimated over the smaller sample. This finding is indicative of a considerable time variation in the parameter estimates, which in turn is reflected in the marked reduction in the estimates of $${\bar R^2}$$ and potentially affect the model's out-of-sample performance.

### 4.2 Economic value calculations

This section reports the results of the economic value calculations discussed in Section 3. Forecasts are generated using parameters that are estimated using information available only at the time the forecast is made. More specifically, we also employ two forecasting schemes: (i) a recursive scheme, which uses all the observations available up to the time of the forecast; and (ii) a rolling scheme, where only a window of past observations is used. We consider a rolling scheme because it is likely that because of changes in the macroeconomic environment (shifts in the Fed policy and so on), parameters estimated using data from very long-ago periods may not be necessarily useful to make current out-of-sample predictions.16 Furthermore, as outlined in Section 3.2, we incorporate the role of parameter uncertainty in the conditional mean and compute the parameter estimates with and without the correction reported in Equations (10) and (11).

The combination of the six models (i.e., FB, CP, and EH with and without time-varying volatility) and the four scenarios (i.e., recursive and rolling estimations with and without parameter uncertainty) yields twenty-four sets of results. The performance measures are calculated for the out-of-sample period January 1970 through December 2010 assuming λ = 5, in line with the value used in previous studies (Barberis 2000; Della Corte, Sarno, and Thornton 2008 and the references therein). We also use two annual targets of portfolio excess returns, $$\zeta^{*}_{p}$$ = 0.01,0.02. The target portfolio excess returns are consistent with reasonable average excess returns obtained by portfolios of Treasury bonds and are higher than the average bond excess returns reported in Table 1, Panel A. The performance measures SR and GISW and the performance fees Φ are reported as annualized. The GISW and Φ measures are expressed in decimals (i.e., 0.01 = 1 annual percentage point). The time-varying variance-covariance matrix of excess returns is computed using α = 0.05, a value within the range of those reported in existing studies (Foster and Nelson, 1996; Fleming et al., 2001, 2003). Finally, the rolling forecasting scheme is implemented using a window of the past 120 months.17

Table 3 shows the results of these exercises in four panels. Panel A presents the results on the recursive forecasting scheme with no parameter uncertainty. The results indicate that the FB and CP predictive models with constant volatility provide no economic value relative to the $$EH_{CVOL}$$ benchmark. Their estimated SR are lower than the one of the benchmark model for either choice of $$\zeta^{*}_{p}.$$ However, in nearly all instances, the difference is not statistically significant at conventional levels. In the few cases where the difference is statistically significant, the $$EH_{CVOL}$$ model generates larger economic gains than the competing predictive models. Qualitatively, the results are identical with the GISW measure. All of the estimates are negative and range between −2.2% $$(EH_{TVOL})$$ and −1.0% $$(CP_{CVOL})$$.

Panel A: Recursive estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.464 | 0.211 | 0.272 | −0.082 | 0.100 | 0.187 |

[0.01] | [0.07] | [< 0.01] | [0.14] | [0.17] | ||

GISW | – | −0.013 | −0.010 | −0.022 | −0.016 | −0.014 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.436 | 0.250 | 0.254 | −0.049 | 0.074 | 0.213 |

[0.02] | [0.21] | [0.02] | [0.13] | [0.25] | ||

GISW | – | −0.011 | −0.012 | −0.022 | −0.018 | −0.012 |

Panel A: Recursive estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.464 | 0.211 | 0.272 | −0.082 | 0.100 | 0.187 |

[0.01] | [0.07] | [< 0.01] | [0.14] | [0.17] | ||

GISW | – | −0.013 | −0.010 | −0.022 | −0.016 | −0.014 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.436 | 0.250 | 0.254 | −0.049 | 0.074 | 0.213 |

[0.02] | [0.21] | [0.02] | [0.13] | [0.25] | ||

GISW | – | −0.011 | −0.012 | −0.022 | −0.018 | −0.012 |

Panel B: Recursive estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.464 | 0.202 | 0.317 | – | 0.035 | 0.169 |

[< 0.01] | [0.03] | – | [0.25] | [0.10] | ||

GISW | – | −0.014 | −0.012 | – | −0.018 | −0.014 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.436 | 0.235 | 0.261 | – | 0.024 | 0.216 |

[0.01] | [0.25] | – | [0.10] | [0.14] | ||

GISW | – | −0.011 | −0.011 | – | −0.020 | −0.012 |

Panel B: Recursive estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.464 | 0.202 | 0.317 | – | 0.035 | 0.169 |

[< 0.01] | [0.03] | – | [0.25] | [0.10] | ||

GISW | – | −0.014 | −0.012 | – | −0.018 | −0.014 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.436 | 0.235 | 0.261 | – | 0.024 | 0.216 |

[0.01] | [0.25] | – | [0.10] | [0.14] | ||

GISW | – | −0.011 | −0.011 | – | −0.020 | −0.012 |

Panel C: Rolling estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.304 | 0.261 | 0.304 | 0.144 | 0.095 | 0.189 |

[0.73] | [0.99] | [0.35] | [0.21] | [0.47] | ||

GISW | – | −0.002 | 0.000 | −0.005 | −0.008 | −0.004 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.355 | 0.205 | 0.279 | 0.128 | 0.076 | 0.174 |

[0.12] | [0.36] | [0.19] | [0.04] | [0.25] | ||

GISW | – | −0.006 | −0.002 | −0.008 | −0.014 | −0.007 |

Panel C: Rolling estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.304 | 0.261 | 0.304 | 0.144 | 0.095 | 0.189 |

[0.73] | [0.99] | [0.35] | [0.21] | [0.47] | ||

GISW | – | −0.002 | 0.000 | −0.005 | −0.008 | −0.004 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.355 | 0.205 | 0.279 | 0.128 | 0.076 | 0.174 |

[0.12] | [0.36] | [0.19] | [0.04] | [0.25] | ||

GISW | – | −0.006 | −0.002 | −0.008 | −0.014 | −0.007 |

Panel D: Rolling estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.304 | 0.288 | 0.157 | – | 0.105 | 0.139 |

[0.90] | [0.31] | – | [0.25] | [0.32] | ||

GISW | – | −0.001 | −0.005 | – | −0.006 | −0.005 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.355 | 0.228 | 0.156 | – | 0.081 | 0.130 |

[0.14] | [0.19] | – | [0.02] | [0.16] | ||

GISW | – | −0.005 | −0.008 | – | −0.014 | −0.008 |

Panel D: Rolling estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

$$\zeta^{*}_{p}$$ = 0.01 | ||||||

SR | 0.304 | 0.288 | 0.157 | – | 0.105 | 0.139 |

[0.90] | [0.31] | – | [0.25] | [0.32] | ||

GISW | – | −0.001 | −0.005 | – | −0.006 | −0.005 |

$$\zeta^{*}_{p}$$ = 0.02 | ||||||

SR | 0.355 | 0.228 | 0.156 | – | 0.081 | 0.130 |

[0.14] | [0.19] | – | [0.02] | [0.16] | ||

GISW | – | −0.005 | −0.008 | – | −0.014 | −0.008 |

The table reports summary statistics of the returns from alternative portfolios constructed using the out–of–sample forecasts from the benchmark of no–predictability model of bond excess returns with constant volatility and the other competing models. FB, CP, and EH denote Fama and Bliss (1987), Cochrane and Piazzesi (2005), and the no–predictability benchmark, respectively. *Recursive* and *Rolling Estimation* denote forecasts that are generated using all past observations available up to the time of the forecast and a window of past 120 months, respectively. *Parameter Uncertainty* denotes the case when the expectations hypothesis prior is imposed to define the distribution of parameter estimates (Kandel and Stambaugh, 1996; Connor, 1997). The subscripts *CVOL* and *TVOL* denote models whose conditional volatility is estimated as a constant and by means of a rolling sample estimator (Foster and Nelson, 1996; Fleming et al., 2001, 2003), respectively. SR values denote Sharpe ratios achieved by each strategy and computed as the ratio of the sample average to the sample standard deviation of portfolios' excess returns. Values in brackets are $$p$$–values of the null hypothesis that the SR of the model is equal to the one of $$EH_{CVOL}$$ (Ledoit and Wolf, 2008). The $$p$$–values are computed using $$V = 1,000$$ bootstrap replications. GISW is a variant of the Goetzmann et al. (2007) manipulation–proof measure of performance computed as portfolios' premium return above the benchmark after adjusting for risk. The asset allocations for all models are carried out using two annual targets of portfolio excess returns: $$\zeta^{*}_{p}$$ = 0.01,0.02. Time–varying variance–covariance matrices of excess returns are estimated using a decay parameter α = 0.05. GISW values are computed using a relative risk aversion (RRA) coefficient λ = 5. The out–of–sample forecasting exercise runs from January 1970 through December 2010. SR values are reported as annualized, and GISW are reported in decimals per annum (i.e., 0.01 = 1 annual percentage point).

The conclusions are unchanged when we allow for parameter uncertainty (Panel B). In this case, however, the $$EH_{CVOL}$$ model is superior to both the $$FB_{CVOL}$$ and $$CP_{CVOL}$$ models at the 5% significance level using the SR measure in three of the four cases reported. As in the case of no parameter uncertainty, all of the estimates of GISW are again negative and are in a range that is only slightly narrower than that of Panel A.

The conclusions are invariant to the rolling forecasting scheme, reported in Panels C and D. The estimates of SR for the $$EH_{CVOL}$$ are generally smaller than those with recursive estimation. However, the equality null hypothesis is rejected in only two instances. The pattern based on SR is confirmed by estimates of GISW. Indeed, they are either zero or negative and of similar magnitude to those obtained with the recursive scheme.

The fact that none of the models with time-varying volatility are superior to those with constant volatility is consistent with the findings of Duffee (2002) and Cheridito et al. (2007). They find that bond excess returns are best captured by constant volatility models, in spite of the fact that such models cannot match the time-series variation in interest rate volatility.18

The results reported in Table 3 are corroborated by the estimates of the performance fees reported in Table 4, Panels A–D. All of the performance fees are negative, reflecting the fact that none of the alternative models are economically superior to the no-prediction benchmark with constant volatility. The magnitude of the performance fees is relatively unaffected by whether parameter uncertainty is taken into account. The performance fees are frequently less negative, however, when the models are estimated using a rolling scheme rather than recursively. Hence, from the perspective of performance fees, there seems to be a gain from focusing on the most recent data.

Panel A: Recursive estimation, no parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.010 | −0.009 | −0.022 | −0.016 | −0.010 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.008 | −0.012 | −0.022 | −0.018 | −0.009 |

Panel A: Recursive estimation, no parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.010 | −0.009 | −0.022 | −0.016 | −0.010 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.008 | −0.012 | −0.022 | −0.018 | −0.009 |

Panel B: Recursive estimation, parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.014 | −0.007 | – | −0.018 | −0.011 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.011 | −0.005 | – | −0.020 | −0.008 |

Panel B: Recursive estimation, parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.014 | −0.007 | – | −0.018 | −0.011 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.011 | −0.005 | – | −0.020 | −0.008 |

Panel C: Rolling estimation, no parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.001 | −0.001 | −0.005 | −0.008 | −0.003 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.006 | −0.004 | −0.008 | −0.014 | −0.007 |

Panel C: Rolling estimation, no parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.001 | −0.001 | −0.005 | −0.008 | −0.003 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.006 | −0.004 | −0.008 | −0.014 | −0.007 |

Panel D: Rolling estimation, parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.001 | −0.004 | – | −0.005 | −0.004 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.005 | −0.005 | – | −0.011 | −0.007 |

Panel D: Rolling estimation, parameter uncertainty | |||||
---|---|---|---|---|---|

$$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

Performance fees Φ | |||||

$$\zeta^{*}_{p}$$ = 0.01 | −0.001 | −0.004 | – | −0.005 | −0.004 |

$$\zeta^{*}_{p}$$ = 0.02 | −0.005 | −0.005 | – | −0.011 | −0.007 |

The table reports out–of–sample performance fees Φ based on out–of–sample forecasts of mean and variance from competing models against the benchmark of no bond excess returns predictability with constant volatility. The measures are computed for two levels of target portfolio excess returns $$\zeta^{*}_{p}$$ = 0.01,0.02. The performance fees denote the amount that the investor with quadratic utility function and a relative risk aversion (RRA) coefficient λ = 5 would be willing to pay for switching from the model with no excess return predictability and constant volatility to the alternative model. Performance fees are reported in decimals per annum (i.e., 0.01 = 1 annual percentage point). See also notes to Table 3.

## 5. Subsample Analysis

This section refines the results reported in the previous section by assessing the predictive ability of predictive models in different subsample periods. This exercise is motivated by a recent literature suggesting that macroeconomic variables, and more specifically interest rates, have been more difficult to predict since the Great Moderation (e.g., Clark and McCracken, 2008; D'Agostino et al., 2006). Specifically, we investigate the economic value of the predictive ability of the FB and CP models over four subsamples: January 1970 through December 1979, January 1980 through December 1989, January 1990 through December 1999, and January 2000 through December 2010. The results of this exercise are summarized in Table 5, Panels A–D, which report the SR from the various models together with the $$p$$-values from Ledoit and Wolf (2008) test. The measures are computed using a level of annual target of portfolio excess returns $$\zeta^{*}_{p}$$ = 0.01, and the other parameters are set equal to the values used to carry out the baseline estimates reported in Tables 3 and 4.

Panel A: Recursive estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.137 | −0.045 | −0.019 | −0.035 | −0.041 | 0.165 |

[0.78] | [0.67] | [0.78] | [0.77] | [0.35] | ||

1980–1989 | 0.385 | 0.199 | 0.219 | −0.067 | 0.019 | 0.196 |

[0.42] | [0.28] | [0.32] | [0.23] | [0.38] | ||

1990–1999 | 0.541 | 0.371 | 0.405 | −0.033 | 0.059 | −0.016 |

[0.64] | [0.40] | [0.41] | [0.21] | [0.15] | ||

2000–2010 | 0.856 | 0.460 | 0.426 | −0.236 | 0.404 | 0.441 |

[0.13] | [0.09] | [0.12] | [0.11] | [0.16] |

Panel A: Recursive estimation, no parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.137 | −0.045 | −0.019 | −0.035 | −0.041 | 0.165 |

[0.78] | [0.67] | [0.78] | [0.77] | [0.35] | ||

1980–1989 | 0.385 | 0.199 | 0.219 | −0.067 | 0.019 | 0.196 |

[0.42] | [0.28] | [0.32] | [0.23] | [0.38] | ||

1990–1999 | 0.541 | 0.371 | 0.405 | −0.033 | 0.059 | −0.016 |

[0.64] | [0.40] | [0.41] | [0.21] | [0.15] | ||

2000–2010 | 0.856 | 0.460 | 0.426 | −0.236 | 0.404 | 0.441 |

[0.13] | [0.09] | [0.12] | [0.11] | [0.16] |

Panel B: Recursive estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.137 | −0.090 | 0.053 | – | −0.072 | 0.190 |

[0.85] | [0.56] | – | [0.82] | [0.26] | ||

1980–1989 | 0.385 | 0.204 | −0.026 | – | −0.142 | 0.043 |

[0.42] | [0.10] | – | [0.18] | [0.31] | ||

1990–1999 | 0.541 | 0.365 | 0.267 | – | 0.061 | 0.140 |

[0.66] | [0.09] | – | [0.30] | [0.06] | ||

2000–2010 | 0.856 | 0.465 | 0.671 | – | 0.396 | 0.354 |

[0.13] | [0.38] | – | [0.07] | [0.21] |

Panel B: Recursive estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.137 | −0.090 | 0.053 | – | −0.072 | 0.190 |

[0.85] | [0.56] | – | [0.82] | [0.26] | ||

1980–1989 | 0.385 | 0.204 | −0.026 | – | −0.142 | 0.043 |

[0.42] | [0.10] | – | [0.18] | [0.31] | ||

1990–1999 | 0.541 | 0.365 | 0.267 | – | 0.061 | 0.140 |

[0.66] | [0.09] | – | [0.30] | [0.06] | ||

2000–2010 | 0.856 | 0.465 | 0.671 | – | 0.396 | 0.354 |

[0.13] | [0.38] | – | [0.07] | [0.21] |

Panel C: Rolling Estimation, No Parameter Uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.243 | −0.126 | −0.103 | 0.226 | 0.124 | 0.082 |

[0.59] | [0.42] | [0.15] | [0.12] | [0.24] | ||

1980–1989 | 0.481 | 0.241 | 0.291 | 0.009 | −0.147 | 0.366 |

[0.06] | [0.30] | [0.25] | [0.02] | [0.77] | ||

1990–1999 | 0.737 | 0.462 | 0.525 | 0.291 | 0.333 | −0.026 |

[0.17] | [0.18] | [0.24] | [0.62] | [0.19] | ||

2000–2010 | 0.709 | 0.455 | 0.462 | 0.155 | 0.359 | 0.285 |

[0.27] | [0.18] | [0.10] | [0.25] | [0.13] |

Panel C: Rolling Estimation, No Parameter Uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.243 | −0.126 | −0.103 | 0.226 | 0.124 | 0.082 |

[0.59] | [0.42] | [0.15] | [0.12] | [0.24] | ||

1980–1989 | 0.481 | 0.241 | 0.291 | 0.009 | −0.147 | 0.366 |

[0.06] | [0.30] | [0.25] | [0.02] | [0.77] | ||

1990–1999 | 0.737 | 0.462 | 0.525 | 0.291 | 0.333 | −0.026 |

[0.17] | [0.18] | [0.24] | [0.62] | [0.19] | ||

2000–2010 | 0.709 | 0.455 | 0.462 | 0.155 | 0.359 | 0.285 |

[0.27] | [0.18] | [0.10] | [0.25] | [0.13] |

Panel D: Rolling estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.243 | −0.129 | 0.304 | – | 0.040 | 0.177 |

[0.64] | [0.04] | – | [0.32] | [0.13] | ||

1980–1989 | 0.481 | 0.263 | −0.054 | – | −0.132 | 0.172 |

[0.06] | [0.01] | – | [0.04] | [0.46] | ||

1990–1999 | 0.737 | 0.519 | 0.021 | – | 0.403 | 0.158 |

[0.69] | [0.06] | – | [0.69] | [0.29] | ||

2000–2010 | 0.709 | 0.452 | 0.253 | – | 0.368 | 0.049 |

[0.20] | [0.11] | – | [0.23] | [0.08] |

Panel D: Rolling estimation, parameter uncertainty | ||||||
---|---|---|---|---|---|---|

$$EH_{CVOL}$$ | $$FB_{CVOL}$$ | $$CP_{CVOL}$$ | $$EH_{TVOL}$$ | $$FB_{TVOL}$$ | $$CP_{TVOL}$$ | |

1970–1979 | −0.243 | −0.129 | 0.304 | – | 0.040 | 0.177 |

[0.64] | [0.04] | – | [0.32] | [0.13] | ||

1980–1989 | 0.481 | 0.263 | −0.054 | – | −0.132 | 0.172 |

[0.06] | [0.01] | – | [0.04] | [0.46] | ||

1990–1999 | 0.737 | 0.519 | 0.021 | – | 0.403 | 0.158 |

[0.69] | [0.06] | – | [0.69] | [0.29] | ||

2000–2010 | 0.709 | 0.452 | 0.253 | – | 0.368 | 0.049 |

[0.20] | [0.11] | – | [0.23] | [0.08] |

The table reports out–of–sample the Sharpe ratios based on out–of–sample forecasts of mean and variance from competing models and the benchmark of bond excess returns' no–predictability with constant volatility. The measures are computed for a level of annual target of portfolio excess returns $$\zeta^{*}_{p} = 0.01$$ over the four different subperiods. Values in brackets are $$p$$–values of the null hypothesis that the SR of the model is equal to the one of $$EH_{CVOL}$$ (Ledoit and Wolf, 2008). See also notes to Tables 3 and 4.

The results for the different subperiods confirm the conclusion obtained using the full sample. For every period except the decade of the 1970s, for all models and for all forecasting schemes the SR from the benchmark $$EH_{CVOL}$$ is higher than that from the competing models. However, there are only a few instances where the economic performance of the $$EH_{CVOL}$$ model is statistically superior at the 10% significance level. During the 1970s, all of the competing models generate an SR larger than that of the $$EH_{CVOL}$$ model; however, there is only one instance where the difference is statistically significant at the 10% level.

It is interesting to note that the performance of the predictive models relative to the $$EH_{CVOL}$$ model deteriorated over time—that is, the difference between the SR of the $$EH_{CVOL}$$ and any competing model generally increased over time. Indeed, the differences are almost always the largest during the 2000s. This performance deterioration is also reflected in the GISW measure for the FB and CP predictive regressions (Figure 3). The deterioration is smaller for the performance fees (Figure 4). Both figures plot the average of the performance measures computed over the decade of reference across all twenty-four specifications, as well as their across-specification standard deviation. The average GISW performance measures and the average performance fees are positive during the 1970s, but they deteriorate quickly during the 1980s, remaining negative until the end of the sample. These results are generally consistent with the notion that forecasting has become more difficult since the Great Moderation. The general pattern of deterioration in these performance measures may be associated with shifts in monetary policy in the late 1980s (see, inter alia, Sims and Zha 2006; Thornton 2006, 2010, the references therein), which generated a greater persistence in the Fed's target and induced less predictability in excess returns.

We also compute the performance measures over the period of the recent financial crisis from January 2007 through December 2009. This analysis is motivated by the evidence that suggests that the predictive ability of the FB model broke down during the recent financial crisis, as also recently pointed out by John Cochrane on his website. Figure 5 plots the GISW performance measures during the crisis period. It is interesting to note that both FB and CP record negative GISW measures, and the values are nearly four times larger than the ones recorded over the full sample period across various specifications, especially when time-varying volatility is taken into account. The rolling forecasting scheme seems to mitigate this negative performance—however, only for FB with constant volatility and CP with constant volatility when parameter uncertainty is not taken into account. During this period of high uncertainty, the predictive models did not provide evident economic gains to investors seeking to rebalance their portfolios.

## 6. Robustness

This section checks the robustness of the baseline results reported in Section 4.2. More specifically, we test whether our results are sensitive to (i) different rolling window sizes; (ii) different values of the RRA coefficient, λ; and (iii) different values of the decay parameter α used to calculate the rolling sample estimator of the variance-covariance matrix of bond excess returns. We show that our main results are robust to all of these issues.19

The first robustness exercise involves the consideration of different window sizes used to carry out the rolling forecasting scheme. Specifically, we consider a rolling window of 240 months. A longer rolling window does not change the conclusions qualitatively. Indeed, the results are quantitatively similar to the ones reported in Table 3. Virtually all of the competing models record SR that are lower than the ones exhibited by the benchmark $$EH_{CVOL}$$. The only exception, as in Table 3, is represented by $$CP_{CVOL}$$ that records (i) SR that are slightly higher than the benchmark and (ii) positive but very small performance fees. However, very few of the differences in SR are statistically significant at conventional levels.

As a second robustness test, we consider two alternative values of the RRA coefficient λ = 2,3. In all cases, the results are qualitatively and quantitatively similar to the ones reported in Tables 3 and 4. The performance measures tend to increase in absolute value for a lower RRA coefficient. As investors become less risk averse, the evidence against the predictive models strengthens in favor of the $$EH_{CVOL}$$ benchmark.

The final test considers two alternative values of the decay parameter α = 0.01,0.10. The results show that our baseline findings do not hinge on the selected value of the decay parameter. Indeed, they are virtually identical to those reported in Tables 3 and 4.

## 7 Conclusions

This study investigates the economic gains accruing to an investor who exploits the predictability of bond excess returns relative to the no-predictability alternative consistent with the expectations hypothesis. In particular, we quantify how much a risk-averse investor is willing to pay to switch from a dynamic portfolio strategy based on a model with no predictable bond excess returns to a model where the forecasts are based on either forward spreads or the term structure of forward rates.

The results show that the no-predictability benchmark is difficult to beat in economic terms by either of the competing forward-rate models. The evidence in favor of naive models replicates the one already recorded in similar or unrelated contexts (see, among others, Goyal and Welch 2008; Guidolin and Thornton 2010 and the references therein). Indeed, the predictive regressions do not record any significant economic value over the no-predictability benchmark. The extent of the underperformance varies across specifications. Generally, it is larger when model parameters are estimated recursively and parameter uncertainty is taken into account. Moreover, the forecasts of the variance-covariance matrix of excess returns computed by a rolling sample estimator generally do not improve upon the performance of the predictive regressions with constant volatility. Importantly, the qualitative conclusions are robust to the sample period as well as the value of the key parameters used in our baseline estimation. We also found that the performance of all predictive models based on forward rates deteriorates over time. Indeed, the relative performance of these models is generally worse in the first decade of the 2000s. Overall, our findings confirm that it is very difficult to improve upon a simple naive benchmark and that the predictability of bond excess returns found in the literature does not necessarily translate into economic gains for investors who rely on forecasts from these models.

This work was partly written while Giorgio Valente was visiting the Federal Reserve Bank of St. Louis, whose hospitality is gratefully acknowledged. The views expressed here are the authors' and do not necessarily reflect the views of the Board of Governors of the Federal Reserve System or the Federal Reserve Bank of St. Louis. The authors are grateful to Rob Bliss, John Cochrane, Pasquale Della Corte, Eric Engstrom, Massimo Guidolin, Mike McCracken, Monika Piazzesi, Lucio Sarno, and Ton Vorst; to participants in the 2010 Workshop on Yield Curve Modeling and Forecasting, Erasmus University Rotterdam; the 6th International Conference on Asia Pacific Financial Markets, Seoul, Korea; 2012 Inquire UK and Inquire Europe Spring Seminar, Budapest; as well as to seminar participants at the Bank for International Settlements, Bank of Italy, City University of Hong Kong, Hong Kong Institute for Monetary Research, Hong Kong Lingnan University, KAIST Business School, National Chengchi University, for useful conversations and comments on earlier drafts of this article. The authors alone are responsible for any errors that may remain.

^{1}However, not all studies are supportive of the predictive power of forward rates. In fact, Hamburger and Platt (1975), Fama (1984), and Shiller et al. (1983) find weak evidence that forward rates predict future spot rates.

^{2}The predictability of excess returns recorded in these studies strongly corroborates the well-documented empirical failure of the expectations hypothesis of the term structure of interest rates (Fama, 1984; Stambaugh, 1988; Bekaert et al., 1997, 2001; Sarno et al., 2007), and it is generally assumed to be the consequence of the slow mean reversion of the spot rate toward a time-invariant equilibrium anchor that becomes more evident over longer horizons (Fama 1984, 2006, and the references therein).

^{3}For a comprehensive overview of portfolio choice problems, see Brandt (2010) and the references therein.

^{4}This argument is echoed in Dai et al. (2004) and Singleton (2006), who show that these predictive regressions are affected by a small-sample bias that causes the

*R*

^{2}statistics to be substantially higher than their population values. The evidence of the near unit-root nature of bond yields is strengthened by other studies that record that the slow mean reversion of the spot rate toward a constant is no longer valid after 1986 (Fama, 2006) and its dynamics are better approximated by a mean-reverting process that is anchored to a nonstationary central tendency that stochastically changes over time (Balduzzi et al., 1998). Duffee and Stanton (2008) also show that the high persistence of interest rates has important implications for the preferred method used to estimate term structure models. Strictly speaking, Ferson et al. (2003a, 2003b) assume that the regressand does not exhibit high serial correlation but the regressor (predictor) does. However, the spurious regression may occur even without a highly autocorrelated regressand if its conditional mean is highly correlated.

^{5}It is interesting to note that Cochrane and Piazzesi (2006) also investigate trading rule profits based on the CP predictive model. Their results are supportive of their in-sample evidence; however, the real-time profits are about half of those obtained over the full sample. Cochrane and Piazzesi (2006, p. 12) point out that “real trading rules should … follow an explicit portfolio maximization problem. They also must incorporate estimates of the conditional variance of returns.” These features are key ingredients of our empirical investigation.

^{6}The new indexation in Equations (4) and (5) reflects the fact that data are sampled at a monthly frequency while bond maturities are of one year and above, and hence a multiple of twelve months.

^{7}It is instructive to note that the FB regression can be obtained from the CP regression by imposing that ζ

_{1}= δ

_{q}= −δ

_{1}for

*q*=

*n*, δ

_{q}= 0 for

*q*≠

*n*and

*q*> 0.

^{8}The estimation problems are strongly exacerbated in the context of bond excess returns, where the high correlations across bond maturities often cause variance-covariance matrices to be near singular.

^{9}In our study, we do not implement an analog parameter uncertainty correction for the conditional volatility model for various reasons. First, it is well known that volatility is highly persistent and therefore predictable. Hence, the approach advocated by Kandel and Stambaugh (1996), which implies a prior of no predictability, is inappropriate. Furthermore, the scope of the article is to assess the economic value of the predictability of bond excess returns by means of forward rates in the conditional mean. The extensions proposed in this study, which consider the case of predictable bond excess return volatilities, are included for completeness, but they do not represent the main focus of the article.

^{10}See Brandt (2010) and the references therein.

^{11}The marginal

*R*

_{j}

^{2}is defined as the full

*R*

^{2}from Equation (6) minus the

*R*

^{2}from the equation where the variable

*j*is dropped from the model. For further details, see Connor (1997, p. 50).

^{12}A critical aspect of this analysis is that it relies, as in previous studies, on the assumptions of the normality of bond returns and quadratic utility function. Although the quadratic utility assumption is appealing for its tractability properties, it not necessary to justify the use of mean-variance optimization (Della Corte, Sarno, and Tsiakas 2009, p. 3501).

^{13}Full details of the bootstrap procedure are reported in the appendix to the Working Paper version of this study (Thornton and Valente, 2012).

^{14}In particular, we have estimated over the full sample period a multivariate GARCH(1,1) model using the residuals from the EH model—that is, where the conditional mean of bond excess returns is equal to a constant. Then we carried out the LM test of conditional correlation by Tse (2000) on the multivariate GARCH model estimated assuming a constant conditional correlation. The result is not reported to save space, but available from the authors upon request.

^{15}We have carried out the symmetry tests assuming that the mean equation contains only an intercept and an intercept and the predictive variables. In both cases, the results lead to the same conclusion.

^{16}We thank the anonymous referee for suggesting this to us.

^{17}The sensitivity of our baseline results to the choice of the relevant parameters is assessed in Section 6.

^{18}However, a notable exception is represented by Almeida et al. (2011), who find, using the information embedded in interest rate options, that the most successful models for predicting excess returns have risk factors with stochastic volatility.

^{19}The full set of results is reported in the appendix to the Working Paper version of this study (Thornton and Valente, 2012).

## References

*Handbook of Financial Econometrics*, vol. 1