تست مدل ساختار مدت تکراری در مورد هزینه های معاملاتی

We empirically analyze the impact of transaction costs on the performance of essentially affine interest rate models. We test the implied Euler restrictions and calculate the specification error bound of Hansen and Jagannathan to measure model misspecification. Using both short-maturity and long-maturity bond return data we find, under the assumption of frictionless markets, strong evidence of misspecification of affine yield models with up to three factors. Next, we incorporate transaction costs in our tests. The results show that the evidence of misspecification of essentially affine yield models disappears in case of monthly holding periods at market size transaction costs.

Nowadays term structure models are used extensively for many purposes, including risk management of portfolios containing bonds and the valuation of interest-rate derivatives. Empirical tests of term structure models have therefore attracted considerable attention in the literature. In line with a large part of the empirical asset pricing literature, the tests are based on the assumption of trading in frictionless markets. In particular, the large literature on affine term structure models1 tests these models using data on Treasury bills and bonds under the assumption of trading in frictionless markets. However, market frictions such as transaction costs are an important fact of life for investors. The implicit assumption when ignoring transaction costs is that these costs are sufficiently small, so that they do not seriously affect the empirical results. In this paper we explicitly take transaction costs into account in the empirical testing of affine term structure models, and show that including market size transaction costs can considerably affect the results of the tests. Our approach is to test whether the stochastic discount factor of a given term structure model satisfies the Euler restrictions. These Euler restrictions are implied by the no-arbitrage assumption, and can be derived in both frictionless markets and markets with frictions. Based on these Euler restrictions, we use two approaches to analyze and test the models. First, we use Wald-type tests to test the Euler restrictions. For the frictionless case, the analysis of Euler restrictions using Wald tests is extensively discussed by Cochrane, 1996 and Cochrane, 2001. In case of transaction costs, we use tests of inequality restrictions adopting the approach developed by Kodde and Palm (1986). A disadvantage of this approach is that, if one rejects a model, there is no clear indication of the direction of misspecification, for example, which individual assets are possibly mispriced by the model and which are not. Also, the Wald test does not allow for a comparison of the degree of misspecification of two non-nested models that are both rejected. To overcome these problems we also consider the specification error bound (SEB) developed by Hansen et al. (1995) and Hansen and Jagannathan (1997). This bound measures the extent to which a model misprices a given set of assets. Hansen and Jagannathan (1997) show that this bound can be interpreted as the maximum pricing error for all portfolios that can be constructed from the assets under consideration. This specification error bound allows for direct comparison across (non-nested) models and the method indicates which (portfolios of) assets contribute most to the misspecification. Hansen et al. (1995) extend the setup of Hansen and Jagannathan (1997) to allow for market frictions. We apply their approach to affine term structure models and compare the results with standard tests using the Euler restrictions. Our work is related to Luttmer (1996) and He and Modest (1995), who both analyze the influence of transaction costs and other market frictions on the size of the volatility bounds of Hansen and Jagannathan (1991), that give a lower bound on the variability of valid stochastic discount factors. Luttmer (1996) finds that small transaction costs greatly influence the size of the volatility bounds; especially the volatility bounds based on T-bill returns are very sensitive to the size of transaction costs. The results of Luttmer (1996) imply that the conclusion of rejection of several asset pricing models in Hansen and Jagannathan (1991), based on the volatility bounds, changes if transaction costs are taken into account. Our work extends the work of Luttmer (1996), because the volatility bound is a special case of the specification error bound, and because Luttmer (1996) focuses on consumption-based asset pricing models, whereas we analyze bond pricing models and bond returns. The bond pricing models that we consider are discrete-time versions of the affine yield models of Duffie and Kan (1996) and the extension to essentially affine models due to Duffee (2002). These latter models deviate from the Duffie and Kan (1996) models—referred to as completely affine yield models in this context—by allowing the market price of risk to depend in a non-affine way on the factors. In our investigation we follow the classification of these models as proposed by Dai and Singleton (2000), consisting of a partitioning into subclasses—for each given number of factors—depending on the way the factors affect the volatility of the process generating the uncertainty. We consider two and three factor models, since these are commonly studied in the literature mentioned above. Given the number of factors, we compare the various subclasses, and select a most preferred subclass, for which we present our results. To relate our tests of Euler restrictions to existing empirical work on affine models, we show that the Euler restrictions can be rewritten into restrictions on unconditional and conditional expected returns on bonds of different maturities. In line with the literature on tests of the expectation hypothesis (Fama and Bliss, 1987 and Campbell and Shiller, 1991), which shows that the spread between a long-term and short-term interest rate predicts future bond returns, we choose as conditioning variable this yield spread. This way, our test restrictions on conditional expected bond returns are related to Dai and Singleton (2002) and Duffee (2002), who analyze to what extent essentially affine models can reproduce the predictability of bond returns by the yield spread.2 However, compared to these two articles, our set of test restrictions is larger since we also include restrictions on the term structure of unconditional expected bond returns in our tests. Before discussing the empirical results, we perform a simulation analysis to analyze the power of the Wald-test on the Euler restrictions. We simulate a three-factor affine model, adding transaction costs, and then test a two-factor affine model (both for the case with frictionless markets and for the case with transaction costs). Although the power of the tests weakens somewhat if the test incorporates transaction costs, we find that the test does have reasonable power to reject the two-factor model in this setup. Our empirical results indicate that, assuming no market frictions, the term structure of average bond returns is less smooth than predicted by the affine models: both the completely affine yield models and the essentially affine yield models significantly misprice the returns on portfolios that contain both extreme long and short positions in near-maturity bonds. In particular, we find that two-factor models are clearly rejected, in line with the existing literature. Furthermore, even the more general essentially affine three-factor models are statistically rejected if we test the Euler restrictions under the assumption of frictionless markets. Dai and Singleton (2001) and Duffee (2002) show that some essentially-affine models are capable of fitting the observed predictability of bond returns. Our results show that, if both restrictions on unconditional and conditional expected returns on bonds with different maturities are tested, all essentially affine models are rejected. Instead of extending the essentially affine three-factor model further, we allow for transaction costs in our tests. We find that, when transaction costs are of market size, the conclusions above need a more carefully balanced appraisal. In case of a monthly holding period, the evidence of misspecification of the considered models disappears when these transaction costs are included. Because of the transaction costs, the portfolios with both long and short positions in T-bills and bonds are no longer mispriced. For quarterly holding periods and market size transaction costs, the results are mixed: the models are not rejected on the basis of data on long-maturity bond returns, but these models do misprice the short-maturity T-bills. However, Duffee (1996) provides evidence that T-bill returns with very short maturities contain a large idiosyncratic component, possibly due to market segmentation. This might partially explain the difficulty the models have in pricing short-maturity T-bills. We supplement our empirical analysis with a second simulation study in order to gain further insights into our findings. We simulate a two-factor affine model with transaction costs, and test three-factor affine models without allowing for transaction costs. The results show that the three-factor models are clearly rejected in this case. This is in line with the empirical results where we also find that three-factor models are rejected in frictionless markets. These results support our conclusion that the presence of transaction costs can lead to a rejection of appropriate models if one tests under the assumption of frictionless markets. The remainder of this paper is organized as follows. In Section 2, we briefly review the literature on affine term structure models. In Section 3, we first summarize the literature on asset pricing in markets with frictions, then we describe a Wald-test of the Euler restrictions in such a market with frictions, and we discuss the specification error bound. In Section 4, after describing our data set and estimation procedures, we present the empirical test results and results from the simulation studies. In Section 5 we summarize and conclude.

In this paper we analyze the bond pricing implications of both completely and essentially affine term structure models, with up to three factors, allowing for the presence of transaction costs. The goal of the paper is to assess the importance of incorporating market frictions for tests of bond pricing models. Our tests focus on Euler equations, which can rewritten into restrictions on expected (conditional) returns on bonds of different maturities. By including the yield spread as conditioning variable, our tests include the implications of the affine models for the predictability of bond returns as studied by Dai and Singleton (2002) and Duffee (2002). However, our set of test restrictions is larger since we also include the term structure of unconditional expected bond returns. We test two- and three-factor preferred models formally for different sizes of transaction costs, using a Wald-test, and we measure the size of misspecification of these models and analyze how sensitive the misspecification size is to the size of the transaction costs. Our analysis can be seen as an extension of Luttmer (1996), because we use the stronger specification error bound test, as opposed to the volatility bound that is used by Luttmer (1996). Also, Luttmer (1996) focuses on consumption-based asset pricing models, whereas we analyze models for the term structure of interest rates. We find that, under the assumption of frictionless markets, completely and essentially affine yield models with up to three factors in general misprice T-bill and bond returns in a significant way. The term structure of average bond returns is less smooth than predicted by the model, so that long-short portfolios of near-maturity bonds are significantly mispriced. However, if we take transaction costs of market size into account, we find that the misspecification of the models disappears, in case of a monthly holding period. For quarterly holding periods and using market size transaction costs, the models fit long-maturity bond returns well, and are only rejected on the basis of short-maturity T-bill returns.