Return-Predicting Factors For Us Treasuries: On The Similarity Of "Tents" And "Bats"

Different authors find optically very different patterns ("tents" and "bats") when excess returns from US Treasuries are regressed against forward rates. A separate source of disagreement is whether the recent tent-shaped factor found by Cochrane & Piazzesi (2004, 2005, 2008) is fundamentally different from the slope (in yields) explanatory factor for the time-dependent term premia that has been identified since the early 1990s. We show that "tent" and "bat" patterns produce predictions of excess returns that are economically indistinguishable, both locally and globally. We also argue that, if the slope (in yield space) is indeed the most important explanatory factor for excess returns, then a simple transformation of the loadings of the forwards rates on the slope factor naturally recovers an approximate tent shape in forward-rate space. However, a transformation from a slope factor in yields to loadings for forward rates does not easily produce a bat pattern. This may provide indirect corroboration to Cochrane & Piazzesi's (2004) claim that the bat factor is a result of near colinearity for smoothed data, and that the "tent" solution is intrinsically different from the "bat" solution.