RDP 2008-09: A Term Structure Decomposition of the Australian Yield Curve 2. Model Overview and Related Literature
December 2008
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The focus of this paper is the estimation of an affine term structure model for Australian interest rates, with the aim of decomposing forward rates into expected future short rates and term premia. While mathematical details of the model are given in Section 3, a brief description of the model here provides the reader with some intuition regarding what is to follow.
We start by estimating zero-coupon yield curves from observed overnight indexed swap (OIS) and government bond data (for further details see Section 4 and Appendix A). These, along with analysts' forecasts of future interest rates, constitute the data used to estimate our term structure model.
Our term structure model describes how the cash rate might evolve. The model assumes that the cash rate can be expressed as a constant plus the sum of three latent factors, which in turn follow the continuous time equivalent of a vector auto-regressive process with normally distributed shocks. Each latent factor is assumed to have zero mean, so that according to our model, the cash rate has a constant long-run steady-state value. The cash rate moves away from this steady-state value when shocks cause the latent factors to move away from zero.
Arbitrage conditions allow us to link bond prices to the evolution of the cash rate. In a world where investors are risk-neutral, the price of a zero-coupon bond would be given by the expectation of the bond's discounted future pay-off, where discounting is with respect to the cash rate process just described. However, investors need not be risk-neutral. If they are risk-averse, they may require extra compensation for holding a bond whose value fluctuates, as opposed to cash whose value does not. This extra compensation can be considered as the term premium.
However, exactly how investors' risk preferences collectively affect term premia is not clear a priori. On the one hand, it is reasonable to think that investors should be compensated for holding long-term bonds over cash, since the value of long-term bonds can fluctuate and thus expose investors to the possibility of mark-to-market losses. On the other hand, for investors who have long-term fixed liabilities, a long-term bond for which the value at maturity is fixed may be less risky than a cash account for which the value will depend on the variable path of short-term interest rates. Term premia could therefore be positive or negative, depending on the mix of investors trading bonds.
Hence, bond prices (and therefore observed yields) depend on both expected future short rates and term premia. Of course observations of bond yields alone are not sufficient to separately identify these two components. We can get information about expected future short rates separate from term premia in two ways. First, we can obtain estimates of the latent factors which can be used to derive expected future short rates. Second, we can augment the zero-coupon yield data with analysts' forecasts of future interest rates when estimating the model – forecasts of the future cash rate are a direct reading of expected future short rates separate from term premia, and so aid in the estimation of the actual short rate process.
The latent factors are not observable, but must be estimated along with the parameters of the model. We use the Kalman filter and maximum likelihood to estimate the latent factors and parameters. The latent factors are estimated so as to provide the best fit possible between the model's implied yields and the actual observed yields. Although no economic structure is imposed on them, the latent factors tend to explain different components of the yield curve. Typically one latent factor is highly correlated with the level of the yield curve, another is correlated with the slope of the yield curve, and the third is correlated with the curvature of the yield curve.
The model of interest rates just described builds on a modelling approach that was first proposed in Duffie and Kan (1996). That work introduces the affine term structure model, an arbitrage-free multifactor model of interest rates in which the yield on any risk-free zero-coupon bond is an affine function of a set of unobserved latent factors. Duffie and Kan also provide a method to obtain the coefficients on the latent factors in the affine function and therefore to price risk-free zero-coupon bonds. The improvement of this model on the previous literature is that it is scaleable, driven by estimable factors which have arbitrary correlation, while at the same time retaining a good level of tractability.
de Jong (2000) implements this model on Treasury yield data from the United States. He estimates one-, two- and three-factor versions of the model, concluding that the one- and two-factor versions are misspecified, but that the three-factor version seems to do a good job of capturing the relevant dynamics of yields. de Jong uses a Kalman filter in estimating the models, which has the advantage that it provides tractable estimation when there are more input yields than factors. Consequently, it has become the most common technique for estimating affine term structure models.
Duffee (2002) generalises the specification of the market price of risk used by Duffie and Kan (1996) and de Jong (2000). He removes the restriction that compensation for interest rate risk must be a multiple of the variance of that risk and suggests a modification which allows it to move independently of the variance. Duffee estimates this new variant (called the ‘A_{0}(3)’ model), the original model and a hybrid model, and demonstrates that the extra flexibility of the A_{0}(3) model provides significant improvements to goodness-of-fit.
Dai and Singleton (2002) implement various specifications of the Duffee (2002) model on US data. They show that while regular yields fail the expectations hypothesis, the ‘risk-premium adjusted’ yields from the A_{0}(3) model satisfy the expectations hypothesis. A further contribution of Dai and Singleton is that they also provide analytical formulae for the coefficients of the affine function, enabling simpler estimation than the method of Duffie and Kan (1996).
Kim and Orphanides (2005) take the A_{0}(3) model of Duffee (2002) but incorporate survey data of analysts' forecasts of short-term interest rates as an additional input to the estimation problem. Using US data, they estimate models both with and without the forecasts and find that those models that incorporate forecasts produce a better fit. Monte Carlo trials suggest that the inclusion of forecasts helps to reduce small-sample problems arising in the estimation of highly persistent factors, especially when data sets of only limited length are available. They find that between the early 1990s and 2003, term premia in the US fell and that the fall was tied to the moderation of macroeconomic volatility seen over the period. The fall in term premia helps to explain the fall in treasury yields also observed. The model used in this paper is a variation of the Kim and Orphanides model, changed slightly to accommodate the different nature of our survey data.
Affine term structure models have also been implemented at other central banks. Kremer and Rostagno (2006) from the European Central Bank use a two-factor affine term structure model to examine the low bond yields observed in the euro area over the first half of this decade. They find a sharp reduction in estimated term premia, indicating that a reduction in risk compensation may have been driving yields lower. In addition, the term premia are found to be related to measures of liquidity, suggesting that excess liquidity may also have been playing a part in driving risk aversion down.^{[3]}
Westaway (2006) also finds falling term premia in the United Kingdom. Given the complexities of the model and the fact that term premia are in effect residuals of the model he is, however, somewhat cautious in interpreting the results. Westaway estimates a dynamic stochastic general equilibrium (DSGE) model of a closed economy and finds that a decline in the volatility of economic shocks should lead to lower term premia, a result consistent with the term structure model. However, the DSGE model does not result in an overall fall in real yields and so cannot fully account for the low level of yields observed.
More broadly, the strategy of incorporating time-varying term premia in modelling long-run interest rates is a response to extensive empirical evidence contradicting the pure expectations hypothesis; that is, evidence that long-run interest rates are not simply an average of the expected path of future short-term interest rates. In particular, studies have found that long-run interest rates display both ‘excess volatility’ (fluctuating more than would be expected given the volatility of the underlying macroeconomy) and ‘excess sensitivity’ (responding to information that might be expected to only influence short-term rates).^{[4]}
The estimation approach used in this paper belongs to the ‘pure-finance’ branch of the term structure literature, where term premia are estimated using observed yield data and perhaps some survey forecast data. This is opposed to the ‘macro-finance’ branch, typified by Rudebusch and Wu (2008), where the interaction between the macroeconomy and the term structure is also modelled. As noted by Kim and Wright (2005), pure-finance models, which rely on latent factors to explain the yield curve, generally have the advantage of being more robust to model misspecification, and provide a better fit to the data, than macro-finance models. Conversely, although macro-finance models generally do not fit the data as well as pure-finance models, they may be easier to interpret from an economic viewpoint given the structure that they impose.
One criticism of the term structure literature is given in Swanson (2007), who argues that different modelling techniques result in different term premia estimates, so that some degree of caution must be placed on any term premia estimate. The criticism is a reasonable one – term premia by their nature are hard to estimate since their effect on observable bond prices is confounded with expectations of future short-term rates. On the other hand, it is not entirely surprising that different modelling techniques, which make different assumptions about financial markets and the economy, should produce different results. A useful survey paper on this topic is that of Rudebusch, Sack and Swanson (2007), who review five alternative term premia estimation methodologies. They find that although different models do produce different term premia estimates, the estimates are generally not too different.^{[5]}
We make some modest contributions to these term structure models; we extend the Kim and Orphanides (2005) model to accommodate a different type of forecast data (cash rate and 10-year bond yield forecasts as opposed to treasury note forecasts), and we extend the zero-coupon yield estimation method to allow the model to account for the actual cash rate prevailing at any given time. Our larger contribution is the estimation of zero-coupon bond yields, and a linear affine term structure model, for Australia.
Footnotes
In this context, excess liquidity refers to the amount of money and liquid assets circulating in the economy. [3]
See Gürkaynak, Sack and Swanson (2003) or Beechey (2004) for an overview of this literature. [4]
Rudebusch et al (2007) consider term premia estimates for US data arising from five different term strucuture models, one of which is equivalent to the term structure model which we use. They find that the model which is equivalent to our model produces term premia which are very similar to those produced by two other models (correlation coefficients of 0.98 and 0.94); that the model which is equivalent to our model produces term premia which are very similar, except for a level shift, to another model (correlation coefficient of 0.96); and finally, that the last model (correlation coefficient of 0.81) produces term premia which are less similar to all other models for theoretical reasons regarding modelling assumptions. [5]