Long-term Forward Rates Research Articles

Bond Risk Premia: The Information in Really Long-Maturity Forward Rates

Long term forward rates contain information that greatly improves the precision with which expectations of future short rates can be distinguished from risk premia in the term structure. Indeed, in affine models, the slope of the term structure of risk premia for long maturities is very closely approximated by the sum of (i) the slope of the forward rate curve and (ii) a term that depends only of yield volatility and maturity (convexity), two quantities that can easily be estimated independently of the details of model specification. Key to extracting the risk premium information in long-term forward rates is capturing the dynamics of convexity, which requires a model with time-varying volatility. Using a four-factor ATSM, we find that risk premia on long term bonds are almost entirely driven by volatility. Short rate expectations in our model account for a much larger fraction of the volatility of yields than is typically reported, a result that we show is mainly the result of econometric bias in some estimates of Gaussian models. We also show - using Monte Carlo simulation - that including data on long-term yields in the estimation of term structure models greatly improves the precision of estimated values of short rate expectations, term premia and risk premia. Compared with benchmark estimates that use yield data up to 25 years, excluding data on yields longer than 10 years results in the standard errors of both estimated means and volatilities roughly tripling in size.

The impact of quantitative easing on the US term structure of interest rates

This paper estimates the impact of the Federal Reserve’s 2008–2011 quantitative easing (QE) program on the US term structure of interest rates. We estimate an arbitrage-free term structure model that explicitly includes the quantity impact of the Fed’s trades on Treasury market prices. As such, we are able to estimate both the magnitude and duration of the QE price effects. We show that the Fed’s QE program affected forward rates without introducing arbitrage opportunities into the Treasury security markets. Short- to medium- term forward rates were reduced ( $$<$$ 12 years), but the QE had little if any impact on long-term forward rates. This is in contrast to the Fed’s stated intentions for the QE program. The persistence of the rate impacts increased with maturity up to 6 years then declined, with half-lives lasting approximately 4, 6, 12, 8 and 4 months for the 1, 2, 5, 10 and 12 years forwards, respectively. Since bond yields are averages of forward rates over a bond’s maturity, QE affected long-term bond yields. The average impacts on bond yields were 327, 26, 50, 70, and 76 basis points for 1, 2, 5, 10 and 30 years, respectively.

The Impact of Quantitative Easing on the U.S. Term Structure of Interest Rates

This paper estimates the impact of the Federal Reserve's 2008 - 2011 quantitative easing (QE) program on the U.S. term structure of interest rates. We estimate an arbitrage-free term structure model that explicitly includes the quantity impact of the Fed's trades on Treasury market prices. As such, we are able to estimate both the magnitude and duration of the QE price eects. We show that the Fed's QE program aected forward rates without introducing arbitrage opportunities into the Treasury security markets. Short- to medium- term forward rates were reduced (less than twelve years), but the QE had little if any impact on long-term forward rates. This is in contrast to the Fed's stated intentions for the QE program. The persistence of the rate impacts increased with maturity up to 6 years then declined, with half-lives lasting approximately 4, 6, 12, 8 and 4 months for the 1, 2, 5, 10 and 12 year forwards, respectively. Since bond yields are averages of forward rates over a bond's maturity, QE aected long-term bond yields. The average impacts on bond yields were 327, 26, 50, 70, and 76 basis points for 1, 2, 5, 10 and 30 years, respectively.

The Forward Rates for Multifactor Model of Term Structure “With Square Root”

The multi-factor model “with square root” is discussed in details. For such model, the representation of state variable process in the integral form is derived and its covariance matrix is found. The special attention to the problem connected with the tendency for the term structure of long-term forward rates to slope downwards is given. For multi-factor models with square root the following results are derived: representations of the forward rate curve through the volatility of the state variable process and through the volatility of zero coupon yield process are obtained; the expectations, variances and co-variances for the forward rates and the yield process volatility are calculated; the expectation and the variance for the derivative of forward rate are found; the Brown − Schaefer approximation for the spread of forward rate is examined. As examples two three-factor models (BDFS and Chen models) are examined. On the basis of the estimates of parameters of these models received by empirical way the numerical analysis including calculation of covariance matrices of process of state variables, calculation of an expectation of the local variance of yield process and calculation of the variance of zero coupon yield have been fulfilled.

The Excess Sensitivity of Long-Term Interest Rates: Evidence and Implications for Macroeconomic Models

This paper demonstrates that long-term forward interest rates in the U.S. often react considerably to surprises in macroeconomic data releases and monetary policy announcements. This behavior is in contrast to the prediction of many macroeconomic models, in which the long-run properties of the economy are assumed to be time-invariant and perfectly known by all economic agents: Under those assumptions, the shocks we consider would have only transitory effects on short-term interest rates, and hence would not generate large responses in forward rates. Our empirical findings suggest that private agents adjust their expectations of the long-run inflation rate in response to macroeconomic and monetary policy surprises. We present an alternative model that captures this behavior. Consistent with our hypothesis, forward rates derived from inflation-indexed Treasury debt show little sensitivity to these shocks, indicating that the response of nominal forward rates is mostly driven by inflation compensation. In addition, we find that in the U.K., where the long-run inflation target is known by the private sector, long-term forward rates have demonstrated little excess sensitivity since the Bank of England achieved independence in mid-1997.

The Excess Sensitivity of Long-Term Interest Rates: Evidence and Implications for Macroeconomic Models

This paper demonstrates that long-term forward interest rates in the U.S. often react considerably to surprises in macroeconomic data releases and monetary policy announcements. This behavior is inconsistent with the assumption of many macroeconomic models that the long-run properties of the economy are time-invariant and perfectly known by all economic agents. Under those conditions, the shocks we consider would have only transitory effects on short-term interest rates, and hence would not generate large responses in forward rates. Our empirical findings suggest that private agents adjust their expectations of the long-run inflation rate in response to macroeconomic and monetary policy surprises. Consistent with our hypothesis, forward rates derived from inflation-indexed Treasury debt show little sensitivity to these shocks, indicating that the response of nominal forward rates is mostly driven by inflation compensation. In addition, we find that in the U.K., where the long-run inflation target is known by the private sector, long-term forward rates have not demonstrated excess sensitivity since the Bank of England achieved independence in mid-1997. We present an alternative model in which agents' perceptions of long-run inflation are not completely anchored, which fits all of our empirical results.