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

Abstract

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.

Appendix

Proof of Theorem 1

From expression (11), for $t\le \tau$, we have

$$\begin{aligned} dF(t,T)=(\mu (t,T)-\psi (T-t)e^{-\lambda (T-t)t})dt+ \sum _{n=1}^{N}\sigma _{n}(t,T)dW_{n}(t) \end{aligned}$$

The HJM condition on $f(t,T)$ implies that

$$\begin{aligned} \mu (t,T)=-\sum _{n=1}^{N}\sigma _{n}(t,T)\left[ \phi _{n}(t) -\int _{t}^{T}\sigma _{n}(t,s)ds\right] \end{aligned}$$

(27)

The HJM condition on $F(t,T)$ implies that

$$\begin{aligned} \mu (t,T)-\psi (T-t)e^{-\lambda (T-t)t}=-\sum _{n=1}^{N} \sigma _{n}(t,T)\left[ \Phi _{n}(t)-\int _{t}^{T}\sigma _{n}(t,s)ds\right] \end{aligned}$$

(28)

where $\Phi _{i}(t)$ ($\phi _{i}(t)$) is the price of risk for factor i with (without) the Fed’s price impact.

From expression. (27) and (28), we obtain the difference in risk premium:

$$\begin{aligned} \sum _{n=1}^{N}\sigma _{n}(t,T)[\Phi _{n}(t)- \phi _{n}(t)]=\psi (T-t)e^{-\lambda (T-t)t}>0 \end{aligned}$$

From expression (11), for $t>\tau$, we have

$$\begin{aligned} dF(t,T)=[\mu (t,T)+\psi (T-t)(e^{\lambda (T-t)\tau }-1) e^{-\lambda (T-t)t}]dt+\sum _{n=1}^{N}\sigma _{n}(t,T)dW_{n}(t) \end{aligned}$$

The HJM condition on $F(t,T)$ implies that

$$\begin{aligned} \mu (t,T)\!+\!\psi (T-t)(e^{\lambda (T-t)\tau }-1)e^{-\lambda (T-t)t} \!=\!-\sum _{n=1}^{N}\sigma _{n}(t,T)\left[ \Phi _{n}(t)\!-\!\int _{t}^{T} \sigma _{n}(t,s)ds\right] \end{aligned}$$

(29)

From expressions (27) and (29), we obtain the difference in risk premium:

$$\begin{aligned} \sum _{n=1}^{N}\sigma _{n}(t,T)[\Phi _{n}(t)-\phi _{n}(t)] =\psi (T-t)(1-e^{\lambda (T-t)\tau })e^{-\lambda (T-t)t}<0 \end{aligned}$$

To sum up, the Fed’s impact on the risk premium is

$$\begin{aligned} \sum _{n=1}^{N}\sigma _{n}(t,T)[\Phi _{n}(t)-\phi _{n}(t)]=\left\{ \begin{array}{l} \psi (T-t)e^{-\lambda (T-t)t},\text { if }t\le \tau \\ \psi (T-t)(1-e^{\lambda (T-t)\tau })e^{-\lambda (T)t},\,\,\text { if }t>\tau \end{array}\right. \end{aligned}$$

In the special case of a one-factor model, we have

$$\begin{aligned} \Phi (t)-\phi (t)=\left\{ \begin{array}{l} \frac{\psi (T-t)e^{-\lambda (T-t)t}}{\sigma (t,T)},\text { if }t\le \tau \\ \frac{\psi (T-t)(1-e^{\lambda (T-t)\tau })e^{-\lambda (T-t)t}}{\sigma (t,T)},\,\,\text { if }t>\tau \end{array}\right. \end{aligned}$$

$\square$