Impacts of Us Quantitative Easing on Financial Assets Finance Essay

On July 24, 2009, Federal Reserve unleashed its recent quantitative easing (QE) campaign. Fed Chairman Ben Bernanke declared that the Fed had an "exit strategy." Interestingly, sixteen months later, Bernanke announced the second phase of QE generally known as QE2.A However, this time insteadA ofA pretending an exit strategy, he designed a plan to expand the program in perpetuity. The market's initial dramatic reaction was most surprising. The Fed is going to be buying (QE2) 600 billion dollars of Treasuries (in the 5-7 year part of the curve) through mid-2011 and another 250-300 billion via coupon reinvestments.A  The "number" that is key for the markets is that 600 billion dollar figure which is about 75 billion per month. That is in the middle of consensus expectations of 50-100 billion dollars.A  For all the excitement, this further expansion of the Fed balance sheet will add between 0.25 and 0.5 pct to real GDP growth -- if it proves to be successful. What the Fed is clearly trying to do is inflate asset values in order to generate a more positive wealth effect on personal spending and pull the cost of debt and equity capital down in order to re-ignite business "animal spirits" and hence corporate investment and hiring.A 

Under the QE program the Federal Reserve is purchasing Treasuries, Agency and Agency Mortgage Backed Securities of different maturity scedule. There is substantial evidence that QE program can influance long-term interest rates. For example, Gagnon, Raskin, Remache, and Sack (2010) present an event-study of QE1 that documents large reductions in interest rates on dates associated with positive QE announcements. Apart from the event-study evidence, there are papers that look at lower frequency variation in the supply of long-term Treasuries and documents causal effects from supply to interest rates (see, for example, Krishnamurthy and Vissing-Jorgensen (2010)).

The main objective of this paper is to evaluate the theoretical channels through which the unconventional monetary policy (QE) works. I review the key channels through which QE operates and then investigate the impacts of QE on different financial assets using the event-study method. In our event-study method I observe the changes in asset prices and yields. I furthermore supplement previous works by adding evidence from QE2 and stochastic forecasting models. Studying daily data allows us to document price reactions after the main announcements.

2. Literature Review

2.1 Theoretical Studies

A monetary policy such as Quantitative Easing program by a central bank would change the supplies of assets held by the market agents and, thereby, may lead to changes in the relative prices of financial assets. This is called portfolio-rebalancing effect. On the other hand if the market agents believe the central bank interference will be successful in stimulating the economic growth by increasing aggregate demand then inflation and dividends on assets are likely to rise in the future which is known as expectation-hypothesis. Doh (2010) refers to imperfect substitutability between assets and explains the expansionary effect of QE announcement decision by arguing that the buy-back program of long-term treasuries by the central bank will tend to lower yields of long-term bonds, because the central bank purchases those securities or assets at a higher price than the market charges. This lower long term yield on treasuries can be explained by theory of portfolio rebalancing effect to other asset markets, such as equity and corporate bond market. Investing in longer-term assets becomes relatively cheaper and hence stimulates the aggregate demand. Essentially, the central bank interferes into the conventional interest rate determination mechanism by directly lowering long-term yields. Being constrained by the zero-lower-bound, the conventional interest rate channel that usually catalyzes lower long-term yields through the expectations hypothesis does not function properly and the central bank outwits this by directly arbitrating in the market for long-term bonds. He found that the central bank's large-scale purchases of the bonds can decrease the term premium on fixed income securities. The magnitude of the decline in term premium depends on the risk aversion nature of the market agents. If the risk aversion is high, large-scale asset purchases can induce more substantial decline. Hence, the success of the portfolio rebalancing theory would depend on the risk aversion level of the market agents. However, he argued that from a theoretical point of view, it is not obvious the yields on long-term treasuries are supposed to drop after announcing Quantitative Easing. From the study of Takeshi Kimura and David Small (2004), we see that QE lowered the key interest rates in Japan. However, the program failed to stimulate the equity market. So, the expectation hypothesis was in effect as far as fixed income securities were concerned. However, because of the high risk aversions of the market agents the portfolio rebalancing effect failed to exert any impact on other asset classes.

2.2 Empirical studies on QE

There is by now a substantial amount of research that studies the impact of unconventional monetary policy on capital market variables like interest rates, yield differential or interest rate spread. Some of the recent studies include Bernanke et al. (2004), Hamilton and Wu (2010) and Stroebel and Taylor (2009), Meier (2009), ECB (2010) for the Euro Zone, and Oda and Ueda (2007) and Ueda (2010). Oda and Ueda (2007) found that that the Bank of Japan's open market operation under the zero-bound rate environment has functioned primarily through the commitment of keeping the rate low, which had reduced the rates across different maturity schedule. Most importantly, the commitment has been effective in lowering the expectations component of interest rates, especially with short to medium-term maturities. Stroebel and Taylor (2009) examined the quantitative impact of the Federal Reserve's mortgage-backed securities (MBS) purchase program. They found evidence of statistically significant effects of the program on financial asset prices and on different interest rates. Without losing generality, I can say that these literatures found negative effects on yield differential of unconventional policies such as monetary or quantitative easing, which implies that the dividends or yields of various assets tend to decline, thereby tightening the risk premium spread to the corresponding risk-free return. Hamilton and Wu (2010) found that a buy-back of 400 billion US dollars of US treasuries in 10-year area of the curve would contribute to a 14 bps fall in the 10-year treasury yield. Gagnon (2010) found that the unconventional monetary policy measure would plunge long-term return by 20 bps across the curve. Meier (2009) observed that the Bank of England's Quantitative Easing program, where asset purchase program lowered inflation-linked-treasuries (gilt) yields by approximately 40 basis points to 1 percentage point (100 bps).

Kamada and Sugo (2006) diagnosed the impact of monetary policy shocks by imposing sign restriction on the IR (Impulse Response) functions. In their analyses, they used several macroeconomic variables, such as the consumer price index, industrial production, the exchange rate, yield on 10 treasury, and a proxy variable for monetary policy. Using the asset purchase date they observed the structural changes between February 1978 and April 2005. They showed that the impacts of asset purchase on prices and output weakened in the 1990s. The decline in the impact of Japanese monetary policy is attributable to the non-negativity constraint on short-term rates.

Baumeister et al. (2010) estimated the impact of a fall in the long-term treasury yield differential within the context of the 2007-2009 financial recession using a sign restrictions on the estimated parameters. They concluded that a 'pure' spread shock which, leaving the short-term rate unchanged by construction, allows us to characterise the macroeconomic impact of a spread tightening induced by central banks' monetary easing program within an environment in which the short-term rate cannot move because it is constrained by the zero lower bound.

2.3 Studies most relevant to my research

Recent literatures on US-QE program have focused only on Treasuries. However, in my opinion, it is inappropriate to focus only on Treasury rates as a policy target because QE works through several channels that affect other financial assets as well. Based on this argument, I build several hypotheses that I testify in this paper. Gagnon, et. al, (2010) showed that by reducing the net supply of assets with long duration, the Federal Reserve's QE programs appear to have been successful in reducing the risk premium. In addition to this reduction in the risk premium, the QE programs has an even more powerful effect on longer term interest rates on agency debt and agency backed mortgage securities by improving market liquidity and by removing assets with high prepayment risk from private portfolios. They found evidence that the asset purchase program led to economically efficient and long-lasting reductions in longer-term interest rates including financial market assets that were not included in the purchase programs. This plunge in interest rates primarily reflects lower term premiums rather than lower expectations of future short-term interest rates. However, in their study they did not investigate the prices of other financial assets that reflect the effectiveness of portfolio rebalancing effect. In my study we will investigate the portfolio rebalancing effect by exploiting an event study method using the price movements of several financial assets. Krishnamurthy et al. (2010) estimated the effect of QE2, assuming a $500bn size, on nominal long-term interest rates. In their primary hypothesis they expected QE2 would result in a 40 basis point drop in interest rates on long-term safe assets (Treasuries). The rationale behind the hypothesis was there is a unique clientele effect for long duration but risk free assets, and the QE2 would reduce the supply of risk free assets of higher duration and hence would increase the risk premium that such market agents would pay for such assets. They expected a much smaller effect on the nominal interest rates on less safe assets such as Baa rated corporate bonds and mortgage rates. These rates are more relevant for long-term financing from market agents' perspective. That is, effects on the duration risk premium, which affects all long-term interest rates, will be much smaller. They concluded that QE1 and QE2 significantly lower nominal interest rates on Treasuries, Agencies and highly-rated corporate bonds, driven mainly by an increase in the safety price premium of assets with near-zero default. To facilitate their research they used credit default swap (CDS) data. In my study I use corporate borrowing spread which directly reflects the health of financial position of the corporations. I also use stochastic interest rate forecasting model to capture the change in evolution of the zero-rate in both pre and post-QE period.

3. Data and Methodology

I evaluate the effect of the Federal Reserve's Quantitative Easing programs of long-term Treasuries ("QE1" in 2008-2009 and "QE2" in 2010-2011) on interest rates and other financial assets. I use an event-study methodology that compares the change in asset prices and yield around the event announcement date. In this paper, I use zero-, three- and seven-day window period for my event-study. I conduct my event-study using the data on treasury and corporate yield, S&P price index, S&P Futures, US-EUR currency rate, Gold price, Inflation Swap rate and VIX index around the event announcement dates. For the event-study method, I use three QE announcement dates for both QE1 (11/25/08, 12/01/08 and 12/16/08) and QE2 (08/10/10, 09/21/10 and 11/30/10) and I compare the relative impacts of these two phases on asset prices. I also use stochastic forecasting method to see the interest rates movement around the QE announcement dates.

The QE strategy entails purchasing long-term assets that increases monetary base. Thus, QE increases the liquidity in the hands of investors and thereby decreases the liquidity premium on the most liquid assets. According to the classical bond pricing formula, QE should decrease the yield on all long-term nominal assets more than the assets with short-term maturities due to the duration risk factor. To the extent that QE is expansionary, it increases inflation expectations, and this can be expected to have an effect on interest rates. This effect should be reflected via inflation swap rate. The entire idea behind the QE is to bolster the economic growth and stability. One of the objectives of the program is to create an interest rate environment where the corporations would be able to borrow their capital at a low cost which would in turn improve the health of income statement through the interest expenses account. Again, this would impact the P/E ratio and hence, would increase the equity prices. Under these circumstances, the debt holders would agree to lend capital at a lower premium. Thus, the spread between treasuries and corporate yields should be narrowed down as a result of QE. If the interest rate goes down due to the QE then I should experience a substantial amount of capital outflows (especially, interbank money market instruments) from US to other financial markets where the liquidity premium is higher. Hence, the US exchange rate should drop.

In this paper, I use Vasicek (1977) and Cox-Ingersoll-Ross (1985) interest rate models to forecast the zero-coupon interest rates for both pre and post QE period. These models are one-factor models that use short rates to forecast the interest rate movements driven by the market risk. They follow a standard Brownian motion to explain the evolution of the interest rate. Here, I use the historical daily data on short rate from 2006 to 2008 to forecast the interest rate environment for the pre-QE period. I then check the stability of the model for the post-QE period. Details of these models are described in the forecasting section.

4. Event Study

As discussed earlier, I am using zero-day (-1,0), three-day (-1,+1) and seven-day (-3,+3) window period in my event-study where I compare the changes in asset prices on the event day as compared to the prices around the event dates using different window period. I use three QE announcement dates for both QE1 (11/25/08, 12/01/08 and 12/16/08) and QE2 (08/10/10, 09/21/10 and 11/30/10). The issue that arises is that I cannot be sure that the identified events are in fact important events, or the dominant events. That is, other significant economic news arrives through this period and potentially create measurement error problems for the event-study. To tackle this problem I have chosen only the first three event dates for each phase. However, inclusion of other dates would not alter my fundamental conclusion.

4.1 Evidence from QE1

Based on our duration risk hypothesis, the yields on longer-term securities in Table 1 drop more than the yields of shorter-term securities. However, I see an exception for the 30 year Treasury bond, where the yield falls less than the 10 year bond. The long Treasury and agency bond yields may fall more than the shorter-term yields because of clientele effect for long-term safe assets. For mortgage backed securities (MBS), long rates may fall more than short rate because of pre-payment nature of the mortgage backed securities.

Table 1

Window (-1,0)

Change in Treasury Yield (bps)

Agency Yield (bps)

Agency MBS Yield (bps)

A 

Term-to-Maturity

Term-to-Maturity

Term-to-Maturity

Announcement Date

30 Year

10 Year

5 Year

1 Year

10 Year

5 Year

10 Year

5 Year

11/25/2008

-24

-36

-23

-2

-76

-57

-75

-147

12/01/2008

-27

-25

-28

-13

-67

-50

-10

58

12/16/2008

-32

-33

-15

-5

-39

-26

-30

-7

Change in yield on fixed income securities with a (0,1) window; Source: Bloomberg

From table 1, we observe that there are differences in the yield changes across the different fixed income securities; for example, Agency bonds demonstrate the largest fall in yields. The duration risk hypothesis cannot explain these effects as it only explains the effects that depend on term-to-maturity. Also, the trade-off between interest rate risk and credit risk is minimal as far as treasuries or agency backed securities are concerned.

The QE strategy involves purchasing long-term securities. Thus, QE increases the liquidity in the hands of investors. With the increased liquidity base and portfolio rebalancing effect, the investors will demand higher premium on liquid assets and will allocate more wealth into risky assets. The most liquid assets in Table 1 are the Treasury bonds. The liquidity premium hypothesis predicts that these yields should increase with QE. However, they do not increase; they actually fall much less than the yields on Agency bonds (these are the bonds issued by government agencies but are not secured by government) which are less liquid. That is the Agency-Treasury spread narrows down with QE1. For example, the 10 year spread falls by 92 basis points. This is interesting because 10 year Agencies and Treasuries have the same default and duration risk based on their underlying covenants and position in the issuers' capital structure. Also, it is evident that portfolio rebalancing effect did not take place in a reaction to an event date. This could be explained by the nature of the utility functions of the market agents but to explore that phenomenon is beyond the scope of this paper.

To assess the effects on real rates, I need information about the impact of QE1 on inflation expectations. Table 2 presents the relevant interest rate swap data. Here, I use inflation swap rate which is also known as plain vanilla swap. This instrument reflects the inflation expectation of the investors on at different point in time. For example the 10-year inflation swap is the fixed rate in the 10-year zero coupon inflation swap. This data suggests that inflation expectations increased by between 1 and 48 basis points (ignoring the drops in swap rate), depending on maturity. It is difficult to explain the drop in the rate on few occasions. However, the market agents were aware that the Federal Reserve was not going to raise the policy rate anytime soon and hence, the agents demonstrated consistent expectation on longer-term maturity schedule (30 yr).

Table 2

A Window (-1,0)

Change in Inflation Swap Rate (bps)

S&P 500 Price Index

A 

Term-to-Maturity

Announcement Date

30 Year

10 Year

5 Year

1 Year

Change in basis points

11/25/2008

1

-6

-28

48

66

12/01/2008

15

27

11

-40

-893

12/16/2008

4

37

35

-17

514

Change in Inflation Swap rate & S&P 500 Index; Source: Bloomberg

QE increases the level of monetary base and consequently lowers the value of the currency. To hedge against the weak currency, large institutions and investors tend to hold gold and this triggers the gold price. The cheaper dollar should stimulate the export demand of locally produced goods. At the same time as QE lowers the corporate borrowing rate which has direct impact on the net income of the corporations' balance sheets. These two effects together improve the Price/Earning ratio which is the key multiple closely followed in stock trading. As a result, I should document higher stock prices in post-QE period. At the same time, due to the spot-future price relationship, economic agents would revise their market expectations and hence, the stock futures should rise as well.

In table 2 I see that S&P 500 reacted immediately to the announcement dates except for one announcement date. On the event day, price index rose from 66 to 514 basis points. Most importantly S&P 500 grew by 47% since the initial announcement date to the end of 2010. However, from table 3, I document that the impact on S&P 500 futures is quite ambiguous as I observe a frequent change in market movements. However, since the initial announcement date to the end of 2010, the index surged by 47% which is consistent with the growth rate of the price index. It is clear that market agents did not react and revised their expectations overnight on equities. However, I notice the reaction in long term market expectation. The volatility in spot and future indices is also reflected in the VIX index where I see a frequent sign change in the market volatility expectation. Same argument applies to the gold spot price.

Table 3

Window (-1,0)

VIX Price Index

USD-EUR Exchange Rate

Gold Spot Price

BBB Mid-Yield Spraed (5 Year)

Announcement Date

(Change in bps)

(Change in bps)

(Change in bps)

(Change in bps)

11/25/2008

-6

-84

-24

-33

12/01/2008

230

65

-448

-28

12/16/2008

-770

-227

148

-2

Change in VIX, USD-EUR, Gold and BBB spread; Source: Bloomberg

On the event day, the gold price actually fell by few basis points but it grew by 75% since the initial announcement day to the end of 2010. Except for one event date, US-EUR exchange rate fell by 84 to 227 basis points. In line with treasury and agency yield movements, the BBB rated corporate borrowing rates also fell on the even day. I document that 5 year yield spread fell by 2 to 33 basis points and the reaction was much larger on the QE1 event dates.

Table 4

Window (-1,+1)

Treasury Yield (5 Yr)

BBB Spread (5 Yr)

S&P 500

Gold Spot Price

11/25/2008

-20

2

4.04%

-1.2%

12/01/2008

-27

-28

6.68%

-4%

12/16/2008

-11

-29

4.13%

-5%

Change in 5-Yr treasury yield, BBB spread, S&P 500 and Gold price; Source: Bloomberg

Using a 3-day window period, from table 4, I see that, the treasury yields drop around all event dates. We also see that the generic spread on BBB rated corporate bonds declined and consistent with the treasury fall. The reaction of S&P 500 price index is much more conclusive when we use 3-day window and they are in line with my primary hypothesis. However, the movement in gold price does not demonstrate its actual long term movement when we observe them in shorter window frame. I observe the similar results when I use a 7-day window period (see table 5). This time around gold price does reflect its trajectory since the initial event day to date.

Table 5

Window (-3,+3)

Treasury Yield (5 Yr)

BBB Spread (5 Yr)

S&P 500

Gold Spot Price

11/25/2008

2

-20

8.48%

10.3%

12/01/2008

-50

-5

-1.3%

-4%

12/16/2008

-18

-55

1.64%

0.9%

Change in 5-Yr treasury yield, BBB spread, S&P 500 and Gold price; Source: Bloomberg

4.2 Evidence from QE2

The QE2 announcement was widely anticipated and one would expect the announcement to have little effect. Prior to the initial announcement for QE2, market expectations were that the Fed would let its MBS portfolio overspill, thereby reducing reserve balances and allowing the Fed to exit from its non-traditional monetary policies. Thus, the announcement of the Fed's intent to continue QE revised market expectations. Moreover, the announcement indicated that the QE would shift towards longer-term Treasuries, and not Agencies or Agency backed securities as in QE1.

In QE2, I do not observe duration risk phenomena. It is not the case that longer term Treasuries or Agency securities move more in yield than shorter term securities.

Table 6

A Window (-1,0)

A 

Change in Treasury Yield (bps)

Agency Yield (bps)

Agency MBS Yield (bps)

Term-to-Maturity

Term-to-Maturity

Term-to-Maturity

Announcement Date

30 Year

10 Year

5 Year

1 Year

10 Year

5 Year

10 Year

5 Year

08/10/2010

-1

-7

-8

-1

-7

-9

1

-5

09/21/2010

-8

-11

-9

0

-11

-9

-7

1

11/30/2010

16

4

-4

0

5

-5

-5

-2

Change in yield on fixed income securities with a (0,1) window; Source: Bloomberg

For example, from table 6 we see that on the first event day, the treasury yield dropped more for the longer-term bonds. We also see that on the third event date, the treasury yield on 10-yr and 30-yr bonds rather increased. There does not appear to be an existence of liquidity premium hypothesis. Treasury and Agency yields fall by nearly the same amounts, and hence the spread also remained flat. Hence, the liquidity premium also remains unchanged.

Table 7

A A Window (-1,0)

Change in Inflation Swap (bps)

S&P 500 Price Index

A 

Term-to-Maturity

Announcement Date

30 Year

10 Year

5 Year

1 Year

Change in basis points

08/10/2010

5

-1

-3

0

72

09/21/2010

6

6

6

-1

26

11/30/2010

6

-3

2

1

37

Change in Inflation swap rate and S&P 500; Source: Bloomberg

From table 7, we see that the inflation expectations rise by 1 to 6 basis points and the impact was smaller as compared to QE1. It is noteworthy that the inflation expectation on 30-yr maturity is in line with the inflation expectation in post-QE1 period.

Table 8

A A Window (-1,0)

VIX Price Index

USD-EUR Exchange Rate

Gold Spot Price

BBB Mid-Yield Spread (5 Year)

Announcement Date

(Change in bps)

(Change in bps)

(Change in bps)

(Change in bps)

08/10/2010

100

-8

-87

-4

09/21/2010

400

-153

-33

-8

11/30/2010

-930

-74

-41

-3

Change in VIX, USD-EUR, Gold and BBB spread; Source: Bloomberg

In line with QE1, the movements in stock and commodity prices remain ambiguous. However, the currency movement was distinct as the exchange rate fell on each QE2 event dates and the magnitude of the change was larger on QE2 event dates. From table 8, we see that, the generic yield spread on BBB graded bonds narrowed by few basis points. The effect is smaller than QE1 as expected. Gold price also fell on each event day of QE2. However, we have seen substantial growth in gold price since the initiation of QE2 to date.

Table 9

Window (-1,+1)

Treasury Yield (5 Yr)

BBB Spread (5 Yr)

S&P 500

Gold Spot Price

08/10/2010

-9

-15

-3.4%

0.2%

09/21/2010

-9

-10

-0.7%

-1%

11/30/2010

13

4

1.5%

-2%

Change in 5-Yr treasury yield, BBB spread, S&P 500 and Gold Price; Source: Bloomberg

Table 10

Window (-3,+3)

Treasury Yield (5 Yr)

BBB Spread (5 Yr)

S&P 500

Gold Spot Price

08/10/2010

-11

-18

-4.1%

1.8%

09/21/2010

-11

-12

2.1%

1.9%

11/30/2010

5

8

2.2%

2%

Change in 5-Yr treasury yield, BBB spread, S&P 500 and Gold Price; Source: Bloomberg

We observe similar movements in treasuries and other asset prices for QE2 when we use 3- and 7-day window event (see table 9 and 10). However, the magnitude of the change is lesser than that of QE1. This is quite plausible as QE2 was well expected before the QE2 announcement day.

5. Stochastic Interest Forecasting Models

5.1 Theoretical Background of Vasicek and CIR Model

I use interest rate models, those of Vasicek (1977) and of Cox, Ingersoll and Ross (CIR; Cox et al., 1985). Both models assume that the risk-neutral process for the (instantaneous) short rate r is stochastic, with one source of uncertainty. The stochastic process includes drift and volatility parameters which depend only on the short rate r, and not on time. The short rate model involves a number of variables, and different parameter choices for these variables will lead to different shapes for the term structure generated from the model.

Both interest rate models feature 'so-called' mean reversion of the short rate, that is, a tendency for the short rate to drift back to some underlying rate. This is an observed feature of the way interest rates appear to vary. The two models differ in the handling of volatility. I start with Vasicek model, and then consider the CIR model.

(i) The Vasicek Model

Vasicek model (1977) assumes that the instantaneous interest rate at time t, r(t), follows the mean-reverting process of the following form:

(1)

Thus a small change (dr) in the short rate in time increment (dt) includes a drift back to mean level at a rate for the short rate. The second volatility term involves uncertainty, dz representing a normally distributed variate with zero mean and variance dt. The short rate r (strictly r(t)) is assumed to be the instantaneous rate at time t appropriate for continuous compounding. I assume that short rate is to be the instantaneous at time t.

Parameter exhibits a mean-reverting feature in the Vasicek model which represents equilibrium level of the short-term interest rate, around which it stochastically evolves.

When the interest rate falls below (above) its equilibrium level, the instantaneous change in interest rate is positive (negative). The short-term interest rate will move toward its long-term value at a greater speed when it is far from it and when the parameter value (speed of adjustment) is high. I assume that the volatility of the short rate is assumed normally distributed.

In the one-factor model, security values are determined by the zero-rate. Bond price in a risk-neutral economy discounted at time t with a maturity of can be represented as follows:

(2)

Given expectation with respect to stochastic process, the spot rate can be estimated as follows:

(3)

Using the expected yield, bond value can be found by the following way:

(4)

The yield to maturity of a bond is defined as , which implies:

(5)

As the maturity increases from , the yield to maturity converges to:

(6)

Vasicek model assumes that the stochastic movement of the zero rate is independent of the level of the zero rate. However, this is not true at a zero-rate environment or at an extreme levels of the zero rate. During high inflation period, the short-term interest rates are very unstable and, as a result, the volatility of the short rate tends to be high. When the short-term rate is very low, its volatility is limited by the fact that interest rates cannot decline much as it approaches zero. In our model simulation, I notice that when the zero-rate approaches to zero, Vasicek model forecast negative interest rate which is not possible in realty. To tackle this problem I use CIR model imposing the non-negativity constraint on its stochastic evolution.

(ii) The CIR Model

The CIR model overcomes the negative interest rate issue that I have in Vasicek model. In this framework, the risk-neutral dynamics of the short rate is described by the equation:

(7)

is the mean reversion parameter, is the mean level, is the proportion of the square root of the level of interest rate and follows a standard Brownian motion. This framework has the same mean reverting drift as the Vasicek model, but the volatility of change in the zero rate in a short period of time is proportional to the square root of the interest rate. It implies that, as the short-term interest rate increases, its standard volatility increases. Under the CIR (1985) model, P(r, t) at time t for all interest rate satisfies the following bond valuation approach:

(8)

and the conditional variance is given by

(9)

With the boundary condition that

(10)

Given the relevant expectation, I can obtain the bond price as follows:

, (11)

where (12)

and

A zero-bond is usually expressed in terms of its yield rather than its price. The yield to maturity on a - period zero-coupon bond, can be estimated from equation (9) as follows:

(13)

The CIR and Vasicek models are single factor equilibrium models of the instantaneous interest rate movement providing equilibrium asset prices and free of arbitrage opportunities. Specially, the CIR model performs better under a zero-rate environment as it imposes the non-negativity constraint on the term structure model. It is interesting to find out which is a better model in estimating market price.

5.2 Term Structure Forecasting

The simulation of the short rate is based on the stochastic model as described in the last section. I run the simulation using the parameters estimated from the first order autoregressive model using the historical time-series daily data on short rate from 2006-2007. We have to remember that the short-rate is explicitly a function of federal fund or over-night target rate. Hence, the model does not give us the flexibility to go beyond two years as the volatility of the key policy rate will get larger. From 2006 to 2007, federal fund rate moved by only 1% and that gives us the rationale for using two years of data points. Table 11 shows the zero-rate on the event dates used in this paper.

Table 11

Announcement Date

Short-Rate

11/25/2008

1 bp

12/01/2008

7 bps

12/16/2008

4 bps

08/10/2010

15 bps

09/21/2010

17 bps

11/03/2010

13 bps

Figure 1: Zero- rate movements since the commencement of QE

The starting point of the simulation process is the zero-rate at January 2008 using the parameters estimated from AR (1) model. I find an average of the simulated (model generated short-rates) short-rates for June 2008 approximately 2.08%, from which the entire term structure can be deduced. In table 12, I represented the parameters used to calibrate the stochastic models.

Table 12

Estimated Parameter for Calibration

Rate

Rate at t=0

1.96%

Drift Factor

0.03%

Equilibrium Rate

0.98%

Volatility

0.73%

The term structure shows a downward sloping average fitted yield curve. The simulated yield curve assumes a variety of shapes through time and downward sloping, hump shaped, and inverted hump shaped. In the term structure below, I represent the model predicted short-rates at different maturity. In the simulated short-rate interest rate curve, I represent the predicted short-rates at different point in time. On the vertical axis I have the simulated short-rates and on the horizontal axis I show year as a unit of time. In the term structure, I see that the short-rates approach to the equilibrium long-rate (derived from the historical rate). However, the Vasicek model predicts a negative long-rate. I have already discussed this problem in the previous section and this is why I deploy the CIR model to have the non-negativity constraint.

Figure 2: Term Structure of zero-rate

As mentioned before, I use three announcement dates for each phase of quantitative easing to testify the stability of our stochastic models. I observe that on the announcement dates, the short-rate ranges between 1 basis point to 17 basis points. Based on our simulated models, the predicted short-rate for June 2008 (month-end) lies in the vicinity of 2%, while the actual rate stood at 1.9%. However, our calibrated model predicts that the interest rate for December 2008 (month-end) would lie somewhere around 2.8% but as we know, since the first announcement date, the interest rate never rose above 32 basis points. Hence, clearly the model fails completely to forecast the future short-rate for any time period after the initiation of the quantitative easing. We have to keep this in mind that in order to make the QE program successful, the Federal Reserve kept its fed fund rate as low as 25 basis points which in turn governed all other rates in the economy as well which I have already documented in our event study approach. If I impose the current policy rate to be the starting point of the simulation, the model indeed diverges to zero-rate and takes the following form:

Figure 3: Simulation of zero-rate movements using current (25 bps) overnight rate as the starting point

In figure 3, I see that for June-2008 (as the initial point of the simulation is jan-2008), the simulated zero-rate is approximately zero percent. However, due the lack of non-negativity constraint in the Vasicek model, the rates become negative for a period of time. The CIR model however keeps the short-rate zero bound because of the non-negativity constraint. The day before the initial QE announcement date, the zero-rate stood at 13 bps and based on that, the trajectory looks like the following:

Figure 4: Simulation of zero-rate movements using the day before the first QE announcement day as the starting point

In figure 4, I see that the Vasicek model again predicted short-rates approaching towards the negative quadrant which is not possible. However, due to the non-negativity constraint, CIR model allows us to forecast quite accurately by keeping the short-rate horizontal asymptote bound. From our models, it is evident that, the models could indeed predict the impact of QE for the post-QE period when the initial starting point of the simulation is anytime in the vicinity of September 2008. In that time period the market agents were aware that the Federal Reserve was going to implement the QE1. However, it was not possible to predict the zero-rates accurately a year ahead or even six months prior to the initial announcement date.

Looking at specific dates, it turns out that inverse humped and upward sloping shapes occur as well. All yields with their respective maturity times are a function of the short rate. Hence, the variance of a yield with an arbitrary maturity time strongly depends on the variance of the simulated short rate. Some of the simulated results have been presented below (see panel a-d).

(a) (b)

(c) (d)

Panel a-d represent the simulation of the zero-rate movements using January 2008 as a starting point. Using the stochastic volatility parameter and long run equilibrium interest rate, I simulate the model assuming a standard Brownian motion. From the simulations, I see that the short-rate for June-2008, the rate ranges from 1.2% to 2.5% as mentioned before; the average of the simulated short-rates was 2.08%, while the actual rate stood at 1.9%. However, no simulation result approaches anywhere close to 32 basis points which was the highest short-rate I have noticed since the initial announcement date of the quantitative easing. This depicts the instability of the forecasting model for the post-QE period and the results remain the same for both QE1 and QE2 as the simulated rates never approach zero as far as one year time frame is concerned starting form January 2008.

6. Conclusion

I document that QE1 and QE2 substantially lower interest rates on treasuries, highly-rated corporate bonds and agency securities across different maturity schedule. The impact of quantitative easing on Agency backed mortgage rates is large when fed purchases MBS along with treasuries (QE1), but not when it only purchases treasuries (QE2). From the movement of inflation swap rates, we see that the expected inflation increased significantly during the first phase of asset purchase program (QE1) and modestly during the second phase. I also see that US currency exchange rate fell both on QE1 and QE2 event dates and the effect was larger for QE2 events. Stock and future prices did not react immediately to the announcement dates. However, I document that the market agents did revise their expectation for the longer term. From our stochastic model, I observe that market models could predict the interest rate movements for the pre-QE period but the model failed to predict the interest rate movements in the post-QE period. This result signifies the significant interest rate movements as a result of QE. Fed's QE program has successfully affected the interest rates and other asset prices. However, we have to wait for years before we know whether QE successfully triggered the wealth effect among the market agents.

Bibliography

Baumeister, C., and Benati, L. (2010). "Unconventional monetary policy and the great recession - estimating the impact of a compression in the yield spread at the zero loIr bound." European Central Bank.

Bernanke, B. S. (1999). "Japanese monetary policy: A case of self-induced paralysis." Institute for International Economics Special Report.

Doh, T. (2010). "The efficacy of large-scale asset purchases at the zero loIr bound." Economic Review.

ECB (2010). "ECB monthly bulletin - October." Monthly bulletin, European Central Bank

Gagnon, J., Raskin, M., Remache, J., and Sack, B. (2010). "Large-scale asset purchases by the federal reserve: did they work?" Federal Reserve Bank of New York.

Hamilton, J. D., and Wu, J. (2010). "The effectiveness of alternative monetary policy tools in azero loIr bound environment." mimeo.

Kamada, K., and Sugo, T. (2006). "Evaluating Japanese monetary policy under the non-negativity constraint on nominal short-term interest rates." Bank of Japan.

Krishnamurthy, A., and Vissing-Jorgensen, Annette, "The Effects of QE2 on Long-term Interest Rates". Kellogg School of Management, NorthIstern University and NBER, October 27, 2010

Meier, A. (2009). "Panacea, curse, or non-event? unconventional monetary policy in the united kingdom." International Monetary Fund.

Oda, N., and Ueda, K. (2007). "The effects of the bank of Japan's zero interest rate commitment and quantitative monetary easing on the yield curve: A macro-finance approach." The Japanese Economic Review.

Stroebel, J. C., and Taylor, J. B. (2009). "Estimated impact of the feds mortgage-backed securities purchase program." National Bureau of Economic Research.

Ueda, K. (2010). "Bank of Japan's experience with non-traditional monetary policy." CIRJE F-Series, Faculty of Economics, University of Tokyo.

Cite this page