Download What Monetary Policy Prevents Financial Chaos?:

What Monetary Policy Prevents Financial Chaos?
James M. Haley1
The continuing debate about what caused the Great Recession
challenges the traditional view of how rational expectations theory
explains the evolution of financial markets. Monetary policy can cause
individuals to chaotically hedge in ways that destabilize financial
markets and the whole economy. It is not possible to make reliable
economic forecasts without knowing how everyone, including the
central bank, corrects their mistakes in a rational way. “If I had a simple
answer, I would be spreading it around the world,” said Professor Sims
after recently winning the Nobel Prize in Macroeconomics (Associated
Press, October 11, 2011). This paper provides an answer.
1. Introduction
There is a simple way to model financial chaos in three dimensions and
continuous time. Specifically, a Sprott nonlinear model perturbed by noise can
represent the evolution of forecast errors of real stock returns, inflation, and
excess bond returns. A strange attractor emerges when monetary policy is
consistent with the Taylor Rule, which targets interest rates to vary directly with
inflation and real output. In order to promote stability the nominal interest rate
and long run bond return should be targeted to equal the same fixed real interest
rate expectation, such that the expectation of inflation is zero. This policy makes
the forecast errors of stock returns behave more normally like a Langevin errorcorrecting, differential equation.
The failure to predict the dynamics of how financial markets correct their
mistakes explains why economists, including central banks, and investors cannot
make reliable forecasts. Central banks can make matters worse by setting
policy, based on inaccurate and skewed forecasts, that disrupts the economy
even more. The resulting forecast and coordination errors can precipitate a
financial crisis, causing chaotic business cycles to emerge.
It is now possible to develop a more complete dynamic model of the evolution of
economic errors, caused by central bank policy. This complexity is due in large
part to the nonlinear feedback in the stock market, which creates speculative
bubbles, when interest rates are set too low to stimulate the economy. But
financial markets can suddenly panic, when interest rates are raised too high to
curtail excessive speculation.
This evolution of all possible errors, especially financial chaos, is more tractable
to analyze in a macro framework. Predicting the collective action of interacting
1
H.J. Heinz Endowed Chair of Management, Ph.D., Economics, and MBA, Finance, the Business School
of Point Park University, Pittsburgh, Pennsylvania, U.S.A., [email protected]
1
heterogeneous agents, as they search for consensus, is the only way to make
more realistic predictions of how financial markets behave. So it should not be
surprising that dynamic stochastic general equilibrium models, based on the
behavior of an isolated representative agent, fail to address the most basic
questions about the economy’s true dynamics (Colander, Howitt, Kirman,
Leijonhufvud, & Mehrling, 2008).
The real reason that new forecasting models should be developed is that
standard financial models have failed to predict the financial crisis that led to the
Great Recession. Furthermore, most macroeconomic forecasting models cannot
predict what happened, because they ignore financial market dynamics
altogether. For example Sargent (2006) and Sims (2006), who both recently won
the 2011 Nobel Prize in Macroeconomics, do not include the behavior of the
stock market and the long-term bond market in their analysis of the
macroeconomy’s response to changes in policy.
Interestingly, these same distinguished economists cannot agree about whether
inflation persistence has declined. Sargent (2002) believes it has, while Sims
(2002) believes it is unchanged. This paper proposes that inflation persists,
when financial markets are chaotic that would be caused by the Taylor Rule
being applied as central bank policy. When the central bank provides prudent
guidance by “pegging” interest rates, to always equal the same real expectation,
inflation no longer persists but mean-reverts to a zero expectation. This prudent
policy should be implemented by central banks and investors, since it eliminates
chaos and reduces risk, making forecasts more reliable.
In fact this policy of pegging short and long interest rates to be the same,
resembles a monetary policy specified by Woodford (2011). He claims that an
equilibrium exists in a deterministic interpretation of a New Keynesian dynamic
stochastic general equilibrium model. He assumes that the dynamics of this
policy are consistent with the Taylor Rule. But this paper proves that only chaos
will emerge in that case. Financial stability requires different policy dynamics.
2. Financial Market Dynamics
Previously, I proved at the January 2007 AMS Conference in New Orleans that
the financial markets can behave like a Rössler nonlinear system, which
predicted the chaos of the Great Recession that started at the end of 2007
(Haley, 2010). There exists a simpler way to model financial chaos with a
nonlinear dynamic system of forecast errors of the real stock return, R * , with
respect to its expectation, Re* , inflation, pˆ , with respect to its expectation, pˆ e ,
and the coordination error of excess bond returns, x B . Specifically, a Sprott
nonlinear model perturbed by noise can be derived in this system,
by targeting

excess demand for real money to be consistent with the Taylor Rule, a monetary


policy that has generally
failed to
stabilize financial markets.

2
There is another bifurcation of policy, which more reliably guides the economy's
search for a rational expectations equilibrium, such that the long-term bond
return, rB , should be targeted to equal a fixed the short-term nominal interest
rate, r , that equals its real expectation, re*, assuming expectation of inflation is
zero. This policy peg, unlike the varying interest rate target of the Taylor Rule,
makes the forecast errors of the real stock return,  * , and inflation, y , to each

differential equation. Consequently, the
 behave like a Langevin error-correcting,

errors of forecasting real stock returns evolve smoothly and quickly converges to
a normal density with a bounded variance 
and a mean error of zero. Whether
 the interest rate
financial markets behave rationally or not, depends on setting
M
target, r, with respect to at any level of real money,
, and real economic
p
output, Q , that is derived from the following assumptions about the macro search
for correcting economic errors:

*
Ý* = R * =  1 x B2 - 

(A1)
2  + ,

such that  *  R* - Re* , x B = rB - r , Re* > re*, 1 ~ N ( 0 , 1),

 

and  2 is sufficiently large to ensure fast convergence;



     
xÝB = - a1  * + a 2  ;
p̂ = b1 x B - b2 y - b3  , such that y = p̂ - p̂ e ;



M
 = - l1 r + l 2 ln Q - ln
, if r = r * + p̂ .
p




(A2)
(A3)
(A4)
What do these financial market dynamics mean? Clearly, the real stock return,
R * , which is the consensus forecast in (A1), mean-reverts to its expectation, Re* ,
which is greater the expectation of the real rate of interest, re*, as long as, Br =
r . Inflation, pˆ , as a consensus forecast in (A3), mean-reverts as well with
respect to random shocks of the excess demand for money,  , as long as Br =

r . Thus real stock returns and rates of inflation are self-correcting, when the

short and long-term interest rates are the same. What distorts this normal

convergence
is when the real stock return exceeds its expectation, making

returns on long-term bonds fall in (A2). Financial stability
requires:
*
rB = r = re , if pˆ e = 0 .
This stabilization policy means that the “real term structure” is flat, such that all
*
interest rates are equal to
the same fixed
real rate expectation, re , based on a


zero inflation expectation,
Interestingly, this deterministic equilibrium
pˆ e .

condition means there are no arbitrage profits in the bond market. Furthermore,


3


in the long run forecast errors in the stock market are normally distributed, such
that the prices of risky assets asymptotically behave like a random walk. Put
simply, financial markets can become a “fair game”. Then no one can beat the
stock market, if real interest rates are pegged by the central bank.
3. Financial Chaos
The evolution of a complex economy's forecasting errors, summarized in the
dynamic system of economics errors, stated above, is best understood by
analyzing its dynamics holistically. One way to see the system's complexity is to
discover how different monetary policies affect the economy's process of
learning from its mistakes. By experimenting with alternative policy rules it can be
proven that a monetary policy exists that efficiently speeds up the convergence
to an unbiased stochastic steady state of forecasting and coordination errors.
One source of noise, which can not be reduced or eliminated, is random news,
specified by  1 in (A1), since the variance of this noise is exogeneous and is not
effected by monetary policy. But endogenous instability can be caused by the
expectation of inflation or deflation that disrupts the economy, according to
Meltzer (1986).
This distortion becomes evident by analyzing the partial
derivative of (A2) with respect to the expectation of inflation, which is positive
and follows from:
xÝB
= a2l1 > 0.
pˆ e
Under these circumstances interest rates can not be easily pegged, which
 as inflation expectations change.
increases instability in financial 
markets

Many economists (Taylor, 1993) believe that maintaining long-term price stability
reduces volatility in the economy. The benefits of this policy become evident by
assuming that the standard deviation of inflation expectations,  2 , is zero, if
and only if the expectation of inflation, p̂ e , is also zero, This assumption about
when inflation expectations are fixed, reduces the uncertainty facing the
economy as follows in assumption (A5):
p̂ e = 0 if and only if  2 = 0.
(A5)
So far, the model specified by assumptions (A1) through (A5) is complete in
terms of describing the evolution of economic errors. If the objective is to predict
the behavior of real stock returns, interest rates, inflation, and money supply for
any given level of output, a specific monetary policy is required to completely
describe alternative monetary regimes of a complex learning economy. Knowing
how the money supply behaves, based on an interest rate target, completes the
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dynamic system at economy's evolution. Consequently, sufficient central bank
intervention makes it possible to avoid indeterminancy in the model (Cass, 1995).
It can be proven that there exists a policy, which approximates how many central
banks currently set interest rates. It requires targeting the short-term nominal
interest rate, r , to vary directly with its expectation, re , the forecast error for
inflation, y , and the forecast error for the natural log of real economic output,  .
Furthermore, r varies inversely with excess real liquidity, L* , which is the
M
difference of the natural log of real money,
, and its expectation, measured by
p

MT
its trend,
. This monetary policy is assumed to be mathematically specified
pe
as follows:
r = re + w1 y + w2  - w3 L* , if w1 =
l
a1
1
, w 2 = 2 , w3 =
l1a2
l1
l1
M
M
*
such that  
= ln Q - ln Q
- ln T .
T , L = ln

p  p e

(A6)

This policy rule implies, that when inflation, pˆ , is greater (less) than its
expectation or real output, Q, is greater (less) than its trend, the target interest
rate should rise (fall), assuming its expectation, re , and excess real liquidity, L ,
are fixed. This policy is consistent with 
the Taylor Rule, which constrains the
economy by raising interest rates when the economy is over-expanding,  > 0,
and inflation is too high, y  0 . Or the central bank, according to Taylor, should
stimulate the economy when real output and inflation are too low,  < 0 and
y  0 , by lowering interest rates. Interestingly, when inflation, output, and real
excess liquidity 
are normal, that is equal to their expectations, then the nominal
interest rate target should equal its expectation as well. Furthermore this rule is
equivalent to the following excess demand for money,  :
 =-
M
a1
y , if ln T = - l1 re + l2 ln QT .
pe
a2
Applying this behavioral rule to a complex learning economy given by the
differential system (A1)
makes
monetary regime emerge.
 a chaotic

through (A6),
 (A2), it easily follows that:
assumption
By substituting  in
xÝB = - a1 (  * + y ), if  = -
 
5 

a1
y.
a2
Clearly, after substituting again for  in assumption (A3), it simply follows that:
y = b1 x B + b y , if  = -
a1
a
y , b = ( 1 b3 - b2 .
a2
a2
Combining the above reduced form equations of (A2), and (A3) with (A1), it
 there exists a
dimensional
follows from (A6),
three
 that
  nonlinear dynamic model


of financial markets unperturbed by noise. For simplicity assume:
 1 =  2 = a1 = b1 = 1, b = 0.5, x B = - x˜ B .
(A7)
Therefore, after making the necessary substitutions, the following theorems can
be derived from assumptions (A1), (A2), (A3) and (A7):
   
 = x˜ B2 -  * ,
(T1)
˜ B = (  * + y ).
xÝ
(T2)
y = - x˜ B + .5 y .
(T3)
*


This three dimensional nonlinear system represents one of the simplest ways to
model chaos in continuous time and is consistent with a chaotic system first

devised by Sprott (1994).
A central bank that changes interest rates too much, makes it impossible to
correct forecasting and coordination errors in the economy. This occurs,
because there is confusion about what interest rates should be. Imprudent
central bank guidance can cause interest rates to rise too high, increasing the
risk of a bear market in stocks. If interest rates are targeted too low, a
speculative bubble can emerge, as well.
4. A Better Way to Stabilize the Stock Market
Whether expectations become rational, meaning that the forecast errors become
unbiased and reliable, critically depends on the stock market being efficient,
such that it quickly converges to a stochastic equilibrium. Then expected excess
returns of a diversified portfolio of stocks equal the market risk premium, and the
probability density of excess returns is asymptotically normal. The stock market
avoids financial chaos, caused by nonlinear feedback, when real stock returns
mean-revert, a necessary condition for rational expectations to emerge. Thus the
variance, caused by the random shocks of news, is reduced asymptotically by
reinforcing mean-reversion, as the financial markets learn to correct errors in a
normal way.
6
In order to stabilize financial markets it is necessary to peg the short-term
nominal interest rate, r , and the long-term bond return, rB , to equal their
expectation, re . This new monetary policy assumption would replace the current
monetary regime stated in (A6):
r = rB = re .

(A6)*
Now it is easily follows that the interest rate peg should equal its real rate
expectation, re* , if price stability is the other necessary condition for further
 
reducing economic uncertainty:
r = re* + p̂ e = re* , if and only if p̂ e = 0 .
(T4)
Furthermore, it follows from (A6)* that the excess demand for real money,  ,
behaves as follows:
 =
a1 *
 .
a2
Then it can be shown that the complex learning economy reduces to a selfcorrecting monetary regime, after substituting the above value of  into the
appropriate error-correcting differential equations. So a different monetary
regime emerges by replacing assumption (A6) with (A6)*, such that a new
system of differential equations, perturbed by noise can be derived. Specifically,
after substituting assumption (A6)* into (A1), the following error correcting,
Langevin equation is implied for the forecast error,  * , of a real stock return, R * ,
with respect to its expectation, Re* , shocked by noise,  1 :
 * = -  2  * +  1 , if x B = p̂ e = 0 ,  1 ~ N ( 0 ,  1 )
(T5)
Furthermore, the long run stochastic equilibrium of this mean-reverting
differential equation can be proven to be
a normal density of  * with a zero mean

and standard deviation of  3 , which is given by:
Asymptotically,  * ~ N ( 0,  3 ), if rB = r ,
(T6)
such that lim E (  * ) = 0 ,  32 =  12 / 2 2 ,  * = R * - Re* .
t 

 
Eventually the long-term variance of the unbiased forecast errors of real stock
returns,  32 , is less than the variance of the continuous random shocks of news,
 12 , assuming that  2 is relatively large (Lasota & Mackey, 1994).
7
Current research tends to model the density of stock returns as not being
normally distributed, especially for the short run returns (Campbell, Lo, &
MacKinley, 1997). In fact actual returns exhibit excess kurtosis or fat tails, as
well as skewness. Thus positive and negative extreme returns are more likely to
occur than standard finance theory would predict. This paper proves that
standard theory is correct, if central banks properly target interest rates to equal
the expectation of the real rate. Then according to theorem (T6) real stock
returns will behave “normally” in the long run, such that extreme returns, resulting
from skewness or excess kurtosis, are not likely.
Finally under certain parameter specifications it can be proven that the whole
economy becomes rational. After making the necessary substitutions it can be
shown that expected forecast errors for real stock returns,  * , real output,  , and
inflation, y , eventually converge to zero as follows (Haley, 2010):
Asymptotically, E (  * ) = E (  ) = E ( y
) = 0,
(T7)
a
b
if x B = p̂ e = 0,  1 ~ N ( 0,  1 ), 1 > 2 .

a2
b3

The complex evolution of the economy converges to a simple state, where stock
returns and inflation mean-revert by evolving as independent Langevin
equations.
Clearly, how a central bank targets interest rates determines whether the whole
economy behaves rationally or not. Guesnerie and Woodford (1995) state it
simply, “... An interest-rate pegging regime provides an anchor for expectations.”
Then it becomes easier to forecast and plan, which eventually stabilizes the
stock market and the whole economy. In fact it speeds up the smooth
convergence to a stochastic steady state, such that expectations become rational
in the long run.
5. Conclusion
This paper proves that the theory of monetary policy can be made more rational,
if it exploits how a complex learning economy learns from its mistakes. This
search for an error correcting stationary process can be disrupted in the short
run, if a central bank distorts the normal relationship between the excess bond
returns with real stock returns. To avoid this confusion the short-term, nominal
interest rate should be pegged to eventually equal its real expectation of 1.8%,
as estimated by Campbell, Lo and MacKinlay (1997), which equals the return of
a long run bond in equilibrium, assuming a zero inflation expectation.
8
Therefore the Federal Reserve's current federal funds target rate of less than
.25% or greater than zero is biased too low. This cheap interest rate policy to
stimulate the economy raises the risks of higher inflation and the emergence of a
chaotic bubble economy in the long run. In the short run investors hoard cash or
speculate in gold to hedge against this uncertainty.
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