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J-/ aJiAW**
WORKING PAPER
^k
Behavioral Theory of Interest Rate Formation
by
James H. Hines,
WP- 177 1-86
Jr.
January 1986
HO
28
^,
NOV 2
1773
1986
U
A
Behavioral Theory of Interest Rate Formation
James H. Hines, Jr.
Sloan School of Managment
Massachusetts Institute of Technology
Introduction, This paper developes and marshals empirical support for a theory of the
nominal, short-term, risk-free interest
Unlike most approaches to the problem, the
rate.
formulation presented here does not assume equality of supply and
the formulation
larger
is
a
mechanism which would help
macroeconomic model. Empirical support
studies of individual
structure
human
two
offered at
for securities. Rather,
and demand
to bring supply
is
demand
into balance in a
levels of aggregation: Empirical
decision making are used to suggest an appropriate structure; the
then estimated using macro-economic data.
is
In the proposed formulation the risk free interest rate
moves
in
response to liquidity
pressures experienced principally by intermediaries. Liquidity pressures cause
interest rate relative to the "underlying interest rate", a construct
movements
which represents the
in the
interest rate
environment to which people have become accustomed.
The formulation has grown from work on
(for a description of the
sector, a financial sector,
cycle, the
and a government
It
inflation
sector.
and
is
a
possesses two production sectors, a household
The Model
is
modes of industriahzed economies including
economic long wave,
the several
System Dynamics National Model
modeling project see Forrester 1984 and 1979). The National Model
simulation model of an industrial economy.
the major behavior
the M.I.T.
deflation,
intended to aid in understanding
the business cycle, the Kuznets
and economic growth. (For discussion of
major economic behavior modes see Schumpeter 1944; Volker 1978; and Forrester
1976, 1982).
In
some
respects the National
The nature of this departure gives
Model
rise to the
represents a departure from prior economic modeling.
need for a new
interest rate formulation.
characteristics are particularly important in this regard. First, the National
exogenous time-dependent inputs l; consequently, exogenous
Three
Model does not permit
interest rates
could not be used as
explanatory variables in the interest rate formulation. Using exogneous interest rates begs the
question of
how
interest rates are formed.
Second, the National Model unfolds
and, therefore, the use of simultaneous equations
1.
is less
precisely, the only
continuous time,
appropriate than in most other economic
exogenous inputs are noise and idealized
disturb the system and elicit its dynamic response.
More
in
test inputs
designed to
models. Finally, model construction has emphasized the portrayal of decision making heuristics
the derivation or application of optimizing decision rules.
more than
The emphasis on
the nominal, short-term, risk-free interest rate
is
one of convention. The
important simplifying strategy in a macro-model examining major movements in the economy (as
movements
opposed
to fine
the rigor
imposed by
from
this
in interest rates) is to
interest rate with strict regard to
the requirements of behavioral modeling. Other rates can then be derived
one as needed. Any
adjustments up or
form one basic
rate
can be regarded as basic; other rates
down depending upon whether
those rates are
may
more or
then be calculated by
less risky than the basic
rate^.
As
in
is
the basic rate,
I
have chosen the nominal, short-term, risk-free
broad keeping with economic
tradition,
although
many
theorists
interest rate.
would focus on
short-term, risk-free interest rate. In the United States, however, there are
the real rate,
and consequendy representing demand and supply pressures
might become problematic were the
real rate used.
on the other hand, corresponds closely
to the interest rate
is
virtually risk-free in terms of default risk
to
pay
its
interest rate
first
section following considers the the
new
model
risk-free, short-term interest rate,
on a federal government
more
traditional equilibrium
upon a modification
note.
The note
approach
to
to Walras' auctioneer.
formulation permits trades out of equilibrium and
own account The
and summarizes
third section of this
is
willing to
paper discusses the behavior of the
new
The
buy and
sell
formulation
statistical tests.
1.
Simultaneous Equations and Tatonnement
Simultaneous Equations. The theory of
interest rate formation
developed
fundamentally a supply and demand formulation. The formulation, however,
from supply-demand formulations
3.
in a behavioral
formation including Walras' taonnement adjustment process. The next section
"auctioneer" of the
2.
no instruments carrying
because the government has the ability to print money
presents an alternative adjustment process based
is
the real,
debt.^
The
for his
The nominal
This choice
in
is
most other econometric work. In such work
in this essay
distinguishable
interest rates are
Translations between short and long rates would also involve expectations regarding changes
in the short or long rates and, perhaps, risk premia for different maturity habitats (Modigliani
and ShiUer 1973, Modigliani and Sutch 1966).
There have been instances in which governments have defaulted on debt denominated in local
currency, although such instances are not common. The default risk on the debt of the
industrialized nations is extremely small.
JHH42.2 1/14/86
2
chosen to equate desired supply and desired demand (see for example, Friedman 1980a, 1980b,
1977; Friedman and Roley 1980, Modigliani, Rasche, and Cooper 1970, Hendershott 1977,
Bosworth and Duesenberry 1973). More
security by lenders
specifically, the desired (primary) purchases of a
and the desired issues of that security by borrowers are specified
as functions
of the interest rate on that security. The condition that desired purchases (demand) equal desired
issues (supply)
interest rate
is
then imposed. With this condition the desired amount of securities and the
can be determined.'*
For example,
and
f(i,z)
and "z"
if g(i,z) is
a function determining desired borrowing
exogenous
is
factors; a
DL (or securities purchases)
a function determining desired loans
DB
(or issues),
where
system of three simultaneous equations
"i" is the interest rate
may be
written:
DL = g(i,z)
DB = f(i,z)
DL = DB.
Equations (1) and (2)
may be
(1)
(2)
(3)
substituted into (3) to yield
g(i,z)=f(i,z)
which may be solved for a unique
rate so discovered
may
(4)
interest rate
i
under suitable restrictions on g and f The
then be substinated into either (1) or (2) to uncover the amount
interest
demanded
and supplied.
It
might be noted
equilibrium desired quantities equal actual quantities.^
be fulfilled only
if
the quantities they each have in
will find itself short of transacting parmers.
4.
5.
approach
that the simultaneous equations
The
mind
is
an equilibrium approach. In
desires of sellers
and buyers can both
are the same, otherwise
one of the groups
The simultaneous equations approach
requires that
Modigliani, Rasche and Cooper (1970) equate the public's demand for demand deposits to the
supply offered by banks. The IS-LM approach makes use of simultaneous equalities between
goods supplied and goods demanded (IS) and between money demanded and money supplied
(LM) to obtain equilibrium figures for GNP and interest rates (Dombusch and Fischer 1981,
Ch. 4, eq. 6a and 12a).
More
precisely, desired quantities equal actual quantities in a stress-free equilibrium.
Most
which aU states (integrations)
of a system are constant or are growing at a constant rate. Such a condition can result when
desires are not fulfilled but where opposing forces in the system conspire to prevent further
movement of actual quantities toward desired. This is termed a "stressed" equilibrium. There
is a pervasive beUef among economists (including economists working in the system
dynamics tradition) that actors in the economic system can analyze a stressed equilibrium and
take action to achieve a more satisfactory equilibrium.
generally, an equilibrium might be considered as a condition in
JHH42.2 1/14/86
3
^
sellers'
desired quantities always equal buyers' desired quantities (equation
Hence, the
3).
simultaneous equations approach carries the assumption that financial markets are always in
equiUbrium.
Real-life interest rates are not literally determined by the solution of simultaneous
equations. This conclusion results both
from observation (people do not appear
to solve
simultaneous equations) and from a recognition of the informational and cognitive limitations
faced by decision makers:
No individual or group of individuals could possibly know all the
functions of all the players in the financial markets. And, even
if all
solution of the set of simultaneous equations resulting therefrom
if
possible to achieve at
would be
known,
the
prohibitively expensive,
all.
Since the process of solving simultaneous equations does not
world process of
functions were
interest rate formation, use
literally
describe the real
of simultaneous equations in models of interest rate
determination must rest upon the assumption that a process exists in the real world which quickly
equates the desired
demand
a modeling context, "quick"
model within which
means quick relative
to the desired
means quick
supply of securities (Cf. Eberlein 1984,
p.
183
ff.),
.
In
in relation to other processes represented in the larger
the interest rate formulation
is
embedded. In an empirical context, "quick"
to the frequency of observations.
The use of simultaneous equations
work of Friedman and
in the present context
others cited above for
two reasons.
seems
First, the
less appropriate than in the
previously cited
intended for quarterly models, while the formulation presented in these pages
continuous-time model. Second, the formulation presented below
is
is
work was
intended for a
intended to account for the
general level of interest rates, not the differences between rates on different securities. Since most
of the previously cited work includes,as explanatory variables, rates on other securities; that work
may be
considered as accounting for interest rate differentials which
may move
faster than the
entire structure of interest rates taken as a whole.
The Adjustment
Process. The present work does not assume simultaneities, but
focusses on the process which
moves
interest rates
toward equilibrium. Walras (1954)
appropriately called this process "a groping" or tatonnement process. In Walras' formulation, an
"auctioneer""^ calls out an interest rate, participants
6-
make
offers to lend
and borrow
at this interest
Modigliani, Rasche and Cooper (1970) do not use exogenous interest rates as explanatory
variables.
7.
The term
"auctioneer"
is
common. "Master of Ceremonies" would
function.
JHH42.2 1/14/86
4
better describe the intended
rate. If
He
desired loans do not match desired borrowing, the auctioneer calls out a
new
interest rate.
continues in this fashion until offers to borrow equal offers to lend, at which point he permits
the participants to transact
While Walras' process
idealization of the
note that
it is
is
considered even by those
mechanism by which
who use
as "a fairly extreme
it
prices are determined" (Malinvaud 1972, 140),
well to
actually only a bit less unwieldly than the solution of simultaneous equations. Instead
of requiring massive amounts of information and computational power,
massive and patient cooperation of institutions and the public
more than
it is
at large.
this
process requires the
The process
is
really
little
a hypothetical computational technique for the solution of a set of simultaneous
equations (Goodwin 1951).
2.
An
Alternative Adjustment Process.
Realism and Computational Ease. The lack of realism
be identified in
price
is
two linked ways.
at least
in Walras'
tatonnement may
cannot take place before an equilibrium
First, transactions
determined, whereas on real-world commodity and stock exchanges "without exception
contracts are
made
at
each of the prices called." (Malinvaud
although a kind of intermediary,
permitted to buy and
sell for his
is
p. 139).
Second, the auctioneer,
not permitted to take a position in the market, that
own account;
is,
unlike his real-life counterparts. Realism
he
is
not
may be
improved, therefore, by permitting transactions out of equilibrium and permitting the intermediary
to
buy and
sell for his
own
account.
A straightforward improvment in realism may be achieved by replacing Walras' auctioneer
with an aggregated financial intermediary. The simplification involved in taking
the reverse of that
made by Walras: The
this step is quite
intermediary, rather than being a party to no transactions,
can be assumed to be a party to every transaction. Such an arrangement decreases the
computational effort required by the auctioneer and points the
way
to a realistic
and
practical
way
of representing interest- rate determination in a dynamic, continuous-time macro-economic model.
If the intermediary stands ready to
to set interest rates encapsulated in
integrate,
and thereby summarize, the
or a small inventory of
money
at a rate greater than investors
money
its
balances are low, that
JHH42.2 1/14/86
borrow and
will find the information necessary
it is
have been selling
securities
illiquid,
it
from
it.
should, and
it
The
inventories
A large inventory of securities
entire history of transactions.
have been buying
when
it
inventory of securities or of money.
indicates borrowers
is
lend,
securities to the intermediary
When
will
the intermediary's
be motivated
to, raise
the
which
price at
it
will
buy or
sell
from
its
own
account^.
The converse holds when
securities are
liquid funds are high.
low and
Assuming
that intermediaries are prepared to
buy and
sell
from
price (as long as their supplies last) implies a decoupling of quantities
intermediaries are wilUng to absorb differences between supply and
inventories
last,
there
is
no
their
own
demanded and
demand,
at least
rigid requirement that desired supply equal desired
intermediary sectors. Hence, the formulation suggested here
is
accounts
at
a
supplied. If
while their
demand in
the non-
a disequilibrium formulation in the
sense that desired supply does not have to equal desired demand.^
It is
important to note that while the formulation does not assume equality between supply
and demand,
it
does not preclude equality. Indeed,
mechanisms tending
demand
to bring a larger
this
formulation would be one of the
macro-economic model
to
an equilibrium in which desired
for securities equals desired supplies.
It is
interesting that the description of interest rate formation in terms of intermediaries
valid for the particular market in
securities dealers are the
which the
risk-free rate is set. In that
is
market government
primary intermediary (often dealing with other intermediaries such as
banks). In giving a first-hand description of the Federal Reserve's open market transactions on a
particular day, Paul
Meek
(1978) writes: "...A last-minute contact with the traders indicates that
banks and others are beginning to
sell treasury bills to dealers in greater
volume than buyers
taking from them. Dealers are raising the rates they are bidding for treasury
that dealers at this point
in
I
might add
have a low "inventory" of liquid funds.
The question remains
manner
bills...".
are
as to
how to formulate,
in a
which intermediaries determine a reasonable
of Walras (Samuelson's (1947,
p.
270),
Goodwin
way
consistent with observation, the
interest rate. In the
modem interpretation
1951, Negishi 1962) price
movement
(dp/dt)
is
proportional to the difference between supply and demand. This formulation possesses the virtue
of simplicity and
grounded
to the
may
be well suited to the analytical needs of stability theory, but
in empirical observation of
how
people make decisions and
is,
it
is
not well
therefore, less well suited
needs of a behavioral macro-economic model. In addition, the formulation assumes
continuity in interest rate movements. Since prices are not physical accumulations, there
8.
An
9.
A
intermediary
further
way
may
in
also restrict
its
no
lending activities.
which the formulation can be a disequilibrium formulation is that there
sectors receive what they desire. This is consistent with the observed
requirement that
occurrence of credit crunches (Sinai 1976).
JHH42.2 1/14/86
is
6
is
no
reason to
make
this
assumption.
continuity and which
An
is
It is
desirable to create a formulation
firmly grounded in observation of
appropriate place to begin thinking about the
individual decision making.
is
a
way people choose rates
common strategy
determination (Hogarth 1980 p.47, Tversky and
adjustment and anchoring process people
come
Kahneman
to a
is
the literature
on
that the process
for arriving at judgments like interest rate
1974, Kleinmuntz 1985). In the
judgment by adjusting a preexisting quantity
of currently available information.
The empirically observed adjustment and anchoring
strategy
undergird the interest rate setting process in the real world. ^^
that in the presence
decision making.
A large and growing body of such work suggests
termed "adjustment and anchoring"
(the anchor) to take account
human
which does not assume
More
is
here hypothesized to
particularly,
it is
hypothesized
of supply and demand imbalance, summarized by the inventory positions of
intermediaries 1 ^ people choose a current interest rate by adjusting up or
termed the "underlying
interest rate".
The underlying
down an anchor here
interest rate represents the interest rate
environment to which people have become accustomed.
A mathematical formulation suitable for use in a simulation model involves clarifying three
issues. First,
it is
necessary to consider the manner in which the current interest rate
above or below the anchor; second, the way people form the anchor
finally, the
manner
in
which inventories are
Multiplicative Adjustment.
example a
five dollar difference in price
dollar difference
is
must be described.
(1982, p. 168) present results
perceived as less important as the price increases. For
on a $15 item
on a $125 item. This suggests
between the anchor and the
adjusted
must be considered; and,
translated into an adjustment tenn
Kahneman and Tversky
suggesting that a given difference in price
itself
is
is
subjectively
that for a given state
more important than a
five
of liquidity the difference
risk-free rate should increase as rates increase in order for the
psychological impact of the difference to remain the same.
A functional form in which the effect of
hquidity enters multiplicatively possesses this characteristic.
TO^
11.
This discussion makes an aggregative leap. Hogarth (1980) and Tversky and Kahneman
(1974) examined individual decision making. Here I am discussing an aggregation of
decisions made by many people. As discussed below, this consideration suggests the need
for empirical validation at a macro level.
possible to formulate this either on the basis of the inventory positions of intermediaries
alone or on the basis of the inventory positions of both intermediaries and non-intermediaries.
For realistic adjustment times in the non-intermediary sector, there is Uttle dynamic difference
since money-inventory imbalances are corrected quickly.
It is
JHH42.2 1/14/86
7
RFIRt=UIRt*ELRt
(5)
ELRt=f(i"tS''"^^i^^s' liquidityt)
(6)
Where
RFIR- Risk Free Interest Rate
UIR - Underlying Interest Rate
ELR -
Effect of Liquidity on the risk free Rate
decreasing function
A
f
In equation 5 the effect of supply and
demand enters
multiplicatively, indicating that the
difference between the risk free rate and the underlying rate depends
Hence, during periods of high
between
UIR and RFIR
will
interest rates, indicated
upon
the underlying rate.
by a high value of UIR (see below), the gap
be greater than during periods of low rates for identical values of
ELR.
The Underlying
Interest Rate.
inflation rate" or the "underlying
at the present time, abstracting
The underlying
The underlying
unemployment rate",
from the
that rate
interest rate is the reference condition to
which people
feel generally holds
on the actual
interest rate.
which people have become accustomed;
become accustomed
to the
it
changed
A weighted average with recent experience weighted most heavily would seem to be
an appropriate mathematical representation of this concept
property and
analogous to the "underlying
effects of transitory pressures
continually adjusts as interest rates change and people
environment
is
rate,
is
easily representable in a simulation
model
An exponential
average possesses
(Forrester 1961, p. 406-4 11).
this
While a
small modification will be considered in the section of this essay dealing with behavior and in
appendix
2, for
most purposes the underlying
rate
may be
defined
as:
UIRt= (RFIRt - UIRt)/TAUIR
(7)
Where UIR
-Underiying Interest Rate
-Risk Free Interest Rate
-Time to Adjust Underlying Interest Rate
TAUIR
and the dot (•) represents a time derivative.
RFIR
The speed of adjustment of UIR toward RFIR depends on
the parameter
diagram below charts the adjustment path of UIR for a step increase
TAUIR.
JHH42.2 1/14/86
in
RFIR
TAUIR. The
for several values of
1
Exponential Adjustment Paths
2 UIR2
UIR1
1.000
il
0.750
k\
0.500
il
0.250
0.0
l)
-
3
UIR3
be the same in an equilibrium with constant nominal rates and no maturity preferences.
rate will
Hence, a notion of a future
normal long
stress-free (hence, equilibrium) short rate
is
very close to a notion of a
Indeed, Modigliani and Sutch (1966) (M-S) suggest that Keynes's normal rate
rate.
can be approximated by a weighted average of past experience where recent experience
most heavily.
An
is
weighted
exponential average possesses this property; hence, (7) conforms to the
computational form of the normal
M-S
rate.
Modigliani and Sutch also considered an extrapolative component in expectations of future
M-S, however, were concerned
rates.
with interest rate expectations over the entire course of the
future, rather than with the very long-run expectation of the future short rate.
that the extrapolative
medium
term.
It
seems possible
component reflects expectations of continuing pressures over the
Such extrapolative terms may not be present
short and
in the expectation of the future short
such time when pressures have relaxed.
rate at
Modigliani and Shiller (1973) reconsidered the problem of interest rate expectations
light
of inflationary expectations. They concluded that interest rate expectations include a separable
effect of inflation.
Whether
this
would hold
which people have become accustomed
component may be unimportant
inflation
and
in the
might be
in
is
true for the underlying rate
less clear. In
which connotes a
rate to
any event, the fact that the extrapolative
forming the underlying rate suggests that a separate impact of
difficult to discern if expectations
of inflation are also regressive
(cf.
Modigliani
Shiller, 20-21).
Effect of Liquidity. While the terminology of "liquidity" tends to restrict attention to
money
balances,
it is
well to note that the effect with which
we are concerned can
function either of the inventories of securities or the inventory of money.
be viewed as a
One can work with
either
inventory because an increase in one implies an equal decrease in the other.
Because the concern here
is
with movements
common
to rates
on
all securities,
a potentially
troublesome aggregation issue can be largely avoided by considering the inventory of money, since
instruments used as
money
explicitly considered
The simplest
desired
money
and the
be quite similar. Consequendy, money inventories will be
"liquidity" terminology is descriptive.
variable measuring the state of Uquidity
holdings.
institutional size,
will
However,
this difference will
which one might expect
is
the difference
between actual and
be sensitive to the price level and
to be unimportant in the determination of interest rates.
Inventory discrepancies, therefore, should be measured relative to
some
base. Either desired
inventories or actual inventories are prime candidates. Choosing inventories as the base yields the
following expression where intermediaries'
JHH42.2 1/14/86
money
inventories are termed "reserves":
10
ELR = f(RAR)
RAR = (R-DR) R
/
considering the relative available reserves of the banking system where, as discussed below,
desired reserves are taken to be identical to required reserves.
1984 and as high as
range of
1
0.
Rates remained between
1
1% and 8%.
RAR went as low as -.25
during
Consequently, a value of a in the
or 2 would seem reasonable.
3.
Behavior. Equations
Behavior and Estimation.
5,7,9,
and 10 constitute a theory of
interest rate formation.
The
equations are reproduced below for convenience.
RFIRt = UIRt*ELRt
(11)
UIRt = (RFIRt - UIRt)/TAUIR
(12)
ELRt = f(RARt) = e-«*RAR^
(13)
RARt = (Rf DRt) / Rt
(14)
Where
UIR
ELR
TAUIR
R
DR
JHH42.2 1/14/86
-
-
-RFIR - Risk Free Interest Rate
Underlying Interest Rate
Effect Liquidity on the risk free Rate
Time to Adjust Underlying Interest Rate
Reserves
Desired Reserves
12
The stock and flow diagram'^ of the
structure
may be drawn
as follows:
A sense for the dynamics implied by the above structure may be gained by considering a
plot,
shown below, from
both set to
2,
a simulation. In this simulation
a reasonable value for each as discussed above. Initially
reserves are set equal to desired reserves, so
structure
is
TAUIR (measured in years)
disturbed from equilibrium by a
UIR
is
and
2% per year, and
RAR begins at its equilibrium value of zero.
10%
a are
step decrease in reserves; after
some
The
years
reserves step up to their original (equilibrium) level. Plotted in the diagram below are the risk free
interest rate
12.
RFIR, the underlying
interest rate
UIR, and
relative available reserves
RAR.
Stock and flow diagrams are in common use in system dynamics. They provide a close
pictorial representation of a system of integral equations. The diagrams appear in several
styles, the above diagram was created using STELLA® on a Macintosh computer. In the
above diagram, the box represents an integration. The "Pipeline" arrow into the box
represents a rate of flow which is controlled by the "valve" symbol termed CUIR (Change in
UIR). CUIR, mathematically, is the time derivative of UIR. Circles represent constants or
functions. Single-line arrows represent information connections and reveal the arguments of
each function. For a good discussion of diagramming conventions in system dynamics see
Richardson andPugh (1981, chapters 1-2).
JHH42.2 1/14/86
13
Interest
1
RAR
1
i)
2
RFIR
Rate Formulation Simulated Behavior
3
UIR
The
analysis of the above behavior provides the logic for uncovering a flaw in the
formulation. Just as the structure causes interest rates to increase exponentially in response to a
maintained decrease
in reserves, the structure will
cause rates to decay exponentially in response to
a maintained increase in reserves. Equation 11, however, indicates that once the underlying rate
reaches zero, the risk free rate will also equal zero. Equation 12 indicates that
identical,
no
further
movement occurs
in the structure.
small risk that rates will get "stuck" at zero.^^
Consequently
The appendix
the
if
two
rates are
this structure carries the
to this essay contains a small
modification to the structure developed above which removes the risk entirely. Since evidence of
would be
the small modification
data,
I
difficult, if
not impossible, to pick up in the available time-series
proceed directly to a discussion of empirical
validity.
Estimation. The structure developed above has been based upon empirical observation
at the individual level,
whereas
it is
intended to represent macro behavior.
present evidence that the formulation
is
It is
now
necessary
consistent with macro-economic observation.
Toward
to
this
end, the structure will be estimated econometrically, significant parameter estimates of reasonable
magnitude and predicted sign will constitute evidence
economic data on
The
fu-st
interest rates
step
is
convenient way to do
solving fcr
that the formulation is consistent with
macro-
and reserves.
to convert equations 11-14 into a
form which can be estimated.
this is to solve the differential equation^'*.
Substituting
1 1
into 12
A
and
UIR yields:
ln(UIRt) = ln(UIRto) + (l/TAUIR)J(ELRs
-
1)
ds
(15)
to
From
(1 1)
we know
that:
UIRt = RFIRt / ELRf
(16)
Substituting 16 into 15 and rearranging yields
continuous-time simulation, UIR will never reach zero. However, since actual
simulations must proceed by finite steps, UIR can hit zero in an actual simulation.
Furthermore, the problem of getting "stuck" does not materialize suddenly; a UIR "very close"
to zero will be "very close" to being stuck at that value.
13. In a truly
would also be possible to approximate the differential equation with a difference equation.
As win be seen, the solution of the differential equation results in a "quasi" linear equation.
The difference equation would be non-linear. Further, there is some evidence that the use of
14. It
difference equations as an approximation to an underlying continuous-time
problems (Richardson, 1981).
JHH42.2 1/14/86
15
model can cause
InCRFDlt) = ln(UIRto) + ln(ELRt) + (l/TAUIR)J(ELRs
-
l)ds
(17).
to
Substituting for
ELR using
13 and reexpressing the integral produces:
t
ln(RFIRt) = In(UIRto)
-
a*RARt + (lArAUIR){je-a*RAR^ds
-
(18)
(t-to)}
to
or for estimating purposes:
ln(RFIRt)= k +
Where z=
The
z*RARt + h*{CELRt
(-a),
integration in equation
in the instantaneous transition
entire equation
1
8
CELR is
a
month by month accumulation.
may be approximated with
be exact
if
a month-by-month accumulation
RAR were constant during the month, changing only
from one month
may be
(19)
(t-to)}
h = (1/TAUIR), and
CELR. The approximation would
The
-
to the next.
estimated by the following method: Start with an estimate of
a
and form the month-by-month accumulation of the exponential CELR. Next, estimate h and the
coefficient z using ordinary least squares.
adjusting the old estimate of
accumulation
The
risk-free
CELR
and
(-z)
a toward
Form a new month- by -month accumulation CELR by
(-z).
Continue
manner
until the
a used to
form the
converge.
question of what data to use in estimating 19 remains to be discussed. Data on the
nominal
interest rate interest rate
poses no significant problem since time series on
treasury bills are readily available (see appendix
Data
in this
1).
Observation of
RAR poses a greater problem.
relating to depository institutions can be used with the assumption that the reserve positions of
depository institutions are highly correlated with the reserve positions of other intermediaries. This
assumption
is
discussed at greater length below.
In forming
RAR for depository institutions, a problem exists.
concerning the reserves of depository institutions
reserves
is
reserves 15
not. Here,
I
is
While information
available (see appendix
will use required reserves (see
appendix
1) as
1),
data on desired
a proxy for desired
7his seems conscionable since required reserves are likely the major determinant of
desired reserves. Other factors, such as changing interest rate spreads (Cf Modigliani, Rasche
15.
The
difficulty in deriving an estimable expression for free reserves is
Rasche, and Cooper (1970).
JHH42.2 1/14/86
16
noted by Modigliani,
.
and Cooper (1970, especially
at
eq. 3.11), fluctuations in the degree of Federal
borrowing from the discount window, the covariance between deposits and withdrawals, and
the degree of risk in the lending portfolio, will result in
away from
desired reserves
It is
instead of
what are
likely to
possible to consider the distortion introduced by using
RAR of all intermediaries.
intermediaries.
one
set
It is
be minor variations
difficult to
It
seems
RAR of depository
institutions
likely that the relative inventory positions of all
may
be quickly traded between
imagine a pool of excess Uquidity obtaining for a significant period
of intermediaries while another class
is illiquid.
reserves of different intermediaries are correlated. This
It
must be
means
true that
tiie
relative available
that^^:
RARt = j*RARtb + et
where RAR''
and j>0. This means
is
in
the level of required reserves.
intermediaries are highly correlated because financial instruments
in
Reserve displeasure
(20)
relative available reserves for depository institutions, e is a disturbance term
that
b
ELRt = e-«*RAR^ = e-a*J*RAR^
and the estimate
positive
j
(-z),
and a, a positive
obtained from 19
estimatj'; (-z)
OLS estimation of equation
No
constitute evidence in favor of
1
from January of 1959
in figure
1
to
the. tiieory.
i"^
June of 1985^^^
below.
20 because such a constant would imply tiiat balanced
depository institutions implies imbalanced liquidity elsewhere in the economy.
This seems unreasonable.
Bias
is
among
also introduced by the error term in (20). In the absence of an obvious instrument
correction has been
18.
would
actually an estimate of a*j. Since theory calls for
intercept constant appears in
Uquidity
17.
is
19, using data
produces the results summarized on line
16.
(21)
made
The time period chosen was determined by
of the data used
JHH42.2 1/14/86
no
for this factor.
may be found
in
appendix
the availability of time series data.
1
17
A description
Figure
1
:
Estimation Results
+
—
.83
a reasonable range.
errors
from
this
The
autocorrelation function and partial autocorrelation functions for the
The proces s
regression appear below.
is
indistinguishable from white nnispl^
1_U JUl^SSS
-_JAi^A:-e«rfck,+«>A.£0ft«Ji4a
J
wv^
• 5"*«Yj**-**««*j«»*»-.w«*« j-tljaBj^aw-
—
4^::
•!
In order to test the stability of these results the data has
performed on each
1959
half.
The
third line of figure
March of 1972, while
to
June of 1985.
It is
1
been divided
of both h (1/TAUIR) and z (-a*j). Taking
and regressions
contains estimates using data from January of
the fourth line contains estimates using data
clear that decreasing the
in half
amount of data increases
this increase in
from March of 1972
the variance of the estimates
variance into account, the estimates
appear to be within tolerable ranges of one another indicating that the coefficients are
Chow
test
(Pindyck and Rubinfeld,
regressions are identical.
The
p.
to
123-126) gives an explicit
F-statistic for this case is 1.0221.
test
stable.
The
of the hypothesis that the
Under
the hypothesis that the
regressions are identical, large values of the statistic are increasingly unlikely. Statistics greater
than 1.0221 would be expected with a probability greater than one half. Consequently, the
hypothesis seems to be quite acceptable.
More
formally, the hypothesis that the regressions are
identical cannot be rejected at the fifty per cent confidence level.
4.
Summary and
Conclusions
In this paper a behavioral theory of interest rate formulation has been developed
The theory
postulates a disequilibrium adjustment process.
intermediary willing to buy and
sell for its
At the heart of the theory
is
and
tested.
an
own account at prices which do not immediately
equate
supply and demand.
The important informational
inventory of money.
input into the price setting formulation
The inventory of money
is
purchases and sales of securities.
financial sectors
have been
19.
As
A low money
inventory indicates that the non-
selling securities at a rate greater than they
is
have been buying them.
An
the appropriate response in this situation. Real-life intermediaries, in
a check, a regression employing a
definitely not white noise.
first
order correction was performed.
were very
JHH42.2 1/14/86
the intermediary's
an accumulation of the history of the non-financial
sectors' realized
increase in the interest rate
is
19
The
error terms
fact,
are motivated to increase rates under these circumstances.
inventories are high and the intermediary
The
rate".
interest rate is adjusted
The underlying
interest rate
is
applies
when money
highly liquid.
up or down
relative to an
the rate to
is
The converse
anchor termed the "underlying interest
which people have become accustomed.
It is
represented mathematically as an weighted average of past values of the actual interest rate with
recent experience weighted most heavily.
Empirical support for the formulation exists
at
two
levels of aggregation.
Research into the
nature of human decision making at the individual level indicates that adjustment and anchoring
common
strategy for
making decisions
Econometric estimation was used
The research reported
more
macroeconomic
level of aggregation.
paper can be extended in two directions: one empirical and the
other theoretic. Better data can be collected
Altematively,
on
the reserves
and desired reserves of intermediaries.
sophisticated estimation techniques might be utilized to correct for biases
introduced by using surrogate measures. Observation of the
determined by those
who actually
way
interest rates are actually
determine them would provide an even more valuable
opportunity to confirm or disconfirm the formulation suggested in this paper. Observation
individual decision
a
like those involved in interest rate determination.
to validate the process at a
in this
is
making
level
would also provide
at the
the opportunity to refine the theory presented
here.
Theory might also be extended through disaggregation. While
this
paper dealt
at a highly
aggregated level, the disequiUbrium process described herein could be applied to a more detailed
model which included several
effort
would involve a
different securities
shift in purpose.
and several different intermediaries. Such an
The overarching goal of the
theory of interest rate formation realistic enough to contribute
obfuscate, an understanding of the major
to,
current research has been a
yet simple
enough not
to
macroeconomic behavior modes. The goal of
disaggregation would be the increased understanding of behavior
Appendix
1:
modes
in the financial markets.
Data Sources
This appendix provides information on the data, and their transformation, used to estimate
equation 19.
three
month
This rate
is
The
risk-free rate is
based on CITIBASE's (Citibank 1985) secondary market rate on
treasury bills (data series
FYGM3, see
calculated on a bank-discownt basis.
following calculation were performed.
DISC = FYGM3
JHH42.2 1/14/86
/
First, the
also Federal Reserve Bulletin, table 1.20)).
To
convert this to a continuous time yield, the
percentage discount (DISC)
is
A 1.1
4.
20
The decimal
(DTY) may now be
discrete-time yield
written as
DTY = DISC/(100-DISC).
This
may be
A1.2
converted to a continuous-time yield (CTY) as
CTY = In (1
DTY).
-t-
A1.3
Finally, prior to performing regressions, this
RFIR = CTY
A1.4
RAR are defined as the difference between reserves and desired
The data
nonborrowed reserves of depository
series
change
The
series are,
institutions,
and for desired reserves
The
in reserve requirements, in particular the
hold reserves associated with the Monetary Control Act of 1980.
between the
correlation
FZRNBA nor FZRQA are seasonally
however, adjusted for changes
in institutions required to
RAR used in this report and an alternate RAR constructed with
non-seasonally adjusted, non-reserve-adjusted time series
alternate definition of
RAR was
FZRNBA,
ClTlBASE's FZRQA,
used for reserves was ClTlBASE's
required reserves of depository institutions. Neither
adjusted.
to a yearly percentage basis:
* (365/90) * 100
Relative available reserves
reserves, divided by reserves.
was converted
(FZRNB and FZRMB)
is
.976.
The
tested over the full range of observations (1959/1 to 1985/6).
Parameter estimates were insignificantly different from those reported in the text using data
adjusted for changes in reserve requirements.
Appendix
As discussed
2:
Modification to Interest Rate Structure
in the text, the structure contains a risk that rates will
A small modification removes this risk.
The change involves introducing a new
indicated underlying rate lUR, toward which the underlying rate adjusts.
rate is identical to the risk-free rate, except the underlying rate
some minimum
The
"stuck" at zero.
variable, the
indicated underlying
does not go to zero, but rather to
value.
Behaviorally such a modification implies there
greater than zero. Economically this
The minimum
intermediation.
expressed as a
become
fiiaction
rate
minimum
may
rate
is
may
some lowest
reference rate which
is
be be determined by the costs of
be interpreted as the average cost of intermediation
of the amount intermediated.
A restatement of the interrest rate formulation using a possible definition of the indicated
underlying rate
20.
is^^:
A
function involving a gradual approach to MUIR, in place of the abrupt change introduced by
max function, would also be a possible assumption. The use of such a function, however,
introduces additional complications and further dynamics without materially enriching the
the
current discussion.
mH42.2
1/14/86
21
RFIRt = UIRt*ELRt
(A2.1)
UIRt = (lURt - UIRt)/TAUIR
lURj = max(RFIRt, MUIR)
(A2.2)
(A2.3)
(A2.4)
Where: RFIR
e
-Risk Free Interest Rate
-Underlying Risk Free Interest Rate
-Effect of Liquidity on the risk free Rate
- Indicated Underlying Interest Rate
- Minimum Underlying Interest Rate, a constant
- Time to Adjust the Underlying Interest Rate
- Relative Available Reserves
- The exponential function
a
-
UIR
ELR
lUR
MUIR
TAUIR
RAR
The new
a constant
definition of the underlying interest rate appears as A2.3.
The modified stock and
flow diagram appears below.
A comparison of the behavior between the original formulation and the modified
formulation appears below. In the simulation,
disturbed from equilibrium by a
step
down
to their original level
JHH42.2 1/14/86
fifty
TAUIR
and
a
are set to 2.
The
structure
per cent step increase in reserves around year
around year 28.
22
1.
is
Reserves
Behavior of Modified vs. Un-Modified Structure
0.025
0.019
0.013
6.250e-3
-
0.0
0.0
30.000
The diagram above
illustrates the possibility that the
(unmodified) Underlying Rate can "stick" at zero.
this risk.
It
It
shows
(unmodified) Risk Free Rate and
that the
modified structure eliminates
should be noted that although the structure keeps the underlying rate from dropping
below MUIR, the
risk free rate
may drop
lower.
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25
Jaff6.
Homewood,
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Date
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