copyright by timothy gordon jones 2008
TRANSCRIPT
The Dissertation Committee for Timothy Gordon Jones
certifies that this is the approved version of the following dissertation:
Essays on Money, Inflation and Asset Prices
Committee:
Russell Cooper, Supervisor
Dean Corbae
Kim Ruhl
Jennifer Huang
M. Fatih Guvenen
Essays on Money, Inflation and Asset Prices
by
Timothy Gordon Jones, B.A.; M.S.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
May 2008
Acknowledgments
I would like to thank my advisor, Russell Cooper, whose guidance and support
have been essential both to this dissertation and to my professional development. I
am also grateful to Dean Corbae, Jennifer Huang, Kim Ruhl, and John Willis for
helpful advice and guidance. This work also benefited from useful comments from
the participants of numerous seminars at the department.
Additionally, Chai Chi, Patrick Cosgrove, Shayne McGuire, Kay Cuclis,
Phillip Auth, and KJ van Ackeren are a few of the excellent investment managers
at the Teacher Retirement System of Texas who assisted with their specific ideas
and general contributions to an open and productive exchange of ideas.
I am most grateful to Conan Crum, Jason DeBacker, Pablo D’Erasmo, Rick
Evans, and Anna Yurko for being supportive friends, excellent officemates, and
invaluable colleagues. This dissertation could not have been completed without
their advice and support.
Timothy Gordon Jones
The University of Texas at Austin
May 2008
v
Essays on Money, Inflation and Asset Prices
Publication No.
Timothy Gordon Jones, Ph.D.
The University of Texas at Austin, 2008
Supervisor: Russell Cooper
This dissertation explores different aspects of the interaction between money and
asset prices. The first chapter investigates how a firm’s financing affects its decision
to update prices: does linking interest rates to inflation alter the firm’s optimal price
updating strategy? Building on the state dependent pricing models of Willis (2000)
and the price indexing literature of Azariadis and Cooper (1985) and Freeman and
Tabellini (1998), this model investigates the financing and price updating decisions
of a representative firm facing state-dependent pricing and a cash-in-advance con-
straint. The model shows the circumstances under which a firm’s financing decision
affects its price updating decision, and how the likelihood of changing prices af-
fects the amount borrowed. It also illustrates how the use of nominal (as opposed
to inflation-linked) interest rates leads to a lower frequency of price updating and
vi
higher profits overall for a firm facing menu costs and sticky prices.
The second chapter extends the bank run literature to present a theoretical
mechanism that explains how money supply can affect asset prices and asset price
volatility. In a two period asset allocation model, agents faced with uncertainty
cannot perfectly allocate assets ex-ante. After income shocks are revealed, they will
be willing to pay a premium over the future fundamental value for an asset in order
to consume in the current period. The size of this premium is directly affected by
the supply of money relative to the asset.
This paper explores the relationship between economy-wide monetary liquid-
ity on the mean and variance of equity returns and in relation to market liquidity.
At an index level, I test the impact of money-based liquidity measures against exist-
ing measures of market liquidity. I proceed to do a stock level analysis of liquidity
following Pastor and Stambaugh (2003). The results indicated that measures of
aggregate money supply are able to match several of the observed relationships in
stock return data much better than market liquidity. At an individual stock level,
monetary liquidity is a priced factor for individual stocks. Taken together, these
papers support the idea that changes in the money supply have consequences for
the real economy.
vii
Contents
Acknowledgments v
Abstract vi
Chapter 1 Price and Interest Rate Indexing 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Sequence of Decisions . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Firm’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 Solving the Firm’s Problem . . . . . . . . . . . . . . . . . . . 8
1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 Price Updating Decision . . . . . . . . . . . . . . . . . . . . . 12
1.4.2 Financing Decision . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Indexing Effects . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Chapter 2 Asset Prices and Money in a Two-Period Equilibrium 39
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
viii
2.2.1 Timing and Uncertainty . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2.3 Market Clearing . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Solving the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.1 First period (t = 1) . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.2 Initial Period . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.3 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.4 Initial Price, p0 . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.1 Impact of liquidity shocks on the mean and variance of p0 . . 55
2.4.2 Monetary Services Index . . . . . . . . . . . . . . . . . . . . . 58
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Chapter 3 Monetary Liquidity and Asset Pricing 65
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Testing Liquidity on Index Aggregates . . . . . . . . . . . . . . . . . 73
3.4 Testing Liquidity on a Cross-Section of Stocks . . . . . . . . . . . . . 76
3.5 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1 Impact of Monetary vs. Market Liquidity on Returns and
Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5.2 Causality between Market and Monetary Liquidity . . . . . . 88
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Bibliography 91
Vita 94
ix
Chapter 1
Price and Interest Rate
Indexing
1.1 Introduction
In an environment where prices are costly to change, firms face a trade-off between
price inefficiency and adjustment costs. In not updating its price, a firm risks a
demand shock from inflation; its real price will be lower than optimal because of an
increase in the aggregate price level. The standard state-dependent pricing solution
is to update prices when the cost of adjusting prices is lower than the profit lost
from having a non-optimal price. Here we consider that a firm’s financing costs can
also be affected by the aggregate price level, and can feed back into the decision to
update prices. The amount borrowed impacts the price updating strategy through
the cash-in-advance constraint. In addition to borrowing, I compare two types of
interest rate regimes, one indexed to inflation (changes in the aggregate price level)
and the other not indexed. The type of interest rate regime alters the optimal
updating strategy of the firm; firms borrowing at non-indexed rates update prices
less frequently and have higher expected profits.
1
Before addressing the role of interest rate regimes, its necessary to estab-
lish how a financing decision can be relevant to a firm’s value in the first place.
Modiglianni and Miller (1958) proposition I, also called the irrelevance proposition,
maintains that under a set of basic market efficiency assumptions financing decisions
should not affect the core business of the firm. In order to consider price and inter-
est rate indexing, I am necessarily relaxing those assumptions in two ways. First,
I’m allowing sticky prices by imposing state-dependent costs on price adjustment.
Secondly, in order for firms to need financing in the first place, I introduce a cash-
in-advance constraint. The interaction between these two distortions results in the
relevance of financing structure to the firm’s price updating strategy and overall
value.
A cash-in-advance constraint is sufficient to contradict the irrelevance propo-
sition by itself; it can be considered a special case of incomplete asset markets. A
menu cost hinders price adjustment, which will affect firm value, though not nec-
essarily through the capital structure. The model shows how the amount a firm
borrows affects its price adjustment strategy by limiting the quantity it can pro-
duce. Moreover, the interaction of those frictions produces an environment where
the non-indexed interest rate regime generates higher expected profits for firms than
indexed ones. The model offers a mechanism to explain this outcome: indexing rates
deters borrowings and causes the cash-in-advance constraint to bind more severely.
Firms that do not index their borrowing have less to gain from price changes and
will therefore update prices less frequently, and borrow more.
Previous literature in this area has focused on the optimal risk sharing prop-
erties of nominal contracts (or the equivalent term, non-indexed prices). These
papers address models that share a common approach. Each of at least two agents
is a buyer of one good or service and a seller of another. Therefore, equilibria exist
when either everyone indexes their price or no one does. For an agent to pay an
2
indexed price but offer a non-indexed one would mean taking on a disproportionate
amount of risk. If ex-ante risk preferences were the same, this extra risk would not
be optimal. Blinder (1977) defines this relationship and the equilibrium between
worker’s wages and firm’s returns; Azariadis and Cooper (1985) for worker’s wages
and the price of goods. Freeman and Tabellini (1998) establishes the conditions
necessary for agents to choose a nominal, and therefore monetary, equilibrium.
This paper contributes to the literature by addressing the link between the
price of goods and interest rates. In what follows, I describe how a firm’s financing
mechanism can affect its overall value at all, and if so, under what conditions the firm
would be better off with non-indexed interest rates. I develop a partial equilibrium
model of the firm assuming some positive cost to adjusting its price. The firm
borrows before the menu cost and shocks to taste and inflation are revealed in order
to fund current production. The firm can change its price by paying the cost of
adjustment, or use last period’s price. I show that for positive costs of adjustment,
firms have higher expected profits for all initial price levels when borrowing at non-
indexed interest rates.
Section 2 details the environment of the model, the sequence of decisions, and
the firm’s problem. Section 3 solves the firm’s problem and defines equilibrium.
Section 4 shows how the financing constraint and non-indexed interest rates increase
price stickiness for a given menu cost, and goes on to prove the primary result, that
firms prefer non-indexed rates. A conclusion follows in Section 5.
1.2 Model
The model environment is one of two-period partial equilibrium, designed to high-
light the pricing behavior of a firm. The firm enters the first period with an initial
price and a unique production technology, converting labor into output. The de-
3
mand for its product is know to the firm and can purchase labor competitively. 1
Due to a cash-in-advance constraint, firms must borrow money in the first period
to produce in the second. If a firm opts to change its price, it has to pay a cost
of adjustment drawn in the second period. When the firm makes its financing
decision, it does so under uncertainty about demand, its real price, and the cost of
adjustment. Consider the choices a firm faces in the event of a high realization of
demand (from inflation or a taste shock). If the firm has borrowed more money
than it expected to use, it has additional capacity to hire workers and meet the high
demand rather than paying the menu cost and raising its price. This increased flex-
ibility comes at the price of higher borrowing costs. If the firm borrowed only what
it expected to need, the same high demand scenario makes it more likely that the
firm will pay the menu cost and raise its price rather than sell its limited quantity at
the initial price, which after the demand shock is much lower than the optimal reset
price. This uncertainty about demand connects the financing decision to the price
updating decision. The firm must decide on the amount to borrow, and thereby
commit to both an upper limit on how much it can produce and lower limit on its
costs. Those limits will affect its decision to update its price in the next period.
1.2.1 Sequence of Decisions
The first period is the financing period, when the firm must decide how much money
to borrow for use in the second period. In the second period, capital markets are
closed, and the firm must use what it has borrowed to hire workers and produce
goods. The interest rate, indexation regime, wages, and productivity are exogenous
and held constant. In the first period, before the shocks and menu cost are revealed,
the firm borrows by issuing one period bonds to finance current second period pro-
duction . After the first period’s borrowing is complete, the taste shock, price level,1The particular form of demand can be derived as an outcome of optimized CES preferences,
which is standard in the price adjustment literature.
4
Two-Period Timeline
Initial Price Draw: p-1
Collect Revenue, Repay Debt
Borrow under uncertainty: B*(S)
If changing, set new price: p*(S)
Hire (and pay) labor to fulfill produce: qs
t=0 t=1
Menu cost draw Ψ, taste shock µ, and aggregate price level P revealed
Decide to Update Price or Not: ΨΨ’(S)
Figure 1.1: Sequence of Decisions by the Firm
and menu cost are realized and the firm decides whether to update prices. After
the firm sets its price, it pays wages and produces output. Lastly, the firm collects
revenue and uses it to repay the bonds. [see Figure 1.1].
1.2.2 Uncertainty
The firm faces three types of uncertainty in the first period when making the fi-
nancing decision. The taste shock that will affect demand in the second period is
exogenous and i.i.d. Similarly, the menu cost that the firm will face when making
its price updating decision is unknown in the first period and drawn from a distri-
bution in the second. This is a similar framework for state dependent adjustment
costs as in Dotsey, King and Wolman (1999). The third type of uncertainty the
firm faces is the new price level, or an inflation shock. The inflation shock is drawn
from a mean zero distribution, and the new price level will affect both demand and
5
the firm’s costs (see Equations 1.1 and 1.2).
logP = logP−1 + ε (1.1)
ε ∼ N (0, σ) (1.2)
1.2.3 Firm’s Problem
In the first period, the firm has an initial price (p−1) and knows the initial price
level (P−1). These two values define the ex-ante state, from which the firm takes
expectations about next period’s demand and costs. The firm then borrows in this
period in order to finance production. In the next period, the aggregate shock is
revealed and the firm decides if it should change its price or not.
Firm’s Optimization Problem
In the first period, the firm chooses an amount to borrow. In doing so, it must
take into account the realization of the shocks and its own strategy for updating the
price Equation (1.3) summarizes the first period problem. S−1 is a vector of state
variables known at the time of the financing decision, including the first period price
and aggregate price level. S includes the realizations of the random variables and
the choice of B.
Π (S−1) = maxBE[{
ΠC (p, q, S) ,ΠNC (q, S)}]
(1.3)
S−1 ={P−1, p−1, r
F , φ, λ}
(1.4)
S = {S−1, P,Ψ, µ,B} (1.5)
The two functions that are considered in the first period, ΠC (p, q, S) and
6
ΠNC (q, S), are the profit functions for the firm in the second period, after the
decision to change its price has been made. If the firm decides to change its
price, the second period firm’s problem is equation (1.6). The firm chooses a price
to maximize profits as well as the quantity to produce, subject to the borrowing
constraint and demand equation (1.8). If the firm chooses not to update its price
it simply chooses a quantity. This is meant to allow the firm the option to exit or
partially fulfill demand, not as a tool for optimization.
ΠC(p, q, S) = max{p,q}
pqs (p)− (1 + r (φ))B −
(wn
(P
P−1
)λ
−B
)−Ψ (1.6)
ΠNC(q, S) = max{q}
p−1qs (S)− (1 + r (φ))B −
(wn
(P
P−1
)λ
−B
)(1.7)
s.t. qs (S) ≤ min{qm (S) , qd (S)
}(1.8)
where qm =
[B
w
(P−1
P
)λ]α
; qdi (pi, P ) =
(pi
P
)−γ (D)P
µ; (1.9)
Demand for good i is CES (1.9), where D is a scaling parameter, µ is the
taste shock, γ is the elasticity parameter, P is the aggregate price level, and pi is
the price of a specific good.
The production technology converts labor to output:
qs = f (n) = nα
The structure of the interest rates facing the firm can depend on the price
level, as see above in r.
r =(1 + rF
) [Pt+1
Pt
]φ
(1.10)
The parameters {φ, λ} (1.6,1.7,1.10) define the indexation in the firm’s factor
contracts. φ determines the extent to which the interest rate changes with inflation,
7
while λ controls the extent to which the wage changes with inflation. if λ is zero,
the nominal wage does not change with the price level. If φ and λ are both 1, the
interest rate and the wage are effectively real costs. Ψ is the menu cost (the cost of
updating prices). Equation (1.8) highlights the effect the borrowing constraint has
on the quantity choice. The less a firm has borrowed in the first period, the lower
the maximum quantity it can produce, and the more likely qm will bind.
1.3 Solution
1.3.1 Solving the Firm’s Problem
The firm maximizes (1.3) subject to the cash-in-advance constraint (1.8). The
firm’s problem is solved backwards through the two periods. Solving the second
period first, the firm takes the realized values of µ, Ψ and P as given, and maximizes
profits conditional on changing the price or keeping last period’s price. With the
second period policy function established, the firm solves the first period problem
of how much to borrow, taking expectations over the realizations of the states and
second period policy functions as given.
Second Period
Firms enter the second period with an amount borrowed and a previous price, both
from last period. Current aggregate price level, P , the taste shock, µ, and the menu
cost Ψ, are revealed at that point. Working backwards, the firm’s last decision is to
choose a price to maximize profits, assuming it chooses to pay the adjustment cost.
Choice of Price when Updating When the firm decides to pay the menu cost,
it chooses a price p, (and implicitly, a quantity) to maximize profits subject to the
limits of the cash-in-advance constraint. Substituting demand (1.9) into (1.6) and
adding a Lagrangian multiplier (δ) on the cash in advance constraint reformulates
8
the firm’s problem as an unconstrained maximization:
ΠC (p, q, S) = maxp
( pP
)1−γDµ−Br (φ)− wP z
(p−γDµ
) 1α −Ψ (1.11)
−δ[qd (p)− qm (S)
]where z = λ+
γ − 1α
and Ψ is known
Solving the first order condition yields a policy rule for price setting, condi-
tional on updating the price. The optimal price, p∗, solves:
if δ = 0 : p∗ (S) =γ
α (γ − 1)w
(P
P−1
)(z−γ+1)
(Dµ)1−α
α (1.12)
if δ ≥ 0 : p∗ (S) = (Dµ)1γ
(B [P−1]
λ
w
)−αγ
Pλαγ
+1− 1γ (1.13)
Choice of Quantity when Keeping Initial Price If the firm opts to keep its
price from last period, then demand is determined by the outcome of the shocks
alone. The firm does not have to fulfill demand however; its finance constraint
could bind, or it might not be able to produce at a profit.
ΠNC (q, S) = maxqs
p−1qs (S)− (1 + r (φ))B − wPn+B
s.t.
qs (S) ≤ qm (S) (1.14)
qs (S) ≤ qd (S) (1.15)
First we consider what happens when the firm produces up to the level
demanded, i.e., when constraint (1.15) binds:
9
f (n) = nα ⇒ n = q1α ; z =
α+ γ − 1α
ΠNC (q, S) =(p−1
P
)1−γDµ− (1 + r (φ))B − wP z
(p−γ−1Dµ
) 1α +B
If the demand is higher than anticipated, and the firm attempts to produce
more than it has borrowed for, constraint (1.14) will bind. In that case, profit is
the following:
ΠNC (q, S) = p−1qm(S)− wP [qm(S)]
1α (1 + r (φ)) (1.16)
Alternatively, the firm may have drawn such back shocks that it chooses to
shut down. Thus, the production strategy for the firm after choosing to keep its
initial price can be summarized as the following:
qs∗ (S) =
max[ΠNC (qm, S) ,ΠNC (0, S)
]if qm ≥ qd
max[ΠNC
(qd, S
),ΠNC (0, S)
]if qd ≥ qm
(1.17)
Price Updating Decision Thus, at the beginning of the second period, the firm
faces a decision to change its price or not, taking the policy functions it uses to set
its price and quantity as given. This decision can be written as a policy function,
Ψ∗(S) that determines the highest menu cost that a firm is willing to pay. If the
menu cost is higher than Ψ∗(S) then the firm will leave prices unchanged.
Ψ∗(S) = ΠC (p∗, q, S)−ΠNC (q, S) (1.18)
An important factor in determining Ψ∗ is B, how much the firm borrowed
in the first period (B is part of S). The amount borrowed will affect the profits
that each pricing strategy will return; the less borrowed, the higher the price will
10
need to be take advantage of the reduced capacity. When the initial price is close
to or less than the optimal price (the price that would be chosen if the menu cost
were paid), under-borrowing raises the value of price adjustment, and increases the
threshold menu cost the firm is willing to pay. When the initial price is above the
optimal price, under-borrowing reduces the value of price adjustment. In that case,
constraining capacity offsets the inefficiency created by an initial price that was too
high relative to an unconstrained optimal price.
First Period
In the previous section, the firm solves its pricing problem conditional on an amount
previously borrowed. In the first period, the firm chooses an amount to borrow
based on the following policy rules for updating and setting its price:
B∗ (S−1) = max{B}
∫S
max{ΠC (p, q, S) ,ΠNC (q, S)
}dS (1.19)
This maximization is somewhat tricky, due to the choice of updating or not.
For each state, the firm must evaluate whether it will change prices or not, i.e., solve
for Ψ∗(S). Equation (1.19) can be rewritten as a strategy if we take Ψ∗(S) as given:
B∗ (S−1) = max{B}
∫S
ΠC (p, q, S) if Ψ ≤ Ψ∗(S)
ΠNC (q, S) if Ψ > Ψ∗(S)
dS (1.20)
Once it knows which profit function will be optimal in each state, it can choose
an amount to borrow that maximizes the expected profit. The problem with this
approach is that Ψ∗(S) itself is dependent upon B. Thus, the problem must be
solved numerically.
11
1.4 Results
The model produces two critical results. First, that the financing amount, B, affects
the price adjustment strategy of the firm. Conversely, the firm takes that strategy
into account when deciding on the amount to borrow. This mechanism is important
to identify as it underpins the second result, that the indexing regime (the value of
φ in the model) also affects the firm’s price adjustment strategy. Related to the
indexing result is the finding that the firm’s expected profits are higher at every
initial price under the non-indexing interest rate regime. In this section, I describe
each result in detail and explain the mechanism driving it.
1.4.1 Price Updating Decision
In order to show how the financing decision affects firm value, I show how the financ-
ing decision B∗ (S−1) affects the price updating decision. Ψ∗ (S) is the reservation
menu cost that determines whether the firm will update prices. If their realized
menu cost is above Ψ∗ (S) , they will not change prices. Ψ∗ (S) depends on the
difference between the profits under a new price and the old price (1.18).
Impact of Amount Borrowed on Price Updating Decision
Figures 1.2-1.4 illustrate how the pricing decision (1.18) changes with the amount
borrowed at different initial prices and various shocks.
Figure 1.2 illustrates the price updating function when the initial price is
low. At all levels of borrowing, a positive inflation shock will increase the chances
of a firm changing its price. A positive taste shock, which increases demand for a
particular firm’s good without altering relative prices between it and other goods,
also increases the likelihood of updating the price as it will increase the optimal reset
price and the gap between the reset price and the initial price. A negative taste
shock, by contrast, lowers the optimal reset price and shrinks the gap between that
12
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Amount Borrowed (Credit Limit)
Val
ue o
f ψ*
Influence of Credit limit on Price Decision, p-1 is 25%
inf -inf +taste -taste +
Figure 1.2: Impact of the amount borrowed on the reservation menu costwhen the initial price is low.
13
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Amount Borrowed (Credit Limit)
Val
ue o
f ψ*
Influence of Credit limit on Price Decision, p-1 is 50%
inf -inf +taste -taste +
Figure 1.3: Impact of the amount borrowed on the reservation menu costwhen the initial price is close to optimal.
and the initial price, thus lowering the probability of a price change. Additionally,
note that as the borrowing amount increases, the probability of changing the price,
or alternatively, the benefit to changing the price, decreases. This increase in
borrowing raises qm (S) (1.9), and raises the value of a price change (1.18).
In Figure 1.3, the initial price is close to the unconstrained optimal (the
price the firm would choose in the absence of a cash in advance constraint or menu
costs). Notice here how the amount borrowed changes the value of updating the
price. When borrowing .06, there is no value to changing prices if the firm receives
14
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Amount Borrowed (Credit Limit)
Val
ue o
f ψ*
Influence of Credit limit on Price Decision, p-1 is 75%
inf -inf +taste -taste +
Figure 1.4: Impact of the amount borrowed on the reservation menu costwhen the initial price is high.
a negative inflation shock. This is the ideal amount to borrow with this initial price
for that shock. With a positive inflation shock, the optimal amount to borrow is .12.
The change in the aggregate price level affects the relative price– demand is higher
with an inflation shock because the initial price looks cheaper. More borrowing is
needed to meet this increased demand.
In Figure 1.4, the initial price is above the ex-ante optimal price. The
effect of the shocks on the value of changing the price reverse. Now a positive
inflation shock brings the initial price closer to optimal, while a negative inflation
15
shock moves it further way. A negative taste shock will reduce demand and make
our higher price closer to optimal, while a positive taste shock will do the opposite.
Additionally, borrowing more increases the value of changing price because any
change will be negative. If the firm is lowering its price to increase quantity, it has
to have borrowed enough to produce additional quantity.
Impact of Initial Price on Price Updating Decision
Figures 1.2-1.4 show how the impact that the amount borrowed has on the price
updating decision depends on the initial price. In order to capture the impact
of borrowing in a more systematic way, we can measure the impact of B on Ψ∗
when different constraints are binding. The amount borrowed creates an upper
limit on the quantity of goods that can be produced. This is represented by qm (S)
(1.9). If demand under the optimal unconstrained price is less than qm (S), then the
financing decision will not affect the expected profits in the change case. Similarly,
if demand under the previous price is less than qm (S), then the financing decision
will not affect profits in the no-change case. Thus, the marginal effect of a change
in financing on price adjustment and value needs to be evaluated for four separate
cases.
The borrowing constraint can affect the price and output decision even when
it doesn’t explicitly bind. I define the impact of the constraint as the difference in
the price and output in the realized second period compared to what the price and
output would have been if the firm had been allowed to borrow ex-post. Figure
1.5 shows how the ex-post and constrained optimal prices differ. Notice how the
constrained price is different from the unconstrained optimal price even when the
borrowing constraint doesn’t explicitly bind. This is due to defining the optimal
price in the second period. At that point, financing costs are fixed, and don’t affect
the marginal cost. Ex-ante, they do matter, so p∗ is higher than the unconstrained
16
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.21.5
2
2.5
3
Amount Borrowed
Opt
imal
Pric
e
Effect of Debt on Price Policy in Change case
C-I-A Constrained Optimal PriceEx-Post Borrowing Optimal Price
Figure 1.5: Compare the optimal price conditioned on an amount borrowedwith an optimal price with ex-post borrowing., N = 3
optimal price (or alternatively, the ex-post borrowing optimal price), p∗unc when
the borrowing constraint is restrictive, but falls off as borrowing increases and the
constraint has less of an impact than the financing costs.
Figure 1.6 details each of the four cases that determine the marginal effects
of borrowing on price adjustment. The cases are divided up by whether or not the
borrowing constraint binds. The lines in Figure 1.6 define the boundaries between
cases, and are the points where the borrowing constraint begins to bind (where
qm(S) = qd(S)) for a given initial price. In the case where the firm changes its
17
Figure 1.6: Marginal effect of B∗ on price adjustment by constraint case.
price (green line), the initial price has no impact on the demand. When the firm
keeps its initial price (blue line), it takes less borrowing to meet demand the higher
the initial price. Whether the borrowing constraint will bind or not critically affects
how additional borrowing will changes the price updating decision. If the firm has a
very high initial price (relative to the expected optimal reset price) then borrowing
less than .06 (case I) will result in the borrowing constraint binding in the change
case but not in the no-change case, thus increasing the likelihood that the firm will
not pay the menu cost. If the firm has a very low initial price, then borrowing above
.06 (case II) will result in the borrowing constraint binding under the no-change case
but not in the change case, thus increasing the likelihood of paying the menu cost.
Here we calculate the marginal effects in each case:
18
Case I: q(p∗unc) > qm (B) , q(p−1) ≤ qm (B)
Ψ∗ (S) = p∗(B
wP
)α
− p−1q(p−1)−B + wPq(p−1)1α
∂
∂BΨ∗ (S) =
∂p∗
∂B
(B
wP
)α
︸ ︷︷ ︸Price Falls
+αp∗
wP
(B
wP
)α−1
︸ ︷︷ ︸Quantity Rises
− 1︸︷︷︸Cost’s Rise
In case I, the borrowing constraint is binding in the change case but not in
no-change case. A firm that has a high initial price and borrows very little will
be in case 1. As the derivative is positive, we see that an increase in the amount
borrowed would increase Ψ∗, increasing the value to a firm of changing its price.
Case II: q(p∗unc) < qm (B) , q(p−1) ≥ qm (B)
Ψ∗ (S) = p∗q(p∗)− p−1
(B
wP
)α
− wPq(p∗)1α +B
∂
∂BΨ∗ (S) = −αp−1
wP
(B
wP
)α−1
+ 1
Case II describes the opposite situation, where the borrowing constraint does
not bind in the change case but does in the no-change case. Here the firm has a
very low initial price and has borrowed a lot. Borrowing more would lower the
value of changing price, as the firm would increase its price if it changes.
Case III: q(p∗unc) > qm (B) , q(p−1) > qm (B)
Ψ∗ (S) = p∗q(p∗)− p−1q(p−1)
∂
∂BΨ∗ (S) =
∂p∗
∂B
(B
wP
)α
+αp∗
wP
(B
wP
)α−1
︸ ︷︷ ︸Change Price
− αp−1
wP
(B
wP
)α−1
︸ ︷︷ ︸No Change
In case III, the constraint binds in both cases. It is unclear what effect an
increase in B would have on Ψ∗.
19
Case IV: q(p∗unc) ≤ qm (B) , q(p−1) ≤ qm (B)
Ψ∗ (S) = p∗q(p∗)− p−1q(p−1)− wP[q(p∗)
1α − q(p−1)
1α
]∂
∂BΨ∗ (S) = 0
Case IV is the unconstrained case. The firm has borrowed enough and has
a high enough initial price that additional borrowing will not affect its decision to
change its price or not.
The table below summarizes the how a change in B∗ (S) affects Ψ∗ through
each price changing scenario:
Table 1.1: Definition of Price Change and Borrowing Cases
Case q(p∗unc) q(p∗−1)
I q(p∗unc) > qm (B) q(p−1) ≤ qm (B)
II q(p∗unc) ≤ qm (B) q(p−1) > qm (B)
III q(p∗unc) > qm (B) q(p−1) > qm (B)
IV q(p∗unc) ≤ qm (B) q(p−1) ≤ qm (B)
Table 1.2: Summary of the Impact of Borrowing on Value
Case ∂∂BC Ψ∗ (θ) ∂
∂BNC Ψ∗ (θ) ∂∂BV (p∗) ∂
∂BV (p−1) ∂∂BV
I ∂p∗
∂B
(B
wP
)α + αp∗
wP
(B
wP
)α−1 − 1 0 + 0 +
II 0 −αp−1
wP
(B
wP
)α−1 + 1 0 + +
III ∂p∗
∂B
(B
wP
)α + αp∗
wP
(B
wP
)α−1 −αp−1
wP
(B
wP
)α−1 + + +/−
IV 0 0 0 0 0
Figure 1.7 shows how the marginal effects of each case change over different
levels of B for a given initial price. We can see that for low levels of borrowing, the
20
0
0.1
0.2
0.3
0.4Overall Effect: ψ* given B
0
5
10
15
ψ*
Marginal Effects: δψ*/δB p* Case
p-1 = 1.36
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-8
-6
-4
-2
0
Amount Borrowed
Marginal Effects: δψ*/δB p-1 Case
Figure 1.7: Marginal effects of additional borrowing on price adjustmentfor each price case given an initial price level.
change case is made more desirable with additional borrowings, and the no-change
case is made less so. This is consistent with case III, as both derivatives are non-
zero. At a critical amount of borrowing, the change case ceases to be affected by
additional borrowing. At this point, the change case borrowing constraint does
not bind. This is consistent with case II. The first panel of Figure 1.7 shows how
changes in B affect the overall value .
In the top of Figure 1.7 we see how the marginal effect of borrowing on
Ψ∗ changes as borrowings increase. At first it increases, as additional borrowings
21
Figure 1.8: Mapping figure 1.7 back into the constraint space.
make a price change more profitable, but that effect begins to slow, and eventually
stop altogether. The middle and bottom panels decompose the marginal effect into
the change case and the no-change case. The middle panel illustrates the effect
of borrowing on the no-change case. While positive, it quickly falls to zero as
the additional constraint stops binding. In the change case, additional borrowing
actually makes changing the price less desirable, as there are more financing costs to
cover. Figure 1.8 shows the same four-case space again, this time with our chosen
initial price shown explicitly. Figure 1.7 maps into Figure 1.8, as the marginal
effects of each level of borrowing change. Note that as borrowing increases the firm
crosses from case III to case II. The borrowing amount, .06, is the same as where
the no-change effect falls to zero in the middle panel of Figure 1.7.
22
0.5 1 1.5 2 2.5 30.05
0.055
0.06
0.065
0.07
0.075
0.08
Relative Price from Last Period
Bor
row
ing
Dec
isio
n
Optimal Borrowing
Figure 1.9: Optimal borrowing given an inital price level.
1.4.2 Financing Decision
As the financing decision occurs in the first period, the firm takes into account its
own decision rule on price changes. If it is likely that the price is going to change,
then the firm can borrow an amount closer to what it requires to maximize the
change case. If its likely the current price is close to optimal, the firm can borrow
based on the no-change case (see Equations 1.21 and 1.22). The following shows
how the pricing decision Ψ∗ (S) affects the financing decision B∗ (S) (see Figure 1.9).
When the initial price (x-axis) is closer to optimal, 1.75, the menu costs will
23
outweigh the benefit of changing the price and the firm will adjust its borrowing
to maximize under the initial price. Thus, the firm borrows more when the initial
price is below optimal, 1.5, in order to meet the higher demand. Conversely, they
can borrow less when the initial price is slightly too high, 2.1, to meet lower demand.
This process is limited by the magnitude of the menu cost however, and we can see
it break down at extreme initial price levels, 1 and 3. When the initial price is so
far away from the optimal that updating is a virtual certainty, the initial price has
little effect on borrowing and the firm borrows to maximize under its newly updated
price.
When faced with the borrowing decision, the firm knows the current price
level and its own price. This is enough for the firm to determine how far its current
price will deviate from the optimal price in the next period for each realization
of the shocks. The likelihood of updating the firm’s price is summarized in the
price change policy function. It is an important factor in the borrowing decision
because the optimal amount of borrowing depends on the demand. To see the
mechanism more clearly, separate the borrowing decision into two policy functions,
each conditional on a particular price updating strategy:
B∗C (S−1) = max{B}
∫S
ΠC (p, q, S) dS (1.21)
B∗NC (S−1) = max{B}
∫S
ΠNC (q, S) dS (1.22)
B∗C (S−1) is the optimal amount to borrow if the firm was changing its price
for sure, B∗NC (S−1) the amount if the firm was definitely not going to change its
price.
Now divide the second period into two states defined by the relative size of
the optimal amount to borrow for each price change scenario. If the initial price is
very low, a decision to raise it will reduce demand and require less borrowing. If
24
the initial price is very high, updating it will increase demand, and necessitate more
borrowing. Define these two situations as cases 1 and 2:
Case 1: B∗C (S−1) < B∗NC (S−1) (1.23)
Case 2: B∗C (S−1) ≥ B∗NC (S−1) (1.24)
There are two channels through which the price change affects the borrow-
ing decision: the benefit of relaxing a binding cash-in-advance constraint, thereby
allowing supply to be optimal, and the cost of paying interest on the borrowing. In
case 1, these two channels reinforce each other; changing the price will move the firm
closer to first-best profits and also mean less borrowing and interest costs. In case
2, the two channels are at odds, as changing the price will yield greater efficiency
but also increased borrowing costs. Figure 1.10 illustrates these two mechanisms
across different values of the initial price.
If we think of the reservation menu cost as the amount of inefficiency the
firm will accept before changing its price, then an increase in Ψ∗ tilts the borrowing
more towards the no-change optimum.
∂
∂Ψ∗B∗ (S−1) =
Case Interest Optimal Net Effect
1 − + +
2 − − −
Figures 1.11 illustrates the net effect.
1.4.3 Indexing Effects
We are now ready to consider the effects of indexing costs to inflation. If firms
choose to set φ greater than zero, then the interest rate will rise and fall with
inflation. Changes in λ, the wage indexation, were also considered but did not
affect the results presented here. While changes in λ impact profits and the policy
25
0.5 1 1.5 2 2.5 3 3.50
0.02
0.04
0.06
0.08
0.1
0.12
Previous price p-1
Am
ount
Bor
row
ed
BCBNC
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Pro
fit
Amount Borrowed Affects Each Price ScenarioPI-CPI-NC
Figure 1.10: Profits and interest cost for both price updating cases.
26
0.5 1 1.5 2 2.5 3 3.5-5
-4
-3
-2
-1
0
1
2
3
4
Previous price p-1
Inte
rest
Exp
ense
Int Exp: C-NC-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Pro
fit D
iffer
ence
Price Change: Interest Cost vs. ProfitPI-C - PI-NC
Figure 1.11: Difference in profits and interest cost between the change andno-change cases.
27
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.484
0.486
0.488
0.49
0.492
0.494
0.496
0.498
Previous price p-1
Ex[
ecte
d P
rofit
Non-IndexedIndexed
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8x 10-3
Diff
eren
ce in
Exp
ecte
d P
rofit
Expected Profits for Alternate IndexingDifference
Figure 1.12: Expected profits for various indexing scenarios and initialprices.
functions, they do not change the effect of φ on those same functions. Thus, in
what follows, λ equals 1. Figure 1.12 shows the main result that expected profits
are lower at all price levels when firms index.
This is somewhat counter intuitive, as indexing rates decreases exposure to
inflation risk. The result follows from the policy functions and the timing of the
decisions. Indexing firms receive a greater benefit from updating their price after
inflation is revealed, as the price realization affects their marginal cost. Taking
that into account, an indexing firm borrows less than a non-indexing firm because
28
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.46
0.48
0.5
0.52
0.54
0.56
0.58
Previous price p-1
Rea
lized
Pro
fit
Non-IndexedIndexed
0.055
0.06
0.065
0.07
0.075
0.08
Diff
eren
ce in
Pro
fit
Realized Profits for Alternate Indexing (Inflation Shock)Difference
Figure 1.13: Realized profits for each indexation scenario when the tasteshock is zero and the inflation shock is positive.
it is less concerned about a low nominal price (1.24). This becomes a self fulfilling
result after price levels are realized. If inflation is positive, the indexed firm will
either have to change prices or produce less, because it did not borrow as much as
the non-indexing firm. Indexing firms manage their exposure to shocks by changing
price, whereas non-indexed firms manage these shocks by borrowing more.
To understand this result more clearly, consider the how each indexation
scenario preforms under the case of a positive inflation shock.
Figure 1.13 shows how the non-indexing firm does much better than the
29
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.5
0.51
Previous price p-1
Rea
lized
Pro
fit
Non-IndexedIndexed
-0.07
-0.065
-0.06
-0.055
-0.05
-0.045
Diff
eren
ce in
Pro
fit
Realized Profits for Alternate Indexing (Deflation Shock)Difference
Figure 1.14: Realized profits for each indexation scenario when the tasteshock is zero and the inflation shock is negative.
indexing firm when inflation increases. This makes sense, as the indexed firm has
higher interest rates. Notice however, that the magnitude of the difference changes
when the initial price is close to the unconstrained optimal price, first rising, then
narrowing. This change in the magnitude is due to the difference in the amounts
borrowed and the probability of a price change from each type of firm. A converse
reaction is clearly seen from a negative inflation shock in Figure 1.14.
To emphasize the change in the difference in profits, consider the effect of a
positive and negative taste shock on indexed and non-index profits.
30
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.475
0.48
0.485
0.49
0.495
0.5
0.505
Previous price p-1
Rea
lized
Pro
fit
Non-IndexedIndexed
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025
Diff
eren
ce in
Pro
fit
Realized Profits for Alternate IndexingDifference
Figure 1.15: Realized profits for each indexation scenario when the tasteshock is positive and the inflation shock is zero.
31
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
0.503
0.5035
0.504
0.5045
Previous price p-1
Rea
lized
Pro
fit
Non-IndexedIndexed
-1.5
-1
-0.5
0
0.5
1
1.5x 10-3
Diff
eren
ce in
Pro
fit
Realized Profits for Alternate IndexingDifference
Figure 1.16: Realized profits for each indexation scenario when the tasteshock is negative and the inflation shock is zero.
Figure 1.15 shows the impact of a demand shock without a corresponding
change in indexed interest rates. This eliminates the large but fixed cost associated
with a higher interest rate bill. There is still a difference in profits right around the
critical optimal reset price, though it slightly favors the non-indexing firm. When
the price is close to the optimal reset price, the non-indexing firm borrows more,
which is an advantage under a positive demand shock. When the taste shock in
negative, as in Figure 1.16, the reverse is true, and the indexing firm does better.
The following example illustrates the asymmetry that leads the non-indexing
32
firm to higher profits. Firm I indexes its financing, while Firm N does not. Firms I
and N have the same initial price in period 1, which is lower than their ex-ante opti-
mal price. When considering how much to borrow, each firm assesses the likelihood
of changing its price. Figure 1.17 illustrates the differences in their assessment.
In the case of positive inflation, the firms have almost identical reservation menu
costs. The positive inflation shock exacerbates the gap between their initial price
and optimal price, and both firms will likely update their price. The case of a
negative inflation shock leads to a much different result. Firm I is willing to pay
a much higher menu cost to update its price. Because its financing is indexed, the
deflation changes the firm’s optimal reset price, moving it further from the initial
price. The deflation has no effect on Firm N’s as its optimal reset price as it is
independent of the price level. The deflation shock merely shrinks the gap between
its initial price and the optimal reset price.
After assessing the likelihood of updating their prices, the firms choose an
amount to borrow. This amount can be thought of as the weighted average of the
amount the firm would borrow if it were certain it was going to change its price and
the amount it would borrow if it were certain it would leave the price unchanged
(see Equations 1.21 and 1.22). Firm I is more certain that it will update its price,
so the amount it borrows will be more heavily weighted to B∗C . Since the initial
price is lower than the ex-ante optimal price, borrowing case 2 applies. In case 2,
B∗C is less than B∗NC , and any change in price will be an increase and will lower
the quantity demanded. Thus, Firm I borrows less than Firm N in this case.
Having made their borrowing decisions, the firms move into the second pe-
riod. In the case of positive inflation, both firms change their price and meet
demand. In that scenario, it is unclear who profits more. Firm I borrowed less
overall, but pays a higher interest cost due to the inflation. If there is negative
inflation, Firm I pays the menu cost and changes its price before Firm N. Firm I is
33
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.4
0.5
0.6
0.7
0.8
0.9
1
Relative Price from Last Period
Pro
babi
lity
of C
hang
ing
Pric
e
Probability of Changing Price for Various Indexing Scenarios
Non-Indexed BorrowingsIndexed Borrowings
Figure 1.17: Expected reservation menu cost for various indexing scenariosand initial prices.
more borrowing constrained than Firm N, and its optimal reset price is increasing.
In this case, Firm N comes out ahead, as it pays the menu cost less frequently.
Summing over all the states, it’s clear that the non-indexed firm is more profitable
than the indexed firm. The non-indexed interest rate helps affects the borrowing
decision by reducing the benefit to changing price and keeps the optimal reset price
fixed over both periods. The same asymmetries that drive the above example are
present in a situation where the initial price is higher than the optimal rest price
(see Figure 1.18).
34
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Realized Price Level
ψ*
ψ* at a low initial price for various indexing
IndexedNot Indexed
Figure 1.18: Realized reservation menu cost (Ψ∗) with a high initial pricefor a firm with indexed financing and without.
35
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.250.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Realized Price Level
ψ*
ψ* at a high initial price for various indexing
IndexedNot Indexed
Figure 1.19: Realized reservation menu cost (Ψ∗) with a low initial pricefor a firm with indexed financing and without.
36
1.5 Conclusion
This paper presents a model of the firm that incorporates state dependent pricing
and a cash-in-advance constraint. This simple framework illustrates how these two
inefficiencies work together to violate the irrelevance proposition and make a firm’s
financing an important factor in determining value.
I show that the cash-in-advance constraint affects both the price updating
decision and the optimal price if the optimal price in either the no-change or the
change case would require additional borrowing. In some cases this influenced the
new optimal price even when the cash-in-advance constraint didn’t appear to be
binding.
Additionally, I show that the type of interest rates alone affect price updating
behavior as well as expected profits. Firm’s paying interest rates indexed to infla-
tion borrowed less and updated their price more frequently, earning lower expected
profits. Financing costs under non-indexed rates move with inflation based demand
shocks. This allows firms to change prices more efficiently and earn higher expected
profits. These results depend critically on the assumption of positive menu costs.
If a firm could change its price costlessly, then there wouldn’t be an advantage to
borrowing at nominal rates.
As a next step in this investigation, I would like to estimate the nature of
the relationship between price stickiness and the degree of indexation in a firm’s
capital structure. I would expect to find the firms that change their price relatively
infrequently would have a high degree of indexation in their capital structure.
37
Appendix A. Parameter ValuesFor the policy functions shown in the paper the following parameter values
were used:
α
γ
β
Ψ
D
w
µ
P−1
rF
λ
φ
=
0.5
1.9
.99
[0, 0.025]
1
.75
(.5, 1.5)
1
0.2
{0, 1}
{0, 2}
description:
Returns to scale
Price elasticity
Discount rate
Menu cost
Money Balance (hh)
Real Wage
Taste shock
Initial price level
Risk Free Rate
Amount Indexing
Interest Rate Indexing
38
Chapter 2
Asset Prices and Money in a
Two-Period Equilibrium
2.1 Introduction
Observers of public securities markets have often documented a level of price volatil-
ity that is much higher than justified by our current understanding of fundamental
value. In attempting to explain this anomaly, liquidity demands have frequently
been proposed as an explanation. If every trade were motivated by new information
about the fundamental value of an asset, we would expect much smaller moves in the
price. However, if some of the trades were motivated simply by a desire to convert
the asset into consumption, price fluctuations could be explained by differences in
agents’ desire to consume or save at a specific point in time.
This paper uses a Diamond and Dybvig (1983) style liquidity model to show
how the money supply relative to the supply of productive assets can affect asset
prices and asset price volatility. The model shows that higher levels of money
relative to assets will result in lower volatility for the price of the asset.
The link between asset price volatility and liquidity has been approached
39
using several modeling techniques. One method of obtaining excess volatility and
increased risk premium is to assume a fixed cost of participation. Models utilizing
this have linked market liquidity to both high and low levels of volatility. Pagano
(1989) uses participation costs to create endogenously thin markets. The thinness
of the market contributes to volatility though the number of agents. The fewer
the number of agents, the greater the impact of idiosyncratic liquidity shocks. The
expected increase in volatility due to idiosyncratic shocks feeds back into the par-
ticipation decision, multiplying the impact of an initial shock on volatility. Orsel
(1995) uses a similar model to Pagano but generates an increase in volatility due to
participation. In his model, the marginal participant only enters after observing a
positive signal about returns. This agent will have the highest cost to participate,
so will accept the largest liquidity cost. Thus, increased participation is correlated
with high level of volatility. Dubra and Herrera (2001) generates a similar result,
using the marginal risk aversion of the agent instead of a signal.
Allen and Gale (1994) model asset price volatility using a Diamond and
Dyvbig style asset market. They introduce a fixed cost of participation to buy the
illiquid asset and uncertainty about the payoff of the asset. The Allen and Gale
model is remarkable in its ability to generate liquidity-based asset price volatility
without the use of thin markets, as in Pagano. At all times, assets and money are
costlessly exchanged. The changes in the level and volatility of the price are driven
solely by agents’ evolving demand for liquidity.
This paper follows the Allen Gale approach of assuming thick markets with
two innovations. First, while I include a liquidity demand shock as in Diamond
and Dyvbig, I also add a supply shock that alters the total supply of money in the
second period. This shock is known to agents in the initial period. I conclude
that shocks to the money supply increase the mean price level of the asset, and
decrease the price volatility. Secondly, I simplify the model by removing the fixed
40
cost of participation. While this innovation has been used to generate excessive risk
premia, I show that it is not necessary to establish the interaction between liquidity
and price volatility in this framework.
In the model, the volatility of the asset price is determined by the relative
value of the liquidity price of the asset, which is determined by the relative supply
of money and the asset at a point in time, and the fundamental value price of
the asset, which is determined by its return if held to maturity. If the liquidity
price is uncertain, the distribution of expected realizations of the price will have
an upper bound at the fundamental value; no one holding money would take a loss
for the privilege of holding the asset. The more of the liquidity price distribution
is truncated by the fundamental value price, the lower the overall volatility of the
price.
Section two presents the household problem in general terms and defines an
equilibrium. Section three derives explicitly policy functions to solve the household
problem and satisfy the equilibrium conditions. Section four utilizes comparative
statics to address the impact of liquidity on asset prices, demand and volatility.
Section five concludes.
2.2 Model
The model follows a simplified version of the Allen Gale portfolio problem. House-
holds face a choice in the initial period of allocating wealth between two assets,
while under uncertainty about which of the remaining two periods they will need to
consume. The model solution is partial equilibrium in the sense that its scope is
limited to the choice of asset allocation; I abstract from specific consumption and
production processes. The liquid asset (money, m) yields no return but can be used
for consumption in either the first or second period. The illiquid asset (“the asset”,
a) yields a positive real return (r), but is only realized in the second period. Money
41
is assumed to be the numeraire good, with consumption denominated in terms of
money. This allows us to avoid a separate aggregate price level between goods and
money. The only price in the model relates to the exchange between money and
assets. Agents face both idiosyncratic and aggregate uncertainty about whether
they are will be early or late consumers. Their liquidity needs will be determined
by an individual draw of a random variable λ. The distribution of λ is partially
revealed; the standard deviation is known, but the mean of the distribution is itself
a random variable (ν). This compound uncertainty creates both idiosyncratic risk
for a consumer about his or her demand timing, and also creates aggregate uncer-
tainty about fraction of total consumers who will consumer early or late. In the
first period, after the liquidity preference shock is revealed, there is a market for
early consumers to sell illiquid assets to late consumers. The aggregate uncertainty
about the distribution of λ creates uncertainty about the market clearing price of
the asset in the first period.
2.2.1 Timing and Uncertainty
t = 0 1 2
a −1 → 1 + r
m1 −1 ε
m2 −1 1
(2.1)
The diagram in (2.1) outlines the performance of each asset by period. Both
money and the asset require an initial outlay of wealth to produce each type of asset.
In the case of money, the initial investment is returned in the first period, where
it can be either reinvested until the second period (in the case of a late consumer)
or consumed (in the case of the early consumer). The proportional money supply
shock, ε, adds money to the initial balance of each consumer at the beginning of the
first period. The asset is more illiquid; after being purchased in the initial period, it
42
cannot be consumed in the first period. However, the asset returns all of the initial
investment in the second period plus the return r.
In the initial period, households make their allocation decision without know-
ing their liquidity preference. This allocation is accomplished through a constant
returns to scale process for both the money and the asset. Thus, the total supply
of money and the asset is determined by the agents’ initial period assessment of
their individual liquidity needs. The liquidity shock is revealed in the first period,
after which the agents attempt to reallocate between money and the asset. Con-
sumption occurs after that reallocation in the first period for early consumers. Late
consumers wait until the second period when the asset and its return converts to
consumption.
The liquidity shock creates both idiosyncratic and aggregate uncertainly. For
household i, the liquidity preference is determine by its draw of λi :
if λi ≥ 0, z = 1
if λi < 0, z = 0
λi is drawn from a normal distribution, where the standard deviation is known to
the households but the mean is not:
λ ∼ N (υ, σ) (2.2)
v ∼ N (ν, 1) (2.3)
The expected mean of the liquidity preference distribution, ν, is know to the con-
sumers in the initial period. Similarly to ε, it will act as a policy parameter and
define the space over which we investigate the changes to the price of the asset and
the volatility of that price.
43
2.2.2 Household Problem
The household essentially faces an optimal insurance problem. If a late consumer
knew his type in the initial period, he would allocate all his wealth to the asset and
earn the superior return. An early consumer would allocate all his wealth to money,
as the asset would be worthless to him. Thus, an allocation to money is insurance
against early consumption. This feature matches a critical aspect of liquidity in
general; the value of liquidity is the ability to access wealth at any time, without
necessarily knowing ahead of time if it will be needed. The premium agents pay
for this insurance is the opportunity cost of a higher return on the less liquid asset.
Stated formally, the household solves the following problem:
V (wi, ε, v, r) = maxci1,ci2
Ez,λ [u (ci1, ci2)] s.t. (2.4)
ci1 ≤ mi0ε+mi1 (2.5)
ci2 ≤ mi0ε+mi1 + (1 + r) (ai0 + ai1) (2.6)
wi ≥ mi0ε+ p0ai0 (2.7)
p1ai1 −mi1 = 0 (2.8)
ai0,mi0ε ≥ 0; ai1 ≥ −ai0, mi1 ≥ −mi0ε, (2.9)
The household attempts maximizes its total utility in each allocation period.
In the first period, it chooses ci1 and ci2 knowing there is a chance it could consume
either early or late.
u (ci1, ci2) =
u (ci1) if z = 1
u (ci2) if z = 0
(2.10)
This choice is subject to a cash-in-advance constraint for each consumption
period (2.5, 2.6) and a budget constraint for each period (2.7, 2.8). Additionally, I
impose a restriction on short selling and net borrowing (2.9).
44
In order to get some meaningful intuition on the comparative statics, and to
attempt a more rigorous answer to the questions posed by the paper, I assume the
following quadratic form to the utility function:
u (c) = c− b
2c2 (2.11)
Any well behaved utility function (continuous, differentiable, u′ (·) > 0,
u′′ (·) < 0) will yield similar results.
2.2.3 Market Clearing
As in the Allen Gale model, the allocation decision in the initial period is actually
a production decision, as agents transform wealth into money and the asset. This
process endogenizes the entire of supply of assets and money in the initial period.
The money supply shock, ε, is applied to balances in the first period, but known to
the agents in the initial period.
∑i
ai = A; ε ·∑
i
mi = M (2.12)
First period market clearing involves only a subset of the total supply of
money and assets. Late consumers who hold assets and early consumers who hold
money will not offer them for exchange as the assets will be held and consumed,
respectively. Thus, the first period market is made up of assets held by early
consumers and money held by late consumers. The proportion of each type can be
determined from the realization of the mean of the λ distribution (ν)
z∗ = Pr(λi ≥ 0|ν = ν) (2.13)
z∗ informs the fraction of total assets and money that are being traded in
45
period 1.
∑i
p1aEi =
∑i
mLi
z∗p1A = (1− z∗)M (2.14)
Its not strictly correct to say that p1 will clear the market for assets in period
1. There are several corner solutions where the market does not clear. In period
1, the asset has a guaranteed return of 1 + r in the next period. Those selling the
asset will be unable to charge more than 1 + r for it no matter how strong the
demand is for late consumption. That is, p1 will be the lesser of either the market
clearing price. or 1 + r. This difference between the market clearing price, which
I referred to in the introduction as the liquidity price, and the true value of the
asset, the fundamental value price, drives the volatility result. If those prices are
close together in expectation, the distribution of the liquidity price will be truncated
more severely than if they’re further apart. The more the distribution is limited,
the lower the volatility.
2.2.4 Equilibrium
Define a set of parameters S = {w, ε, v, r} . An equilibrium for this model is defined
as a series of policy functions a0 (S) , m0(S), aE1 (p1) , mE
1 (p1) , aL1 (p1) , mL
1 (p1)
that solve the household problem (2.4) subject to the constraints (2.5-2.9) and a
price, p1, that clears the market in period 1 (2.14). The next section will use the
assumed functional form of demand to develop a solution for the equilibrium, and
derive expressions for the expected price and volatility of the first period asset.
46
2.3 Solving the Model
The household faces two sequential decisions, with the second depending critically
on the first. The solution to the household’s problem begins in the first period
and works back to the initial allocation. In what follows, I have dropped the ”i”
subscripts on the policy functions, as the only difference between consumers is their
ex-post type, which is labelled where appropriate.
2.3.1 First period (t = 1)
At the beginning of the first period, the household has already allocated its wealth
between money and assets. At the time of the allocation, the household did not
know if it would be an early or late consumer; liquidity preference is revealed at
the start of the first period. Thus, while in the initial period our agents were
homogenous, the first period sorts them into two groups, early and late consumers,
who each solve a separate problem.
Early Consumer
Early consumers experience utility from consuming in the first period, but none
from consuming in the second, when the asset pays off.
u(c1, c2; z = 1) = maxa1,m1
1 · u(c1) + 0 · u(c2)
c1 ≤ m0ε+m1
c2 ≤ m1 + (1 + r) (a1 + a2)
Thus, the early consumer will want to exchange his asset holdings for money in
order to maximize his current period consumption and overall utility. Substituting
the first period budget constraint (2.7), which we can assume will bind, into the
household problem (2.4), the early consumer will have the following demand for
47
money and assets:
mE∗1 = p1a0 (2.15)
aE∗1 = −a0 (2.16)
The market will always clear for the early consumer, as they will accept any
price for their initial asset allocation.
Late Consumer
Late consumers experience utility from consuming in the second period, when the
asset pays off and returns the initial investment plus the return in terms of con-
sumption. In the first period, upon learning that they are late consumers, these
households no longer have a need to insure against early consumption, and would
like to put the wealth that they allocated towards money in the initial period (m0)
to a more productive use, i.e., buy the asset. Rewriting the early consumer problem
to account for z = 0 :
u(c1, c2; z = 0) = maxa1,m1
0 · u(c1) + 1 · u(c2) (2.17)
c1 = m0ε+m1 (2.18)
c2 = m0ε+m1 + (1 + r) (a0 + a1) (2.19)
The late consumers problem is slightly more complex than the early consumer’s.
By substituting the second period budget constraint (2.8) into (2.19), we can write
48
the late consumers problem as a Lagrange maximization:
L(a,m, µ) = maxa1
m0 − p1a1 + (1 + r) (a0 + a1)− µ [m0ε− p1a1]
FOC :∂L
∂a1: −p1 + (1 + r) + µp1 = 0
µ : m0ε− p1a1
µ [m0ε− p1a1] ≤ 0
Unlike the early consumers, for whom the allocation to assets in the initial period
is now worthless if the hold onto it, late consumers value their money holdings to
the extent they can hold them until the second period. Where the early consumers
would accept any p1 ≥ 0, late consumers are only willing to pay p1 ≤ 1+r, the price
where they are indifferent between holding money and the asset. That is, when the
liquidity price is less than or equal to the fundamental value price.
Case 1: p1 ≤ 1 + r , µ > 0 (2.20)
Case 2: p1 > 1 + r , µ = 0 (2.21)
Therefore, the demand for the asset depends on the price:
aL∗1 =
m0εp1
if µ > 0
0 if µ = 0
(2.22)
Thus, by the first period budget constraint (2.7):
mL∗1 =
−p1a1 if µ > 0
0 if µ = 0
(2.23)
49
Market Clearing
Early consumers are natural buyers of money, late consumers will demand assets
as long as the price is not above the fundamental value of the asset, 1 + r. From
the market clearing condition (2.14) we can derive an expression for the first period
price:
p1 = min{
1− z∗
z∗M
A, 1 + r
}(2.24)
The late consumers each demand aL∗1 . They are willing to exchange all of the
money holdings, and there are a total of 1−z∗ of them. Conversely, early consumers
each demand mE∗1 , and there are z∗ of them (2.15 and 2.13). The distribution of
the expected realization of p1 in period 0 will be negatively skewed, as any value of
z∗ that yields a price above 1 + r will be truncated to 1 + r. For every realization
of the shock where the true value dominates, the price equals the true value. This
discrete choice drives the key result that volatility decreases as liquidity increases.
The more states yield market clearing prices above 1 + r, the more the distribution
is truncated, and the lower the volatility.
2.3.2 Initial Period
In contrast to the first and second periods, agents in the initial period do not know
whether they will be consuming earlier or later, and thus all face the same prob-
lem. Agents solve the household allocation problem by taking expectations over the
probability of drawing an early or late liquidity preference and the expected mean
of that distribution.
2.3.3 Household Problem
Rewrite the household problem (2.4) after substituting in the demand functions
from the first period (2.15, 2.22). As each consumption period is exclusive to the
50
other, c1 is optimized assuming an early liquidity preference, c2 assuming a late
liquidity preference.
V (wi, ε, v, r) = maxc1,c2
∫ν
∫λ[λu (c1) + (1− λ)u (c2)] dλdν (2.25)
c1 = m0ε+mE∗1 (p1) (2.26)
c2 = (1 + r)(a0 + aL∗
1 (p1))
if µ > 0 (2.27)
c2 = (1 + r) a0 +m0ε if µ = 0 (2.28)
Substituting in the budget constraint (2.7), the expectation of p1 (2.24) and
the specific demand functions in for c1 and c2 yields the following first order condition
for each λi and υ realization:
∂V
∂a0: −λ (1− p1)u′ (c1) + (1− λ)u′ (c2) (1 + r)
(1− 1
p1
)= 0 (2.29)
If we assume quadratic utility (2.11), demand in the initial period is a function of
the parameters and the shocks:
a∗0 = g (wi, ε, v, r) (2.30)
Figure’s 2.1 and 2.2 show how demand for the asset changes with the expected
fraction of early consumers and with the money shock.
Demand for the asset falls off as liquidity demand increases, as agents find
money to be more valuable relative to the asset as the chance of consuming early
increases. As the money shock increases, the return for owning money increases
and eventually overtakes the return on the asset. Hence zero demand for the asset
when ε is above 1.17.
51
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.60.41
0.42
0.43
0.44
0.45
0.46
0.47
0.48
0.49
0.5
Expected Mean of z*
Dem
and
for
the
Ass
et in
Per
iod
0
Figure 1: Demand vs. z*
Figure 2.1: Demand for the initial asset holding wealth, interest rates, andthe money supply shock constant and changing z∗, the fraction of peopleconsuming early.
52
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Money Supply Subsidy (Tax), ε
Dem
and
for
the
Ass
et in
Per
iod
0
Figure 2: Demand vs. Money Shock
Figure 2.2: Demand for the initial asset holding wealth, interest rates, andthe fraction of early consumers constant while changing ε, the moneysupply shock.
53
2.3.4 Initial Price, p0
While there is no exchange occurring in the initial period, the allocation decision
that the households are making implicitly requires an expectation of the price of
assets in terms of money in period 1 (demand equation 2.15). This expectation
can be calculated as a shadow price in order to explore the implications of shocks
to liquidity demand (υ) and shocks to the aggregate stock of money (ε), to that
shadow price.
p0 = p1 (z∗ (ν)) (2.31)
p0 = E [p1] =∫
νp1 (z∗ (ν)) dv
The mean and variance of p0 are calculated based on the market clearing
price at each and every possible realization of ν. The expected value of p1 is:
E [p0] = min{
1− z∗
z∗M
A, 1 + r
}= µp (2.32)
The exists a value of z such that 1−zz
MA = 1 + r :
z =M
A (1 + r) +M(2.33)
Thus, the variance of the truncated distributions is:
V ar [p0] =∫ z
0(1 + r − µp)
2 dz +∫ 1
z
(1− z∗
z∗M
A− µp
)2
dz (2.34)
An increase in MA will affect variance through two channels. On the intensive
margin, it will increase the variance of the of p1 for values below (1 + r). On the
extensive margin, it will increase the mean of the distribution and move a higher
portion of it into values above (1 + r), where the variance is very small.
54
2.4 Results
In order to asses the impact of liquidity demand and liquidity supply on the shadow
price, I show a series of comparative statics on the mean and variance of p0. This
exercise shows unambiguously how an increase in liquidity will raise the price of the
asset. The volatility of the price of an asset declines as the money shock increases,
and increases with liquidity demand. Additionally, in order to provide a theoretical
link between this model and the empirical measure used in chapter 3, I define the
monetary services index in the context of this model. An alternate to a more
traditional measure of aggregate liquidity (M), I show how the total expenditure of
monetary services is in fact of measure of illiquidity.
2.4.1 Impact of liquidity shocks on the mean and variance of p0
In the Allen and Gale framework, they exclusively consider a liquidity shock to be
a shock to liquidity demand (λ). As in their model, a positive liquidity demand
shock decreases p0 (2.32):
∂E [p0]∂z
= −wεzA
(2z− 1 +
A′z
zA
)if z < z (2.35)
∂E [p0]∂z
= 0 if z ≥ z (2.36)
Because A′z ≤ 1 by quadratic utility (2.11), ∂E[p0]
∂z ≤ 0 ∀z. If the aggregate
liquidity demand increases, its more likely that a household will be an early con-
sumer. The mean price will decrease unless its already at 1 + r. The change in
price with respect to z∗ is observed in Figure 2.3.
55
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.60.81
0.815
0.82
0.825
0.83
0.835
0.84
0.845
Expected Mean of z*
Initi
al P
rice
of th
e A
sset
in P
erio
d 0
Figure 3: Initial Price vs. z*
Figure 2.3: The expected price of the initial asset holding wealth, interestrates, and the money supply shock constant and changing z∗, the fractionof people consuming early.
56
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Money Supply Subsidy (Tax), ε
Initi
al P
rice
of th
e A
sset
in P
erio
d 0
Figure 4: Initial Price vs. Money Shock
Figure 2.4: Expected price for the initial asset holding wealth, interestrates, and the fraction of early consumers constant while changing ε, themoney supply shock.
Now consider a shock to liquidity supply, ε.
∂E [p0]∂ε
=1− z
Az
(W −A′
εε− 1)
if z < z (2.37)
∂E [p0]∂ε
= 0 if z ≥ z (2.38)
Thus, because W ≥ 1 and A′ε < 0 (see Figure 2.2), ∂E[p0]
∂ε ≥ 0 ∀ε. If the
aggregate supply of money increases, the mean price will increase unless its already
at 1 + r. Figure 2.4 illustrates this graphically.
57
The volatility of the asset price increases with liquidity demand. On the
extensive margin, an increase in z∗ raises z (2.33) and decreases the states where
the liquidity price is realized:
∂z
∂z=
−A′z
rA+W
((1 + r)WrA+W
)≥ 0 (2.39)
This effect is overshadowed by the intensive margin, which is strongly posi-
tive:∂var [p0]
∂z= var
(1− z∗
z∗
)−WA2
A′z ≥ 0 (2.40)
Increasing liquidity demand increases the liquidity price and the fundamental
price of the asset in the first period. This increases the variance of the liquidity
price, as illustrated in Figure 2.5.
The effect of an increase in money supply is the opposite, as we can see from
its positive impact on p0. As the liquidity price increases, it gets closer to 1 + r,
and more of the distribution is truncated. This will reduce the overall variance of
p0, as seen in Figure 2.6.
2.4.2 Monetary Services Index
The monetary services index is an alternate way to calculate money aggregates and
expenditures on liquidity. Used extensively in the next chapter, the goal of this
index is to quantify the value of liquidity in terms of opportunity cost. The total
expenditure measure is the specific MSI series analyzed in chapter 3. It is calculated
by adding up the various components of a given money aggregate, weighted by the
opportunity cost of holding each of them. In this model, that cost is the spread
58
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.60.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Expected Mean of z*
Var
ianc
e of
Initi
al P
rice
Figure 5: Variance of the Initial Price vs. z*
Figure 2.5: The volatility of p0 increases with the fraction of early con-sumers. As the price of the asset falls, the range of potential realizationswidens, driving up the volatility.
59
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Money Supply Subsidy (Tax), ε
Var
ianc
e of
the
Initi
al P
rice
Figure 6: Variance of the Initial Price vs. Money Shock
Figure 2.6: The volatility of p0 decreases with the money supply shock. Asthe price of the asset rises, the range of potential realizations narrows,shrinking down the volatility until it converges to zero.
60
between the return on m and a:
ra − rm = 1 + r − p0 (2.41)
To express the total expenditure, I multiply this cost by aggregate money holdings
and substitute in the mean of p0 (when its less than 1 + r).
MSI : (1 + r − p0) (W −A) ε (2.42)
The MSI reacts positively to a shock to money demand:
∂MSI
∂z= − (1 + r − p0)
∂a0
∂z− (W −A)
∂p0
∂z(2.43)
As both asset demand and the price decline with respect to z, the price and
amount of money, the total expenditure on monetary assets, is increasing The MSI
increasing with a shock to money demand is consistent with a measure of illiquidity.
This is shown in Figure 2.7.
The opposite is true for an increase in money supply via a money shock:
∂MSI
∂ε=∂ (1 + r − p0)
∂εM +
∂M
∂ε(1 + r − p0) (2.44)
The quantity of money is increasing with ε, but its getting more expensive
per unit as well; p0 is falling, driving up the opportunity cost. Figure 2.8 shows
how the MSI decreases with ε.
The MSI increases with a liquidity demand shock and decreases with a liq-
uidity supply shock. Thus, it is a measure of illiquidity.
61
0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 0.60.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
Expected Mean of z*
Var
ianc
e of
Initi
al P
rice
Figure 7: MSI vs. z*
Figure 2.7: The MSI increases with respect to the fraction of early con-sumers. The total resources spent on monetary services (our ”liquidityinsurance”) increases with the chance of needing liquidity (z∗).
62
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Money Supply Subsidy (Tax), ε
Var
ianc
e of
the
Initi
al P
rice
Figure 8: MSI vs. Money Shock
Figure 2.8: The MSI decreases with respect to the money supply shock.The total resources spent on monetary services decreases as the oppor-tunity cost of holding money falls.
63
2.5 Conclusion
This paper introduces a simple asset allocation model to explore the impact of a
change in liquidity demand and supply on the price level and volatility of an asset.
I find that an increase in money supply or a decrease in liquidity demand raises the
price of the asset. Additionally, I find that volatility decreases as the liquidity price
of the asset increases. The decrease in volatility is driven by the increasingly severe
upper bound on the liquidity price, the fundamental value price. This asymmetry is
based on the idea that those with liquid assets but no immediate needs will not part
with them unless they will earn a better return, while those with illiquid assets and
immediate needs have no lower bound on what they’ll accept for their illiquid goods.
The model abstracts from transaction costs, participation costs, or other frictions,
and focuses on the interaction between the most critical component of liquidity, the
cost of converting wealth into consumption.
In preparation for the next chapter, I investigated some of the theoretical
properties of a monetary services index within the context for the model and show
that it is a measure of illiquidity. As an extension to this paper, introducing a
second type of agent, an infinitely lived market maker, may reduce the liquidity
premium but amplify the impact of a shock.
64
Chapter 3
Monetary Liquidity and Asset
Pricing
3.1 Introduction
In an effort to solve numerous finance puzzles such as excess volatility and the
equity risk premium, much attention has been directed at relaxing the assumption
of costless exchange in asset markets. Liquidity factors, as they have come to be
called, are now commonly cited indicators of the state of the market and contribute
significantly to explaining returns and volatility. While widely discussed, liquidity
can be a nebulous concept, and has been applied to areas ranging from explaining
the behavior of intraday volume of a particular stock to assessing global monetary
conditions.
In this chapter, I propose monetary liquidity as the best and most fundamen-
tal measure of the underlying distortion that liquidity factors are trying to capture.
Using a GARCH-in-mean estimation model, I show that monetary factors more
accurately match existing stylized facts of liquidity and asset returns than market
based measures of liquidity. Moreover, while this result is focused on returns and
65
volatility of an equity index over time, it is robust to cross-sectional analysis at an
individual stock level such as the decile return methodology used by Pastor and
Stambaugh (2003).
Monetary liquidity measures the cost of converting capital to consumption.
When the quantity of money is high, the cost of liquidity is low, and the difference
between selling an asset today and the value of holding it forever is very small.
Similarly, in an asset market, the cost of liquidity is interpreted as the discount in
price that a seller gives up to hold cash instead of the asset.
To the extent liquidity costs are non-zero, their will be a liquidity premium
on every transaction. This could accrue to the buyer or the seller, depending on
whose needs are greater. The seller may need to convert the asset to cash in order
to consume, the buyer may have too much cash earning a low return. Thus, the
observed transaction price is the true value of the asset plus or minus the liquidity
premium. When liquidity is high, the prices observed will be in a tight range around
true value. When liquidity is low, that range will widen out. If the distribution of
observed prices widens out about the true value, we would expect to observe higher
price volatility in periods of low liquidity. Chapter 2 of this dissertation offers a
two-period model to illustrate this mechanism.
Empirical studies linking monetary conditions, market liquidity, and asset
prices have found several common observations across different measures of market
liquidity.
F1. Illiquidity is associated with lower contemporaneous returns and higher
expected returns.
F2. Assets that are on average more illiquid have higher expected returns
than assets that are more liquid.
F3. Illiquidity is associated with higher contemporaneous price volatility.
The first two stylized facts address the link between equity returns and liq-
66
uidity. Empirical evidence for these two facts has been found using a broad array of
market liquidity measures. Amihud (2002) found that excess returns vary positively
with illiquidity, both cross-sectionally and over time. He employs an information
flow based measure of liquidity, defined as the magnitude of the change in price over
the total dollar volume. Pastor and Stambaugh (2003) found strong evidence for F2
using a liquidity measure based on order flow, which is essentially volume but only
counts the trades that change the direction of the price. They construct an order
flow index at daily frequency for most stocks on major US exchanges. In order to
find aggregate market liquidity, they average individual stock liquidity over a month
and across all stocks.
Empirical studies relating liquidity and volatility have found evidence of F3,
and most have used a GARCH based estimation methodology. Lammoureux and
Lastrapes (1990) used a GARCH model and volume based liquidity measure to test
the impact of liquidity on stock volatility. By comparing a GARCH estimation
conditioned on past volatility to a GARCH estimation conditioned on volume, they
concluded that much of the change in volatility can be attributed to the amount of
volume traded.
Chordia and Subrahmanyam (2005) is one of the few papers that considers
the role of monetary liquidity in asset markets. Using a VAR approach to jointly
estimate stock and bond liquidity, volatility, and returns, they find that monetary
policy only affects markets in extreme ”crisis periods”. The authors use a pro-
portional spread measure of liquidity, the difference in the bid/ask spread over the
average price.
Watanabe and Watanabe (Forthcoming) closely followed Lammoureux and
Lastrapes’s methodology and tested other market based measures of liquidity such as
Amihud’s price impact, Chordia’s proportional spread, and Pastor and Stambaugh’s
order flow based aggregate liquidity. They find strong support for stylized F3 for
67
all liquidity measures.
Previous finance literature has focused almost exclusively on market mea-
sures of liquidity. While these measures are excellent measures of the cost of
buying or selling an asset, they do not encompass the complete range of impacts
that the cost of liquidity can have on asset prices. Low market liquidity may oc-
cur in periods of high price volatility, and may also be a contributing factor, but
because both observations are the result of decisions made by price-taking agents,
a more fundamental process must be the source of the distortion. I am proposing
that monetary liquidity is that deeper process for two reasons. First, money is an
excellent numeraire good. It is the most common link between the asset and goods
market, and as such the cost of holding money has been argued to be a summary
statistic for economy wide liquidity costs1
In section 2, I describe the Monetary Services Index (MSI), a monetary
aggregate designed to capture the service flow provided by the money supply. This
index is designed to capture the expense of holding money, and as such is ideally
suited to measure the state of monetary liquidity in the economy.
Section 3 tests the MSI against the other, more traditional market based
liquidity measures in a GARCH-in-mean framework outlined in Lamoureux and
Lastrapes. This technique has the advantage of testing stylized facts F1 and F3
at the same time. I find that the MSI is the only measure that matches both
simultaneously.
As a further robustness check, Section 4 presents a test of the MSI using
Pastor and Stambaugh’s methodology for individual stocks. Using a test period of
5 years, I measure each stock’s sensitivity to liquidity (a liquidity ”beta”) and sort
stocks into portfolios based on their liquidity beta decile. In this method, the most1For a good survey of the literature on money aggregates as a summary of interest rates, see
Nelson (2002) and Nelson (2003). Second, disturbances in money supply are initiated by the centralbank, and are therefore not subject to stabilizing market forces and do not have to be in equilibrium.
68
important statistic is the excess return from a portfolio constructed from being long
decile 10 and short decile 1. I find that the 10-1 portfolio for the MSI is comparable
in magnitude to the 10-1 portfolio for Pastor and Stambaugh, despite their liquidity
measure being based on market data. Additionally, the market liquidity measure
doesn’t earn any excess return after controlling for monetary liquidity.
Section 5 addresses the question of which liquidity measure is more funda-
mental, and causing the other. Similar to Chordia, Sarkara and Subrahmanyam, I
estimate a VAR on returns, volatility, and both liquidity measures. Contrary to
their results, I find that monetary liquidity is the only liquidity factor that explains
return volatility, and that while there appears to be minor feedback loop through
market liquidity, monetary liquidity provides the bulk of the fundamental shock.
Section 6 concludes.
3.2 Data
Data for the liquidity metrics and returns came from several sources. For the time
series estimates of market returns and liquidity I used the monthly total return
(including dividends) index published by Standard and Poors. To calculate Ami-
hud’s measure of price impact, I took the change in the price of the index divided
by the dollar volume of shares traded. For the cross-sectional analysis, I obtained
monthly data on individual securities from CRSP. This included both total returns
and variables used to calculate market liquidity. Chordia’s measure of proportional
spread was calculated for each stock as the difference in the Ask and Bid price di-
vided by last month’s price. These values were then aggregated across stocks to
generate a value-weighted index level for stocks that are in the S&P 500. Pastor
and Stambaugh’s aggregate order flow index and liquidity innovation is available for
download through WRDS. Their index is constructed using a three step, bottom up
procedure. At the stock level, the author’s run a regression on daily stock returns
69
over a the period of a month:
ri,d+1,t = θi,t + φi,tri,d,t + γi,tsign(rei,d,t
)· vi,d,t + εi,d+1,t (3.1)
ri,d,t = ri,d,t − rm,d,t (Excess return over the market)
vi,d,t = dollar volume
The coefficient γi,t is designed to capture the ”order flow” effect. That is,
the volume of shares traded in excess of the market move for a given day should be
partially reversed if a stock is not perfectly liquid. The next step is to aggregate
γi,t, across stocks to generate an market wide measure:
γt =1N
N∑i=1
γi,t (3.2)
An AR(1) process is then parameterized to first difference of this series in
order to generate what the authors term a ”liquidity innovation”, µt :
∆γt = a+ b∆γt−1 + c
(mt−1
mt
)γt−1 + µt (3.3)
The residual of the above process, µt, is the liquidity factor used throughout
their paper and in the following comparisons.
For empirical proxies of monetary liquidity, I downloaded the Monetary Ser-
vices Index and associated indices from the St. Louis Federal Reserve. Also called
a Divisia or Fisher aggregate, these indices are constructed to weight each compo-
nent of the money aggregate by the ”moneyness” characteristics it possesses. For
example, currency is 100% money, as it generates no return and is perfectly liquid.
An interest bearing checking account is close to money, as it is a demand deposit,
but not 100%. A CD or treasury bill would only be a little like money, as it has a
fixed term. Traditional money aggregates tend to be swamped by their most illiquid
70
components, as the illiquid components carry a notional value that is much higher
than the more liquid components. I am using the MZM (money, zero maturity)
aggregate, which contains all the zero duration components of M1 and M2. I also
include the simple sum MZM aggregate to compare with the services index.
The Divisia aggregation technique creates several indices from the aggregate
components. I choose to use the total expenditure on monetary assets (referred
to henceforth as the MSI) both because it’s constructed in a more transparent way
than some of the others and it accounts for both changes in the cost of liquidity and
the quantity. The MSI computes the total expenditure of monetary assets as the
sum across all MZM components of the opportunity cost of holding the component
times the amount held (3.4).
Total Expenditure on Monetary Assets: yt =n∑i
πrealit mnom
it (3.4)
The total expenditure is measured in nominal terms, and because it is a
measure of the total economy-wide cost of liquidity services, it should be interpreted
as a measure of market illiquidity. In the following analysis, the total expenditure
on monetary assets is divided by nominal GDP to scale it across time.2
To get a sense of how the different measure of liquidity relate to one another,
and to returns, a correlation matrix is shown in Figure 3.1
The two metrics that will be the focus of this paper, Pastor and Stambaugh’s
market liquidity measure and total expenditure measure of monetary liquidity, have
almost no correlation. The low correlation between total expenditure and the
simple sum aggregation of the same MZM components highlights the impact of
the aggregation technique on the liquidity measure. The high correlation of price2For a complete description of the Monetary Services Index and its contructions, see Anderson,
Jones and Nesmith (1997). Constructing the Index has been an ongoing project at the St. LouisFederal Reserve.
71
Name Abrev spr P&S prim psprd MSI SSS&P 500 Return spr 1.00P&S Liquidity Innovation P&S 0.06 1.00Price Impact prim 0.63 -0.01 1.00Proportional Spread psprd 0.08 -0.03 0.18 1.00MSI Total Expenditure MSI -0.15 -0.07 -0.02 0.19 1.00Simple Sum MZM Aggregate SS 0.01 -0.01 -0.05 0.31 0.16 1.00
Figure 3.1: Correlation between Stock Returns and empirical proxies forboth monetary and market liquidity.
72
impact and S&P returns raises some concern that this has more to do with the
lagged price term in the calculation of price impact than any measure of liquidity.
3.3 Testing Liquidity on Index Aggregates
In testing liquidity on the index I follow the methodology developed by Lammoureux
and Lastrapes (1990). They use a GARCH model to evaluate how different measures
of liquidity can explain both returns and volatility of a particular stock. I first
establish a baseline by estimating a GARCH model on a market return series, in
this case the S&P 500.
rmt − rf
t = µ+ ψσ2t−1 + εt (3.5)
εt ∼ N(0, σ2
t
)(3.6)
σ2t = α0 + α1ε
2t−1 + α2σ
2t−1 (3.7)
The first equation, (1.5), defines the process for generating excess returns. Because
this is a GARCH-in-mean estimation, returns are a basic MA(1) process with the
addition of an explicit variance term. The error of that process, εt, is drawn from
a different normal distribution each period, (3.6), which is parameterized by the
variance. The third equation, (3.7), defines the process by which the variance
evolves. The estimated coefficient for the ARCH term is α1, for the GARCH term
α2. These coefficients represent the unattributed change in volatility. Having
established a baseline, I estimate a similar GARCH model, adding a measure of
liquidity (xt) to the volatility portion of the estimation.
σ2t = α0 + α1ε
2t−1 + α2σ
2t−1 + βxt (7’)
In order to simplify the interpretation of the results, any measures that are
73
conceptually measures of illiquidity (Pastor and Stambaugh’s order flow, the MSI,
and the proportional spread) I multiply by −1 to put everything in terms of a
liquidity measure.
Comparing the α1 and α2 terms in both estimations reveals the contribu-
tion of the liquidity term to explaining the volatility. This comparison is seen in
Figure 3.2. Column II is the baseline GARCH-in-mean test, columns III-V are the
GARCH-in-mean with market based liquidity measures, VI and VII are monetary
based liquidity measures. In order to determine whether the liquidity factors con-
tribute to explaining volatility compare the size and significance of α1 and α2 in the
baseline case to the cases with a liquidity factor. The ARCH term, α1, is much
reduced in both size and statistical significant by the each of the market liquidity
factors.3 The monetary factors had no impact on the ARCH term.
The GARCH term, α2, reveals more of an impact. Models IV-VI, price
impact, proportional spread, and the Monetary Services Index, reduce α2 to the
point of being statistically insignificant. Model II, Pastor and Stambaugh’s order
flow measure, as well as model VII, the simple sum money aggregate, fail to explain
any of the GARCH term.
Having established that the monetary services index is statistically significant
in explaining volatility, I can use the results of the estimation to evaluate each model
on how closely it matches the two time-series based stylized facts (F1 and F3).
To test stylized F1, we need to determine the impact of liquidity on the
excess return. In the estimation model, this is calculated by multiplying ψ, the
GARCH-in-mean coefficient, by βn, the coefficient of liquidity in the expression for
variance. The product of these two coefficients is seen in the last row of Figure
3.2. In order to confirm F1, ψβn needs to be positive, as these are all measures of
liquidity. Of the market liquidity measures, only Amihud’s measure of proportional3This is a similar result to Watanabe and Watanabe (Forthcoming).
74
Tab
le 2
: G
AR
CH
Res
ults
with
Var
ious
Pro
xies
for
Liq
udity
(t-s
tatis
tics a
re in
par
enth
eses
)
Para
met
erSy
mbo
lI
IIII
IIV
VV
IV
IISo
urce
of L
Q F
acto
r
Ret
urn
Mea
n0.
0033
-0.0
014
0.09
050.
1385
-0.0
088
0.12
71-0
.001
(2.3
7)(-
0.39
)(2
.46)
(2.9
2)(-
2.03
)(3
.09)
(-0.
30)
Gar
ch-in
-mea
n─
4.23
85-8
6.34
5-1
66.3
810
.40
-7.9
227
3.98
9(1
.33)
(-2.
37)
(-2.
76)
(2.6
4)(-
2.01
)(1
.26)
Var
ianc
e M
ean
0.00
010.
0001
-8.1
507
-7.0
910
-8.4
051
-8.0
713
-9.0
81(2
.33)
(2.2
5)(-
44.6
8)(-
59.5
2)(-
20.0
3)(-
28.7
0)(-
20.3
4)
Arc
h Te
rm0.
1425
0.14
13-0
.017
60.
0050
0.02
880.
1629
0.14
16(4
.09)
(4.1
3)(-
2.15
)(1
.96)
(0.7
3)(4
.12)
(4.1
5)
Gar
ch T
erm
0.76
520.
7730
0.73
27-0
.018
20.
0612
-0.2
672
0.77
62(1
2.28
)(1
2.75
)(1
5.84
)(-
.016
)(0
.35)
(-1.
95)
(12.
94)
Past
or In
nova
tion
──
4.92
65─
──
─Pa
stor
and
Sta
mba
ugh
(200
3)(I
lliqu
idity
)(3
.99)
Pric
e Im
pact
──
─9.
01E+
08─
──
Am
ihud
(200
2)(L
iqui
dity
)(3
.89)
Prop
ortio
nal S
prea
d─
──
─0.
0000
1─
─C
hord
ia, e
t al.
(200
3)(I
lliqu
idity
)(4
.36)
MSI
MZM
──
──
─-1
7.03
─M
onet
ary
Serv
ices
Inde
x(I
lliqu
idity
)(-
5.54
)
MSM
──
──
──
-0.0
0004
Sim
ple
Sum
Mon
ey A
ggre
gate
(Liq
uidi
ty)
(-0.
42)
Coe
f of L
Q o
n R
etur
ns─
─-4
25.3
829
-1.5
0E+1
10.
0001
134.
91-0
.000
1
0α
1α
2α 1β
2β 3β 4β 5βµ ψ
() (
)t
tt
t
tt
tt
f tm t
x
Nrr
βσ
αε
αα
σ
σε
εψσ
µ
++
+=
++
=−
−−
−
21
22
11
02
2
21
,0~
nψβ
Figure 3.2:
75
spread delivers this result. Of the monetary liquidity measures, the Monetary
Services index matches F1 but the simple sum aggregate does not.
To test F3, that volatility increases in illiquid environments, we need only to
determine the βn coefficient. For measures of liquidity, a negative βn would confirm
F3. On this basis, none of the market measures of liquidity match. βn for models
III, IV, and V, (order flow, price impact, and proportional spread), is significant but
negative. The result suggests volatility is higher in liquid environments, contrary
to F3. The monetary models do better: the MSI is both negative and significant.
The simple sum aggregate is negative, which is consistent with liquidity reducing
volatility, though it’s not significant.
The GARCH estimation produces two results. First, that monetary mea-
sures of liquidity contribute to explaining return volatility beyond that of market
liquidity. We infer this from the reduced size and significance of the generalized
ARCH and GARCH terms, α1, and α2. Secondly, that monetary liquidity is seen to
reduce volatility and increase returns simultaneously. None of the market liquidity
measures were observed doing both at once.
3.4 Testing Liquidity on a Cross-Section of Stocks
The previous section established that monetary liquidity preforms as well or better
than market liquidity in matching the stylized facts of returns and volatility at
an aggregate level and over time. In order to show this result to be robust, the
monetary measures should explain a liquidity based excess return over a cross-section
of stocks, thereby matching F2. A major strength of Pastor and Stambaugh (2003)
is their result that excess returns can be attributed to a particular stock’s sensitivity
to the aggregate liquidity factor. They show this using the decile ranking approach
pioneered by Fama and Macbeth (1973). In order to evaluate how well monetary
factors match F2, I replicate Pastor and Stambaugh’s procedure of checking decile
76
rank returns, substituting the MSI in for their order flow liquidity index. In this
section and the next, I use the raw series for both Pastor and Stambaugh and
the MSI. This makes them both measure of illiquidity, which will allow a direct
comparison with the historical results from Pastor and Stambaugh’s 2003 paper.
Beginning at the end of each year from 1966 to 2005, I identify stocks with
a 5 year monthly return history through the end of the current year. Stock return
data comes from CRSP. I follow Pastor and Stambaugh by restricting the stocks
to ordinary shares (codes 10 and 11) that are listed on the NYSE, AMEX and
NASDAQ stock exchanges. The price must be between 5 and 1000 in the final
month of the observation period. I then estimate the historical liquidity beta for
each stock over that period using a multi-factor model:
ri,t − rft = β0
i + βLi Lt + βM
i MKTt + βSi SMBt + βH
i HMLt + εi,t (3.8)
In order to create the deciles, I sort the stocks by their estimated liquidity betas.
The holding period is for 1 year, upon which a new estimation and sorting takes
place. Decile portfolios are linked across years, and estimated against different
specifications of CAPM: the basic model, the Fama French 3 factor model, and
Fama French plus a momentum factor (3.9-3.11).
rp,t − rft = α1 + βM
p MKTt + εi,t (3.9)
rp,t − rft = α2 + βM
p MKTt + βSp SMBt + βH
p HMLt + εi,t (3.10)
rp,t − rft = α3 + βM
p MKTt + βSp SMBt + βH
p HMLt + βUp UMDt + εi,t(3.11)
In this regression the α term is the unexplained excess returns that accrue to
a particular portfolio. Because we have constructed the portfolios to correspond to
77
a particular decile ranking of liquidity sensitivity, comparing the alphas reveals any
returns to liquidity exposure. By slowly increasing the number of dependent vari-
ables, other potential risk factors that may be correlated with liquidity are filtered
out. For example, larger firms tend to be more liquid, so putting just the market
return on the right hand side wouldn’t rule out the impact from size. In their
paper, Pastor and Stambaugh preformed this analysis with stocks equal weighted
and value weighted. The results were not significantly affected. All results shown
here are value-weighted.
Figure 3.3 shows the alphas (or unexplained excess returns) that result from
estimating each decile portfolio on different versions of a CAPM model (3.9-3.11).
The most important column in these tables is the results for the 10-1 portfolio, as it
arguably cancels out unexplained noise, and allows for the most precise estimation
of the alpha associated with a particular model. The 10-1 alphas are attempting
to capture exposure to the liquidity factor without any other factor returns mixed
in. None of the alphas at the four factor (3.11) level were significant for the MSI.
Compare these results with the Figure 3.4, which shows the results from an identical
procedure using the Pastor and Stambaugh measure of liquidity (Table 8 in their
paper). The 10-1 level alpha’s were positive and statistically significant for even
the four risk factor model.
One surprising result from this analysis is that the excess return on the MSI
10-1 alphas are negative. Investors usually demand a higher return to take on more
exposure to a risk factor. It’s possible that because the MSI is an economy wide
measure, it’s correlated with investors fluctuating demand as well. If liquidity costs
rise with sudden changes in wealth, it may be more desirable to own a stock that
goes up at the same time an agent would like to sell it. This would reduce the risk
premium on decile 10 stocks and increase in for decile 1 stocks.
The decile ranking analysis of the MSI factor yielded interesting results, but
78
TAB
LE 3
: M
SI T
otal
Exp
endi
ture
Bas
ed L
iqui
dity
Alp
has
Alp
has o
f Val
ue-W
eigh
ted
Portf
olio
s Sor
ted
on H
isto
rical
Liq
uidi
ty B
etas
Dec
ile P
ortfo
lio
12
34
56
78
910
10-1
A. J
anua
ry19
67-D
ecem
ber 2
006
CA
PM a
lpha
0.01
0.02
0.00
-0.0
10.
000.
000.
020.
01-0
.01
-0.0
4-0
.03
(.02)
(1.3
2)(1
.03)
(1.2
6)(.9
4)(1
.28)
(1.8
8)(1
.23)
(-.7
6)(-
1.81
)(-
1.46
)Fa
ma-
Fren
ch a
lpha
-0.0
80.
730.
130.
720.
260.
460.
620.
83-0
.70
-1.4
1-1
.34
(-.0
5)(.6
9)(.1
3)(.8
4)(.3
2)(.5
0)(.8
1)(.9
1)(-
.59)
(-.9
4)(-
.62)
Four
-fac
tor a
lpha
1.25
0.95
0.72
0.84
0.03
0.56
1.08
0.89
0.10
-2.0
6-3
.28
(.84)
(.87)
(.69)
(.95)
(.03)
(.59)
(1.3
6)(.9
5)(.0
8)(-
1.34
)(-
1.52
)B
. Jan
uary
1967
-Dec
embe
r 198
2C
APM
alp
ha-0
.41
1.36
-0.0
81.
030.
360.
342.
49-0
.08
-1.4
3-3
.41
-3.0
1(-
.21)
(.87)
(-.0
5)(.7
4)(.3
1)(.2
8)(2
.07)
(-.0
7)(-
.85)
(-1.
44)
(-1.
00)
Fam
a-Fr
ench
alp
ha-2
.87
0.76
-1.0
51.
270.
581.
242.
300.
32-1
.32
-3.1
5-0
.29
(-1.
59)
(.47)
(-.7
1)(.8
8)(.5
0)(1
.01)
(1.9
6)(.2
5)(-
.76)
(-1.
36)
(-.1
0)Fo
ur-f
acto
r alp
ha-1
.62
1.02
-0.2
32.
861.
720.
792.
64-1
.28
-1.4
0-5
.47
-3.9
0(-
.87)
(.61)
(-.1
5)(1
.96)
(1.4
8)(.6
2)(2
.16)
(-1.
02)
(-.7
8)(-
2.36
)(-
1.32
)
C. J
anua
ry19
83-D
ecem
ber 2
006
CA
PM a
lpha
0.45
1.62
2.19
1.42
1.08
2.20
0.75
2.00
-0.6
5-3
.31
-3.7
5(.2
1)(1
.14)
(1.4
7)(1
.20)
(.87)
(1.4
7)(.6
9)(1
.58)
(-.4
0)(-
1.40
)(-
1.29
)Fa
ma-
Fren
ch a
lpha
2.09
0.94
1.37
0.66
-0.0
60.
12-0
.73
1.24
-0.5
1-0
.92
-2.9
5(.9
7)(.6
8)(.9
9)(.6
3)(-
.05)
(.10)
(-.7
2)(.9
7)(-
.31)
(-.4
7)(-
1.01
)Fo
ur-f
acto
r alp
ha3.
451.
041.
820.
09-0
.93
0.46
-0.1
72.
130.
80-0
.78
-4.1
0(1
.58)
(.73)
(1.2
8)(.0
8)(-
.84)
(.37)
(-.1
6)(1
.65)
(.50)
(-.3
9)(-
1.38
)
Figure 3.3: Excess returns attributed to the MSI sensitivity.
79
TAB
LE 4
: Pa
stor
and
Sta
mba
ugh
Ord
er F
low
Bas
ed L
iqui
dity
Alp
has
Alp
has o
f Val
ue-W
eigh
ted
Portf
olio
s Sor
ted
on H
isto
rical
Liq
uidi
ty B
etas
Dec
ile P
ortfo
lio
12
34
56
78
910
10-1
A. J
anua
ry19
67-D
ecem
ber 2
006
CA
PM a
lpha
-2.8
3-0
.76
0.06
0.65
0.16
1.70
1.69
0.98
0.85
2.74
5.71
(-1.
93)
(-.7
1)(.0
6)(.7
4)(.1
9)(2
.00)
(1.9
0)(1
.05)
(.87)
(2.2
0)(2
.95)
Fam
a-Fr
ench
alp
ha-1
.49
-1.1
1-0
.05
0.29
-0.3
80.
870.
480.
070.
442.
984.
54(-
1.07
)(-
1.02
)(-
.05)
(.36)
(-.4
7)(1
.12)
(.57)
(.08)
(.45)
(2.3
8)(2
.32)
Four
-fac
tor a
lpha
-2.2
2-1
.62
-0.8
10.
210.
790.
680.
550.
841.
553.
515.
85(-
1.57
)(-
1.47
)(-
.87)
(.26)
(.97)
(.86)
(.64)
(.89)
(1.5
6)(2
.73)
(2.9
2)
B. J
anua
ry19
67-D
ecem
ber 1
982
CA
PM a
lpha
-0.8
10.
231.
22-0
.23
0.41
-0.1
30.
90-0
.32
-0.6
53.
063.
90(-
.34)
(.16)
(1.0
2)(-
.19)
(.36)
(-.1
0)(.8
1)(-
.23)
(-.4
7)(1
.76)
(1.4
1)Fa
ma-
Fren
ch a
lpha
-0.5
61.
312.
120.
460.
05-0
.66
0.38
-1.4
2-1
.02
2.04
2.61
(-.2
4)(.8
8)(1
.84)
(.39)
(.05)
(-.5
2)(.3
5)(-
1.01
)(-
.73)
(1.2
2)(.9
3)Fo
ur-f
acto
r alp
ha-2
.95
1.41
1.21
-0.1
11.
00-0
.24
0.56
-0.0
70.
741.
624.
70(-
1.29
)(.9
0)(1
.03)
(-.0
9)(.8
6)(-
.18)
(.49)
(-.0
5)(.5
3)(.9
3)(1
.62)
C. J
anua
ry19
83-D
ecem
ber 2
006
CA
PM a
lpha
-4.1
1-1
.01
-0.6
71.
31-0
.15
3.12
2.60
1.87
2.03
2.63
7.00
(-2.
21)
(-.6
9)(-
.47)
(1.0
5)(-
.13)
(2.8
2)(2
.02)
(1.5
1)(1
.53)
(1.5
1)(2
.61)
Fam
a-Fr
ench
alp
ha-1
.98
-2.2
2-1
.35
0.23
-0.8
92.
171.
020.
931.
523.
775.
86(-
1.13
)(-
1.55
)(-
1.01
)(.2
1)(-
.78)
(2.2
0)(.8
5)(.7
5)(1
.11)
(2.1
3)(2
.16)
Four
-fac
tor a
lpha
-2.0
8-2
.94
-2.0
50.
380.
361.
701.
111.
402.
324.
606.
81(-
1.16
)(-
2.03
)(-
1.52
)(.3
3)(.3
2)(1
.70)
(.91)
(1.1
0)(1
.68)
(2.5
5)(2
.46)
Figure 3.4: Excess returns attributed to the order flow sensitivity.
80
did not confirm F2, certainly not to the degree that Pastor and Stambaugh’s liquid-
ity metric was able to. In order to more clearly identify the source of the market
liquidity factor, I reestimated the decile ranking analysis on their published liquidity
innovation series with an additional factor. The three versions of the second re-
gression (3.9-3.11) are designed to produce a return attributed to the liquidity betas
alone. They successively remove influences from the market, the Fama French fac-
tors, and finally momentum. I added a final specification to filter decile portfolios
on the Fama French factors, momentum, and the MSI index. Thus, the alpha’s
remaining should measure the return to the Pastor and Stambaugh liquidity factor
that is unrelated to the MSI. These results are seen in Figure 3.5.
Including the MSI as a risk factor causes the alphas at all decile levels (in-
cluding 10-1) to become insignificant, and the clear pattern in their magnitude is
no longer evident. While I was not able to directly measure the impact of the MSI
index on the stock return, this result clearly shows that the MSI index underpins
the systemic factor that Pastor and Stambaugh are measuring with their order flow
aggregate.
In order to provide some additional intuition about what sort of industries
are sensitive to each measure of liquidity, Figure 3.6 sorts the liquidity beta scores
by SIC sector group.
Despite their low correlation, and large disparity in results, many of the
industries have similar sensitivities to each type of liquidity. Construction, Man-
ufacturing, and Retail Trade all posted similar relative sensitivities to liquidity.
Agriculture and Mining were sensitive to both in opposite ways, with Agriculture
sensitive to monetary liquidity but not market liquidity, and the reverse for Mining.
81
TAB
LE 5
: Pa
stor
and
Sta
mba
ugh
Ord
er F
low
Bas
ed L
iqui
dity
Alp
has,
Filte
red
on th
e M
SIA
lpha
s of V
alue
-Wei
ghte
d Po
rtfol
ios S
orte
d on
His
toric
al L
iqui
dity
Bet
a
Dec
ile P
ortfo
lio
12
34
56
78
910
10-1
A. J
anua
ry19
67-D
ecem
ber 2
006
CA
PM a
lph a
-2.8
3-0
.76
0.06
0.65
0.16
1.70
1.69
0.98
0.85
2.74
5.71
(-1.
93)
(-.7
1)(.0
6)(.7
4)(.1
9)(2
.00)
(1.9
0)(1
.05)
(.87)
(2.2
0)(2
.95)
Fam
a-Fr
ench
alp
ha-1
.49
-1.1
1-0
.05
0.29
-0.3
80.
870.
480.
070.
442.
984.
54(-
1.07
)(-
1.02
)(-
.05)
(.36)
(-.4
7)(1
.12)
(.57)
(.08)
(.45)
(2.3
8)(2
.32)
Four
-fac
tor a
lpha
-2.2
2-1
.62
-0.8
10.
210.
790.
680.
550.
841.
553.
515.
85(-
1.57
)(-
1.47
)(-
.87)
(.26)
(.97)
(.86)
(.64)
(.89)
(1.5
6)(2
.73)
(2.9
2)FF
4 +
MSI
Liq
uidi
ty A
lpha
4.60
2.30
-4.8
42.
095.
39-4
.00
2.79
1.80
-8.5
0-3
.56
-7.8
3(.5
9)(.3
7)(-
.99)
(.46)
(1.2
0)(-
.95)
(.59)
(.35)
(-1.
66)
(-.5
3)(-
.78)
B. J
anua
ry19
67-D
ecem
ber 1
982
CA
PM a
lpha
-0.8
10.
231.
22-0
.23
0.41
-0.1
30.
90-0
.32
-0.6
53.
063.
90(-
.34)
(.16)
(1.0
2)(-
.19)
(.36)
(-.1
0)(.8
1)(-
.23)
(-.4
7)(1
.76)
(1.4
1)Fa
ma-
Fren
ch a
lpha
-0.5
61.
312.
120.
460.
05-0
.66
0.38
-1.4
2-1
.02
2.04
2.61
(-.2
4)(.8
8)(1
.84)
(.39)
(.05)
(-.5
2)(.3
5)(-
1.01
)(-
.73)
(1.2
2)(.9
3)Fo
ur-f
acto
r alp
ha-2
.95
1.41
1.21
-0.1
11.
00-0
.24
0.56
-0.0
70.
741.
624.
70(-
1.29
)(.9
0)(1
.03)
(-.0
9)(.8
6)(-
.18)
(.49)
(-.0
5)(.5
3)(.9
3)(1
.62)
FF4
+ M
SI L
iqui
dity
Alp
ha-1
.13
1.79
-7.7
24.
438.
71-4
.01
1.01
4.51
-12.
77-9
.07
-8.0
2(-
.10)
(.22)
(-1.
35)
(.70)
(1.4
2)(-
.61)
(.17)
(.61)
(-1.
93)
(-1.
07)
(-.5
7)
C. J
anua
ry19
83-D
ecem
ber 2
006
CA
PM a
lpha
-4.1
1-1
.01
-0.6
71.
31-0
.15
3.12
2.60
1.87
2.03
2.63
7.00
(-2.
21)
(-.6
9)(-
.47)
(1.0
5)(-
.13)
(2.8
2)(2
.02)
(1.5
1)(1
.53)
(1.5
1)(2
.61)
Fam
a-Fr
ench
alp
ha-1
.98
-2.2
2-1
.35
0.23
-0.8
92.
171.
020.
931.
523.
775.
86(-
1.13
)(-
1.55
)(-
1.01
)(.2
1)(-
.78)
(2.2
0)(.8
5)(.7
5)(1
.11)
(2.1
3)(2
.16)
Four
-fac
tor a
lpha
-2.0
8-2
.94
-2.0
50.
380.
361.
701.
111.
402.
324.
606.
81(-
1.16
)(-
2.03
)(-
1.52
)(.3
3)(.3
2)(1
.70)
(.91)
(1.1
0)(1
.68)
(2.5
5)(2
.46)
FF4
+ M
SI L
iqui
dity
Alp
ha9.
8212
.26
2.61
1.96
3.72
-9.7
05.
48-4
.54
-9.2
9-6
.59
-15.
05(.8
8)(1
.31)
(.32)
(.28)
(.54)
(-1.
71)
(.72)
(-.6
0)(-
1.19
)(-
.64)
(-1.
01)
Figure 3.5: Excess returns attributed to the order flow sensitivity net ofMSI.
82
Pastor and Stambaugh Liquidity Factor Beta MSI Liquidity Factor BetaSIC
Industry MeanStandard
Dev Obs RankSIC
Industry MeanStandard
Dev Obs RankAgriculture 1 0.06 0.45 2652 2 1 -0.24 1.33 2652 9Mining 2 -0.07 0.48 41211 10 2 0.26 1.61 41211 1Construction 3 -0.04 0.51 12921 9 3 -0.33 1.54 12921 10Manufacturing 4 0.01 0.45 492675 5 4 0.13 1.53 492675 4Transportation & Utilities 5 -0.01 0.32 118102 6 5 -0.14 1.03 118102 8Wholesale Trade 6 0.02 0.46 38356 4 6 0.17 1.67 38356 3Retail Trade 7 0.03 0.45 73475 3 7 0.22 1.40 73475 2Financial Services 8 -0.03 0.35 201694 7 8 -0.10 1.11 201694 7Services 9 -0.03 0.53 97292 8 9 0.06 1.88 97292 6Public Administration 10 0.13 0.37 404 1 10 0.08 1.28 404 5
Figure 3.6: Liquidity Betas by industry group. Observations are in termsof stock-month pairs.
3.5 Causality
From the GARCH analysis we can conclude that both monetary liquidity and market
liquidity have an impact on market returns and volatility. Repeating Pastor and
Stambaugh’s decile ranking analysis shows that the two types of liquidity are related,
as accounting for one removes the effect of the other. This interrelatedness is
more complicated than a simple linear relationship, as the correlation between the
two is almost zero, and the decile ranking on monetary liquidity alone yielded no
statistically significant results. To understand the relationship between the two
types of liquidity better, it’s important to establish if changes in one are causing
changes in the other. From a conceptual perspective, it’s reasonable to believe
that monetary liquidity is the fundamental cause of market liquidity. The Federal
Reserve and the banking system are concerned with the economy as a whole, and
routinely change or distort the price and quantity of monetary instruments. Market
83
liquidity is the result of individual agents choosing to trade in a competitive market
where agents are price takers with respect to both assets and liquidity. However, it
can be argued that the Federal Reserve is influenced by market events, particularly
when market liquidity is low. In an attempt to measure the competing sources of
causation, I estimate a Vector Auto Regression of S&P 500 returns, the monthly
volatility of those returns, the Pastor and Stambaugh market liquidity innovation
and the MSI total expenditure. The monthly volatility of the S&P 500 returns is
estimated from the base GARCH case. Both liquidity measures are of illiquidity.
The results are presented in Figure 3.7.
3.5.1 Impact of Monetary vs. Market Liquidity on Returns and
Volatility
I specify a VAR with four lags based on the AIC lag selection criteria. With
both types of liquidity accounted for, the MSI is significant in explaining returns
and volatility, while the market liquidity measure in only significant in explaining
returns. Figure 3.8 shows the cumulative impulse response function for a 1% shock
to liquidity on returns. The panel on the left is the response of returns to an
innovation in market liquidity, the right to monetary liquidity. Both are significant
at the 95% level.
Figure 3.9 shows a similar impulse response function for monthly return
volatility. In the left panel, we see that market liquidity is not significant at the
95% level while on the right, monetary liquidity is.
These results are at odds with a similar analysis in Chordia and Subrah-
manyam (2005). They ran a similar VAR with fed funds as the proxy for monetary
liquidity. They found it to only be significant in ”crisis periods” of severe market
illiquidity. I attribute the difference to the broader scope of the MSI total expen-
diture series. It will pick up not only the change in fed funds, but also how the
84
Table 7: VAR(4) of Returns, Volatily, Market and Monetary Liquidity
RMSE R-sq Chi2S&P 500 Return 0.41 0.979 24343.7 Sample Period Feb-62 to Dec-05S&P 500 Volatility 3.04 0.950 9923.1 Observations 524MSI Total Exp. 0.01 0.915 5649.1P&S LQ Innovation 0.05 0.047 25.7
Coefficient Std. Error t-stat Coefficient Std. Error t-statS&P 500 Return MSI Total Expenditure
L (1) S&P 500 Ret. 1.30 0.044 29.58 L (1) S&P 500 Ret. 0.001 0.001 2.28L (2) S&P 500 Ret. -0.26 0.096 -2.73 L (2) S&P 500 Ret. -0.002 0.001 -1.33L (3) S&P 500 Ret. 0.16 0.120 1.34 L (3) S&P 500 Ret. 0.000 0.002 -0.02L (4) S&P 500 Ret. -0.20 0.073 -2.72 L (4) S&P 500 Ret. 0.000 0.001 0.39L (1) S&P 500 Vol. 0.02 0.006 3.58 L (1) S&P 500 Vol. 0.000 0.000 0.41L (2) S&P 500 Vol. 0.00 0.007 0.56 L (2) S&P 500 Vol. 0.000 0.000 -1.87L (3) S&P 500 Vol. -0.01 0.005 -1.22 L (3) S&P 500 Vol. 0.000 0.000 2.05L (4) S&P 500 Vol. -0.01 0.004 -3.49 L (4) S&P 500 Vol. 0.000 0.000 0.00L (1) MSI Tot. Exp. -27.84 3.145 -8.85 L (1) MSI Tot. Exp. 1.182 0.044 26.80L (2) MSI Tot. Exp. 28.58 4.877 5.86 L (2) MSI Tot. Exp. -0.379 0.068 -5.54L (3) MSI Tot. Exp. 1.71 5.090 0.34 L (3) MSI Tot. Exp. 0.084 0.071 1.17L (4) MSI Tot. Exp. -4.58 3.459 -1.33 L (4) MSI Tot. Exp. 0.071 0.049 1.47L (1) P&S LQ Innv. -1.16 0.387 -3.00 L (1) P&S LQ Innv. 0.011 0.005 2.04L (2) P&S LQ Innv. -0.56 0.389 -1.45 L (2) P&S LQ Innv. -0.001 0.005 -0.10L (3) P&S LQ Innv. -0.33 0.386 -0.86 L (3) P&S LQ Innv. 0.005 0.005 0.98L (4) P&S LQ Innv. 0.40 0.386 1.05 L (4) P&S LQ Innv. -0.011 0.005 -2.05Constant 0.11 0.099 1.12 Constant 0.003 0.001 2.47
S&P 500 Volatility P&S Liquidity InnovationL (1) S&P 500 Ret. -10.11 0.327 -30.89 L (1) S&P 500 Ret. 0.013 0.005 2.58L (2) S&P 500 Ret. 12.60 0.720 17.50 L (2) S&P 500 Ret. -0.015 0.011 -1.42L (3) S&P 500 Ret. -2.25 0.899 -2.50 L (3) S&P 500 Ret. 0.002 0.014 0.18L (4) S&P 500 Ret. -0.38 0.549 -0.69 L (4) S&P 500 Ret. 0.001 0.008 0.14L (1) S&P 500 Vol. 0.72 0.043 16.51 L (1) S&P 500 Vol. 0.001 0.001 0.86L (2) S&P 500 Vol. 0.35 0.052 6.78 L (2) S&P 500 Vol. 0.000 0.001 -0.02L (3) S&P 500 Vol. -0.04 0.037 -0.97 L (3) S&P 500 Vol. -0.001 0.001 -1.26L (4) S&P 500 Vol. -0.08 0.031 -2.72 L (4) S&P 500 Vol. 0.000 0.000 0.87L (1) MSI Tot. Exp. 6.04 23.486 0.26 L (1) MSI Tot. Exp. 0.238 0.357 0.67L (2) MSI Tot. Exp. 136.12 36.422 3.74 L (2) MSI Tot. Exp. -0.046 0.553 -0.08L (3) MSI Tot. Exp. -156.17 38.009 -4.11 L (3) MSI Tot. Exp. -0.644 0.577 -1.12L (4) MSI Tot. Exp. 31.96 25.830 1.24 L (4) MSI Tot. Exp. 0.180 0.392 0.46L (1) P&S LQ Innv. -2.56 2.889 -0.89 L (1) P&S LQ Innv. -0.010 0.044 -0.23L (2) P&S LQ Innv. -2.38 2.908 -0.82 L (2) P&S LQ Innv. 0.029 0.044 0.66L (3) P&S LQ Innv. 0.49 2.882 0.17 L (3) P&S LQ Innv. 0.082 0.044 1.88L (4) P&S LQ Innv. 0.35 2.879 0.12 L (4) P&S LQ Innv. -0.083 0.044 -1.90Constant -1.91 0.743 -2.57 Constant 0.026 0.011 2.31
Figure 3.7: Results of a vector autoregression including both the MSI andthe order-flow measure of liquidity
85
Figure 3.8: Cumulative response of S&P 500 returns to a 1% shock tomarket liquidity (left) and monetary liquidity (right).
86
Figure 3.9: Cumulative response of S&P 500 return volatility to a 1%shock to market liquidity (left) and monetary liquidity (right).
87
P-val of Column GranName Abrev S&P500 S&P VolS&P 500 Return S&P500 0.00S&P 500 Volatility S&P Vol 0.00MSI Total Expenditure MSI 0.25 0.18P&S Liquidity Innovation P&S 0.11 0.52
Figure 3.10: P-values associated with a granger causality test on theVAR(4) structure from Figure 3.7.
rest of the transmission mechanism (out to zero duration) responds to the fed funds
shock.
3.5.2 Causality between Market and Monetary Liquidity
Returning attention to the VAR results in Figure 3.7, both measures of liquidity are
significant in at least one lag in explaining the other liquidity measure. In order
to account for all the lags at once, I preform a granger causality test to determine
if a model with one of the liquidity measures is statistically different from a model
without. Figure 3.10 shows the results of the granger causality test.
The granger causality test confirms the results of the VAR. Both measure
of liquidity are statistically significant in explaining the other. Market liquidity to
93% confidence, Monetary liquidity to 87%.
To test the causal relationship in a more dynamic setting, Figure 3.11 shows
the cumulative impulse response functions of each liquidity measure in response to
a 1% innovation in the other.
In the left panel, market liquidity reacts strongly, to an innovation in mon-
etary liquidity. On the right, an innovation in market liquidity has almost no
88
Figure 3.11: Cumulative impulse response of monetary liquidity to a 1%shock in market liquidity (left). Response to market liquidity to a 1%shock in monetary liquidity.
89
measurable impact on monetary liquidity.
From the VAR results we can conclude that while both market and monetary
liquidity have a measurable source of fundamental shocks to returns, market liquidity
does not on its own impact market volatility. Further, while there is statistical
evidence of joint causation between the two measures, the magnitude of the impacts
are such that monetary liquidity is a much larger fundamental shock.
3.6 Conclusion
The cost of liquidity is an important state variable for investors and consumers.
Previous attempts to measure the impact of liquidity on asset prices have focused
on market liquidity instead of monetary liquidity. This paper has shown that mar-
ket liquidity, while easily observable, overlooks a significant portion of fundamental
shocks that impact equity returns and volatility and are measured by monetary
liquidity. I have shown that using the monetary services index as a proxy for
liquidity can match the stylized facts of liquidity and asset prices as well or better
than existing market based measures of liquidity. While both measures explain
a significant portion of excess returns, only monetary liquidity contributes to ex-
plaining asset price volatility over time. Cross-sectionally, market liquidity is a
more easily measured systemic risk factor, but is completely undetectable after con-
trolling for changes in monetary liquidity. The impulse response function for a
Vector Auto Regression involving both measure of liquidly confirm the relative im-
portance of monetary liquidity. Additionally, they show that over time, innovations
to monetary liquidity contribute significantly to market liquidity, and the bulk of
causality should be attributed to monetary liquidity. The success of the MSI with
US data suggests that constructing a monetary services index for other countries
or across regions would provide an important measurement critical state variable of
the market.
90
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Vita
Timothy Gordon Jones was born in Santa Monica, California on November 30th,
1978, son of Kenneth Moody Jones and Eileen Ann Jones. After completing his
studies at Piedmont High School, Piedmont, California, in 1996, he matriculated at
Yale University, New Haven, Connecticut. His senior thesis at Yale examined the
policy implications of a search model of unemployment. During the following years
he worked as a financial analyst for JPMorgan and NERA Economic Consulting.
In the Fall of 2003 he entered the graduate economics program at the University of
Texas.
Permanent Address: 4017 Vineland Drive, Austin, TX 78722
This dissertation was typeset with LATEX2ε4 by the author.
4LATEX2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademark ofthe American Mathematical Society. The macros used in formatting this dissertation were writtenby Dinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extendedby Bert Kay, James A. Bednar, and Ayman El-Khashab.
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