Biing-Shen Kuo

Department of Money and Banking

Department of International Trade

National Chengchi University

National Chengchi University

Ming-Chin Lin∗ Department of Finance Chaoyang University of Technology

Abstract

We model the effects of monetary liquidity shocks on asset prices and market liquidity in an economy with borrowing constraints. Asset prices are determined not only by fundamentals but also by liquidity shocks. The prices set by market makers under the economy with borrowing constraints would be different from those without constraints. In the economy with borrowing constraints, there always exist a proportion of informed traders who will engage in borrowing-and-buying strategy in equilibrium, given exogenous liquidity shocks and private information about fundamentals. The main results are that the higher the probability of liquidity shocks hits the economy, the more likely the informed traders will borrow; and that as the probability of liquidity shocks increases, the market deepens and the proportion of informed traders increases at first, then turn around to decrease later. In contrast to the findings by Chordia, Roll and Subrahmanyam (2001), this paper analyzes the impacts of monetary liquidity shocks on bid-ask spreads and market depth and find out borrowing constraint is the main drive for the asymmetric pattern of impacts. Key Words: Liquidity Shocks, Borrowing Constraints, Market Depth, Asset Prices,

∗

Correspondence: Ming-Chin Lin, Department of Finance, Chaoyang University of Technology, Taichung County

413, Taiwan. Tel: (04)2332-3000 ext:4589; Fax: (04)2374-2333; E-mail: [email protected].

1. Introduction How the anticipation of future monetary policy affects the prices in financial markets has been widely studies 1 . Through adjusting the supply of reserves, a central bank can effectively control the short-term interest rate. In financial markets the expectations about the future monetary policy significantly affect the decisions of market participants and, thus, the prices of financial assets. This forward-looking nature plays an important role in explaining the volatility of asset prices and market depth. The findings in the literature of monetary economics and finance reinforce the need for developing market microstructure models that explicitly allow volume, volatility, and information acquisition to interact in an environment with aggregate liquidity shocks arising from monetary policy. This paper addresses a basic, yet unresolved, question: How does the anticipated monetary liquidity shock affect the liquidity of assets? Here the term “monetary liquidity” is used in the exact sense a l´a Grossman and Weiss (1983) and Rotemberg (1984). These papers worked out the effects of open market operations on interest rate behavior in the settings in which the agents on the opposite side of a government sale (or purchase) of bonds hold only a fraction of the economy’s money supply. The liquidity of assets is often defined as how quickly the assets can be sold without a price discount, which is measured by the bid-ask spread and market depth. Notwithstanding the importance of research in integrating monetary factors with asset pricing, for instance, Lucas (1990) and Chan, Forest, and Lang (1996), so far not much has been done about the relationship between anticipated monetary policy and market liquidity. To explore this relationship, we ask: What are impacts of aggregate liquidity shocks on the financial market? How will the market microstructure evolve after shocks? Do prices of assets become more volatile? Does the market depth decrease and the bid-ask spreads become wider? Is the informed investor deterred from trading? 1

Lange, Sack and Whitesell (2003) show that the predictability of FOMC actions can be associated

with stronger serial correlation in changes in the federal fund rate. Data from the federal funds futures market suggests a probability of tightening or easing of monetary policy. Fama (1984), Mishkin (1988), Hardouvelis (1988), and Longstaff (2000), have found that the yield curve does contain some information regarding future interest rate changes over particular horizons.

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In our set-up, the price of assets are set by a market maker who receives buying and selling orders from different investors. A number of investors choose whether to become informed by paying an information cost, or to stay as uninformed. The informed will receive a piece of private information of the dividend payoff of investment. All investors knows the probability about liquidity shock. (We interpret this probability as a proxy for the anticipated monetary policy.) Assume investors need to borrow when they encounter with liquidity shock and impose a borrowing cost to make borrowing less attractive. As a result, one would expect that investors who are willing to pay borrowing costs are those who have greater expected benefits from borrowing. We find out that the higher probability of liquidity shocks leads to a narrower bid-ask spread and greater market depth. The market can absorb liquidity trading without large changes in prices. A greater market depth will induce more informed investors to enter into markets to benefit from arbitrage because their information advantage, even though they have to pay borrowing costs. However such a negative relation between probabilities of liquidity shock and bid-ask spreads will eventually reverse as the probability gets higher and higher such that the financial market cannot absorb liquidity trading anymore. A central idea of market microstructure in information contents is that, due to market frictions, asset prices need not equal the prices in a setup with full information. When an aggregate liquidity shock enforces investors to borrow funds, the presence of borrowing costs causes an asymmetric pattern in which the probability of liquidity shocks affects borrowing and trading activities. Specifically, when the probability is greater, the intention to borrow (and thus trading activities) increases at a slower pace. On the contrary, the need for borrowing decreases when the probability of liquidity shock is smaller. The change in the intention to borrow decreases more rapidly. In sum, market liquidity reacts to changing exogenous shocks asymmetrically, and the core of this asymmetry is borrowing constraints and costs. In addition, one further argument deserves a detailed account. Plenty evidence shows that sample estimates of skewness for daily US stock returns tend to be negative for stock indexes but close to zero or positive for individual stocks. Sample estimates of excess kurtosis for daily US stock returns are large and positive for both indexes and individual stocks, indicating that returns have more mass in the tail areas than would be predicated by a normal distribution. Based on our

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model, imposing borrowing constraints can generate the skewness and fat-tailed distribution of stock prices. Even though not being accommodated with return distribution directly, this findings still provide insight into the behavior of asset prices. We conclude that the relationship between market liquidity and exogenous liquidity shocks is not monotonic and the role of borrowing constraints can hardly be ignored. This paper proceeds as follows. Section 2 reviews the existing literature on the relation between stock returns and market liquidity. Section 3 describes the main features of the model, and the solution method for it. Section 4 provides numerical analysis for the case of symmetric solution. Section 5 discusses the empirical implication and concludes.

2. Discussion of the Literature Several papers have relevance for our work, although they do not address in an unified way the main questions we are interested in, namely the transmission mechanism from monetary liquidity to market liquidity. Let us take up these themes in order. Microstructure research is especially interested in liquidity and trading activity which are important characteristics of financial market. Recent extensive literature provides distinguished progress not only by rigorously theoretical but empirical studies. For example, Amihud and Mendelson (1986), Brennan and Subrahmanyam (1996), Jacoby et al. (2000) implies a direct link between liquidity and corporate cost of capital. Those studies present a model showing that liquidity, marketability or transaction costs influence investors’ portfolio decisions. There is a large body of research that supports the view that the liquidity of securities affects their expected returns. Amihud and Mendelson (1989) conduct cross-sectional analyses of US stock returns and show that risk-adjusted returns as decreasing with respect to liquidity, as measured by the bid-ask spread. Brennan et al. (1998) investigate the relation between expected returns and several firm characteristics including market liquidity, as measured by trading volume. They find a significant negative relation between returns and trading volume for both NYSE and

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NASDAQ stocks, thus linking expected returns and liquidity. Datar et al. (1988) use turnover rate as a measure of liquidity, and provide evidence for a negative correlation between liquidity and stock returns. This proposition that a negative correlation between liquidity and stock returns has been empirically supported in various studies on mature capital markets. Since rational investors require a higher risk premium for holding illiquid securities, cross-sectional risk-adjusted returns are lower for liquid stocks. It is worth to emphasize that the notion of liquidity of an overall equity market is quite different from the notion of liquidity for individual assets. The liquidity of a country’s equity market is largely determined by macroeconomic factors that are systematic to the economy. Chordia, Roll and Subrahmanyam (2001) shows in full detail that how aggregate market liquidity behaves over time. In contrast with previous studies of liquidity spanning short time period and focus on the individual security, in the research they study aggregate market spreads, depths, and trading activity for U.S. equities over an extended time sample. They find that short-term interest rates and the term spread significantly affect market liquidity as well as trading activity. A growing number of studies in the monetary economics literature follow an event-study approach, in that they are mainly concerned with measuring the financial markets’ reaction to monetary policy action. A typical example is the early work of Cook and Hahn (1989), who examined the Treasury securities market’s reaction to monetary policy actions. More recently, instead of simply looking at observed changes in the target rate, Roley and Sellon (1998) used federal funds futures rates to estimate the element of surprise in each policy decision. Other studies undertaken in the finance literature assess the relationship between news and market volatility. Bomfim (2003) presents new statistical evidence that US stock prices do respond reliably to macroeconomic news conveyed by monetary policy decisions regarding the target federal funds rate. The paper shows that the element of surprise in monetary policy decisions tneds to boost stock market volatility significantly in the short run. Chen, Mohan and Steiner (1999) examines the effect of discount rate changes on stock returns, volatility, and trading volume. The changes in unexpected discount rate contribute to higher market volatility although the volatility

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is short-lived. Our paper takes the issue that the potentially significant effect of monetary policy on the informational efficiency of asset price into closer consideration by the market microstructure model. Most relative to our work is Diamond and Verrecchia (1987) among microstructure literature. Their model takes into account the effects of short-sale constraints on the speed of adjustment to private information of asset prices. Imposing a cost on short-selling obviously makes it less attractive. Constraints eliminate some informative trades and reduces the adjustment speed of prices to private information. Contrary to the exogenous probability that trader is informed and wants to short sell in their model, the borrowing probability of informed traders is endogenously determined in our model.

3. A General Framework To explore the relationship between monetary policy shock which is called the liquidity shock and the behavior of stock prices, such as market liquidity and price volatility, we use standard workhorses and follows Kyle (1985) and Glosten and Milgrom (1985) for modeling the market structure. Based on market microstructure model, we can explicitly discuss with the depth of market and bid-ask spread which measure market liquidity and informational efficiency.

3.1 The Model There are three time points t = 0, 1, 2. All agents are born at t = 0. Assume that the economy has one risky asset paying off v˜, v˜ ∈ {H, L} The number of shares is normalized to 1 and held by a continuum of investors. All investors have the same share-holdings, and the total measure of investors is 1. These investors are risk neutral. They can also invest in a risk-free asset. The return on the risk-free asset is zero.

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The investors suffer liquidity shocks at t = 1 that force them to sell their assets, and these shocks are correlated across investors. That is, these shocks are aggregate uncertainty, not idiosyncratic shocks. In particular, there is a total probability of 1 − h that investors are subject to such a exogenous shock, since h is the probability of no liquidity shocks. As viewed from the ex post time point t = 1, if liquidity shocks realized, all of investors encountered with the problem of liquidity shortage, they have to deal with their security holding for raising funds.

Table 1: Timing of Events Date Date 0

Stage 1

Events Informed investors decide to pay information costs c I ,and buy α shares; passive investors buy 1 − α shares

Date 1

2

Informed investors receive private information of dividend payoff

3

All investors experience liquidity shocks with probability 1 − h

4

Informed investors who have received good signals and suffered liquidity shocks will decide whether to buy stock via borrowing or to sell out holdings; Informed investors receiving bad signals will sell out holdings directly; Passive investors will sell stocks if they suffer from liquidity shocks; otherwise they will continue to hold stocks.

Date 2

5

Market maker receives orders from investors and set stock price P 1

6

Dividend payoff realized

Investors can decide whether to pay information costs c I at t = 0. If paying the costs, they can acquire the exact information how much dividends the risky asset will pay at the interim (t = 1) date. Ex ante the probability of receiving fundamental information the dividends payoff will be H or L is 1/2 respectively. Investors who pay the information costs will become the informed investors, while the ones not paying will be the passive investors. Investors who have experienced a sudden need for liquidity will be lack of funds to hold securities, and these investors are able to engage in borrowing for investment purposes. The risky asset can be traded in a secondary market at t = 1. The structure of the market is 6

like Kyle(1985). Investors submit their orders to a market maker who can observe the total net order flow y. The price of the security is then set by the market maker according to its expected value, conditional on the order flow: P 1 = E(˜ v | y). Thus, risk neutral market maker earns on average zero profits. All mentioned before are common knowledge.

PSfrag replacements Buy No Liquidity Shock h

Buy

1−h

H Informed Investor

Borrowing q 1−q

Liquidity Shock

1/2

Non-Borrowing

Sell

1/2 α

L

1−α

Sell

No Liquidity Shock

No-Trade

Passive Investor h

1−h Liquidity Shock Sell Figure 1: Tree Diagram

Initially, the informed investors owns a stock α, and passive investors own the remaining 1 − α shares. Table I summarizes these events and Figure 1 depicts it abviously by way of tree diagram.

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3.2 Analysis The analysis proceeds by solving the model backward. Initially, informed investors’ shareholdings are taken exogenously in order to solve the subgame starting with a given allocation of shares. Later, the equilibrium size of informed investors’ investment is derived endogenously. The portfolio allocation at the initial stage 1 is taken as given. Suppose that α are owned by informed investors and 1 − α shares are owned by passive investors. If passive investors were subject to a liquidity shock, they sell a total of 1 − α shares to the market maker. In contrast to the passive investors’ actions, informed investors can decide whether to borrow money and buy stocks when they suffer from liquidity shocks, since they have the private information of dividend payoff. Informed investors are homogeneous, having the same information sets and borrowing costs. Once informed investors have engaged in borrowing in order to make their investments, they have to pay the borrowing costs c˜B . c˜B are informed investors’ private information, the other players only know that c˜B is an uniform distribution, and c˜B ∈ [cL , cH ]. Then informed investors will borrow if and only if her costs are below a critical level cˆB . The market maker and all other investors will perceive informed investors’ strategy as random because they do not know her costs. Without loss of generality assume that informed investors’ strategic decision makings are as follows 2 : (1)With probability q: Borrow money and buy shares. In this case informed investors buy a quantity xB > 0 and q = P rob(˜ cB ≤ cˆB ), (2)With probability 1 − q: Not borrow and sell initial holdings. In this case informed investors buy a quantity xS < 0. 2

The informed traders will borrow and buy if and only if their borrowing costs are below the critical

level cˆB . Then the market maker and all other investors will perceive the strategy of informed traders as random because they do not know their costs. It is easy to see that there is a Bayesian equilibrium in pure strategies.

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Figure 1 illustrates five different combination resulting from simultaneous trading decision of informed investors and passive investors: (1) The news of fundamental value is good and liquidity shocks does not occur. Because the liquidity shock is aggregate shock, informed trader must continue to buy and passive investors do nothing. The probability of this event is h/2; (2) The news of fundamental value is good and liquidity shocks does occur. If informed investors decide to borrow, then informed trader will buy stock and passive investors have to sell. The probability of this event is (1 − h)q/2; (3) The news of fundamental value is good and liquidity shocks does occur. But informed traders decide not to borrow, then both informed traders and passive investors will sell. The probability of this event is (1 − h)(1 − q)/2; (4) The news of fundamental value is bad and liquidity shocks does not occur. Informed investors will sell and passive investors do nothing. The probability of this event is h/2; (5) The news of fundamental value is bad and liquidity shocks does occur. Then informed investors and passive investors will all sell. The probability of this event is (1 − h)/2. Table 2 gives the possible combinations of transactions and their probabilities as well as the prices set by the market maker. In order to be able to gain from trading, informed investors must confound the information contained in the order flow with the liquidity trading from passive investors. This requires that the market maker cannot distinguish between the case where informed traders buy and passive investors sell, and the case where informed traders sell and passive investors are not subject to a liquidity shock. This gives the following conditions 3 :

xS + 0 = xB − (1 − α) ⇐⇒ xB − xS = (1 − α)

(1)

Hence, the order flow has three possible realizations: x B , xB − (1 − α) and xB − 2(1 − α) If the order flow is xB , then the market maker can infer the value of fundamental perfectly, and the stock price is fully revealing. If the order flow is x B − (1 − α) and xB − 2(1 − α), no information is revealed to the market maker, and the price is uninformative. 3

There are no short sales constraints, hence it is not required that xS + α ≥ 0.

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Table 2: Order Flows and Market Prices Order Flow

Transactions

Probability

Value

Price

xB

Informed traders buy xB

h 2

H

1

(1−h)q 2

H

(1−h)q (1−h)q+h

h 2

L

(1−h)q (1−h)q+h

(1−h)(1−q) 2

H

(1−h)(1−q) (1−h)(1−q)+(1−h)

(1−h) 2

L

(1−h)(1−q) (1−h)(1−q)+(1−h)

(1)

Passive investors sell 0 (2)

xB − (1 − α)

Informed traders buy xB Passeive investors sell (1 − α)

(3)

xB − (1 − α)

Informed traders sell xB − (1 − α) Passive investors sell 0

(4)

xB − 2(1 − α)

Informed traders sell xB − (1 − α) Passive investors sell (1 − α)

(5)

xB − 2(1 − α)

Informed traders sell xB − (1 − α) Passive investors sell (1 − α)

Note: 1. Order flow (1) correspond to investors not facing with liquidity shocks, and informed investors receiving good news. 2. Order flow (2) correspond to investors encountering with liquidity shocks, and informed investors buying stocks through borrowing. 3. Order flow (3) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. 4. Order flow (4) correspond to investors encountering with liquidity shocks, and informed investors receiving good news but not borrowing and buying stocks. 5. Order flow (5) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. Due to the existence of borrowing constraints, borrowing strategy of informed investors create another source of noise trading. The trading combination (4) and (5) in Table 2 can not be discriminated. This viewpoint is what our model be different from other papers. That is to say if without borrowing constraints, then q will equal to 1, and the forth trading combination in Table 2 will not occur, the fifth trading combination will be fully revealing. Besides, informed investors can not always take advantage of passive investors, they will lose when the forth trading combination emerges. 10

For solving the probability q of borrowing and buying strategy, we have to know the expected payoff of alternative strategy. Letting these expected payoff equalized, we can derive the critical value of borrowing cost cˆB , such that the value of q can be calculated using cˆB . Denote by E(P | B) the expected price per share at which informed investors buy and by E(P | S) the expected price per share at which they sell. If informed investors want to buy stocks, the price set by market maker conditionally on the actions of informed investors will possibly be P 1 = 1, which coming from the first trading combination in Table 2; or P 1 =

(1−h)q (1−h)q+h

coming from the second

combination. Their probability are h/[h + (1 − h)q] and (1 − h)q/[h + (1 − h)q] respectively. Similarly, conditional on selling strategy of informed investors, expected price can be calculated from the pricing schedule too. Hence, E(P | B) =

h[h + (1 − h)q] + [(1 − h)q]2 [h + (1 − h)q]2

E(P | S) =

(1 − h)(1 − q) h(1 − h)q + [1 + (1 − h)(1 − q)][h + (1 − h)q] 1 + (1 − h)(1 − q)

and,

These expressions can now be used to calculated the expected payoff from borrowing and buying as (2)

xB {1 − E(P | B) − cB } + α

Similarly, the expected payoff from non-borrowing and selling can be calculated as −(1 − α − xB ){1 − E(P | S)} + α − (1 − α − xB )

(3)

Inserting E(P | B) and E(P | S), equation (2) and (3) can be further expressed as xB {1 −

h[h + (1 − h)q] + [(1 − h)q]2 − cB } + α [h + (1 − h)q]2

(4)

and, −(1−α−xB ){1−

h(1 − h)q (1 − h)(1 − q) − }+α−(1−α−xB ) [1 + (1 − h)(1 − q)][h + (1 − h)q] 1 + (1 − h)(1 − q) 11

(5)

Then, in equilibrium, the randomizing probability q must equal the market maker’s beliefs about the distribution of v˜ ∈ {H, L}. The market maker set prices so as to make informed investors indifferent between borrowing-buying and nonborrowing-selling. Given α determined at date 0, one can solve the critical value cˆB which makes equation (4) and (5) equalized. As a result, for any given α and xB , the optimal value of q chosen by informed investors can be implemented with a continuum of combinations of α and x B .

3.3 Equilibrium The important observation for determining α endogenously is that informed investors makes profits from two sources: First, from trading because of asymmetric information and, second, from the purchase of the initial stake at price P 0 relative to the expected price for which informed investors would liquidate their initial stake. The expected price per share informed investors receive when liquidating their holding is the intrinsic ex ante value, namely 1/2. Passive investors also receive this price per share if they do not have to liquidate their stock holdings in the intermediate period and hold them until the final period. This is intuitive as the initial market in shares will only clear if the shares are fairly priced from the point of view of passive investors who are the marginal investors. Hence, passive investors’ loss can be calculated as follows:

(1 − h)q e (1 − h)(1 − q) e 1−h e (E (P | S) − 1) + (E (P | S) − 1) + E (P | S) 2 2 2

(6)

E e is the expected price per share at which passive investors sell. Because the initial share price is determined by the valuation of passive investors, P 0 is lower 12

than 1/2 for this reason:

P0 =

1 1−h − E e (P | S)(1 − h) + 2 2

If passive investors expected to make a loss on purchasing shares, they would not buy them, and the market would not clear at P0 . Conversely, if passive investors expected to profit from buying shares, there would be excess demand for shares. Informed investors buy α shares at P 0 and expects to liquidate them at H and L with probability 1/2 respectively. Then, total benefits from the initial purchase are 1 α( − P0 − cI ) 2

(7)

Informed investors make profits from trading against passive investors because of asymmetric information. Define the probability of each event as follows: the probability of receiving good news and without liquidity shock is P rob(H, Buy) =

h 2;

the probability of receiving good news

but facing with liquidity shock and then buying and borrowing stocks is P rob(H, Borrow) = (1−h)q ; 2

the probability of receiving good news but facing with liquidity shock and having to sell

is P rob(H, Sell) =

(1−h)(1−q) ; 2

the probability of receiving bad news and selling stocks holding

is P rob(L, Sell) = 21 . The expected net trading profits are therefore xB {[1 − E(P | B)][P rob(H, Buy) + P rob(H, Borrow)] − c B P rob(H, Borrow)} −(1 − α − xB )[1 − E(P | S)]P rob(H, Sell) + (1 − α − x B )E(P | S)P rob(L, Sell) (8) The first item in equation (8) is the gain from receiving good news and buying stocks; the second item is the trading loss from receiving good news but having to sell; the last is the gain from receiving bad news and selling.

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Adding the trading gains from (7) to the profits on initial purchases from equation (8) gives 1 α( − P0 − cI ) + xB {[1 − E(P | B)][P rob(H, Buy) + P rob(H, Borrow)] 2 −cB P rob(H, Borrow)} − (1 − α − xB )[1 − E(P | S)]P rob(H, Sell)

(9)

+(1 − α − xB )E(P | S)P rob(L, Sell)

According to the above analysis, we can find out two equations and derive the solution. One is the equation arising from letting equation (4) and (5) equalized, denoted by F (q, α, h, x B ) = 0; The other is the equation of letting equation (9) equal to 0, denoted by G(q, α, h, x B ) = 0. Solving the simultaneous equations, the equilibrium solution will be q ∗ and α∗ . q ∗ and α∗ are all the functions of h and xB . We summarize the results in the following proposition. Proposition 1. There exists a set of parameter values for which α, q ∈ [0, 1] ,and 1.F (q, α, h, xB ), G(q, α, h, xB ) ∈ C 1 2.F (q0 , α0 , h0 , xB0 ) = G(q0 , α0 , h0 , xB0 ) = 0 3.∆ 6= 0, Fq Fα ∆ = Gq Gα

=⇒ there exists an equilibrium solution (q ∗ , α∗ ), 0 < α∗ < 1, 0 < q ∗ < 1, such that F (q, α, h, xB ) = G(q, α, h, xB ) = 0

By implicit function theorem, if Jacobian matrix is not equal to 0, that is, ∆ 6= 0, then one can prove theorem 1 hold.

4. Special Case The possibility of multiple equilibria resulting from highly degree polynomial functions F and G appears to rule out a fully analytic solution. In order to understand the characteristics and intuition of equilibrium outcomes, we have to select one special case, that is symmetric solution. 14

4.1 Symmetric Solution Assume for simplicity that informed investors chooses symmetric trading quantities x B = −xS = (1 − α)/2. This assumption does not affect the results because informed investors have two variables that determine her trading intensity: the amount she trades x B (orxS ) and the randomizing probability q. These two variables are constrained by only one equation, namely the indifference condition for randomization. Hence, one variable can be arbitrarily chosen. For notational convenience, define u ≡ (1 − α)/2. Table 3 gives the possible combinations of transactions and their probabilities as well as the prices set by the market maker. Equation (2) and (3) can be rewritten as follows:

u{1 − E(P | B) − cB } + α

and

−u{2 − E(P | S)} + α

Letting the foregoing equations equalized, we can derive randomizing probability q independently with α 4 . We shall go backwark to date 0 and solve the problem. Hence, equation (9) can be expressed as 1 α( − P0 − cI ) + u{[1 − E(P | B)][P rob(H, Buy) + P rob(H, Borrow)] 2 −cB P rob(H, Borrow) − [1 − E(P | S)]P rob(H, Sell) + E(P | S)P rob(L, Sell)} (90 )

As a result of equation (10) equal to 0, the size of the initial stake α can be solved: α=

Ω Ω − 2[(E e (P | S)(1 − h) − (1/2)(1 − h) − cI ]

(10)

in which Ω = [1 − E(P | B)]((1/2)h + (1/2)(1 − h)q) − cˆB (1/2)(1 − h)q − (1/4)(1 − q) + E(P | S)(1/2)(1 + (1/2)(1 − q)), this item corresponds to profit of informed investors in period 4

For a wide range of real parameter values there are four numeric solutions — one solution with positive

prices, one pair of solutions that are complex conjugates, and a forth solution that includes a negative prices. In such cases, the latter three solutions are inadmissible, leaving a unique equilibrium with positive prices.

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1; 2[(E e (P | S)(1−h)−(1/2)(1−h)−cI ] represents profit in period 0. Equation (10) shows that if Ω increases, that is profit in period 1 become higher, then the vale of α will increase either. The intuition of this result is that investors’ expectation of gain from asymmetric information become more higher, more investors will choose to be informed. Holding other conditions unchanged, if information cost cI increases, resulting in the profit of informed investors decreasing, then the value of α will decrease.

Table 3:Order Flows and Market Prices Order Flow

Transactions

Probability

Value

Price

u

Informed traders buy u

h 2

H

1

(1−h)q 2

H

(1−h)q (1−h)q+h

h 2

L

(1−h)q (1−h)q+h

(1−h)(1−q) 2

H

(1−h)(1−q) (1−h)(1−q)+(1−h)

(1−h) 2

L

(1−h)(1−q) (1−h)(1−q)+(1−h)

(1)

Passive investors sell 0 (2)

−u

Informed traders buy u Passeive investors sell 2u

(3)

−u

Informed traders sell u Passive investors sell 0

(4)

−3u

Informed traders sell u Passive investors sell 2u

(5)

−3u

Informed traders sell u Passive investors sell 2u

Note: 1. Order flow (1) correspond to investors not facing with liquidity shocks, and informed investors receiving good news. 2. Order flow (2) correspond to investors encountering with liquidity shocks, and informed investors buying stocks through borrowing. 3. Order flow (3) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. 4. Order flow (4) correspond to investors encountering with liquidity shocks, and informed investors receiving good news but not borrowing and buying stocks. 5. Order flow (5) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. 16

Investigating equation (90 ) and equation (10) we will know that: First, if profit in period 0, α( 12 − P0 − cI ), is higher, then α will increase. This is very intuitive, as it should become more attractive to be informed investors. The larger the initial stake α, the more important to the gain from trade is the payoff on the portfolio holding. A higher α reduces the liquidity of the market in period 1 because a smaller amount of shares held by passive investors implies that fewer shares are traded, this is the liquidity effect from trading. Second, if h is higher, there will be three different effects. (1) P rob(H, Buy)+P rob(H, Borrow) in equation (90 ) becomes larger, then it is more easily for informed traders to make profit from passive investors providing camouflage which enables informed traders to confuse market maker; (2) the possibility of fully revealing will increase, then 1 − E(P | B) becomes smaller and profit decreases; (3) the possibility of giving up trading for good news which is arising from lack of capital will be lower, then [1 − E(P | S)]P rob(H, Sell) + E(P | S)P rob(L, Sell) becomes larger and expected profit increases. If the net effect is larger as h increases, then expected profit will increase in period 1, this is the liquidity effect from money.

4.2 Numerical Results In this section we resort to numerical calibrations in order to compare equilibrium outcomes across alternative tightness degree of liquidity in which the probability of monetary liquidity shocks move from high to low. We seek to understand under what circumstances one would expect to see the relationship between monetary liquidity and liquidity of assets will respond to change. We have computed equilibrium for the grid of parameter values below: 1.[cL , cH ] = [0.1, 0.4]; 2.{h} ∈ {0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6}, with{1 − h} ∈ {0.9, 0.85, 0.8, 0.75, 0.7, 0.65, 0.6, 0.55, 0.5, 0.45, 0.4}.

17

A. The relation between monetary liquidity and market depth In Table 4, cˆB is the critical value, if borrowing cost is smaller than cˆB , informed investors will decide to buy stocks via borrowing. q is the probability informed investors borrow money and buy stocks. q =

cˆB −CL CH −CL .

α is the proportion of informed investors. λ can be defined by the difference

between the price at which informed investors expects to sell and the price at which she expect to buy, relative to the spread between prices if they were fully revealing, H − L(= 1):

λ=

E(P1 | B) − E(P1 | S) = E(P1 | B) − E(P1 | S) H −L

In an infinitely deep market E(P1 | B) = E(P1 | S), and λ = 0; whereas if trades always move prices to the fully revealing level, then λ = 1. Hence, define market depth as 1 − λ, that is, as the probability that market prices are not fully revealing. This relation is represented in parallel to Kyle (1985) in which 1/λ measures the depth of the market. Kyle’s measure of market liquidity is proportional to a ratio of the amount of noise trading to the amount of private information the informed trader is expected to have. In this sense, it captures the intuition that the more the noise trading is, the deeper the market liquidity is. From investigating Table 4, we see that:

(1 ) The higher the probability of liquidity shocks hits the economy, the more likely the informed traders will borrow. That is, when 1 − h becomes larger (h become smaller), the value of q continues to grow upwards. We might explain this result as if monetary liquidity become more tighter, the willingness to borrow will become stronger at the same time. (2) As sketched in Figure 1, h and q represent the probability of buying stocks. From observing the definition of λ, h and q are two channels by which trading activity reflects the direction of moving prices to revealing information. When the exogenous liquidity shock occurs, informed investors will decide whether to borrow-and-buy or not depending on information revealing to what extent. Table 4 indicates while the value of h rises, the value of q falls. h × q reach the 18

highest point at h = 0.4 given the interval of [0.2, 0.6]. That is to say, price revealing (λ) is at the least level. The explanation of this result is straight that informed investors become more willing to trade when the degree of information revelation is lower.

Table 4: Simulation results for different exogenous parameter values h

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

1−h

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

cˆB

0.399

0.374

0.351

0.331

0.312

0.292

0.272

0.249

0.223

0.192

0.155

q

0.999

0.912

0.837

0.769

0.705

0.640

0.572

0.496

0.410

0.308

0.182

α

0.266

0.267

0.134

0.210

0.251

0.274

0.283

0.281

0.262

0.216

0.103

λ

0.520

0.512

0.571

0.493

0.439

0.408

0.400

0.415

0.454

0.517

0.608

1−λ

0.480

0.488

0.429

0.507

0.561

0.592

0.600

0.585

0.546

0.483

0.392

Note: 1.h is the probability of no liquidity shocks, 1 − h is the probability of liquidity shocks. 2.ˆ cB is the critical value of borrowing cost that informed investors decide to whether borrowing and buying. 3.q is the probability of informed investors receiving good news and buying stocks through borrowing. 4.α is the proportion of informed investors. 5.1 − λ is the measurement of market liquidity. 6.The uniform distribution of borrowing cost is [c L , cH ] = [0.1, 0.4]. (3) The patterns of market depth (1−λ) and the proportion of informed traders (α) will reverse when the probability of liquidity shocks continues rising. In Table 4 as the probability of liquidity shocks increases from 0.4 to 0.6, the market deepens from 0.392 to 0.6; while the probability of liquidity shocks increases from 0.6 to 0.8, the market depth declines from 0.6 to 0.429. Similarly, as the probability of liquidity shocks increases from 0.4 to 0.6, the proportion of informed traders increases from 0.103 to 0.283; while the probability of liquidity shocks increase from 0.6 to 0.8, the proportion of informed traders declines from 0.283 to 0.134. At first glance, in table 4 the highest market depth takes place at h = 0.4. Meanwhile the 19

proportion of informed trader is also the largest. For the reason that more deeper the market is, lesser information will be revealed, the uninformed traders provide camouflage which enables the insider to make profits more easily. (4) The results in Table 4 appear to show that there are certain kinds of asymmetries. Dividing the range of h into two intervals [0.2, 0.4] and [0.4,0.6], the changes of level in the two intervals which 1 − λ and α belong to are not symmetric. The key driving force of asymmetries have to do with borrowing constraints. Comparing with benchmark model in which no borrowing constraints are binding, that is q = 1, 1 − α will be equal to 2h − 2h 2 . Obviously, its shape of the above function is symmetric around the point h = 0.5. It gives evidence for this result that imposing borrowing costs matters. If no borrowing constraints bind, there will be not any asymmetries. B. Monetary liquidity and asset price distribution Financial economists have long considered the effects of release of economic data on the volatility of asset markets by examining what happens to market volatility on news arrival dates. This paper looks at the question of how monetary policy can be linked to stock market volatility. Especially by constructing market microstructure model, this paper explicitly allows to analyze information revealing as well as asset price distribution. We model anticipated monetary policy as a liquidity shock to an individual’s initial capital. Market participants perceive the probability of liquidity shock, and price schedule can be determined ex ante given the structure of model. Table A in Appendix illustrates the calculation of price distribution. From investigating Table A, we can see that: (1) if order flow is −u, the probability of event informed traders buy and passive investors sell becomes higher and price will be adjusted upward as probability of liquidity shock increases, since we can expect the greater probability that fundamental value is H. (2)if order flow is −3u, the probability of event informed traders sell and passive investors sell becomes higher and price will be adjusted downward as probability of liquidity shock increases, since the greater probability that fundamental value is L will be expected. (3) The greater probability that large price changes are downward suggests, roughly, that higher probability of liquidity shock makes the distribution of the change in price more skewed to the left. The final implication of our analysis is that how the asset volatility response to liquidity shock. 20

Volatility are usually represented by variance of return distribution in the literature. In Table 5, we report the mean and variance of price distribution set up by market makers. The lowest variance takes place at h = 0.4 where prices are not revealing too much, it turns out that the distribution of price is more tigher. On the contrary the variances will be higher on the two opposite sides, where prices are more revealing and then the distributions are dispersed further. Moreover, it is consistent with the intuition that the shapes of λ and variance are similar in Table 4 and 5. This result implies that the bid-ask spreads are more wider, the asset prices are more volatile.

Table 5: Mean and Variance h

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

1−h

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

M ean

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

V ariance

0.204

0.153

0.117

0.093

0.077

0.069

0.068

0.073

0.085

0.103

0.128

V ariance∗

0.205

0.186

0.17

0.156

0.145

0.136

0.13

0.126

0.125

0.126

0.13

Note: 1.h is the probability of no liquidity shocks, 1 − h is the probability of liquidity shocks. 2.Mean and Variance are calculated from Appendix A. 3.V ariance∗ are calculated under no borrowing constraints.

In Table 5 V ariance∗ represents the variance of price distribution set up by market makers if there are no borrowing constraints. That is q = 1 in Table 3. We can find out mean will be same whether with borrowing constraints or not. But variance under borrowing constraints binding will be smaller than variance under no borrowing constraints. The main reason is that borrowing constraints make information hidden by informed investors when they cannot borrow enough funds to buy stocks, and then price distributions are more concentrated.

21

5. Conclusion This paper studies how monetary liquidity interacts with market liquidity. The main findings are:

1. The informed investor realizes capital gains from her information advantages. However, such gains are just large enough to cover information costs and/or borrowing costs. In equilibrium, no agent makes excess returns and there is no opportunity for arbitrage. 2. The informed trader takes the possibility of liquidity shock into account in her decision on investment and borrowing. Through this channel, the anticipated monetary liquidity shock has an impact on the liquidity of assets in the market and, thus, the market depth. 3. Both market depth and asset volatility are asymmetric in probabilities of liquidity shock. More specifically, in our numerical examples, the distribution of bid-ask spreads against the probability of aggregate shock is skewed toward the right. The asymmetry is caused by the presence of borrowing costs. This theoretical finding is similar in spirit to the empirical study by Chordia, Roll and Subrahmanyam (2001). They found out both quoted and effective spread increases dramatically in the down markets, while decrease only marginally in up markets. This paper analyzes the asymmetric impacts on spreads, depth and volatility from the liquidity shocks. In modeling uninformed trading it is quite often — in order to avoid unrealistic fully revealing Rational Expectation Equilibrium (REE) and profitable trading for the informed trader — to postulate some portion of the market demand for securities arising from unmodeled “noise traders” 5 . Suffering from liquidity shocks the noisy trader has to sell assets to raise funds. This is called “liquidity trading” in literature, distinguished from “informed trading.” An important issue remains largely unresolved — why noise traders, not informed traders, encounter with liquidity shocks. Our goal in this paper is to rectify this problem by modeling both noise traders and informed traders facing liquidity shocks simultaneously. These shocks are aggregate uncertainty about liquidity shortage. In the literature some papers focus on this issue, such as: Allen and Gale (1994) show that aggregate uncertainty about the proportion of early consumers generates price volatility. 5

For example: Grossman and Stiglitz (1980) and Leland (1992).

22

Bhattacharya and Nicodano (2001) compare equilibrium trading outcomes with and without participation by an informed trader. Noise trading arises from aggregate uncertainty regarding other agents’ intertemporal consumption preference. Contrary to these papers, this paper follows Kyle (1985) and uses microstructure model for our purposes. One limitation of this model is that we focus on the liquidity shock arising from anticipated monetary policy. Therefore we cannot analyze how monetary liquidity (i.e., the amount of funds in the market) changes and how sensitive equilibrium prices are to the change in monetary liquidity. These issues are beyond the scope of this paper and left for further research.

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Appendix A Table A: The distribution of asset prices h: the probability of no liquidity shocks 0.1

h

0.15

0.2

0.25

0.3

OrderF low

Prob.

Price

Prob.

Price

Prob.

Price

Prob.

Price

Prob.

Price

(1)u

0.05

1

0.075

1

0.1

1

0.125

1

0.15

1

(2) − u

0.448

0.900

0.388

0.838

0.3346

0.770

0.2885

0.698

0.2467

0.622

(3) − u

0.05

0.900

0.075

0.838

0.1

0.770

0.125

0.698

0.15

0.622

(4) − 3u

0.002

0.003

0.037

0.080

0.0654

0.140

0.0866

0.187

0.1033

0.227

(5) − 3u

0.450

0.003

0.425

0.080

0.4

0.140

0.375

0.187

0.35

0.227

Note: 1. Order flow (1) correspond to investors not facing with liquidity shocks, and informed investors receiving good news. 2. Order flow (2) correspond to investors encountering with liquidity shocks, and informed investors buying stocks through borrowing. 3. Order flow (3) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. 4. Order flow (4) correspond to investors encountering with liquidity shocks, and informed investors receiving good news but not borrowing and buying stocks. 5. Order flow (5) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news.

24

Table: The distribution of asset prices–continued h:the probability of no liquidity shocks h

0.35

OrderF low

Prob.

Price

Prob.

Price

Prob.

Price

(1)u

0.175

1

0.2

1

0.225

1

(2) − u

0.2081

0.543

0.1715

0.462

0.1365

0.377

(3) − u

0.175

0.543

0.2

0.462

0.225

0.377

(4) − 3u

0.1169

0.265

0.1285

0.299

0.1385

0.335

(5) − 3u

0.325

0.265

0.3

0.299

0.275

0.335

h

0.4

0.5

0.45

0.55

0.6

OrderF low

Prob.

Price

Prob.

Price

Prob.

Price

(1)u

0.25

1

0.275

1

0.3

1

(2) − u

0.1025

0.291

0.0693

0.201

0.0364

0.109

(3) − u

0.25

0.291

0.275

0.201

0.3

0.109

(4) − 3u

0.1475

0.371

0.1557

0.409

0.1636

0.449

(5) − 3u

0. 25

0.371

0.225

0.409

0.2

0.449

Note: 1. Order flow (1) correspond to investors not facing with liquidity shocks, and informed investors receiving good news. 2. Order flow (2) correspond to investors encountering with liquidity shocks, and informed investors buying stocks through borrowing. 3. Order flow (3) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news. 4. Order flow (4) correspond to investors encountering with liquidity shocks, and informed investors receiving good news but not borrowing and buying stocks. 5. Order flow (5) correspond to investors encountering with liquidity shocks, and informed investors receiving bad news.

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17. Leland, H. (1992), “Insider Trading: Should It Be Prohibited?” Journal of Political Economy 100, 859-887. 18. Longstaff, F.A. (2000), “The Term Structure of Very Short-Term Rates:New Evidence for the Expectations Hypothesis” Journal of Financial Economics 58, 397-415. 19. Lucas, Robert E. (1990), “Liquidity and Interest Rates” Journal of Economic Theory 50, 237-264. 20. Maug, E. (1998), “Large Shareholders as Monitors: Is There a Trade-Off between Liquidity and Control?”Journal of Finance 53, 65-98. 21. Mishkin, F.S. (1988), “The Information in the Term Structure: Some Further Results” Journal of Applied Econometrics 3, 307-314. 22. Rotemberg, J.J. (1984), “A Monetary Equilibrium Model with Transactions Costs” Journal of Political Economy 92, 40-58.

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