# 2: Theory.

In this section, we first relate the theory of liquidity and asset pricing to the standard theory of asset pricing in frictionless markets. We then show how liquidity is priced in the most basic model of liquidity, where securities have exogenous trading costs and identical, risk-neutral investors have exogenous trading horizons (Section 2.2). We then extend this basic model to take into account clientele effects (Section 2.3), time-varying trading costs and liquidity risk (Section 2.4), uncertain trading horizons (Section 2.5) and endogenous trading horizons (Section 2.6). We also briefly review the sources of illiquidity and consider models of asset pricing with endogenous illiquidity (Sections 2.7-2.8).2.1 Liquidity and standard asset pricing theory

To study how liquidity affects asset pricing, it is useful to place it in the context of standard asset pricing theory. Readers may, however, choose to skip directly to Section 2.2, where we start discussing the actual theories of liquidity and asset pricing.

2.1.1 Background: Standard asset pricing

Standard asset pricing (1) is based on the assumption of frictionless (or, perfectly liquid) markets, where every security can be traded at no cost all of the time, and agents take prices as given. The assumption of frictionless markets is combined with one of the following three concepts: no arbitrage, agent optimality, and equilibrium.

No arbitrage means that one cannot make money in one state of nature without paying money in at least one other state of nature. In a frictionless market, the assumption of no arbitrage is essentially equivalent to the existence of a stochastic discount factor [m.sub.t] such that the price process [p.sub.t] of any security with dividend process [d.sub.t] satisfies

[p.sub.t] = [E.sub.t] ([p.sub.t+1] + [d.sub.t+1]) [m.sub.t+1]/[m.sub.t]. (2.1)

Equation (2.1) is the main building block of standard asset pricing theory. It can also be derived from agent optimality: if an insatiable investor trades in a frictionless market, his optimal portfolio choice problem only has a solution in the absence of arbitrage--otherwise he will make an arbitrarily large profit and consume an arbitrarily large amount. Further, the first-order condition to the investor's problem has the form (2.1). In particular, if the investor's preferences are represented by an additively separable utility function [E.sub.t] [[summation].sub.s] ([c.sub.s]) for a consumption process c, then [m.sub.t] = [u'.sub.t]([c.sub.t]) is the marginal utility of consumption.

Finally, in a competitive equilibrium with complete markets and agents i = 1,..., I with separable utility functions [u.sup.i], (2.1) is satisfied with [m.sub.t] = [u'.sub.t[lambda]]([c.sub.t]), where [u.sub.t[lambda]] = [[summation].sub.i][[lambda].sup.i][u.sup.i.sub.t] is the utility function of the representative investor and [[lambda].sub.i] are the Pareto weights that depend on the agents' endowments.

2.1.2 On the impossibility of frictionless markets

One could argue that, if there were a friction that led to large costs for agents, then there would be an institutional response that would profit by alleviating this friction. According to this view, there cannot be any (important) frictions left in equilibrium.

Alleviating frictions is costly, however, and the institutions which alleviate frictions may be able to earn rents. For instance, setting up a market requires computers, trading systems, clearing operations, risk and operational controls, legal documentation, marketing, information and communication systems, and so on. Hence, if frictions did not affect prices then the institutions that alleviated the frictions would not be compensated for doing so. Therefore, no one would have an incentive to alleviate frictions, and, hence, markets cannot be frictionless.

Grossman and Stiglitz (1980) use a similar argument to rule out informationally efficient markets: market prices cannot fully reveal all relevant information since, if they did, no one would have an incentive to spend resources gathering information in the first place. Hence, investors who collect information must be rewarded through superior investment performance. Therefore, information differences across agents is an equilibrium phenomenon, and this is another source of illiquidity.

There must be an "equilibrium level of disequilibrium," that is, an equilibrium level of illiquidity: the market must be illiquid enough to compensate liquidity providers (or information gatherers), and not so illiquid that it is profitable for new liquidity providers to enter.

2.1.3 Liquidity and asset pricing: The point of departure

If markets are not frictionless, that is, if markets are beset by some form of illiquidity, then the main building blocks of standard asset pricing are shaken. First, the equilibrium aggregation of individual utility functions to a representative investor may not apply. Second, individual investor optimality may not imply that (2.1) holds with [m.sub.t] = [u'.sub.t]([c.sub.t]) at all times and for all securities. This is because an investor need not be "marginal" on a security if trading frictions make it suboptimal to trade it. Indeed, Luttmer (1996, 1999) shows that trading costs can help explain the empirical disconnect between consumption and asset returns. Hence, illiquidity implies that we cannot easily derive the stochastic discount factor from consumption, much less from aggregate consumption. Then, what determines asset pricing?

Some people might argue that the cornerstone of standard asset pricing is the mere existence of a stochastic discount factor, not necessarily its relation to consumption. Indeed, powerful results--such as the theory of derivative pricing--follow from the simple and almost self-evident premise of no arbitrage. It is, however, important to recognize that the standard no-arbitrage pricing theory relies not only on the absence of arbitrage, but also on the assumption of frictionless markets.

To see why the assumption of frictionless markets is crucial, consider the basic principle of standard asset pricing: securities, portfolios, or trading strategies with the same cash flows must have the same price. This simple principle is based on the insight that, if securities with identical cash flows had different prices, then an investor could buy--with no trading costs--the cheaper security and sell--with no trading costs--the more expensive security, and, hence, realize an immediate arbitrage profit at no risk. Another way to see that standard asset pricing implies that securities with the same cash flows must have the same price is to iterate (2.1) to get

[p.sub.t] = [E.sub.t] ([[infinity].summation over s=t+1] [d.sub.s] [m.sub.s]/[m.sub.t]), (2.2)

which shows that the price [p.sub.t] only depends on the pricing kernel and the cash flows d.

With trading costs, however, this principle need not apply. Indeed, with transaction costs, securities with the same cash flows can have different prices without introducing arbitrage opportunities.

Do real-world securities with the same cash flows have the same price? Perhaps surprisingly, the answer is "no," certainly not always. As discussed in Section 3, on-the-run (i.e. newly issued) Treasuries often trade at lower yields than (almost) identical off-the-run Treasuries, and Treasury bills and notes of the same cash flows trade at different prices (Amihud and Mendelson, 1991). Shares that are restricted from trade for two years trade at an average discount of about 30% relative to shares of the same company with identical dividends that can be traded freely (Silber, 1991). Chinese "restricted institutional shares," which can be traded only privately, trade at a discount of about 80% relative to exchange-traded shares in the same company (Chen and Xiong, 2001). Options that cannot be traded over their life trade at large discounts relative to identical tradable options (Brenner et al., 2001). The put-call parity is sometimes violated when it is difficult to sell short, implying that a stock trades at a higher price than a synthetic stock created in the option market (Ofek et al., 2004). Further, in so-called "negative stub value" situations, a security can trade at a lower price than another security, which has strictly lower cash flows (e.g. Lamont and Thaler, 2003).

The existence of securities with identical cash flows and different prices implies that there does not exist a stochastic discount factor m that prices all securities, that is, there does not exist an m such that (2.2) holds for all securities.

Another important difference between standard asset pricing and liquidity asset pricing is that the latter sometimes relaxes the assumption of price-taking behavior. Indeed, if prices are affected by the nature of the trading activity, then agents may take this into account. For instance, if an agent is so large that his trades significantly affect prices, he will take this into account, or if agents trade in a bilateral over-the-counter market, then prices are privately negotiated. Further, the liquidity literature relaxes the assumptions that all investors have the same information and that all investors are present in the market at all times.

2.1.4 Liquidity and asset pricing: Where it will take us (in this survey)

The prices of securities are determined by the general equilibrium of the economy. Hence, the price of a security is some function of the security's cash flow, the cash flows of other securities, the utility functions of all agents, and the agents' endowments. In an economy with frictions, the price depends additionally on the security's liquidity and the liquidity of all other securities.

One strength of a frictionless economy is that a security's cash flows and the pricing kernel are sufficient statistics for the pricing operation described by Equation (2.1). This means that the pricing kernel summarizes all the needed information contained in utility functions, endowments, correlations with other securities, etc.

In some liquidity models, there still exists a pricing kernel m such that (2.1) holds. In this case, illiquidity affects m, but the pricing of securities can still be summarized using a pricing kernel. This is the case if certain agents can trade all securities all of the time without costs. For instance, in the models of demand pressure and inventory risk that follow Grossman and Miller (1988), competitive market makers can trade all securities at no cost (whereas customers can only trade when they arrive in the market). Garleanu et al. (2004) show explicitly how m depends on demand pressure in a multi-asset model. The empirical analysis of Pastor and Stambaugh (2003) is (implicitly) based on an assumption that there exists an m that depends on a measure of aggregate liquidity (but this does not rely on an explicit theory).

In other models of liquidity, however, there is no pricing kernel such that (2.1) applies. For instance, in transaction-cost-based models, securities with the same dividend streams have different prices if they have different transaction costs. Hence, a security's transaction cost not only affects the nature of the equilibrium, it is a fundamental attribute of the security.

When there does not exist a pricing kernel, then the computation of equilibrium asset prices becomes more difficult. Indeed, the general equilibrium prices with illiquidity may depend on the fundamental parameters in a complicated way that does not have a closed-form expression. Nevertheless, we can derive explicit prices under certain special assumptions such as risk neutrality, some structure of trading horizons, partial equilibrium, normally distributed dividends, and so on. While the difficulty of general equilibrium with frictions often forces us to use such special assumptions to get closed-form results, we can still gain important insights into the main principles of how liquidity affects asset prices.

2.2 Basic model of liquidity and asset prices

It is important to understand the effect of liquidity on asset prices in the most basic model. Hence, we consider first a simple model in which securities are illiquid due to exogenous trading costs, and investors are risk neutral and have exogenous trading horizons. This model is a special case of Amihud and Mendelson (1986a).

The basic idea is as follows. A risk-neutral investor who buys a security and expects to pay transaction costs when selling it, will take into account this when valuing the security. She knows that the buyer will also do that, and so on. Consequently, the investor will have to consider, in her valuation, the entire future stream of transaction costs that will be paid on the security. Then, the price discount due to illiquidity is the present value of the expected stream of transaction costs through its lifetime.

Translating this into the required return on the security which is costly to trade, we obtain that the required return is the return that would be required on a similar security which is perfectly liquid, plus the expected trading cost per period, i.e., the product of the probability of trading by the transaction cost.

We consider a simple overlapping generations (OLG) economy in discrete time t [member of] {..., -2, -1,0,1,2, ...}. There is a perfectly liquid riskless security and agents can borrow and lend at an exogenous risk-free real return of [R.sup.f] = 1 + [r.sup.f]. Further, there are I illiquid securities indexed by i = 1,..., I with a total of [S.sup.i] shares of security i. At time t, security i pays a dividend of [d.sup.i.sub.t], has an ex-dividend share price of [P.sup.i.sub.t], and has an illiquidity cost of [C.sup.i.sub.t] = [C.sup.i]. The illiquidity cost [C.sup.i] is modeled simply as the per-share cost of selling security i. Hence, agents can buy at [P.sup.i.sub.t] but must sell at [P.sup.i.sub.t] - [C.sup.i]. We assume (for now) that [d.sup.i.sub.t] are independent and identically distributed (i.i.d.) with mean [[bar.d].sup.i].

Agents are risk neutral and have a discount rate of 1 / [R.sup.f], and the market prices are determined in a competitive equilibrium. In a competitive equilibrium, agents choose consumption and portfolios so as to maximize their expected utility taking prices as given, and prices are determined such that markets clear. We are looking for a stationary equilibrium, that is, an equilibrium with constant prices [P.sup.i].

To fix ideas, suppose first that agents live for two periods and new agents are born every period. If an agent buys a share of security i in the first period of his life, then he must sell the share in the following period, realizing an expected revenue of [[bar.d].sup.i] + [P.sup.i] - [C.sup.i]. The agent will buy an arbitrary large number if shares of the price is lower than the expected discounted revenue ([[bar.d].sup.i] + [P.sup.i] - [C.sup.i]) / [R.sup.f], or short an arbitrarily large amount if the price is higher. Hence, we must have

[P.sup.i] = ([[bar.d].sup.i] + [P.sup.i] - [C.sup.i]) / [R.sup.f], (2.3)

implying that

[P.sup.i] = ([[bar.d].sup.i] - [C.sup.i]) / [r.sup.f]. (2.4)

We see that the price is equal to the present value of all future expected dividends [d.sub.i] minus the present value of all future transaction costs [C.sub.i]. This is intuitive. The investor foresees receiving a dividend and paying a transaction cost next period, and he must sell to another investor who foresees the following dividend and transaction cost, and so on.

Alternatively, we can express this result by looking at the effect of liquidity on the required gross return defined as

E([r.sup.i]) := E([d.sup.i] + [P.sup.i] / [P.sup.i]) - 1 = [[bar.d].sup.i] / [P.sup.i]. (2.5)

We see that the required return is the risk free return increased by the relative transaction cost:

E ([r.sup.i]) = [r.sup.f] + [C.sup.i] / [P.sup.i]. (2.6)

Equivalently, the liquidity-adjusted expected return is the risk free rate,

[[bar.d].sup.i] - [C.sup.i] / [P.sup.i] = [r.sup.f]. (2.7)

We can easily generalize this result to an economy in which an agent lives for more than one period. Specifically, suppose that, in any period, an agent must exit the market with some probability [mu]. This exit event captures the notion of a "liquidity shock" to the agent, for instance, a sudden need for cash. Then, at any time t, the equilibrium price must be the present value of dividends until the random exit time T plus the liquidation value, that is,

[P.sup.i] = [E.sub.t]([T.summation over (s=t+1)] 1 / [([R.sup.f]).sup.s-t] [d.sup.i.sub.t] + 1 / [([R.sup.f]).sup.T-t] ([P.sup.i] - [C.sup.i])) (2.8)

= [E.sub.t]([[infinity].summation over (s=t+1)] [(1 - [mu]).sup.s-t+1] / [([R.sup.f]).sup.s-t] [d.sup.i.sub.t]

+ [[infinity].summation over (s=t+1)] [mu][(1 - [mu]).sup.s-t+1] / [([R.sup.f]).sup.s-t] ([P.sup.i] - [C.sup.i])) (2.9)

= 1 / [r.sup.f] + [mu] ([[bar.d].sup.i] + [mu] ([P.sup.i] - [C.sup.i])). (2.10)

Rearranging, this implies the following

Proposition 1. When investors are risk neutral with identical trading intensity [mu], the equilibrium price of any security i is given by

[P.sup.i] = [[bar.d].sup.i] - [mu][C.sup.i] / [r.sup.f]. (2.11)

or, equivalently, the required return on security i is

E([r.sup.i]) = [r.sup.f] + [mu] [C.sup.i] / [P.sup.i]. (2.12)

Intuitively, (2.11) shows that the price is the expected present value of all future dividends, minus the expected present value of all future transaction costs, taking into account the expected trading frequency [mu]. The equivalent equation (2.12) shows that the required return is the risk free return plus the per-period percentage transaction cost, that is, the relative transaction cost [C.sup.i] / [P.sup.i] weighted by the trading frequency [mu].

2.3 Clientele effects

Suppose that investors differ in the likelihood that they need to trade in any period, or in their expected holding period. For example, some investors expect a greater likelihood of a liquidity shock that will force them to liquidate, or a greater likelihood of arrival of a good investment opportunity that will make them want to liquidate their investment and switch to another. Consequently, each investor considers differently the impact of transaction costs on the return that he requires. Since investors require compensation at least for their expected per-period trading costs, a frequently-trading investor requires a higher return than does an infrequently-trading one. A long-term investor who can depreciate the trading cost over a longer (expected) holding period requires lower per-period return than does the short-term investor. Long-term investors can thus outbid the short-term investors on all assets. However, investors have limited resources and cannot buy all assets, and therefore they specialize in investments that are most beneficial for them. While all investors prefer assets with low transaction costs, these assets are most valued by short-term investors who incur transaction costs most frequently. Long-term investors then opt for assets in which they have the greatest advantage--those that are most costly to trade. These illiquid assets are shunned by the frequent traders and are heavily discounted by them. As a result, long-term investors, who bear the costs less frequently, earn rent in holding these assets which exceeds their expected transaction costs. Put differently, the liquidity premium on these assets will be greater than their expected trading costs. And, in equilibrium, liquid assets are held by frequently-trading investors while the illiquid assets are held by investors with long expected holding period.

This idea is presented by Amihud and Mendelson (1986a), who study the effect of having different types of investors with different expected holding periods. In particular, suppose that an agent of type j, j = 1, ..., J receives a liquidity shock with probability [[mu].sub.j] that forces him to sell and leave the market. We number the agent types such that type 1 has the highest risk of a liquidity shock, type 2 has the second highest, and so on, [[mu].sup.1] [greater than or equal to] [[mu].sup.2] [greater than or equal to] ... [greater than or equal to] [[mu].sup.J]. Also, we number the securities such that security 1 is most liquid, security 2 is second most liquid, and so on, [C.sup.1] / [[bar.d].sup.1] [less than or equal to] [C.sup.2] / [[bar.d].sup.2] [less than or equal to] ... [less than or equal to] [C.sup.I] / [[bar.d].sup.I]. (2)

Agents of type J are the natural buyers of illiquid securities because they have the longest expected holding period and, hence, the smallest per-period transaction costs. Hence, without borrowing constraints, the equilibrium is that type-J agents buy all of the illiquid assets and the model reduces to the single-type model of Section 2.2. In this case, transaction costs would matter little as the long-term investors can amortize the trading costs over a long time period. Further, without borrowing constraints, investors could achieve a long holding period by postponing liquidation of assets when facing a cash need and instead financing consumption by borrowing. Hence, borrowing frictions are important for market liquidity to affect prices. See Brunnermeier and Pedersen (2005a) for a discussion of the interaction between market liquidity and borrowing constraints (so-called funding liquidity). In reality, unconstrained borrowing is infeasible. Instead, we make here a convenient (extreme) assumption that agents have limited wealth and cannot borrow. Specifically, agents of type j are born with a wealth of [W.sup.j], and there is a mass [m.sup.j] of agents of type j at any time.

The optimal trading strategy of agent j is to invest all his wealth in securities with the highest liquidity-adjusted return,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2.13)

and the agent is indifferent among the securities with this maximal liquidity-adjusted return. Note that the liquidity-adjusted return depends both on the security and the agent type.

Amihud and Mendelson (1986a) show that the equilibrium has the following form: The agents with the shortest holding period, i.e., type 1, hold the riskless security and the illiquid securities with the lowest trading cost, i.e. securities 1, ..., [i.sup.1] where [i.sup.1] is a nonnegative integer. (If [i.sup.1] = 0 then this means that type 1 agents hold no illiquid securities.) Agents with the next shortest holding period hold a portfolio of securities with the next lowest trading costs, [i.sup.1], ..., [i.sup.2] and so on. Hence, agents of type j holds securities [i.sup.j-1], ..., [i.sup.j] where 0 [less than or equal to] [i.sup.1] [less than or equal to] [i.sup.2] [less than or equal to] ... [less than or equal to] [i.sup.J] = I. The equilibrium cutoff levels, [i.sup.1], ..., [i.sup.J] depend on the total wealth of each type of investors. Specifically, type j investors must invest all their wealth in securities [i.sup.j-1], ..., [i.sup.j], and total demand must equal supply.

Securities with low trading costs are priced such that type 1 agents are indifferent between holding these securities or the risk free asset. Hence, these securities must offer a liquidity adjusted return equal to the risk free rate,

[r.sup.f] = [[bar.d].sup.i] - [[mu].sup.1][C.sup.i] / [P.sup.i], (2.14)

that is,

[P.sup.i] = [[bar.d].sup.i] - [[mu].sup.1][C.sup.i] / [r.sup.f] (2.15)

for i = 1, ..., [i.sup.1].

With these prices, agents of type 2 earn a return higher than the risk free rate because they have a longer holding period and, hence, pay the trading cost less often. To see this, consider the liquidity-adjusted return of a type-2 agent for a security i [member of] {1, ..., [i.sup.1]}:

[[bar.d].sup.i] - [[mu].sup.2][C.sup.i] / [P.sup.i] = [[bar.d].sup.i] - [[mu].sup.2][C.sup.i] / [[bar.d].sup.i] - [[mu].sup.1][C.sup.i] [r.sup.f] > [r.sup.f], (2.16)

since [[mu].sup.2] < [[mu].sup.1]. Note that this return is increasing in [C.sup.i] / [[bar.d].sup.i] since the larger is the trading cost, the larger is type-2 agents' comparative advantage. Hence, the largest liquidity-adjusted return is that of security [i.sup.1]. We denote this return by

[r.sup.*2] := [[bar.d].sup.[i.sup.1]] - [[mu].sup.2][C.sup.[i.sup.1]] / [P.sup.[i.sup.1]]. (2.17)

To make type 2 agents hold securities i = [i.sup.1], ..., [i.sup.2], these securities must offer a liquidity-adjusted return of

[r.sup.*2] := [[bar.d].sup.i] - [[mu].sup.2][C.sup.i] / [P.sup.i], (2.18)

that is,

[P.sup.i] := [[bar.d].sup.i] - [[mu].sup.2][C.sup.i] / [r.sup.*2] (2.19)

for [i.sup.1], ..., [i.sup.2].

We can continue this iterative process. In general, a security that is held in equilibrium by type-j investors has a price of

[P.sup.i] := [[bar.d].sup.i] - [[mu].sup.j][C.sup.i] / [r.sup.*j] (2.20)

where [r.sup.*j] is type-j investors' required liquidity-adjusted return, which is determined by

[r.sup.*j] := [[bar.d].sup.[i.sup.j-1]] - [[mu].sup.j][C.sup.[i.sup.j-1]] / [P.sup.[i.sup.j-1]]. (2.21)

Clearly, investors with long horizon earn higher liquidity-adjusted returns (or "rents"), that is, [r.sup.*1] [less than or equal to] ... [less than or equal to] [r.sup.*j].

Competition among investors implies that the gross return on any security is determined by the minimum required return across possible investors:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.22)

Hence, the expected gross return [[bar.d].sup.i] / [P.sup.i] is the minimum of a finite number of increasing linear functions of the relative transaction cost [C.sup.i] / [P.sup.i]. Since the minimum operator preserves monotonicity and concavity, we have

Proposition 2. When investors are risk neutral and clientele j has trading intensity [[mu].sup.j] and limited capital, the equilibrium has the following properties:

(i) Securities with higher transaction costs are allocated to agents with longer (or identical) investment horizons.

(ii) If type j agents are marginal investors for security i, then security i has an expected gross return of

E([r.sup.i]) = [r.sup.f] + ([r.sup.*j] - [r.sup.f]) + [[mu].sup.j] [C.sup.i] / [P.sup.i] (2.23)

which is the sum of the risk free rate [r.sup.f], investor j's "rent" ([r.sup.*j] - [r.sup.f]), and his amortized relative trading cost [[mu].sup.j] [C.sup.i] / [P.sup.i].

(iii) The expected gross return E([r.sup.i]) is an increasing and concave function of the relative transaction cost [C.sup.i] / [P.sup.i].

Note that liquid securities are allocated in equilibrium to agents with short investment horizons. Since these agents are the least capable of dealing with illiquidity, they earn lower rents. Therefore, the liquidity premium ([r.sup.*j] - [r.sup.f]) + [[mu].sup.j] [C.sup.i] / [P.sup.i] for relatively liquid securities arise mostly from the amortized spread [[mu].sup.j] [C.sup.i] / [P.sup.i].

Agents with long investment horizons, however, can earn rents because patient capital is in short supply. Hence, according to this clientele theory, the liquidity premium ([r.sup.*j] - [r.sup.f]) + [[mu].sup.j] [C.sup.i] / [P.sup.i] for relatively illiquid securities arise largely from the rents ([r.sup.*j] - [r.sup.f]) and less from the amortized spread because the relevant [[mu].sup.j] is small. (3)

2.4 Time-varying transaction costs and liquidity risk

Liquidity varies over time. (4) This means that investors are uncertain what transactions cost they will incur in the future when they need to sell an asset. Further, since liquidity affects the level of prices, liquidity fluctuations can affect the asset price volatility itself. For both of these reasons, liquidity fluctuations constitute a new type of risk that augments the fundamental cash-flow risk. This section presents a model of the effect of a security's liquidity risk on its expected return, thus extending the standard Sharpe-Lintner-Mossin effect of risk on expected return. The exposition follows Acharya and Pedersen (2005) dynamic OLG model of the effect of variations in liquidity on asset prices under risk aversion. The model gives rise to a liquidity-adjusted capital asset pricing model that shows how liquidity risk is captured by three liquidity betas, and how shocks to liquidity affect current prices and future expected returns. A static asset pricing model with uncertain trading costs is presented by Jacoby et al. (2000).

To make the model tractable, we assume that agents live for only one period (that is, [mu] = 1). Generation t consists of N agents, indexed by n, who live for two periods, t and t + 1. Agent n of generation t has an endowment at time t and no other sources of income, trades in periods t and t + 1, and derives utility from consumption at time t + 1. He has constant absolute risk aversion [A.sup.n] so that his preferences are represented by the expected utility function -[E.sub.t] exp(-[A.sup.n][x.sub.t+1]), where [x.sub.t+1] is his consumption at time t + 1.

Uncertainty about the illiquidity cost is what generates the liquidity risk in this model. Specifically, we assume that [d.sup.i.sub.t] and [C.sup.i.sub.t] are autoregressive processes of order one, that is:

[d.sub.t] = [bar.d] + [[rho].sup.D] ([d.sub.t-1] - [bar.d]) + [[epsilon].sub.t] (2.24)

and

[C.sub.t] = [bar.C] + [[rho].sup.C] ([C.sub.t-1] - [bar.C]) + [[eta].sub.t], (2.25)

where [bar.d], [bar.C] [member of] [[??].sup.I.sub.+] are positive real vectors, [[rho].sup.D], [[rho].sup.C] [member of] [0, 1], and ([[member of].sub.t],[[eta].sub.t]) is an independent identically distributed normal process.

We are interested in how an asset's expected gross return,

[r.sup.i.sub.t] = [d.sup.i.sub.t] + [P.sup.i.sub.t] / [P.sup.i.sub.t-1] - 1, (2.26)

depends on its relative illiquidity cost,

[c.sup.i.sub.t] = [C.sup.i.sub.t] / [P.sup.i.sub.t-1], (2.27)

on the market return,

[r.sup.M.sub.t] = [[summation].sub.i] [S.sup.i] ([d.sup.i.sub.t] + [P.sup.i.sub.t] / [[summation].sub.i] [S.sup.i][P.sup.i.sub.t-1] - 1, (2.28)

and on the relative market illiquidity,

[c.sup.M.sub.t] = [[summation].sub.i] [S.sup.i][C.sup.i.sub.t] / [[summation].sub.i] [S.sup.i][P.sup.i.sub.t-1]. (2.29)

To determine the equilibrium prices, consider first an economy with the same agents in which asset i has a dividend of [D.sup.i.sub.t] - [C.sup.i.sub.t] and no illiquidity cost. In this imagined economy, standard results imply that the CAPM holds (Lintner, 1965; Markowitz, 1952; Mossin, 1966; Sharpe, 1964). We claim that the equilibrium prices in the original economy with frictions are the same as those of the imagined economy. This follows from two facts: (i) the net return on a long position is the same in both economies; and, (ii) all investors in the imagined economy hold a long position in the market portfolio, and a (long or short) position in the risk-free asset. Hence, an investor's equilibrium return in the frictionless economy is feasible in the original economy, and is also optimal, since positive transactions costs imply that a short position has a worse payoff than minus the payoff of a long position.

These arguments show that the CAPM in the imagined frictionless economy translates into a CAPM in net returns for the original economy with illiquidity costs. Rewriting the one-beta CAPM in net returns in terms of gross returns, we get a liquidity-adjusted CAPM for gross returns. To capture this, Acharya and Pedersen (2005) introduce three liquidity betas [[beta].sup.L1], [[beta].sup.L2], and [[beta].sup.L3], which complement the standard market beta [beta]:

Proposition 3 (Liquidity-Adjusted CAPM). When investors are risk averse and liquidity and dividends are risky as specified above, the conditional expected net return of security i in the unique linear equilibrium is

[E.sub.t] ([r.sup.i.sub.t+1] - [c.sup.i.sub.t+1]) = [r.sup.f] + [[lambda].sub.t] [cov.sub.t] ([r.sup.i.sub.t+1] - [c.sup.i.sub.t+1], [r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]) / [var.sub.t] ([r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]), (2.30)

where [[lambda].sub.t] = [E.sub.t] ([r.sup.M.sub.t+1] - [c.sup.M.sub.t+1] - [r.sup.f]) is the risk premium. Equivalently, the conditional expected gross return is:

[E.sub.t] ([r.sup.i.sub.t+1]) = [r.sup.f] + [E.sub.t] ([c.sup.i.sub.t+1]) + [[lambda].sub.t] ([[beta].sub.t] + [[beta].sup.L1.sub.t] - [[beta].sup.L2.sub.t] - [[beta].sup.L3.sub.t]), (2.31)

where

[[beta].sub.t] = [cov.sub.t] ([r.sup.i.sub.t+1],[r.sup.M.sub.t+1]) / [var.sub.t] ([r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]), (2.32)

[[beta].sup.L1.sub.t] = [cov.sub.t] ([c.sup.i.sub.t+1],[c.sup.M.sub.t+1]) / [var.sub.t] ([r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]), (2.33)

[[beta].sup.L2.sub.t] = [cov.sub.t] ([r.sup.i.sub.t+1],[c.sup.M.sub.t+1]) / [var.sub.t] ([r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]), (2.34)

and

[[beta].sup.L3.sub.t] = [cov.sub.t] ([c.sup.i.sub.t+1],[r.sup.M.sub.t+1]) / [var.sub.t] [r.sup.M.sub.t+1] - [c.sup.M.sub.t+1]). (2.35)

Equation (2.31) is simple and natural. It states that the required excess return is the expected relative illiquidity cost, [E.sub.t] ([c.sup.i.sub.t+1]) as in the basic model above, plus four betas (or covariances) times the risk premium. These four betas depend on the asset's payoff and liquidity risks. As in the standard CAPM, the required return on an asset increases linearly with the market beta, that is, the covariance between the asset's return and the market return. This model yields three additional effects which could be regarded as three forms of liquidity risks.

The first liquidity beta [[beta].sup.L1] is positive for most securities due to commonality in liquidity. (5) The model implies that expected return increases with the covariance between the asset's illiquidity and the market illiquidity, because investors want to be compensated for holding a security that becomes illiquid when the market in general becomes illiquid.

The second liquidity beta [[beta].sup.L2], which measures the exposure of asset i to marketwide illiquidity, is usually negative (6) in part because a rise in market illiquidity reduces asset values. This beta affects required returns negatively because investors are willing to accept a lower return on an asset with a high return in times of market illiquidity. Consequently, the more negative is the exposure of the asset to marker illiquidity, the greater is the required return.

The third liquidity beta [[beta].sup.L3] is also negative for most stocks. (7) This liquidity beta has a negative sign in the pricing model, meaning that the required return is higher if the sensitivity of the security's illiquidity to market condition is more negative. The negative effect stems from the willingness of investors to accept a lower expected return on a security that is liquid in a down market. When the market declines, investors are poor and the ability to sell easily is especially valuable. Hence, an investor is willing to accept a discounted return on stocks with low illiquidity costs in states of poor market return.

Empirically, liquidity is persistent over time, (8) meaning that if a market is illiquid today, then it is more likely to not fully recover next month. Mathematically, this means that [[rho].sup.C] > 0.

Acharya and Pedersen (2005) show that persistence of liquidity implies that liquidity predicts future returns (Equation (2.36) below) and co-moves with contemporaneous returns (Equation (2.37) below). Intuitively, as stated in Amihud (2002), a high illiquidity today predicts a high expected illiquidity next period, implying a high required return, which is achieved by lowering current prices. This result relies on the realistic assumption that cash flow shocks and shocks to the trading costs are not too highly correlated.

Proposition 4. Assuming that liquidity is persistent and certain technical conditions are satisfied for a portfolio q, then an increase in illiquidity implies that the required return increases:

[partial derivative] / [partial derivative][C.sup.q.sub.t] [E.sub.t] ([r.sup.q.sub.t+1] - [r.sup.f]) > 0 (2.36)

and contemporaneous returns are low

[cov.sub.t-1] ([c.sup.q.sub.t], [r.sup.q.sub.t]) < 0. (2.37)

2.5 Uncertain trading horizons and liquidity risk

In the previous section, we considered the risk that it suddenly becomes very costly to liquidate a portfolio. Another way of considering liquidity risk is to focus on the trading horizon, while keeping the trading costs constant.

The net-of-transaction-cost rate of return (per unit of time) of an asset is increasing with the holding period since the transaction cost is depreciated over a longer period and thus its per-period effect is smaller. If the holding period becomes stochastic due to liquidity shocks, the net return becomes random as well even if both the gross return and the transaction costs are deterministic. In the basic model of Section 2.2, this risk is ignored since agents are assumed risk neutral. If agents are risk averse, then this liquidity-induced risk will be priced. Huang (2003) analyzes this problem, assuming two console bonds that are identical except that one is liquid and the other is illiquid, i.e., it incurs proportional transaction costs. Investors are risk averse (with CARA utility function) and have a constant income stream. Each investor is hit by a negative "liquidity shock" with a Poisson arrival rate and, when this happens, the investor must liquidate his securities and exit. Importantly, there is a constraint on short selling and on borrowing against future income. Thus, the transaction costs that the investor incurs upon liquidation negatively affect his immediate consumption and he cannot alleviate this effect by borrowing. This effect would be particularly costly (in terms of utility of consumption) if the investor has just recently acquired the illiquid asset, in which case the return that has accumulated on it is small relative to the liquidation costs. Huang shows that in equilibrium, the illiquid security whose net return becomes stochastic will have a premium over the liquid security which exceeds the magnitude of expected transaction costs, reflecting the liquidity-induced risk premium. This can help explain why the return premium on illiquid stocks, estimated in empirical studies, is so large relative to expected per-period transaction costs.

Vayanos (2004) considers a model in which investors' risk of needing to liquidate is time varying and shows that the liquidity premium--that is, the return compensation for illiquidity--is also time varying. Indeed, when investors have a high likelihood of needing to sell, the liquidity premium is high. Further, Vayanos (2004) links the risk of needing to liquidate to the market volatility.

2.6 Endogenous trading horizons

The models considered so far have assumed that the trading horizon was exogenous, that is, [mu] was exogenous. In many cases, however, trading horizons are the outcome of optimal investor behavior, and investors can trade off cost and benefits of delaying trades. To capture this effect, Constantinides (1986) studies a continuous-time model in which an investor with constant relative risk aversion holds a risk free and a risky security, the latter having proportional exogenous trading cost. (9) Absent trading costs, theory suggests that the investor will hold a fixed ratio of the assets and trade continuously to balance his portfolio in response to the risky asset's price changes. With trading costs, the investor faces a tradeo.: frequent portfolio rebalancing when the risky/riskless assets ratio deviates from its optimal value entails high trading costs, while refraining from rebalancing renders the portfolio suboptimal and imposes a utility cost. The solution is setting a boundary around (above and below) the optimal asset ratio within which there is no trading, and when the ratio is outside of the boundary, the investor transacts to the nearest boundary. (10) The width of the no-trade region increases in the risky security's trading costs, and thus higher trading costs lead to less trading, less demand for the risky asset, and a higher utility loss for the investor. The liquidity premium on the risky asset with trading costs is then defined as the decrease in its expected return that would leave the investor indifferent between this asset and an identical asset with no trading costs. Using calibrated parameter values, Constantinides finds that the liquidity premium in terms of per-annum return is an order of magnitude smaller than the trading cost. This is because investors in his model can largely alleviate the cost by reduced trading and because the utility costs of not trading are small.

Vayanos (1998) constructs a general equilibrium model with endogenous trading horizons. Overlapping generations of investors trade and consume continuously over their deterministic lifespan and have access to a perfectly liquid riskless asset and to a general number of risky assets that are costly to trade. In equilibrium, investors buy the risky assets when born, and slowly sell them as they become older and more risk averse. A calibration of the model finds that the effects of transaction costs are small because life-cycle is the only trading motive (similar to Constantinides (1986)). Further, Vayanos shows that a general equilibrium model with endogenous horizons can lead to surprising results. For instance, in certain special cases, transaction costs can actually raise an asset's price. This can happen because, while trading costs make investors buy fewer shares, they induce them to hold the shares for longer periods, which can raise the total asset demand.

Some of the observed evidence in the market is consistent with these models' predictions. We observe that in the last decade there has been a significant decline in trading costs in U.S. stock markets, partly due to reducing the minimum tick from $ 1/8 to $ 1/16 and then to a penny. At the same time, stock turnover in the New York Stock Exchange has risen from 54% in 1994 to 99% in 2004. (11) This relationship is predicted, e.g., by Constantinides's model, where the width of the no-trade region increases in trading costs. However, the liquidity premium predicted by these models is low relative to the empirical results documented in Section 3, reflecting the models' assumptions that the only reason for trading is portfolio rebalancing where the parameters of the risky asset's process are constant. (12) Further, due to this minor need for trade, the annual turnover predicted, e.g., by Vayanos is around 3%, which is quite low relative to observed turnover.

In the real world, many investors have larger needs to trade in spite of the significant trading costs. A simple reduced-form way of capturing a large trading need is to use the models of Sections 2.2-2.5 with a trading intensity calibrated to match the observed volume. Further, the following studies introduce additional motives for trade in models with endogenous trading horizon and, consequently, obtain that the effect of trading costs on the liquidity premium is greater. Lo et al. (2004) consider an equilibrium with two completely "opposite" agents who have perfectly negatively correlated endowment risk. The agents face fixed per-trade trading costs so they refrain from trading when the security position is within certain boundaries and, when the position reaches a boundary, they trade to the optimal position somewhere in the middle of the no-trade region (as oppose to trading to nearest boundary as is as the case with proportional trading costs). Lo et al. (2004) find that the liquidity premium can be large when the investors have high-frequency trading needs. Also, the partial equilibrium approach of Constantinides (1986) has been extended by Jang et al. (2005) who introduce a time-varying investment opportunity set, and Lynch and Tan (2004) who consider return predictability, wealth shocks, and state-dependent transaction costs. (13) These motivations for trading in excess of Constantinides (1986) portfolio rebalancing motive increase the resulting trading frequency and the impact of transaction costs on price. These authors show through numerical calibration that the liquidity premium can be large for certain parameters that they find realistic.

Vayanos and Vila (1999) study an OLG model with a risky asset with proportional trading costs and a liquid riskless asset in fixed supply, thus endogenizing the riskless interest rate. If the risky asset has a higher trading cost then the risk-free asset becomes a more attractive alternative. Therefore, the equilibrium price of the risk-free asset is increasing (i.e., the risk-free interest rate is decreasing) in the trading cost of the risky asset. Heaton and Lucas (1996) solve an equilibrium model of incomplete risk sharing numerically and find that trading costs increase the equity premium and lower the riskfree rate. Heaton and Lucas (1996) find sizeable effects only if trading costs are large or the quantity of traded assets is small.

2.7 Brief aside: Sources of illiquidity

As discussed in the introduction, illiquidity can arise from exogenous trading costs, private information, inventory risk for market makers, and search problems. While the illiquidity related to exogenous costs and search are straightforward, we briefly review how information and inventory problems can also lead to illiquidity. In Section 2.8 we discuss how these sources of illiquidity affect security prices.

2.7.1 Illiquidity deriving from private information

Certain investors or corporate insiders can have superior information (or information processing ability) about the fundamental value of a security. This creates an adverse selection problem: informed traders with bad news are likely to sell, and informed traders with good news have an incentive to buy (Akerlof, 1970).

Grossman and Stiglitz (1980) show that information asymmetries are fundamental to market equilibrium for, if all information were contained in prices, no one has an incentive to gather information in the first place. Hence, they consider a noisy rational expectations equilibrium (REE) in which investors are competitive price takers who learn from prices. In equilibrium, some investors refrain from collecting information while others incur cost in gathering information and get compensated in the form of superior expected investment performance such that the two groups of investors have the same overall expected utility. The literature on how information is revealed through prices in REE also includes Grossman (1976), Hellwig (1980), Admati (1985), and others.

Investors with private information have an incentive to strategically take into account the price effect of their trades, and market makers strategically protect themselves against informed traders. Bagehot (1971) proposes that the market maker gains from trading with uninformed liquidity traders and loses money to informed traders. This gives rise to the bid-ask spread, which is necessary to compensate the market maker for his losses to the informed traders. Copeland and Galai (1983) model the quoting decision of a profit-maximizing market maker, with profit defined as the difference between the gain from liquidity traders and the loss to informed traders. Copeland and Galai view the quoted bid and ask prices as strike prices on two free options with a very short expiration period written by the market maker to the informed trader. The ask and bid prices are, respectively, the strike (exercise) prices of the call and of the put, straddling the current security's price. The model's implication is that increased uncertainty (volatility) widens the spread, which concurs with the empirical evidence.

A standard way of modeling the market maker's strategy when trading with informed investors is to assume that the market maker is competitive and risk neutral with a discount rate equal to the risk-free rate, which is normalized to zero. Such a competitive market maker sets the price [p.sub.t] at time t according to

[p.sub.t] = E([upsilon]|[[??].sub.t],[OF.sub.t]), (2.38)

where v is the fundamental value, [[??].sub.t] is the public information, and [OF.sub.t] is the order flow at time t. Hence, the market maker sets a price equal to his best estimate of the asset's fundamental value, given what he learns from the order flow. With this price-setting principle, execution prices follow a martingale.

This general modeling approach is applied in the context of various market structures. Glosten and Milgrom (1985) consider a market structure in which competitive market makers must quote binding bid and ask prices and investors arrive sequentially and can decide whether to buy one share at the ask, sell one share at the bid, or refrain from trading. In this case, the bid is the expected value of the fundamental given that the next trade is a sell order, and similarly for the ask, leading to the following "regret free" prices:

[bid.sub.t] = E([upsilon]|[[??].sub.t], sell), (2.39)

and

[ask.sub.t] = E([upsilon]|[[??].sub.t], buy). (2.40)

The quoted bid price reflects the risk that a seller is informed of bad news, and the ask reflects the risk that a buyer is informed of good news. If the market maker were sure that the counterparty is informed, she would not trade at all since as long as the informed trader wishes to sell, the price is too high. What makes the market maker willing to trade is the possibility that the counterparty is uninformed, and it may gain by selling to him at a "high"-ask-price or buying from him at a "low"-bid-price. Thus, the market maker gains from trading with uninformed traders and looses to informed ones. Since in a competitive market the market maker ends up with zero profit, the gains of the informed traders are at the expense of the uninformed trade. Clearly, the model implies a bid-ask spread (bid<ask) which is greater if the probability of trading with informed traders is larger.

Kyle (1985) considers a market where an informed and an uninformed "noise" trader each submit a market order for an asset, and the market maker sets the price depending on the aggregate order flow such that he ends up with zero gain (as is the case in a competitive market). A large demand will make the market maker raise the price since it may reflect demand by an informed investor who knows that the asset value is high. The noise trader submits an exogenous normally distributed order u, while the informed trader optimally decides on his order x given his signal about the value v, where x is constrained by the informed trader's knowledge that a large order will reveal his information to the market maker and will cause the price to be set closer to v, leaving him with a smaller per-unit gain. (14) Kyle shows that there exists a linear equilibrium in which the market maker sets the price as

[p.sub.t] = E(v|[[??].sub.t],[u.sub.t] + [x.sub.t]) = [p.sub.t-1] + [[lambda].sub.t] ([u.sub.t] + [x.sub.t]), (2.41)

where [lambda] describes the price change per unit of net order flow, that is, the market impact, which is a measure of illiquidity. Kyle shows that [lambda] increases in the variance about [upsilon], i.e., in the extent of asymmetry in information, and it declines in the variance of u, the uninformed investors' order flow. (15) Thus, both the bid-ask spread and the market impact are measures of market illiquidity that can result from information asymmetry. Mendelson and Tunca (2004) extend the Kyle (1985) model to the case of endogenous liquidity trading. Like Kyle (1985) and the other papers briefly reviewed in this subsection, they do not address how information asymmetry affects required return.

Whereas the above-mentioned papers deal with the effects of private information about fundamental news, there is also a more recent literature that recognizes the importance of private information about order flow; for example, a trader might be using his knowledge about someone else moving a large block of shares. This literature includes Madrigal (1996) who considers non-fundamental speculation, Attari et al. (2005) and Brunnermeier and Pedersen (2005b) who study predatory trading (trading the exploits or induces other traders need to liquidate a position), (16) Vayanos (2001) and Cao et al. (2003) who consider strategic trading due to risk sharing, and Gallmeyer et al. (2004) who study uncertainty about the preferences of potential counterparties.

2.7.2 Illiquidity deriving from inventory risk

A fundamental source of illiquidity is the fragmentation of investors and markets due to the fact that not all investors are present in the same market all of the time. For instance, a seller may arrive to the market at a time when a natural buyer is not present. This gap between the seller and buyer is bridged by market makers who provide immediacy through their continuous presence in the market and thus enable continuous trading by any trader who so wishes. In particular, the market maker can buy from the seller and later resell to the buyer. However, the market maker faces a risk of fundamental price changes in the meantime and must be compensated for this risk. This has been pointed out by Stoll (1978a). Garman (1976) introduces a model with a monopolist market maker whose quoted prices affect the intensity of arrival of buyers and sellers. If quoted prices are constant, the market maker will be surely ruined. Amihud and Mendelson (1980) and Ho and Stoll (1981) resolve this problem by having the quoted bid-ask prices depend on the market maker's inventory of the traded security. Amihud and Mendelson assume a market maker who constrains his inventory position (due to capital constraint and risk) and manages inventory to avoid the constraints, and Ho and Stoll assume a risk-averse market maker who manages inventory to reduce his risk exposure. Demand-pressure models with competitive market makers are considered by Ho and Stoll (1983) and Grossman and Miller (1988). Brunnermeier and Pedersen (2005a) relate variations in liquidity over time and cross-sectionally to market makers' capital constraints.

2.8 Asset pricing with endogenous illiquidity

In Section 2.7 we described how illiquidity can arise endogenously due to various fundamental frictions. We are interested in determining how these frictions ultimately affect asset prices. One approach is to take the endogenously derived liquidity costs and "plug them into" the asset pricing models with exogenous trading costs (Sections 2.2-2.6). As we show below, we can sometimes get additional insights by considering asset pricing directly in a model of endogenous illiquidity.

2.8.1 Private information and the required return

The effect of information asymmetry on the required return is studied in dynamic REE models by Wang (1993, 1994) and in a strategic model by Garleanu and Pedersen (2004). Wang (1993) considers a dynamic infinite-horizon model in which all investors observe a dividend process and the corresponding stock price, but only a fraction of the investors observe the dividend process's stochastic growth rate [PI]. The price does not fully reveal [PI] since the supply of shares is random. Wang shows that if there is a larger fraction of less-informed investors who do not observe [PI], then the required return is higher. One reason for this is that when dividends increase, less-informed investors increase their expectations of dividend growth, thus pushing prices up. This process raises the correlation of prices and dividends, thus raising total return volatility, which reduces consumption smoothing and risk sharing and increases the average risk premium. (17)

Garleanu and Pedersen (2004) consider a model in which a finite number of agents trade repeatedly by submitting market or limit orders. Each period, one agent may receive a signal about the next dividend, and potentially a "liquidity shock." Garleanu and Pedersen show that, if agents are symmetric ex ante, then future bid-ask spreads due to private information are not a direct trading cost. That is, their present value does not directly reduce the price--unlike the case of exogenous trading costs. This result obtains because, in expectation, the future losses an agent will incur when trading due to liquidity reasons are balanced by the gains he will make when trading based on information. If the agents differ ex ante, though, in that some agents are more likely to make liquidity trades than others, then the marginal investor does not break even on average and her expected net trading losses augment the required return. Importantly, the adverse-selection problems lead to an indirect cost associated with allocation inefficiencies caused by trading-decision distortions. This indirect allocation cost (further) increases the required return.

2.8.2 Illiquidity due to inventory-risk and demand pressure

From the investor's viewpoint, illiquidity due to demand pressure and inventory-related costs can be treated as exogenous illiquidity cost, whose effect on asset prices are as derived in Sections 2.2-2.6. The smaller the inventory position that the market maker is willing to assume due to reasons such as the risk of his position or limits on his capital (Amihud and Mendelson, 1980; Brunnermeier and Pedersen, 2005a), or the greater the price volatility of the security traded (Ho and Stoll, 1981), the greater is the bid-ask spread that the market maker sets. In addition, variations in demand pressure that cause variations in the market maker inventory change the prices at which he is willing to trade. These are short-term, transitory effects of inventory on prices, but the permanent effect on prices and expected return flows through the effect on trading costs. For example, in market systems with better capacity to absorb inventory shocks, the models would predict smaller illiquidity costs and consequently there would be smaller price discount due to illiquidity.

2.8.3 Search, bargaining, and limits on trading

Liquidity problems often play a role in "over the counter" (OTC) markets, that is, when there is no centralized market and investors trade bilaterally, for instance over the phone. In such markets, illiquidity arises because of search and bargaining problems. For instance, when a trader needs to sell her position, she must search for a counterparty willing to buy, and, once a potential counterparty is located, the trader must negotiate the price--a negotiation that reflects each trader's outside option to find other counterparties. Further, due to the bilateral trading in OTC markets, intermediaries can have market power, allowing them to earn fees, which translate into trading costs for investors. Duffie et al. (2003, 2005) model such search and bargaining features and study how these sources of illiquidity affect asset prices. They find that, under certain conditions, search frictions increase the liquidity premium (i.e., lower prices) and increase bid-ask spreads. Further, higher bargaining power to buyers leads to lower prices. Also, Duffie et al. (2003) link volatility to liquidity and thus to prices.

Weill (2002) and Vayanos and Wang (2002) extend the model of Duffie, Garleanu and Pedersen to the case of multiple illiquid securities and show, among other things, that search frictions lead to cross-sectional differences in the liquidity premium. In particular, securities with larger float (or supply) of securities are predicted to have less severe search problems and correspondingly lower liquidity premia. Lagos (2005) shows that search frictions can help explain the risk-free rate and equity-premium puzzles.

Vayanos and Wang (2002) further show how search externalities can lead to concentration of trade in one security among several substitutes. Duffie et al. (2002) consider a model in which shortsellers must search for lendable securities, and must negotiate the lending fee with the lender, thus capturing the real-world OTC institution for short-selling. They show how the lending fee initially increases the value of the security. Vayanos and Weill (2005) develop a multi-asset model in which both the spot markets (for buying and selling securities) and the securities lending markets (for borrowing shares to shortsell) are OTC search markets. In equilibrium, one security is "special": it is liquid, has a high price, and has a large lending fee. Boudoukh and Whitelaw (1993) show that it can be in the issuer's interest to maintain segmented markets in which one security is special.

Hopenhayn and Werner (1996) consider a matching model in which certain assets have payoffs that are "non-verifiable" to uninformed agents. When uninformed agents are matched, they do not trade assets with non-verifiable payoffs and, therefore, these assets become less liquid and have a higher expected return.

Longstaff (1995) considers the liquidity premium in a partial equilibrium model with limited access to counterparties using a different approach. A hypothetical investor can perfectly predict the future price movements of a security over a certain time period but cannot trade the security during this period. If the investor could time the sale of the asset optimally, it would be worth more than if he has to hold it until the end of the period. The difference--the value of the foregone option to sell the asset optimally--is an upper bound on the value of liquidity. The value of liquidity can also be likened then to the payoff from an option on the maximum value of a security whose exercise (strike) price is the value of the security when the liquidity restriction expires. This option is also in the money, meaning that there is a positive liquidity discount. Longstaff then obtains that the maximum value of this option when the restriction period is 2 years and volatility is 30% is 38.6% of the asset value--quite close to the discount observed for restricted stock (see Section 3.2.4 below). It is of course questionable whether investors can perfectly time their trades, therefore the price discount in this model is the upper bound on the cost of illiquidity. But the method can be used for any strategy that investors wish to apply based on observables, and then the cost of illiquidity is the opportunity cost of foregoing this strategy.

Longstaff (2001) considers a continuous-time model in which an investor must limit his trading intensity, thus capturing the idea that investors cannot unwind a position immediately. Consequently, the investor must avoid shorting or taking a leveraged positions, i.e., his investment in the risky asset is limited (the model is in partial equilibrium). Longstaff derives the optimal portfolio choice under this constraint, and shows numerically that the implied liquidity premium can be substantial. This trading friction is also used by Brunnermeier and Pedersen (2005b) who study the asset pricing effects of illiquidity deriving from predatory trading. They show that a trader who needs to sell a large position is exploited by other traders, and this endogenous illiquidity raises the required return ex-ante.

(1) See, for instance, Duffie (1996) or Cochrane (2001).

(2) Of course, what matters is the transaction cost relative to the fundamental value. In fact, one could use "stock splits" to achieve constant expected dividends for all securities.

(3) See Kane (1994) for an alternative proof of the clientele effect. In a static mode without clientelle effects, Jacoby et al. (2000) show that short-lived securities' required return can be convex in trading costs.

(4) Brunnermeier and Pedersen (2005a) offer a model that explains the variations over time in liquidity, linking it to variations in the funding conditions of market makers.

(5) See evidence in Hasbrouck and Seppi (2001), Huberman and Halka (2001), Chordia et al. (2002).

(6) See evidence in Amihud (2002).

(7) See evidence in Acharya and Pedersen (2005) and Chordia et al. (2005).

(8) See evidence in Amihud (2002).

(9) See also Liu (2004a) who determine the optimal trading strategy for an investor with constant absolute risk aversion and many independent securities with both fixed and proportional costs.

(10) Constantinides (1986) assumes that the investor continuously consumes a constant proportion of riskless wealth. Davis and Norman (1990) prove formally that the optimal investment indeed is bang-bang as assumed by Constantinides (1986) and derive the optimal consumption, which is, however, not a constant proportional of riskless wealth. While the investor behavior considered by Constantinides (1986) is therefore not fully optimal, the order of magnitude of Constantinides's calibration results has not been questioned.

(11) Source: NYSE Fact Book 2004.

(12) Novy-Marx (2005) describes how liquidity can appear to be priced (as in the empirical literature) even when its not, because liquidity can proxy for unobserved risk factors.

(13) See also Balduzzi and Lynch (1999).

(14) In this model, the informed trader is a monopolist on the information on v and he thus acts as a monopolist who considers the consequences of his action on price.

(15) Admati and Pfleiderer (1988) further show how private information leads to endogenous concentration of trade since all traders prefer to trade at the time of highest liquidity, Kyle (1989) considers the case of imperfect competition between market makers, and Easley and O'Hara (1987) study the information content of trade side and its consequent effect on prices.

(16) See also Pritsker (2003)

(17) Recent static models of asymmetric information and asset prices include Easley and O'Hara (2004) and O'Hara (2003).

Yakov Amihud (1), Haim Mendelson (2) and Lasse Heje Pedersen (3)

(1) Stern School of Business, New York University, yamihud@stern.nyu.edu

(2) Graduate School of Business, Stanford University

(3) Stern School of Business, New York University

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Title Annotation: | Liquidity and Asset Prices |
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Author: | Amihud, Yakov; Mendelson, Haim; Pedersen, Lasse Heje |

Publication: | Foundations and Trends in Finance |

Geographic Code: | 1USA |

Date: | Jul 1, 2005 |

Words: | 11017 |

Previous Article: | 1: Introduction. *. |

Next Article: | 3: Empirical evidence. |

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