Liquidity Premia and Transaction Costs

Standard literature concludes that transaction costs only have a second-order effect on liquidity premia. We show that this conclusion depends crucially on the assumption of a constant investment opportunity set. In a regime-switching model in which the investment opportunity set varies over time, we explicitly characterize the optimal consumption and investment strategy. In contrast to the standard literature, we find that transaction costs can have a first-order effect on liquidity premia. However, with reasonably calibrated parameters, the presence of transaction costs still cannot fully explain the equity premium puzzle. TRANSACTION COSTS ARE PREVALENT in almost all financial markets. Extensive research has been conducted on the optimal consumption and investment policy in the presence of transaction costs since the seminal work of Constantinides (1979, 1986). The presence of transaction costs significantly changes optimal consumption and investment strategies. For example, in the presence of transaction costs, continuous trade incurs infinite transaction costs, and thus even a small transaction cost can dramatically decrease the frequency of trade. However, most studies find that the utility loss due to the presence of transaction costs is small. In particular, Constantinides (1986) finds that the liquidity premium (i.e., the maximum expected return an investor is willing to exchange for zero transaction cost) is small relative to the transaction cost, even for a suboptimal trading strategy, and hence concludes that transaction costs only have a second-order effect for asset pricing. Indeed, in this framework it seems unlikely that transaction costs can have an important role in explaining the cross-sectional patterns of expected returns, the equity premium puzzle, or the small stock risk premia. This finding contrasts with many empirical studies that highlight the importance of transaction costs or related measures such as turnover in influencing the cross-sectional patterns of expected returns.1 ∗Jang is from the Financial Supervisory Service of Korea; Koo is from the Department of Business Administration of Ajou University, Korea; Liu is from the Olin School of Business of Washington University in St. Louis; and Loewenstein is from the Robert H. Smith School of Business of the University of Maryland. We thank George Constantinides; Phil Dybvig; Jingzhi Huang; Jiang Wang; and seminar participants at the 2006 American Finance Association conference, the 2005 China International Conference in Finance, the University of Illinois at Chicago, and Washington University in St. Louis for helpful comments. We are especially grateful to an anonymous referee and Rob Stambaugh (the Editor) for their useful suggestions. All errors are our responsibility. 1 See, for example, Eleswarapu (1997) and Brennan, Chordia, and Subrahmanyam (1998). 2329 2330 The Journal of Finance One of the common assumptions of the existing literature on optimal consumption and investment with transaction costs is that the investment opportunity set stays constant. For example, Constantinides (1986), Vayanos (1998), Liu and Loewenstein (2002), and Liu (2004) all assume that the expected stock return, the return volatility, and the liquidity (transaction costs) are constant throughout the investment horizon. Empirical research, however, documents a great deal of evidence that is inconsistent with the constant investment opportunity set hypothesis. For example, Schwert (1989) and Campbell and Hentschel (1992) conclude that the volatilities of stock returns vary substantially over time. Campbell (1991) and Lewellen (2004) find that expected returns on equities also change over time. In addition, the existence of large liquidity shocks (e.g., the 1987 crash, the 1998 Long Term Capital Management event) suggests that transaction costs can also vary over time. Taking into account the stochastic nature of the investment opportunity set may qualitatively change the conclusion in the standard literature (e.g., Constantinides (1986)) that transaction costs only have a second-order effect. Intuitively, the presence of transaction costs reduces an investor’s welfare through two mechanisms, namely, the transaction cost payment and the deviation from the optimal strategy in the no-transaction-cost case. If investment opportunity set parameters change over time, the optimal stock investment target in the no-transaction-cost case should also change over time. Thus, as market conditions change over time, compared to the constant-investmentopportunity-set case, an investor should rebalance more often to avoid being too far away from the target if the transaction cost rate is small. The relative impact of the transaction cost should therefore increase, mainly because of a greater transaction cost payment resulting from more frequent trading. In addition, if liquidity can vary dramatically over time, an investor should significantly change his trading strategy to minimize the loss from a bad liquidity shock, and thus the impact of transaction costs should also increase significantly, mainly due to the substantial deviation from the optimal strategy in the no-transaction-cost case. To quantify this intuition, we build a model similar to that of Constantinides (1986) and Davis and Norman (1990), but with regime switching for fundamental parameters. Specifically, we consider the optimal consumption and investment problem for a small investor (i.e., one with no price impact) who derives constant relative risk aversion (CRRA) utility from intertemporal consumption and bequest.2 The investor can invest in one risky asset and one risk-free asset. In contrast to most of the existing literature, we assume that the investment opportunity set is not constant and that there are two regimes (“Bull” and “bear”) with different fundamental parameters such as expected return, volatility, and liquidity. One regime switches to the other regime at the first jump time of a market-independent but possibly regime-dependent Poisson process.3 2 The bequest can also be interpreted as an exogenous need for cash. 3 The investor we consider in this paper can be an institutional investor who does not have any price impact and who updates the estimates of fundamental parameters from time to time. Liquidity Premia and Transaction Costs 2331 We explicitly characterize the solution to the investor’s problem in a general setting.4 Using parameter estimates from Ang and Bekaert (2002), our extensive numerical analysis demonstrates that in contrast to the standard conclusion that transaction costs only have a second-order effect, transaction costs can have a first-order effect if the investment opportunity set varies over time. Specifically, the liquidity premium to transaction cost (LPTC) ratio could be well above one. The consideration of a stochastic investment opportunity set makes this ratio typically more than 4 times and in many cases 10 times higher than what Constantinides finds, even for the optimal policies.5 In addition, we find that the LPTC ratio increases as the Bull regime investment opportunity set becomes more and more attractive than the bear regime investment opportunity set. Intuitively, as the difference in, for example, expected returns increases, an investor invests more (less) in the risky asset in the Bull (bear) regime, revises the portfolio more dramatically, and thus incurs higher transaction costs. Our analysis therefore suggests that consideration of a time-varying investment opportunity set is an important factor in explaining the high volume of trade and the relation between transaction costs and the cross-sectional patterns of expected returns. However, our analysis also suggests that even when the investment opportunity set is time-varying, the magnitude of liquidity premia cannot be large enough to fully explain the equity premium puzzle (see Mehra and Prescott (1985)). Unlike the no-transaction-cost case, smoothing of trading strategies across regimes is optimal in the presence of transaction costs. Without transaction costs, the optimal investment policy in one regime is independent of parameters in the other regime. In contrast, our analysis shows that in the presence of transaction costs, an investor optimally responds to changes in one regime by altering investment behavior in both regimes. For example, as the transaction cost in one regime increases (all else equal), the investor trades less in this regime and trades more in the other regime because the latter regime becomes relatively cheaper to trade in. This finding of cross-regime smoothing suggests that the presence of transaction costs can lead to patterns of optimal investment behavior that would seem suboptimal if only the current market conditions were considered. As far as we know, in the literature with regime switching, this paper is the first to provide a verification theorem for a candidate solution, explicit bounds on the no-transaction regions, and the steady-state distribution function for the portfolio holding. Our theoretical analysis suggests that extending the two-regime model to a multi-regime setting is straightforward but requires significantly more intensive computation. However, the qualitative results we obtain in the paper 4 We also derive closed-form solutions up to some constants in special cases such as the nointertemporal-consumption case. Results are not reported in this paper to save space, but are available from the authors. 5 Recall that Constantinides (1986) uses a suboptimal consumption policy to emphasize how small the liquidity premium is. 2332 The Journal of Finance should stay the same. For example, as long as the transaction cost is small relative to the changes in the optimal portfolio target across regimes, we expect an investor to optimally incur transaction costs when a regime switches. This fundamental intuition also applies to the case in which the investment opportunity set depends on a continuous state variable. Hence, jumps in the fundamental parameters in the financial market are not critical for our results. Our paper is related to a large body of literature that addresses the optimal transaction policy for an investor facing transaction costs. Dumas and Luciano (1991) derive a closed-form solution for an investor with a long-term growth objective. Schroder (1995) uses a numerical method to solve for the optimal transaction policy in the presence of fixed costs. Vayanos (1998) derives the asset pricing impact of transaction costs in an overlapping generations framework. Leland (2000) examines a multiasset investment fund that is subject to transaction costs and capital gains taxes. He develops a relatively simple numerical procedure to compute the multidimensional no-transaction region. Lo, Mamaysky, and Wang (2004) study the effect of fixed transaction costs on asset prices and find that even small fixed costs can give rise to significant illiquidity discounts on asset prices. Shreve and Soner (1994) provide important theoretical results and an analysis of the liquidity premium. All these papers assume that the investment opportunity set is constant. Our analysis suggests that a stochastic investment opportunity set is an important consideration in generating both a higher volume of trade and a greater impact of transaction costs. While some previous results characterize optimal policies in more general models (e.g., Koo (1992) and Loewenstein (2000)), they do not lead to transparent statements concerning liquidity premia and transaction frequency. Lynch and Balduzzi (2000) examine the impact of stock return predictability and transaction costs on portfolio rebalancing rules by discretizing both time and states to obtain a numerical approximation. However, due to some nonstandard modelling choices, it is difficult to interpret their results. For example, for some of their analysis they assume that consumption is financed by costlessly liquidating the stock and the riskless asset in proportions given by the pre-rebalancing portfolio weights. As a result, the post-consumption but prerebalancing portfolio weights are unchanged. When the investor rebalances these portfolio weights, a transaction cost proportional to post-consumption wealth is then incurred and paid by costlessly liquidating the stock and the riskless asset in proportions given by the post-rebalancing portfolio weights. Thus, an investor who does not want to change the dollar amount invested in the stock would first need to sell stock to finance consumption and then incur a transaction cost to buy back the stock. In contrast, in our model and the standard literature (e.g., Constantinides (1986)), the transaction cost payment and consumption withdrawals are financed using an optimal trading strategy. In addition, we provide a transparent and tractable model in which we can explicitly compute bounds on the transaction boundaries, liquidity premia, trading frequency, and expected lifetime transaction cost expenditures. The rest of the paper is organized as follows. Section I presents the model with transaction costs and regime switching and provides characterizations Liquidity Premia and Transaction Costs 2333 of the solution. Numerical and graphical analysis is presented in Section II. Section III closes the paper. All of the proofs are in the Appendix. I. Optimal Consumption and Investment A. The Basic Model Throughout this paper we assume a probability space ( , F , P ), where uncertainty and the filtration {Ft} are generated by a standard one-dimensional Brownian motion w and two independent Poisson processes representing the regime-switching risk and the mortality risk. We assume that all stochastic processes in this paper are adapted. An investor can trade two assets, a money market account (“the bond”) and a risky investment (“the stock”). There are two regimes, “Bull” (regime B) and “bear” (regime b). The fundamental parameters in the financial market may be regime dependent. We assume that regime i switches into regime j at the first jump time of an independent Poisson process with intensity λi, for i, j ∈ {B, b}. In regime i, the risk-free interest rate is ri, and the investor can buy the stock at the ask St = (1 + θi)St or sell the stock at the bid St = (1 − αi)St, where θi ≥ 0 and 0 ≤ αi ≤ 1 represent the proportional transaction cost rates and St satisfies dSt St = μi dt + σi dwt , (1) where all parameters are positive constants and μi > ri. In regime i ∈ {B, b}, when θi + αi > 0 the above model gives rise to equations governing the evolution of the dollar amount invested in the bond, xt, and the dollar amount invested in the stock, yt: dxt = rixt dt − (1 + θi) dIt + (1 − αi) dDt − ct dt, (2) dyt = μi yt dt + σi yt dwt + dIt − dDt , (3) where ct is the consumption rate and the processes D and I represent the cumulative dollar amount of sales and purchases of the stock, respectively. These processes are nondecreasing and right-continuous adapted processes with D(0) = I(0) = 0. Let x0 and y0 be the given initial positions in the bond and in the stock, respectively. Let (x0, y0) denote the set of admissible trading strategies (c, D, I) such that (2) and (3) are satisfied, ct ≥ 0, ∫ t 0 cs ds < ∞ for all t, and the investor is always solvent, that is,6 xt + (1 − αi) yt ≥ 0, ∀t ≥ 0 and i ∈ {B, b}. (4) Similar to Constantinides (1986), we consider a CRRA investor who derives von Neumann–Morgenstern time additive utility from intertemporal consumption c with weight 1 − k and bequest at death with weight k, with a time discount 6 The assumption that μi > ri implies that the investor never shorts the stock and thus that yt ≥ 0. 2334 The Journal of Finance rate of ρ. For simplicity, we assume that death occurs at the first jump time of an independent Poisson process with intensity δ. Thus, after integrating out the mortality risk, the investor solves sup (c,D, I )∈ (x0, y0) E [∫ ∞ 0 e−(ρ+δ)t ( (1 − k) c 1−γ t 1 − γ + kδ (xt + (1 − αi) yt)1−γ 1 − γ ) dt ] . (5) B. Optimal Policies with No Transaction Costs In this section, we solve the optimal consumption and portfolio selection problem in the absence of transaction costs, that is, θi = αi = 0, under the regimeswitching model presented in the previous section. The results in this section serve as a benchmark for the subsequent analysis. In this case, the cumulative purchases and sales of the stock can be of infinite variation. Let τi be the first jump time since the beginning of regime i. The investor’s problem in regime i ∈ {B, b} can be rewritten as Vi(W ) = sup { yt :t≥0} E [∫ τi 0 e−(ρ+δ)t ( (1 − k) c 1−γ t 1 − γ + kδ W 1−γ t 1 − γ ) dt + e−(ρ+δ)τi V j (Wτi ) ] (6) subject to dWt = riWt dt + (μi − ri) yt dt + σi yt dwt , (7) where Wt ≡ xt + yt ≥ 0 and Vj(W) is the value function in regime j = i. Under regularity conditions on Vi and Vj, the Hamilton–Jacobi–Bellman (HJB) equations take the form sup (ci , yi ) { 1 2 σ 2 i y 2 i ViWW + riW ViW + (μi − ri) yiViW − ciViW − (ρ + δ + λi)Vi + λiV j + (1 − k) c 1−γ i 1 − γ + kδ W 1−γ 1 − γ } = 0, (8) where i, j ∈ {B, b}, i = j. We conjecture Vi(W ) = Mi W 1−γ 1 − γ , for a constant Mi > 0, i ∈ {B, b}. (9) By the first-order conditions, we have ci = ( ViW 1 − k )− 1 γ and yi = − (μi − ri)ViW σ 2 i ViWW . Liquidity Premia and Transaction Costs 2335 Then, plugging (9) into (8), we have that Mi and Mj satisfy the system of equations −(ηi + λi)Mi + γ (1 − k)1/γ M 1−1/γ i + λi M j + kδ = 0, (10)