THE WORLD PRICE OF LIQUIDITY RISK
3.3 Liquidity-adjusted capital asset pricing model
As discussed in section 3.2, it has been empirically shown that liquidity is a priced factor both as a characteristic and as a systematic risk factor. However, a theoretical asset pricing model that includes both of these aspects of liquidity was proposed only recently. The liquidity-adjusted capital asset pricing model (LCAPM, henceforth) of Acharya and Pedersen (2005) is derived from a framework similar to CAPM in that risk-averse investors maximize their expected utility under wealth constraint, but by replacing the cost-free stock price, Pi,t, with a stochastic trading-cost-adjusted stock price, Pi,t−Ψi,t, where Ψi,t is a trading cost in absolute amount, in an overlapping generation economy. The LCAPM is in equation (3.1).
Et(Ri,t+1−Ci,t+1) =Rf +λt
Covt Ri,t+1−Ci,t+1, RDt+1−Ct+1D
V art RDt+1−Ct+1D (3.1) where Ri is a gross return of stock i, Rf is a gross risk-free rate and Ci,t is a trading cost per price at time t (Ci,t ≡ Ψi,t/Pi,t−1). Subscript t for expectation, covariance, and variance denotes that these operators are conditional on the information set available up to time t. Superscript D denotes that the variable is defined in terms of the local market portfolio (D from ‘domestic’).
As a result of the study of the liquidity-adjusted price, LCAPM has three addi- tional covariance terms related to stochastic trading costs other than the traditional market risk component. It is clear that without trading cost terms of CD and Ci, the LCAPM in (3.1) is equivalent to the traditional CAPM.
By assuming constant conditional variance or constant premia, the unconditional version of the model is derived as:
E(Ri,t−Rf,t) =E(Ci,t) +λDβi1,D+λDβi2,D−λDβi3,D−λDβi4,D (3.2)
where,12
βi1,D = Cov Ri,t, RDt
V ar(RDt −[CtD−Et−1(CtD)])
βi2,D = Cov Ci,t−Et−1(Ci,t), CtD −Et−1 CtD V ar(RtD−[CtD−Et−1(CtD)]) βi3,D = Cov Ri,t, CtD−Et−1 CtD
V ar(RDt −[CtD−Et−1(CtD)]) βi4,D = Cov Ci,t−Et−1(Ci,t), RDt
V ar(RDt −[CtD−Et−1(CtD)]). The risk premium is defined as λD =E λDt
=E RtD−CtD−Rf,t
. Additionally, I define:
βi5,D ≡ βi2,D−βi3,D−βi4,D (3.3) βi6,D ≡ βi1,D+βi2,D−βi3,D−βi4,D.
βi5,D is defined as a linear combination of the three liquidity betas excluding market beta, while βi6,D contains all four covariance terms in it. Henceforth, I will call βi5,D the liquidity net beta and βi6,D the net beta. It is worth noting that the net beta corresponds to the covariance term in equation (3.1) and the liquidity net beta helps distinguish the pricing effect of liquidity risks from that of market risk. As shown in Acharya and Pedersen (2005), each beta has an associated economic interpretation:
• βi1,D is similar to traditional market beta of CAPM except for the terms related to trading cost in the denominator.
• βi2,D is a liquidity risk arising from the comovement of individual stock liquidity with market liquidity (Chordia, Roll, and Subrahmanyam (2000), Hasbrouck
12Since liquidity is persistent (Chan (2002), Pastor and Stambaugh (2003), Acharya and Pedersen (2005), Korajczyk and Sadka (2006)), trading cost terms are denoted in terms of their innovation.
and Seppi (2001), Huberman and Halka (2001), Coughenour and Saad (2004)).13 βi2,D is expected to be positively related to asset returns since investors require compensation for a stock whose liquidity decreases when the market liquidity goes down. For a similar reason, a stock whose liquidity negatively comoves with market liquidity will be traded at a premium since such stock is easier to sell when the market is highly illiquid.
• An unexpected decrease in stock market liquidity will bring a potential wealth reduction for investors who hold stocks that are highly sensitive to market- wide liquidity and need to liquidate them immediately since liquidation of such stocks would be costlier under low market liquidity (Pastor and Stambaugh (2003), Sadka (2004)). βi3,D captures this liquidity risk and negatively relates to expected returns since investors are willing to accept low returns on stocks whose expected return is high when the market is illiquid.
• The fourth beta, βi4,D, is newly proposed by Acharya and Pedersen (2005) and is negativelyrelated to asset returns since stocks that become more liquid in a down market will be preferred by investors, thus will be traded at a premium.
The negative sign forβi4,D is due to investors’ willingness to accept low returns on such stocks.
If the world financial markets are fully segmented, the local market version of LCAPM in equation (3.2) should be able to explain the cross-sectional dispersion of asset returns. However, in perfectly integrated world financial markets, countries are irrelevant to investors and individual stocks should co-move with world-market
13Throughout this paper,β2i,D will sometimes be referred to as acommonality beta.
factors rather than with local-market factors (Karolyi and Stulz (2003)). Thus, I have the following unconditional version of LCAPM under perfectly integrated financial markets.
E(Ri,t−Rf,t) = E(Ci,t) +λWβi1,W +λWβi2,W −λWβi3,W −λWβi4,W. (3.4) where,
βi1,W = Cov Ri,t, RWt
V ar(RWt −[CtW −Et−1(CtW)])
βi2,W = Cov Ci,t−Et−1(Ci,t), CtW −Et−1 CtW V ar(RtW −[CtW −Et−1(CtW)]) βi3,W = Cov Ri,t, CtW −Et−1 CtW
V ar(RWt −[CtW −Et−1(CtW)]) βi4,W = Cov Ci,t−Et−1(Ci,t), RWt
V ar(RWt −[CtW −Et−1(CtW)]).
Superscript W denotes that the variable is defined in terms of the world market (W denoting ‘world’). The liquidity net beta and the net beta are defined in a similar way to equation (3.3).
βi5,W ≡ βi2,W −βi3,W −βi4,W (3.5) βi6,W ≡ βi1,W +βi2,W −βi3,W −βi4,W
I investigate both of these versions of the LCAPM.