Motivation Outline Model Empirical Results
Asset Pricing with Liquidity Risk
Viral V. Acharya and Lasse Heje Pedersen
2005
Presented By: Farhang Farazmand
November 6th, 2007
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Goal
Pose a model that takes liquidity concerns into account.
Examine the impact on returns.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity
Features of liquidity we would like to capture
Riskiness(availability) of liquidity.
Commonality in liquidity.
Time variation in liquidity.
Channels through which it affects asset returns.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Outline
The Model.
Calibration.
Various results.
Conclusion.
Discussion.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
The Agent
The model is set within an OG-Model framework where t ∈ Z
The agents live for two periods and the number of agents in
each generation is constant and set to N.
Income is given by the endowment.
The agent trades at time t and t + 1 but has
consumption,xt+1 , at time t + 1 only.
Utility exhibits CARA
nx
−Et exp−A t+1
where An is agent n’s absolute risk aversion.
Borrowing and lending is done at a constant real risk-free rate
rf .
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Securities
There are I securities, with S i representing the shares of
security i.
Dti and Pti are the time t dividend respectively price of
security i.
Cti is assumed to be the cost of Illiquidity for asset i and is
defined as the per-share cost of selling security i. So you buy
at Pti but sell at Pti − Cti
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Variables
we assume the following dynamics for the exogenous variables
Dt ¯ ¯
= D + ρD Dt−1 − D + εt
Ct ¯ ¯
= C + ρC Ct−1 − C + ηt
¯ ¯
D, C ∈ RI , ρD , ρC ∈ [0, 1], (εt , ηt ) is i.i.d. Gaussian with
+
mean zero and var(εt )=ΣD , var(η)=ΣC and E (η D εt ) = ΣCD
Note that both dividend and cost of illiquidity are persistent, i.e.
have predictable growth rates.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
Define
The asset return
Dti + Pti
rti = i
Pt−1
.
The relative illiquidity cost
Cti
cti = i
Pt−1
Market return
S i (Dti + Pti )
rtM = i
i i
i S Pt−1
relative market illiquidity
S i Cti
ctM = i
i i
i S Pt−1
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
CAPM
In equilibrium it holds
i i f
i i M
cov t (rt+1 − ct+1 , rt+1 −M )
t+1
Et (rt+1 − ct+1 ) = r + λt
M M
vart (rt+1 − ct+1 )
M M
where λt = Et (rt+1 − ct+1 − r f ). This can be re-written as
i M i M
i f i covt (rt+1 , rt+1 ) covt (ct+1 , ct+1 )
Et rt+1 = r + Et (ct+1 ) + λt +λt
M M
vart (rt+1 − ct+1 ) M M
vart (rt+1 − ct+1 )
¡ ¢ £ ¡ ¢ £
market beta comovement in illiquidity beta
i M i M
cov t (rt+1 , ct+1 ) covt (ct+1 , rt+1 )
−λt −λt
M M
vart (rt+1 − ct+1 ) M M
vart (rt+1 − ct+1 )
¡ ¢ £ ¡ ¢ £
beta with market illiquidity beta of asset illiquidity and market return
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
Channels of Liquidity
i M
covt (ct+1 , ct+1 ): Comovement with market illiquidity.
Investor demand compensation for holding assets that do not
hedge against overall illiquidity.
i M
covt (rt+1 , ct+1 ): Investors cherish assets that payoff when
market liquidity is low.
i M
covt (ct+1 , rt+1 ): Investors prefer assets that are liquid when
markets are down.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
Other Implications
Given a persistent cost of liquidity, i.e. ρ C > 0 it is shown
(under a mild technical condition) that conditional expected
returns increase with illiquidity. In essence if higher illiquidity
today predicts higher illiquidity tomorrow then expected
returns must rise.
If the demand in returns is rising due to a higher cost of
illiquidity then prices today must be decreasing. Hence
p p
covt (ct+1 , rt+1 ) < 0
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
The Unconditional CAPM
Can obtain unconditional CAPM given independence of
dividends and cost of illiquidity across time.
However, empirical studies have documented that illiquidity is
persistent.
So instead of having independence across time it is assumed
that the conditional covariances between innovations in
illiquidity and returns are constant.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Liquidity Adjusted CAPM
The Unconditional CAPM
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Estimation
(i) For each month measure cti for each security. We do so by
using
i
Dayst i
1 Rtd
ILLIQit =
Daysti d=1
i
Vtd
where i i
is the dollar return in millions and V td is the dollar
Rtd
volume in millions on day d of month t. This is in percent per
dollar. ct is in dollar cost per dollar invested, so
M
cti = min(0.25 + 0.3 · ILLIQti · Pt−1 , 30)
M
Pt−1 is the ratio of capitalization of the market portfolio at
end of month t − 1 and the end of July 1962.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Estimation
(ii) To lessen the effect of having a noisy measure of illiquidity for
each stock the authors consider portfolios instead of individual
securities.
Using data covering the period July 1st 1962 to December
31st 1999 they form a market portfolio for each month. For
each year they also form 25 sets of portfolios sorted by their
previous year’s illiquiidty, last year’s variation in illiquidity and
size. Finally 25 portfolios are formed by first creating five
portfolios based on book-to-market value and then within
each quintile sorting by size.
Portfolio returns and costs of illiquidity measures are
computed. For the market portfolio and value of illiquidity
equal weighted values are preferred due to possible
over-representation of highly liquid assets in the sample.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Estimation
(iii) Then the innovations in illiquidity c tp − Et−1 ctp are estimated
as the residuals of an AR(2) specification of the unnormalized
version of the market portfolio’s measure of cost of illiquidity.
The same approach is used to compute the innovations for the
test portfolios. Note that the coefficients are those estimated
for the market.
(IV) Given the previous measures the βs are computed. Focus will
be on the value-weighted illiquidity portfolios.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Results
Let
β 1p : beta with market portfolio.
β 2p = covt (ct+1 , ct+1 ): Comovement with market illiquidity.
i M
β 3p = covt (rt+1 , ct+1 ): pays off when market liquidity is low.
i M
β 4p = covt (ct+1 , rt+1 ): portfolio is liquid when markets are
i M
down.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Results
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Results
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Liquidity Risk and Expected Returns
Run cross-sectional regressions on the test portfolios while
accounting for pre-estimation of βs and autocorrelations.
To impose the same premium, λ, on all measures of risk, the
betas, they consider the regression
E (rtp − r f ) = α + κE (ctp ) + λβ net,p
where β net,p = β 1p + β 2p − β 3p − β 4p . κ is an adjustment factor
for the difference between holding period and estimation period.
Sometimes κ is set to the average monthly turnover 0.034 implying
a holding period of appr. 29 months.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Liquidity Risk and Expected Returns
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Liquidity Risk and Expected Returns
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Liquidity Risk and Expected Returns
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Liquidity Risk and Expected Returns
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
The Economics
What is the required premium to hold illiquid rather than
liquid securities.
Within the model we can estimate this by looking at the
difference in premiums between the highest and lowest liquid
portfolios.
Using the estimated value of λ from Table 4 and the betas
from Table 1 we get.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
The Economics
Comovement with market illiquidity.
λ(β 2,p25 − β 2,p1 ) = 0.08%
Sensitivity of portfolio return to market liquidity.
−λ(β 3,p25 − β 3,p1 ) = 0.16%
Sensitivity of portfolio illiquidity to market returns
−λ(β 4,p25 − β 4,p1 ) = 0.82%
With a total effect of 1.1%. Evidently, agents especially care
about the option of being able to liquidate in recessions. Not
too surprising when you think about agents wanting to
smooth consumption.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
The Robustness of the results are studied by replicating the
results in Table4 for portfolios sorted on different
characteristics.
In Table 5 they show that the choice of equal weighted versus
value-weighted portfolios does not ruin the results.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness:Equal verses Value
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness:Equal verses Value
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
For portfolios sorted according to size we see a reduction in
significance.
However, for portfolios sorted on B/M-size the model fairs
poorly as does the simple CAPM.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
Now to control for size and B/M while running the
regressions.
The results for the illiquidity sorted portfolios are similar to
the earlier ones and the coefficient on the control variables are
for the most part insignificant.
Unfortunately, the same does not hold for B/M-size portfolios.
Liquidity risk is still not able to explain the cross-section of
expected returns.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Robustness
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Specification Tests
They cannot reject the restrictions the model imposes. I.e.
same λ for all risks,α = 0, and κ equal to the calibrated value.
They also test whether or not the liquidity adjusted CAPM
has zero pricing error. Compared to the simple CAPM the
model fairs better as illustrated by the graphs shown earlier.
Similarly, they test to see whether or not their estimated
premium equals the observed one. Again their model fairs
better than the simple CAPM.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Conclusion
Within a simple framework resulting in a liquidity adjusted
CAPM we have seen that liquidity risk matters.
We have seen that agents value the option of being able to
liquidate in down markets heavily.
Considering different sorted portfolios we have seen that
compared to the simple CAPM the model performs much
better at explaining the cross-section of expected returns.
Although, B/M-size sorted portfolios still pose a challenge.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk
Motivation Outline Model Empirical Results
Results
Discussion
Model the timing of asset purchases/sales.
Study the effect of short-selling on asset prices.
Viral V. Acharya and Lasse Heje Pedersen 2005
Asset Pricing with Liquidity Risk