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The Synthetic Cost of Liquidity Nicolas Mougeot∗ 1st June 2018

Abstract We propose a new approach to model the cost of liquidity based on the synthetic replication of the sale of a liquid asset. Assuming that markets are complete, the sale of a liquid asset can be replicated by a) taking the decision to sell an illiquid asset and eectively sell it at a later date at its fair value b) borrowing cash until the sale is completed and c) short-selling an equivalent liquid asset or portfolio of assets in order to cancel out the economic exposure over the same period. The synthetic replication implies that the cost of liquidity should equal the investor's credit spread plus the securities lending cost or cost of borrow, accounting for the time to liquidate and the holding period. The model produces annualized liquidity cost that can range from a few basis points for liquid stocks to 60bps for real estate and up to 3% for illiquid assets such as infrastructure or private equity. This approach links market and funding liquidity measures and can explain contagion of liquidity crisis among asset classes as well as the absence of liquidity premium in the presence of investors with little funding constraints such as pension funds and soverign wealth funds.

Nicolas Mougeot is at John Molson School of Business, Concordia University ([email protected]) ∗

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Electronic copy available at: https://ssrn.com/abstract=3189052

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Introduction

The 2008 debt crisis raised the awareness of illiquidity of a number of asset classes and investment vehicles.

Some investors invested in instruments such as commercial

papers that were deemed to be liquid only to discover that they weren't in the midst of the crisis.

Gating, which consists in forbidding investors from exiting a fund, became

the norm for a number of real estate funds, even though they were marketed as liquid instruments with daily or weekly liquidity. Yet, illiquid assets such as hedge funds, real estates, infrastructure or private equity, have since then beneted from strong inows, in particular from long-term investors such as pension funds and sovereign wealth funds. BarclayHedge for example estimates that $3.35 trillion were invested in hedge funds in 2017 vs. a previous peak of $2.14 trillion in 2007. Liquidity risk can also impact liquid asset classes such as xed income and equities. The 2010 sovereign debt crisis has even put into question the liquidity of one of the most liquid instrument, sovereign debt. And since then, regulators are trying to tackle the issue of liquidity going forward by constraining banks to own safer and more liquid assets on their balance-sheet while maintaining a steady access to credit to their clients. As institutional investors rely more and more on illiquid assets to beat their benchmark or meet their liabilities, measuring liquidity across asset classes using a unied framework becomes an important question to be addressed. Getmansky, Lo and Makarov (2004) summarize well the diculty of measuring liquidity.

They consider that a liquid asset is one that can be:

(1) traded quickly; (2)

traded in large quantities; and (3) traded with little impact on the prevailing price. But behind this simple denition hides a bigger challenge: how to measure it. To measure liquidity, one needs to understand what makes an asset liquid or illiquid. Is it its own characteristics or is it a function of investor's preference?

or both?

A stock may be

illiquid simply because it is overlooked by investors. Tech stocks may trade substantially

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Electronic copy available at: https://ssrn.com/abstract=3189052

more than railways stocks as retail investors may consider them as attractive, in line with the visibility argument advanced by Gervais, Kaniel and Mingelgrin (2001). A stake in a long-short equity hedge fund may be illiquid not because of the stocks traded by the hedge fund but just because the hedge fund imposes a number of clauses that prevents its investors from getting out too quickly. Even if stocks are traded on a daily basis, the hedge fund only allows for monthly or quarterly liquidity exit. Theses two examples highlight two radically dierent causes of liquidity: stock characteristics or legal covenant. The investor needs however in both cases to value the lack of liquidity, or illiquidity. There is a large literature on the liquidity of equity markets and a number of liquidity measures have been proposed that rely on stock's characteristics such as volume or bidoer spread.

Amihud and Medelson (1987) for example show that there is a positive

relationship between market-observed average returns and bid-oer spread.

Brennan,

Chordia and Subrahmanyam (1998) nd a signicant and negative relationship between returns and trading volume, even after accounting for the Fama-French factors. Amihud (2002) created an illiquidity index based on the ratio of absolute return to volume and found a strong relation between his illiquidity measure and stock returns.

Pastor and

Stambaugh (2003) construct an aggregate measure of liquidity based on signed volume of single stocks. They show that stocks with low sensitivity to their market-wide measure of liquidity under-perform stocks with high sensitivity by 7.5% annually on average over a 34 years period. A number of studies complement these empirical ndings by oering a theoretical framework to the impact of liquidity on asset pricing. For example, Acharya and Pedersen (2005) proposes a liquidity-adjusted CAPM according to which a security's return should depend on its own liquidity as well as on the covariances of its return and liquidity with the market return and market liquidity. Beber, Driessen and Tujip (2012) develop an asset pricing model accounting for heterogeneous horizons. They show that the introduction of short and long-term investors signicantly impact the properties of

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the liquidity premium. The standard measures of measuring liquidity risk present a number of drawbacks. First, Grossman and Miller (1988) raise the issue they may not perfectly reect liquidity. Some liquid stocks experienced a high turnover associated with high volume traded and negative returns in the midst of the 2008 crisis. A signed volume measure such as the one suggested by Pastor and Stambaugh (2003) could take this event as a liquidity shock. What happened however was a ight-to-quality which increased the volume of safer stocks but their returns were still negative given the large negative returns of the market overall. Lou and Shu (2014) also advance that the Amihud (2002) illiquidity measure is mainly driven by its volume component instead of its ratio of absolute return to volume. Therefore, it could capture non-liquidity eect linked to volume such as visibility, investor disagreement, or information uncertainty for example. Furthermore, all the aforementioned measures of (il)liquidity are independent from the demand for liquidity.

In other words, they are supply-side model focusing on the

liquidity provided by securities in opposition to a demand-side model analyzing liquidity requirements by investors. Therefore, a second stream of the literature has been initiated by Grossman and Miller (1988) in particular who link market liquidity and the supply and demand of immediacy, hence introducing a link between securities liquidity and the demand for liquidity. Brunnermeier and Pedersen (2009) expand on Grossman and Miller's work by introducing real-world funding constraints. Their model link asset liquidity and trader's funding liquidity: A trader is able to provide liquidity to the market if he can access funding himself. When funding conditions become tighter such as in 2008, the ability of the trader to provide market liquidity on a stock falls. Their model has the advantage of explaining a number of market-facts about liquidity such as the fact that liquidity can suddenly dry up, has commonality across securities, is related to volatility, is subject to ight to quality and co-moves with the market.

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Despite the large literature on the liquidity of equity markets and to some extent xed income instruments, there has been far less attention given to the liquidity of illiquid assets such as large real estate development, private equity and infrastructure transactions. One of the main reasons resides in the fact that volume or bid-oer spread data are poorly available if not nonexistent for illiquid markets. The second stream of literature linking market liquidity and funding liquidity is based on the presence of a market-maker which is usually not the case for illiquid assets such as infrastructure and can not therefore be easily extended to illiquid instruments. Getmansky, Lo and Makarov (2004) have proposed an alternative approach to measure the premium to hold illiquid assets by assuming that the more illiquid assets are the higher auto-correlation can be found in their returns. This is due to the fact that illiquidity allows portfolio managers to smooth their returns, thus introducing serial correlation. Auto-correlation however has been found in the most liquid assets such as the S&P 500

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which would indicate that liquidity is not the sole factor explaining auto-correlation . In spite of the lack of approaches that can equally be applied to both liquid and illiquid assets, we propose a new and simple approach to measure the cost of liquidity by synthetic replication. First of all, we dene liquidity as the time necessary to liquidate an asset at its fair value. In other words, liquid and illiquid assets dier with respect to the time it takes to sell them at their fair value. A small position in a liquid security may be liquidated at its fair value in a matter of hours or even seconds by an investor. That same investor however may be stuck in a hedge fund with tight lock-up restrictions for months if not a year before he can get his cash back. If an investor requires cash, he can choose between immediately selling a liquid asset and synthetically replicating the sale of a liquid asset. The synthetic replication consists in selling an illiquid asset which by denition takes time. In order to meet liquidity needs and oset economic exposure to the illiquid asset, the investor then borrows cash for the

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See for example Lo and Mackinlay (1988) who use a variance ratio test to show that US stocks do not follow a random walk and display serial correlation 5

period required to sell it at its fair value and short-sell an equivalent liquid asset. By equivalent, we assume completeness of nancial markets and imply that we can construct a portfolio of liquid securities that provides the same future stream of cash ows as the illiquid asset.

Borrowing cash allows to access funding immediately while selling the

illiquid asset and short-selling the liquid one allows to eliminate any economic exposure to the risky asset. The idea is therefore to synthetically replicate the sale of the liquid asset by a combination of a forward sale of the illiquid one, a cash loan and a short position in the liquid asset. The cost of liquidity in our model arises from the demand for immediacy: an investor may require cash now and needs to sell assets immediately. To that extent, our model is related to the previous works of Grossman and Miller (1988) and Brunnermeier and Pedersen (2009). We also link market-wide liquidity to funding liquidity but we dier with Brunnermeier and Pedersen (2008) as funding needs are met by the seller and not an intermediary such as a market-maker. The model can therefore be applied to a broader spectrum of assets, including those which do not trade on a public exchange. Our approach has links with Beber, Driessen and Tujip (2012) as we also show the impact of heterogeneity in investor's time horizon on the cost of liquidity. At the time the investor requests funding, we show that the dierence in value between the illiquid and liquid assets should be the investor's credit spread plus the stock lending cost. Our model provides a theoretical framework for the cost of liquidity and allows to empirically measure this cost across asset classes in a coherent way. It allows to measure the extra cost of owning illiquid assets and show that this should dier with investor's holding horizon. While we start the demonstration in a two-stage economy, we show that the model is also valid in an economy with

n

periods and highlight the importance of

the marginal investor on the cost of liquidity priced in the market. As our approach can be extended to any asset classes, it allows portfolio managers, insurance companies or pension funds to incorporate it within their asset allocation.

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Further, the model is simple and could encompasses some existing methods to assess liquidity cost. For example, market impact models are in practice used to measure the cost of of immediacy, i.e. the cost associated with trading large blocks without waiting. But the same models could also provide the time it would take to trade without market impact. The paper is structured as follows. The next section introduces the model and its basic assumptions. Section 3 investigates the cost of liquidity in a multi-period economy and discuss how our approach can explain stylized market-facts about liquidity such as contagion and the potential absence of liquidity premium for some illiquid assets. In section 4, we show how heterogeneity in credit risk and investment horizon among investors can materially impact the liquidity premium paid by the market. In section 5, we propose a scenario-based analysis to assess the liquidity cost produced by the synthetic replication. In section 6, we model dynamically the cost of liquidity and show that it encompasses both funding and market liquidity risks as proxied by the Fed's Senior Loan Oce Opinion Survey and Pastor and Stambaugh (2003) liquidity measures respectively. Section 7 concludes.

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The Model

2.1

The Market

The market is constituted of two types of risky assets that dier in terms of liquidity:

1. Liquid asset

ALiq

fair value noted

2. Illiquid asset

that can be sold at its fair value immediately at any time

t

at its

νtA

AIlliq

that requires a period

7

tL

to be liquidated at its fair value

We assume that markets are complete and that any illiquid asset has either an equivalent liquid asset with same stream of future cash ow and volatility or can be replicated by a portfolio of liquid assets with that same properties. In practice, the equivalent liquid asset could exist such as on-the-run US treasuries which are more liquid than o-the-run US treasuries despite quasi-identical cash ows (see for example Fontaine and Garcia (2012)). When such liquid asset does not exist, the Arbitrage Pricing Theory of Ross (1976) tells us that the future cash ows of such asset

ALiq

could be mimicked by a

portfolio of assets. Some investment banks for example mimic the payo of hedge fund indices with a portfolio of liquid instruments as hedge funds may be hard to short-sell and to trade at a daily frequency. We assume that investors can borrow or lend money to a nancial institution (e.g. a bank). Per unit of time, the borrowing cost is equal to a constant risk-free rate credit spread

csi

rf

plus a

where the i subscript refers to the investor i. We initially evaluate the

cost of liquidity with one representative investor but we will further relax this assumption by allowing for

J

investors with heterogeneous access to credit. The investor can also

invest cash in a risk-free investment that pays the risk-free rate the credit spread relaxed. period

csi

We initially model

as an independent variable but this assumption could be further

The price of the bond sold by the investor

(T − t)

rf .

must therefore follow

i

to obtain cash at time

t

for a

e−(rf +csi )(T −t) .

There is also a securities lending market where investors can short sell a liquid asset

ALiq

for a period

tL

in exchange of a borrow cost equal to

bcA,tL

. We assume that the

cash proceeds from the short-sale is used as collateral and yields the risk-free rate

8

rf

.

2.2

Cash Requirement and the Choice between the Liquid and Illiquid Assets

Let assume that an investor is seeking cash. This cash could be required to buy a new asset, increase an existing position in another asset or simply meet cash obligation such as investor's redemption for a hedge fund or pension payment for a pension fund. To meet this cash obligation, the investor could choose between divesting a liquid asset

ALiq

or an illiquid one

AIlliq .

Let assume that these two assets provide the same stream

of future cash ow (dividend, coupon, rent,...) dier.

ALiq

and terminal value but their liquidity

is highly liquid and can be liquidated instantaneously without incurring any

market impact.

AIlliq

however requires a period

tL

tL

is an important

AIlliq

at its fair value.

to be liquidated.

factor as it is dened as the time it takes to sell the illiquid asset

Therefore, tL does not represent the time it would take for a re sale but instead the time to sell the asset in an orderly manner. This is an important distinction as for example a large equity block trade could nd a buyer immediately at a discount price but could also be sold at its fair price by allowing for more time. As both assets are identical in all respects but their liquidity, if they trade at the same price, any investor should prefer

ALiq

. Indeed, in practice, the illiquid asset should

trade at a discount vs. the liquid one. But by how much less? Assume now that there is a loan market. If an investor requires cash at time he owns

AIlliq

(rf + csi ) tL

T − tL ,

then he could sell

. If

instead, the investor could then borrow cash to meet its liquidity, paying

where

rate that investor

i

rf

is the risk-free rate and

must pay for a period

csi

is the credit spread above the risk-free

tL .

However, from the decision to sell the illiquid asset at time sale at time

ALiq

T,

to the eective

the investor would be exposed to the return of the illiquid asset. In order

to cancel out the exposure to the illiquid asset between time could short-sell

T − tL

ALiq ,

paying a borrow cost equal to

9

bcA

T − tL

and

T,

the investor

. The proceeds to the short-sale

is used as collateral and yields the risk-free rate indierent at time

T − tL

Selling



Synthetically replicating the sale of

1. Selling

AIlliq

Therefore the investor should be

between:



ALiq

rf .

immediately or

with eective sale date

ALiq

by:

A

Illiq T , yielding a log-return rT −t L ,T

as the investor

remains holder of the illiquid asset until it is eectively sold

2. Borrowing cash from

3. Short-selling

ALiq

T − tL

until

T

to

T,

yielding

− (rf + csi,tL ) tL

and put the cash as collateral, yielding

A

Liq + −rT −t L ,T

(rf − bcA ) tL

The replication strategy allows the investor to obtain cash before the denitive sale of the illiquid asset while canceling the exposure to the risky asset, eectively replicating the sale of a liquid asset. Given that

tL

is the time it takes to sell the illiquid asset at

T

should equal its fair value which

PT Illiq = νTA = PT Liq

(2.1)

its fair value, the price of the illiquid asset at time should also equal the price of the liquid asset:

A

At time

T,

A

the price of the illiquid asset should therefore be such that the investor

is indierent between selling the liquid asset or opting for the synthetic replication. The cash proceeds from the sale of the liquid asset yields nothing, keeping cash available for

2

whatever need the investor has . The replication strategy produces the following P&L:

2

Indeed, introducing a return on the available cash would not change anything to the result as cash is available in both cases 10

A

A

Illiq rT −t ,T | {zL }

P &Lreplication =

Long position in illiquid asset

liq + (rf − bcA ) tL − (rf + csi,tL ) tL − rT −t {z } | L ,T {z | }

cash loan

short sale of the liquid asset

(2.2) Assuming that all investors have the same credit risk represented by the credit spread

csi ,

tting equation 2.1 into equation 2.2 leads to our rst proposition:

Proposition 1.

If all investors have the same credit risk, the price of the illiquid asset

must follow:

A

A

Liq −(bcA +csi )tL Illiq e = PT −t PT −t L L

Proof.

(2.3)

The price of the illiquid asset must be such that an investor is indierent between

selling a liquid asset or implementing the replication strategy. The P&L of the synthetic replication must then be equal to zero:

A

A

liq Illiq + (rf − bcA ) tL = 0 − (rf + csi ) tL − rT −t rT −t L ,T L ,T

which is equivalent to:

ln

As



ALiq

A

PT Illiq/P

and

AIlliq T −tL

AIlliq



− (rf + csi ) tL − ln



A

PT Liq/P ALiq

T −tL

have the same fair value at time

simplied to equation 2.3.

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T,

+ (rf − bcA ) tL = 0

the above equation can be

According to equation 2.3, the log return of the illiquid asset between time

T

T − tL

and

is equal to the log return of the liquid asset minus the sum of the investor's credit

spread and the liquid asset's borrow cost multiplied by the time necessary to liquidate the illiquid asset:

A

A

Illiq Liq RT −t = RT −t + (csi + bcA ) tL L ,T L ,T

2.3

(2.4)

The Impact of the Holding Period and the Decision to Buy

We now turn our attention to the holding period. The holding period, denoted tH , is dened as the time between the purchase of the asset and the eective selling date. Let assume now that the investor is standing at time one of his assets at time

T

T − tH .

He's planning to eectively sell

and knows the cost associated with selling the illiquid assets.

In absence of arbitrage, the relationship between the liquid and illiquid assets as stated in proposition 1 should also hold at time

Corollary 2.

t:

In in a complete market without transaction cost and assuming the cost

of borrow, the credit spread and the time to liquidate to be all constant, the absence of arbitrage and equation 2.3 imply that the price of the illiquid asset at the time T − tH of purchasing it , with T − tH < T − tL must follow:

A

A

Illiq Liq PT −t = PT −t e(bcA +csi )tL H H

(2.5)

which implies that the log-return between the decision to buy the illiquid asset at time

T − tH

and the eective date of selling it at time

12

T

should equal:

A

A

Illiq Liq RT −t = RT −t + (bcA + csi ) tL H ,T H ,T

Corollary 3.

(2.6)

In a market with buy-and-hold investors, investors should be indierent

between holding a liquid or an illiquid asset and therefore the price of both assets should be equal.

According to equation 2.6, the longer the asset is held, the lower the impact of liquidity cost. By dividing both sides of equation 2.6 by the holding period tH , we obtain the relationship in terms of average annualized expected log-returns:

A

Illiq RT −t H ,T

tH As

tH

tends to innity,

A

=

Liq RT −t H ,T

tH

(bcA,tL +csi,tL )/tH

+

(bcA + csi ) tL tH

(2.7)

tends to zero and the returns of the liquid

and illiquid assets converge.

Corollary 4.

equal to

The annualized cost of liquidity for an investor with holding period tH is

(bcA +csi )tL tH

as it is the extra return an investor should require to bear the lack of

liquidity when he will need to liquidate the illiquid asset.

This corollary provides a measure of the annualized cost of liquidity which indicates what an investor should require to be indierent between an illiquid asset and its equivalent liquid portfolio. This measure can be used to assess the cost of liquidity of a broad set of assets and investment vehicles as we show in section 5.

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Proposition 5.

Everything else being equal, an investor with a high turnover should

prefer liquid assets over illiquid assets

Proof.

A

and

and

tB H,i .

Assume two assets

and holding periods

tA H,i

B

with respective liquidating time equal to

Subscript

i

and

tB L

is added to signify that the holding period

is an investor's decision. If the liquid equivalent of assets

A

R

and

and same volatility, the average expected returns of

to

tA L

A

B

and

B

yield the same return

should in turn be equal

A

RAIlliq

=

RBIlliq =

Illiq RT,T −t

H,i

=

tH,i BIlliq RT,T −t H,i

=

tH,i

Let assume that

A

¯ R tA H,i ¯ R tB H,i

+

(bcA +csi )tA L tA H,i

+

(bcA +csi )tB L tB H,i

is less liquid than

have positive expected returns.

B,

implying that

B tA L ≥ tL

An investor should prefer

B

to

and that both assets

A

if its return is the

greatest, which implies:

¯ R tB H,i

+

¯ (bcA + csi ) tB (bcA + csi ) tA R L L > + A A tB t t H,i H,i H,i

(2.8)

Re-ordering, we get:

tB H,i tA H,i as

B tA L ≥ tL .

¯ − (bcA,t + csi,t ) tB R L L L ≤1 < ¯ R − (bcA,tL + csi,tL ) tA L

(2.9)

Finally:

A tB H,i < tH,i

(2.10)

Equation 2.10 implies that the holding period of illiquid assets has to be longer than for liquid investments in order to compensate for the cost of liquidity. An investor with

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a high turnover will therefore be unlikely to invest in an illiquid asset as he will face the cost of liquidity each time he closes a position. Pension funds and sovereign wealth funds are likely to hold assets for a longer period than a speculative hedge fund for example and their presence in a market should impact the market-price of liquidity as we will see in a latter section.

3

The Cost of Liquidity in a Multi-period Economy

3.1

Expanding the Fair Value to

n

Periods

We have shown the relationship between the liquid and the illiquid assets returns in a two-stage economy where there is only one decision to buy and sell. This result can be extended to an economy with

n

periods. First, assume that there is a date

T

where the

liquid and illiquid assets have the same price. In the case of xed income instrument,

T

could be the maturity date: While o-the-run treasuries usually trade at a discount

vs. on-the-run treasuries because of liquidity dierential, their price should converge at maturity to the par. In the case of a private investment in public equities (or PIPE), at which the investor is free to sell his stake in the market.

T

T

could be the date

could also be the date

at which a private equity manager anticipates to put back his investment on the market through an IPO. For real estate investments or equities where there is no nal date,

T

could be equal to innity. Let assume that an illiquid asset is traded every takes a period and

T − tH ,

tL

T − tH

periods and that it always

to nd an investor and sell the asset at its fair value. At time

T − tL

we have shown that the discount between the price of the illiquid asset and

the liquid one is the same and the price of the illiquid asset must respect:

15

A

A

Illiq Liq PT −t = PT −t e−(bcA +csi )tL H H

This is therefore the fair value of the illiquid asset at time that

νTA

is the fair value at time

constraint at time

T

(3.1)

T − tH ,

to the same extent

. Assume now that an investor was facing liquidity

T − tH − tL . As in the previous section, this investor could have chosen

between selling a liquid asset or implementing the synthetic replication strategy. This replication strategy would have again consisted in 1) deciding to sell the illiquid asset, taking a period tL to sell, b) borrowing cash and c) short-selling the liquid asset to cancel out the economic exposure. As in the previous section, the price of the illiquid asset at time

ln

T − tH − tL



A

Illiq PT −t −t L

must be given by the replication strategy:

  A  AIlliq ALiq Liq PT −t / P − (r + cs ) t − ln / P + (bcA − rf ) tL = 0 −t i L f T T H L H

However, at the time of unwinding the transactions (i.e.

(3.2)

the date of the eective

sale of the illiquid asset), the fair value of the illiquid asset is not the price of the liquid one but is instead given by equation 3.1. Replacing

A

Illiq PT −t H

by

A

Liq e−(bcA +csi )tL PT −t H

into

equation 3.2 implies that:

A

A

Liq Illiq = PT −t e−2∗(bcA +csi )tL PT −t H −tL H −tL

Again, at the time of purchasing the asset, i.e. at

T − 2 ∗ tH

(3.3)

, the relationship must

hold too:

A

A

Illiq Liq PT −2∗t = PT −2∗t e−2∗(bcA +csi )tL H H

This recursive approach can then be extended to any proposition:

16

(3.4)

n period, leading to the following

Proposition 6.

In a multi-period economy, where investors trade for n periods of equal

time tH , the price of the illiquid asset must follow:

A

A

Illiq Liq PT −n∗t = PT −n∗t e−n∗(bcA +csi )tL H H

For each period

tH ,

(3.5)

the log-return of the illiquid asset is given by:

A

A

rtHIlliq = rtHLiq + (bcA + csi ) tL with date

T

n = T /tH . n

is the frequency at which an asset is traded until the maturity

and can be interpreted as a measure of turnover to link it to previous studies of

liquidity. A high value of

n

(3.6)

n

means a frequent trading of the asset. For a given

T,

a high

should also imply a high cost of liquidity as per our denition the cost of liquidity is

associated with trading/liquidating the asset. This is conrmed by Equation 3.5 where a high

n

generates a high discount factor.

For illustration purposes, Figure 8.1 displays the log price of an illiquid asset since 1871 whose liquid equivalent would be the S&P 500, by assuming a terminal date equal to year 2300, a stock lending cost of 50 bps, a credit spread of 1% and a holding period of 2 years. Indeed, as Figure 8.1 shows log-prices, the spread between the liquid asset, i.e. the S&P 500 and the illiquid one is constant and should be equal to

3.2

−n ∗ (bcA + csi ) tL .

Interpreting the Cost of Liquidity

According to Equation 3.5, the price of the illiquid asset is a function of:

1. The price of the liquid asset

A

Liq PT −t L

17

2. A characteristic of the asset itself as well as market conditions, the time to liquidate,

tL 3. A characteristic of the investor, its net cost of credit, measured by the credit spread

csi 4. A characteristic of the asset itself, the security lending cost

bcA

The fact that the illiquid asset price is a function of the liquid one is obvious.

In-

vestors should always compare illiquid assets with their liquid counterparts should they are available.

The cost of liquidity is a function of the investor's credit spread which

shows similarity with Brunnermeier and Pedersen (2009) who link the cost of liquidity with funding conditions of the market makers.

Our model diers from the approach

of Brunnermeier and Pedersen as we do not assume the presence of a market makers, allowing our approach to be used to model the cost of liquidity of markets where there is no intermediaries. The cost of liquidity is also a function the security lending cost. This cost could vary from one asset class to another.

Data from brokers show that the cost of borrow can

vary from 20bps for liquid stocks to a few percentage points for hard-to-borrow stocks. While our model assumes market completeness, further research could look into relaxing this hypothesis and introduce a tracking error risk that could be added to the standard borrow cost in the event that an illiquid asset's future cash ows could not be perfectly replicated with cash ows of available liquid assets. In practice though, this may not be a major issue as the seller owns the illiquid asset. Therefore, if he needed to short-sell it in a derivatives format with a bank for example, he could put his illiquid asset as collateral with the bank and the bank could use it to

3

adjust its delta .

3

This method is often used by corporate derivatives traders who are asked to price a put on a large 18

The time to liquidate, tL , could be interpreted as a liquidity beta as it is the sensitivity of the illiquid asset to the liquidity cost per unit of time. Highly illiquid assets should take time to nd buyers at their fair value and should therefore display a high

tL .

The fact that the cost of liquidity is a function of asset characteristics (time to liquidate and borrow cost) as well as general credit conditions can explain a number of stylized fact about liquidity. First, a shock on the credit market should impact all asset classes equally and increase the cost of liquidity. In 2008, when conditions to access to the credit markets became tighter, most liquidity measures increased from LIBOR-OIS spread on the xed income repo markets to bid-ask spreads in equity markets to the discount required to enter into an illiquid investment. Second, the cost of liquidity is a function of the asset's own characteristics which can explain why some assets may face liquidity shocks that do not aect other asset classes. Further, it can explain why a strong demand for a specic asset class can signicantly reduce the required cost of liquidity. Real estate markets in some specic areas such as London for example have known a huge demand from sovereign funds and pension funds pushing the required cost of liquidity close to zero despite the fact that real estate is one of the least liquid asset class.

4

Liquidity Cost in an Economy with Heterogeneous Investors

In practice, investors have heterogeneous access to credit and can have dierent asset's holding periods.

Long-term investors such as pension funds or sovereign wealth funds

equity position. If the bank's client owns the large position, it can use it as collateral and signicantly reduces the cost of the put given that the trader has access to the position to adjust his delta.

19

usually hold asset for the long-run. Other types of investors ranging from retail investors to hedge funds have much shorter time horizon. All these investors also dier in their access to credit.

The Government Pension Fund Global of Norway, managed around

4

NOK 5'547 bio as of October 2014, which is equivalent to around USD 854 bio .

It

received around USD 2-3bio of new inow each month from the revenue generated by the sale of Norwegian oil. Given the size of the new inow, the fund can somewhat lend to itself if it needs capital as it takes time to allocate any new dollar across asset classes. Its own credit spread, while not published by the Fund, is likely to be close to zero. The cost of liquidity that the Government Pension Fund Global of Norway should price should therefore be dierent from the cost required by a short-term investor facing a high credit spread such as a retail investor or a hedge fund in a dire situation. Indeed, while these radically dierent types of investors may face each other in the equity market chasing the same stock, an average retail investor is unlikely to look at the same real estate investment as the Norwegian fund. Therefore, the question is how dierence in credit worthiness and time horizon may impact dierent markets.

4.1

Heterogeneous Investors and a Single Asset

In various markets such as real estates or private equity, it is not rare to see a number of investors competing for the acquisition of a single asset, be it a specic building or a new equity tranche in an existing investment.

In practice, the investors will assess

the current value of the potential investment through various methods (CAPM, NPV of future cash ows, etc...) but we consider here for simplicity that they have all access to the fair value of the investment. This could for example be the case of a private investment in public equities where a few institutional investors are invited to bid for a block of securities. They are oered a discount price vs. the current listed value in exchange of being locked up for a period

4

See its web page for market value and inow data at http://www.nbim.no/en/the-fund/

20

of time. In such a case, the fair value of the illiquid investment is known and is simply the value of the liquid security. However, at the time of bidding for the investment, each investor may have dierent access to the credit market and they therefore dier in terms of credit spread. They could also dier in terms of holding period. While an investor may expect to sell its shares as soon as the lock-up period ends, others may consider to keep their investment for a much longer period. Therefore, while they all share the same time to liquidate

tL

and credit spread

csi .

and the same fair value, they dier in terms of holding period

tH

Now recall that the expected return is given by:

A

A

ILLiq RT −t H,i ,T

tH,i

=

Liq RT −t H,i ,T

tH,i

+

(bcA + csi ) tL tH,i

Assume for example that two investors bid for the same asset has a much lower credit spread than investor this case, investor

i

j

than investor

should accept a lower return for

j

AIlliq

but investor

i

and a much longer holding period. In

AIlliq

credit spread and long holding period. In turn, investor

AIlliq

(4.1)

i

than investor

j

given his low

should oer a higher price for

and win the bid as the seller should prefer to sell it at the highest

price. This leads to the following proposition:

Proposition 7.

Everything else being equal, the market cost of liquidity of a single illiquid

asset should be set by the investor with the lowest required cost of liquidity

(bcA +csi )tL tH,i

The impact of the marginal investor is therefore material as he xes the market's cost of liquidity. Indeed, in practice,

(bcA +csi )tL might be lowest for an investor with a high tH,i

credit spread and a very long holding period or the other way round. Another way to look at the impact of the marginal investor and his holding period is to recall equation 3.5 which links the price of the illiquid asset at each period to the price of the liquid

21

asset but to express it as a function of the investor

A

i:

A

Illiq Liq PT −n = PT −n e−ni ∗(bcA +csi )tL i ∗tH,i i ∗tH,i

n,

which is the trading frequency until the terminal date

holding period as it is de facto dened as:

T,

(4.2)

is also the inverse of the

ni = T /tH,i . e−ni ∗(bcA +csi )tL

is the discount

factor of the illiquid asset to the illiquid asset. Proposition 7 can therefore be expressed in terms of discount factor:

Corollary 8.

Everything else being equal, the market cost of liquidity of a single illiquid

asset should be set by the investor with the lowest required cost of liquidity expressed in terms of the liquidity discount factor e−ni ∗(bcA +csi )tL .

The mechanism of the impact of the access to credit is simple to comprehend. Easy access to credit with low credit spread allows some investors to bear a low liquidity cost. Those same investors should in turn require a lower cost of liquidity and agree on a higher price for the illiquid asset. To explore the impact of the holding period, let's take two investors with trading frequency

j

ni

trades only half the time investor

and

i

nj , with nj = 0.5 ∗ ni .

trades. If they both trade at time

use as fair value of the illiquid asset at time fair value of the illiquid asset at

t + tH

t + 2 ∗ tH .

while investor

4.2

i

j

t,

investor

i

will

should consider the

From that simple holding time dierence,

investor j should be at an advantage compared to investor investor

This implies that investor

i and oer a higher price than

even if they both pay the same credit spread.

Heterogeneous Investors and

X

Equivalent Assets

The equity market can't fall into the single asset category as investors have access not

22

to a single share in a company but to a large number and equivalent assets, i.e.

X

X.

In the presence of

X

identical

shares of the same public company, the overall impact of

the investor with the lowest required cost of liquidity is not the same as in the case of a single share in a private company. In practice, a market is constituted of a series of buyers as well as a series of sellers with various prices for liquidity. We now analyze the impact of heterogeneous access to credit on the cost of liquidity by assuming that that there are

I

sellers, all facing the

same holding period. For simplicity, we assume a single buyer, who could be represented by a market-maker with a credit spread equal to

csb .

Let's rank sellers according to their

credit spread such that:

• cs1 < cs2 < · · · < csb < · · · < csI−1 < csI

.

According to Proposition 1, sellers have a fair price for the illiquid asset equal to

A

A

Liq −(bcA +csi )tL Illiq e . = PT −t PT −t L L

Sellers should consider selling their illiquid assets if and

only if they nd a buyer who oers at least

A

Liq −(bcA +csi )tL . e PT −t L

This fair price is meant

to be a reserve price at which an investor should be willing to sell his illiquid asset. If another investor bid less than

A

Liq −(bcA +csi )tL e , PT −t L

the current owner of the asset should

be able to apply the synthetic replication strategy which consists in borrowing cash and short-selling the liquid asset for the time its fair value.

A

tL

necessary to liquidate the illiquid asset at

However, nothing prevents him from selling it at a price greater than

Liq −(bcA +csi )tL PT −t e L

if another investor is willing to pay above the reserve price of the

current asset owner. If an auction mechanism is put in place to decide the number of shares bought from each sellers and the associated share price, the seller with the lowest credit spread should have his order lled completely as long as the number of shares he oers to sell is less than what the buyer oers to buy. Then the seller with the second lowest credit spread

23

should see his order lled and so on and so forth as long as the buyer has not lled his order and that sellers have a credit spread lower than the buyer's. When it comes to a seller

i

with a greater credit spread than the buyer, we should

have:

A

A

Liq −(bcA +csi )tL Liq −(bcA +csb )tL PT −t e > PT −t e L L

and no transaction should take place. The buyer should prefer to buy the equivalent liquid asset as he would not be rewarded enough for the lack of liquidity of the illiquid asset. The seller should prefer to implement the synthetic replication strategy in order to obtain cash and cancel out his exposure to the risky asset. The seller could also choose to sell a more liquid asset to meet his liquidity requirement. Such behavior could explain the August 2007 quant crash when large caps have

5

dropped more than small caps as investors choose to sell the most liquid assets rst .

5

Measuring Liquidity Cost across Asset Classes

5.1

A Scenario-based Approach

In order to put an estimate on the cost of liquidity produced by our model, we use a scenario-based approach. We simply assume that the market can be in two states, normal or agitated. In a normal market, liquidity is plentiful, the time to liquidate asset is low and credit spreads are tight. In an agitated market, the situation is obviously the reverse where assets take longer to be divested at their fair value and credit spreads increase dramatically. The borrow cost however remains relatively constant for liquid assets which is in line with what has been observed empirically. In period of crisis such as 2008 for example,

5 See for example Cont and Wagalath (2014) for a good review of the August 2007 quant crash and the dynamic of re sale

24

banks` prop trading desks and hedge funds tend to reduce risk or face margin calls, reducing their appetite for short-selling. Therefore, the demand for borrowing securities is signicantly reduced, lowering the fees paid for such activity.

However, we assume

that the borrowing cost for illiquid asset increases as their replication must be function

6

of volatility . For the following empirical exercise, we assume that the market is agitated in 15% of the time while it is normal in remaining 85%. This proxy is in line with occurrence of crisis over the past 50 years which have tended to happen on average every seven years: 1966, 1973, 1979, 1987, 1994 (sole negative year for S&P 500 in the nineties), 2001 and 2008.

5.2

Equities

The classical approach of market impact models consists in assessing the cost of liquidity associated with selling an asset in a short period of time, potentially instantaneously. pressure.

The larger the position the higher the cost as it would increase the selling Liquidity cost is therefore associated with the potential haircut an investor

may face if he decides to divest a portion of his portfolio. Standard market impact models

7 are usually based on volume and volatility. Com-

paring the size of the trade with the average daily volume provides a measure of the relative importance of the trade compared to what the market can cope with on average. Liquidity can also dry up when markets become more agitated and therefore higher market volatility may imply lower liquidity. One caveat of market impact models is the fact that they rely on volume data while these ones are often unavailable for assets such as infrastructure transactions or hedge funds for example. Our approach is dierent from

6

Tracking error tends to increase as volatility increases. In other words, while correlation increases during agitated period, it becomes harder to perfectly replicate an asset 7 See for example Keim and Madhavan (1998) or Damodaran (2005) for introductory surveys of price impact models

25

market impact model yet very close: We assume that an investor facing cash requirement can either sell immediately a stake of an asset or take the time to sell his stake in an orderly manner. The cost of liquidity is a function of the size of the transaction too. the size, the longer time to liquidate the position will be long.

The larger

The model is in that

respect similar to the market impact model where the liquidity cost is a function of the size of the transaction relative to the average daily volume. For equity markets,

T

could therefore be measured by a market impact model. Instead of measuring the cost of liquidating immediately a position, an investor can use a market impact model to assess the maximum size any investor can liquidate in one day without impacting the market. Let us assume for illustration purposes that any order larger than 5% is likely to impact a stock price. Therefore, if an investor holds a position that represents one average daily volume, it should take an average 20 days to liquidate the position without selling pressure.

This number is likely to be higher in a volatile market, especially as

most investors are likely to be seller in a down market. If the credit spread of the investor is 150bps, the liquidity cost should therefore be equal to: 150bps * 20/252 = 11.9bps. Indeed, if the position was twice as large, the cost would be doubled too.

Figure 8.2

and 8.3 compare the cost of liquidity for a diversied equity portfolio with high turnover with a concentrated portfolio with a low turnover. as

The cost of liquidity is calculated

(bcA +csi )tL as highlighted by Corollary 4 and produces liquidity cost ranging from 8 tH

to 20bps for equities depending on the relative size of the transaction and the trading frequency.

5.3

Private Equity, Hedge Funds and Infrastructure

Investors enter into private equity deals as well as infrastructure transactions generally for the long-term, from 6 years to up to 50 years for some infrastructure transactions. The holding period, tH , is therefore very long. These transactions can also involve a long

26

time to liquidate the position,

tL .

In a private equity transaction, investors, or limited

partners (LP's) can be locked up for the rst two years in order to allow general partners (GP's) time to restructure the company they are buying. Infrastructure transactions can also impose very long lock-up periods, where the investors may even not be able to get out before the end of the deal, implying

tH = tL .

This is the case for example for some Private-Public Partnerships (PPP) where a longterm investor brings in the nancing for a project co-managed with a public authority such as a municipality.

Projects could be building a new opera or a football stadium

and the public authority will pay back the nancing throughout a long-term lease. Our framework is exible enough to account for these kind of contractual engagements which can constraint investors from exiting a transaction early.

tL

can also be a function of the

size of the transaction as this could also impact the time to liquidate. Figure 8.4 shows the expected cost of liquidity in our simple two-state framework in a private equity transaction with a two-year lock-up period (and up to three years in a volatile market). The average liquidity cost, 0.98%, is signicantly higher than for equities, and this despite a much longer holding period. Figure 8.5 reports a simulation for the cost of liquidity of a 10-year infrastructure transaction where the investor is locked up until the maturity of the deal. This could be for example a road concession or a government-led project where exit could be impossible before maturity.

tL

is therefore equal to 10 years and is totally independent from market

conditions and the size of the deal. In such a case, the cost of liquidity is predicted to be 3.33% p.a. which is comparable with the 2.9% liquidity return inferred by Franzoni, Nowak and Phalippou (2012) from private equity transactions from 1975 to 2006. Our liquidity framework can also be applied to hedge funds and their complex liquidity covenants. Even in normal market conditions, a majority of hedge funds impose liquidity constraints. They can limit the amount an investor can divest in any period (month or year) or use lock-up periods that implies a minimum time to liquidate. For hedge funds

27

that trade illiquid instruments such as structured products or private debt, lock-ups

8 reports average lock-up periods that can vary from

can last for up to a year. Prequin

2.3 months for managed futures and CTA hedge funds to 10.4 months for event-driven strategies with an average for all hedge funds close to half a year (5.9 months). Regarding the holding period, some recent surveys turnover of around 15%, implying tH

9 tend to show an average hedge funds portfolio

= 6.7. Under these assumptions, the cost of liquidity

for holding hedge funds would vary from 20bps for very liquid ones to 70bps for the least liquid hedge funds. These gures are below the liquidity premium found in hedge funds by previous studies such as Getmansky, Lo and Makarov (2004). The dierence could come a number of factors: First, by accounting for a turnover of 25% instead of 15%, the range becomes 55 to 110bps, already more in line with previous results. Second, previous studies use auto-correlation as a proxy for illiquidity: Getmansky, Lo and Makarov (2004) for example assume that the less liquid a fund is the higher is its propensity to smooth returns, which implies auto-correlation in its reported returns.

However, even

the S&P 500 index shows strong evidence of auto-correlation in its return (See Lo and Mackinlay (1988) for example) which may indicate that auto-correlation is not only due to illiquidity. Therefore, the liquidity premium previously estimated using this method may actually accounts for liquidity but also for other factors, yet to be determined.

5.4

Real Estate

The real estate market can be decomposed into a public side and a private side. The public side is mainly constituted of REITs which can be traded on stock exchanges. Its size though is relatively small compared to the private real estate market. As a comparison, the US REIT market was totaling around USD 670 bio at the end of 2013 (USD 271bio at the end of 2009) while the US commercial real estate market was estimated

8 9

See https://www.preqin.com/docs/newsletters/hf/Preqin_HFSL_Dec_2012_Liquidity_Structures.pdf See for example Deutsche Bank Eleventh Annual Alternative Investment Survey, February 2013

28

being worth around USD 11.5 trillion at the end of 2009. Size, however, is not the right factor to measure liquidity of real estate markets as its private side is by denition less liquid

10 . The time to liquidate a real estate portfolio

can be signicantly longer than for a stock portfolio, be it constituted of REITs or other common stocks. But buying and selling a residential or a commercial property takes time as it usually requires to value the asset then to advertise it and then the completion can take weeks as the buyer needs to secure nancing. Figure 8.6 shows for example the time on market of residential properties in the US.

11 , a US property web-listing company. It shows that the average

as provided by Zillow

time on the market of a property as declined over the past four years but it also shows a clear seasonality as individuals tend to prefer to move houses over the Summer. Using our simple two-state framework, we can assess the liquidity cost associated with trading real estates. First, real estates is an asset category where there is actually buy-and-hold investors. A number of individuals buy properties once in their lifetime or at least initially with the view of living forever in it. In that case, as

tH

would tend to

be very high, their required liquidity cost should converge towards zero. Figure 8.7 and 8.8 report the simulation of the cost of liquidity for real estates held for four and eight years respectively. The borrow cost is similar to the one paid for US REITs. The average simulated liquidity cost ranges from 34 to 69bps depending on the holding period.

10 The fact that REITs are more liquid is debatable actually. Large pension funds hold real estate portfolios that can amount for more than USD 50bio, which would represent almost 10% of the US REIT market. If they were invested in REIT instead of real estates directly, they would likely face liquidity issues should they need to ooad all of their portfolio. 11 Zillow real estate data is available on: http://www.zillow.com/research/data/

29

6

The Dynamics of the Cost of Liquidity

6.1

Modeling the Liquidation Time

We now turn our attention to the dynamics of the cost of liquidity by relaxing the hypothesis of constant credit spread and time to liquidate. The credit spread is modeled as the spread between Moody's corporate bond yields rated BAA and the US government 10-year bond yield. Following the market impact model literature and market practices, we assume that the time to liquidate is a function of market conditions. For example, Rashkovich and Verma (2012), discuss a market impact model where the permanent impact is modeled as a linear function of volatility and a ratio of the trade size over the average daily volume. We also model here the time to liquidate as as linear function of volatility. Assume for example that it would take 10 days to ooad a large equities stake in a low volatility at its fair value, i.e. without market impact and that low volatility is considered to be 12%. If the volatility rises fourfold to 48%, then we assume that it would take 40 days to liquidate the same equities position. The relative size of the trade is also accounted for as the number of days (or hours) necessary to liquidate an equities position would also vary with respect to the size of the transaction. This approach has the advantage to be exible enough to model the cost of each asset class using its own volatility. . The cost of liquidity is calculated as follow:

(creditspreadt + borrowcostt ) ∗ (timetoliquidatet ) / (holdingperiod) Where

creditspreadt

(6.1)

is the credit spread over the quarter using the spread between

Moody's corporate bond yields rated BAA and the US government 10-year bond yield. The borrow cost,

borrowcostt ,

is assumed constant.

The liquidation time is a linear

function of realized volatility and takes the following form:

30

timetoliquidatet = tL ∗ V ol3M /V olbase Where

V olbase

(6.2)

is the base volatility which corresponds to the time to liquidate

tL

in normal market conditions. Figure 8.9 shows the evolution of the cost of liquidity for small caps using the Russel 2000 Small Caps index as benchmark for the period 1986 to 2014 and assuming stocks are held by a long-term investor for a period of ve years. The simulated cost of liquidity would have increase during the October 1987 crash as well as in the aftermath of the TMT bubble burst from 2001 to 2003. Figure 8.10 compares the required cost of liquidity of large caps vs. a private equity investment. Large caps, modeled by the Russell 1000 index, are assumed to be held for a period of 5 years while the private equity investment is held for 8 years. We also assume that the large cap position can be disposed of at its fair value in 10 days in normal market conditions and linearly increases with Russell 1000 index 3-month realized volatility. The private equity investment would require two years in normal market conditions, in line with common private equity transactions where limited partners are locked up for at least the rst two years in the transaction.

We extend the initial two-year lockup to

three years in a stressed environment (modeled as high volatility here) to account for the fact that it would take longer to sell a private equity stake at its fair value in a highly volatile market. We therefore impose a cap on the time to liquidate, which becomes:

timetoliquidatet = M in (2 ∗ V ol3M /V olbase ; 3)

(6.3)

Equation 6.1 implies that the time necessary to liquidate a private equity at its fair value is an increasing function of volatility but can't exceed three years.

The holding

period is xed and equal to 8 years. We proxy the volatility of private equity by the 3month realized volatility of small caps stocks using by the Russell 2000 index and assume that the base volatility is 15%.

31

Indeed, all these assumptions could be adjusted to better approximate the cost of liquidity of certain types of investment where for example the holding period could be shorter (or longer) or the time to liquidate be shorter (or longer). Figure 8.10 shows that the spread between a liquid investment and an illiquid one would have signicantly increased during the 2008-9 crisis, highlighting diculty to sell illiquid assets at their fair value in a reasonable time. The cost of liquidity of private equity investment peaked at 3% and needs to be thought as an annualized cost. Therefore, the discount at which an investor would have been ready to dispose of his private equity stake would have been around 24% given that the holding period considered is 8 years. Our approach is exible enough to consider further developments. For example, while we have estimated the cost of liquidity using realized volatility, one could consider using implied volatility instead to obtain a forward-looking measure of the cost of liquidity. In such a case, investors could also calculate a term structure of liquidity cost using credit spread with dierent maturities as well as the term structure of implied volatility.

6.2

The Synthetic Cost of Liquidity and Existing Measures of Liquidity: The Case of Private Equity

We now assess the relationship between the synthetic cost of liquidity and existing measures for the case of private equity. Following Franzoni, Nowak and Phalippou (2012), we compare the synthetic cost of liquidity cost with Pastor and Stambaugh (2003) measures of liquidity. Pastor and Stambaugh (2003) have developed three measures of liquidity which are all available on Pastor's website

12

12 :



Levels of aggregate liquidity, Equation 5



Innovations in aggregate liquidity, Equation 8

See http://faculty.chicagobooth.edu/lubos.pastor/research/liq_data_1962_2013.txt

32



Traded liquidity factor, 10-1 portfolio return

Pastor and Stambaugh (2003) dene the level of aggregate liquidity as the cost of a $1 million trade distributed equally across stocks.

The synthetic liquidity we have

developed is a measure of the cost of liquidity, not the returns of assets or portfolios sensitive to liquidity. Therefore, the synthetic cost of liquidity should be compared with the level of aggregate liquidity as dened by Equation 5 in Pastor and Stambaugh (2003). Still following Franzoni, Nowak and Phalippou (2012), we also compare our cost of liquidity with changes in access to credit using data from the Federal Reserve Board's Senior Loan Oce Opinion Survey on Banking Lending Practices. This quarterly survey with data available since 1990 reports if loan ocers at major US banks are tightening standards for commercial and industrial loans compared to the previous quarter.

As

noted by Franzoni, Nowak and Phalippou (2012), it provides a proxy of funding liquidity similar to the liquidity theory of Brunnermeier and Pedersen (2009). In our context too, the level of access to credit is a key element of our cost of liquidity as it directly impacts the credit spread paid by investors. We then model the cost of liquidity of private equity following Equation 6.1, with a time to liquidation varying from two to three years, a borrowing cost equal to 1.5% and a holding period of eight years. Figure 8.11 reports the correlation between the cost of liquidity measures using quarterly data from Q2:1990 to Q3:2013.

As expected, the synthetic cost of liquidity for

private equity shows a strong correlation with the aggregate liquidity measure of Pastor and Stambaugh (2003) and to a lesser extent with their innovations in liquidity. The correlation with the credit access measure as given by the Fed survey is even stronger but can be easily explained by the presence of the credit spread in our liquidity measure. The funding liquidity measure (Fed) shows little to low correlation with the various market liquidity measures proposed by Pastor and Stambaugh (2003), emphasizing that they in

33

eect proxy two dierent types of liquidity risks. Figure 8.12 shows that the dynamics of our cost of liquidity has been closely linked to the movement in Pastor and Stambaugh (2003) aggregate liquidity as well as the change in credit standards as polled by the Fed. Figure 8.13 conrms the relationship between the synthetic cost of liquidity and the market (Pastor and Stambaugh (2003)) and the funding (Fed survey) liquidity measures. Both contribute to explain our cost of liquidity for private equity with signicant tstat and a R-square of 57.24%. The appendix also contains the same analysis with the innovations in aggregate liquidity as calculated by Pastor and Stambaugh (2003) and shows similarly strong explanatory power in terms of t-stat and R-square.

7

Conclusion

We derive a model of the cost of liquidity based on a simple idea: The cost of liquidity arises from the inability to sell an asset quickly at its fair value. Liquidity should therefore be measured as a function of the time it takes to sell any asset at its fair value. Based on this simple idea, we have developed a new approach to model the cost of liquidity by synthetic replication. While investors may not be able to sell instantaneously an illiquid asset, they can replicate the sale of a liquid asset. This involves taking the time to sell the illiquid asset at its fair value, borrowing cash to meet liquidity needs and short-selling an equivalent liquid asset in order to cancel out the economic exposure to the illiquid asset until it is eectively sold. We rst develop our model in a two-stage economy and expand it to an economy with

n periods.

We show that the cost of liquidity is a function

of investor's credit spread, the stock lending cost of the liquid asset, the time to liquidate the illiquid asset and the investor's holding period. We discuss how heterogeneity in access to credit and holding period can aect the

34

market price of liquidity. In particular, we show that in the presence of a single asset, the marginal investor with the lowest credit spread and longest holding period should set the price of liquidity. In the nal sections, we show how the model could be used in practice to model the cost of liquidity of a wide range of assets from listed equities to real estate and private placements. Our model is exible enough to account for liquidity issues that are specic to a market (small caps less liquid than large caps) as well as liquidity cost that could arise from legal obligations (lock-up periods in a hedge fund). We nally highlight how the cost of liquidity could be modeled dynamically and compare it with previous empirical studies on liquidity. It shows that the synthetic cost of liquidity encompasses both funding as well as market liquidity risks, conrming the hypothesis of the model. Our approach has several advantages over previously available methods.

First, it

oers a unied framework to compare the cost of liquidity across asset classes and investment vehicles. Second, the approach is exible enough to be applied to a single asset, say a building, or to an asset class as a whole. Third, our model links liquidity to funding requirements as well as asset characteristics which can explain some market facts such as the absence of liquidity for illiquid assets that are in high demands by long-term investors with little funding constraints. This measure of cost of liquidity could be developed in a number of ways. The time to liquidate and the credit spread could be properly modeled to account for the stochastic nature of these two variables as well as their correlation with the asset itself. Further, it would be interesting to integrate this cost of liquidity into an asset allocation framework to assess its impact on the optimal allocation to illiquid assets. This could be of interest for asset allocators as well as regulators looking into transmission mechanism of liquidity crisis from one asset class to another. Finally, as the cost of liquidity cost is a function of credit spread and asset volatility, nancial engineers could consider instruments based on CDS and VIX to hedge liquidity risk.

35

References [1] Acharya, V. and L. Pedersen, 2005, Asset Pricing with Liquidity Risk, Journal of Financial Economics 77, 375-410

[2] Amihud, Y., 2002, Illiquidity and Stock Returns: Cross-sections and Time-series Eects, Journal of Financial Markets, 5, 31-56

[3] Amihud, Y. and H. Medelson, 1986, Asset Pricing and the bid-ask Spread, Journal of Financial Economics 17, 223-249.

[4] Beber, A., Driessen, J. and P. Tujip, 2012, Pricing Liquidity Risk with Heterogeneous Investment Horizons, CEPR working paper

[5] Brennan, M., Chordia, T. and A. Subrahmanyam, 1998, Alternative Factor Specications, Security Characteristics, and the Cross Section of Expected Stock Returns, Journal of Financial Economics 49, 345373.

[6] Brunnermeier, M. and L. Pedersen, 2009, Market Liquidity and Funding Liquidity, Review of Financial Studies 22, 2201-2238.

[7] Cont, R. and L. Wagalath, 2014, Fire Sale Forensics: Measuring Endogenous Risk Mathematical Finance, forthcoming.

[8] Damodaran, A., 2005, Marketability and Value: The Illiquidity Discount, Stern School of Business working paper

[9] Fontaine, J.F. and R. Garcia, 2012, Bond Liquidity Premia, Review of Financial Studies, 25, 1207-1254

[10] Franzoni, F., Nowak, E. and L. Phalippou, 2012, Private Equity Performance and Liquidity Risk, Journal of Finance 67, 2341-2373.

36

[11] Gervais, S., Kaniel. R. and D. Mingelgrin, 2001, The High-Volume Return Premium, Journal of Finance 56, 877-919.

[12] Getmansky, M., Lo, A. and I. Makarov, 2004, An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns, Journal of Financial Economics 74, 529609.

[13] Glosten, L. and L. Harris, 1988, Estimating the Components of the bid-ask Spread, Journal of Financial Economics 21, 123-142

[14] Grossman, S. and M. Miller, 1988, Liquidity and Market Structure, Journal of Finance 43, 61733.

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[17] Lou X. and T. Shu, 2014, Why is the Amihud (2002) Illiquidity Measure Priced?, working paper.

[18] Pastor, L. and R. Stambaugh, 2003, Liquidity Risk and Expected Stock Returns, Journal of Political Economy 111, 642685.

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8

Appendix

Figure 8.1: SP 500 vs. Illiquid Asset (in log price)

Note:

SP

500

data

from

(http://www.econ.yale.edu/~shiller/data.htm).

Robert Illiquid

Shiller's asset

prices

web

page

calculated

as-

suming T=year 2300, borrow cost = 50bps, credit spread = 1% and holding period = 2 years.

38

Figure 8.2: Diversied equity portfolio with high turnover

Market conditions

tL

Probability

tH

(in

Credit

Borrow

(in

year)

spread

cost

year)

Normal

85%

0,04

1%

0,25%

1

Stressed

15%

0,16

6%

0,25%

1

Average liquidity cost 0,19%

Hypothesis: 10 and 40 days to divest the portfolio in normal and stressed market conditions respectively.

Figure 8.3: Concentrated equity portfolio with low turnover

Market

Probability

condi-

tL

(in

Credit

Borrow

year)

spread

cost

tH

(in

year)

Average liquidity cost

tions Normal

85%

0,08

1%

0,25%

5

Stressed

15%

0,32

6%

0,25%

5

0,08%

Hypothesis: 20 and 80 days to divest the portfolio in normal and stressed market conditions respectively.

Figure 8.4: Private equity portfolio with restriction to sell in rst two years

Market

Probability

condi-

tL

(in

Credit

Borrow

year)

spread

cost

tH

(in

year)

tions

Average liquidity cost

Normal

85%

2

1%

1.5%

8

Stressed

15%

3

6%

2%

8

0.98%

Hypothesis: We assume that the investor can't exit the investment in the rst two years for contractual reasons and that this period is doubled during adverse market conditions

39

Figure 8.5: Infrastructure portfolio with restriction to sell before maturity

Market

Probability

tL

condi-

(in

Credit

Borrow

year)

spread

cost

tH

(in

year)

Average liquidity cost

tions Normal

85%

50

1%

1.5%

10

Stressed

15%

50

6%

2%

10

3.33%

Hypothesis: We assume that the investor can't exit the investment until maturity of the transaction for contractual reasons. This is independent from market conditions.

Figure 8.6: Time on Market

Data from http://www.zillow.com/research/data/, representing the average time on market of houses in the USA.

40

Figure 8.7: Real-estate portfolio

Market

Probability

condi-

tL

(in

Credit

Borrow

year)

spread

cost

tH

(in

year)

tions

Average liquidity cost

Normal

85%

0.5

1%

1.5%

4

Stressed

15%

1.5

6%

1.5%

4

0.69%

Hypothesis: We assume that the investor can't exit the investment in the rst two years for contractual reasons and that this period is doubled during adverse market conditions

Figure 8.8: Real-estate portfolio

Market

Probability

condi-

tL

(in

Credit

Borrow

year)

spread

cost

tH

(in

year)

Average liquidity cost

tions Normal

85%

0.5

1%

1.5%

8

Stressed

15%

1.5

6%

1.5%

8

0.34%

Hypothesis: We assume that the investor can't exit the investment in the rst two years for contractual reasons and that this period is doubled during adverse market conditions

41

Figure 8.9: Small Caps Cost of Liquidity

The small caps benchmark is the Russel 2000 Small Caps index from which a 3-month realized volatility is calculated. The borrow cost is assumed constant and equal to 1%. The credit spread is the dierence between Moody's corporate bond yields rated BAA and the US government 10-year bond yield. the position is assumed to held for 5 years (tH ) and the baseline time to liquidate is 20 days.

42

Figure 8.10: Private Equity Cost of Liquidity

The large caps benchmark is the S&P 500 index from which a 3-month realized volatility is calculated. The borrow cost is assumed constant and equal to 0.20%. The credit spread is the dierence between the single A US corporate bond yield and the US Government bond yield with similar maturities. the position is assumed to held for 5 years (tH ) and the baseline time to liquidate is 20 days. For private equity, the transaction is held for 8 years and the baseline time to liquidate is 2 years. This increases with volatility and is capped at 3 years. The borrow cost is 1.50% for private equity.

43

Figure 8.11: Correlation between Liquidity Measures

Cost of

Levels of

Innovations

Traded

Fed Survey

Liquidity

Aggregate

in

Liquidity

of Private

Liquidity

Aggregate

Factor

Equity

(PS)

Liquidity

(PS)

(PS) Cost of Liquidity

100%

-58.14%

-39.75%

-0.63%

65.94%

100%

73.03%

5.84%

-35.58%

100%

8.99%

-27.36%

100%

7.50%

of Private Equity Levels of Aggregate Liquidity (PS) Innovations in Aggregate Liquidity (PS) Traded Liquidity Factor (PS) Fed Survey

100%

Hypothesis: quarterly data, from Q2:1990 to Q3:2013. PS stands for Pastor and Stambaugh (2003)

44

Figure 8.12: Private Equity Cost of Liquidity vs. Alternative Measures of Liquidity

The quarterly cost of liquidity is calculated following equation 6.1, where

creditspreadt

is the average credit spread over the quarter using the spread between Moody's corporate bond yields rated BAA and the US government 10-year bond yield. cost,borrowcostt , is assumed constant and equal to 1.5%.

The borrow

The time to liquidate is a

function of the 3-month realized volatility of small caps stocks as proxied by the Russell 2000 index. We assume that volatility in normal time is 15% and the time to liquidate is therefore given by:

M in (2 ∗ V ol3M /15%; 3).

The aggregate liquidity measure shown

is equal to Pastor and Stambaugh (2003) aggregate liquidity measure divided by -10.

Figure 8.13: Regression on existing Liquidity Measures

Aggregate

Fed Survey

Constant

Liquidity (PS) Coecient

-0.0459

0.0001

0.0080

T-Stat

-5.41

7.06

20.27

R-Square

57.24%

Hypothesis: quarterly data, from Q2:1990 to Q3:2013. PS stands for Pastor and Stambaugh (2003)

45

Figure 8.14: Regression on existing Liquidity Measures

Innovations

Fed Survey

Constant

in Aggregate Liquidity (PS) Coecient

-0.0340

0.0001

0.0090

T-Stat

-3.00

7.61

22.70

R-Square

48.57%

Hypothesis: quarterly data, from Q2:1990 to Q3:2013. PS stands for Pastor and Stambaugh (2003)

46