Analytic Model of Illiquidity Risk and Return or

The Risk-Adjusted Cost of Illiquidity

Emilian Belev, CFA, ARPM Newport, RI June 12, 2015

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The Effects of Illiquidity on the Buy Side

Slide 2

• Illiquidity: We cannot trade an asset X at the interim time T1

• By definition, the problems of Illiquidity concerns (at least) a two period horizon, T0 -7 T1 , and T1 -7 T2

• Modern finance suggest that we will carry an implicit cost of illiquidity due to:

• Not being able to re-balance shifted asset weights in the portfolio at time T1 (Ang et al, 2014)

• Not being able to re-optimize due to shifted wealth and hence risk aversion per Discretionary Wealth Hypothesis (Wilcox 2003 ) at time Tl

• Not being able to meet a liability cash outlay due to insufficient liquid --.&.. --- I --- -L- -.&. .&.~--- IT'I

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Two Birds with One Stone

Slide 3

• To isolate the Cost of Ill iquidity in terms of Return and Risk performance we need to compare two cases:

1. All assets are liquid 2. All assets have the same characteristics as in the first case, but some

of them are illiquid (cannot be traded at time T1)

The difference in Risk and Return metrics indicates the imposed cost of being Illiquid.

• To get the answer, let's employ a simulation of the capital market performance up to time T17 and measure the dispersion of differences in liquid/illiquid performance over the points of the distribution

• An optimization at simulated future states of the world take care of both the

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Capturing Risk Aversion Shifts

Slide 4

• Under exponential utility, investor's risk aversion is given by the DWH as:

Assets A = 2(Assets -Liabilities)

• Let's presume investor's portfolio with Beta p has a return over T0 ~ T1 of Rp

• It can be shown that investors changed risk aversion at time T1 changes to:

A = A * __ ( l_+_R___:._p_)_ Tl ( 1 + 2 * A * Rp)

• The modified weights of the investor portfolio and the modified Ar1 requires an optimization at time T1 to realign the portfolio with investor objectives.

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The Price of Constraint

Slide 5

• An unconstrained optimization at time T1 will produce a fully optimal portfolio of Expected Return Roptand Expected Volatility aopt with:

UtilitYopt = Ropt - Ar1 * a2 opt

• But investor is constrained due to illiquid portion of his portfolio. Constrained optimization at time T1 will produce a sub-par optimal portfolio with:

UtilitYsubopt = Rsubopt - Ar1 * a 2 subopt

where UtilitYsubopt < UtilitYopt

• Our intuitive estimate of the cost of illiquidity is:

UtilitYopt - UtilitYsubopt

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The Price of Constraint (cont’d)

Slide 6

•

•

•

Comparing UtilitY opt to UtilitYsubopt might be exaggerating the actual cost of illiquidity. There might be other levels of risk aversion on the efficient frontier that have shorter distance to the suboptimal portfolio in utility space. Let's denote with U tilitYopt_mod

The piece-wise segment slope of the efficient frontier indicates various levels of risk aversion. Using efficient risk aversion, variance and return we can find UtilitYopt_mod which is closest to UtilitYsubopt·

If the suboptimal portfolio were to be liquidated at time T1 the UtilitYopt_mod investor (the one tangent to efficient frontier point UtilitYopt_mod) would require a return equal to:

Rreq = Rsubopt + (UtilitYopt_mod- UtilitYsubopt)

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The Price of Constraint (cont’d)

Slide 7

• Then the Cost of Illiquidity (CI) will be expressed in T1 dollars as:

•

( ) ( l+Rsuopt)

CI = Portfolio Valuero * 1 + Rp * [1 - C ) ] l+Rreq

If we simulate the distribution of R13 , then we can capture the distribution return component up to time T1 :

[ ] CI

E R· · ·· · = E -1 LllLquudLty [Port[ olio Valuer)

with standard deviation equal to (JR .lt ' .d.t l. l.qUl. l. y

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The Cost to Borrow

Slide 8

• If investor has set liquidity targets (expenditure or explicit liability cash outflows), a scenario in the simulation where he ends up with insufficient liquid assets creates, poses an additional problem.

• Scenario A [for Armageddon] : Investor does not find the funds to cover and defaults, eventually filing for bankruptcy. The legal cost of proceedings comes in addition to illiquidity cost calculated previously

• Scenario B [for Borrow] : Investor has access to credit and borrows, which has effects that compound to sub optimality at time T1

• Projected Expected Return at time T1 decreases with additional interest to be paid at time T2

• Standard Deviation at at time T1 increases with the leverage ratio

• Risk Aversion also increases with leverage (DW H)

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The Buy Side Liquidity Framework in Practice

• Take a portfolio of a number of assets, some of which illiquid • Using Northfield’s SIENS (EENIAC’s simulation technology) generate 100 scenarios for the portfolio • For each scenario run two optimizations: (a) with constraint on trading illiquid assets, and (b)

without constraint – all liquid; optimizations reflect modified risk aversion levels due to changed leverage

• Under each scenario, capture fully optimal expected return (all liquid), and sub-par optimal return (constrained optimization); we can then compute the cost of illiquidity under each scenario, and calculate moments of the “illiquidity cost distribution”

Slide 9

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Buy Side Framework in Practice (cont’d)

Slide 10

• In a separate but related exercise, assume a mandatory liability cash flow at the end of the time horizon (e.g. 60°/o redemption of the weight of the initial portfolio)

• Assume borrowing for the part of liability that is over the amount of liquid assets in the portfolio at the end of the time horizon under each scenario; assume borrowing interest rate (e.g. 5°/o)

• Capture the distribution of this extra borrowing cost which is a drain on expected return at time T1 .

• The borrowing cost effect is cumulative on top of the sub-par optimality effect of illiquidity

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Sub-Optimal Returns and Illiquidity Weight

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0

1

2

3

4

5

6

10 20 30 40 50 60 70 80 90

percent illiquid

Scenario Sub-optimal Expected Return Due to Illiquid Content Sub-optimal expected return in %

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Expected Sub-Optimal Return and Illiquidity Weight

Slide 12

-2.0%

-1.8%

-1.6%

-1.4%

-1.2%

-1.0%

-0.8%

-0.6%

-0.4%

-0.2%

0.0% 10 20 30 40 50 60 70 80 90

percent illiquid

Illiquidity Mean Return and Level of Illiquid Content Mean return due to illiquidity %

Illiquid weight is almost like a linear exposure to the illiquidity premium, when no borrowing occurs

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Illiquidity Weight and Scenario Borrowing Cost

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10 20 30 40 50 60 70 80 90

Percent Illiquid

Cost % Scenario Illiquidity Cost Due to Borrowing Dependence on Illiquid Content

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Illiquid Weight and Mean Borrowing Impact

Slide 14

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

10 20 30 40 50 60 70 80 90

Percent Illiquid

Expected Cost % Expected Illqiquidity Cost Due to Borrowing Dependence on Illiquid Content

Borrowing adds non-linearity to the illiquidity premium required by the investor

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Meanwhile, on the Sell Side

Slide 15

• Market Makers specialize in particular securities and hold specialized portfolios in those specific securities. This is very appetizing from an estimation point as they can imply possible illiquidity triggers for securities that are currently liquid, but may become illiquid at time T1

• Their bid-ask spread consists of:

• A set business profit margin set by the supply and demand of their service (same as a car dealer)

• A compensation for:

• Loss for temporarily holding the security, due to market moves

• Loss for holding the security due to security becoming illiquid

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Sell Side (cont’d)

Slide 16

• We can perform a cross sectional study of bid-ask spreads of stocks and observe the following relationship (with often quoted drivers of B-A-S):

BAS = a+ y1 *Volume + y2 *Number of Market Makers+ y3 * f3Market

• Split in three essential components this equation will give us: • a - the equilibrium profit of the dealers

• Y3 * f3Market- the compensation for market loss

• sub8 As = y1 *Volume+ y2 *Number of Market Makers

sub8 As is the compensation for the loss due to security potentially becoming illiquid

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Sell Side (cont’d)

Slide 17

• Let's assume that the security becomes illiquid under a certain value barrier

• The value of sub8As is a sum of barrier knock-in options exercisable at different time horizons, each dependent whether the barrier is breached at the prior horizon.

• Using the value of sub8 As and its equivalence with the barrier option representation we can solve for the barrier- the imputed security price where the security becomes and remains illiquid under any of the observed times.

• Essentially the barrier is a equivalent risk-neutral representation of the ~xn~r.t~tion of BAS wid~nina due to illiauiditv

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Sell Side (cont’d)

Slide 18

• The relatiionship is:

Where:

• Ti is the observed time horizon

• K is the strike price, which is also the barrier B

• S Bid is the spot bid price

• Oi is the value of the knock-in option expiri:ng at time Ti

• N is the assumed longest time by which the specialist would want to liquidate the inventory acquired at time zero

• We can solve the above expression for KIB • A fast and accurate method for dealing with path dependency in

barrier options is developed by Northfield with the online EENIAC r1 mc:tnm nrnr~c:c:inn C:\Jc:t~m

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Back to the Buy Side

Slide 19

•

•

•

•

•

Once we have found Kj we can observe directly in the simulation and determine the scenarios where certain securities become illiquid, while they remain liquid in more favorable market scenarios.

This will incorporate the effect of illiquidity risk and return from positions that are currently not illiquid but may become illiquid over horizon Ti

This analysis can be repeated over different horizons. Normally, over longer horizons the value of the knock-in options will decrease, making illiquidity to have less of an impact on portfolio performance. This will reflect time decay of illiquidity.

Bid and ask for the investment are available - EASY; use this procedure; good for stocks, bonds, options, exchange traded private equity

Bid and ask are not available - EASY: assume illiquid under all scenarios; good for real estate, infrastructure, private debt, private equity

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Ramblings on General Market Illiquidity • We label illiquid assets (real estate, infrastructure) as such because the market has evolved so that they are

illiquid.

• What if more asset classes become illiquid (e.g. bonds in GFC 2008). Then the investment world gets divided in two: • The “Warren Buffets” with predominantly liquid assets who can stay on the static efficient frontier and

assume the usual IID world in optimality • The “Lehman Brothers” with more of the troubled assets who gravitate to sub-optimality

• Our framework will dictate that the cost of illiquidity to the Lehmans of the world will be categorically larger due to larger portion of portfolio being suboptimal and unable to meet cash needs. Rising demand for borrowing will further increase the expected illiquidity cost.

• The probability of a larger proportion the market turning illiquid will not only entail the “Warren Buffets” will decrease in number, but also that credit (an asset to someone liquid at the time of crisis) will also disappear, which would make Global Scenario A more likely.

• Simulating with various degree of general market illiquidity will give us a way to estimate the probability and magnitude of this impact.

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Enter the Primary Dealers

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Primary Dealers and the REPO market

• Primary Dealers: approved major security dealer companies that assist the Federal Reserve in

conducting open market operations (buy and sell securities)

• Required by the Fed to disclose routinely, among other things, their long and short positions in the REPO market. • A REPO (repurchase agreement) – “sell securities today, buy back tomorrow” – is essentially a

short term loan collateralized by bonds • A “reverse REPO” is being on the lending side of the REPO

• The REPO market is the routine place where financial institutions seek short term liquidity apart from

outright borrowing from each other and the Fed

• The REPO market is non-centralized, arcane, and lacks visibility bar the requirements of the Fed imposed on Primary Dealers to report REPO positions

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Primary Dealers and REPOs

• The REPO market, as much visible, is a convenient experimental lab to test our assumptions on

market moves, leverage, and liquidity

• As a first test, let’s observe the increase of the proportion of short terms borrowing (overnight REPO) to the proportion of medium term borrowing (term REPO) and its relationship with the contemporaneous stock market

• Increase of short term borrowing will indicate an increase in current cash needs above expectations; our Buy Side framework would have us believe that this would happen with the decrease in assets (stock market downward move) and vice versa.

• Let’s see what happens in practice:

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Short Term Borrowing and The Market

Slide 24

-0.5

0

0.5

1

1.5

2

2.5

1 5 9 13 17 21 25 29 33

Stock Market (S&P 500)

Short Term Borrowing Ratio (REPO)

Quarterly Return %

Quarters since 2006 Q1

ρ = -0.53

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More on REPOs and Primary Dealers

• Another interesting experiment is to test the relationship between the balance sheet of the Primary

Dealers (Long minus Short) in overnight REPOs and the stock market

• An increase in the balance sheet will indicate an increased reliance of third parties (most market players) on short term borrowing as the modus operandi of doing business. Operating under high leverage precipitates liquidity driven market slumps down the road.

• Indeed, it turns out this signal is predictive to the stock market one year out…

• ….until QE 2 comes along

• (QE2: Fed moves QE from “toxic” MBS assets, to treasuries, prime MBS, and potentially any fixed income asset)

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More on REPOs and Primary Dealers

Slide 26

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

1 5 9 13 17 21 25 29

Primary Dealer Long less Short REPO

Market Return Lag 4 Quarters

Return

Start of QE 2

ρ = -0.72 ρ = -0.06

Quarters since Q1 2006

Primary Dealer Providing of Short Term Liquidity and the Stock Market Return a Year Later

QE clouds everything

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Conclusions

• A comprehensive liquidity framework

• Intuitive

• Rigorous

• Supported empirically

• Tractable risk and return estimation

• Tractable optimization: based on usual exponential utility function

• Technologically available using existing Northfield investment tools

• Q & A

Slide 27