-Zentrum, Ziirich, Switzerland.46 1 Modeling Feedback Effects with Stochastic Liquidity
 
Hisata, Y. and Y. Yamai (2000): Research toward the practical applica-tion of liquidity risk evaluation methods, Monetary and Economic Studies,December, 83-128.
Jarrow, R. (1992): Market manipulation, bubbles, corners, and short
squeezes, Journal of Financial and Quantitative Analysis 27, 311-36.
Kampovsky, A. and S. Trautmann (2000): A large trader’s impact on price
processes, Johannes-Gutenberg- Universitat, Mainz, Germany.
Kyle, A. (1985): Continuous auctions and insider trading, Econometrica 53,1315-35.
Liu, H. and J. Yong (2004): Option pricing with an illiquid underlying assetmarket, Washington University, USA and University of Central Florida,Orlando, USA.
Ménch, B. (2003): Optimal liquidation strategies, Goethe University, Frank-furt am Main, Germany.
Pastor, L. and R. F. Stambaugh (2002): Liquidity risk and expected stockreturns, Graduate School of Business, University of Chicago, USA.
Pritsker, M. (2002): Large investors and liquidity: A review of the literature,Federal Reserve Board, Washington, USA.
Ranaldo, A. (2000): Intraday trading activity on financial markets: TheSwiss evidence, Ph.D. thesis, Université de Fribourg, Switzerland.
Schénbucher, P. J. and P. Wilmott (2000): The feedback effect of hedgingin illiquid markets, STAM Journal of Applied Mathematics 61(1), 232-72.
Sircar, K. R. and G. Papanicolaou (1998): General Black-Scholes modelsaccounting for increased market volatility from hedging strategies, AppliedMathematical Finance 5, 45-82.2 Optimal Liquidation Strategies
2.1 Introduction
This chapter analyzes optimal liquidation strategies for large security hold-ings. Order submission decisions are among the most important choicestraders make. For many institutional investors (like insurance companies orpension funds) even a moderately sized position in a stock may represent alarge fraction of this stock’s daily trading volume. Liquidating such port-folios can incur significant costs that directly influence the return on the
investment.
Nevertheless, compared with other fields of research in financial engineering,little theoretical and empirical work has been done in this area. The lack oforder book data, which are necessary to calibrate appropriate models, maybe one crucial reason for this fact. However, the introduction of electronicorder book systems at more and more stock markets around the world and thekeen competition among the large trading platforms will probably induce theexchanges in the near future to provide customers with suitable informationabout the instant liquidity of the listed products. The model proposed in thischapter exploits this kind of information and proposes an optimal trading
strategy for a large investor.
The following sections focus on the case where a large trader has a positive
initial investment in an asset and wants to close this position within a trad-48 2 Optimal Liquidation Strategies
 
ing window of one day (the analysis for a purchasing strategy is symmetric).For this purpose he or she can submit market orders basically continuouslythroughout the day. Thereby the trader has to balance different issues. Ob-viously, the investor can liquidate the portfolio all at once immediately afterthe opening of the stock exchange. In this case he or she has to bear virtuallyno market risk, since the entries in the order book are known with certainty.If there are fixed costs on each trade, such a strategy would also minimizetransaction costs of this kind. However, selling large amounts aggressivelyagainst the orders in the book will presumably have a substantial price im-pact and consequently lead to large liquidity discounts. Thus, the liquidationof a large portfolio is often broken up into smaller packages. Furthermore, itseems advisable to liquidate more in times when market liquidity is typicallyhigh (around noon) and less in periods where markets are usually illiquid (inthe morning and before the close of the exchange). However, this procedurecomes also at a cost. Forgoing the immediacy increases the market risk of theliquidation revenues. Additionally, as the number of trades increases fixedtransaction costs, this component of total costs will increase as well. Themodel proposed in this chapter is a first approach to assess these differentissues simultaneously in order to derive an optimal sales trajectory.
There is a growing theoretical literature on optimal liquidation strategiesfor large portfolios. Bertsimas and Lo (1998), Almgren and Chriss (2000),Hisata and Yamai (2000), and Dubil (2002) are prominent examples. Thecharacter of those papers is mainly theoretical with the intention to deriveclosed or semi-closed form solutions for the optimal liquidation trajectories.As mentioned above, empirical work has been hampered until recently by thelack of order book data. The main ideas of these papers are briefly outlinedin Section 2.2.
This chapter focuses on a realistic modeling of intraday liquidity patternsand price impact functions for large transactions in markets, which are notperfectly liquid. Many empirical papers that investigate high-frequency or-der book data or transaction prices (among others see Wood, Mclnish, andOrd (1985), Harris (1986), Jain and Joh (1988), McInish and Wood (1990),2.1 Introduction 49
 
Mclnish and Wood (1992), Kirchner and Schlag (1998), and Ranaldo (2000))find that intraday time series of volatility, trading volume, order flow, andtransaction costs follow a U-shape pattern, i.e. these variables start at a highvalue at the opening of the trading day, fall to lower levels over the day andthen rise again towards the close of trading.
The possible reasons for these intraday patterns are discussed controversiallyin the literature, and it seems they can be explained at least partly by theinformation flow through trades, resulting in progressively smaller adverse se-lection costs as the day evolves (see Wei (1992), Hasbrouck (1991), Foster andViswanathan (1993), Lin, Sanger, and Booth (1995)). The decrease in liqui-dity before the overnight trading halt may reflect the cost of holding inventoryover the upcoming nontrading period (see Amihud and Mendelson (1982),Bessembinder (1994), Lyons (1995), and Huang and Masulis (1999)). Thischapter does not aim to provide yet another explanation, but rather takesthese patterns as exogenous.
A recent study conducted by Linnainmaa (2003) analyzes order book data forthe 30 largest stocks traded at the Helsinki Exchanges. The author providesempirical evidence that institutional investors often demand liquidity andemploy market orders more intensively than less sophisticated retail investors
who may be classified as net suppliers of liquidity.
Modeling and hedging aspects introduced by illiquidity and the presence ofone or more large traders are discussed by Cvitanié and Ma (1996), Cuocoand Cvitanié (1998), Sircar and Papanicolaou (1998), Frey (1998, 2000),Schénbucher and Wilmott (2000), Kampovsky and Trautmann (2000), Freyand Patie (2001), Liu and Yong (2004), Bank and Baum (2002), and Esserand Monch (2002).
The study provided in this chapter is based on an approach similar toFrey’s (2000) model with constant liquidity, but introduces atime-dependent factor to model the U-shape pattern.
The rest of the chapter is organized as follows: Section 2.2 summarizes thekey ideas of related papers. The dataset used throughout this chapter is50 2 Optimal Liquidation Strategies
 
introduced in Section 2.3. The general framework is presented in Section 2.4.In Section 2.5 the modeling of the impact function is discussed. The proposedsetup to determine the optimal sales trajectory is calibrated in Section 2.6 forone example to give an impression of the magnitude of the liquidity discountsand the liquidation intervals. The chapter concludes in Section 2.7 with abrief summary and a discussion of issues for further research.
2.2 Related Literature
Since research conducted by Bertsimas and Lo (1998), Almgren andChriss (2000), Hisata and Yamai (2000), and Dubil (2002) is closely relatedto this chapter, the results of these studies are summarized briefly in this
section.
Bertsimas and Lo (1998), as well as Almgren and Chriss (2000), consider anoptimization problem of an investor who has to liquidate a large position ina security within an exogenously given period of time, whereas Dubil (2002)solves for the optimal length of the period over which an investor should close
a certain position in a stock.
Bertsimas and Lo (1998) derive dynamic trading strategies using stochasticdynamic programming techniques that minimize the expected cost of pur-chasing a large block of shares. The authors do not consider the volatility ofcosts for different trading strategies. They show that naive strategies of sell-ing a constant number of shares in equally spaced time intervals are optimalonly if the price impact is linear and permanent and if the stock prices followan arithmetic random walk. The authors also consider alternative formu-lations of the stock price dynamics and the impact function and show thatoptimal liquidation strategies vary through time and may depend on marketconditions and the number of shares that remain to be liquidated.
Almgren and Chriss (2000) incorporate the market risk associated with theliquidation of a position over a longer period by using a mean-variance ap-proach. The authors construct the efficient frontier in the space of time-2.2 Related Literature 51
 
dependent liquidation strategies. They derive tractable analytical resultsfor static strategies assuming an arithmetic random walk for the stock priceand linear impact functions. Like Dubil (2002), they distinguish betweena permanent and a temporary market impact, a concept that is discussed,for example in Holthausen, Leftwich, and Mayers (1987). The permanentprice effect changes the equilibrium price due to the transaction of the largetrader. The temporary impact is caused by an exhaustion of the liquiditysupply when the market order of the large investor is executed. It does notinfluence the stock price in the next liquidation interval.
Hisata and Yamai (2000) and Dubil (2002) extend the framework of Alm-gren and Chriss (2000) in order to endogenize the final liquidation horizon.Both approaches consider the risk of a liquidation strategy in a Value-at-Risk(VaR) setup and assume a constant speed of trading. The two frameworksbasically differ in the way they specify the dynamics of the asset price processand in the parameterization of the impact function, yet they derive similarresults. In addition to the standard case, where the market impact is linearin trading volume, Hisata and Yamai (2000) provide closed-form solutions ifthe market impact is defined by a square root function. They also consider astochastic market impact model and a portfolio model where the solution forthe optimal liquidation horizon can be obtained numerically. Dubil (2002)assumes two parameterizations for the impact function: general power func-tions uncorrelated with the share price, and linear functions correlated withthe price process. The first choice of the impact functions allows for thederivation of closed-form solutions for certain specialized parameterizations,while the second formulation can model a feedback loop between price dislo-
cations and liquidity.
The general setting of this chapter is similar to the approaches of Hisataand Yamai (2000) and Dubil (2002) as the overall time needed to completethe liquidation process is determined endogenously. However, the new modelallows us additionally to specify an upper bound for the final liquidationhorizon. The new framework forgoes the assumption of a constant speedof trading. This enables the investor to adapt the liquidation behavior to52 2 Optimal Liquidation Strategies
 
varying levels of liquidity throughout the trading day. Furthermore, the newmodel considers explicitly the resiliency of the order book.
2.3 Description of the Market and Dataset
Important properties of the liquidation model proposed in this chapter will beillustrated by using real-world order book data from the German automatedtrading system XETRA.
XETRA is an order-driven market where investors, by placing limit orders,establish prices at which other participants can buy or sell shares. A tradetakes place whenever a counterpart order hits the quotes. The system was in-troduced in 1997 by the German Stock Exchange. At the time the data werecollected, XETRA was open for trading from 9:00 a.m. to 8:00 p.m. Thetrading day starts with an opening auction, followed by continuous trading,which can be interrupted by one or several intraday auction(s). At the endof the day there is either a closing or an end-of-day auction. On XETRA, allof approximately 6,000 equities listed on the Frankfurt Stock Exchange aretradable. The minimum trading volume is one share. Market participantscan see all non-hidden entries on each side of the order book, but trading inXETRA is anonymous, i.e. market participants cannot identify the counter-parts. On XETRA there are no dedicated providers of liquidity for blue chipstocks. For small and mid cap stocks, designated sponsors (banks and secu-rity firms) are given incentives to provide sufficient liquidity by respondingto a quote request within a fixed period of time. Floor trading with marketmakers on the Frankfurt Stock Exchange still takes place but loses more andmore market share. In the blue chip segment merely every tenth share is stilltraded on the floor.
Based on event histories for 61 trading days (January 03 - March 28, 2002),which were provided by the Trading Surveillance Office of the Deutsche BérseAG, order book sequences were reconstructed. Therefore, by starting froman initial state, each change in the order book depth caused by entry, filling,2.4 The Basic Setup 53
 
cancelation or expiration of orders was considered as prescribed by the marketmodel of XETRA. This allows us to estimate the price impact function of theliquidation model. Due to the huge amount of data only the blue chip shareMAN is considered as a representative example of the stocks that are tradedin XETRA. The MAN corporation is one of Europe’s leading suppliers ofcapital goods and systems in the fields of commercial vehicle construction,and mechanical and plant engineering. Over the sample period the dailyturnover in MAN on XETRA ranged from 500,000 to 1,000,000 shares, andthe order book depth from 80,000 to 150,000 shares.
2.4 The Basic Setup
This section introduces the basic model and provides the motivation for the
key assumptions.
The large trader holds @ units of a security, which have to be liquidatedbefore time T*. The variable T’ denotes the actual time needed to sell theinitial position ¢g completely if the investor follows a certain strategy. Con-sequently a strategy is admissible if T < T*. The period from time to = 0 totime T is divided into n intervals 7; = t:4; — t;, whereas i = 0,...,n—1, sothat
nol
T=t,= > Tj.
#=0
In contrast to Almgren and Chriss (2000) and Dubil (2002), the intervals
can have different length, but at the end of each interval the large trader
liquidates some positive amount of shares Ad,, > 0, so that
i-lAdu S$ b0- >> Ady, i=l,....n
j=l
and
> Ad, = 0.i=l54 2 Optimal Liquidation Strategies
 
To obtain comparable flexibility one could alternatively assume equallyspaced time intervals (as is done, for example in Almgren andChriss (2000)) and allow the investor not to trade in some intervals. In thiscase, in order to model a representative set of liquidation trajectoriesattainable in practice, the length of the time intervals has to be sufficientlysmall. However, for 7 — 0 such an assumption blows up the optimizationproblem, since for every time interval one has to determine the optimal
quantity the investor should trade.Similarly to Frey (2000), it is supposed that the dynamics of the asset priceprocess S; follow the stochastic differential equation:
dS; = 0S;_dW, — S;_avt, (2.1)
where S;_ = bm S, and dW} denotes the price impact of the investor’s trad-ing strategy. In this setup the stock holdings of the large investor vary dis-cretely. Therefore, the jumps caused by the trading activity will be denotedby AUT := Uf — o.
To model the price impact AW;, a product approach is used:AW, = p(Ad) 8 (8),
so that the price impact depends both on the trade size Ad; and on the timeof day ¢, which is measured in minutes since the opening of the exchange.
The stock price immediately before and after the jump i can be written asSt, = [1 — p(Agi,) 8 (ts)] Si- = St, (2.2)
where 7; := 1 — p(A¢d,) 0 (ti).
The stock price after the trade must not be confused with the average priceper unit S?, realized by the large investor who liquidates Ag, stocks at timet;. The average price per unit Sis bounded by
Sy- 2 Si, 2 Si2.4 The Basic Setup 55
 
 
 
S.&,Ho tFig. 2.1: Stock prices immediately before and after a jump due to liquidation by
the large trader.
and can be calculated asAgi;1
Si. = ae / (1— p(w) 0(t:)) Sy du
Adi;
i
1- x2 J etweau Sy-0a1-3 tT plu
reo " & = 5S, (2.3)
I- aiey * f * olu)du1- ~atae
Furthermore, suppose that there are also fixed transaction costs of k per
where 6; := . Figure 2.1 shows this setup.
trade. While variable transaction costs could easily be introduced by anadditional multiplicative factor in equation (2.3) and would not significantlyinfluence the choice of the optimal trading trajectory, fixed transaction costs56 2 Optimal Liquidation Strategies
 
certainly do. For example, assume that the objective function of the largeinvestor considers only the expected net liquidation value, EG, given by
EG = —nk + > Agi, 5; ESi,. (2.4)
i=]
For k = 0 and a sensible definition of p, ie. for 9p/0¢ > 0 and 62p/d¢" > 0,one gets an optimal sales trajectory with n — oo and Ad; — 0. If k > 0,the investor has to balance, on the one hand, the advantage of tradingsmall amounts that cause only a small price impact and, on the other hand,the overall fixed transaction costs, increasing in the number of tradingintervals. At this point the question may arise, which costs can be assumedto represent fixed transaction costs? Clearly, fixed brokerage fees and anyfixed charges incurred by the exchange belong to this category. However,more important in daily life are the opportunity costs for handling thetransactions in the front and back offices. These costs do not depend on
trading volume, but predominantly on the number of transactions.
For the sake of simplicity, the risk-reward trade-off is modeled via theexpected net liquidation value EG and the respective standard deviationSTD (G). The large trader then maximizes the objective function with a as
the degree of risk aversionmax [EG — aSTD (G)] (2.5)by choosing the optimal sales trajectory {7;, Ad:,, n}".
Proposition 2.1 (Expectation and standard deviation of the net liq-uidation value) Suppose the stock prices satisfy the stochastic differentialequation (2.1). Then the expected value and the standard deviation of the net
liquidation value G are given by
EG = —nk + Si, > AG: 5: Ila (2.6)i=l jal2.5 Modeling the Impact Function 57
 
andvar(@) = 37 (Ad, 6)*var(S,) (2.7)i=1+2 > Adu, 5: Adu, 5; cov (S1,, 5;),i<jwherecov(S:,51,) = Sz, [exp (07(t; — to)) - 1] I * il MWm=1 1Si+1
The proof is given in Appendix 2.8.
In the proposed model, G is not exactly normally distributed, especiallyfor small n. However, to keep the problem handy for demonstration, thehigher moments of the distribution will not be considered in the objectivefunction (2.5). A moderate number of liquidation intervals is sufficient tojustify the approximation of the distribution of G by a Gaussian. This willbe demonstrated by a simulation study in Section 2.6.3.
2.5 Modeling the Impact Function
The framework proposed in this chapter incorporates distinctive features inorder to model stock market liquidity in a sensible way. Specifically, the
model contains:
e a U-shape for intraday stock market liquidity,power price impact functions,
¢ periods that allow the order book to be rebuilt, as boundary conditionsfor the time intervals between subsequent trades.
The relevance of these issues will be discussed briefly in this section.58 2 Optimal Liquidation Strategies
 
Averageliquiditydiscount ~#— 10,0000.015 —k- 20,000—# 30,0000.010.005
 
 
t
10:00 12:00 14:00 16:00 20:00Time of day
Fig. 2.2: Relative liquidity discounts for sell market orders (averages over 61 trad-ing days) as a function of the time of day. Quantities: 10,000 (20,000,30,000) MAN shares.
2.5.1 U-shape of Stock Market Liquidity
In the dataset used for the implementation of the proposed model one canclearly identify a U-shape pattern in the relative liquidity discount V over thetrading day (see Figure 2.2). This phenomenon is typical for stocks tradedon XETRA. The relative liquidity discount is the difference between the bestbid price and the average execution price per unit of a hypothetical marketorder, divided by the best bid price.
From the opening at 9:00 a.m. to 11:00 a.m. there is a sharp decline inthe liquidity discount. From 11:00 a.m. to 5:00 p.m. it moves in a narrowrange at a low level. A significant reaction of the liquidity discount to thestock market opening at 3:30 p.m. in New York cannot be identified. From5:00 p.m. the average liquidity discount rises fast and remains at a high levelfrom 6:00 p.m. to 8:00 p.m., when the stock exchange closes.
Why is it important to consider the U-shape pattern in an intraday liquida-tion model? Obviously, if the investor is not too impatient, he or she shouldliquidate more in periods of high liquidity (around noon) than in situations2.5 Modeling the Impact Function 59
 
of low liquidity (immediately after the opening and before the closing of theexchange).
A functional form that is able to model this U-shape pattern is given by 0 (2)withdp= 4th. (2.8)t+te g-t
Here ¢ measures the time in minutes elapsed after the opening of the exchangeand d,e, f, and g are positive parameters with e < #™¢* < g. The variablet™* denotes the time in minutes between the opening and the close of the
exchange.
2.5.2 Power Price Impact Function
The dotted line in Figure 2.3 shows the relative price impact (averaged over61 trading days) at noon as a function of the trading volume for the datasetunder consideration. It is apparent that a linear approximation (dashed line)
of this function provides only a poor fit.
Exponential and power functions seem more suitable to model this relation-ship. For numerical convenience a simple power function is employed in the
following:p(Ad:) =aAg?, a20. (2.9)
As one can see in Figure 2.3, this function fits the data very well.
2.5.3 The Resiliency of the Order Book
The introduction of a deterministic relationship between the price impact pand the trade size Ad implies that between two transactions the large traderhas to allow the market to rebuild the order book completely. Ignoring thisboundary condition in the optimization problem can result in misleading
conclusions. Imagine a situation where the completion of a sell market order60 2 Optimal Liquidation Strategies
 
 
 
de0 20,000 40,000 60,000 80,000 100,000
Fig. 2.3: Relative liquidity discounts at noon (averages over 61 trading days) asa function of the number of MAN shares to be liquidated. Dotted line:raw data. Dashed line: linear fit. Solid line: fitted power function.
by the large investor completely clears the order book on the bid side. Thenthe large trader has to wait until the bid side of the order book has recoveredbefore he or she can sell another package of shares. Also in less extremesituations, if the market order of the large investor absorbs the order bookdepth only partly, being too impatient between subsequent transactions willresult in deteriorating path-dependent liquidity discounts.
To operationalize this idea, two assumptions are made in the following. Theorder book is considered completely rebuilt after a transaction by the large
investor as soon as:
e the bid-ask spread has again narrowed to the level immediately beforethe last transaction of the large investor, and
e the same volume that was traded by the large investor can be tradedwith the same or a smaller liquidity discount compared with the onethe large trader had to bear in the previous transaction.
Instead of using the resiliency of the order book as a boundary conditionfor the length of the trading intervals, one could also separate the price im-2.6 Numerical Results 61
 
pact p(A¢;) into a permanent and a temporary component, as proposed byAlmgren and Chriss (2000), Hisata and Yamai (2000), and Dubil (2002).However, if market orders are observed very frequently in the dataset avail-able to calibrate the model, the permanent and the temporary componentmay be difficult to separate, especially if the temporary price impacts ofdifferent market orders overlap.
2.6 Numerical Results
The model described above was implemented using the MAN dataset for61 trading days. We focus our analysis on continuous trading and do notconsider intraday auctions. The results are presented below.
2.6.1 Parameter Specification
Assume that f in equation (2.8) measures the time in minutes elapsed after9:00 a.m. The parameter values for the functions 0 (2) and p (A¢@,), obtainedvia least-squares estimation, are shown in Table 2.1.
The proposed functional form is able to model the empirical data quite well(Rey = 0.964). Figure 2.4 shows the surface of the fitted function.
In order to estimate the minimum time that has to elapse between subse-quent trades, all large market sell orders (order volume > 1,000 shares) inthe sample were grouped into seven intervals, assuming that the time themarket needs to recover is independent of the time of day. Table 2.2 andFigure 2.5 show the average 7™”"Although the relationship between Ad and r™* seems to be slightly concave
empirically, r™™" is modeled as a linear function of Ag; in order to avoid
(measured in minutes) for every interval.
nonlinear constraints in the optimization problem:Th” = BAG. (2.10)
The estimation of the parameter @ in equation (2.10) via least squares yieldsan estimate of # = 0.00098 and an R? of 0.83. If accuracy is crucial and62
2 Optimal Liquidation Strategies
Table 2.1: Estimated parameters for the functions 9 (¢) and p (Ag).
 
 
 
 
 
 
Estimate Asymp. SE t-statisticsa 0.42322 1.34534 *107-!? 3.14583 «101!b 1.45806 0.00911 159.995d 4.22391 «10-7 4.24458 «10-8 9.95129e€ 17.68900 0.29029 60.93590f 2.34585 *10~° 2.35274 «10-7 9.97072g 837.27800 3.13531 267.04800
 
0 (Ad) 9(E)0.05
 
Fig. 2.4: The fitted surface p(Ad:) # (#).
 2.6 Numerical Results 63
 
Table 2.2: Average time 7™" in minutes required for recovery of the order bookafter the execution of a large market order.
                        
    
    
 
 
Midpoint of interval
<A¢< 2,000 1,5002,000 < A@<_ 3,000 2,5003,000 <A¢g< 4,000 3,5004,000 <A¢d< 5,000 4,5005,000 < Ad < 10,000 7,50010,000 < A¢é < 15,000 12,50015,000 < Ad < 25,000 20,000
 
minuy
 
 
on
&0 5,000 10,000 15,000 20,000
Fig. 2.5: The resiliency of the order book. Fitted function to model the time inminutes to be elapsed between subsequent trades.64 2 Optimal Liquidation Strategies
 
computational time is cheap, one should define 7™" as a nonlinear function,
which may provide a higher R?.
For the estimation of the volatility parameter in the stock price dynamicsgiven in equation (2.1) best bid quotes, collected at intervals of 15 minutesfrom 9:00 a.m. to 8:00 p.m., were used and 15-minute returns for the bestbid price were computed. Overnight price changes were not taken into con-sideration. The estimation for the volatility parameter yields o = 0.7. Fur-thermore, So is set to € 28.55 which is the closing price on March 28, 2002,the last day of the sample period.
2.6.2 Numerical Examples
For the first example, suppose that the large trader does not care about thetisk of the liquidation revenues (a = 0). Consequently, the investor willchoose the sales trajectory that maximizes the expected liquidation value.Furthermore, assume that the investor has