Dynamic programming models and algorithms for the mutual fund cash balance problem

Transcription

1 Submied o Managemen Science manuscrip Dynamic programming models and algorihms for he muual fund cash balance problem Juliana Nascimeno Deparmen of Operaions Research and Financial Engineering, Princeon Universiy, Princeon, NJ 08540, jnascime@princeon.edu Warren Powell Deparmen of Operaions Research and Financial Engineering, Princeon Universiy, Princeon, NJ 08540, powell@princeon.edu Fund managers have o decide he amoun of a fund s asses ha should be kep in cash, considering he radeoff beween being able o mee shareholder redempions and minimizing he opporuniy cos from los invesmen opporuniies. In addiion, hey have o consider redempions by individuals as well as insiuional invesors, he curren performance of he sock marke and ineres raes, and he paern of invesmens and redempions which are correlaed wih marke performance. We formulae he problem as a dynamic program, bu his encouners he classic curse of dimensionaliy. To overcome his problem, we propose a provably convergen approximae dynamic programming algorihm. We also adap he algorihm o an online environmen, requiring no knowledge of he probabiliy disribuions for raes of reurn and ineres raes. We use acual daa for marke performance and ineres raes, and demonsrae he qualiy of he soluion (compared o he opimal) for he op 10 muual funds in each of nine fund caegories. We show ha our resuls closely mach he opimal soluion (in considerably less ime), and ouperform wo saic (newsvendor) models. The resul is a simple policy ha describes when money should be moved ino and ou of cash based on marke performance. Key words : Muual fund cash balance, approximae dynamic programming Hisory : 1. Inroducion Muual fund managers have o deermine how much cash o keep on hand, sriking a balance beween he cos of meeing redempion requess agains he opporuniy cos of holding cash. The academic lieraure ypically ignores several dimensions of he problem, such as he characerisics of he demands for redempions. For example, he muual fund has o consider boh individual and insiuional invesors. No only are redempion requess by insiuional invesors much larger, 1

2 Auhor: Muual fund cash balance problem 2 Aricle submied o Managemen Science; manuscrip no. hey have o be saisfied righ away, producing shor-erm borrowing coss if here is insufficien cash. The manager has o consider ransacion coss and bid-ask spreads. He also has o ake ino accoun wheher he marke is under or over-performing long-erm averages, as well as he paern of deposis and redempions which are correlaed wih boh marke performance and ineres raes. In Wermers (2000) and Edelen (1999), he auhors presen empirical evidence ha suppors he value of acive fund managemen, as expenses and ransacions coss are considerable facors in he fund ne reurns. In paricular, hese papers poin ou he significan cos of holding cash in a muual fund. Variaions of he cash balance problem have been sudied in he academic communiy in differen forms. A survey of deerminisic cash balance problems can be found in Srinivasan & Kim (1986). The deerminisic problem assumes demands, marke reurns and ineres raes are known, bu inroduces issues such as fixed coss for a ransacion, and he modeling of coninuous ime processes. A similar deerminisic version of his problem is he cash balance problem based on Sehi & Thompson (1970) and Sehi (1973) and laer described in Sehi & Thompson (2000). The problem is represened by a sysem of differenial equaions and he opimal policy is deermined sudying he associaed Hamilonian and adjoin funcions. Golden e al. (1979) and Jorjani & Lamar (1994) propose nework flow opimizaion models o deal wih efficien managemen of cash, invesmens and shor-erm loans. Myopic and saionary soluions for a sochasic cash balance problem are presened in Peninen (1991). A relaed problem is corporae cash holding which is addressed in Kim e al. (2001), Almeida e al. (2004) and Faulkender & Wang (2006). These papers presen a qualiaive analysis indicaing ha he more volaile he cash flow and he he higher he cos of finance, he more cash holding is observed. Almeida e al. (2004) also analyzes cash flow sensiiviy. A cash holding problem specifically applied o muual funds is discussed in Yan (2006). Like he papers on corporae cash holdings, his paper only deals wih qualiaive aspecs. The auhor develops a simplified saic model for he problem and hen proceeds wih a regression analysis on acual muual fund daa ha validaes he model predicions. The auhor only hins, based on

3 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 3 Consaninides (1986), ha he holding of cash should be kep wihin a cerain range and i should be adjused only when he cash level is oo low or oo high. Hinderer & Waldmann (2001) presen a broader cash managemen model which includes a basic model of he muual fund cash holding problem as a special case. The auhors propose a dynamic programming formulaion bu do no address he compuaional problems ha arise as a resul of he curse of dimensionaliy. The muual fund cash holding problem can be viewed as a more complex invenory conrol problem (Graves e al. (1981)). The complicaions arise due o he presence of side variables no presen in a sandard invenory conex, such as ineres raes and marke raes of reurn. Even invenory problems wih one-dimensional sae spaces may lead o prohibiive compuaional requiremens, due o he cardinaliy of he sae space. In he conex of a single-iem sochasic lo-sizing problem, Halman e al. (2006) develops approximaion algorihms o deal wih i. A relaed problem in he economic lieraure is he commodiy sorage problem (Wrigh & Williams (1982, 1984)). This problem assumes an infinie planning horizon where a crop is harvesed each year and a decision regarding he allocaion beween consumpion and changes in invenory mus be aken a each period. Boh a dynamic programming analysis and a compeiive equilibrium analysis are discussed in Judd (1998). We provide a model of he muual fund cash-managemen problem ha capures several dimensions of a realisic seing. In fac, our model is based on an sen by an acual fund manager describing he issues he was facing and could no find he answers in he lieraure o approach hem. We propose boh finie and infinie horizon models. We hen describe several mehods for solving hese models: a) we give wo soluions based on newsvendor models suggesed by he muual fund manager in his , b) we give an exac algorihm using backward dynamic programming (he mos deailed version requires hree days o solve), and c) we provide an approximae dynamic programming algorihm. Our ADP algorihm, dubbed he SPAR-Muual algorihm is based on a procedure described in Powell e al. (2004) which adapively learns a piecewise linear funcion giving he value of cash as a funcion of marke reurn and ineres raes. The SPAR-Muual

4 Auhor: Muual fund cash balance problem 4 Aricle submied o Managemen Science; manuscrip no. algorihm is provably convergen for finie horizon problems (he proof is given in Nascimeno & Powell (2008), based on he proof echnique in Nascimeno & Powell (o appear)) which uses a pure exploiaion sraegy. The algorihm can be used in a model-based implemenaion (where we assume ha we know he disribuion of marke reurns and ineres raes) over a finie horizon, or a model-free implemenaion if we are willing o assume seady sae, using acual (raher han simulaed) observaions of he exogenous informaion. 2. Problem Descripion and Assumpions Our problem is o deermine how much cash o keep on hand o srike a balance beween having enough cash o handle redempions, and maximizing he reurn on invesed asses. We consider boh finie and infinie horizon versions of his problem. The objecive is o minimize discouned expeced coss, where coss include early liquidaion coss (if we have o liquidae asses o mee a redempion reques), borrowing coss (if a redempion reques has o be saisfied before asses can be liquidaed), ransacion coss and he los reurn on funds ha are no invesed in he marke. During periods of marke decline, he los reurn is negaive. We disregard he use of cash o pay managemen fees and oher expenses and o make dividends and capial gains disribuions, since hey are ypically a fixed percenage of he oal asses under consideraion. There are wo ypes of shareholders, namely large insiuional invesors and small reail cusomers. We denoe by D l and D s he demand for redempion a ime from insiuional (large) and reail (small) invesors, respecively. We denoe by D i he inflow of money from new invesors, reaed as a single, aggregae quaniy. We assume ha he sochasic processes describing D l, D s and D i are Markovian. Moreover, hey are ineger, bounded and posiive. We denoe by R he cash level a period afer new deposis D i have been added. If he oal demand for redempion D l + D s a is larger han R, hen par of he fund porfolio mus be liquidaed in order o mee he demand. The cos involved will be denoed by ρ sh, he shorfall cos, given by a deerminisic fracion of he liquidaing amoun. We noe ha we can easily accommodae increasing coss for larger ransacions which migh arise from he cos of liquidaing

5 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 5 illiquid asses. If he insiuional demand alone is higher han R, hen a financial cos (ineres rae), denoed by P f, is also charged, as he demand mus be saisfied immediaely hrough shor erm loans while securiies are liquidaed (a process ha can require several days). On he oher hand, if oo much cash is mainained, he porfolio is losing invesmen opporuniies. This loss is measured using he fund s porfolio rae of reurn, denoed by P r. P f and P r are Markov processes (no necessarily independen) which are exogenous o he sysem. We also assume hey have finie suppor. Coninuous processes could be discreized appropriaely in order o apply he algorihms. We le W = (P r, P f, D i, D l, D s ) be he vecor of exogenous informaion ha becomes available a every ime period. We assume ha W is independen of he cash level. Moreover, (P r, P f ) is condiionally independen of (D i, D l, D s ) given W 1, ha is, IE [ P r P f D i D l D s W 1 ] = IE [ P r P f W 1 ] IE [D i D l D s W 1 ]. We noe ha more general srucures, oher han he Markovian assumpion, could be considered for he processes. As a consequence, we would have o augmen W wih addiional feaures. Knowing W and R, he cash rebalancing decision x = (x 1, x 2 ) is made, where x 1 is he amoun of money o ransfer from he porfolio o he bank accoun, while x 2 is he amoun of money o ransfer from he bank accoun o he porfolio. There is a limi on he amoun of he porfolio ha can be liquidaed in one period, so we impose he consrain 0 x 1 M, where M is a deerminisic bound. Normally, M would simply be chosen large enough ha i never consrains he opimal soluion, bu i can also be jusified by he 1940 Invesmen Company Ac 1 which allows redempions o be saisfied using shares of sock insead of cash if a redempion reques is oo large. I is obvious ha 0 x 2 max(0, R D l D s ). We denoe by X (W, R ) he feasible region for x = (x 1, x 2 ). The fund incurs ransacion coss ρ r for each dollar moved ino or ou of he fund, which means ha oal ransacion coss are given by ρ r (x 1 + x 2 ). Moreover, we assume ha he shorfall cos ρ sh is larger han he ransacion cos ρ r. Finally, we assume ha IE [ P f P f 1] is posiive and IE [ P r P r 1] is greaer han ρ sh and ρ r. Clearly, ρ sh and ρ r mus be posiive. 1 hp://

6 Auhor: Muual fund cash balance problem 6 Aricle submied o Managemen Science; manuscrip no. We denoe by R x he cash level a he end of period afer he decision is aken. This means ha R x = max(0, R D l D s ) + x 1 x 2 and R +1 = R x + D i +1. Noe ha since we assume ha D l, D s and D i are ineger and bounded, hen R, x 1, x 2 and R x are also ineger and bounded. The sae of he sysem before he decision is made is denoed by S = (W, R ), while he sae of he sysem afer he decision is made is denoed by S x = (W, R x ). The one period cos is given by C (S, x ) = ρ sh (D l + D s R )1 {D l +D s R } + P f (D l R )1 {D l R } + P r (R D l D s )1 {D l +D s <R } + ρr (x 1 + x 2 ). The firs erm is he shorfall cos, while he second erm is he financial cos. The hird erm is he opporuniy cos and he las erm is he ransacion cos. Le X π (S ) be a decision funcion which deermines x given he informaion in S. We assume ha we have a family of funcions X π, π Π. We consider boh saionary and nonsaionary policies in his paper, so for he purpose of properly represening nonsaionary policies, he decision (1) funcion is indexed by ime. When we refer o a policy π, we mean he decision funcion X π for some policy π Π. Laer, we provide more specific meaning o a specific policy. Our problem is o solve T inf IE γ C (S, X π (S )) π Π =0 where γ is a discoun facor beween 0 and 1. There are wo sraegies we can use o solve he problem (we es boh in our experimenal work). The firs is an offline sraegy where we use prior hisory o updae our forecass a each ime period, from which we generae wha is ypically a nonsaionary forecas of he fuure. For example, we may feel ha a sudden drop in he marke will be followed by a fairly fas reurn o normal levels over he nex few days. We would implemen such a sraegy using a finie horizon model (T migh be five or 10 days). The second sraegy is an online implemenaion where we assume ha all processes are saionary. In his case, we would use an infinie-horizon objecive (T = ). We demonsrae how a seady-sae model such as his can be implemened very easily, wihou requiring an explici updae o shor-erm forecass.

7 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no Muual Fund Cash Holding Models We firs inroduce a dynamic programming model where decisions a ime consider heir impac on he fuure. This model is compared o wo saic models which use myopic policies. The firs one is a sraighforward simplificaion of he dynamic model, in he sense ha he only difference beween hem is ha he value funcion will no play a role when a decision is aken. This way, we are able o measure he imporance of he role of he value funcion, since i adds significan compuaional burden o solve he problem. The second saic model, proposed by Yan (2006), is even simpler. I does no consider ransacion coss and here is no disincion beween he wo ypes of demand. The policy obained from minimizing he cos is he same as he one obained from maximizing he profi. Since he original approximae algorihm deals wih profi maximizaion, we use his erminology when describing he models and algorihmic sraegies The Dynamic Model We sar his secion discussing he opimal value funcions for he cash holding problem. In secion 2, we inroduced he sae of he sysem, a ime, before and afer he decision is aken, denoed by S = (W, R ) and S x = (W, R x ), respecively. We le V (S ) be he opimal value of being in sae S, where V (S ) = max x X (W,R ) C (W, R, x) + sup IE π Π [ T =+1 γ C (S, X π (S )) (S, x) Similarly, we le V,x (S x ) be he opimal value of being in pos-decision sae S x, where [ T ] V,x (S x ) = sup IE π Π =+1 γ C (S, X π (S )) S x Equivalenly, he opimal value funcions can be defined recursively. Using he value funcions around he pre- and pos-decision saes, we break Bellman s equaion ino wo seps:. ]. V,x 1(W 1, R x 1) = IE [ V (W, R ) (W 1, R x 1) ], (2) V (W, R ) = max x X (W,R ) C (W, R, x) + γv,x (W, R x ), (3)

8 Auhor: Muual fund cash balance problem 8 Aricle submied o Managemen Science; manuscrip no. A ime = T, since i is he end of he planning horizon, V,x T (W T, R x T ) = 0. A ime 1, for = T,..., 1, he value of being in any pos-decision sae (W 1, R x 1) does no involve he soluion of an opimizaion problem, i only involves an expecaion, since he nex pre-decision sae (W, R ) only depends on he exogenous informaion ha firs becomes available a, as in (2). On he oher hand, he value of being in any pre-decision sae (W, R ) a does no involve expecaions, since he nex pos-decision sae (W, R x ) is a deerminisic funcion of W, i only requires he soluion of an opimizaion problem, as in (2). If we subsiue (3) ino (2), we ge [ V 1(W,x 1, R 1) x = IE max x X (W,R ) ] C (W, R, x) + γv,x (W, R x ) (W 1, R 1) x. (4) For algorihmic reasons, hroughou he paper, we only use (4), insead of (3) and (2), i.e., we only consider he value funcion around he pos-decision sae. Is main feaure is he inversion of he opimizaion/expecaion order in he value funcion formula. See Powell (2007), Chaper 4, for a complee discussion of pos-decision saes. In order o simplify noaion, we will jus drop he superscrip x in he value funcion noaion. We perform a qualiaive analysis of (4) in order o provide insighs abou he opimal policy. This analysis is also he foundaion for he SPAR-Muual algorihm. Even wihou compuing he expecaion in (4), given he inegraliy assumpions on D i, D l and D s, he funcion V 1(W 1, ) : [0, ] IR is piecewise linear wih ineger break poins. Thus, disregarding is value a (W 1, 0), he funcion can be idenified uniquely by is slopes { v 1 (W 1, 1), v 1(W 1, 2),..., v 1(W 1, B 1 ) }, where B 1 is he upper bound on he cash level R x 1 and v 1(W 1, R x 1) = V 1(W 1, R x 1) V 1(W 1, R x 1 1). Assuming he slopes are monoone decreasing in he cash level dimension, ha is, v (W, R x ) v (W, R x + 1), (5) for all saes (W, R x ), we give a simple and inuiive proof ha he opimal cash holding policy only rebalances he cash holdings when he cash level goes ouside a cerain range.

9 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 9 To describe he policy, we define he following ranges: R 1 (W ) = {R : γv (W, R) > ρ r }, R 2 (W ) = {R : ρ r γv (W, R) ρ r } and R 3 (W ) = {R : γv (W, R) < ρ r }. Theorem 1. Given a pre-decision sae (W, R x ), assume (5) for all possible cash levels. Regions R 1 (W ), R 2 (W ) and R 3 (W ) deermine a decision rule for he opimizaion problem max C (W, R, x ) + γv (W, R + x 1 x 2 ), (6) x X (W,R ) where R = max(0, R D l D s ) and V (W, R) = R v (W, r). The opimal policy is: Rule 1: If R R 1 (W ): x 1 = min ( M, max { x 1 : R + x 1 R 1 (W ) }), x 2 = 0; Rule 2: If R R 2 (W ): x 1 = 0, x 2 = 0; Rule 3: If R R 3 (W ): x 1 = 0, x 2 = max ( R, min { x2 : R x 2 R 2 (W ) }). r=1 Proof: Proof of heorem 1. For any pre-decision sae (W, R ), he opimal decision x for (6) is he same as he opimal decision x of he following linear problem: subjec o max x,y,z ρr (x 1 + x 2 ) + γ M y i = x 1 i=1 M v (W, R + i)y i + γ i=1 R j=1 R j=1 z j = x 2 0 y 1 0 z 1. v (W, R j)(1 z j ) Noe ha even wihou imposing inegraliy, each componen of y and z is eiher equal o 0 or 1. Each y i = 1, for i = 1,..., M, represens he decision o ransfer one dollar from he porfolio o he bank accoun. On he oher hand, each z j = 1, for j = 1,..., R represens he ransfer of one dollar from he bank accoun o he porfolio. Moreover, given (5), if for some i, y i = 1 hen y i = 1 for all i < i, y i = 0 for all i > i and he enire vecor z is equal o 0. Furhermore, if for some j, z j = 1 hen z j = 1 for all j < j, z j = 0 for all j > j and he enire vecor y is equal o 0. This follows he logical reasoning ha if money is ransferred from he porfolio o he bank accoun i makes no sense o ransfer money from he bank accoun o he porfolio. Clearly, if R R 1 (W ), hen he opimal vecor z is equal o zero and y i = 1 for all i such ha γv (W, R + i) > ρ r, ha is, we keep ransferring money o he bank accoun while he ransacion cos is smaller han he marginal value of having one more dollar in cash or unil he upper bound

10 Auhor: Muual fund cash balance problem 10 Aricle submied o Managemen Science; manuscrip no. M is reached. Symmerically, if R R 3 (W ), hen he opimal vecor y is equal o zero and zj = 1 for all j such ha γv (W, R j) < ρ r, ha is, we keep ransferring money o he porfolio while he ransacion cos is smaller han he marginal value of having one less dollar in cash or unil here is no more cash o be ransferred. Finally, if R R 2 (W ), hen y = 0 and z = 0, since he ransacion cos is oo high o jusify any ransfer of money, proving rules 1-3. We nex give an expression for he opimal slopes and show ha hey are monoonically decreasing, a propery ha we exploi in he SPAR-Muual algorihm. This propery no only acceleraes he rae of convergence of he algorihm, bu i is also cenral o is pure exploiaion naure, resuling in a simpler and faser procedure. Theorem 2. For = T,..., 1 and all saes (W 1, R x 1), he opimal slopes are given by ] v 1(W 1, R 1) x = IE [Ĝ(W, R, v ) (W 1, R 1) x, (7) where Ĝ(W, R, v ) = ρ sh 1 {D l +D s R } + P f 1 {D l R } P r 1 {D l +D s <R } + max ( ρ r, γv (W, R D l D s + M ) ) 1 {γv (W,R D l Ds )>ρr } 1 {D l +Ds <R } + γv (W, R D l D s )1 { ρ r γv (W,R D l Ds ) ρr } 1 {D l +Ds <R } ρ r 1 {γv (W,R D l Ds )< ρr } 1 {D l +Ds <R }. Moreover, he opimal slopes are monoone decreasing in he cash level dimension, ha is, v 1(W 1, R x 1) v 1(W 1, R x 1 + 1), implying concaviy of he opimal value funcion. The proof of Theorem 2 is given in he appendix The Saic Models In our firs saic model, given he ime period and he pre-decision sae (W, R ), he objecive is o find a decision ha maximizes he expeced profi in he nex ime period, where he profi

11 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 11 is he negaive of he cos given by (1). The second model is similar o he one proposed in Yan (2006). I does no consider ransacion coss and here is no disincion beween he wo ypes of demand. Because of he laer simplificaion, he finance and he shorfall coss are collapsed ogeher. As in he firs saic model, given he ime period and he pre-decision sae (W, R ), he objecive is o find a decision ha maximizes he expeced profi in he nex ime period, where he expression for he profi akes ino consideraion he simplificaions menioned above. Le D +1 = D+1 l + D+1 s D+1 i be he ne flow of money. For he firs and second model, he opimal decision x is hus found by solving, respecively, he maximizaion problems: max IE [ C +1 (W +1, R x + D+1, i x ) (W, R ) ], (8) x X (W,R ) max IE [ (ρ sh + P+1)(D f +1 R x )1 {D+1 R x x X (W,R ) } P+1(R r x D +1 )1 {D+1 <R x} (W, R ) ], (9) where R x = max(0, R D l D s ) + x 1 x Algorihmic Sraegies In his secion, we presen algorihms o acually compue he decisions implied by each cash holding model. For he dynamic seing, once we compue he slopes of he value funcions, a soluion is easily deermined following he decision rule described in Theorem 1. We propose wo approaches o find he slopes. The firs one is radiional backward dynamic programming. Even hough his approach is simple and sraighforward, i is compuaionally very demanding. The second approach is hrough approximae dynamic programming (ADP). The ADP algorihm replaces he compuaion of he expeced value by ieraively observing sample pah realizaions and, mos imporanly, by exploiing he srucural propery of he opimal slopes, namely he monoone decreasing propery. Of course, ADP replaces he compuaional burden of he exac soluion wih he saisical errors of a Mone-Carlo based procedure. The upside is ha he slopes do converge o he opimal ones in he limi. The idea is ha if he number of ieraions is large enough, he approximae soluions are very close o he opimal ones. The ADP approach is discussed in full in he nex secion.

12 Auhor: Muual fund cash balance problem 12 Aricle submied o Managemen Science; manuscrip no. The opimal decision for he saic models is obained in a similar fashion as he opimal decision for he dynamic model a ime period T 1, since no downsream effecs are aken ino consideraion a he end of he planning horizon T. Therefore, our procedure o deermine he opimal decision for he saic models follows he reasoning of he proof of Theorem 1, i.e., we make he decision comparing he ransacion cos wih he marginal value of ransferring one dollar o/from he bank accoun. Given he pre-decision sae (W, R ), le f 1 (W, R x ) and f 2 (W, R x ) denoe he marginal value of ransferring one dollar o he bank accoun for he firs and second saic models, respecively, when he cash level is R x = max(0, R D l D s ) + x 1 x 2. From (8) and (9), we can easily obain f 1 (W, R x ) = E f IP{D l +1 D i +1 R x (W, R )} + (ρ sh + E r )IP{D +1 R x (W, R )} E r, f 2 (W, R x ) = (E f + ρ sh + E r )IP{D +1 R x (W, R )} E r, where E f = IE[P f +1 W ], E r = IE[P r +1 W ] and D +1 = D l +1 + D s +1 D i +1. Symmerically, he marginal value of ransferring one dollar from he bank accoun is given by f 1 (W, R x ) and f 2 (W, R x ), respecively. The opimal decision for he firs model can be found using he procedure: STEP 0: Iniialize x 1 = x 2 = 0. STEP 1: While f 1 (W, R x ) > ρ r and x 1 < M do x 1 = x STEP 2: While f 1 (W, R x ) < ρ r and x 2 < max(0, R D l D s ) do x 2 = x Since here is no ransacion cos involved in he second model, is soluion is deermined using a similar procedure, replacing ρ r by zero and f 1 (W, R x ) by f 2 (W, R x ). 5. The Approximae Dynamic Programming Approach The main idea of he algorihm is o consruc concave and piecewise linear funcion approximaions V n (W, ), learning is slopes v n (W, R x 1),..., v n (W, R x N) over he ieraions. A each ieraion, our decision funcion looks like X π (S n ) = arg max C (S n x X (W n,rn ), x ) + γ n 1 V (W n, R x )

13 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 13 where n 1 V (W, ) is he piecewise, linear value funcion approximaion compued using informaion up hrough ieraion n 1. We now see ha our policy is parameerized by he slopes v n 1 (W, r), r = 1, 2,... for each possible value of W. Thus, when we refer o a policy π, we are acually referring o a specific se of slopes. The cach is ha he algorihm does no ry o learn he slopes for he whole sae space, only for pars close o opimal cash levels, which are deermined by he algorihm iself. Figure 1 illusraes he idea. V (W, ) Opimal - Unknown V n (W, ) Approximaion - Consruced v (W, R x 1 ) v (W, R x 2 ) v n (W, R x 1 ) v n (W, R x 2 ) v (W, R x 3 ) v n (W, R x 3 ) v (W, R x 4 ) v n (W, R x 4 ) Imporan Region v n (W, R x 5 ) v (W, R x 5 ) R x 1 R x 2 R x 3 Asse Dimension Figure 1 Opimal value funcion and he consruced approximaion A each ieraion n and ime, insead of compuing he expecaion in (4), he algorihm observes one sample realizaion of he informaion vecor. Afer ha, he sample realizaion and he curren value funcion approximaion are used o ake a decision x n, leading he sysem o a pos-decision sae. Sample informaion around he new pos-decision sae is gahered and is used o updae he approximae slopes v n 1 1. A projecion operaion is hen performed in case a violaion of he concaviy propery occurs The SPAR-Muual Algorihm Before we presen he algorihm, some noaion is necessary. A general pos-decision sae a is denoed by S x or (W, R x ). The wo of hem are used inerchangeably. We use S x,n o denoe he acual pos-decision sae visied by he algorihm a ieraion n and ime. The same noaion convenion holds for he pre-decision saes. A ieraion n and ime, he acual decision aken by he algorihm is denoed by x n, while he value funcion approximaion is denoed by V n (W, ). The

14 Auhor: Muual fund cash balance problem 14 Aricle submied o Managemen Science; manuscrip no. STEP 0: Algorihm iniializaion: STEP 0a: Iniialize v 0 (W, R x ) for all and (W, R x ) monoone decreasing in R x. STEP 0b: Pick N, he oal number of ieraions. STEP 0c: Se n = 1. STEP 1: Planning horizon iniializaion: Observe he iniial cash level R x,n 1. Do for = 0,..., T : STEP 2: Sample/Observe P f,n, P r,n, D i,n, D l,n and D s,n. STEP 3: Compue he pre-decision cash level: R n = R x,n 1 + D i,n. STEP 4: Slope updae procedure: If > 0 hen STEP 4a: Observe ˆv n (R x,n 1) and ˆv n (R x,n 1 + 1). STEP 4b: For all possible saes S x 1: z n 1(S x 1) = (1 ᾱ n 1(S x 1)) v n 1 1 (S x 1) + ᾱ n 1(S x 1)ˆv n (R x 1). STEP 4c: Perform he projecion operaion v n 1 = Π C,W n 1,R x,n 1 (zn 1). See (10). STEP 5: Find he opimal soluion x n of max x X (W n,rn ) C (S n, x) + γ STEP 6: Compue he pos-decision cash level: R x,n STEP 7: If n < N increase n by one and go o sep 1. Else, reurn v N. n 1 V (W n, R x ). = max(0, R n D l,n D s,n ) + x n 1 x n 2. Figure 2 SPAR-Muual Algorihm corresponding slopes are denoed by v n (W ) = ( v n (W, 1),..., v n (W, B )). Clearly, V n (W, R x ) = R x r=1 vn (W, r). We refer inerchangeably o he value funcion iself and o is slopes. The SPAR-Muual algorihm is presened in figure 2. As described in Sep 0, he algorihm inpus are piecewise linear value funcion approximaions represened by heir slopes v 0 (W, R x ). The iniial slopes mus be decreasing in he cash level dimension. A slope vecor ha is equal o zero for all saes and ime periods is a valid inpu. Since we know ha v T (W T, R x T ) = 0 for all saes (W T, R x T ), hen we use v n T (W T, R x T ) = 0 for all ieraions n. A each ieraion n, he algorihm sars by observing he iniial cash level, denoed by R x,n 1, as

15 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 15 in Sep 1. The iniial cash level mus be a posiive ineger. Afer ha, he algorihm proceeds over ime periods = 0,..., T. A he beginning of ime period, he algorihm generaes a sample of he ineres rae P f,n, rae of reurn P r,n, money inflow D i,n, insiuional redempion D l,n reail redempion D s,n, as in Sep 2. These are Mone Carlo samples following he probabiliy disribuion of he informaion process W given ha W n 1 = (P f,n 1, P r,n 1, D i,n 1, D l,n 1, D s,n 1). Afer ha, he pre-decision cash level R n is compued, as in Sep 3. Before he decision a ime period is aken, he algorihm uses he sample informaion o updae he slopes of ime period 1. Seps 4a-4c describes he procedure and figure 3 illusraes i. We firs observe slopes relaive o he pos-decision saes (W n 1, R x,n 1) and (W n 1, R x,n 1 + 1) (see Sep 4a and figure 3.a). Then, hese sample slopes are used o updae he approximaion and slopes v n 1 1 hrough a emporary slope vecor z n 1 (see Sep 4b and figure 3.b). This sep requires he use of a sepsize rule ha is sae dependen, denoed by ᾱ n (S x ). We have ha ᾱ n (S x ) = α n 1 {W =W n} (1 {R x =R x,n } + 1 {R x =R x,n +1}), where 0 < α n 1 and α n can depend only on informaion ha became available up hrough ieraion n and ime. For example, i is valid o use α n = 1 N(S x,n ), where N(S x,n ) is he number of visis o sae S x,n up unil ieraion n. The updaing scheme may produce slopes ha violae he propery of being monoonically decreasing. In his case, a projecion operaion is performed o resore he propery and obain he updaed approximaion slopes v n 1 (see Sep 4c and figure 3.c). Nex, a decision x n is made given he curren sae a ime. This decision is he opimal soluion wih respec o he curren pre-decision sae (W n, R n ) and value funcion approximaion V n 1 (W n, ), as saed in Sep 5. This decision can be easily calculaed following he decision rule described in Theorem 1. We jus need o consider he curren pre-decision sae (W n, R n ) and he curren slope approximaion v n 1 (W n ). Finally, he pos-decision sae R x,n is compued, as in Sep 6, and we advance he clock o ime + 1. As he algorihm reaches he planning horizon = T, if he number of ieraions has no reached is limi N, hen he ieraion couner is incremened, as in Sep 7, and a new ieraion is

16 n 1, V (x) Auhor: Muual fund cash balance problem 16 Aricle submied o Managemen Science; manuscrip no. F W n,r n 1 R n 1 ˆv n +1 A T x n R n ˆv n +1+ Exac x Slopes of v n 1 (W n, R n ) n 1 V (W n ) 3.a: Curren approximae funcion, opimal decision and sampled slopes Concaviy violaion Exac z n (W n, R n ) Slopes of z n (W n ) 3.b: Temporary approximae funcion wih violaion of concaviy (x, yn ) Fˆr n,p n,rn 1, V n Exac x v n (W n, R n ) Slopes of V n (W n ) 3.c: Level projecion operaion: updaed approximae funcion wih concaviy resored Figure 3 Slopes updae procedure, where F 1(S 1, n n 1 V 1, x) = C 1(Sn n 1 1, x) + γ V 1 (W 1, n R 1). x sared from Sep 1. Oherwise, he algorihm is finished reurning he curren slope approximaion v N (W, R x ) for all and (W, R x ). We obain sample slopes by replacing he expecaion and he opimal slope v in (7) by he sample realizaion W n he sample slope is ˆv n (R) = Ĝ(W n, R, v n 1 ). and he curren slope approximaion v n 1, respecively. Thus, for = 1..., T, The projecion operaor Π C,W n,r x,n maps a vecor z n ha may no be monoone decreasing in

17 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 17 he cash level dimension, ino anoher vecor v n ha has his srucural propery. The operaor imposes he propery by forcing he newly updaed slope a (W n, R x,n ) o be greaer han or equal o he newly updaed slope a (W n, R x,n + 1) and hen forcing he oher violaing slopes o be equal o he newly updaed ones. For any sae (W, R x ), he projecion is given by z(w n,rx,n )+z(w n,rx,n +1), if C1 2 Π C,W n,r x,n (z)(w, R x Π ) = C,W n,r x,n (z)(w n, R x,n ), if C2 Π C,W n,r x,n (z)(w n, R x,n + 1), if C3 z(w, R x ), oherwise, (10) where he condiions C1, C2 and C3 are C1: W = W n, R x = (R x,n or R x,n + 1), z(w n, R x,n ) < z(w n, R x,n + 1); C2: W = W n, C3: W = W n, R x < R x,n, z(w, R x ) Π C,W n,r x,n (z)(w n, R x,n ); R x > R x,n + 1, z(w, R x ) Π C,W n,r x,n (z)(w n, R x,n + 1) Convergence Analysis We sae wo heorems ha prove ha he algorihm converges o an opimal policy, ha is, he algorihm does learn he opimal decision o be aken a each sae ha can possibly be reached by an opimal policy. The proofs are based on more general resuls in Nascimeno e al. (2007), hence only a skech is provided here. We sar wih some remarks. We denoe by {S n } n 0 = {(W n, R n )} n 0 he sequence of predecision saes visied by he algorihm a ime. Likewise, we denoe by {x n } n 0 he sequence of decisions aken by he algorihm and by {S x,n } n 0 = {(W n, R x,n )} n 0 he sequence of posdecision saes visied by he algorihm. Each one of hese sequences has a leas one accumulaion poin. This resul derives from he fac ha he informaion process and he cash level have finie suppor. Moreover, he decisions are consrained o compac ses. We denoe by S, x and S x, he respecive accumulaion poins. We denoe by { v n (W )} n 0 he sequence of slopes generaed by he algorihm. Noe ha here is one such sequence for each vecor W. This sequence also has a leas one accumulaion poin, since he projecion operaion guaranees, for all n, ha v n (W ) belongs o a compac se, namely, he

18 Auhor: Muual fund cash balance problem 18 Aricle submied o Managemen Science; manuscrip no. se of vecors ha are monoone decreasing in he cash level dimension and are bounded (due o he finie suppor assumpion of he informaion process). We denoe by v (W ) an accumulaion poin of his sequence. Theorem 3. On he even ha (W, R x, ) is an accumulaion poin of {(W n, R x,n )} n 0, if n=0 ᾱ n (W, R x, ) = and (ᾱ n (W, R x, )) 2 < B < a.s., n=0 hen v n (W, R x, ) v (W, R x, ) and v n (W, R x, As a byproduc of he previous heorem, we obain he nex heorem: + 1) v (W, R x, + 1) a.s. (11) Theorem 4. For = 0,..., T, on he even ha (W, R, v, x ) is an accumulaion of {(W n, R n, v n 1, x n )} n 1, if he sepsize condiion of Theorem 3 is saisfied, hen, wih probabiliy one, x is an opimal soluion of max F (W x X (W,R ), R, V, x ), where F (W, R, V, x ) = C (W, R, x ) + γv (W, max(0, R D l, D s, ) + x 1 x 2 ). (12) Equaion (12) implies ha he algorihm has learned an opimal decision for all saes ha can be reached by an opimal policy. This implicaion can be easily jusified as follows. We sar wih = 0. For each accumulaion poin (W 0, R 0) of {(W n 0, R n 0 )} n 0, (12) ells us ha he accumulaion poins x 0 of {x n 0 } n 0 along he ieraions wih iniial pre-decision sae (W 0, R 0) are in fac an opimal decision for period 0 when he pre-decision sae is (W 0, R 0). This implies ha all accumulaion poins R x, 0 = max(0, R 0 D l, 0 D s, 0 ) + x 01 x 02 of {R x,n 0 } n 0 are pos-decision cash levels ha can be reached by an opimal policy. When = 1, for each R x, 0 and each accumulaion poin W 1 of {W n 1 } n 0, R 1 = R x, 0 + D i, 1 is a pre-decision cash level ha can be reached by an opimal policy. Once again, (12) ells us ha he accumulaion poins x 1 of {x n 1 } n 0 along he ieraions wih (W n 1, R n 1 ) = (W 1, R 1) are indeed an opimal decision for period 1 when he pre-decision sae is (W 1, R 1). As before, he accumulaion poins R x, 1 = max(0, R 1 D l, 1 D s, 1 ) + x 11 x 12 of {R x,n 1 } n 0 are pos-decision cash levels ha can be reached by an opimal policy. The same reasoning can be applied for = 2,..., T.

19 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no The Infinie Horizon Cash Holding Problem The infinie horizon problem arises when we believe ha he exogenous informaion processes (prices, ineres raes, deposis and redempions) are saionary. The only change is ha we drop he indexing by ime of all variables. We adjus he algorihms o he infinie horizon case in a fairly sraighforward way. For he saic models, we use he procedure described in secion 4, dropping he ime index. The ADP algorihm for he infinie horizon case is similar o he SPAR-Muual algorihm described in figure 2, excep ha we do no loop over he differen ime periods and again he ime index is dropped. The modified SPAR-Muual algorihm for he infinie horizon mehod brings a nice wis. I can be considered an online algorihm, in he sense ha probabiliy disribuions do no need o be known or esimaed. For he finie horizon case, he sample realizaions for he ineres rae, rae of reurn, deposis and wihdrawals are obained from a sample generaor ha follows a known disribuion. On he oher hand, acual daily hisorical values of he ineres rae, rae of reurn, inflow money and demand for redempion can be used for he infinie horizon case, wihou any need o esimae he probabiliy disribuion underlying hem. Moreover, as new daily informaion becomes available, i can be used o updae he curren slope approximaions. We do no have a proof of convergence for his online algorihm, bu we show ha he policies produced by his algorihm ouperform he saic ones. 7. Numerical Experimens The purpose of his secion is o analyze he behavior of he differen algorihmic approaches. We sudy he effec of discreizaion on CPU ime and soluion qualiy for he exac and he ADP algorihms. Moreover, we quanify how much we gain by considering he impac a decision has on he fuure, insead of jus using a myopic policy. Finally, for he infinie horizon case, we can observe he behavior of an online algorihm and he errors incurred in consrucing probabiliy disribuions and esimaing heir parameers. We wan o make sure we compare he algorihmic sraegies when hey are applied o realisic environmens. To achieve his goal, he probabiliy disribuions are esimaed using real daa,

20 Auhor: Muual fund cash balance problem 20 Aricle submied o Managemen Science; manuscrip no. including bank prime loan raes, raes of reurn of op performing funds, oal asse values and redempion raes of a broad range of funds. We sar by describing he insances considered and he daa used o consruc hem. Afer ha, we discuss implemenaion deails. We hen presen and analyze he resuls for he finie and infinie horizon cases Problem Insances We sar wih he ineres rae models. The Vasicek and he Cox-Ingersoll-Ross (CIR) processes are commonly used o model ineres raes (Cairns (2004)). The processes are described by dr = α(µ r )d + σdb and dr = α(µ r )d + σ r db, respecively, where r represens he ineres rae process, B is a sandard Brownian moion and α, µ and σ are parameers ha mus be esimaed. We used discree ime versions of hese models, capuring he behavior ha decisions are made once each day. Thus, we use r +1 = r + α(µ r ) + σ Z and r +1 = r + α(µ r ) + σ r Z, respecively, where Z is normally disribued wih mean 0 and variance 1, and is he lengh of a single day (in whaever unis we are measuring ime). We applied he maximum likelihood esimaion (MLE) mehod o fi hese parameers, using weekly prime rae hisorical daa (from July, 2001 o June, 2006) from he Federal Reserve Board websie 2, as bank prime loan is one of several base raes used by banks o price shor-erm business loans. For he porfolio rae of reurn, we used he Vasicek process o model he rae of reurn. We consider nine differen ses of funds. Each se represens a marke capializaion caegory (large, mid, small cap) and a syle (blend, growh, value). Moreover, each se is composed of he 10 bes performing funds in he given caegory/syle over he pas hree years according o Yahoo Finance 3. We idenify each se using he firs leer of he corresponding caegory/syle. For example, LB represens he se of large blend funds. 2 hp:// 3 hp://finance.yahoo.com/

21 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 21 Daily raes of reurn (from July, 2001 o June, 2006) for each fund were colleced from he CRSP Muual Fund Daabase 4 and were averaged across he en funds on each syle/caegory. The resuling daily raes and he MLE mehod were used o esimae he parameers of he Vasicek process. For each ime period, we assume he arrival of shareholders invesing (redeeming) money follows a compound Poisson process wih rae λ i (λ o ). Moreover, s i percen of hese shareholders are insiuions invesing d i (redeeming d o ) unis of dollars, while 100 s i (100 s o ) percen are reail shareholders invesing (redeeming) one uni of dollar. Daa from 8460 differen muual funds ha have boh insiuional and reail invesors and ha did no have any mergers or acquisiions from July, 2005 o June, 2006 were colleced from he CRSP Muual Fund Daabase. They show ha i is reasonable o consider λ i (λ o ) as a linear funcion of he rae of reurn P r, see figure 4.a. The redempion rae, as repored by he Invesmen Company Insiue websie 5, is calculaed by aking he sum of regular redempions and redempion exchanges for a given period as a percen of he average oal ne asses a he beginning and end of his period. Considering a group of 4623 sock funds, he redempion rae for he July, 2005-June, 2006 period was 25.2%. Therefore, we approximae he redempion amoun for each insance over his period by muliplying heir oal ne asses by he redempion rae. The resuling redempion amoun is displayed in figure 4.b. Since he cash holding problem objecive value is direcly relaed o he redempion amoun involved, we can use he numbers in 4.b o quanify he acual amoun of dollars ha can be saved when each algorihmic sraegy is employed o produce a policy. Finally, we picked he ransacion cos ρ r and he shorfall cos ρ sh o be equal o.1% and.2%, respecively. For he finie horizon case, he planning horizon is se o T = 5 working days Implemenaion Issues One of our goals is o compare he performance of an ADP-based algorihm o an exac soluion obained using radiional backward dynamic programming mehods. This requires ha we address 4 hp:// 5 hp://

22 Auhor: Muual fund cash balance problem 22 Aricle submied o Managemen Science; manuscrip no. Monhly Ne Flow Acual Daa Linear Regression Redempion (in millions of dollars) Monhly Rae of Reurn 50 0 LB LG LV MB MG MV SB SG SL Insance 4.a - Average Ne Flow 4.b - Approximae redempion amoun Figure 4 Ne Flow and Redempions he fac ha he ineres rae and rae of reurn processes are unbounded and coninuous, bu an exac soluion requires ha hey be bounded and discree. To obain an exac soluion, we had o creae a discreized version of he ineres rae and reurn processes. We did his by firs discreizing and runcaing he exogenous changes o ineres rae and reurn processes. Using his modified process, we creaed a probabiliy disribuion for hese processes ha refleced boh he discreizaion and he runcaion. We hen chose he fines level of discreizaion ha could be solved wih reasonable execuion imes (we allowed run imes o span days), and defined his o be he exac soluion. This soluion could hen be compared o soluions obained using backward dynamic programming for coarser levels of aggregaion, as well as soluions obained using he SPAR-Muual algorihm. When using he SPAR-Muual algorihm, we simulaed he original coninuous sae (ha is, he ineres rae and reurns movemens of cash were always assumed o occur in discree quaniies). Sample observaions of changes in ineres raes and reurns were generaed by eiher using he original coninuous disribuions or sample observaions from hisory (wihou discreizaion). Only he value funcion approximaion was discreized. This sraegy allowed us o esimae he errors produced using approximae dynamic programming in a realisic seing, bu we sill compared our performance agains an exac model which assumed a very fine level of discreizaion.

23 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 23 A second imporan implemenaion issue was he choice of he righ sepsize rule which is a key ingredien o faser raes of convergence. Even hough he sepsize rule α n = 1/N(S x,n ) produces a provably convergen algorihm, he associaed rae of convergence is poor, since his rule goes o zero oo quickly. The problem is ha while learning he slope for one sae, he updaing procedure depends on anoher slope approximaion, which is seadily changing, since he algorihm is also learning his oher slope, leading o a nonsaionary process. Moreover, he rae of convergence for each slope is differen and, ideally, he sepsize rule would reflec ha. For example, he slower he convergence of a given slope, he larger is sepsize should be. Since we canno une a sepsize rule for each slope, an adapive sepsize rule works bes. In our implemenaion, we use he adapive sepsize rule proposed in George & Powell (2006). I is given by α n = 1 (ˆσn (Sx,n he number of visis o sae S x,n )) 2 δ m(sx,n ), where m is up o ieraion n. (ˆσ n (S x,n )) 2 is an esimae of he variance of he observaion error, and δ m (S x,n ) is an esimae of he oal squared variaion beween he observaion and he esimae which includes boh bias (if our esimaes are consisenly high or low) as well as he observaion error. These are compued using δ m (S x,n ) = (1 ν m 1 ) δ m 1 β m (S x,n ) = (1 ν m 1 ) (ˆσ n (S x,n )) 2 = δ m (S x,n 1 + β m 1 (S x,n (S x,n ) ( β m (S x,n λ m 1 Here, ν m = 10/(10 + m) is a sepsize formula. ) + ν m 1 (ˆv n +1(R x,n )) 2 (S x,n ) ) + ν m 1 (ˆv n +1(R x,n. ) v n 1 (S x,n )) 2 ; ) v n 1 (S x,n )); The oher ingredien o faser raes of convergence is he choice of discreizaion incremens. I is inuiive o expec ha coarser discreizaion incremens (smaller sae space) lead o faser raes of convergence in he iniial ieraions bu poorer resuls in he long run, while finer discreizaion incremens (larger sae space) lead o slower raes of convergence in he iniial ieraions bu more accurae resuls in he long run. The reasoning behind his inuiion is ha, when he discreizaion is fine, in he iniial ieraions, mos of he sae space (in he informaion dimension W ) does no have enough observaions o produce a reasonable slope approximaion. Since he SPAR-Muual

24 Auhor: Muual fund cash balance problem 24 Aricle submied o Managemen Science; manuscrip no. algorihm depends on he slope approximaion o make a decision and his decision deermines he furher course of he algorihm, he rae of convergence can suffer from his lack of informaion. We sar he SPAR-Muual algorihm (for he infinie horizon case) considering a coarse discreizaion incremen for he rae of reurn. Then, afer N 1 ieraions, we swich he discreizaion incremen increasing by a facor of wo he sae space. We repea he same procedure afer N 2 and N 3 ieraions. The values of N 1, N 2 and N 3 are a rough esimae of he number of ieraions necessary o observe a wide specrum of values of W. We close his secion describing how he numerical experimens were conduced. All he algorihms were implemened in Java, on a 3.06 GHz Inel P4 machine wih 2 GB of memory running Linux. In order o evaluae he policies generaed by he differen algorihmic sraegies, we creaed, for each insance, a unique esing se. For he finie horizon case, he esing se consiss of 1000 differen sample pahs ha were randomly generaed following he processes described in secion 7.1. For he infinie horizon case, he esing se consiss of acual daily prime rae and rae of reurns from July, 2005 o June, I is worh menioning ha he esing se is no used as par of he sample daa required o learn he slopes. Unless oherwise noed, we adop as opimal he policy obained using radiional dynamic programming wih discreizaion incremens.001 and.0001 for he ineres rae and rae of reurn, respecively. For i = 1,..., 1000, le F π i be he value of following policy named π for he i h sample pah ω i in he esing se, given by Le F i F π i = T γ C (S (ω i ), X π (S (ω i ))). =0 be he value of following he opimal policy labeled by π. Moreover, when he approximae algorihms are considered, we add he superscrip n o he noaion, indicaing ha F n i is measuring he policy obained afer n ieraions of he algorihm. In order o ake ino accoun he randomness involved in he approximae approaches, he policy considered o obain F n i is in fac an average over 10 runs of he SPAR-Muual algorihm, each saring wih a differen random seed.

25 Auhor: Muual fund cash balance problem Aricle submied o Managemen Science; manuscrip no. 25 Finally, we measure our disance from he opimal policy using η n = i=1 ( F n i 1000 F i=1 i F ) i. (13) We noe ha i is possible for F n i o be negaive since one of he coss of holding cash is he los reurn from no having he money properly invesed. This cos can be negaive during a marke downurn, when we acually make money relaive o he marke Finie Horizon Resuls We sar by presening he opimal value funcion (figure 5.a) and is corresponding slopes (figure 5.b) for ime period = 0 and a fixed ineres rae when insance LB is considered. 5.a - Opimal Value Funcion 5.b - Corresponding Opimal Slopes Figure 5 Iniial ime period and fixed ineres rae - Insance LB Figure 5.b also shows he hree regions considered in heorem 1 as well as he planes ha define hem. Noe ha as he rae of reurn decreases he hresholds beween Regions I (R 1 (W )) and II (R 2 (W )) and beween Regions II and III (R 3 (W )) increase, indicaing ha more cash should be held. This is consisen wih he inuiion ha lower opporuniy coss and marke iming when he reurn is low lead o more cash holdings. Region II represens he main role played by he ransacion cos ρ r. When he cash level is in his region, i means ha moving money from/o