Invesmen Decisions and Falling Cos of Daa Analyics Hong Ming Tan Insiue of Operaions Research and Analyics Naional Universiy of Singapore Jussi Keppo NUS Business School Naional Universiy of Singapore Chao Zhou Deparmen of Mahemaics Naional Universiy of Singapore January 15, 218 1s Draf Absrac We sudy how he cos of daa analyics and he characerisics of invesors and invesmen opporuniies affec invesmen decisions and heir daa analyics. We show ha he falling cos of he daa analyics raises invesors leverage, financially consrained or highly risk-averse invesors use less daa analyics, and he demand of daa analyics is highes wih high expeced reurn opporuniies. Due o he increased leverage, he falling cos of daa analyics leads o higher losses during he crises. We have benefied from commens a he Fifh Asian Quaniaive Finance Conference Seoul, 217 Supply Chain Finance & Risk Managemen Workshop Washingon Universiy in S. Louis, Elevenh Annual Risk Managemen Conference Singapore, 217 INFORMS Houson, Leeds School of Business Universiy of Colorado Boulder and Roman School of Managemen Universiy of Torono, and more specifically from Ashish Agarwal, Vlad Babich, Diego Garcia, Seven Kou, Kai Larsen, Pauli Muro, Juuso Valimaki, and Zhiwen Wang. All errors are ours. 1
1 Inroducion Informaion reduces uncerainy and enables beer decisions. Therefore, companies and individuals are willing o pay for daa and is analyses we call his as daa analyics. For insance, companies inves in informaion sysems and build analyics eams, hey pay marke research firms o esablish he likelihood ha a new produc is well received in he marke, and hey pay recruimen agencies o find producive new employees and o collec informaion on hem. New echnologies, digializaion of daa, and daa analyics mehods have decreased he cos of informaion subsanially during he las few decades. Figure 1 below illusraes his rend. In his paper, we sudy how he cos of daa analyics affec firms and individuals invesmen decisions, and which ype of invesors and invesmen opporuniies benefi he mos from he declining cos of daa analyics. USD/MBye 1 8 1 6 1 4 1 1.1 196 197 198 199 2 21 Figure 1: Cos of daa sorage. Daa is from [1], and y-axis is US$ per megabye in a log scale. Year We develop a model where a risk-averse invesor faces a join decision of he level of daa analyics and invesmen amoun. More specifically, we model daa acquisiion, collecion, and analyics wih Bayesian learning. The invesor has a CARA or exponenial uiliy funcion and she has an opion o buy daa and analyze ha before her invesmen decision. Informaion is cosly due o he price of daases and effors in collecing and processing he daa. The more he invesor 1
buys and analyze daa, he more signals she receives on he payoff of he invesmen opporuniy. Afer he daa analyics he decision maker decides how much o inves and, if needed, she uses leverage in he invesmen. We derive several key resuls. Firs, he more daa analyics he invesor uses, he more on average she invess. This is because he analyics lowers he riskiness of he invesmen and, hus, uncondiionally on he analyics oucome, i makes he invesmen opporuniy more aracive. This indicaes ha he daa analyics is a risk managemen device. Second, he lower he cos of daa analyics, naurally he more he invesor uses analyics, which decreases he riskiness of he invesmen opporuniy and his way raises he invesmen amoun and leverage. We show ha he higher leverage driven by he falling cos of daa analyics leads o higher losses during he crises. On he oher hand, if he invesor canno ake leverage hen naurally she invess less, bu also uses less daa analyics. Tha is, financially consrained firms have less incenive o learn before heir invesmens. Third, he lower he risk aversion of he invesor, he more she uses daa analyics. This is because an invesor wih a low risk aversion invess more in he risky invesmen and, herefore, she has a higher incenive o collec new informaion on he invesmen opporuniy. Fourh, he demand of daa analyics is highes wih high expeced reurn opporuniies. Wih hese opporuniies he invesor uses a high level of leverage and, herefore, she decides o analyze horoughly he invesmen opporuniy. Our model applies o, e.g., companies real asse invesmens, privae equiy and venure capial firms invesmen decisions, and muual funds porfolio decision on publicly raded asses wih a rare even risk ha canno be learned from marke for free. In hese examples he invesors need o do cosly daa analyics if hey wan o ge informaion on he invesmen opporuniy, and hen inves according o heir beliefs and precision on he invesmen payoff. We uilize value of informaion model in [8], which is relaed o [2], [11], [12] and [15]. In conras o hese papers, we also opimize he invesmen level which is no a binary decision. Our work is also relaed o [5], where raders maximize CARA uiliy and may buy informaion abou he payoff of a sock, which is raded in a compeiive marke. The raders privae informaion is only parially observed from he equilibrium price because in he equilibrium here are also some uninformed raders. [16] exends ha model o include diverse informaion acquisiion and derives an equilibrium in such a marke. By uilizing [5] and [16], [14] shows ha as long as absolue risk aversion falls in wealh, here is rising reurns o acquiring privae informaion even hough i ges revealed by public signals. [4] shows ha 2
daa abundance rises asse price informaiveness in he shor run bu no necessarily in he long run, because profis from rading on more precise signals fall. Our paper is also relaed o sudies ha consider porfolio opimizaion when he expeced reurns of risky asses are unknown, bu can be learned from he realized reurns of he asses see e.g. [6], [7], and [9]. In conras o hese sudies, in he presen paper we opimize he level of informaion and focus on he falling cos of informaion. The res of he paper is organized as follows: Secion 2 presens he decision problem and he quaniy of daa analyics. Secion 3 derives he value of daa analyics and Secion 4 solves he opimal quaniy of daa analyics. Secion 5 analyzes he opimal invesmen. Appendixes A B exend he model o consider a nonlinear cos of daa analyics and muliple invesmen opporuniies. 2 Model We assume a wo period model, and ha he decision maker DM has a CARA uiliy funcion uκ = exp γκ, where γ > is he risk-aversion parameer. Hence, u κ = γ exp γκ >, u κ = γ 2 exp γκ <, and he derivaives absolue values fall in κ. The DM has iniial wealh κ and decides on he level of risky invesmen w and risk-free invesmen κ w. Each uni of he risky invesmen has an iniial sunk cos of $1 and a payoff θ which is unknown o he DM. When κ w <, he DM uses leverage. The borrowing and lending raes are boh equal o r. Hence, borrowing $1 gives a payoff of $ e r in he nex period. We assume ha boh risky invesmen and deb payoffs are already he presen values r and θ can be calibraed o give presen value payoffs. Therefore, he DM s uiliy in he nex period is exp γwθ 1 + κ we r 1. Le Ω, F, P be a probabiliy space, where Ω is a se, F a σ-algebra of subses of Ω, and P a probabiliy measure on F. This space capures all he uncerainies relaed o he payoff θ. The quaniy of daa analyics corresponds o observing a noisy signal Z on payoff θ from o noe ha ime is fixed, is quaniy. The noisy signal follows dz = θd + db, where B is a sandard Wiener process and payoff θ is consan bu unknown. F is he informaion level from observing Z up o quaniy. Thus, he DM can learn abou he payoff θ by obaining quaniy of daa analyics a a marginal cos c, i.e. quaniy of daa analyics coss c. For a posiive cos of daa analyics, using quaniy of daa analyics causes he DM s iniial wealh κ o decrease o κ c. In his case, he leverage is given by max{κ c w, }. A larger quaniy of daa analyics always allows he DM o 3
learn more abou he payoff. So if 2 1, hen he DM wih a quaniy of 2 knows more han wih a quaniy of 1. In he firs period, informaion is firs bough and analysed, and hen he invesmen amoun and is funding own money and leverage are decided. In he second period, he invesmen payoff is realized and he loans if any are paid back. Wih respec o he firs period s informaion, he payoff is a normally disribued random variable. Thus, decisions in he firs period are quaniy of daa analyics variable, which gives informaion on he invesmen payoff, and he invesmen level variable w, which gives also he leverage. We illusrae he srucure of he model in Figure 2. Z w uw w uw Z Sep 1: Sep 2: Quaniy of analyics Invesmen amoun Period 1 Uiliy of invesmen payoff Period 2 Figure 2: Model srucure. In period 1, he DM performs wo seps. Firs, he DM opimizes he quaniy of daa analyics. This corresponds o observing he noisy signal Z from o. Second, he DM chooses he invesmen amoun w based on he realized signal from o. In period 2, he invesmen payoff is realised, which wih w gives uiliy uw. In period 1, he payoff θ is consan bu unknown and wih zero analyics quaniy we have θ F N ˆθ,. More generally, ˆθ = E[θ F ] and s = E[θ ˆθ 2 F ] are he expeced payoff and variance corresponding o analyics quaniy decided by he DM. So, θ F N ˆθ, s, i.e, θ F = ˆθ + s ɛ 1, where ɛ 1 N, 1. Parameers ˆθ and s are calculaed by Kalman-Bucy filer see e.g. [13]. If he DM decides o obain analyics quaniy hen he payoff can be wrien as follows. 4
Lemma 1 Uncondiional and condiional payoffs Le he DM obain analyics quaniy. Uncondiionally on he signal oucome, we have θ = ˆθ s 2 + 1 + ɛ 2 + s ɛ 1, and condiional on he signal oucome, we have θ = ˆθ + s ɛ 1, where s = 1+, ɛ 1, ɛ 2 N, 1, and ɛ 1 ɛ 2. Proof. See Appendix C.1. Here, ˆθ and are he DM s prior mean and variance. By Lemma 1, iniially he payoff θ follows a normal disribuion wih mean ˆθ and variance ; and wih analyics quaniy >, variance s <, i.e. analyics decreases he uncerainy of he payoff. Afer he DM learns he firs uncerainy erm ɛ 2 hrough observing he signal Z from o, he DM has belief ˆθ on he payoff and, hus, he second uncerainy erm ɛ 1 sill remains. This second uncerainy erm disappears as he DM learns more since s vanishes a infiniy. Tha is, he DM learns he rue payoff wih no uncerainy a infinie quaniy of analyics. 3 Value of daa analyics We begin firs by deermining he value of daa analyics o he DM. Daa analyics has value if i raises he DM s expeced uiliy. So, he value of daa analyics is he difference beween he expeced uiliies wih a posiive quaniy of daa analyics and wih zero quaniy of daa analyics, and if his difference is posiive hen he daa analyics is valuable o he DM. Noe ha presumably he value of daa analyics is posiive since i lowers he uncerainy of he payoff. In his secion, we do no ye solve he opimal quaniy of daa analyics. Insead we analyze he value of any given quaniy of daa analyics o he DM. However, his is he firs sep in deermining he opimal quaniy of daa analyics used by he DM. 5
3.1 Opimal invesmen level Recall ha our model is a wo-period model consising of a single DM wih a CARA uiliy funcion uκ = exp γκ wih a coefficien of absolue risk aversion γ and wealh κ. We firs perform invesmen opimizaion for a given analyics quaniy. The opimal invesmen level of he DM is given in he following lemma. Lemma 2 Opimal invesmen Given analyics quaniy and CARA uiliy funcion wih γ >, he opimal invesmen level of risky invesmen is given by w = max { } ˆθ e r,. 1 γs Proof. Uiliy of DM in he second period is exp γwθ 1+κ c we r 1. This gives he maximizaion problem: max E [ exp γwθ 1 + κ c w wer 1 F ] = max γwˆθ exp 1 γκ c we r 1 + γ2 w 2 w 2 s, which is sricly concave in w. The firs order condiion gives γˆθ 1 + γe r 1 + γ 2 ws = and, herefore, w = ˆθ e r γs. By he sric concaviy of he objecive funcion and he consrain on w, we ge 1. By equaion 1, he higher he expeced payoff ˆθ, he more he DM invess in he risky opporuniy. Furhermore, he higher he uncerainy s, he less he DM invess. A more risk averse invesor invess less, as expeced. When rises, s decreases and, uncondiionally on he signal oucome, he DM invess more. So, he more he DM uses analyics on he risky opporuniy, he more she invess. Noe also ha, as e.g. in [5] and [1], due o he CARA uiliy he opimal invesmen 1 is independen of wealh κ, which means ha he DM s leverage level max{κ c w, } is given by κ. We address his issue in subsecion 4.2, where we consider he case of financial consrain. Tha is, in ha subsecion he leverage is bounded and we show ha our key resuls hold also in ha seing. 6
3.2 Value and marginal value of daa analyics Subsiuing he opimal invesmen ino he uiliy funcion gives he expeced uiliy for any quaniy of daa analyics. This allows us o analyze how he expeced uiliy of he DM changes wih respec. Lemma 3 Expeced uiliy Given he quaniy of analyics, he expeced uiliy of he opimal invesmen before using he analyics is given by u =E [ exp γw θ 1 + κ c w e r 1] Φ d = e γκ c1 er s exp +1 Φd s + 1 ˆθ 2+e2r 2ˆθ e r +2γ κ ce r 1 2 where Φ is cumulaive sandard normal disribuion and d := e r ˆθ s +1. s 2 The expecaion is wih respec o F, so uncondiional on he signal oucome. Proof. See Appendix C.2. We use he uncondiional expecaion in Lemma 3 because ha is needed o decide he quaniy of analyics. More specifically, he value of daa analyics is he difference of he uncondiional expeced uiliies wih a posiive quaniy of daa analyics and zero quaniy of daa analyics. The following heorem gives he value of daa analyics. Proposiion 1 Value of daa analyics The value of daa analyics v := u u is e γκ c1 er Φd + 1 eγcer 1 e γκ1 er 1 e γcer 1 Φd Φ Φ ˆθ e r s +1 ˆθ e r exp exp ˆθ 2+e2r 2ˆθ e r +2γ κe r 1 2 ˆθ 2 +e2r 2ˆθ e r +2γ κ ce r 1 2 s +1 Noe ha when ˆθ = e r, we have v = e γκ1 er 1 eγcer 1 1 + 1 2 s. +1 if ˆθ e r if ˆθ < e r Proof. By Lemma 3, we ge u and u, which give he resul. See Appendix C.3. We calculae he marginal value of daa analyics from Proposiion 1. This helps us in solving for he opimal quaniy of daa analyics. 7
Corollary 1 Marginal value of daa analyics For >, he marginal value of daa analyics is given by [ 1c κ v = eγer 2cγ e r 1 s 2 + 1 3/2 + 1 3/2 Φ d 2cγ e r 1 + 1e e r ˆθ 2 2 Φ d s + 1 + e e r ˆθ 2 2 Φ d ] s + 1 Proof. v = u u = u. The resuls follows from Lemma 3 by aking he parial derivaive. A firs, obvious quesion o ask is wha happens when he cos of daa analyics is equals o. We consider his in he following. Corollary 2 Zero cos When c =, he DM buys an infinie amoun of daa analyics. Proof. If c =, hen v = 2 +1 3/2 exp er ˆθ 2 +2 κγe r 1 2 Φ d s +1 which is always greaer han. Hence, each exra uni of daa analyics always increases he uiliy of he DM. This is an inuiive resul as if here is no cos of using daa analyics, hen he DM should use an infinie amoun o updae his beliefs abou he payoff. The correcness of his resul serves as a simple es ha our model is correc abou i s predicions. Corollary 3 Small quaniy of analyics The DM does no buy small quaniies of analyics when c > and i ˆθ < e r or ii ˆθ = e r and < 22cγe r 1 or iii ˆθ > e r and < 2cγe r 1. Proof. Le c >. As, if ˆθ < e r, v 1 e r cγ expκγ1 e r <, if ˆθ = e r and < 4cγe r 1, hen v 1 expκγ1 4 er 4cγe r 1 <, and if ˆθ > e r and < 2cγe r 1, hen v 1 expκγ1 2 er er ˆθ 2 2 2cγe r 1 <. Since v = u u = and v is coninuous, hen v < implies ha here exiss a region, ɛ such ha for all ɛ, ɛ, v ɛ <, i.e. small amouns of analyics reduces he expeced uiliy of he DM. 8
We noe here ha he marginal value funcion is no coninuous when = and e r = ˆθ. However, we will always consider he case where e r > ˆθ or e r < ˆθ, and he value and marginal value funcions are coninuous in hese regions. Corollary 4 Choke-off cos There exiss cos c CH such ha if c < c CH, hen v > for some >. We call s c CH he choke-off cos, which is given by c CH = if ˆθ 2γe r 1 > e r. For ˆθ < e r, c CH, 2γe r 1. Proof. For ˆθ > e r, by Corollary 3, v if c 2γe r 1 concave and v =, v < for all >. For ˆθ < e r, if c =. Since v is sricly 2γe r 1 v < for all >. Since v is coninuous when ˆθ < e r, here exiss an ɛ > such ha v < for all c ɛ, + ɛ, so ha c = 2γe r 1 2γe r 1 2γe r 1 canno be c CH. When c =, v >. Hence by he Inermediae Value Theorem, c CH, 2γe r 1 Hence, by Corollary 4, he value of daa analyics has a leas a posiive par if c < c CH. This is he par ha is of imporance o our analysis because i is he region where using daa analyics provides a ne increase in he uiliy of he DM. I is also naural for he choke-off cos for invesmens wih ˆθ < e r o be lower han he choke-off cos for invesmens wih ˆθ > e r, since for low expeced reurn invesmens, a DM requires lower marginal cos c o ge value from analyics. Corollary 5 Risk aversion When c < c CH hen v falls in risk-aversion γ for all such ha v > Proof. From Proposiion 1, we ge v γ = 1 er κv cu which is negaive when v > as u is always negaive. Corollary 5 ells us ha he value of daa analyics falls in risk aversion. This is because, by Lemma 2, he higher he risk aversion he less he DM invess in he risky opporuniy and, herefore, informaion on he invesmen is less valuable o her. This resul is couner-inuiive as we would hink ha a more risk-adverse invesor would value informaion more. An inuiive way o hink of his is if an invesor holds none of he risky invesmen, hen having more informaion on he invesmen offers no value o he invesor. We illusrae his resul laer in he secion. Proposiion 2 Shape of he value of daa analyics The value of daa analyics funcion is 9
i no globally concave if ˆθ < e r and c < c CH, so ha he value of daa analyics is posiive in some inerval of. ii sricly concave if ˆθ > e r. Furhermore, if > 2cγe r 1, hen v as ˆθ. Proof. See Appendix C.4. When he expeced payoff is higher han e r, and > 2cγe r 1, hen Proposiion 2 says ha he value funcion becomes flaer as expeced payoff increases. Informaion on invesmens ha have high expeced payoffs offers less value for he DM. We illusrae his in Figure 3. As ˆθ > e r increases, we see ha each of he value of daa analyics curve becomes less hill-shaped and flaer compared o he previous. v.2 θ =1.15.15.1.5 θ =1.2 θ =1.25 θ =1.3 5 1 15 2 -.5 Figure 3: Value of daa analyics v wih respec o for increasing ˆθ. By Proposiion 2, he value of daa analyics becomes fla as ˆθ increases. By Theorem 1 he posiive value of daa analyics falls for ˆθ ha is large enough. Parameer values: r =.5, γ =.1, =.1, κ = 1, c = 1, and = 1. Theorem 1 Inerval value The value of daa analyics v behaves as follows. i If ˆθ < e r hen v rises in ˆθ for all >. ii If ˆθ > e r and c < c CH hen here exiss ˆθ such ha v falls in ˆθ for all ˆθ > ˆθ > e r for all > where v >. 1
Proof. See Appendix C.5. So when ˆθ < e r, he value of daa analyics always increases wih he expeced payoff, regardless of he values of oher parameers. Beween a low expeced payoff invesmen and a lower expeced payoff invesmen, informaion on he lower expeced payoff invesmen is no as valuable o he DM as compared o he oher. For ˆθ > e r, if he value of daa analyics is posiive, hen when he expeced payoff is large enough, he value of daa analyics always decreases wih ˆθ. This is illusraed also in Figure 3. Observe he poins on he curves corresponding o = 1; we see ha he value of daa analyics is posiive for all he curves, and as ˆθ increases, he value of daa analyics falls. As anoher example, look again a poins corresponding o = 18; again, he value of daa analyics is posiive for all he curves, bu for ˆθ = 1.15, he value of daa analyics rises as ˆθ increases o 1.2. However, for he curves corresponding o ˆθ = 1.2, 1.25, and 1.3, we see ha he value of analyics here falls afer ˆθ > 1.2. By Theorem 1, when he expeced payoff is high hen v falls wih respec o ˆθ. Hence, he value of daa analyics is low for invesmens wih high expeced payoff. Similarly, for invesmens wih low expeced payoff less han e r, v rises in ˆθ. Therefore, he value of daa analyics is also low for invesmens ha have low expeced payoff. The highes value of daa analyics is hence achieved afer ˆθ > e r, bu no for very high expeced payoff invesmens. We see an illusraion of his in Figure 4. 11
v.15.1.5.9 1. 1.1 1.2 1.3 1.4 θ -.5 Figure 4: Value of daa analyics v wih respec o ˆθ. Doed line represens ˆθ = e r. We see ha he peak value occurs no oo far away from e r, as in Theorem 1. Parameer values: r =.5, γ =.1, =.1, κ = 1, c = 1, and = 1. An example can be he screening for cancer. Cancer screening can be viewed as a form of informaion acquisiion. When i is done for young people ha are usually healhy, he odds are ha hey would no benefi from he informaion high expeced payoff, hus giving he informaion lile o no value. Similarly, if done for cancer paiens ha are already sick low expeced payoff, his informaion is no valuable a all. So in his sense, screening done for average people offers he mos valuable informaion. For more on cancer screening, see e.g. [3]. We nex plo he value and marginal value of daa analyics wih respec o he quaniy of daa analyics under differen parameer values. Since he value of daa analyics depends on ˆθ Proposiion 1, we consider wo cases: ˆθ e r and ˆθ < e r. Case ˆθ e r : Expeced reurn higher han he risk-free reurn Figure 5 plos he value and marginal value of daa analyics agains analyics quaniy for four differen increasing values of risk aversion, γ =.1,.2,.5, and.1. Consisen wih Corollary 5, Figure 5a shows ha he value of daa analyics falls in risk aversion. Figure 5b shows ha he marginal value falls wih risk aversion as well. 12
v.4.2 -.2 γ=.5 γ=.1 γ=.1 γ=.2 2 4 6 8 1 -.1 a Value of analyics v.2.1 -.2 -.3 v γ=.1 γ=.2 1 2 3 4 γ=.1 γ=.5 b Marginal value of daa analyics v Figure 5: Value and marginal value of daa analyics wih differen risk aversion γ. Parameer values: r =.5, =.1, ˆθ = 1.15, κ = 1, and c =.1. Figure 6 plos he value and marginal value of daa analyics agains analyics quaniy for four differen increasing values of uncerainy, =.5,.1,.2, and.4. The figure shows ha he marginal value of daa analyics falls in he uncerainy. v.48.46.44.42.4.38.36 =.1 =.2 =.5 =.4 -.2 1 2 3 4 5 -.3 a Value of daa analyics v v.4.3.2.1 -.1 =.5 =.1 =.2 1 2 3 4 5 =.4 b Marginal value of daa analyics v Figure 6: Value and marginal value of daa analyics wih differen iniial uncerainy level. Parameer values: r =.5, γ =.1, ˆθ = 1.15, κ = 1, and c =.1. Boh Figures 5a and 6a shows also ha when ˆθ e r, he value of daa analyics is sricly concave wih respec o, which we proved in Proposiion 2. Case ˆθ < e r : Expeced reurn lower han he risk-free reurn Figure 7 plos he value and marginal value of daa analyics agains analyics quaniy for four differen increasing values of risk aversion, γ =.1,.2,.5, and.1. Again, he value and marginal value of daa analyics falls in risk aversion see Corollary 5. 13
v v.1 γ=.1.4 γ=.1 γ=.2.5 -.5 γ=.2.2 γ=.5 1 2 3 4 5 γ=.1 -.2 γ=.5 1 2 3 4 5 γ=.1 a Value of daa analyics v b Marginal value of analyics v Figure 7: Value and marginal value of daa analyics wih differen risk aversion γ. Parameer values: r =.5, =.1, ˆθ = 1, κ = 1, and c =.1. Figure 8 plos he value and marginal value of daa analyics agains analyics quaniy for four differen increasing values of he uncerainy, =.5,.1,.2 and.4. Here, he value of daa analyics rises wih uncerainy and he relaionship beween he marginal value of analyics and he uncerainy is more complicaed..4 v =.4 v.35.3.25.2.15 =.2 =.1 =.5 1 2 3 4 5 a Value of daa analyics v.2 -.2 -.4 =.4 =.5 =.1 =.2 1 2 3 4 5 b Marginal Value of daa analyics v Figure 8: Value and marginal value of daa analyics wih differen iniial uncerainy level. Parameer values: r =.5, γ =.1, ˆθ = 1, κ = 1, and c =.1. 4 Opimal quaniy of daa analyics In his secion, we consider he opimal quaniy of daa analyics. We consider wo cases. Firs, in subsecion 4.1 we analyze he case where he DM can ake leverage, and hen in subsecion 4.2 we sudy he case where he DM canno ake leverage. 14
4.1 Opimal quaniy wih leverage Theorem 2 Opimal quaniy of daa analyics The value of daa analyics v has a global maximum a, where = arg min{ v is a global maximum}. Noe: We call he opimal quaniy of daa analyics. Proof. When ˆθ e r, v is sricly concave, and hence admis a unique global maximum. When ˆθ < e r, lim v =. Hence, M R such ha > M, v <. Se > M for some and consider he inerval [, ]. Then, by he Weiersrass Exreme Value Theorem, v achieves maximum and minimum in [, ] a leas once. Se as he minimum value of ha v achieves maximum a. By Proposiion 2, For ˆθ e r, he value of daa analyics is sricly concave. Hence, one candidae for he opimal quaniy is given by he firs order condiion v =, which, by Corollary 1, can be wrien as follows 2cγ e r 1 + 1 3/2 Φ d 2cγ e r 1 + 1e e r ˆθ 2 2 Φ d s + 1 + e e r ˆθ 2 2 Φ d =. 2 s + 1 Lemma 4 Choke-off cos The opimal quaniy of daa analyics = if c c CH or solves v = if c < c CH. Proof. If c c CH, hen v is always negaive for all > and v =. Hence he maximum is a = and =. If c > c CH, hen he maximum occurs no on he boundary, and so is given by he firs order condiion v =, and solves v = Corollary 3 says ha he DM does no buy small quaniies of daa analyics when 2cγe r 1. In fac, by Lemma 4, no only does he DM no buy small quaniies, she never buys any quaniy of daa analyics. Nex we derive he properies of he opimal quaniy of daa analyics from heir respecive derivaives. Since he value funcion is sricly concave for when e r < ˆθ, we are able o use he Implici Funcion Theorem on he firs order condiion o deermine some properies of he opimal quaniy. 15
Theorem 3 Properies of he opimal quaniy The opimal quaniy is independen of iniial wealh κ. For c < c CH and ˆθ > e r, we have: i Opimal quaniy falls in cos c. ii Opimal quaniy falls in risk aversion γ. Proof. See Appendix C.6. By Theorem 3, he opimal quaniy of analyics rises when he cos of analyics or he DM s risk aversion falls. The opimal quaniy of analyics is independen of he iniial wealh κ as he firs order condiion 2 is independen of κ. Hence, a change in iniial wealh does no change he opimal quaniy of analyics. We illusrae he opimal quaniy of daa analyics numerically in Figures 9 11. * 2 15 1 5 * 12 1 8 6 4 2 2 4 6 8 a ˆθ = 1.15 > e r. c.2.4.6.8 1. 1.2 1.4 b ˆθ = 1 < e r. c Figure 9: Opimal quaniy of daa analyics wih respec o c. Parameer values: r =.5, γ =.1, =.1, and κ = 1. Figure 9 shows ha he opimal quaniy of daa analyics falls in cos, and i gives he demand curve for daa analyics. Hence, he DM uses more analyics he less cosly i is. The jump disconinuiy in Figure 9b is due o he cos reaching he choke-off cos value. When ˆθ < e r, he value of daa analyics funcion is firs negaive, and herefore, align wih Corollary 3 he choke-off quaniy T CH is posiive and he DM never buys small quaniies of informaion. When ˆθ > e r, here is no disconinuiy as he value funcion is sricly concave and since v is coninuous, v ends coninuously o and here is a always a posiive opimal quaniy as long as c < c CH. 16
* * 2 12 1 15 8 1 6 4 5 2.2.4.6.8 1. γ.2 a θ = 1.15 > er..4.6.8.1.12 γ b θ = 1 < er. Figure 1: Opimal quaniy of daa analyics wih respec o γ. Parameer values: r =.5, c =.1, s =.1, and κ = 1. Figure 1 shows he opimal quaniy of daa analyics falls in risk aversion. This is because, by Lemma 2, a highly risk averse DM invess less in he risky opporuniy han a less risk averse DM, ceeris paribus, and herefore, he highly risk averse DM has less incenive o use daa analyics. The jump disconinuiy in Figure 1b is again due o he cos reaching he choke-off cos value. * 1 8 6 4 2 1 2 3 4 5 θ Figure 11: Opimal quaniy of daa analyics wih respec o θ. Parameer values: r =.5, c =.1, s =.1, κ = 1 and γ =.1. Figure 11 shows he opimal quaniy of daa analyics is non-decreasing in expeced payoff. This is a surprising resul and we conjecure ha i is driven by he amoun of leverage he DM akes. We performed a numerical analysis for 17
1 equally spaced values of ˆθ,, γ [e r, 5] [.1,.5] [.1,.5] and found he opimal quaniy o be non-decreasing. * 2 15 1 5 * 12 1 8 6 4 2.1.2.3.4.5 a ˆθ = 1.15 > e r..2.4.6.8.1 b ˆθ = 1 < e r. Figure 12: Opimal quaniy of daa analyics wih respec o. Parameer values: r =.5, γ =.1, c =.1, and κ = 1. 4.2 Opimal quaniy wihou leverage In his subsecion, we consider a financial consrain so ha he DM is no able o ake leverage. Hence, he DM is unable o inves more han his iniial wealh afer buying analyics κ c. We also assume ha he DM is unable o obain more han = κ quaniy of daa analyics. The opimal risky invesmen in his case is given c by he following lemma. Lemma 5 Opimal invesmen wihou leverage Given analyics quaniy [, κ ] and he CARA uiliy funcion wih γ, he c opimal level of risky invesmen wihou leverage is given by w NL = min { max { } } ˆθ e r,, κ c. γs Proof. By Lemma 2, he opimizaion problem wihou leverage: max E [ exp γwθ 1 + κ c w [,κ] wer 1 F ]. By he concaviy of he objecive funcion, if w of Lemma 2 is inside [, κ c] hen w NL = w, if w < hen w NL =, and if w > κ c hen w NL = κ c. As in Secion 3, we firs find he expeced uiliy of he opimal invesmen wihou leverage. 18
Lemma 6 Expeced uiliy wihou leverage The expeced uiliy of he opimal invesmen under given analyics quaniy : u NL = exp ˆθ e r 2 +2γκ c e r 1 2 e Φ r ˆθ s + 1 e γ 2 κ c 2 s s + 1 ˆθ e r +γκ c1 ˆθ 2 Φ e γκ c1 er Φ s + 1 e r ˆθ Proof. The expeced uiliy wihou leverage: + γκ c e Φ r ˆθ γκ c s + 1 u NL = E [ exp γw NLθ 1 + κ c w NLe r 1]. The proof is similar o ha of Lemma 3. The inegraion of he firs ieraed inegral is he same as in Lemma 3. For he second ieraed inegral, we consider he regions where w <, < w < κ c, and κ c < w, respecively. The inegraion limis becomes d + D +, where d := +1e r ˆθ d D and D := d+ κ cγ. s +1 Simplifying he inegrals direcly using he normal CDF gives he resul. The expeced uiliy of daa analyics wihou leverage is lower han he expeced uiliy of daa analyics wih leverage. This is illusraed by Figure 13. -.5 E[u] E[u] -.8 1 2 3 4 5 -.9-1. -1. -1.1-1.5-1.2-2. a ˆθ = 1.15 > e r. -1.3 1 2 3 4 5 b ˆθ = 1 < e r. Figure 13: Expeced uiliy of daa analyics wih and wihou leverage. The blue solid line is wih leverage and he orange doed line is wihou leverage. Parameer values: r =.5, γ =.1, =.1, c =.1, and κ = 1. 19
We sae his resul in Corollary 6 below. Corollary 6 Expeced uiliy The expeced uiliy wihou leverage is lower han he expeced uiliy wih leverage. Proof. See Appendix C.7. From he expeced uiliy wihou leverage, we nex calculae he value of daa analyics. Proposiion 3 Value of daa analyics wihou leverage If ˆθ e r hen he value of daa analyics v NL = u NL u NL is given by exp ˆθ e r 2 +2γκ c e r 1 2 e Φ r ˆθ s + 1 e γ 2 κ c 2 s s + 1 ˆθ e r +γκ c1 ˆθ 2 Φ + e γκ1 er 1 e γcer 1 Φ s + 1 e r ˆθ + γκ c e Φ r ˆθ γκ c s + 1. If e r < ˆθ κγ + e r hen he value of daa analyics is given by 2 ˆθ e r + 2γκs e r e 1 Φ r ˆθ exp 1 e γcer 1 2 + γκ c Φ s + 1 e γ 2 κ c 2 s s + 1 ˆθ e r +γκ c1 ˆθ 2 Φ γκ c s + 1 s + 1 e r ˆθ e γκ c1 ert Φ. e r ˆθ 2
If κγ + e r < ˆθ hen he value of daa analyics equals exp ˆθ e r 2 +2γκ c e r 1 2 e Φ r ˆθ s + 1 + γκ c e Φ r ˆθ + e γ 2 κ 2 s +γκ1 ˆθ 2 γ 2 κ c 2 s s + 1 ˆθ e r e +γκ c1 ˆθ 2 Φ e γκ c1 er Φ s + 1 e r ˆθ. γκ c s + 1 The key difference beween Proposiion 1 and 3 is he one addiional region in Proposiion 3. This is due o he financial consrain of Lemma 5 ha prevens leverage and requires us o consider he region where he DM s budge consrain is no igh e r < ˆθ κγ + e r. The marginal value of daa analyics is complicaed, bu we are sill able o make a comparison when ˆθ > e r + κγ and 2cγe r 1 >. When ˆθ > e r + κγ, his is he region where ˆθ e r γ > κ, so ha before acquiring daa analyics =, he DM s holding is more han her iniial wealh and her invesmen is bounded by her financial consrain. 2cγe r 1 > is he condiion required o have posiive opimal quaniy in he leverage case. We show his in Corollary 7 and Figure 14. Corollary 7 Marginal values wih and wihou leverage If ˆθ > e r + κγ and 2cγe r 1 > hen he marginal value of daa analyics wihou leverage is lower han he marginal value of daa analyics wih he leverage opion where i is posiive. Proof. See Appendix C.8. 21
v.5.4.3.2.1 2 4 6 8 1 Figure 14: The marginal value of daa analyics wih and wihou leverage. The blue solid line is wih leverage and he orange doed line is wihou leverage. Parameer values: r =.5, γ =.1. =.1, κ = 1, c =.1, and ˆθ = 1.15. We nex show ha he value of daa analyics is lower wihou he leverage opion. Corollary 8 Value of daa analyics wih and wihou leverage For ˆθ < e r + κγ, or ˆθ > e r + κγ and 2cγe r 1 >, he value of daa analyics wihou leverage is lower han he value of daa analyics wih he leverage opion. In he second case, he value wihou leverage is firs lower han wih leverage. Proof. When ˆθ e r + κγ, u = u NL, u > u NL, hen v = u u > u NL u NL = v NL. When ˆθ > e r + κγ, v = v NL =. Le V := v v NL. Then V = and V is coninuous and V = v v NL > for all ha saisfies 2cγe r 1 + 1 > by Corollary 7. s i.e. v > v NL for all < < 1. 2cγe r 1 The value of daa analyics funcion for a financially consrained DM is bounded above by he value of daa analyics funcion for a DM who is no financially consrained. Hence Theorem 2 applies oo for he value of daa analyics funcion wihou leverage. When ˆθ > e r + κγ he region where ˆθ e r γ > κ, since he marginal value of daa analyics is lower for a financially consrained DM, he opimal quaniy of daa analyics used by a financially consrained DM is also lower, ceeris paribus. This is because he opimal quaniy is deermined by he firs order condiion, which is firs posiive. Thus a lower marginal value implies ha he funcion is equal o earlier. We illusrae he opimal quaniy wihou leverage in Figures 15 and 16 o ge an overall picure of how he opimal quaniy changes wih respec o cos and risk aversion. 22
* 2 * 1 * 1 15 1 5 8 6 4 2 8 6 4 2.15.2.25.3.35.4 c a ˆθ = 1.15 > e r + κγ. 1 2 3 4 5 c b ˆθ = 1.6 < e r + κγ..15.2.25.3 c c ˆθ = 1 < e r. Figure 15: Opimal quaniy of daa analyics wihou leverage respec o c. Parameer values: r =.5, γ =.1, =.1, and κ = 1. wih Figure 15 shows ha he opimal quaniy of daa analyics falls in cos, and i gives he demand curve for daa analyics. Hence, he DM uses more analyics he less cosly i is. The jump disconinuiy in Figures 15a and 15c is due o he cos reaching he choke-off cos value. There is no jump disconinuiy in Figure 15b because in his region, he DM is no consrained. * 12 1 8 6 4 2 * 1 8 6 4 2.2.4.6.8 1. γ a ˆθ = 1.15 > e r..2.4.6.8.1.12 γ b ˆθ = 1 < e r. Figure 16: Opimal quaniy of daa analyics wihou leverage respec o γ. Parameer values: r =.5, c =.1, =.1, and κ = 1. wih Figure 16 shows he opimal quaniy of daa analyics wihou leverage is firs rising hen falling in risk aversion. The rising par is due o he leverage consrain. By Lemma 5, increasing risk aversion does no firs change he invesmen because he leverage consrain is igh, bu i raises he DM s incenive o decrease he uncerainy by daa analyics. When he risk aversion is high enough, he invesmen amoun sars o fall because he leverage consrain is no igh anymore see Lemma 5, and his decreases he incenive o use daa analyics. There are only wo regions ploed here as e r + κγ is always increasing wih γ. Hence for a fixed ˆθ ha is firs greaer han e r + κγ will be less han e r + κγ once γ is large 23
enough. This explains why here is no disconinuiy on he righ side of Figure 16a when γ is near 1. 5 Expeced opimal invesmen In his secion we solve for he DM s expeced invesmen amoun and a he same ime he expeced leverage. More specifically, his is he uncondiional expeced invesmen amoun and leverage before he DM observes he noisy signal on he invesmen payoff. Since we do no have an explici formula for he opimal quaniy of daa analyics, he opimal invesmen is implicily defined by. Noneheless, we can sill draw some conclusions and comparaive saics. We firs consider he opimal invesmen when he DM is allowed o ake leverage. Proposiion 4 Expeced opimal invesmen wih leverage If c c CH hen he expeced opimal invesmen is given by [ { }] ˆθ E [w e r ] = E max, γs = ˆθ e r 1 + Φ ˆθ e r 1 + s + + 2 φ er ˆθ 1 + s. γ γ If c > c CH hen he expeced opimal invesmen is given by if ˆθ E [w e r ] = ˆθ e r if ˆθ γ > e. r The expeced leverage is given by E [max {w + c κ, }] = ˆθ e r 1 + + c κ Φ b + + 2 φ b, γ γ where b = κ cγs 1+ + e r ˆθ s +1 s 2. Proof. The proof is in Appendix C.9. Nex we derive he expeced opimal invesmen when he DM is no allowed o ake leverage. 24
Proposiion 5 Expeced opimal invesmen wihou leverage If c c CH hen he expeced opimal invesmen is given by [ { { } }] ˆθ E [wnl e r ] =E min max,, κ c γs = ˆθ e r 1 + Φ d + γ + + 2 φ d φ d + γ + κ c 1 Φ d + κ cγ Φ d 1 + κ cγ 1 + κ cγ 1 + where d = er ˆθ 1 +. If c > c CH hen he expeced opimal invesmen is given by if ˆθ e r E [wnl] = ˆθ e r if e r < ˆθ γ < κγ + e r κ if κγ + e r < ˆθ Proof. The proof is in Appendix C.1. The expeced opimal holding is lower for a financially consrained DM, because her invesmen is bounded. We are no able o compare direcly he expeced opimal invesmens in Proposiions 4 and 5 because heir opimal quaniies of daa analyics are differen and we do no have explici formulas for. Therefore, we compare hem numerically in Figure 17, where we plo he expeced opimal invesmens wih and wihou he leverage opion agains cos of daa analyics. 25
E[w * * ] 2 15 1 5 E[w * * ] 1 8 6 4 2.2.4.6.8 1. 1.2 1.4 a ˆθ = 1.15 > e r. c.2.4.6.8 1. 1.2 1.4 b ˆθ = 1 < e r. c Figure 17: Expeced opimal invesmen wih respec o c. The blue solid line is wih leverage and he orange doed line is wihou leverage. Parameer values: r =.5, γ =.1, =.1, and κ = 1. As c ges larger, he expeced opimal invesmen falls. We see from Figure 17 ha he expeced opimal invesmen wih leverage falls in cos of daa analyics c. Therefore, lower coss of daa analyics increases he expeced holding of he DM. When c is low, he DM acquires more daa analyics which hen lowers he risk and raises he invesmen amoun as shown in Figure 17. This srongly implies ha if he DM s expeced holding is such ha she needs o ake leverage, lower coss of daa analyics should resul in her increasing her expeced holding and hus increasing he amoun of leverage she akes. From Proposiion 4 he derivaive of he expeced opimal invesmen wih leverage opion wih respec o he cos of analyics is given by E[max{w + c κ, }] c if c > c CH = ˆθ e r + cφ b + φ γ c b 1+2s + c Φ b oherwise 2γ + 2 where b = κ c γ 1+ + e r ˆθ s +1 s 2 Since here is a no shor selling consrain, a DM only akes leverage if he expeced payoff is more han e r, and her holding is more han her iniial wealh. Hence, we consider he case where ˆθ is much larger han e r for he numerical analysis of he leverage. There is a small probabiliy ha afer acquiring informaion, he new ˆθ is less han e r, bu his probabiliy is small. The DM also will be less likely o ake leverage if her iniial wealh is high. We performed a numerical analysis 26
wih 1 evenly spaced values of κ on [1, 1], 1 evenly spaced values of on [.1,.5], and 1 evenly spaced values of γ on [.1,.5]. Oher parameer values: r =.5, ˆθ = 1.15, r =.5, and c = 1. Noe ha he invesmen sunk cos is 1. We were unable o find any case wih E[max{w +c κ}] c >. Due o he increased leverage, he falling cos of daa analyics in Figure 1 may lead o higher losses during he crises. This is illusraed in Figure 18a. As can be seen, he losses are higher under he low cos of analyics. This is because under he low cos, he DM acquires more daa analyics and, herefore, her expeced leverage is higher. In Figure 18b we analyze furher he effec of analyics quaniy on he invesmen losses. We plo he expeced Profi and Loss given by E[w ]θ 1 + κ c E[w ]e r 1. The realized payoff θ is a cerain number of sandard deviaions below he expeced payoff. Noe ha under he high cos of daa analyics, he sandard deviaion is higher. As can be seen, he losses are again higher under he low cos of analyics. This is again driven by he higher leverage. Profi Profi.7.8.9 1. θ -2 1.5 2. 2.5 3. 3.5 4. n -5-4 -1-6 -8-15 -1-2 a Profi and loss wih respec o he realized payoff. The blue solid line is wih high cos c = 1 and he orange doed line is wih low cos c =.1. -12 b Profi and loss wih respec o he number of sandard deviaions below he expeced payoff. Realized payoff θ = ˆθ n s. The blue solid line is wih high cos c = 1 and he orange doed line is wih low cos c =.1. Figure 18: Profi and loss of an invesmen under differen analyics coss and quaniies of analyics. Oher parameer values: ˆθ = 1.6 =.1, r =.5, γ =.1, and κ = 1. Le us define wo differen meanings of shock. Definiion 1 Shocks A shock is given by eiher of he following. i The rue payoff θ is below e r. 27
ii The rue payoff θ is a number of sandard deviaions below he expeced payoff of he DM. Our main conclusion involves he following proposiion, which we prove for ˆθ > e r, since his is he case where leverage is aken. Theorem 4 Higher losses There is a level of shock such ha when eiher i θ falls below e r enough, or ii θ is enough sandard deviaions away from ˆθ, lower cos of daa analyics gives higher losses. Proof. See Appendix C.11 The preceding analysis considers he firs momen of he profis. We consider he second momen of he profis in he following analysis. Corollary 9 Second momen The second momen of he profi is given by E[P rofi 2 ] = θ e r 2 E[w 2 ] + 2e r 1κ c θ e r E[w ] + e r 1 2 κ c 2 where E[w ] is as given in Proposiion 4 and E[w 2 ] is s 2 1 + + 1 + ˆθ e r 2 s 2 Φ d + γ 2 s 2 and d := e r ˆθ s +1 s 2. ˆθ e r 1 + 3 2 γ 2 φd We can hen calculae he variance of he profi, given by θ e r 2 var w = θ e r 2 E[w 2 ] E[w ] 2. In Figure 19, we solve numerically for he opimal quaniy for values of cos of analyics c beween.1 o 1.5, and for γ =.1 and.5. Using he opimal, we simulaed 1 pahs of he observed signal and calculaed he opimal holding w for each of he sample pahs. The figures show he mean and variance of he profi under rue payoff θ = 1.1 and expeced payoff ˆθ = 1.6. The graphs show ha he when here is low cos of analyics, he expeced profi and sandard deviaion are high, while high cos of analyics resuls in low expeced profi and sandard deviaion. 28
E[Profi] 4 3 2 1 2 4 6 8 1 12 sdprofi Figure 19: Simulaed indifference curves for differen risk aversion γ. The blue solid line is wih high risk aversion γ =.5 and he orange doed line is wih low risk aversion γ =.1, wih c [.1, 1.5]. For a given γ, low high cos of analyics produces a high low expeced profi and high low sandard deviaion. Parameer values: r =.5, =.1, κ = 1, ˆθ = 1.6, and θ = 1.1, for 1 simulaions of he signal. 29
A Nonlinear cos We assume he same srucure as in he main discussion, and change only he cos of using daa analyics. Insead of paying c for using quaniy of daa analyics, he DM now pays c for using 2 quaniy of daa analyics. In his case, i is sraighforward o obain he expeced uiliy by changing c o c 2 in he nex period uiliy, and we have he following. Lemma 7 Expeced uiliy Given he quaniy of analyics, he expeced uiliy of he opimal invesmen before using he analyics is given by u =E [ exp γw θ 1 + κ c 2 w e r 1 ] Φ d = e γκ c2 1 e r s exp +1 Φd s + 1 ˆθ 2+e2r 2ˆθ e r +2γ κ c 2 e r 1 2 where Φ is cumulaive sandard normal disribuion and d := e r ˆθ s +1. s 2 The expecaion is wih respec o F, so uncondiional on he signal oucome. The opimal invesmen remains he same as in he main discussion. We can also differeniae o obain he marginal value of daa analyics. Corollary 1 Marginal value of daa analyics For >, he marginal value of daa analyics is given by v = eγer 1c2 κ [ 4cγ e r 1 s 2 + 1 3/2 + 1 3/2 Φ d 4cγ e r 1 + 1e e r ˆθ 2 2 Φ d s + 1 + e e r ˆθ 2 2 Φ d ] s + 1 We can check he concaviy of he value funcion. Similar o he case wih linear cos, he value funcion is sricly concave for ˆθ > e r. Proposiion 6 Shape of he value of daa analyics The value of daa analyics funcion is sricly concave if ˆθ > e r. 3
Proof. The second derivaive of he value funcion v = eγer 1c2 κ 16c 2 γ e r 1 s 8 2 + 1 7/2 + 1 7/2 2cγ e r 1 2 + 1 Φ e r ˆθ + 1 s 2 2 2 + 1e e r ˆθ 2 2 8cγ e r 1 + 1 2c 2 γ e r 1 + 1 + 1 ˆθ e r + 3s 2 Φ + 2 π e r ˆθ + 1 2 e er ˆθ 2 +1 2s 2 is < if ˆθ > e r. So when ˆθ > e r, we have a unique global maximum given by he firs order condiion v =, which can be wrien as follows 4cγ e r 1 + 1 3/2 Φ d 4cγ e r 1 + 1e e r ˆθ 2 2 Φ d s + 1 + e e r ˆθ 2 2 Φ d =. 3 s + 1 Again, he opimal quaniy is independen of wealh since we did no change he uiliy funcion. We also show ha he opimal quaniy increases wih falling cos of daa analyics. Theorem 5 Properies of he opimal quaniy The opimal quaniy is independen of iniial wealh κ. For ˆθ > e r, he opimal quaniy falls in cos c. Proof. We can see from 3 ha he firs order condiion ha i is independen of κ, hus he opimal quaniy is also independen of kappa. Le he lef hand side of 3 be f. We have: f = e r ˆθ φ e r ˆθ s +1 s 2 2 3/2 4cγ e r 1 2 + 1e e r ˆθ 2 2 Φ 2cγ e r 1 5 + 2 + 1Φ e r ˆθ + 1 s 2 f c = 4γ er 1 + 1 e e r ˆθ 2 ˆθ e r 2 Φ + + 1Φ ˆθ e r e r ˆθ + 1 s 2 31
and by he Implici Funcion Theorem, we ge c ˆθ > e r. = f c f which is < when Since nohing changes for he opimal holding formula, he form of he expeced opimal holding funcion remains he same and ogeher wih Theorem 5, Theorem 4 holds for non-linear cos. B Muliple Invesmens We exend he resuls o n independen invesmen opporuniies. We assume ha signals on differen invesmens are independen of each oher. The parameers of he DM remains he same. For each of he i-h invesmen, we have he unknown payoff θ i, expeced payoff ˆθ,i and variance,i. The cos of obaining and using quaniy i of daa analyics for he i-h invesmen is c i. Hence, he uiliy of he DM in he nex period is exp γ [ n i=1 w iθ i 1 + κ n i=1 c i i n i=1 w ie r 1] = exp γ[w θ 1 + κ c w 1e r 1]. Lemma 8 and Lemma 9 are derived direcly from heir counerpars in he single invesmen case and heir proofs are exacly he same, excep he variables are indexed by he i-h invesmen. Lemma 8 Uncondiional and condiional payoffs for he i-h invesmen For he i-h invesmen, uncondiionally on he signal oucome, we have θ i = ˆθ s 2,i,i + i ɛ 2,i + s,i ɛ 1,i 1 +,i i and condiional on he signal oucome, we have θ i = ˆθ,i + s,i ɛ 1,, where s,i =,i 1+ i,i, ɛ 1,i, ɛ 2,i N, 1, and ɛ 1,i ɛ 2,i. Proof. See Lemma 1. Lemma 9 Opimal invesmen for muliple invesmens Given analyics quaniy = 1, 2,..., n and CARA uiliy funcion wih γ >, he opimal invesmen level of risky invesmen is given by w = w1 1, w2 2,..., wn n, where { } ˆθi wi,i e r i = max,. 4 32 γs i,i
Proof. See Lemma 2. Lemma 1 Expeced uiliy The expeced uiliy of he opimal invesmen for muliple invesmens u =E [ exp γ[w θ 1 + κ c w 1e r 1]] d = e γκ n n Φ i i exp ˆθ,i e r 2 i=1 c i i 1 e r Φd s,i i +1 i i + s,i i + 1 i=1 where Φ is cumulaive sandard normal disribuion and d i i := 2,i e r ˆθ,i s,i i +1 s 2,i i Proof. Similar o he single invesmen case, we ake find he expeced uiliy by inegraing he uiliy in he nex period over R 2n wih respec o he 2n normal PDFs and replacing he ɛ 1,i s and ɛ 2,i s by dummy variables x i and x i+1 for he i-h invesmen: exp γ[w θ 1 + κ c w 1e r 1 1] 2π n e R 2n 1 2n x 2 i i=1 We hen inegrae wih respec o he dummy variables associaed wih ɛ 1,i o ge R n e γκ n i=1 c i i 1 e r exp n i=1 γw i e r + γ2,i w i 2 2,i i +2 γw i x i 2π s 2,i i 2n i=1,i i +1 γ ˆθ,i w i x2 i 2 Observing ha each i-h erm in he produc is similar o ha in he single invesmen case and from he independence of he normal disribuions, we ge: e γκ n i=1 c i i 1 e r n Φ Φd i i + i=1 d i i s,i i +1 exp ˆθ,i e r 2 s,i i + 1 2,i dx i n i=1 dx i Proposiion 7 Value of daa analyics Le i j be he index for invesmens such ha ˆθ,ij > e r and assume here are 33
m n such invesmens. Then he value of daa analyics v := u u is i m e γκ1 er exp ˆθ,ij e r 2 2s i j =i,ij 1 e γκ n n Φ i=1 c i i 1 e r Φd i i + i=1 Noe: if m =, hen i m i j =i 1 exp ˆθ,ij e r 2 2,ij = 1. d i i s,i i +1 Proof. Observe ha lim d i i i = lim e r ˆθ s,i i +1,i i s 2,i i exp ˆθ,i e r 2 s,i i + 1 2,i = + if e r > ˆθ,i, if e r ˆθ,i e r < ˆθ,i or if e r = ˆθ d,i and lim i i = lim 1 = if e r > ˆθ i s,i i +1 i s 2,i,,i i + if e r < ˆθ,i or if e r = ˆθ,i. Hence, for he n m invesmens ha have ˆθ,i < e r, heir corresponding erms simplifies o 1 in he produc, and he erms involving he m invesmens ha have ˆθ,ij > e r reduce o exp ˆθ,ij e r 2 2,ij. This gives us he resul. Corollary 11 Marginal value of daa analyics The marginal value of daa analyics for he j-h invesmen is given by [ v = eγer 1cjj κ 2c j 2,j j + 1 3/2 j γ e r 1,j j + 1 3/2 Φ d j 2c j γ e r 1,j j + 1e e r ˆθ,j 2 2,j Φ d j s,j j + 1 e γ n c i i e r 1 i=1,1 j n i=1,i j Φd i i + Φ d i i s,i i +1 +,j e e exp ˆθ,i e r 2 s,i i + 1 2,i r ˆθ,j 2 2,j Φ d j s,j j + 1 ] Proof. Since we ake derivaive wih respec o only he j-h invesmen, we can rea he oher variables as consans. Thus he resul is obained direcly from he single invesmen case in Corollary 1, bu muliplied by he oher variables associaed wih he res of he invesmens. Our resuls in he single invesmen case hold in he case of muliple invesmens for each invesmen because he marginal value of daa analyics for each invesmen 34
is jus a posiive scalar muliple of he marginal value of daa analyics in he single invesmen case. Hence, following he same argumen as he case for a single invesmen, he opimal quaniy of daa analyics for he j-h invesmen is given by he firs order condiion v j = c j. To solve for requires solving he sysem of non-linear equaions where v 1 = v 2 =. v n = f 1 1 = f 2 2 =. f n n = f i i = 2c i γ e r 1,i i + 1 3/2 Φ d i 2c i γ e r 1,i i + 1e e r ˆθ,i 2 2,i +,i e e r ˆθ,i 2 2,i Φ d i s,i i + 1 Φ d i s,i i + 1 Noe ha due o he marginal value for each invesmen being a muliple of he marginal value in he single invesmen case, each equaion for he i-h invesmen is independen of he oher n 1 invesmens, and f i i is jus he indexed version of 2 in he single invesmen case. Hence, he resuls abou he properies of he opimal quaniy of daa analyics, Theorem 3, all hold for each of he i-h invesmen, and are idenical. Fuhermore, since he opimal invesmen for each of he i-h invesmen given by Lemma 9 has he same formula as he single invesmen case in Lemma 2, he corresponding expeced opimal invesmen wih leverage is given by Proposiion 4 wih he appropriae indices, and he oal expeced opimal invesmen E[w ] is given by he sum n E[wi i ]. By he discussion in he single invesmen case, i=1 he falling coss of daa analyics lead o increase in leverage, and higher losses during he crises. 35