THE RETURN ON INVESTMENT FROM PROPORTIONAL PORTFOLIO STRATEGIES

Transcription

1 THE RETURN ON INVESTMENT FROM PROPORTIONAL PORTFOLIO STRATEGIES Si Browne Columbia Universiy Final Version: November 11, 1996 Appeare in: Avances in Applie Probabiliy, 30, , 1998 Absrac Dynamic asse allocaion sraegies ha are coninuously rebalance so as o always keep a fixe consan proporion of wealh invese in he various asses a each poin in ime play a funamenal role in he heory of opimal porfolio sraegies. In his paper we suy he rae of reurn on invesmen, efine here as he ne gain in wealh ivie by he cumulaive invesmen, for such invesmen sraegies in coninuous ime. Among oher resuls, we prove ha he limiing isribuion of his measure of reurn is a gamma isribuion. This limi heorem allows for comparisons of ifferen sraegies. For example, he mean reurn on invesmen is maximize by he same sraegy ha maximizes logarihmic uiliy, which is also known o maximize he exponenial rae a which wealh grows. The reurn from his policy urns ou o have oher sochasic ominance properies as well. We also suy he reurn on he risky invesmen alone, efine here as he presen value of he gain from invesmen ivie by he presen value of he cumulaive invesmen in he risky asse neee o achieve he gain. We show ha for he log-opimal, or opimal growh policy, his reurn ens o an exponenial isribuion. We compare he reurn from he opimal growh policy wih he reurn from a policy ha invess a consan amoun in he risky sock. We show ha for he case of a single risky invesmen, he consan invesor s expece reurn is wice ha of he opimal growh policy. This ifference can be consiere he cos for insuring ha he proporional invesor oes no go bankrup. Key wors: Porfolio Theory; Diffusions; Saionary Disribuions; Convergence in Disribuion; Limi Theorems; Sochasic Orer Relaions; Logarihmic Uiliy; Opimal Growh Policy. AMS 1991 Subjec Classificaion: Primary: 90A09, 60F99. Seconary: 90A10, 60J60, 60H10. Posal aress: 402 Uris Hall, Grauae School of Business, Columbia Universiy, New York, NY USA. sb30@columbia.eu. Acknowlegmen: This paper has benefie grealy from he very careful reaing, correcions an helpful suggesions of an anonymous referee, o whom he auhor is mos graeful.

2 1 Inroucion Consan proporions invesmen sraegies play a funamenal role in porfolio heory. Uner hese policies, an invesor follows a ynamic raing sraegy ha coninuously rebalances he porfolio so as o always allocae fixe consan proporions of he invesor s wealh across he invesmen opporuniies. These sraegies are quie wiely use in pracice an are also someimes referre o as consan mix, or coninuously rebalance, sraegies (see e.g., Perol an Sharpe [25]). Furhermore, for cerain objecives an uner some specific assumpions abou he sochasic behavior of he invesmen opporuniies, i is well known ha hese policies have many opimaliy properies associae wih hem as well. These properies are reviewe in he nex secion. Given he funamenal naure of such policies in heoreical as well as acual porfolio pracice, i is of ineres o know wha he sochasic behavior of he rae of reurn on invesmen (RROI) efine here as he ne gain ivie by he cumulaive invesmen is for such policies. In his paper we suy his imension of he porfolio problem. We ake as our seing he coninuous ime financial moel inrouce in Meron [22] an use in Black an Scholes [3]. For his moel we obain some limi heorems for he RROI which allow us o compare an erive some specific opimaliy properies for cerain porfolio sraegies. A summary of our main resuls an he organizaion of he paper is as follows: in he nex secion, we review he coninuous-ime moel an some well known opimaliy properies associae wih consan proporion invesmen policies. In Secion 3 we provie our main resul (Theorem 3.1): ha he reurn on invesmen for such policies converges o a limiing isribuion which is a gamma isribuion. This resul provies a basis upon which o compare ifferen sraegies an o explore an ienify various opimaliy crieria. For example, wih his isribuional limi heorem in han we show in Secion 3 ha he policy ha maximizes logarihmic uiliy of wealh generaes a RROI ha has some sochasic ominance properies over oher policies. The logarihmic uiliy funcion has cerain oher objecive opimaliy properies ha are reviewe in Secion 2. In Secion 4 we prove Theorem 3.1. We show in paricular ha he limiing behavior of he RROI is in fac eermine by he limiing behavior of a relae iffusion process, which is compleely analyze. In Secion 5 we move on o consier he excess reurn from invesmen above he risk-free rae. We call his he rae of reurn on risky invesmen (RRORI). We show ha his measure converges o a ifferen gamma isribuion, an in paricular, for he case of logarihmic uiliy, o an exponenial isribuion. However, for he RRORI measure, he logarihmic uiliy funcion has only limie sochasic ominance properies over oher policies, an we show ha he mean RRORI is in fac maximize by a ifferen class of sraegies, namely, by sraegies ha inves only 1

3 a consan amoun (as oppose o a consan proporion) in he risky asses. For such sraegies he RRORI follows a Gaussian process ha is inepenen of he consan amoun invese in he risky asses. Furhermore, i urns ou ha ha he mean RRORI for such consan amoun invesmen sraegies is wice he mean RRORI for a logarihmic uiliy funcion. (These las wo saemens are specific o he case wih a single risky sock, an o no hol for he more general case reae in Secion 6.) Since bankrupcy is possible uner such sraegies, his halve reurn can be consiere he price he consan proporional invesor mus pay for he insurance of never going bankrup, since in coninuous ime bankrupcy is impossible uner a proporional invesmen sraegy (in he absence of any wihrawals an oher consrains). The precise isribuional resuls obaine in Secion 5 allow us o compue explicily various comparaive probabiliies. Finally, in Secion 6 we exen all our resuls o he muliple risky sock case. Our suy was moivae by he simulaing paper of Ehier an Tavare [9] who suie he reurn on invesmen in a iscree-ime gambling moel, where he reurn on he iniviual gambles is assume o follow a ranom walk. They showe ha he asympoic isribuion of he reurn, as he mean incremen in he ranom walk goes o zero, is a gamma isribuion. Since here is only one invesmen opporuniy in he moel of Ehier an Tavare [9], heir resuls have counerpars in our reamen of he RRORI, bu hey i no obain he iscree-ime analogs of our more general resuls for he RROI, an hence of he opimaliy of he logarihmic uiliy policy, iscusse in Secions 3 an 4. 2 Opimal Properies of Proporional Invesmen We recall here some basic facs abou cerain opimal properies associae wih invesmen policies ha inves a fixe proporion (possibly greaer han 1) of curren wealh in he risky asse. These policies are commonly referre o as consan proporions or consan mix sraegies. While for exposiional purposes we concenrae on he case of a single sock in Secions 3-5, we inrouce here he moel wih an arbirary number of risky socks since we will reurn o he muliple sock case laer in Secion 6. The moel we rea is ha of a complee marke wih consan coefficiens, wih k (correlae) risky socks generae by k inepenen Brownian moions. The prices of hese socks will be enoe by {S (i) : i = 1,..., k}, where i is assume ha he prices evolve accoring o S (i) = µ i S (i) + k j=1 σ ij S (i) W (j), for i = 1,..., k (2.1) where {µ i : i = 1,..., k} an {σ ij : i, j = 1,..., k} are consans, an W (j) enoes a sanar 2

4 inepenen Brownian moion, for j = 1,..., k. There is also a risk-free securiy, a bon, available for invesmen. The price of he bon, {B, 0}, evolves accoring o B = rb, (2.2) where r > 0 is a consan. Thus he sock prices follow a muli-imensional geomeric Brownian moion, as inrouce in Meron [22]. An invesmen policy is a (column) vecor conrol process {π : 0} wih iniviual componens π i (), i = 1,..., k, where π i () is he proporion of he invesor s wealh invese in he risky asse i a ime, for i = 1,..., k, wih he remainer invese in he risk-free bon. Thus, uner he policy π, he invesor s wealh process, X π ( k X π = X π π i () S(i) i=1 S (i) = X π ( r + π (µ r1) upon subsiuion from (2.1) an (2.2), where π evolves accoring o ) k + X π (1 π i ()) B B i=1 ) k k + X π i=1 j=1 π i ()σ ij W (j), (2.3) = (π 1 (),..., π k ()), µ = (µ 1,..., µ k ), an 1 = (1,..., 1) (ransposiion is enoe by he superscrip ). I is of course assume ha π is an amissible conrol process, i.e., i is a nonanicipaive aape vecor saisfying T 0 π π <, for all T <. We place no furher resricions on π. For example we allow π i () < 0, in which case he invesor is selling he ih sock shor, as well as π i () > 1, for any an all i = 1,..., k. Le σ enoe he marix σ = (σ) ij, an le Σ = σσ. We will assume for he remainer ha he marix σ is of full rank, an hence σ 1 as well as Σ 1 exis. For he sequel, le µ := µ r1 (2.4) enoe he vecor of he excess reurns of he risky socks over he risk-free reurn. 2.1 Opimaliy of consan proporions Of paricular ineres o us is he case where π is a consan vecor for all 0. Such a policy is calle a consan proporions policy an is in fac he opimal invesmen policy for many ineresing objecive funcions. For example, i is well known ha if he invesor s objecive is o choose an amissible invesmen sraegy o maximize expece uiliy of erminal wealh for a uiliy funcion ha is eiher a power funcion, or logarihmic, hen he opimal policy is a consan vecor. Furhermore, a consan vecor is also he opimal policy for oher objecive crieria, such as minimizing he expece ime o reach a given level of wealh, as well. We summarize hese main properies in he following: 3

5 Proposiion 2.1 For an invesor whose wealh evolves accoring o (2.3), consier he following four problems: Problem 1. Given a uiliy funcion u(x) of he form δ δ 1 x1 1 δ for x > 0, δ > 0, δ 1 u(x) = ln(x) for x > 0, δ = 1 (2.5) choose an amissible invesmen policy o maximize expece uiliy of erminal wealh a a fixe erminal ime T. Le π (δ) = {π (δ), 0 T } enoe his opimal policy, i.e., π (δ) = arg supπ E x [u(xt π )], where he funcion u( ) is given by (2.5) an he ynamics of {X π, 0} is given in (2.3). Problem 2. For a given iscoun rae λ > 0, choose an amissible invesmen policy o maximize he expece iscoune rewar of reaching a given arge level, say b, of wealh. Specifically, for X 0 = x < b, le τ π b := inf{ 0 : X π b}, (2.6) enoe he firs passage ime o he wealh level b, an le π (b) (λ) = {π (b) (λ), 0} enoe ] he opimal policy, i.e., π (b) (λ) = arg supπ E x [e λτ b π. Problem 3. Choose an amissible invesmen policy o minimize he expece ime o reach a given level, say b, of wealh. Specifically, for X 0 = x < b an τb π as in (2.6), le π(b) = {π (b), 0} enoe he opimal policy, i.e., π (b) = arg infπ E x [τb π]. Problem 4. Choose an amissible invesmen policy o maximize he acual (asympoic) rae a which wealh compouns, efine as lim inf (1/) ln [X π ], an le π = {π, 0} enoe he opimal policy, i.e., π = arg supπ {lim inf (1/) ln [X π ]}. Le f enoe he consan vecor given by f := Σ 1 (µ r1) Σ 1 µ. (2.7) Then he opimal policies for he four problems are given respecively by he consan vecors π (δ) = δf, for all 0, an all δ > 0 (2.8) π (b) 1 (λ) = f, for all 0, an all λ 0, an any b > x, (2.9) 1 η π (b) = f, for all 0, an any b > x, (2.10) π = f, for all 0, (2.11) 4

6 where in (2.9), η = η(λ) saisfies 0 < η < 1 an is he (smaller) roo o he quaraic equaion η 2 r η(γ + λ + r) + λ = 0, where he parameer γ := (1/2) µ Σ 1 µ. Problem 1 an is opimal policy (2.8) were firs consiere in Meron [22]. The ienical policy is in effec if he invesor maximizes uiliy obaine from consumpion as well (see Meron [22, 23] for more eails). Problem 2 an is opimal policy (2.9) follows from a somewha more general resul of Browne [6, Theorem 4.3] (see also Orey e al.[24]). Problem 3 an is opimal policy (2.10) was firs consiere in Heah e al.[15] for he case k = 1 (see also Meron[23, Secion 6.5]). The mulivariae case follows irecly from he more general resul in Browne [6]. I is ineresing o noe ha he opimal policies for Problems 2 an 3 are inepenen of he arge level of wealh b. Problem 4 was consiere in Karazas [17, Secion 9.6]. 2.2 Logarihmic uiliy an opimal growh For logarihmic uiliy, he opimal policy for Problem 1 is π (1) f. A comparison wih he resuls for Problems 3 an 4 hen shows an ineresing corresponence beween he logarihmic uiliy funcion an objecive crieria relae o growh; specifically, π (1) = π (b) = π f. For his reason he log-opimal policy, f of (2.7), is someimes referre o as he opimal growh policy (e.g., Meron [23, Chaper 6]). (While he equivalence beween Problem 1 wih logarihmic uiliy an he asympoic compouning rae of Problem 4 hols for more general price processes han ha given by (2.1), i appears ha he exac equivalence of his policy o he policy ha minimizes he expece ime o a arge level of wealh for Problem 3 hols only for he consan coefficiens moel reae here.) These opimaliy resuls an he connecion beween logarihmic uiliy an he objecive crieria of growh have heir preceens in earlier iscree-ime resuls. In paricular, Kelly [20] was he firs o observe he relaionship beween maximizing he logarihm of wealh an he expece asympoic rae a which wealh compouns. (In ligh of his, maximizing he logarihm of wealh is someimes referre o as he Kelly crierion.) This was amplifie o he porfolio seing in he papers of Hakansson [14], Thorp [30] an Markowiz [21] an ohers. Breiman [4] grealy expane on he resuls on [20] an among oher funamenal resuls esablishe ha he policy ha maximizes he logarihm of wealh is asympoically (as b ) opimal for he objecive of minimizing he expece ime o b, however i is only in coninuous-ime ha he relaionship is exac. Thorp [29], Hakansson [14], Finkelsein an Whiely [10] an Ehier an Tavare [9] among ohers conain peneraing analyses of opimaliy properies of consan proporional invesmen policies in iscree-ime. A Bayesian version of he opimal growh policy in boh iscree an coninuous-ime is suie in Browne an Whi [7]. 5

7 A comparison of (2.9) wih (2.5) an (2.8) shows ha jus as he logarihmic uiliy funcion is relae o opimal growh, a power uiliy funcion wih δ 1 is also relae o cerain growh objecives relae o ha of Problem 2 (in paricular for δ = (1 η) 1 in (2.5)). (Relaionships beween objecive crieria oher han growh an paricular uiliy funcions for some oher porfolio problems are iscusse in Browne [5].) Remark 2.1: Risk aversion. The Arrow-Pra measure of relaive risk aversion for a uiliy funcion u( ), enoe here by Λ( ), is efine as Λ(x) := xu (2) (x)/u (1) (x), where u (i) (x) := i u(x) x i, see [26]. Thus for he power uiliy funcion in (2.5) we have Λ(x) = δ 1 for all x > 0, corresponing o consan relaive risk aversion. Since he logarihmic case correspons o a relaive risk aversion Λ(x) = 1, i is seen from (2.8) ha for he case where f 1 > 0, an invesor who is more risk averse han a logarihmic invesor (i.e., δ < 1) will in fac unerinves in he risky asses relaive o he logarihmic invesor while an invesor who is less risk averse (δ > 1) will over-inves in he risky asses relaive o he logarihmic invesor. Since consan proporion invesmen policies play such a funamenal role in porfolio heory, an is he focus of suy here, we summarize is main properies nex. For moels where he opimal policy is a consan proporion of he surplus of wealh over some level insea of wealh iself see Golieb [11], Sunaresan[27], Black an Perol[2], Grossman an Zhou[13], Dybvig[8] an Browne[6]. 2.3 Wealh uner Proporional Invesmen Le f = (f 1,..., f k ) enoe a fixe consan vecor, where f i is he percenage of he invesor s wealh invese in risky securiy i, for i = 1,..., k. The invesor s wealh uner his policy, enoe by X f, hen evolves as Xf = X f r + f µ + X f ki=1 kj=1 f i σ ij W (j). Equivalenly, X f is he geomeric Brownian moion ( X f = X 0 exp r + f µ 1 ) 2 f Σf + As such, i is hen well known (e.g., [18, Pg. 349]) ha k k i=1 j=1 f i σ ij W (j). (2.12) inf 0 < Xf > 0, an lim X f =, a.s. (2.13) so long as r + f µ 1 2 f Σf > 0. (2.14) For he sequel, we will assume ha (2.14) hols. 6

8 Recognize of course { ha he resuling wealh process (2.12) is isribuionally equivalen o } he process X 0 exp r + f µ 1 2 f Σf + f Σf W, where {W, 0} is a one-imensional orinary Brownian moion. In his paper, our focus is on he rae of reurn on invesmen, which is efine here o be he ne gain in wealh ivie by he cumulaive invesmen. Since boh of hese are one-imensional processes, for exposiional purposes we will firs rea he case k = 1, where here are only wo invesmen opporuniies: a single risky sock whose price process follows S = µs + σs W, an he bon of (2.2), an hen in a laer secion (Secion 6) simply ouline how o exen he resuls obaine for single risky sock o he case of muliple socks, an highligh he ifferences beween hem. For he single sock case, le f enoe a fixe consan, which is he percenage of he invesor s wealh invese in he risky sock. This invesor s wealh, enoe by X f hen evolves as X f = X f (r + f µ) + fσ Xf W, where µ := µ r is he excess reurn of he risky sock over he reurn from he risk-free bon. Since f is consan, he wealh process is he geomeric Brownian moion X f = X 0 exp {(r + f µ 1 2 f 2 σ 2) } + fσw. The coniion (2.14) becomes equivalenly, r + f µ 1 2 f 2 σ 2 > 0, (2.15) µ σ 2 1 µ 2 σ σ 2 + 2r < f < µ σ µ 2 + 2r. (2.16) σ σ2 For he sequel we will assume ha he consan f is such ha coniion (2.15) is me an hence ha (2.13) hols, i.e., inf 0 < X f > 0, an lim X f =, a.s. Noe ha we allow boh f > 1 as well as f < 0. In he former hols, he invesor is borrowing money a he risk-free rae o inves long in he sock, while if he laer hols, hen he invesor is selling he sock shor an puing he procees ino he risk-free asse. Since logarihmic uiliy an he opimal growh policy play a prominen role in he sequel, for he remainer of his paper we will enoe i by f, i.e., for he sequel we will ake f = (µ r)/σ 2. In erms of µ, his gives f := µ/σ 2. In he nex secion we urn our aenion o he rae of reurn on invesmen. 3 The Rae of Reurn from Toal Invesmen Our ineres here is he rae of reurn from oal invesmen (RROI), which for he fixe policy f will be enoe by he process {ρ f (), 0}. This is efine here as he raio of he ne gain o he cumulaive invesmen (see Ehier an Tavare [9]), i.e., since X f s is he invesor s wealh a ime s, an his wealh is compleely invese in he risky sock an he riskless bon, he cumulaive 7

9 invesmen unil ime is hen 0 Xf s s, an he RROI is hen efine by ρ f () := Xf X 0 0 Xf s s, for > 0. (3.1) Thus ρ f () is a measure of he amoun of wealh i akes o finance a gain. If ρ f () is large, hen he invesor is accumulaing gains a a faser rae han if i is small (see Remark 3.1 below for a iscussion of oher measures of rae of reurn). Noe ha if we ivie he numeraor an enominaor of he raio in (3.1) by, we may also inerpre he measure ρ f () as he average ne gain over he average wealh level. Subsiuion of (2.3) ino (3.1) shows ha for a fixe f, ρ f () = exp {( r + f µ 1 2 f 2 σ 2) + fσw } 1 0 exp {(r + f µ 1 2 f 2 σ 2 ) s + fσw s } s. (3.2) Placing f = 0 in (3.2) shows ha for ha policy, uner which oal wealh is always invese in he risk-free asse, we o have ρ 0 () := er 1 0 ers s = r he risk-free ineres rae, as expece. However, for f 0, he RROI process {ρ f (), > 0} is quie complicae an oes no yiel o a simple irec analysis. Neverheless, we will show ha for any fixe proporion f such ha (2.15) hols, he process {ρ f (), > 0} amis a unique limiing isribuion, which is a gamma isribuion. We make his more precise in he nex heorem. Remark on noaion: For reference, we noe ha hroughou he remainer of he paper, when we say ha a ranom variable X gamma(α, β), we mean ha X is a ranom variable wih ensiy funcion ψ(x) = e βx x α 1 β α /Γ(α), an so E(X) = α/β, an V ar(x) = α/β 2. Our main resul can now be sae as he following: Theorem 3.1 For any fixe proporional sraegy f which saisfies coniion (2.15), he RROI process {ρ f (), > 0} converges (as ) in isribuion o a ranom variable which has a gamma isribuion. Specifically, as, where ρ f () ρ f gamma enoes convergence in isribuion. As such ( 2 (r + f µ) σ 2 f 2 1, ) 2 σ 2 f 2, (3.3) E[ρ f ] = r + f µ 1 2 f 2 σ 2 > 0. (3.4) 8

10 The proof of his heorem is given in he nex secion. However, since he precise isribuional naure of he resul allows us a basis upon which o compare ifferen invesmen sraegies, we will invesigae some of is ramificaions irecly. Remark 3.1: Our efiniion of ρ f () as he rae of reurn on invesmen, while base irecly on measures suie in Ehier an Tavare [9] (see also Griffin [12]), is no quie he rae of reurn ha is ypically use in corporae finance. In coninuous-ime he rae of reurn unil ime from he porfolio sraegy f is more usually efine in he financial lieraure as he value of r f () such ha X 0 e r f () = X f. (Recall ha in iscree ime, if an invesmen yiels a reurn r i in perio i, hen an iniial invesmen of X 0 grows (if all gains are reinvese) afer n invesmen perios o X n = X 0 ni=1 (1 + r i ). The rae of reurn is hen ypically efine as ha value of r n such ha [(1 + r n) n 1] = [X n X 0 ] /X 0, from which i follows ha r n = [ n i=1 (1 + r i )] 1/n 1. I shoul be clear ha he analog of his for our coninuous-ime moel wih coninuous reinvesmen is inee r f (). The rae of reurn measure suie by Ehier an Tavare [9] is efine in iscree-ime by R n := [X n X 0 ] / n 1 i=0 X i, he coninuous-ime analog of which is clearly our ρ f ().) Using (2.12), i is seen ha r f () = r + f µ 1 2 f 2 σ fσw, an so for any > 0, E[r f ()] = r + f µ f 2 σ 2 /2, which by (3.4) is equivalen o E[ρ f ]. However, he measure r f () has a egenerae sochasic limi, since by he law of large numbers r f () a.s. r + f µ f 2 σ 2 /2 E[ρ f ], as. Thus he more usual measure r f () oes no convey any informaion abou he variaion of he asympoic reurn aroun he mean. By Theorem 3.1 above, we see ha such (more refine) informaion is inee provie by he measure ρ f. Remark 3.2: The expecaion in (3.4) shoul no be confuse wih he raio of he expece gain o he expece oal invesmen, which for any > 0, is equal o [ ] E[oal gain] E[oal invesmen] E X f X 0 [ E = r + f µ. (3.5) 0 Xf s s] (This can be esablishe by recognizing ha EX f heorem o compue he expecaion in he enominaor.) = X 0 e (r+f µ), an hen by using Fubini s The isincion beween measures relae o (3.4) an (3.5) for simple iscree-ime gambling moels is iscusse in Griffin [12]. 9

11 3.1 Opimal Growh Policy an Sochasic Dominance Noe ha while r+f µ is unboune in f, hence implying ha he quaniy in (3.5) is maximize by a sraegy ha invess as much as possible in he risky asse, he mean RROI of (3.4), r+f µ 1 2 f 2 σ 2, is maximize a a finie value. In paricular a he value f = µ/σ 2, which is he same policy ha is opimal for maximizing logarihmic uiliy of wealh a a fixe erminal ime an hence for maximizing he asympoic (exponenial) rae of growh, an by he resuls of [15, 23, 6] is also opimal for minimizing he expece ime o a goal, i.e., he mean RROI is maximize by he log-opimal or orinary opimal-growh sraegy of (2.7). Thus, we fin ha a byprouc of our analysis is ha i provies ye anoher objecive jusificaion for he use of logarihmic uiliy. We make his precise in he following. Corollary 3.2 The mean of he limiing isribuion of he RROI process is maximize a he value f = µ σ 2, (3.6) wih resuling mean E[ρ ] = r + γ, where γ := 1 2 ( µ/σ)2. For his sraegy he RROI, ρ (), saisfies ( r + γ ρ () ρ gamma, 1 ). (3.7) γ γ In fac, he precise isribuional characerizaion of he limiing RROI allows for a somewha sronger saemen, in erms of sochasic orerings. Recall firs (see e.g. Soyan [28]) ha for wo ranom variables, X, Y, we say ha X icx Y if E(X x) + E(Y x) + for all x. This is equivalen o saying ha E[g(X)] E[g(Y )] for all increasing convex funcions g, an is referre o as he increasing convex orering. We also say ha X icv Y if E(x X) + E(x Y ) +, for all x (provie he expecaions are finie). This is equivalen o saying ha E[h(X)] E[h(Y )] for all increasing concave funcions h, an is hence referre o as he increasing concave orering. (This las orering is also referre o as secon egree sochasic monoonic ominance (see [16, Secion 2.9]).) The following corollary shows ha he RROI obaine from using he opimal growh policy ominaes in he increasing convex sochasic orer he RROI from any oher proporional sraegy ha uner-invess relaive o i, an ominaes in he increasing concave sochasic orer he RROI for any proporional sraegy ha over-invess relaive o i. Corollary 3.3 Le ρ enoe he RROI obaine from using he policy f efine in (3.6), an le ρ f be he RROI from any oher consan proporional sraegy f := cf, where c is an arbirary consan. Then he following hol: ρ f icx ρ for c 1, (3.8) 10

12 ρ f icv ρ for c 1. (3.9) Proof: See Appenix A.1. Comparing Corollary 3.3 wih equaions (2.5) an (2.8), an he earlier iscussion of risk aversion in Remark 2.1, shows clearly ha (3.8) is in effec for an invesor wih greaer relaive risk aversion han a log invesor, while (3.9) is in effec for an invesor wih less relaive risk aversion. Remark 3.3: I shoul be noe ha while he previous wo corollaries highligh some opimal properies of he log-opimal sraegy, i is no claime ha he log-opimal sraegy, f, is in fac he sraegy ha maximizes he mean RROI over all amissible sraegies only among all consan proporional ones. I is an open quesion as o wha he acual (global) opimal sraegy is for his crierion. Remark 3.4: I is ineresing o observe ha we canno exen Corollary 3.3 o ge a sanar sronger sochasic orering in general wih respec o over or uner invesing relaive o he log-opimal policy. Raher, we can only obain a (sharp) characerizaion of he ype of consan proporional sraegies ha yiel RROIs ha are sochasically ominae by he RROI from he log-opimal or opimal growh policy. However, before we procee we recall he likelihoo raio orer, enoe by LR. This sochasic orer is sronger han (an hence implies) he sochasic orer s, which is also known as firs egree sochasic ominance (see [16]). (Recall ha X s Y if P (X z) P (Y z), for all z.) For wo coninuous nonnegaive ranom variables, X 1, X 2, wih ensiies ψ 1 (x) an ψ 2 (x), we say ha X 2 LR X 1 if ψ 1 (y) ψ 1 (x) ψ 2(y) ψ 2 (x) for all x y. (3.10) Corollary 3.4 For a proporional sraegy f = cf, wih c 1, where f is he opimal policy of (3.6), he relaionship ρ f LR ρ (which implies he weaker relaionship ρ f s ρ ) hols if an only if c saisfies 1 r + γ γ < c < r r + 2γ. (3.11) Proof: See Appenix A.2. Corollary 3.4 shows ha he only ype of consan proporional policy (oher han f ) for which he RROI is sochasically ominae by he opimal growh policy is one ha is shoring he sock o he egree require by (3.11). 11

13 4 Proof of Theorem 3.1 an a Relae Diffusion Process In his secion we provie he proof of Theorem 3.1. While he process {ρ f (), 0} oes no ami a simple irec analysis, here is a relae Markov process amenable o analysis which hols he key for he limiing behavior of {ρ f ()}. Specifically, he process {R f (), 0} efine by R f () := X f X Xf s s. (4.1) (Noe ha R f (0) = 1.) We will firs show ha he limiing behavior of {ρ f ()} is equivalen o he limiing behavior of he process {R f ()}. Since our main ineres here is in fac on he limiing behavior of he RROI process {ρ f ()}, his is quie convenien, since as we will show below {R f ()} is in fac a iffusion process whose limiing behavior we can analyze precisely. Theorem 4.1 Suppose ha for some ranom variable R f, we have R f () for any f ha saisfies (2.15) we have, as, ρ f () R f as. Then R f. (4.2) Proof: Observe ha [ X f ρ f () = R f () X ] [ 0 X0 + ] 0 Xf s s X f 0 Xf s s [ ] [ R f () 1 e B 1 + where {B s, s 0} is he linear Brownian moion efine by ( B s := r + f µ f 2 σ 2 ) s + fσw s, wih B 0 = 0. 2 ] 1 0 ebs s (4.3) If (2.15) hols, hen {B s } has posiive rif, which hen implies ha lim e B = 0 a.s., as well as lim ( 0 ebs s) 1 = 0 a.s., which in urn implies he isribuional limi (4.2) as a consequence of Theorem 4.4 of Billingsley [1]. As such, Theorem 3.1 will be compleely esablishe if we can now prove ha R f () R f, for some ranom variable R f wih R f = ρ f, i.e., we mus esablish ha {R f ()} has he limiing gamma isribuion posie earlier. To ha en, we will firs show ha he process {R f ()} is in fac a emporally homogeneous iffusion. This enables us o use sanar echniques from he heory of iffusion processes o esablish is limiing saionary isribuion exacly. We will also prove ha {R f ()} is uniformly inegrable. We begin by firs noing he following. Proposiion 4.2 For a fixe proporional invesmen policy f, he process {R f ()} follows he sochasic ifferenial equaion R f () = [ ] (r + f µ)r f () Rf 2 () + fσr f ()W, (4.4) 12

14 i.e., {R f ()} is a emporally homogeneous iffusion process wih rif funcion b(x) = (r+f µ)x x 2 an iffusion funcion v 2 (x) = f 2 σ 2 x 2. Proof: Defining A := 0 Xf s s as he cumulaive wealh (invesmen) process, we may wrie R f () = X f /(X 0 + A ). Since A is a process of boune firs variaion, is quaraic variaion is zero, an so Io s rule shows ha A = X f. Applying Io s rule o Xf /(X 0 + A ) gives which upon subsiuion gives X f = X 0 + A which is equivalen o (4.4). X f = X 0 + A 1 X 0 + A X f Xf X 0 + A (r + f µ) + X f (X 0 + A ) 2 A Xf X 0 + A fσw X f 2 X 0 + A The fac ha he {R f ()} follows he sochasic ifferenial equaion (4.4) allows us o conclue ha i is in fac a srongly ergoic Markov process wih a gamma saionary isribuion. We sae his as a heorem nex. The proof will follow as an applicaion of he more general lemma ha follows irecly. Theorem (r + f µ) 2 R f () R f gamma σ 2 f 2 1, σ 2 f 2. (4.5) Moreover he he process R f () is uniformly inegrable, an hence Proof: lim E(R f ()) = E[R f ] = r + f µ 1 2 f 2 σ 2. (4.6) Since we will have oher occasions o examine sochasic ifferenial equaions similar o (4.4), we will esablish he resul for he more general sochasic ifferenial equaion Z = [ az bz 2 ] + cz W, of which (4.4) is jus a special case wih a = r + f µ, b = 1, c = fσ. Therefore, Theorem 4.3 (an hen in ligh of Theorem 4.1, by implicaion Theorem 3.1) is jus a consequence of he following lemma: Lemma 4.4 Suppose Z evolves as he sochasic ifferenial equaion [ ] Z = az bz 2 + cz W (4.7) wih a > 0, b > 0. Then (i) Z = } Z 0 exp {(a 1 2 c2) + cw 1 + Z 0 b { } 0 exp a 1 2 c2 s + cw s s. (4.8) 13

15 (ii) (iii) Furhermore, for a > c 2 /2, b > 0, he process Z is srongly ergoic wih 2a 2b Z Z gamma 1, c2 c 2. (4.9) Moreover, Z is uniformly inegrable, so ha in paricular we have lim E(Z ) = E(Z ) = 1 (a 1 ) b 2 c2. (4.10) Proof: (i) (ii) Le Y := Z 1. Then an applicaion of Io s formula shows ha Y evolves accoring o he sochasic ifferenial equaion Y = [ b + (c 2 a)y ] cy W, which is a linear sochasic ifferenial equaion, an so can be solve by sanar mehos (see e.g.[18, Secion 5.6.C]) o yiel { Y = exp (a 1 ) } [ 2 c2 cw Y 0 + b exp {(a 1 ) } ] 0 2 c2 s + cw s s. Using Z = 1/Y hen gives (4.8). The saionary isribuion for Z can be recovere by recognizing ha Z is a emporally homogeneous iffusion wih rif funcion µ(z) = az bz 2 an iffusion funcion σ 2 (z) = c 2 z 2. As such, is scale ensiy, s(z) is given by ([19, Chaper 15]) { z 2(ax bx 2 } ) s(z) exp c 2 x 2 x = z 2a/c2 e z2b/c2. Leing S(z) = z s(x)x, we see ha S(0+) = while S(z) as z. As such, i is well known (see e.g. [19, Secion 15.6]) ha Z is a srongly ergoic process wih unique saionary isribuion given by m(z)/ 0 m(x)x, where m(z) is he spee ensiy efine by [ 1 m(z) := σ (z)s(z)] 2 c 2 z 2a/c2 2 e z2b/c2, which is he kernel of he gamma ( 2a/c 2 1, 2b/c 2) ensiy, which esablishes (4.9). (iii) I remains o prove ha Z is in fac uniformly inegrable. A sufficien coniion for his is ha sup E [ Z 2 ] <. We can esablish ha his hols by leing H = Z 2, an noing ha by Io s rule, H saisfies [ H = (2a + c 2 )H 2bH 3/2 ] + 2cH W, wih H 0 = Z

16 Since b > 0, we may now choose consans α > 0 an β > 0 so ha an consruc a new process H where Noe ha H is always nonnegaive. α βx > (2a + c 2 )x 2bx 3/2, for x 0, H = (α β H ) + 2c H W, wih H 0 Z 2 0. (4.11) By he comparison heorem for iffusions (e.g.[18, Proposiion ]) i follows ha H H, a.s., an hence P z (H > x) P z ( H > x), from which i follows ha E(H ) E( H ). Noe now ha since (4.11) is a linear equaion, we can solve i exacly o fin from which i is follows ha ] H = e (β+2c2 )+2cW [H 0 + α e (β+2c2 )s 2cW s s 0 E( H ) = H 0 e β + α β (1 e β ), an for which i is herefore evien (since β > 0) ha sup E( H ) <. Since E(Z 2 ) = E(H ) E( H ) for every 0, i follows ha Z is uniformly inegrable. 5 Rae of reurn from invesmen in he risky asse alone. In he previous secions we concenrae on he reurn from oal invesmen, which inclue boh invesmen in he risky sock as well as invesmen in he risk-free asse. Here we focus on he excess reurn above he risk-free rae, i.e., he gain an reurn from he invesmen in he risky sock alone. To ha en, recognize ha he excess gain in wealh above wha coul have been obaine by invesing in he risk-free asse is he iscoune, or presen value, of he gain, e r X f X 0. Noe ha if f = 0, whereby all he wealh is always invese in he risk-free asse, hen his quaniy is simply 0. (This follows since if all he money is invese in he risk-free asse, hen X 0 = e r X 0.) Since f is he proporion of wealh invese in he risky sock, he oal amoun invese in he risky sock a ime is fx f, an so he cumulaive amoun of money invese in he risky sock unil is f 0 Xf s s. The iscoune, or presen value, of he cumulaive amoun of money invese in he risky sock unil ime is herefore f 0 e rs Xs f s. Define now he reurn on he risky invesmen (RRORI) o be he iscoune gain ivie by he iscoune cumulaive invesmen in he risky sock, i.e., le ρ f () enoe he RRORI, which is 15

17 efine by ρ f () = e r X f X 0 f 0 e rs X f s s. (5.1) As will be seen presenly, if µ > 0 we require f > 0 for { ρ f ()} o have a nonegenerae limi, i.e., shorselling he sock is prohibie, as hen he iscoune gain ens o 0. The RRORI is hus he presen value of he gain from risky invesmen ivie by he presen value of he oal amoun of wealh invese in he risky sock ha is neee o obain he gain. As such, i is again a measure of he effeciveness of an invesmen sraegy, wih larger values signifying a beer sraegy. We will show ha he limiing isribuion of ρ f () is again a gamma isribuion, specifically, Theorem 5.1 For any consan f such ha 0 < f < 2 µ σ 2, (5.2) he RRORI process ρ f () converges as o a gamma isribuion, i.e., 2 µ ρ f () ρ f gamma fσ 2 1, 2 fσ 2, (5.3) an hence Remark 5.1: E[ ρ f ] = µ 1 2 fσ2. (5.4) Noe ha he mean RROI in (5.4) is a sricly ecreasing funcion of f, he proporion invese in he risky sock (for f > 0). However, i is ineresing o noe ha if we look a he raio of he expece values of he iscoune gain o he iscoune cumulaive invesmen, we ge a value ha is inepenen of he proporion invese, for any > 0. Specifically, noe ha E(e r X f ) = X 0e f µ, an so by Fubini we have E 0 e rs Xs f s = X 0 (f µ) 1 e f µ 1. Hence, i follows ha ( ) e r X f X 0 E(iscoune gain) E( iscoune cumulaive risky invesmen) E fe 0 e rs X f s s = µ, (5.5) for any value of f > 0, an > 0. (See Griffin [12] for a iscussion of he ifferen inerpreaion of measures relae o (5.4) an (5.5) for simple gambling moels.) 5.1 Proof of Theorem 5.1 To suy he limiing isribuion of ρ f (), we firs efine he process R f () := e r X f ( f X 0 + ). (5.6) 0 e rs Xs f s 16

18 In he nex heorem we show ha he limiing behavior of he process { ρ f ()} is eermine by he limiing behavior of he process { R f ()}. Theorem 5.2 If for some ranom variable R f ha saisfies (5.2), we have, as, we have R f () R f, hen for any consan f ρ f () Proof: I is easies o firs efine he iscoune wealh process Y f formula applie o Y shows ha Y f = f µy f f + σfy W, an so Noing now ha ρ f () = Y f R f. (5.7) = X 0 exp {(f µ 1 ) } 2 f 2 σ 2 + fσw. ( Y f Y 0 / f ) 0 Y s f s, i is clear ha we can wrie := e r X f, since hen Io s [ ] [ ρ f () := R Y f Y 0 Y0 + f () 0 Y f ] s s Y f 0 Y s f. (5.8) s From classical resuls on geomeric Brownian moion we know ha so long as (5.2) hols, inf 0 Y f > 0 an Y f (5.8) ha if (5.2) hols we have lim a.s., which also implies ha 0 Y f s s a.s.. Therefore i follows from [ ] [ Y f Y 0 Y0 + Y f = 1 a.s, an lim 0 Y f which hen implies (5.7) as a consequence of Theorem 4.4 of Billingsley[1]. 0 Y s f ] s = 1 a.s, (5.9) s s Since (5.7) shows ha he limiing behavior of he RRORI process { ρ f ()} is eermine by he limiing behavior of { R f ()}, we now urn o he suy of he process R f (). We will show ha R f () is an ergoic iffusion process, which will allow us o ienify is limiing isribuion from previous resuls, an exrac is limiing momens accoringly. To begin, we noe he following, which shoul be compare wih Proposiion 4.2. Proposiion 5.3 The process R f () saisfies he sochasic ifferenial equaion R f () = ( f µ R f () f R f () 2) + fσ R f ()W, (5.10) i.e., R f () is a emporally homogeneous iffusion process wih rif funcion b(x) = f µx fx 2, an iffusion funcion v 2 (x) = f 2 σ 2 x 2. 17

19 Proof: Noe ha R ( f () = Y f /D, where D := f Y 0 + ) 0 Y s f s. Recognize ha D = fy f f, an apply Io s formula o (Y /D ) o ge Y f D which is equivalen o (5.10). = D 1 Y f Y f D 2 D Y f (f µ + fσw ) f D Y f 2, D Since for f > 0 he process R f () follows he sochasic ifferenial equaion (5.10), we may now ienify a = f µ, b = f an c = fσ in (4.7), (4.10) an herefore appeal o Lemma 1 o conclue he following: Theorem 5.4 For 0 < f < 2 µ/σ 2, he process R f () is srongly ergoic an has a limiing gamma isribuion. Specifically, as R f () R 2 µ f gamma fσ 2 1, 2 fσ 2 Furhermore, we have uniform inegrabiliy, an so (5.11) [ ] lim E Rf () = E[ R f ] = µ 1 2 fσ2. (5.12) This in conjuncion wih Theorem 5.2, an he fac ha R f = ρf, proves Theorem 5.1. We nex move on o suy some consequences of Theorem Opimal Growh Policy I is clear from (5.4) ha here, where our ineres is in he reurn from he risky invesmen, logarihmic uiliy an he opimal growh policy, f, no longer gives he maximal mean RRORI. In fac, E( ρ f ) := µ fσ 2 /2 is a sricly ecreasing funcion of he (fixe) proporion f an so has no maximizer for f > 0. This woul seem o preclue any opimaliy properies for he opimal growh policy uner his performance measure. Furhermore i urns ou ha he RRORI uner he opimal growh policy has a limiing exponenial isribuion, which agrees wih he asympoic resuls obaine in Ehier an Tavare [9] for he iscree-ime gambling moel suie here. Recall firs ha he opimal growh proporional invesmen sraegy is f := µ/σ 2. Therefore by aking f = cf in (5.3), where f is he opimal growh policy an 0 < c < 2, we fin ha 2 ρ f gamma c 1, 2 (5.13) c µ which is inepenen of σ. 18

20 Remark 5.2: This now allows us o recover he asympoic resuls for he iscree-ime gambling moel in Ehier an Tavare [9], since (5.13) implies ha ( ρ 2 f µ gamma c 1, 2 ), c which shoul be compare o (1.9) of Ehier an Tavare [9]. Taking c = 1 in (5.13), an recognizing ha a gamma(1, β) Exp(β), where Exp(λ) enoes he exponenial ensiy funcion wih mean 1/λ, hen allows us o euce irecly ha he RRORI for he opimal growh policy ens o an exponenial isribuion. Corollary 5.5 For he case where f = f, he opimal growh policy, he associae RRORI, say ρ (), has a limiing exponenial isribuion. Specifically, ρ () ρ Exp (2/ µ), an so E[ ρ ] = µ 2. (5.14) Checking he coniions given in he appenices for sochasic ominance of gamma ranom variables, we fin ha no orerings are possible in general, excep for he following, which shows ha he RRORI from he opimal growh policy ominaes in he increasing concave sochasic orer he RRORI from any oher consan proporional sraegy which over invess in he risky sock relaive o he opimal growh policy, i.e., for an invesor who is less risk averse han an invesor wih a logarihmic uiliy funcion. (This is quie limie compare o wha we obaine earlier in Secion 3 for he rae of reurn on oal invesmen, he RROI.) Corollary 5.6 For f > f, where f is he opimal growh policy, we have ρ f icv ρ. Proof: The coniions neee for he sochasic orer relaion X 2 icv X 1, for X i gamma(α i, β i ), are β 1 > β 2 an α 1 /β 1 α 2 /β 2 (see Appenix A.1). Taking X 2 = ρ f an X 1 = ρ, (5.13) an he fac ha ρ gamma(1, 2/ µ), shows ha hese wo coniions are equivalen o he requiremen c > 1, i.e., f > f. I is ineresing o noe ha he coniions for he oher forms of sochasic ominance (see Appenix A.1 an A.2), such as icx, LR an s here give conraicory coniions on c, so ha no ominaion can be esablishe. I is easily checke ha he previous resul generalizes o arbirary consan proporional sraegies as Corollary 5.7 Consier wo proporional sraegies, wih f i = c i f for i = 1, 2, an le ρ i enoe heir respecive RRORIs. Then c 2 > c 1 implies ρ 2 icv ρ 1. 19

21 5.3 Consan invesmen For µ > 0, he mean RRORI, E( ρ f ) is a ecreasing funcion of f, an we see ha he maximal RRORI is aken a f = 0 wih corresponing value µ. As noe earlier, his of course oes no correspon o an invesor who invess all of his wealh in he risk-free asse, since for such an invesor ρ 0 () 0, for all > 0. Insea, as we now show, his maximal reurn is insea achieve for an invesor who invess a fixe consan amoun (as oppose o a consan proporion) in he risky asse. Such invesmen policies are opimal when he invesor has an exponenial uiliy funcion (see e.g., Meron[22] or Browne[5]). Ineresingly enough, he RRORI for such an invesmen policy is inepenen of he paricular consan amoun invese. Proposiion 5.8 Consier an invesmen sraegy ha always keeps a fixe consan oal amoun of money, say $κ, invese in he risky sock, regarless of he wealh level wih he remainer invese in he risk-free bon. Then he RRORI for his sraegy, { ρ κ (), 0}, is an ergoic Gaussian process wih consan mean funcion E[ ρ κ ()] = µ, an covariance funcion, for s E [( ρ κ () µ)( ρ κ (s) µ)] = rσ2 2 ( 1 e 2rs ) (1 e r )(1 e rs. (5.15) ) Therefore, as ρ κ () ρ κ N µ, rσ2, (5.16) 2 where N(α, β 2 ) enoes a normal isribuion wih mean α an variance β 2. Proof: If he invesor always keeps $κ invese in he risky sock, regarless of his wealh level, wih he remainer invese in he bon, hen his invesor s wealh, say X κ, evolves as X κ = κ S S + (X κ κ) B B = (rx κ + κ µ) + κσw, (5.17) which is a simple linear sochasic ifferenial equaion, someimes calle compouning Brownian moion. I follows ha (see e.g.[18, Secion 5.6]) e r X κ = X 0 + κ µ r (1 e r ) + κσ e rs W s (5.18) 0 an so he iscoune gain for a consan invesor, e r X κ X 0 is a Gaussian process wih mean funcion κ µ(1 e r )/r an variance funcion κ 2 σ 2 (1 e 2r )/(2r). 20

22 The cumulaive amoun invese in he risky sock for a consan invesor is 0 κs = κ, an he iscoune cumulaive amoun invese is 0 κe rs s = κ(1 e r )/r. Therefore, he RRORI for a consan invesor, ρ κ () is given by ρ κ () := e r X κ X 0 κ r (1 e r ) = µ + rσ 1 e r e rs W s (5.19) 0 where he secon equaliy follows from (5.18). I is clear from (5.19) ha for any consan κ, ρ κ () is a Gaussian process wih mean µ an covariance funcion given by (5.15). I is also clear from ) (5.19) ha ρ κ () ρ κ, where ρ κ N ( µ, rσ2 2. Remark 5.3: Comparing he RRORI for he opimal growh policy wih he RRORI for a consan invesmen policy shows ha he mean RRORI uner he opimal growh policy is half he mean RRORI for a consan amoun invesmen policy, i.e., (5.14) an (5.16) shows E( ρ ) = 1 2 E( ρ κ). This raher isurbing resul can bes be unersoo in he conex of he fac ha he consan invesor can go bankrup uner his invesmen policy, while a proporional invesor will never go bankrup. Thus he halve reurn can be consiere he insurance premium for never going bankrup. This is iscusse in Ehier an Tavare [9] as well. I urns ou ha his comparison is specific o he case of a single risky sock, an oes no generalize o he muliple sock case, as we will see in he nex secion. The isribuional resuls in Theorem 5.1 an Proposiion 5.8 allow for he explici compuaion of various probabiliies. The probabiliy ha ρ κ is posiive is irecly seen o be P ( ρ κ > 0) = ( ) Φ 2 µ r σ, where Φ( ) is he CDF of he sanar normal variae. This quaniy is of course no he probabiliy ha he consan invesor never goes bankrup. This laer quaniy is in fac ( ) ( ) ( ) P inf 0 Xκ > 0 = Φ 2r κσ (X 0 + κ µ) Φ 2r σ µ ( ) 1 Φ 2r σ µ. (5.20) This las resul can be esablishe from he fac ha (5.17) shows ha X κ is a iffusion process wih rif b(x) = rx + κ µ an iffusion funcion v 2 (x) = κ 2 σ 2. Hence, i follows ha is scale funcion is given by { s κ (x) = e r µ/σ2 exp r } κ 2 (x + κ µ)2, σ2 an ha herefore P (inf 0 X κ > 0) = X 0 0 s κ (x)x/ 0 s κ (x)x, which reuces o (5.20). 21

23 Anoher ineresing quaniy is he probabiliy ha he (opimal) proporional reurn ρ excees he reurn on consan invesmen ρ κ, P ( ρ > ρ κ ). This can be compue by recognizing ha which hen implies P ( ρ > ρ κ ) = P ( ρ κ < 0) + 6 Muliple Risky Socks P ( ρ > ρ κ ρ κ = x) = e 2x/ µ, 1 { [ exp πrσ 2 0 2x µ + 1 ]} (x µ)2 x rσ2 ( ) { 2 µ µ 2 rσ 2 µ 3 } 2 µ = 1 Φ + exp r σ rσ 2 µ Φ 2 rσ 2 µrσ 2. As promise earlier, in his secion we reurn o consier he case wih k risky socks. Since he wealh process, X f of (2.12), is sill one imensional i is no har o exen he previous analysis o his case. As such, we merely highligh he resuls, leaving he eails for he reaer. I oes urn ou ha he resuls for he RRORI measure iffer somewha from he case wih a single risky sock. 6.1 RROI For any fixe proporional sraegy f which saisfies coniion (2.14), he RROI process ρ f () converges in isribuion o a ranom variable which has a gamma isribuion. I.e., as, ρ f () ρ f gamma 2 r + f µ 2 1,. f Σf f Σf I is clear once again ha he mean of his isribuion, E[ρ f ] = r+f µ 1 2 f Σf, is again maximize a he log-opimal, or opimal growh policy, which is now f = Σ 1 µ. For his sraegy he RROI, ) ρ where ρ gamma, an hence wih mean E[ρ ] = r + γ, ρ (), sill saisfies ρ () bu now we have γ := (1/2) µ Σ 1 µ. The comparisons in Corollaries 3.3 an 3.4 sill hol in erms of f = cf. All of his can be prove in essenially he ienical manner as in Secion 4, wih he only proviso being ha now {R f ()} is a emporally homogeneous iffusion process wih rif funcion b(x) = (r + f µ)x x 2 an iffusion funcion v 2 (x) = f Σf x 2. As such, no new ifficulies are encounere. ( r+γ γ, 1 γ 22

24 6.2 RRORI Here he siuaion is slighly ifferen. The amoun invese in he risky sock a ime is (f 1)X f, an so he RRORI, ρ f (), is efine by ρ f () := e r X f X 0 (f 1) 0 e rs Xs f s. The analog of Theorem 5.1, which hols so long as f µ (1/2)f Σf > 0, is ( ) 2f µ ρ f () ρ f gamma f Σf 1, 2, f Σf ( ) an hence E[ρ f ] = f µ 1 2 f Σf / f 1. For his case we also noe ha he raio of he expece values is no longer inepenen of he policy (see Remark 5.1), an is in fac ( ) E e r X f X 0 (f 1)E 0 Xf s s = f µ f 1. The RRORI uner he opimal growh policy, f is sill exponenial, bu is mean now epens on he covariance marix as well: specifically, E ρ = µ Σ 1 µ 2 µ Σ 1 1. (6.1) Comparisons wih a consan amoun invesmen sraegy no longer seem relevan. For a consan amoun policy, say κ := (κ 1,..., κ k ), where κ i is he amoun of money invese in sock i, he wealh process is isribuionally equivalen o he process X κ X κ = (rx κ + κ µ) + κ Σκ W. which evolves as Since he amoun invese a ime is jus κ 1, he cumulaive iscoune invesmen is jus (κ 1/r)(1 e r ), an hence he RRORI for his policy, ρ κ () is he Gaussian process ρ κ () := e r X κ X 0 (κ 1/r)(1 e r ) κ µ κ κ 1 + Σκ r κ 1 1 e r e r W s, which epens on he specific value of κ. I follows ha ρ κ () ρ κ, where κ µ ρ κ N κ 1, 2κ Σκ 2(κ 1) 2. (6.2) Wha is apparen from (6.1) an (6.2) is ha if he consan amoun invesor chooses κ f, hen he mean RRORI for he consan amoun invesor is wice he mean RRORI for a log-opimal, or opimal growh policy invesor. 0 23

25 A Appenix A.1 Proof of Corollary 3.3 Le f = cf, where c is an arbirary consan. Noe ha i hen follows from Theorem 3.1 ha uner his parameerizaion we have ( r + 2cγ c 2 ) γ 1 ρ f gamma c 2, γ c 2. (A.1) γ Recall firs (Soyan[28]) ha if X i gamma(α i, β i ) for i = 1, 2, hen α 1 < α 2 an α 1 β 1 α 2 β 2 implies X 2 icx X 1. (A.2) Taking X 1 = ρ an X 2 = ρ f, we see from he parameerizaion in (3.7) an (A.1) ha he wo coniions for he ominance ρ f icx ρ in (A.2) are equivalen o (i) r + γ γ < r + 2cγ c2 γ c 2 γ, an (ii) r + γ r + 2cγ c 2 γ. Coniion (ii) hols rivially by he fac ha f was chosen o maximize he mean RROI. Coniion (i) is equivalen o he requiremen ha Q(c) < 0 where Q(c) is he quaraic efine by Q(c) := c 2 (r + 2γ) c2γ r. (A.3) I is easily checke ha he wo roos o he equaion Q(c) = 0 are c = 1 an c = r r+2γ, wih Q(c) 0 only for c c or c c, an corresponingly wih Q(c) < 0 for c < c < c. Hence for any f < f we have c < 1, for which ρ f icx ρ. Similarly, i is well known (ibi) ha for X i gamma(α i, β i ) for i = 1, 2, β 1 > β 2 an α 1 β 1 α 2 β 2 implies X 2 icv X 1. (A.4) Uner he parameerizaion of (A.1) he wo coniions for he ominance ρ f icv ρ in (A.4) hen become (ii), which hols by consrucion of f, an he new coniion 1 γ > 1. Bu his rivially c 2 γ hols for c > 1 which is he same as f > f. A.2 Proof of Corollary 3.4 Taking X i gamma(α i, β i ), i is clear ha he inequaliy in (3.10) is equivalen o ( y x ) α1 α 2 e (β 1 β 2 )(y x), for all x y, for which a necessary an sufficien coniion is obviously α 1 α 2 an β 1 β 2. These wo coniions are in fac he coniions given in Soyan[28] for he (weaker) relaionship X 2 s X 1. 24

26 Hence, a sufficien coniion for he likelihoo raio orer o apply o wo gamma ranom variables is ha hey be sochasically orere. (This is obviously no he case in general.) Using (3.7) an (A.1) we see ha hese wo coniions (i.e., α 1 α 2 an β 1 β 2 ) for he ominaing relaionship ρ f LR ρ (which implies ρ f s ρ ) are equivalen o he requiremen ha c simulaneously saisfy he inequaliies (iii) r + γ γ r + 2cγ c2 γ c 2 γ, an (iv) 1 γ 1 c 2 γ. Coniion (iv) is obviously equivalen o c 2 1, while (iii) reuces o he requiremen ha Q(c) 0, where Q(c) is he quaraic of (A.3). However, as noe above, he wo roos o he equaion Q(c) = 0 are c = 1 an c = r r+2γ, wih Q(c) 0 only for c c or c c. Hence Q(c) 0 implies ha eiher c 1 or c r/(r +2γ). The former is impossible (excep a equaliy c = 1) by (iv). The requiremen ha r + 2cγ c 2 γ > 0, which is jus (2.15), is equivalen o he requiremen (recall (2.16)) 1 r + γ γ r + γ < c < 1 +. γ Since r/(r + 2γ) > 1 r+γ γ, we obain he resul. I shoul be noe ha Corollary 3.4 exens o arbirary consan proporional policies as in he following. Proposiion A.1 Consier wo proporional sraegies, wih f i = c i f, for i = 1, 2, where f is he opimal growh policy of (3.6), an le ρ i, i = 1, 2 enoe heir corresponing RROIs. Then a necessary an sufficien coniion for ρ 2 LR ρ 1 (an by implicaion, also ρ 2 s ρ 1 ) is 1 c2 1 c 2 2 r + 2c 1γ c 2 1 γ r + 2c 2 γ c 2 2 γ. (A.5) Proof: Using (A.1) he wo coniions (α 1 α 2 an β 1 β 2 ) for he relaion ρ 2 LR ρ 1 become r + 2c 1 γ c 2 1 γ c 2 1 γ r + 2c 2γ c 2 2 γ c 2 2 γ, an 1 c 2 1 γ 1 c 2 2 γ, (A.6) which ogeher imply he simulaneous inequaliy (A.5). References [1] Billingsley, P. (1968). Convergence of Probabiliy Measures. Wiley, NY. [2] Black, F., an Perol, A.F. (1992). Theory of Consan Proporion Porfolio Insurance. Jour. Econ. Dyn. an Cnrl. 16,