OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

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1 OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected value of the utlty of an nvestor s fortune are presented for nvestment models n whch there are m nvestment opportuntes exactly one of whch wll pay off, smlar to bettng on a wnner n a horse race. It s assumed the nvestor knows for =,...,m the probabltes p that outcome s the wnner, and the respectve odds z that wll be pad f outcome wns, where z represents the number of dollars returned on a dollar bet. The nvestor s problem s to choose the amount to nvest on each outcome n order to maxmze the expected utlty of hs resultng wealth. Ths model combnes and extends the models of Kelly and Murphy. Usng the Karush-Kuhn-Tucker Theorem, the optmal strateges for the log, power and exponental utlty functons are derved.. Introducton. In Ferguson and Glsten [6], condtons were gven under whch the myopc rule s optmal for makng nvestment decsons n a general multstage nvestment model. Usually, evaluaton of the myopc rule requres approxmaton methods assocated wth concave programmng problems. In ths paper, we examne one class of nvestment models for whch the myopc rule may be evaluated explctly. The dstngushng feature of ths class of models s that of the m 2 possble nvestment opportuntes exactly one wll pay off, a stuaton referred to as the horse race n Baldwn [2]. The nvestor s gven the opportunty to bet on m possble outcomes, one and only one of whch wll occur, each payng odds of a specfed amount. The nvestor knows the probabltes p =,...,m of the outcomes and the respectve odds z that wll be pad f outcome occurs, =,...,m. Here, z represents the number of dollars returned on a dollar bet. We assume p > 0and z > 0for all snce an outcome gvng zero return wll not be bet on and can be removed from the set of bettng opportuntes. The nvestor s problem s to choose the amount to nvest on each possble outcome n order to maxmze the expected utlty of hs resultng wealth. The problem of maxmzng the expected value of a log utlty functon was consdered n models of Kelly [7] and Murphy [8], both n the context of ths problem. The model consdered here brdges these two models and generalzes ther solutons to the power and exponental utlty functons. We descrbe the optmal rules for log, power, and exponental utlty functons. We frst consder the class of log and power utlty functons { (x U γ (x = γ /γ for γ 0, ( log(x for γ =0. where x 0represents the nvestor s fortune. The parameter γ measures the nvestor s averson to rsk; the value γ = beng the lnear utlty functon and smaller values of

2 γ ndcatng greater averson to rsk. These are the utlty functons that have constant relatve rsk averson, where the relatve rsk averson of a utlty, u, s defned by Arrow[] and Pratt [9] to be xu (x/u (x. They are farly representatve, contanng the lnear utlty (γ = and the logarthmc utlty (γ = 0 as specal cases. Snce utlty functons are determned only up to change of locaton and scale, the form of U γ (x has been chosen so that lm γ 0 U γ (x =U 0 (x. As was noted by Kelly [7] and Bellman and Kalaba [3], the optmal nvestment for an nvestor wth a utlty functon n ths class s proportonal to the nvestor s fortune. The second class of utlty functons we consder s the exponental class W θ (x = e θx, (2 where the fortune x may assume negatve as well as postve values and θ>0.modelswth ths utlty functon were consdered n Ferguson [5] for nvestors whose man concern was not gong bankrupt. Larger values of θ lead to more cautous nvestment decsons. The optmal nvestments for an nvestor wth a utlty functon n ths class are ndependent of fortune. In Secton 2, we descrbe the nvestment models and the optmal nvestment polces for both the U γ utlty functons and the W θ utlty functons. In Secton 3, we descrbe the optmal nvestment polces for several dfferent utlty functons n each class. In Secton 4, we present the proofs of the theorems gven n Secton Optmal Investment Strateges. 2. The Utlty Functons, U γ. As mentoned above, for the utlty functons U γ, the optmal nvestments are proportonal to the nvestor s wealth. Thus, the nvestor must choose an nvestment polcy c =(c 0,c,...,c m, where for =,...,m, c 0represents the proporton of the nvestor s wealth bet on outcome, andc 0 = m c 0represents the porporton of the nvestor s wealth not nvested. If the nvestor uses nvestment polcy c and f outcome occurs, the return V per unt wealth s assumed to be of the form V = α + αc 0 + c z, for =,...,m. The parameter α, 0 α, represents a bound on the downsde rsk of the nvestment and provdes a connecton between the nvestment model of Kelly and that of Murphy. Regardless of the outcome, the nvestor cannot lose more than α tmes hs fortune. Kelly s model occurs when α =:V = c 0 +c z. The amount not nvested s held at no nterest and money bet on outcomes that dd not occur s lost. The case α = 0corresponds to Murphy s model: V =+c z. In ths case, the nvestor may nvest n several dfferent opportuntes, one and only one of whch wll show an ncrease, the others remanng constant. Thus, the fnal holdng s equal to the ntal holdng plus the ncremental ncrease due to nvestment n outcome. 2

3 The utlty of V s taken to be U γ (V whereu γ s gven by (. Snce outcome occurs wth probablty p and p =, the expected utlty of nvestment polcy c s equal to m Φ γ (c ={ γ p ( α + αc 0 + c z γ γ for γ 0 m p (3 log( α + αc 0 + c z, for γ =0. The problem s to choose c to maxmze Φ γ (c subect to the constrants c 0for =0,,...,m, and m 0 c =. (4 The nvestor may nvest only nonnegatve amounts and cannot nvest more than he has. Ths model may be generalzed to nclude an addtonal parameter, r, representngthe nterest rate on money not nvested. The equaton for the return gven outcome becomes V = α + rαc 0 + c z,for =,...,m. However, by factorng out S = α + rα and changng varables, we may wrte V /S = β + βc 0 + c (z /S, where β = rα/s. Forthe utlty functons U γ, the optmal nvestment vector for maxmzng the expected utlty of V s the same as that for maxmzng the expected utlty of V /S. Thus, the generalzed model may be solved by solvng the specalzed model wth the returns z replaced by z /S, andα replaced by rα/s. A smlar substtuton may be used to derve the optmal nvestments for the more general utlty functons U(x (A+ x/( γ γ, A>0, γ 0, (see Rubensten [0]. If γ, the functon Φ γ s convex on ts doman of defnton so that t assumes ts maxmum at one of the extreme ponts of the constrant set. The soluton of the problem s then trval. Fnd a subscrpt such that p ( α + z γ s a maxmum and put c = (nvest everythng n opportunty, unless p ( α + z γ <, n whch case put c 0 =, (nvest nothng. The soluton for γ< s gven below n Theorem. The statement of the theorem s facltated by the ntroducton of some notaton. Frst, we assume that the nvestment opportuntes have been ordered n order of decreasng expected returns per unt nvested, p z p 2 z 2... p m z m. (5 There s an optmal rule that chooses nvestments accordng to ths order; more precsely, t has the property that f no nvestment s made n opportunty, then no nvestment s made n opportuntes wth >. When α m z <, the optmal c 0 wll be zero,.e. the whole fortune wll be nvested. Ths s because the nvestor can gan a postve amount surely by nvestng c = z / m z n opportunty for =,...,m. For =0,,...,m,letQ be defned as Q = α( p α z f α z <, + f α z 3. (6

4 Note that Q 0 = α. Wth the varables ordered as n (5, let 0 k m be the smallest nteger such that Q k = mn Q. (7 0 m Note that f α =0thenk = 0. In Lemma of Secton 4, we wll see that the Q are decreasng for k and nondecreasng for k. Ths mples that f Q α then k =0. For γ<, let R be defned by R = (p z γ /( γ +( α, for =0,,...,m. (8 z Note that R 0 =0. Let0 m be the smallest nteger such that R = max 0 m R. (9 In Lemma 2, we wll see that the R are ncreasng for and nonncreasng for. Theorem. Gven probabltes p,...,p m > 0, m p =, odds z,...,z m > 0, 0 α, andγ<, the functon Φ γ (c of (3 s maxmzed subect to the constrants (4 by the vector c = c chosen as follows: Order the subscrpts as n (5 and fnd k as n (7 and as n (9. Then, A. If (( αr k γ <Q k,thenα>0 and T/( + αt for =0, c = (/( + αt [(p z /Q k /( γ ]/z for =,...,k. 0 for = k +,...,m, where T = k = [(p z /Q k /( γ ]/z. B. If (( αr k γ Q k,then { 0 for =0, c = (p z γ /( γ /R ( αz for =,...,, 0 for = +,...,m. It s nterestng to note as a corollary that no resources are nvested, that s c 0 =,f and only f max p z α. Snceα represents a bound on the downsde rsk to the nvestor, the proporton of the nvestor s fortune at rsk s equal to α, so unless an nvestment opportunty has expected return n excess of α, the nvestor should nvest nothng. The use of ths theorem s llustrated by examples dscussed n Secton 3 and the proof s gven n Secton The Utlty Functons, W θ, θ>0. As before, the nvestor has m nvestment opportuntes one and only one of whch wll pay off. Opportunty has probablty p of payng off at odds z for =,...,m. The nvestor wshes to choose a vector of bets, b =(b,...,b m, that wll maxmze the expected 4

5 utlty of return. However, we no longer requre that the total amount bet, m b,beno greater than hs present fortune, X 0. That s, we allow the nvestor to borrow unlmted amounts of money at prevalng nterest rates. The return, V, when the nvestor uses the bet vector b and opportunty pays off s assumed to be of the form V =(X 0 b r + b z, = where r>0represents the growth factor so that r s the nterest rate, taken to be known. The case r< may be used to treat nvestment problems n perods of hgh nflaton. Note that f m b exceeds X 0, the frst term on the rght s negatve. Ths represents nterest and captal the nvestor must pay on borrowed money. The expectaton of the utlty of the of the return may be wrtten as EW θ (V = Ee θv. The problem of maxmzng the expectaton of the utlty of the return s equvalent to the problem of choosng the vector b to mnmze Ee θv = = p e θ[(x 0 m b r+b z ] = e θx 0r e θr m b p e θb z, subect to the restrcton b 0for =,...,m. Note that the factor exp{ θx 0 r} s ndependent of b so that the optmal nvestment wll be ndependent of the nvestor s fortune. The soluton to ths problem s gven n the followng theorem. Theorem2. Let θ, r, p,...,p m and z,...,z m be postve numbers such that m p =, and defne Φ(b =e θr m b p e θb z. If r m z <, thenφ(b can be made arbtrarly close to zero by choosng b = Nz for =,...,mwth N arbtrarly large. If r m z, thenφ(b s mnmzed subect to b 0 by b = b obtaned as follows: Order the subscrpts as n (5 and let k be the = = largest postve nteger less than or equal to m such that ( p k z k r k = z >r ( k = p. If no such k exsts, let k =0.Thenk<mand ( p z ( r k z b = θz log r( k p and b =0for = k +,...,m.inpartcular,b m =0. for =,...,k, As a corollary, t may be noted that no money s bet, that s b =0forall, fandonly f max p z r. In addton, there s a dscontnuty n the optmal return as a functon 5

6 of r at the pont r 0 =/ m z.forr<r 0,nf b Φ(b = 0. For r = r 0, k of Theorem 2 becomes m and the optmal nvestment polcy reduces to b = θz log(p z /p m z m for =,...,m. Ths leads to the optmal return of Φ(b =r m 0 exp{r 0 z log(p z } > 0. In ths formula for the b, p mz m can be replaced by any smaller number wthout changng the value of Φ(b. Ths gves other optmal strateges. An example llustratng the use of ths theorem s gven n the next secton and the proof s gven n Secton Examples. It s worthwhle statng Theorem for the case α =,andγ = 0(log utlty, as a corollary. Ths result s found n Kelly s orgnal paper n a slghtly dfferent form. The resultng optmal polcy s known as the Kelly bettng system. It s partcularly mportant because t s a system of bets whch, used n repeated play, gves the maxmal rate at whch the nvestor s fortune tends to nfnty. Moreover, the result has a smple statement that gves one a better dea of what s gong on. Corollary. Assume n Theorem that α =and γ =0.Let Q = p z f z <, + f z, and let k be the smallest nteger such that Q k =mn Q. Then, { Qk for =0, c = p z Q k for =,...,k. 0 for = k +,...,m. The optmal expected return per unt bet s k p log(p z. Both cases A and B of Theorem can occur. Case B occurs f and only f k = m or equvalently, Q k =0,orequvalently, m z. The optmal nvestment polcy n Case Bsc 0 =0andc = p for =,...,m. Thus t s optmal to nvest the whole fortune n proporton to the probabltes of success, ndependent of the z! Ths partcular result for the horse race model was also notced n Proposton 2 of Breman [4] as well as n the paper of Kelly. Here s an example to llustrate ths phenomenon. Suppose m = 3, wth probabltes p =., p 2 =.3, and p 3 =.6, and wth odds z =00,z 2 =0andz 3 =.5. The subscrpts have been ordered accordng to (5. Snce 3 z = <, we have case B where our entre fortune s nvested. Ten percent s nvested n the most favorable outcome, thrty percent n the next most favorable and sxty percent n the least favorable outcome. Sxty percent of the tme we wll end up wth only nnety percent of our orgnal 6

7 fortune (snce p 3 z 3 =0.9. Moreover, f someone tells us that they can rase the return on the frst outcome to 200 or even 000, we smply say Thank you! ; we do not ncrease our nvestment n that outcome! Let us also consder a more typcal example of Theorem for varous values of α and γ. Table contans the values of the optmal nvestment vector, c, for utlty functon U γ for the case p =0.2 p 2 =0.3 p 3 =0.4 p 4 =0. z =7.0 z 2 =4.0 z 3 =2.0 z 4 =6.0 and several values of α and γ. Note that the varables have been ordered n decreasng value of p z. As an example of how Table was computed, consder the case α =0.5 and γ =0.9. Frst note that the k found n (7 s equal to 4 snce α 4 z =.47 > 0and so Q 4 = 0whle Q 3 > 0. Next we compute R 4 =.874 and note that R γ 4 >Q 4 =0,so that case B s applcable and c 0 = 0. To fnd, one computes say R =.928, R 2 =2.374, and R 3 =.982. So =2andc 3 =0andc 4 =0. Thenc and c 2 are computed usng the formula for case B. Table. Optmal Investments for U γ Utlty Functons γ = γ =0.0 γ =0.9 c c α =0.0 c c c c c α =0.5 c c c c c α =.0 c c c Snce a greater value of γ ndcates a smaller aversonto rsk, as γ ncreases the nvestor nvests more of hs wealth and a greater proporton of hs nvestment s on outcomes wth 7

8 the hgher expected return. Also as α ncreases the nvestor chooses to nvest less of hs wealth snce more s at rsk. If m z < /α, the nvestor wll nvest all of hs wealth snce n ths case k = m and Q m = 0. It s easy to see that f mn p z >α,then m z < /α. Notce that n the example, snce mn p z =0.6, the nvestor wll nvest all of hs wealth for any value of α<0.6 regardless of how rsk averse the nvestor s. Whle the above condton s suffcent for the nvestor to nvest all hs wealth, t s not necessary. The condton n Theorem whch separates Case A from Case B s a necessary and suffcent condton for the nvestor to nvest all of hs wealth. The optmal b for the utlty functon W θ and the same values of the p and z as above may be found n Table 2 for θ = and for varous value of r. Forr max p z =.4, nothng s nvested (.e. all c = 0. For arbtrary θ>0, the optmal b can easly be found from ths by dvdng the results by θ. Note that these nvestments are n absolute dollar terms and the amount nvested s ndependent of the nvestor s fortune. Table 2. Optmal Investments for W θ Utlty Functons r r b b b b r 0 =/ 4 z =0.9438, θ = 4. Proofs of Theorems and 2. In ths secton we gve the proofs of the two theorems presented n Secton 2. The proofs nvolve verfyng the condtons of the Karush-Kuhn-Tucker Theorem whch gves suffcent condtons under whch a pont s a local maxmum of a nonlnear functon subect to equalty and nequalty constrants. Snce the functon consdered here s convex, a local maxmum s also a global maxmum. 4. Proof of Theorem. Let m be defned as the largest nteger m m such that α m z <, 0 m m. Then Q has the form α( Q = p for =0,,...,m α z = for = m +,...,m. The proof of Theorem s facltated by the use of two lemmas 8

9 Lemma. Wth the varables ordered as n (5, there s a unque k, 0 k m,such that a. Q <Q for =,...,k and Q Q for = k +,...,m, b. Q <p z for =,...,k and Q p z for = k +,...,m, c. Ifk<m,thenp k+ z k+ Q k. Proof. Easy algebra gves the equatons for =,...,m Q Q = α(q p z z ( α z (0 and α(q p z Q Q = z ( α ( z If Q <Q, then from (8 and (5 Q <p z p z,sothatfrom(9wth replaced by, Q <Q 2. Ths gves statements a and b for =,...,m.therest of a and b follows easly from (9 and the defnton of Q for >m. To show statement c, suppose that p k+ z k+ >Q k.then α k+ z =( α k z αz k+ = α( k p αz k+ Q k > α( k p p k+ z k+ α z k+ 0. Hence m >k, and snce (8 then holds for = k +, we have Q k+ <Q k, a contradcton. Lemma 2. Wth the varables ordered as n (5, there exsts a unque nteger, m, such that a. R >R for =,..., and R R for = +,...,m. b. (( αr γ <p z for and (( αr γ p z for >. c. If <m,thenp k +z k + (( αr γ. d. If(( αr k γ Q k,wherek s as n Lemma, then k. Proof. Easy algebra gves the equatons for =,...,m, R R = ( αr (p z /( γ z ( + ( α z = ( αr (p z /( γ z ( + ( α z 9

10 From these equatons, statements a, b, and c follow as n the proof of Lemma wthout the complcaton of m. Futhermore, cannot be zero snce R 0 =0andR > 0.Toprove statement d, assume (( αr k γ Q k and consder two cases. If (( αr k γ p k z k,then k from parts b of Lemmas and 2. If (( αr k γ <p k z k,then (( αr k γ Q k p k+ z k+, whch mples k = by b. Proof of Theorem. Snce c 0 = m c,φ γ (c may be wrtten Φ γ (c = { γ m = p ( α m = c + c z γ γ for γ 0 m = p log( α m = c + c z for γ =0, Where we take c =(c,c 2,...,c m as the varable. Snce the functons Φ γ (c areconcave over ther convex doman, c,...,c n 0, m c, there exsts a maxmum that satsfes the condtons of the Karush-Kuhn-Tucker (KKT Theorem. From ths, t s suffcent to show: ( c 0, for =,...,m. (2 m = c. (3 There exsts a number λ 0such that for =,...,m, Φ γ (c λ, wth c c equalty f c > 0,andwhereλ =0f m c <. The partal dervatves of Φ γ may be wrtten where c Φ γ (c =p z ( α m = c + c z γ αµ(c, µ(c = m = p ( α m = c + c z γ Suppose the subscrpts are ordered as n (5 and that k s chosen as n Lemma. Part A: (( αr k γ <Q k.inthscase,q k and hence α are postve. Frst we show that c 0 > 0, or equvalently that ( αt > 0. Ths s trval f k = 0(snce then T =0,andfk>0, T = Q /( γ k (p z γ /( γ < (( αr k (p z γ /( γ z z =/( α from the defnton of R k. From ths, we can see that the frst KKT condton s satsfed snce /( + T > /( + = α>0and for =,...,k, α α p z p k z k >Q k 0

11 from Lemma. The second KKT condton follows from the computaton m c =( α( c 0 [ Q /( γ k k (p z γ /( γ k ] z = [ αt ] T = c +αt 0 <. Ths mples that n the thrd KKT condton we must have λ = 0. Ths condton breaks nto two parts Φ γ (c =0for =,...,k, (2 c c and Φ γ (c 0for = k +,...,m. (3 c c Snce for =,...,k, p z ( α k c + c z γ = p z ( α( c 0 +c z γ =( α( c 0 γ p z (p z /Q k =( α( c 0 γ Q k and µ(c = p ( α = =( α( c 0 γ Q k c + c z γ + = =( α( c 0 γ Q k /α, z =k+ p ( α +( α( c 0 γ c γ =k+ p we see that (2 follows. For = k +,...,m, Φ γ (c = p z ( α( c c c 0 γ αµ(c =( α( c 0 γ (p z Q k ( α( c 0 γ (p k+ z k+ Q k snce the p z are nonncreasng. Now (3 follows from Lemma c. Part B: (( αr k γ Q k.ifα = n ths case, then R k s clearly ncreasng, whch mples that = m. Let be chosen as n Lemma 2, whch mples here k. Now snce for =,...,, [ c = (p z /( γ ( + ( α ] z z k (p z γ ( α, /( γ

12 we have c > 0for =,...,k from the defnton of. Thus c 0for all and the frst KKT condton s proved. The second KKT condton follows from the computaton = k c =(+( α ( α z =. z The thrd KKT condton agan breaks nto two peces. For some λ 0, Φ γ (c c c = λ for =,...,, (4 and Φ γ (c c c for = k +,...,m. (5 For =,...,, Φ γ (c = p z ( α + c c c z γ αµ(c, where for α< m µ(c = p ( α + c z γ +( α γ p. = For α =, the last term does not appear because = m, so the formula s vald n ths case also. Now for =,...,, so that and hence + p z ( α + c z γ = p z (z (p z γ /( γ /R γ = R γ, Φ γ (c c µ(c =R γ c = R γ z +( α γ ( p, ( α z ( α γ α( p =( α γ ( α z [(( αr γ Q ] s a constant ndependent of, calltλ. Ths gves (4. Fnally, note that for = +,...,m, Φ γ (c = p z ( α γ αµ(c c c = p z ( α γ R γ λ 2 + λ

13 snce p z s nonncreasng n and p +z + (( αr γ from Lemma 2c. Ths gves (5 and completes the proof. Proof of Theorem2. If r m z <, and b = Nz for =,...,m,then Φ(b =exp{ Nθ( r z } 0as N. Assume now that r m z. Snce 2 η Φ(ηb +( 2 ηb 0for b 0, b 0,and0 η, Φ s convex over ts doman. Therefore, t s suffcent to show that b satsfes the KKT condtons. Ths amounts to showng (a b 0, for =,...,m. (b Φ(b 0wth equalty f b b b > 0, for =,...,m. Suppose the subscrpts are ordered as n (5 and that k s chosen as n the statement of the theorem. Note that k<msnce r m z. The nequaltes (a are automatcally satsfed for = k +,...,m.for =,...,k, b s postve provded p z ( r z >r( p. But p z p k z k for k, so (a follows from the defnton of k. To check condton (b, note that for =,...,m, Φ(b =θ exp{θr m b b }(r m p e θb z p z e θb z. We must show that ths, evaluated at b, s zero for =,...,k, and s nonnegatve for = k +,...,m.for k, Hence, r = exp{ θb z } = p e θb z = r = r( k np p z ( r k z. r( k p z ( r k z + r =k+ = r( ( k k r p z r + k z = r( k p r, k z 3 p

14 so that Φ(b =0,for =,...,k.for = k +,...,m, b b Φ(b b b 0,f or equvalently f r = p e θb z p z = r( k p r k z p z ( r k By the defnton of k, wemusthave whch reduces to z r( k p. p z 0, p k+ z k+ ( r k+ z r( k+ p p k+ z k+ ( r k z Snce the p z are nonncreasng,, we must have for = k +,...,m. p z ( r k z r( k p. r( k p References. [] K. J. Arrow (97. Essays n the Theory of Rsk Bearng. Markham Publshng Co., Chcago. [2] J. F. Baldwn (972. Gamblng and curve fttng problems n dynamc programmng. J. Opt. Theory Appl [3] R. Bellman and R. Kalaba (957. Dynamc programmng and statstcal communcaton theory. Proc. Nat. Acad. Sc. 43, [4] L. Breman (96. Optmal gamblng systems for favorable games. Proc. Fourth Berkeley Symposum on Probablty and Mathematcal Statstcs, [5] T. S. Ferguson (965. Bettng systems whch mnmze the probablty of run. J. SIAM, [6] T. S. Ferguson and C. Z. Glsten (985. A General nvestment Model. Preprnt at tom/papers/unpublshed/zach.pdf [7] J. L. Kelly, Jr. (956. A new nterpretaton of nformaton rate. Bell System Techncal J [8] R. E. Murphy (965. Adaptve Processes n Economc Systems, Academc Press, New York. [9] J. W. Pratt (964. Rsk averson n the small and n the large. Econometrca, [0] M. Rubensten (974. Aggregaton theorems for securtes markets. J. Fnancal Econ