The Risk and Rewards of Minimizing Shorfall Probabiliy The risk may be worhwhile. Sid Browne 76 SID BROWNE is vice presiden of firmwide risk a Goldman, Sachs and Co. in New York (NY 10005), and a professor of business a Columbia Universiy in New York (NY 1007). * * * IT IS ILLEGAL TO REPRODUCE THIS ARTICLE IN ANY FORMAT * * * Many differen invesmen objecives and crieria have been suggesed for choosing invesmen sraegies. In a saic seing, Markowiz [195] suggess he meanvariance approach. Economic heory more formally posulaes ha an individual invesor would choose an invesmen sraegy o maximize expeced uiliy of wealh and or consumpion. In oher seings, oher crieria migh be more relevan. Raher han maximizing uiliy, invesors in cerain circumsances migh be more concerned abou minimizing he probabiliy of a shorfall, where he shorfall is measured relaive o a arge reurn or a specific invesmen goal. Roy [195] suggess his crierion in a saic (one-period) framework and applies Chebyshev s inequaliy o obain a crierion ha is closely relaed o he mean-variance framework of Markowiz. Invesmen sraegies ha minimize he probabiliy of a shorfall can be more opimisically referred o as probabiliy-maximizing sraegies, in ha hey maximize he probabiliy of reaching he invesmen goal. We consider he performance and implemenaion of such sraegies in a dynamic muliperiod seing. We show ha dynamic probabiliy-maximizing invesmen sraegies have a variey of posiive feaures ha are aracive in a variey of economic seings, alhough here is also subsanial invesmen risk. This laer poin seems no ha well undersood by many praciioners, and an elaboraion of his and oher properies of probabiliy-maximizing objecives is he main THE RISK AND REWARDS OF MINIMIZING SHORTFALL PROBABILITY SUMMER 1999
objecive of his aricle. (See Williams [1997] and Leibowiz, Bader, and Kogelman [1996].) To illusrae our basic idea mos direcly, we adop a simple coninuous-ime model, as in Black- Scholes [1973] or Meron [1971], where asse prices are lognormally disribued, and coninuous rading is allowed. While our resuls hold for any number of asses, here we consider an economy wih jus wo asses: a risky sock or equiy index whose price a ime is denoed by S, and a riskless asse whose price a ime is denoed by B. As in he Black-Scholes model, we assume ha a a fixed ime, hese prices can be wrien as 1 S = S0 exp[( µ σ / ) + σ Z] B = B0 exp( r ) where S 0 and B 0 are he iniial prices, µ and σ are drif and volailiy parameers of he risky sock, Z is a sandard normal variable, and r is he insananeous riskfree rae of ineres. Noe ha he naural logarihm of he annual reurn relaive of he sock has a normal disribuion wih mean µ σ / and variance σ. For illusraive purposes, we firs consider wo differen arge reurns: beaing an all-cash sraegy by 10%, and beaing an all-sock sraegy by 10%. (The choice of he level of 10%, while reasonable, is arbirary and jus for illusraive purposes. Oher levels are considered laer.) The all-cash sraegy is of course he sraegy ha jus buys and holds he riskless asse; similarly, he all-sock sraegy jus buys and holds he risky asse. To illusrae some of he advanages and he possible pifalls associaed wih shorfall probabiliy minimizaion as an objecive, we will compare is performance o he class of consan allocaions (or mix) sraegies, where he porfolio is coninuously dynamically rebalanced so as o always hold a consan percen of wealh in he wo asses. To enable exac calculaions, we use he specific consan allocaion sraegy ha maximizes a logarihmic uiliy funcion. In paricular, if in he Black-Scholes model we ake for illusraive purposes he numerical values µ = 0.15, σ = 0.3, and r = 0.07, hen he log-opimal consan allocaion sraegy always rebalances he porfolio so ha 89% of wealh is held in he risky asse wih he remaining 11% held in he risk-free asse, or cash. (1) () As we will show, i akes niney-eigh years before he probabiliy ha his consan allocaion sraegy beas an all-cash sraegy (by 10%) reaches 90%. Moreover, i will ake 6,50 years for he probabiliy ha he consan mix sraegy will ouperform he sock iself (again by 10%), o reach 90%. For he dynamic probabiliy-maximizing sraegy ha we analyze, we will show ha he 90% confidence or probabiliy limis for beaing he all-cash and he all-sock sraegies by 10% fall o 0.04 years (fifeen days) and.60 years, respecively (see Browne [1997a] for a full mahemaical reamen). While hese numbers for he probabiliy-maximizing sraegy are orders of magniude beer han he resuls of he consan mix sraegy, here is an imporan dimension overlooked here: namely, ha he probabiliy-maximizing sraegy is in general riskier han he consan allocaion sraegy. This can be illusraed mos direcly by noing ha he expeced ime i akes for he consan allocaion sraegy o bea an allcash sraegy by 10% is.7 years, and he expeced ime for he consan mix sraegy o bea an all-sock sraegy is 17 years. Alhough hese numbers are no exacly a heary endorsemen of he consan mix sraegy, hey look quie good when compared wih he resuls for he probabiliy-maximizing sraegy, since he expeced ime for a probabiliy-maximizing sraegy o bea an all-cash (or all-sock) sraegy is infinie. This occurs because he dynamic probabiliy-maximizing sraegy is essenially equivalen o he replicaing or hedging sraegy associaed wih a European digial opion on he sock, and hus enails he risk of no meeing he arge reurn, or invesmen goal, by he associaed ime (see Browne [1997a]). 3 This would occur if he sock finishes ou of he money, and hence he digial opion becomes worhless. We also show ha here is anoher dimension of risk associaed wih probabiliy maximizaion; here is a significan amoun of leveraging necessary o underake a probabiliy-maximizing sraegy relaive o oher sraegies. In some cases, probabiliy maximizaion is enirely appropriae, and when used judiciously may even be risk-reducing. For he mos par, his happens mainly when here is no finie deadline, or when he invesor has an exernal source of income ha can miigae he invesmen risk associaed wih a probabiliymaximizing objecive. 4 SUMMER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 77
Even when probabiliy maximizaion does enail more risk, our comparisons above show ha his risk migh be well worh aking, especially given ha he implemenaion of a dynamic probabiliy-maximizing sraegy is compleely equivalen o he saic sraegy ha simply purchases a digial opion. The implicaion of his equivalence is ha an invesor wishing o maximize he probabiliy of achieving a prese arge reurn or invesmen goal by a given ime need only purchase a (European) digial opion on he sock, wih a specific exercise price deermined by he underlying parameers, as we discuss below. This objecive opimaliy propery of opions has imporan consequences for risk managemen. 5 CONSTANT ALLOCATION PORTFOLIO STRATEGIES In a consan allocaion porfolio sraegy, he porfolio holdings are coninuously rebalanced so as o hold a consan percenage of wealh in he risky asse wih he remaining percenage held in he risk-free asse. To use such a sraegy, he invesor needs o sell he risky asse as is price increases and buy i as is price falls. Such porfolio sraegies are known o be opimal for a variey of economic invesmen objecives. 6 For he model reaed here, he wealh associaed wih a consan allocaion porfolio sraegy has a lognormal disribuion, which allows exac probabiliy calculaions o be made. In paricular, if θ is he rebalancing consan, hen he porfolio holdings are coninuously rebalanced so as o hold a consan θ of wealh in he risky asse and he remaining 1 θ in he risk-free asse. The value of he porfolio, or wealh, associaed wih his consan allocaion sraegy a he fixed ime, X (θ), can be wrien as he lognormal random variable X (θ) =X 0 e N1, where N 1 is a normal random variable wih variance θ σ and mean G(θ), where G(θ) is he quadraic funcion of he allocaion fracion θ given by: 7 G( θ) = r+ θ ( µ r) θ σ / The funcion G(θ) plays he role of he coninuous compounding rae a which he wealh obained from using sraegy θ grows. The choice of he rebalancing consan, θ, is deermined in general by he invesor s individual uiliy funcion and risk aversion parameer, and hus is 78 (3) invesor-specific. To ge a numerical value for θ, we use he value ha maximizes he coninuous compounding rae funcion G(θ). This value will be denoed by θ, and is given by The fac ha wealh under any consan mix sraegy has a lognormal disribuion enables some exac performance analysis via probabiliy comparisons. In paricular, for any given exceedence level ε, he probabiliy ha he growh-opimal sraegy, θ, ouθ = µ r σ This value of θ maximizes he logarihmic uiliy funcion, which plays a cenral role in he coninuous-ime financial heory, and minimizes he expeced ime unil a given arge level of wealh is reached. The specific consan allocaion sraegy θ in (4) is ofen referred o as he opimal growh porfolio sraegy. For he numerical values used here, µ = 15, σ = 0.3, and r = 0.07, we obain θ = 0.89. 8 Two oher cases of alernaive consan allocaion sraegies are of ineres as comparaive benchmarks: θ = 0, and θ = 1. In he firs case, wealh is he porfolio associaed wih an all-cash sraegy, and in he second case, wealh is he value of a porfolio fully invesed in he sock iself. The raio of he wealh associaed wih any wo consan allocaion sraegies is again lognormally disribued. The raio of he wealh associaed wih he arbirary consan allocaion sraegy θ o he wealh associaed wih he opimal growh sraegy θ, X (θ)/x (θ ), can be wrien probabilisically, for any fixed, as he lognormal random variable e N, where N is a normal random variable wih mean σ (θ θ) / and variance σ (θ θ). This allows us o calculae some performance measures. 9 Moreover, i can be shown ha as varies, he raio process is a maringale, which has profound significance for he pricing of opions and oher coningen claims (his maringale propery holds in much greaer generaliy for he raio of any porfolio sraegy o he opimal growh sraegy). In fac, he Black-Scholes formula for pricing coningen claims can be expressed solely in erms of he log-opimal wealh process. 10 PROBABILITY CALCULATIONS THE RISK AND REWARDS OF MINIMIZING SHORTFALL PROBABILITY SUMMER 1999 (4)
performs any oher consan allocaion sraegy θ by ε percen, by a fixed ime T, is given explicily by: 11 Φ 1 ln( 1 + ε) M M where Φ denoes he cumulaive disribuion funcion for a sandard normal disribuion, and where he value of M is given by M = σ ( θ θ) T The variable M is he sandard deviaion of he raio of he differen wealhs a he ime T. Observe ha M is a monoonically increasing funcion of he ime horizon T, and herefore increases wihou bound as T increases wihou bound. The probabiliy given in Equaion (5) hus ends o one as T increases wihou bound. Tha is, he probabiliy ha he growh-opimal sraegy ouperforms any oher sraegy θ by ε percen, for any given exceedence level ε, ends o one as he horizon increases wihou bound. This of course is one of he well known long-run opimaliy properies of he growhopimal sraegy. As observed by Rubinsein [1991], however, he long run may be very long indeed. Specifically, observe ha for any given porfolio sraegy θ, any given exceedence level ε, and any fixed given probabiliy level 1 α, he probabiliy funcion in Equaion (5) can be invered in order o obain he associaed ime T, which is given explicily by: 1 q + q + + α α ln( 1 ε) T = σθ ( θ) where q α is he (1 α)-h percenile or quanile of he sandard normal disribuion. Formally, q α is defined as he roo o he equaion Φ(q α )=1 α, or equivalenly, q α = Φ 1 (1 α), where Φ 1 is he inverse of he cumulaive sandard normal disribuion funcion. Thus for example, he niney-fifh percenile is denoed by q 0.05 = 1.645. Exhibi 1 shows he value of T, using an exceedence level of 10% (i.e., ε = 0.1), for various values of α and for he comparaive sraegies of all-cash and allsock. The resuls are somewha unseling. I would (5) (6) (7) EXHIBIT 1 YEARS FOR OPTIMAL-GROWTH TO BEAT COMPETING STRATEGIES BY 10% Probabiliy Time Needed (in years) for θ o Bea θ by 10% 1 α All-Cash (θ = 0) All-Sock (θ =1) 0.900 98 6,50 0.950 158 10,080 0.990 310 19,84 0.999 54 34,70 ake niney-eigh years for he opimal-growh sraegy o have a 90% probabiliy of beaing an all-cash sraegy, while i would ake 6,50 years o be 90% cerain of beaing an all-sock sraegy. The probabiliy-maximizing sraegy ha we provide below will bea hese numbers handily, bu i incurs exra risk. For any exceedence level ε > 0, he expeced ime for he wealh of he opimal-growh sraegy o bea he wealh associaed wih any oher consan allocaion sraegy θ by ε percen is finie, and is given by: ( ) σ θ θ ln( 1 + ε) Evaluaing (8) for ε = 0.1 wih he illusraive numbers used previously gives he resuls in Exhibi. DYNAMIC PROBABILITY- MAXIMIZING STRATEGIES Suppose insead of following a consan allocaion sraegy, he invesor dynamically changes he rebalancing level so as o minimize he probabiliy of a shorfall by a given ime. Equivalenly, he invesor wans o maximize he probabiliy of reaching a given invesmen goal, say, b, by he given ime T. The opi- EXHIBIT EXPECTED TIME FOR OPTIMAL-GROWTH TO BEAT COMPETING STRATEGIES BY 10% Sraegy E (ime o bea by 10%) All-Cash (θ = 0) All-Sock (θ = 1).7 years 17.0 years SUMMER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 79 (8)
mal invesmen sraegy for his problem in he simple Black-Scholes model considered here is equivalen o he replicaing, or hedging, sraegy of a European digial call opion on he risky sock iself, wih payoff b and some specific srike price. This equivalence follows from a direc economic argumen based on pricing and valuaion in a complee marke. Specifically, since he objecive of he invesor is only o reach he goal b, a probabiliy maximizer would never choose a sraegy for which erminal wealh would exceed b, since doing so would increase he cos of achieving ha erminal wealh wihou increasing he associaed rewards. Likewise, a probabiliy-maximizing invesor would never choose a sraegy for which erminal wealh would give a value sricly beween 0 and b, since he invesor would gain resources by seing erminal wealh equal o 0 in ha sae. Thus, he mos efficien sraegy for maximizing he probabiliy of reaching he saed goal b is o purchase, or dynamically replicae, a digial opion. The srike price of his opimal digial opion is now obained by equaing he invesor s iniial wealh wih he Black-Scholes value of a digial call opion, and hen solving for he srike price. (See Browne [1997a] for a complee analysis of relaed probabiliy-maximizing sraegies for more complicaed models.) Specifically, he value a ime of a digial opion on a sock wih price S, payoff b, and srike price K is, from he sandard Black-Scholes equaion, given by C(, S ) where 80 C(, S ) = be r(t ) Φ(d ) (9) where d is given by d (10) Seing he iniial price C(0, S 0 ) equal o iniial wealh X 0 gives he appropriae value for he srike price K for he probabiliy-maximizing objecive. 14 The probabiliy-maximizing sraegy is hus compleely equivalen o he dynamic replicaing sraegy for his paricular digial opion. This is he sraegy ha is coninuously rebalanced so as o hold: 15 σ ln( S / K) + ( r σ / )( T ) = σ T 1 φν ( ) T Φ( ν) (11) percen of wealh in he risky sock a ime wih he remainder held in he risk-free asse, where φ and Φ denoe he densiy and he cumulaive disribuion funcion of a sandard normal variae, and where he quaniy ν is he value of d for his paricular probabiliy-maximizing srike price. I urns ou ha ν is he quanile of he sandard normal disribuion associaed wih he percenage of he goal reached by ime. Explicily, ν = Φ 1 (z ), where z, he percenage of he goal reached, is given by z = x/[be r(t ) ], where x is he wealh level a ime. Equaion (11) represens he dynamic probabiliy-maximizing sraegy as he produc of wo separae posiive effecs. The firs componen is a purely imedependen effec 1 σ ( T ) which increases (wihou bound) as he ime unil he deadline, T, decreases. The second quaniy is a scalar deermined solely by he percenage of he (effecive) invesmen goal currenly achieved, φ(ν)/φ(ν), which decreases as he raio of curren wealh o he effecive goal, z, increases from 0 o 1. Since i is equivalen o a digial opion, he probabiliy-maximizing sraegy incurs he risk ha wih some probabiliy he invesmen goal will no be reached, and he iniial wealh X 0 will have been compleely los by ime T. This possibiliy mus be conrased wih he fac ha under a consan allocaion sraegy, a complee loss of he iniial sake is no possible in finie ime in he coninuous-ime model. Neverheless, he risk of a complee loss can be small enough over a paricular horizon ha he approach may be an aracive invesmen sraegy. (We analyze he amoun of leverage necessary o underake such a sraegy laer.) Maximizing he Probabiliy of Beaing Anoher Sraegy The probabiliy-maximizing sraegy can be generalized o rea a comparaive raher han consan level b invesmen goal. We call his case acive probabiliy maximizaion, where performance is measured relaive o anoher benchmark porfolio. Suppose he invesor wans o bea a given compeing porfolio sraegy by a given exceedence level ε (e.g., 10%). The compeing porfolio sraegy is THE RISK AND REWARDS OF MINIMIZING SHORTFALL PROBABILITY SUMMER 1999
assumed o allocae θ % o he risky sock a ime, wih θ known a each ime. For his problem, he corresponding acive probabiliy-maximizing sraegy is no longer equivalen o a simple digial opion, alhough he generalizaion is no hard o work ou. 16 If he compeing sraegy is a consan allocaion sraegy, wih allocaion consan θ, hen i urns ou ha he bes you can do agains he given sraegy θ (wih respec o maximizing he probabiliy of beaing i by ε%) is given by Φ Φ 1 + 1 + ε (1) where Φ 1 denoes he appropriae quanile of he sandard normal disribuion. For a given sraegy θ and a given exceedence level ε, we may now se he maximal probabiliy of (1) equal o a given probabiliy level, say, 1 α, and hen solve for he corresponding ime, T, ha is needed o achieve he given probabiliy. The resul is given explicily by: 17 Φ T = 1 ( 1 α) Φ ( 1/[ 1+ ε]) σθ ( θ) 1 1 σ ( θ θ) T (13) where again he appropriae quaniles of he sandard normal disribuion are given by Φ 1. We may now evaluae his for various values of θ, ε, and α. By aking θ = 0 we ge he relevan ime o bea an all-cash sraegy, and by aking θ = 1 we ge he relevan ime o bea an all-sock sraegy. Exhibi 3 presens resuls for an exceedence level EXHIBIT 3 YEARS FOR PROBABILITY-MAXIMIZING STRATEGY TO BEAT COMPETING STRATEGIES BY 10% Probabiliy Time Needed (in years) o Bea θ by 10% 1 α All-Cash (θ = 0) All-Sock (θ =1) 0.900 0.04 (15 days).6 0.950 1.35 86 0.990 14.00 884 0.999 43.00,77 of 10% wih he numerical values µ = 0.15, r = 0.07, and σ = 0.3. For ε = 0.1, 1/(1 + ε) = 1/1.1 = 0.91, and Φ 1 (0.91) = 1. As Exhibi 3 shows, he acive probabiliy-maximizing sraegy gives resuls ha are orders of magniude beer han he comparaive resuls for he consan allocaion (opimal-growh) sraegy analyzed. The downside of course, is ha under his sraegy, he erminal value of he porfolio a ime T has posiive probabiliy of being 0, as i is essenially an opions sraegy. As Exhibi 3 shows, however, hese probabiliies can be made arbirarily small by increasing he ime horizon. Risk-Taking and Leveraging There is a side o he sory ha Exhibi 3 does no show: he amoun of leverage underaken by a probabiliy maximizer. Leveraging or borrowing akes place when he percenage of wealh invesed in he risky asse exceeds uniy. The consan allocaion sraegy has no enailed leveraging, since θ for our example is less han one. (Of course, if θ > 1, hen, unless oherwise consrained, he invesor mus always leverage.) For he probabiliy-maximizing sraegy, we see from Equaion (11) ha leveraging occurs when φν ( )/ Φ( ν) τ where τ = σ (T ), or he risk-adjused remaining ime. The leveraging necessiaed by a probabiliymaximizing sraegy can be analyzed by looking a he borrowing region, denoed in Exhibi 4 by he curve z (τ). Borrowing is necessiaed wih τ risk-adjused ime remaining only if z(τ) <z (τ), where z(τ) is he acual proporion of he goal aained wih τ risk-adjused ime unis o go, and z (τ) is he calculaed roo o he equaion φ(ν)/φ(ν) = τ. 18 Some selec values of of τ and z (τ) are given in Exhibi 5. As Exhibi 5 shows, if τ = 0.05 risk-adjused ime unis remain unil he deadline, he probabiliymaximizing invesor mus borrow (a he risk-free rae r) unless he invesor is already 88% of he way o he goal. As he remaining risk-adjused ime o go increases, a probabiliy-maximizing invesor needs o borrow only a lower percenages. For example, if here is τ =1 uni of ime lef o go, he invesor will need o borrow unless wealh a ha ime is more han 38% of he way o he invesmen goal. I is imporan o noe ha increasing he risk fac- SUMMER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 81
EXHIBIT 4 z (τ) PLOTTED AGAINST τ EXHIBIT 5 BORROWING REGION or of he sock, σ, has he same effec as increasing he acual ime lef o play, T. Therefore, for a higher risk facor, one would borrow less in he hopes of reaching he invesmen goal laer. CONCLUSIONS 8 τ z (τ) τ z (τ) τ z (τ) 0.001 0.99 0.50 0.56 1.00 0.38 0.050 0.88 0.55 0.54 1.50 0.7 0.100 0.8 0.60 0.51.00 0.19 0.150 0.77 0.65 0.49.50 0.14 0.00 0.73 0.70 0.48 3.00 0.10 0.50 0.70 0.75 0.46 3.50 0.08 0.300 0.67 0.80 0.44 4.00 0.06 0.350 0.64 0.85 0.43 4.50 0.04 0.400 0.61 0.90 0.41 5.00 0.03 0.450 0.58 0.95 0.39 τ = risk-adjused ime o go; z (τ) = criical percenage of he disance o he goal aained wih τ risk-adjused ime unis remaining. Borrowing occurs if he acual z(τ) is less han z (τ). We have provided a comprehensive analysis of he dynamic probabiliy-maximizing sraegy, and compared is performance o ha of a consan allocaion sraegy. While probabiliy maximizaion necessiaes some exra risk, his risk migh be worhwhile when we look a some of he performance disadvanages of he consan allocaion sraegy. We have also shown ha i is quie simple o underake a probabiliy-maximizing sraegy since i is compleely equivalen o he purchase of a digial opion, wih a specific srike price. In many scenarios, such as in corporae risk managemen seings, here is no finie deadline. A manager may raher be ineresed in minimizing he probabiliy of ever going below a given shorfall level. The major problem wih probabiliy maximizaion is ha he payoff funcion is binary valued (1 a he invesmen goal and 0 elsewhere). Therefore, if here is a finie deadline, significan risk-aking occurs near he deadline if wealh is far from he invesmen goal. This is he case when here is no oher source of income available o he invesor. If indeed here is an exernal source of income, his income sream can be used o miigae he invesmen risk associaed wih a probabiliy-maximizing objecive. This conrass wih uiliy maximizaion objecives, where income is used o ake exra risk by borrowing agains fuure earnings. ENDNOTES 1 In Black and Scholes [1973] and Meron [1971], he price of he risky asse, S, is assumed o evolve according o he sochasic differenial equaion ds = µs d + σs dw where W is a sandard Brownian moion, or Wiener process, and he price of he riskless asse, B, is assumed o evolve according o he differenial equaion: db = rb d. (A sandard Brownian moion is a process wih coninuous sample pahs ha has independen incremens. These incremens are normally disribued wih mean zero, and, in paricular, for any, W has a normal disribuion wih mean 0 and variance.) The soluion of he sochasic differenial equaion is given by he geomeric Brownian moion S = S 0 exp[(µ σ /) + σw ] Since W is normally disribued wih mean 0 and variance, for any fixed his is probabilisically equivalen o he lognormal random variable given in Equaion (1). Equaion () is he soluion o he differenial equaion solving for db. See Perold and Sharpe [1988] for performance analysis of his and oher sraegies. 3 Specifically, if S denoes he sock price in he Black- Scholes economy, a digial call opion wih srike price K and payoff b is an opion ha pays $b o he bearer a ime T if and only if he sock price exceeds he srike price a T, i.e., if S T K. Sandard resuls on opion pricing show ha he (fair) price a ime for his opion is THE RISK AND REWARDS OF MINIMIZING SHORTFALL PROBABILITY SUMMER 1999
C(, S ) = be r(t ) Φ(d ) where d is given by d The associaed hedging, or replicaing, sraegy is he dynamic rading sraegy ha holds C/ S shares of he sock a ime, wih he remaining wealh held in he riskless asse. 4 See, for example, Browne [1995, 1997b] for infinie horizons and Browne [1997a] for he finie-horizon problem wih exernal income. 5 In more general models wih muliple socks, he probabiliy-maximizing sraegy is sill equivalen o a digial opion, bu on a paricular index of he socks, as discussed in Browne [1997a]. 6 In paricular, wih no furher conribuions or wihdrawal of funds, a consan allocaion sraegy maximizes expeced uiliy of erminal wealh for a concave uiliy funcion wih consan relaive risk aversion, as discussed in Meron [1971, 1990] or Hakansson [1970]. 7 In he Black-Scholes model, if θ is he consan fracion of wealh invesed in he risky sock a ime, wih he remainder held in he riskless asse, he oal amoun of money invesed in he sock is θx (θ), and herefore he oal number of shares held in he sock is θx (θ)/s. Similarly, he oal number of shares of he riskless asse held a ime is given by (1 θ)x (θ)/b. The insananeous change in wealh a ime can be wrien as ds db dx( θ) = θx( θ) + ( 1 θ) X( θ) S B When ds and db from endnoe 1 are subsiued ino he equaion, we obain he sochasic differenial equaion dx (θ) = X (θ)[r + θ(µ r)]d + X (θ)σθdw whose soluion is he geomeric Brownian moion given by X (θ) = X 0 exp[g(θ) + θσw ] where X 0 is he iniial wealh level, and W is a sandard Brownian moion. Since W is normally disribued wih mean 0 and variance, for any fixed we can wrie his as X( θ) = X0 exp[ G( θ) + θσ Z] where Z is a sandard normal variable wih mean 0 and variance 1. 8 See Hakansson [1970], Meron [1990, Chaper 6], and Browne [1998] for furher properies and analysis of he opimal growh and oher consan allocaion rules. Meron discusses he imporance of he logarihmic uiliy funcion. 9 Specifically, for any fixed, we have σ X( θ)/ X( θ ) exp ( θ θ ) σ ( θ = θ ) Z where Z is a sandard normal variable. This follows direcly from he las equaion in endnoe 7, since for any fixed, we can wrie he raio as X ( θ) / X ( θ ) = exp [ G( θ ) G( θ)] σ( θ θ) Z 1 ln( S / K) + ( r σ )( T ) = σ T [ ] where G is he funcion defined in Equaion (3). Using Equaion (3) and he definiion of θ given in (4), we can wrie G(θ ) = r + (θ ) σ / and, for an arbirary θ, we can also wrie G(θ) = r + θθ σ θ σ / Thus: G(θ ) G(θ) = (θ ) σ / + θ σ / θθ σ which is equivalen o σ (θ θ) /. 10 Consider a derivaive securiy ha, for a given prespecified payoff funcion H(x), and a given prespecified deadline T, will pay he owner H(S T ) a ime T, where S is he price of he sock a ime. Some sandard examples of payoff funcions are: 1) H(x) = max[0, x K]; ) H(x) = max[0, K x]; and 3) H(x) = b if x K, 0 oherwise. These are he erminal payoff funcions associaed wih 1) a European call opion wih srike price K; ) a European pu opion wih srike price K; and 3) a European digial opion wih payoff b and srike price K. The raional or fair price a ime for his coningen claim can now be expressed, in erms of he log-opimal wealh X (θ ), he sock X (1), and he risk-free asse X (0), as X E HX [ T( 1)] () 0 XT( θ ) where E denoes he condiional expecaion on he relevan hisory a ime. The Black-Scholes pricing formula of his equaion holds in more general models of securiy prices and for much more general (someimes pah-dependen) payoff srucures. For a more complee reamen, see Meron [1990]. 11 The probabiliy ha porfolio sraegy θ beas porfolio sraegy θ is P[X T (θ ) > (1 + ε)x T (θ)] Some direc manipulaion using he firs equaion in endnoe 9 shows ha his probabiliy can be wrien as 1 P[ σθ ( θ) TZ > σ ( θ θ) T+ ln( 1 + ε)] where Z has a sandard normal disribuion wih mean 0 and variance 1, so σθ ( θ) TZ has a normal disribuion wih mean 0 and variance σ (θ θ) T. This probabiliy is equal o he probabiliy in Equaion (5). 1 Specifically, seing he probabiliy in Equaion (5) equal o 1 α gives Φ 1 ln( 1 + ε) M = 1 α M which can be invered o 1 ln( 1 + ε) 1 M = Φ ( 1 α) M Muliplying by M, subracing appropriaely, and using he sandard normal quanile noaion q α = Φ 1 (1 α), he value of M ha achieves he probabiliy of 1 α is given as he posiive roo o he quadraic equaion 1 M qαm ln( 1+ ε) = 0 This posiive roo is given by α α ε M = q + q + ln( 1+ ) When we subsiue for M as i is defined in Equaion (6), we ge σ ( θ θ) T = q + q + ln( 1 + ε) SUMMER 1999 THE JOURNAL OF PORTFOLIO MANAGEMENT 83 α α
and finally when we square boh sides and solve for T, we obain Equaion (7). 13 The expeced ime formula in Equaion (8) follows from he well-known fac ha if Y is he geomeric Brownian moion given by Y = exp[γ + βw ] where W is a sandard Brownian moion, hen for any γ > 0, and any β, he expeced ime for Y o grow 100 λ% (i.e., he ime unil Y =1+λ) is given by 1 ln( 1 + λ) γ From he las equaion in endnoe 7 we observe ha he relevan growh rae γ for a consan allocaion sraegy is he quadraic funcion G(θ) of Equaion (3). I also follows ha for any ε > 0, he expeced ime for he wealh of he opimal-growh sraegy o bea he wealh associaed wih any oher consan allocaion sraegy θ by ε is in fac given by Equaion (8). 14 The relevan srike price (K ) is given by K = S exp ( r σ / ) T σ TΦ ( X e / b) where S 0 denoes he iniial sock price, and X 0 denoes he iniial wealh level. For more general models, he probabiliy-maximizing sraegy is again equivalen o he replicaing sraegy of a European digial call opion, bu now he opion is on he log-opimal wealh for ha model, raher han he sock iself. See Browne [1997a] for he deails. 15 The dynamic sraegy in (11) is explained as follows. Observe ha for a digial opion, we have S Noe furher ha d of (10) evaluaed a he srike price K as in endnoe 14 is given by ν. 16 As Browne [1997a] shows, he dynamic acive probabiliy-maximizing sraegy invess θ S CS CS 1 φ( d) (, )/ (, ) = σ T Φ( d) + 0 1 rt [ 0 ] 1 φ( v ) σ T v ( ) Φ( ) % of is wealh in he risky sock a ime wih he remainder invesed in he riskless asse. The acive probabiliy-maximizing porfolio sraegy decomposes ino wo addiive pars. The firs par mimics he racking porfolio sraegy θ, and he second par hen overinvess in he risky sock according o he regular probabiliymaximizing sraegy. The quaniy ν is given by Φ 1 (z ) where z is defined as he raio of he probabiliy-maximizing wealh o he compeing porfolio sraegy s wealh, divided by 1 + ε; i.e., a ime he quaniy z is given by w /(1 + ε), where w is he raio of he probabiliy-maximizing sraegy o he wealh of he compeing allocaion sraegy θ. The maximal probabiliy associaed wih he acive probabiliy-maximizing sraegy for a compeing consan allocaion sraegy θ is given by Φ Φ 1 w + σ ( θ θ) ( T ) 1 + ε where θ is he opimal-growh sraegy of (3). 17 Specifically, seing Equaion (1) equal o 1 α gives Φ Φ 1 + 1 + ε which can be invered o Solving his las equaion for T gives he resul in Equaion (13). 18 If ν (τ) denoes he unique roo o he Equaion φ(ν)/φ(ν) = τ, hen z (τ) =Φ[ν (τ)]. These roos are decreasing in τ, so borrowing occurs wih τ risk-adjused ime remaining only when z(τ) <z (τ), where z(τ) is he proporion of he goal aained wih τ risk-adjused ime unis o go. REFERENCES Black, F., and M. Scholes. The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy, 81 (1973), pp. 637 659. Browne, S. Opimal Invesmen Policies for a Firm wih a Random Risk Process: Exponenial Uiliy and Minimizing he Probabiliy of Ruin. Mahemaics of Operaions Research, 0 (1995), pp. 937 958.. Reaching Goals by a Deadline: Digial Opions and Coninuous-Time Acive Porfolio Managemen. Proceedings of he 1997 IAFE Annual Conference, 1997a. (To appear in Advances in Applied Probabiliy, 1999.). Survival and Growh wih a Fixed Liabiliy: Opimal Porfolios in Coninuous Time. Mahemaics of Operaions Research, (1997b), pp. 468 493.. The Reurn on Invesmen from Proporional Porfolio Sraegies. Advances in Applied Probabiliy, 30 (1998), pp. 16-38. Hakansson, N.H. Opimal Invesmen and Consumpion Sraegies Under Risk for a Class of Uiliy Funcions. Economerica, 38 (1970), pp. 587 607. Leibowiz, M.L., L.N. Bader, and S. Kogelman. Reurn Targes and Shorfall Risks. Burr Ridge, IL: Irwin, 1996. Markowiz, H.M. Porfolio Selecion. Journal of Finance, 7 (195), pp. 77 91. Meron, R. Coninuous Time Finance. Cambridge: Blackwell, 1990.. Opimum Consumpion and Porfolio Rules in a Coninuous Time Model. Journal of Economic Theory, 3 (1971), pp. 373 413. Perold, A.F., and W.F. Sharpe. Dynamic Sraegies for Asse Allocaion. Financial Analyss Journal, January-February 1988, pp. 16 7. 84 THE RISK AND REWARDS OF MINIMIZING SHORTFALL PROBABILITY SUMMER 1999 Φ σ ( θ θ) T = 1 α 1 1 + σ ( θ θ) T = Φ ( 1 α) 1 + ε 1 1
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