Leveraged Stock Portfolios over Long Holding Periods: A Continuous Time Model. Dale L. Domian, Marie D. Racine, and Craig A.

Transcription

1 Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Dale L. Domian, Marie D. Racine, and Craig A. Wilson Deparmen of Finance and Managemen Science College of Commerce Universiy of Saskachewan Saskaoon, SK S7N 5A7 Canada domian@commerce.usask.ca; (306) racine@commerce.usask.ca; (306) cwilson@commerce.usask.ca; (306) (306) fax May 2003

2 Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Absrac We use a coninuous ime model o derive reurn and wealh disribuions for leveraged porfolios over long holding periods. These heoreical disribuions closely mach empirical disribuions obained from a resampling procedure. The expeced annualized reurn is a concave funcion of he degree of leverage. Wih hisorical parameer values, he funcion is maximized a 241% sock, borrowing an amoun equal o 141% of ne wealh. This maximal sock proporion is considerably reduced if he borrowing rae is higher han he hisorical lending rae.

3 Leveraged Sock Porfolios over Long Holding Periods: A Coninuous Time Model Asse pricing models ypically allow he use of leverage. For example, in he diagram illusraing he radiional Sharpe-Linner CAPM, he capial marke line is angen o he efficien fronier a he marke porfolio. Poins beyond he angency are achieved by borrowing a he risk-free rae. The diagram can be easily modified o illusrae differenial borrowing and lending raes. However, few academic papers examine he effecs of using leverage. Numerous sudies in he asse allocaion lieraure consider mixes beween socks and bonds (or Treasury bills), wih he mos exreme case being a 100% sock porfolio. These sudies do no give any insighs ino he effecs of using leverage. In a sudy from he praciioner-oriened lieraure, Ferguson (1994) describes how leverage magnifies he volailiy of sock reurns and leads o a posiive probabiliy of bankrupcy. He demonsraes ha long-run reurns decline as leverage increases, even when bankrupcy is no possible. These reurn declines are confirmed in a recen empirical sudy by Domian and Racine (2002). We use a coninuous ime model o derive reurn disribuions for leveraged porfolios over long holding periods. Ending wealh disribuions are easily obained from he reurn disribuions. We compare he heoreical disribuions o empirical disribuions obained from a resampling procedure. The paper begins wih a derivaion of he heoreical disribuions followed by he developmen of he empirical disribuions. The resuls secion includes pracical implicaions for

4 2 invesors while he conclusion summarizes he close associaion found beween he wo disribuions. Theoreical Disribuions from a Coninuous Time Model We begin wih some marke assumpions: 1. We have a small invesor ha canno influence he markes for he sock or he borrowing rae and does no face ransacion fees or any oher marke fricions. 2. The sock is infiniely divisible (for he ownership of fracional shares), and does no pay dividends The price or value per uni of he sock, S, has log-normal dynamics (i.e. in coninuous ime i follows a geomeric Brownian moion). 2 Wihou affecing he resuls, we could suppose ha his price or value has been adjused for inflaion, provided we also adjus he borrowing rae. 4. The invesor faces a consan cos of borrowing i does no change over ime, nor does i depend on he degree of leverage. This rae can also be adjused for inflaion as required in he previous assumpion. We also assume ha he ineres is paid a he erminaion of he invesmen, i.e. in coninuous ime a uni of deb, B, grows exponenially a he coninuously compounded rae, r. These assumpions allow he following characerisaion of he economy in coninuous ime as described by Samuelson (1965) and employed by Black and Scholes (1973) and Meron (1973). The sock follows dynamics described by he sochasic differenial equaion (SDE) ds = µ S d + σs dw, (1) 1 The assumpion of no dividends can be relaxed by assuming ha hey are reinvesed in he sock so ha he sock price is cum dividend and remains coninuous. 2 This is characerized by an efficien marke wih consan mean and variance of reurns (see Jarrow and Turnbull 2000, for insance).

5 3 where µ is he insananeous expeced reurn per uni ime, σ is he reurn volailiy per uni ime, and W is a sandard Brownian moion. 3 I is well known (see, for insance, Ellio and Kopp, 1999), ha his SDE has he soluion S 2 = S exp{( µ σ / 2) + σw }. (2) 0 Since W has a normal disribuion wih mean 0 and variance, S has log-normal disribuion wih mean (condiional on knowing he curren price), E S S ] = S exp( µ ). (3) [ 0 0 The uni of deb has dynamics described by he ordinary differenial equaion (ODE) db = rbd, (4) where r is he coninuously compounded borrowing rae. If we suppose ha iniially, a ime 0, he value of he deb is $1, hen he soluion o his ODE is B = exp(r). (5) We suppose ha ime,, has unis of years, so ha he parameers µ, σ, and r, are in annual erms. This gives us a mahemaical descripion of he wo asses in his simple economy. The nex sep is o describe a porfolio, or more correcly, a dynamic invesmen sraegy in his world. We do his as a vecor ha is allowed o change hrough ime indicaing he number of unis of deb invesed or borrowed and he number of socks held, (H b, H s ). A negaive number of unis signifies a shor posiion, and we are paricularly ineresed in he case where H b is negaive and H s is posiive. As his vecor describes an invesmen rule, i is 3 We assume he exisence of an underlying probabiliy space, (Ω, F, P), large enough o suppor he Brownian moion, W.

6 4 imperaive ha i doesn rely on or make use of informaion ha is no known a ime. 4 The value, V, of he invesmen a any ime is b s V = H B + H S. (6) We impose wo resricions on he invesmen sraegy, he firs being sandard in his ype of problem, and he second enabling us o explore he effecs of leverage in a sysemaic way: 1. The invesmen sraegy is self-financing, so ha no funds are added or removed via income or consumpion. This means ha changes in porfolio value come only from changes in asse prices, and he mahemaical form of his consrain is 2. The porfolio has a consan sock proporion, c: b s dv = H db + H ds. (7) H s V S = c. (8) Since we are ineresed in he effecs of leverage, we usually ake c > 1, so ha more han 100% of he invesor s wealh is invesed in he sock and ha porion above 100% is financed by a margin loan. However, if he borrowing and lending rae are he same and here are no resricions on shor sales or margin raios, hen c can be any real number. I is imporan o noice ha in order o mainain a consan sock proporion in a coninuous ime seing we require he invesor o coninuously rebalance her porfolio hrough rading and adjusing deb. Condiion 2 implies he dual condiion ha he porfolio has a consan deb proporion: 4 This means ha we require he sochasic processes H b and H s o be adaped o he filraion generaed by S. We also require hem o saisfy some echnical condiions so ha he SDE defined below in condiion 1 is well defined. These are 0 H b u du < and 0 2 s H u du < wih probabiliy 1, for any.

7 5 H b V B = 1 c. (9) Solving for H b and H s give H b (1 c) V s cv =, and H =. (10) B S Subsiuing his ino condiion 1 gives and subsiuing he dynamics for B and S gives (1 c) V cv dv = db + ds, (11) B S dv (1 c) V cv cv = rb d + µ Sd + σs dw, (12) B S S or simplifying dv = r + ( µ r) c) V d + σcv dw. (13) ( As above, he soluion o his SDE is V 2 2 = V exp{( r + ( µ r) c σ c / 2) + σcw }, (14) 0 which has log-normal disribuion wih mean 5 E V V ] = V exp{( r + ( µ r) c) }. (15) [ 0 0 From his we can see ha he expeced wealh of an invesor is increasing in boh he amoun of leverage, c, and he invesmen horizon,, provided he risk premium µ r is posiive. This indicaes ha a risk neural invesor would opimally choose c as large as possible, regardless of her invesmen horizon. Bu how should he invesmen horizon affec he leverage 5 Wih a soluion o V we can explicily wrie H s and show ha i also has log-normal dynamics; i is hen easy o show ha he condiions of foonoe 3 are saisfied. The same is rue for H b.

8 6 choice of a ypical risk-conscious invesor? To examine he issue we firs look a he annualized reurn, R, from he invesmen sraegy, R 1 V 1 2 σc log r ( r) c c 2 W. = + µ σ + V (16) 0 2 From his we see ha he annualized reurn has a normal disribuion wih mean 1 E[ R ] r ( r) c 2 c 2 = + µ σ, (17) 2 and variance var[ R 2 2 σ c ] =. (18) Noe ha his is a coninuously compounded rae of reurn so i is sensible o allow he possibiliy of he reurn o be less han 1 wihou violaing a limied liabiliy consrain. Using he expressions for he mean and variance, Equaions (17) and (18), leads o he following densiy funcion of he annualized reurn: 2 1 ( ) exp 2 2 f x = ( ) for < < x r µ r c + σ c x (19) πσc 2 2 σ c Graphs of his funcion are discussed in he Resuls secion below. We observe several imporan poins: 1. The mean of he annualized reurn does no depend on he invesmen horizon, and he variance decreases wih invesmen horizon on he order of Furhermore, wih any fixed degree of leverage, he annualized reurn converges wih probabiliy 1, and in mean squared o is expeced value as he invesmen horizon increases o infiniy. This is someimes used (incorrecly) as an argumen for advocaing

9 7 people wih longer invesmen horizons inves more aggressively. The fallacy of his sraegy is discussed laer. 3. For any fixed invesmen horizon, he variance of he annualized reurn is increasing wih he degree of leverage, which is no surprising. In paricular i increases on he order of c 2, so variance is acually increasing a an increasing rae. 4. Perhaps mos imporan is he observaion ha he expeced annualized reurns is a concave funcion of he degree of leverage, c, aking a maximum a µ r c * =. (20) 2 σ This indicaes ha he requiremen of mainaining a given margin raio has imporan implicaions on he efficiency of highly leveraged porfolios. Indeed, a porfolio wih a sock proporion greaer han c* is dominaed in he Markowiz sense by he c* porfolio, which has higher expeced reurn and lower reurn variance. Empirical Disribuions In he previous secion he heoreical reurn disribuion and he maximum degree of leverage were derived. We now urn our aenion o he empirical esimaion of leveraged reurn disribuions for 5- and 20-year holding periods. This poses some mehodological problems because here are very few independen observaions of long holding periods. For example, sixy years of daa conain jus hree independen 20-year holding periods. Buler and Domian (1991) overcome his difficuly wih a resampling approach o esimae long-run disribuions from monhly sock reurns. Domian and Racine (2002) use his approach for leveraged porfolios of US socks. We use wo monhly series from he CFMRC daabase, value-weighed sock index reurns and 91-day Governmen of Canada Treasury bill

10 8 reurns. These series conain 623 values over he period February 1950 hrough December We conver hese o real reurns using inflaion daa from CANSIM. The real sock reurns have a monhly mean of 6447 and a sandard deviaion of 4895, while he real Treasury bill reurns have a 1636 monhly mean and a 4619 sandard deviaion. These correspond o coninuously compounded annual reurns of 7.71% on socks and 1.96% on Treasury bills, and an annual sock sandard deviaion of 15.44%. For a 5-year holding period, he resampling procedure is implemened as follows: 1. Randomly selec one of he 623 monhs. Record he observed real sock and Treasury bill reurns for his monh. 2. Compue he porfolio reurns for six asse allocaions ranging from 50% sock up o 300% sock. For sock proporions below 100%, he remainder of he allocaion is in Treasury bills. Sock proporions exceeding 100% are achieved wih funds borrowed a he Treasury bill rae. Noe ha he curren 30% margin requiremen on mos large socks would allow a Canadian invesor o reach a 333% sock proporion. 3. Repea he previous seps 60 imes wih replacemen and compound he monhly reurns o consruc one represenaive 5-year holding period reurn for each asse allocaion. 4. Perform he enire procedure 100,000 imes o generae 5-year holding period reurn disribuions from he observed hisory of real monhly reurns. For a 20-year holding period, he firs and second seps are repeaed 240 imes. The resuling empirical disribuions are compared in he nex secion o he heoreical disribuions.

11 9 Resuls Our graphs begin wih he heoreical densiy funcions f(x) of he annualized reurn, as given in Equaion (19). Hisorical parameer values, as presened in he previous secion, are used: µ = , r = , and σ = Figure 1 uses an invesmen horizon = 5 years and six differen sock proporions. The densiy funcion f becomes flaer and more spread ou as he sock proporion is increased from he lowes value c = 50% (a porfolio wih half in sock and half in Treasury bills) o he highes value c = 300%. Figure 2 shows he relaionship beween expeced reurn and he sock proporion. The figure includes all six of he densiy funcions ha are displayed separaely in Figure 1. The darker line races he pah of expeced reurn as he sock proporion is varied coninuously from 50% o 300%. Noe ha only he horizonal axis is relevan for ha darker line. Iniially expeced reurn rises as he sock proporion is increased. Afer reaching is maximum a 241% sock (c* from Equaion 20), expeced reurn declines. This figure could also be inerpreed as a relaionship beween expeced reurn and variance, since higher sock proporions have larger variances. Figure 3 compares he heoreical disribuions o he empirical disribuions obained by he resampling procedure. The heoreical disribuions of Figure 1 are shown wih darker lines. The shaded regions show he empirical disribuions. Noe ha he scale has been adjused o reflec discree compounding used o obain he empirical disribuions. I is apparen ha he empirical disribuions have he same shape as he heoreical, alhough wih a sligh lefward shif a higher amouns of leverage. Table 1 repors he firs four momens of he disribuions. Any discrepancy beween he momens of a given heoreical and empirical disribuion inensifies as he sock proporion increases. This is paricularly rue for skewness and kurosis.

12 10 Resuls for he 20-year holding period are displayed in Figures 4, 5, and 6. These densiies are more concenraed han for he 5-year horizon. Neverheless, expeced reurn as a funcion of he sock proporion follows exacly he same paern as for he shorer holding period. Table 2 repors he momens of he heoreical and empirical reurn disribuions. The resuls are similar o he findings for he 5-year holding period. In paricular, he difference beween he heoreical and empirical hird and fourh momens grows as he sock proporion increases. The expeced reurn as a funcion of he sock proporion is independen of he holding period. Therefore he reurn-maximizing sock proporion is always 241%. As Figure 7 illusraes, expeced reurn iniially increases bu declines for sock proporions above 241%. This figure begins wih a 0% sock allocaion a he lef edge, and exends o sock proporions well above 300%. Alhough margin requiremens preclude such excessive leverage, i is neverheless ineresing o see he dramaic decline in expeced reurn. Even when a modes amoun of leverage enhances he expeced reurn, here is sill a rade-off wih risk. This can be illusraed on graphs ploing expeced reurn and sandard deviaion, analogous o he capial marke line in he radiional CAPM. Figure 8 presens hese relaionships for he 5-year and 20-year holding periods wih sock proporions ranging from 50% o 300%. As noed in Figure 7, annualized reurn does no depend on he holding period, so a 50% sock porfolio has a 4.54% expeced reurn over boh 5 and 20 years. However, he sandard deviaion declines from 3.45% over 5 years o 1.73% over 20 years. The fourfold increase in he holding period reduces he sandard deviaion by half for any sock proporion. Thus, he sandard deviaion of he 300% sock porfolio falls from 20.72% o 10.36%, wih he expeced reurn unchanged a 8.48%.

13 11 As we remarked earlier, annualized reurns can be misleading when used as a crierion relaing invesmen horizon and aggressiveness of invesmen sraegy (here represened by he degree of leverage, c). This is bes illusraed by examining he disribuion of wealh direcly. The densiy funcion of porfolio value, wih an iniial invesmen of $1, is ( ln( y) { r + ( µ r) c σ c / 2} ) for 0 < <. 1 1 g ( y) = exp y (21) 2 2 2πσcy σ c The remaining graphs, Figures 9 o 14, explore he properies of his wealh densiy. Figure 9 shows he wealh densiies over he 5-year holding period for six sock proporions. The wealh disribuions become more highly skewed as he sock proporion is increased. Due o he skewness of he wealh disribuions, we summarize hem using he median. The median of he porfolio value is 2 2 m[ V ] = V exp{( r + ( µ r) c σ c / 2) }. (22) 0 The median is concave in he degree of leverage, c, as is he expeced annualized reurn. The median also akes is maximum a c*, given by Equaion (20) and repeaed here as Equaion (23). µ r c * =. (23) 2 σ As before, i is independen of he invesmen horizon,. Figure 10 shows he relaionship beween median wealh and sock proporion. The op panel includes he six disribuions illusraed in Figure 9 while he boom panel considers a range of sock proporions from zero o 1000%. The median increases as he sock proporion increases o 241% and hen seadily declines, consisen wih he resuls for he mean reurn. Figure 11 compares he heoreical and empirical disribuions. The heoreical disribuions of Figure 9 are shown wih darker lines. The shaded regions show he empirical disribuions. In he figure, he empirical disribuions appear o be shifed slighly lefward, paricularly a higher

14 12 sock proporions. Table 3, which presens he firs four momens of he heoreical and empirical wealh disribuions, confirms his. Alhough he differences are small, he mean empirical wealh is always less han he mean heoreical wealh. Furher, he disance beween he wo increases along wih he degree of leverage. This is also rue for he skewness and he kurosis. Figures 12, 13 and 14 repea he analysis for he 20-year horizon. The lower panel of Figure 13 shows higher wealh amouns han Figure 10, simply due o he longer holding period. The empirical disribuions in Figure 14 are more seeply peaked for he highes sock proporions. The momens are given in Table 4. The same general paerns ha were observed in all previous comparisons of he heoreical and empirical disribuions remain. The differences beween momens is posiively correlaed wih an increase in leverage and hese differences are mos pronounced for skewness and kurosis. Implicaions for Invesors Our analysis migh appear o sugges ha risk-oleran invesors should embrace he use of leverage. Boh expeced reurn and median ending wealh go up as he sock proporion is raised owards 241%, he reurn-maximizing sock proporion, c*, in our sudy. However, here are some imporan pracical limiaions. The sandard heoreical assumpion of a borrowing rae equal o he lending rae is no saisfied in he real world. Margin borrowing raes are well above Treasury bill raes. Canadian brokerage firms use he charered banks prime rae as he benchmark for margin loans. Invesors are ypically charged a borrowing rae of 1% above prime. Since he prime rae iself is normally abou 1% above Treasury bills, his implies a margin borrowing rae ha is 2% above he Treasury bill rae.

15 13 I is easy o deermine he impac of he higher borrowing rae. The derivaive of c* wih respec o r is jus 1/σ 2. Wih our parameer value of σ =.1544, he derivaive is An increase of 2% in he borrowing rae would lower c* by abou 84%, from 241% o 157%. Thus, a small increase in he borrowing rae produces a large reducion in he maximum leverage an invesor should consider. One inerpreaion of c* is ha i is an upper bound for risk-averse invesors. While he mos risk-oleran invesors could conemplae approaching c*, more risk-averse invesors will prefer o choose much lower sock proporions, perhaps well below 100%. Despie he lower expeced reurn, many invesors prefer he lower variance associaed wih unlevered porfolios. Conclusion Our findings provide some useful insighs ino he behaviour of leveraged porfolios. In he single-period CAPM, expeced reurn is an increasing linear funcion of he degree of leverage. Bu in our coninuous ime model, expeced annualized reurn is a concave funcion of leverage. This funcion achieves is maximum a 241% sock wih hisorical parameer values. Empirical disribuions esimaed by resampling wih monhly rebalancing provide a close approximaion o he heoreical disribuions. This is evidence ha he heoreical findings are useful for invesmen decisions. The wealh-maximizing degree of leverage should be viewed as an upper bound for he mos risk-oleran invesors. More risk-averse invesors may prefer o avoid leverage.

16 14 References Black, F., & Scholes, M. (1973). The pricing of opions and corporae liabiliies. Journal of Poliical Economy, 81, Buler, K., & Domian, D. (1991). Risk, diversificaion, and he invesmen horizon. Journal of Porfolio Managemen, 17, Domian, D., & Racine, M. (2002). Wealh and risk from leveraged sock porfolios. Financial Services Review, 11, Ellio, R., & Kopp, P. (1999). Mahemaics of financial markes. New York: Springer-Verlag. Ferguson, R. (1994). The danger of leverage and volailiy. Journal of Invesing, 3, Jarrow, R., & Turnbull, S. (2000). Derivaive securiies, 2 nd ediion. Souh-Wesern. Jorion, P. (2001). Value a Risk: The new benchmark for managing financial risk, 2 nd ediion, New York: McGraw-Hill. Meron, R. (1973). Theory of raional opion pricing. Bell Journal of Economics and Managemen Science, 4, Samuelson, P. (1965). Raional heory of warran prices. Indusrial Managemen Review, 6,

17 15 Figure 1 Theoreical Reurn Densiy, 5-Year Holding Period The probabiliy densiy funcions are for he annualized coninuously compounded real reurn. The probabiliy disribuions are based on he normal disribuion wih parameers calculaed from Canadian real monhly sock and Treasury bill reurns over 1950 o % Sock Proporion -50% -25% 0% 25% 50% 100% Sock Proporion -50% -25% 0% 25% 50% 150% Sock Proporion -50% -25% 0% 25% 50% 200% Sock Proporion -50% -25% 0% 25% 50% 250% Sock Proporion -50% -25% 0% 25% 50% 300% Sock Proporion -50% -25% 0% 25% 50%

18 16 Figure 2 Theoreical Expeced Reurn, 5-Year Holding Period The expeced annualized coninuously compounded real reurn is superimposed upon he reurn densiies o illusrae he relaionship beween expeced reurn and variance for varying degrees of sock proporion. The figure includes all six of he densiy funcions ha are displayed separaely in Figure 1. The darker line races he pah of expeced reurn as he sock proporion is varied coninuously from 50% o 300%. Noe ha only he horizonal axis is relevan for ha darker line. The lower poins are associaed wih flaer disribuions wih higher variances. The sock proporion ranges beween 50% a he op of he graph and 300% a he boom, maximizing expeced reurn a a 241% sock proporion. -50% -25% 0% 25% 50%

19 17 Figure 3 Empirical and Theoreical Reurn Disribuions, 5-Year Holding Period The empirical annualized discreely compounded reurn disribuion hisograms are calculaed from real monhly sock and Treasury bill reurns over 1950 o 2001, as are he parameers for he overlain heoreical reurn densiies. The scale for he reurn densiies has been adjused o reflec he discree compounding used o obain he empirical disribuion. 50% Sock Proporion 100% Sock Proporion -50% -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75% 150% Sock Proporion 200% Sock Proporion -50% -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75% 250% Sock Proporion 300% Sock Proporion -50% -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75%

20 18 Figure 4 Theoreical Reurn Densiy, 20-Year Holding Period The probabiliy densiy funcions are for he annualized coninuously compounded real reurn. The probabiliy disribuions are based on he normal disribuion wih parameers calculaed from real monhly sock and Treasury bill reurns over 1950 o % Sock Proporion % Sock Proporion % -25% 0% 25% 50% -50% -25% 0% 25% 50% 150% Sock Proporion -50% -25% 0% 25% 50% 200% Sock Proporion -50% -25% 0% 25% 50% 250% Sock Proporion -50% -25% 0% 25% 50% 300% Sock Proporion -50% -25% 0% 25% 50%

21 19 Figure 5 Theoreical Expeced Reurn, 20-Year Holding Period The expeced annualized coninuously compounded real reurn is superimposed upon he reurn densiies o illusrae he relaionship beween expeced reurn and variance for varying degrees of sock proporion. The figure includes all six of he densiy funcions ha are displayed separaely in Figure 4. The darker line races he pah of expeced reurn as he sock proporion is varied coninuously from 50% o 300%. Noe ha only he horizonal axis is relevan for ha darker line. The lower poins are associaed wih flaer disribuions wih higher variances. The sock proporion ranges beween 50% a he op of he graph and 300% a he boom, maximizing expeced reurn a a 241% sock proporion % -25% 0% 25% 50%

22 20 Figure 6 Empirical and Theoreical Reurn Disribuions, 20-Year Holding Period The empirical annualized discreely compounded reurn disribuion hisograms are calculaed from real monhly sock and Treasury bill reurns over 1950 o 2001, as are he parameers for he overlain reurn densiies. The scale for he reurn densiies has been adjused o reflec he discree compounding used o obain he empirical disribuion % Sock Proporion % Sock Proporion % -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75% 150% Sock Proporion 200% Sock Proporion -50% -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75% 250% Sock Proporion 300% Sock Proporion -50% -25% 0% 25% 50% 75% -50% -25% 0% 25% 50% 75%

23 21 Figure 7 Expeced Reurn This graph depics he behaviour of he heoreical annualized coninuously compounded reurn for various sock proporions including hose beyond 300%. As before, he maximum is a 241% sock proporion. This resul does no depend on he paricular holding period. 20% 10% Expeced Reurn 0% -10% -20% -30% -40% -50% 0% 100% 200% 300% 400% 500% 600% 700% 800% 900% 1000% Sock Proporion

24 22 Figure 8 Mean versus Sandard Deviaion These graphs depic he heoreical rade-off beween he expeced value and he sandard deviaion of he annualized real reurn on porfolios wih sock proporions ranging from 50% o 300%. 10% 5 Year Holding Period Expeced Reurn 8% 6% 4% 2% 0% 0% 5% 10% 15% 20% 25% Sandard Deviaion 10% 20 Year Holding Period 8% Expeced Reurn 6% 4% 2% 0% 0% 5% 10% 15% 20% 25% Sandard Deviaion

25 23 Figure 9 Theoreical Wealh Densiy, 5-Year Holding Period The probabiliy densiy funcions are for end-of-period real wealh from a $1 iniial invesmen wih a 5-year holding period. The probabiliy disribuions are based on he lognormal disribuion wih parameers calculaed from real monhly sock and Treasury bill reurns over 1950 o The horizonal axes measure ending wealh in dollars. 50% Sock Proporion % Sock Proporion % Sock Proporion 200% Sock Proporion 250% Sock Proporion 300% Sock Proporion

26 24 Figure 10 Theoreical Median Wealh, 5-Year Holding Period The median end-of-period real wealh from a $1 iniial invesmen wih 5-year holding period is superimposed upon he ending wealh densiies o illusrae he relaionship beween hese variables and variance for varying degrees of sock proporion. The median ending wealh firs moves slighly righ, hen lef as he sock proporion increases. The sock proporion ranges beween 50% a he op of he graph and 300% a he boom, maximizing median wealh a a 241% sock proporion. The second graph shows he behaviour of he heoreical median end-of-period wealh wih a 5-year holding period if he sock proporion were allowed o increase beyond 300%. Noe ha his resul does depend on he paricular holding period; for comparison see Figure $6 $4 $2 $- 0% 200% 400% 600% 800% 1000%

27 25 Figure 11 Empirical and Theoreical Wealh Disribuion, 5-Year Holding Period The empirical end-of-period real wealh disribuion hisograms from a $1 iniial invesmen wih a 5-year holding period are calculaed from real monhly sock and Treasury bill reurns over 1950 o 2001, as are he parameers for he overlain wealh densiies. 50% Sock Proporion % Sock Proporion % Sock Proporion 200% Sock Proporion 250% Sock Proporion 300% Sock Proporion

28 26 Figure 12 Theoreical Wealh Densiy, 20-Year Holding Period The probabiliy densiy funcions are for end-of-period real wealh from a $1 iniial invesmen wih a 20-year holding period. The probabiliy disribuions are based on he lognormal disribuion wih parameers calculaed from real monhly sock and Treasury bill reurns over 1950 o The horizonal axes measure ending wealh in dollars. 50% Sock Proporion 100% Sock Proporion 150% Sock Proporion 200% Sock Proporion 250% Sock Proporion 300% Sock Proporion

29 27 Figure 13 Theoreical Median Wealh, 20-Year Holding Period The median end-of-period real wealh from a $1 iniial invesmen wih 20-year holding period is superimposed upon he ending wealh densiies o illusrae he relaionship beween hese variables and variance for varying degrees of sock proporion. The median ending wealh firs moves slighly righ, hen lef as he sock proporion increases. The sock proporion ranges beween 50% a he op of he graph and 300% a he boom, maximizing median wealh a a 241% sock proporion. The second graph shows he behaviour of he heoreical median end-ofperiod wealh wih a 5-year holding period if he sock proporion were allowed o increase beyond 300%. Noe ha his resul does depend on he paricular holding period. $6 $4 $2 $- 0% 200% 400% 600% 800% 1000%

30 28 Figure 14 Empirical and Theoreical Wealh Disribuion, 20-Year Holding Period The empirical end-of-period real wealh disribuion hisograms from a $1 iniial invesmen wih a 20-year holding period are calculaed from real monhly sock and Treasury bill reurns over 1950 o 2001, as are he parameers for he overlain wealh densiies. Noice ha he scale differs from ha of he densiies in Figure 12 so as o illusrae some of he finer deails. 50% Sock Proporion 100% Sock Proporion % Sock Proporion 200% Sock Proporion % Sock Proporion 300% Sock Proporion

31 29 Table 1 Theoreical and Empirical Reurn Disribuions, 5-Year Holding Period Sock Proporion Momen Disribuion 50% 100% 150% 200% 250% 300% Mean Theoreical Empirical Variance Theoreical Empirical Skewness Theoreical Empirical Kurosis Theoreical Empirical

32 30 Table 2 Theoreical and Empirical Reurn Disribuions, 20-Year Holding Period Sock Proporion Momen Disribuion 50% 100% 150% 200% 250% 300% Mean Theoreical Empirical Variance Theoreical Empirical Skewness Theoreical Empirical Kurosis Theoreical Empirical

33 31 Table 3 Theoreical and Empirical Wealh Disribuions, 5-Year Holding Period Sock Proporion Momen Disribuion 50% 100% 150% 200% 250% 300% Mean Theoreical Empirical Variance Theoreical Empirical Skewness Theoreical Empirical Kurosis Theoreical Empirical

34 32 Table 4 Theoreical and Empirical Wealh Disribuions, 20-Year Holding Period Sock Proporion Momen Disribuion 50% 100% 150% 200% 250% 300% Mean Theoreical Empirical Variance Theoreical , ,248 Empirical , ,869 Skewness Theoreical Empirical Kurosis Theoreical ,853 29,585,551 Empirical ,021.72