Investors have traditionally equated volatility. Volatility Harvesting: Why Does Diversifying and Rebalancing Create Portfolio Growth?

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1 Volume 5 o. Fall 0 Investment management s a hghly f ckle dscplne. There s plenty of room for successful nvestors to prosper. Those who do, have learned the need for humlty and adopted nvestment processes whch rely on measured decsons and possess dscplne. Jean Brunel - Edtor s Letter Volatlty Harvestng: Why Does Dversfyng and Rebalancng Create Portfolo Growth? PAUL BOUCHEY, VASSILII EMTCHIOV, ALEX PAULSE, AD DAVID M. STEI L BOUCHEY nagng drector search at Parametrc attle, WA. hey@paraport.com SILII EMTCHIOV ector of research, y strateges, at Parac chnov@paraport.com X PAULSE esearcher at Parac en@paraport.com ID M. STEI ef nvestment er at Parametrc attle, Look at market fluctuatons as your frend, rather than your enemy; proft from folly rather than partcpate n t. Warren Buffet Investors have tradtonally equated volatlty wth rsk and vewed t as unavodable. However, volatlty also affects how returns compound over tme, whch rases the queston: Is t possble to proft from volatlty? The answer s a defntve yes. In ths artcle, we explore the concept of volatlty harvestng, or the extra growth generated from systematcally dversfyng and rebalancng a portfolo. In contrast to huntng for securtes wth hgh return potental, we use the term harvestng because the actvty s akn to farmng, where seeds are spread wdely and results are patently harvested over tme. The excess return from volatlty harvestng s not an expected arthmetc excess return derved from forecastng skll, secuvde nsght nto the mathematcal deas derved n Appendx. ext, we use market data to evaluate a smple rebalancng strategy for equal-weghted portfolos of stocks n U.S. developed, and emergng markets. We also examne portfolos of stocks selected at random to show that the excess return s ndependent of an actve stock selecton process. We show that roughly half of the excess return from volatlty harvestng comes from a dversfcaton beneft and half from rebalancng. We focus on equal weghtng because of ts smplcty and because t provdes a clear llustraton of the underlyng theory. In practce, a more nuanced approach s requred, one that takes nto consderaton lqudty, tradng costs, taxes, and other frctons. In ths artcle, we present theoretcal The Voces of Influence and emprcal support for volatlty harvestng the dea that, for assets that are volatle and lqud, dversfyng and rebalancng creates excess

2 Volatlty Harvestng: Why Does Dversfyng and Rebalancng Create Portfolo Growth? PAUL BOUCHEY, VASSILII EMTCHIOV, ALEX PAULSE, AD DAVID M. STEI PAUL BOUCHEY s managng drector of research at Parametrc pbouchey@paraport.com VASSILII EMTCHIOV s drector of research, equty strateges, at Parametrc vnemtchnov@paraport.com ALEX PAULSE s a researcher at Parametrc apaulsen@paraport.com DAVID M. STEI s chef nvestment offcer at Parametrc dsten@paraport.com Look at market fluctuatons as your frend, rather than your enemy; proft from folly rather than partcpate n t. Warren Buffet Investors have tradtonally equated volatlty wth rsk and vewed t as unavodable. However, volatlty also affects how returns compound over tme, whch rases the queston: Is t possble to proft from volatlty? The answer s a defntve yes. In ths artcle, we explore the concept of volatlty harvestng, or the extra growth generated from systematcally dversfyng and rebalancng a portfolo. In contrast to huntng for securtes wth hgh return potental, we use the term harvestng because the actvty s akn to farmng, where seeds are spread wdely and results are patently harvested over tme. The excess return from volatlty harvestng s not an expected arthmetc excess return derved from forecastng skll, securty selecton, or an nformatonal advantage. Rather, t s the excess compounded return generated from rebalancng volatle assets over multple tme perods. Ths excess growth s avalable n lqud markets wth assets that have volatltes greater than zero and correlatons less than one. However, only nvestors wth the dscplne to trade systematcally wll harvest ths extra growth. We begn the artcle wth two thought experments to stmulate the topc and provde nsght nto the mathematcal deas derved n Appendx. ext, we use market data to evaluate a smple rebalancng strategy for equal-weghted portfolos of stocks n U.S. developed, and emergng markets. We also examne portfolos of stocks selected at random to show that the excess return s ndependent of an actve stock selecton process. We show that roughly half of the excess return from volatlty harvestng comes from a dversfcaton beneft and half from rebalancng. We focus on equal weghtng because of ts smplcty and because t provdes a clear llustraton of the underlyng theory. In practce, a more nuanced approach s requred, one that takes nto consderaton lqudty, tradng costs, taxes, and other frctons. In ths artcle, we present theoretcal and emprcal support for volatlty harvestng the dea that, for assets that are volatle and lqud, dversfyng and rebalancng creates excess portfolo growth. THOUGHT EXPERIMET : APPLE AD STARBUCKS Assume only two stocks are avalable for nvestment: Apple and Starbucks. We selected these two stocks to hghlght the value of rebalancng; both are volatle, uncorrelated, and have smlar growth rates. Generally, any equty par n whch one stock does not VOLATILITY HARVESTIG: WHY DOES DIVERSIFYIG AD REBALACIG CREATE PORTFOLIO GROWTH? FALL 0

3 domnate over the entre perod wll show a beneft from rebalancng. Hstorcally, both Apple and Starbucks were outstandng nvestments. From 994 to 0, a dollar nvested n Starbucks grew to $8, and a dollar nvested n Apple grew to $43. The annualzed growth rates were.7% and 4.8%, respectvely. A buy-andhold portfolo that started wth half a dollar n each stock would have grown to $35.5, halfway between $8 and $43. For a buy-and-hold nvestor wth foreknowledge of future growth rates, the hghest growth portfolo would have been 00% n Apple. However, f the portfolo were rebalanced back to constant weghts, then an even hgher growth rate could have been acheved. For the rebalancng nvestor, the best portfolo was 59% Apple and 4% Starbucks. Ths portfolo would have grown to $7 substantally more than an nvestment n ether stock alone! Exhbt plots the fnal wealth for a dollar nvested n several dfferent portfolos wth weghts of 0%, %, %, and so on, up to 00% n Apple, wth the remander nvested n Starbucks. The dashed lne represents the endng balance for the buy-and-hold portfolos, the sold lne, for rebalanced portfolos. For portfolos that held both stocks, rebalancng led to a hgher growth rate than drftng. Of course, t s possble for a rebalanced portfolo to underperform a buy-and-hold portfolo. For example, E XHIBIT Growth of $ for Apple and Starbucks Portfolos (994 0) n the frst four years of our sample from 994 to 998, Starbucks had steady postve growth whle Apple had steady negatve growth. In ths subperod, rebalancng hurt performance relatve to the buy-and-hold strategy, whch allowed the weght of Starbucks to buld up n the portfolo. However, a concentrated portfolo s desrable only f the dfference n expected future growth rates s very large relatve to the volatlty of the securtes. In practce, future growth rates are unknown and allowng concentraton to buld up n a portfolo s undesrable. Exhbt shows the pattern of cumulatve excess return from rebalancng by takng the rato of portfolo values through tme. Values above one ndcate that a rebalancng portfolo s outperformng a buy and hold. The frst four years fall below one, ndcatng underperformance. However, over the whole sample, the rebalanced portfolo had two tmes the growth of the buy-and-hold portfolo. The trend of outperformance by the rebalancng strategy s predcted wth remarkable accuracy by a rebalancng premum formula represented by the dashed lne (see Equaton A-8 n Appendx). In some perods, the rebalancng premum s less than theory predcts, and other tmes, t s greater. Apple and Starbucks both have hgh volatltes 50% and 43%, respectvely and a correlaton of 0.7. The hgh volatlty and low correlaton between these stocks provdes a dramatc llustraton of the extra growth that can be produced by volatlty harvestng. In the next experment, we show that the extra growth s not only a stock phenomenon but s general to any nvestment facng uncertanty. THOUGHT EXPERIMET : COI FLIPPIG FOR FU AD PROFIT Imagne a game of chance that depends on the flp of a con: Heads, and the player can double her money; tals, she loses half. The expected return of a sngle flp s 5%. If the con s f lpped twce wth one result heads and one tals, then the fnal compounded amount of wealth s the same as the ntal amount zero percent return. If all proceeds are renvested and the con s flpped multple tmes, then a run of good luck can earn a lot of money. Ten heads n a row wll turn $00 nto $0,400. Ten tals n a row makes $00 turn nto a few pennes. Ths game would be attractve to gamblers, snce the expected return for a sngle flp s postve and the FALL 0 THE JOURAL OF WEALTH MAAGEMET

4 E XHIBIT Cumulatve Rebalancng Premum of an Equal-Weghted Portfolo of Apple and Starbucks (994 0) wealth dstrbuton has a hghly postve skew (you can wn a lot, but you can lose only the ntal nvestment). However, for a rsk-averse nvestor, the game s not very attractve. Wth a suffcently large number of flps, the expectaton s an equal number of heads and tals. Thus, the game has zero long-term expected growth. ow magne that a player always holds half of her money n reserve: Heads, and she can place a porton of the proceeds nto her pocket; tals, and she can replensh her stake. Half of her money s always at rsk. If the con s flpped twce, wth one result heads and one tals, then ths player wll earn a.5% return. For example, f she put $50 at rsk and $50 n her pocket, her rsk money would double after the frst flp and $5 would be put back n her pocket, so that $75 would be at rsk and $75 would be safe. The subsequent loss would apply only to half the assets, leavng $.50 at the end of two flps. Ths strategy has a lower expected return on a sngle flp.5% nstead of the 5% for the full-nvestment case but over many flps, t has a hgher long-term growth rate, about 6% on average. 3 Exhbt 3 shows a smulated return path for random con tosses n the full-nvestment and half-nvestment cases (both strateges usng the same sequence of flps). For ths game, the wealth levels can quckly get very large or very small, so the charts are plotted on a logarthmc scale n order to observe the growth patterns more clearly. Each asset the rsky game and the rskless pocket has zero long-term expected growth. However, the act of tradng n the presence of volatlty creates a postve growth rate. Ths s a surprsng result when frst encountered: Rebalancng creates growth out of no growth, thereby harvestng an expected return from volatlty. 4 ITUITIO: WHY DOES REBALACIG WORK? The above examples llustrate how rebalancng can mprove returns, but to understand why t works we must turn to captal growth theory. 5 The lterature n ths area tends to be qute mathematcal, but some ntuton can be ganed by examnng the formulas. One of VOLATILITY HARVESTIG: WHY DOES DIVERSIFYIG AD REBALACIG CREATE PORTFOLIO GROWTH? FALL 0

5 E XHIBIT 3 Wealth Smulaton for Con-Flppng Example the most basc fndngs s that, for any asset, the growth rate s lower than the average return because volatlty s a drag on the compoundng effect. Mathematcally, the growth of $ return s gven by the expected arthmetc return mnus one-half the varance: g = μ σ Ths relatonshp apples to contnuously compounded returns that follow a normal dstrbuton, but s also approxmately true for dscrete compoundng ntervals and non-normal returns. For example, an nvestment wth 0% volatlty faces a drag on return of 0.50% (snce 0. / = 0.005). However, because the functon s exponental, the drag grows quckly as volatlty ncreases. Investments wth 0%, 40%, and 60% volatltes create a %, 8%, and 8% drag per year, respectvely. In Appendx, we extend the volatlty drag formula to a portfolo of securtes wth weghts, w, and derve the followng relatonshp for portfolo growth: M, g = w p w g + w σ j j w σ = = j= = Averag e Growth + Averag e Var ance Portfolo Varance eb () = Average Grow th +Re alancng Premum () Ths equaton has been descrbed as the dversfcaton return by Booth and Fama [99] and the rebalancng premum by Sten, emtchnov, and Pttman [009]. Wllenbrock [0] observes that f dversfcaton s the only free lunch of nvestng, then the dversfcaton return from rebalancng s the only free dessert. Indeed, rebalancng s closely lnked to dversfcaton. A buy-and-hold portfolo, although ntally dversfed, can drft and become a more concentrated portfolo over tme. The control over portfolo concentraton, plus the extra growth, makes a strong case for rebalancng. Manufacturng return out of thn ar seems too good to be true. Where do these extra returns come from? There are two dstnct components, whch we valdate emprcally n the next secton: extra return from dversfcaton and extra return from rebalancng. The dversfcaton return s due to reweghtng the portfolo s long-term exposures. For example, an equalweghted portfolo has less weght n large-cap stocks and more weght n small-cap stocks than the market-cap ndex. Thus, t has a natural small-cap bas. If small-cap stocks outperform, then an equal-weghted strategy wll beneft. However, n addton to creatng long-term exposures, another way to earn return s through a pattern of tradng. If you can consstently buy low and sell hgh, you can create postve portfolo growth, even f the overall asset growth s flat. 6 Intutvely, t s easy to see that trendng hurts a rebalancng strategy and large reversals help. 7 Rebal- FALL 0 THE JOURAL OF WEALTH MAAGEMET

6 ancng, however, s not merely related to momentum and reversal. A deeper reason exsts for the observed outperformance. The con-flppng thought experment ponts ths out. In that scenaro, there s no concept of momentum or reversal snce there s no seral correlaton between the returns. The probablty dstrbuton at each pont n tme s dentcal and ndependent from what has occurred n the past. However, there s stll mean reverson n ths example. If 0 tals n a row are flpped, the mean return s 50%. The sample mean wll revert to the long-term average as more flps are made. Even n ths smple case of no seral correlaton, the rebalancng premum s evdent. In the more complex case usng actual returns, there may be tme dependences. If asset prces experence boom perods followed by bust perods, then rebalancng wll be even more valuable than theory predcts. Who s on the other sde of the trade? The other trader s someone who buys after prces have gone up and sells after prces have gone down. There are a few possbltes: An nvestor who chases postve performance (greed) but becomes rsk averse when returns become negatve (fear); A quanttatve trader who uses momentum to predct returns; An nvestment manager who prefers wnners to losers ; An nsttuton seekng downsde protecton through a dynamc portfolo nsurance tradng strategy; An nsttuton that receves new cash to nvest n good economc tmes but requres lqudty from ts portfolo n bad economc tmes; It s hard to magne the rebalancng premum beng arbtraged away. If many nvestors n the market swtched to an equal-weght and rebalance scheme, then volatlty n the market would be suppressed. If technology stocks started to boom, nvestors would sell some of ther shares and nhbt prce growth. If bank stocks started to crash, nvestors would step n and buy, supportng the prce. Even n ths type of market, there would be mbalances n supply and demand, or other exogenous shocks to the system that create volatlty and opportuntes for rebalancng. We suspect that the lack of rebalancng s the reason that the captalzaton-weghted ndex underperforms a broad range of portfolo dversfcaton strateges equal weghtng, mnmum varance, mean varance, fundamental weghtng, dversty weghtng, maxmum dversty, and others. Chow et al. [0] observe that, on average, each of the approaches studed outperforms by % to % per year over 45 years n the U.S. stock market and by a smlar amount over years n the global equty markets. Part of these returns can be explaned by exposure to value, sze, and momentum effects, but there s stll a resdual excess return. Despte dfferent approaches to portfolo constructon, all these strateges have one thng n common: They systematcally rebalance. Ths s lkely the source of ther resdual return. The key dea s that the growth of a portfolo s the weghted-average growth of the securtes plus a rebalancng premum. Ths premum s always postve because the portfolo varance s always lower than the weghted-average varance of the ndvdual assets when correlatons are less than one. Hgher volatlty and lower correlaton among the assets wll lead to a hgher rebalancng premum. Another nterestng nsght to be ganed from ths formula s that concentrated portfolos wll produce a smaller rebalancng premum. For example, f the fxed weghts are 99% and %, then the rebalancng premum wll be lower because only a small porton of the portfolo s beng rebalanced. The shape of the plot n Exhbt reveals ths result. The most dverse portfolo, equal weght, s not always the hghest growth, but t s often close to the hghest growth (Platen and Rendek [00]). Portfolos wth more assets and more evenly dstrbuted weghts should garner a hgher beneft from rebalancng. Overall, dversfyng and rebalancng s a valuable dscplne and can be used to explot volatlty. Theoretcally, rebalancng reduces concentraton rsk, downsde rsk, and volatlty, whle ncreasng the longterm growth rate of the portfolo. In practce, t creates a contraran tradng pattern that trades aganst natural nvestor tendences and takes advantage of volatlty, reversals, and other return characterstcs. EMPIRICAL EVIDECE How does theory hold up when we use actual market returns? Rebalancng s often thought of n the VOLATILITY HARVESTIG: WHY DOES DIVERSIFYIG AD REBALACIG CREATE PORTFOLIO GROWTH? FALL 0

7 context of asset allocaton, partcularly wth respect to the relatve weghts of stocks and bonds. A recent paper by Anderson, Banch, and Goldberg [0] compared rsk-party, fxed-weght, and captalzaton-weghted portfolos of stocks and bonds. One of ther results was that a rebalanced constant mx of 60% stocks and 40% bonds, after transacton costs, outperformed a buy-andhold mx by 74 bass ponts per year from 96 to 00, wth sgnfcantly lower volatlty. It also outperformed durng each of the subperods examned: pre-946, the post-war perod between 946 and 98, the bull market between 983 and 000, and from 00 to 00. The captalzaton-weghted strategy had an average weght of stocks of 68%. Durng bull markets, the allocaton to equty drfted as hgh as 95%, and n bear markets, down as low as 30%. It had a hgher volatlty, suffered more durng the crashes, and benefted less durng recoveres, because t became hghly concentrated. The fxed-weght strategy holdng the weghts at 60% stocks, 40% bonds was more rsk controlled. It pulled money out durng bull markets and put money n after bear markets (n the same way as n the con-flppng example) and thus created a hgher growth rate. So even n the two-asset case, stocks and bonds, there s return to be created by rebalancng. Dversfcaton and rebalancng can also be appled at the nvestment manager, country, ndustry, or securty level. As granularty and volatlty ncrease, the potental for excess growth ncreases. As a smple emprcal test, we examne stock portfolos usng monthly hstorcal data from the Russell Global Index, whch extends from January 997 to March 0. 8 Exhbt 4 shows the performance of global, U.S., developed ex-u.s., and emergng market stock portfolos. For each regon, three strateges are tested: captalzaton weghted (CAP), equal weghts allowed to drft wth no rebalancng (EWD), and equal weghts rebalanced monthly (EWR). 9 Dfferences between CAP and EWD capture the beneft of dversfcaton, whle dfferences between EWD and EWR capture the rebalancng premum. Dversfcaton, n ths context, refers to the lack of concentraton n the portfolo weghts. Over ths 5-year perod, the EWD portfolos tended to outperform the CAP, except n emergng markets where ther performances were approxmately equal. However, of greater nterest s the addtonal return generated by rebalancng the equal-weghted portfolo. For example, for the U.S. market, EWD resulted n a 0.6% outperformance over CAP, whle the rebalancng to equal weghts earned a.68% outperformance. The dfference of.4% s the rebalancng premum. The rebalancng premums for global, developed, and emergng markets were 0.7%, 0.34%, and.4%, respectvely. E XHIBIT 4 Characterstcs of Stock Portfolos (January 997 March 0) for CAP, EWD, and EWR FALL 0 THE JOURAL OF WEALTH MAAGEMET

8 In these examples, volatlty s hgher for the equalweghted portfolos. Ths s true for most equty portfolos, as stocks tend to be hghly correlated wth one another and smaller stocks tend to be more volatle. In asset classes wth lower cross-correlatons, equal-weghted portfolos are less volatle than the more concentrated ndexes. For example, an equal-weghted portfolo of commodtes has a lower volatlty than ts ndex because commodtes have lower cross-correlatons and the ndexes are concentrated n hghly volatle energy contracts. As expected, turnover s hgher n the rebalancng portfolos. In practce, turnover and lqudty ssues would need to be addressed to ensure that benefts from rebalancng are not eroded by tradng frctons. Some possble ways to address these ssues are: To reduce the frequency of rebalancng, allow the portfolo to drft wthn specfed bounds To reduce the overall amount of tradng, rebalance at the country or sector level nstead of the stock level To allevate lqudty ssues, use a dversfcaton functon to create weghts that are a compromse between captalzaton weghts and equal weghts Despte the strong theoretcal support, numerous engneerng problems need to be addressed n a real portfolo. As a fnal emprcal llustraton, we examne smulated portfolos of random stocks. Smulaton provdes an avenue to solate dversfcaton and rebalancng prema whle controllng for selecton and weghtng effects. For a large number of trals, we selected 00 stocks at random from the Russell Global Index, thus smulatng an actve stock selecton strategy. Usng the same tme perod, we agan examne three strateges: captalzaton weghted (CAP), equal weghts allowed to drft wth no rebalancng (EWD), and equal weghts rebalanced monthly (EWR). 0 Each tral holds the same 00 stocks across portfolo constructon strateges. As before, the dfference between CAP and EWD captures the beneft of dversfcaton, whle the dfference between EWD and EWR captures the rebalancng premum. The frst panel of Exhbt 5 llustrates the dstrbuton of smulated annualzed return outcomes by strategy. On average, EWR contrbutes almost.80% of annual excess return above CAP, and EWD posts nearly a.40% return mprovement. Thus, we can attrbute the excess annual return n almost equal parts to dversfcaton and rebalancng. The second panel of Exhbt 5 depcts the dstrbuton of annualzed volatlty outcomes by strategy. Both the level and the range of annualzed volatlty outcomes show meanngful mprovement compared wth the captalzaton-weghted strategy. Whle all of strateges post an average annualzed volatlty of almost 0%, allowng weghts to drft ncreases the lkelhood E XHIBIT 5 Return and Volatlty Dstrbutons for Random 00-Stock Portfolos by Portfolo Constructon Strategy (January 997 March 0) VOLATILITY HARVESTIG: WHY DOES DIVERSIFYIG AD REBALACIG CREATE PORTFOLIO GROWTH? FALL 0

9 E XHIBIT 6 Excess Returns over Market-Cap Weghtng for Random 00-Stock Portfolos (January 997 March 0) of hgher-volatlty outcomes. Equal weghtng sgnfcantly decreases the lkelhood of outszed hgh-volatlty outcomes n the rght tal (n ths context the rght tal s bad). Monthly rebalancng magnfes ths effect by keepng the ntal portfolo and future portfolos dversfed. In other words, hgher concentraton can lead to hgher rsk. Measurng the excess returns of each strategy wthn each tral provdes addtonal nsght nto the nature of dversfcaton and rebalancng effects. Exhbt 6 depcts dstrbutons of returns n excess of the captalzaton-weghted strategy. The dversfcaton benefts of EWD lead t to outperform CAP nearly 74% of the tme. Monthly rebalancng rases the lkelhood of outperformance to over 90% of trals. The symmetrc, smooth dstrbuton of excess returns suggests results hold consstently across trals. When lookng at concentrated portfolos of stocks, regardless of whch securtes are selected, we fnd that nearly half of the excess return from an equal-weghted portfolo comes from rebalancng. We also fnd that drftng portfolos tend to experence a buldup n concentraton, volatlty, and rsk. COCLUSIO In the end, our advce s smple: dversfy and rebalance. Ths prescrpton not only provdes a framework for managng rsk, but also enhances returns n the long term. In a real portfolo, the turnover generated by rebalancng can be costly, partcularly when transacton costs are hgh. Unconstraned rebalancng could result n transacton costs that outwegh the rebalancng benefts. Of course, of key nterest to the practtoner s the queston of how to reduce these costs and measure how much would be gven up n performance. In an upcomng artcle, we plan to dscuss mplementaton ssues and crcumstances n whch costs can be carefully controlled. Specfcally, we wll show that a number of pragmatc methods can substantally reduce the frequency of rebalancng and tradng costs. Ths can be acheved, for example, by allowng the portfolo to drft wthn bounds, by enforcng dversty at the country and sector level, and by selectng weghts that seek the rght compromse between captalzaton weghtng and equal weghtng. The prncples presented here are mathematcal n nature and apply to any set of suffcently lqud nvestments that are volatle and uncorrelated; therefore, they can be appled at the asset allocaton level and wthn subsegments of the portfolo. Investors should consder ther portfolos n a multperod framework and realze that volatlty s more than just a rsk measure t represents an opportunty that can be exploted through thoughtful rebalancng. Just as t s possble to harness energy from waves n the ocean, t s possble to harvest return from volatlty n the market. A PPEDIX DERIVATIO OF VOLATILITY DRAG AD DIVERSIFICATIO RETUR Followng the opton-prcng and captal growth lterature, we assume that returns follow a geometrc Brownan moton process: ds S = μ dt + σ dz (A-) where S s the asset prce, μ s the expected return, σ s the volatlty, dt s the tme ncrement, and dz s a normal random varable (0,). From Ito s Lemma we know that the above stochastc dfferental equaton has the followng soluton: G dg = S G G t μs + + σ S S dt + dg ds σ Sd z (A-) FALL 0 THE JOURAL OF WEALTH MAAGEMET

10 If we let prces follow a lognormal process, G = ln(s), then: G = S S ; G = S S ; Substtutng back nto Ito s formula: dg = dt dz μ σ + σ G t = 0 (A-3) (A-4) Thus, the contnuously compounded return dg = dln(s) = ds/s s a geometrc Brownan moton process wth drft parameter: σ g = μ (A-5) When assets are held wthn a fxed-weght portfolo, the long-term growth from Equaton (A-5) becomes M g w p w μ, σ w = j, = w j j (A-6) Where w s the portfolo weght allocated to asset, σ j s the return covarance of asset and j, and g p s the contnuously compounded portfolo return. To emphasze the beneft of dversfcaton on portfolo return, we solve for µ from Equaton (A-5) and substtute nto Equaton (A-6) to obtan where M g w, σ g + w w p σ We can rewrte ths as = j= g = w g d p w = M, d w σ σ w w = j j j= j j (A-7) (A-8) (A-9) Equaton (A-8) expresses the portfolo growth rate as the sum of the ndvdual asset growth rates plus the premum d derved from dversfcaton and rebalancng. Ths value s postve for correlatons less than one, mplyng that the beneft of rebalancng to fxed weghts s postve. The frst term of d s the weghted sum of the component asset varances, and the second s the portfolo varance. An ncrease n asset volatlty ncreases the frst term (ncreases growth potental from rebalancng) but also ncreases portfolo varance (decreases growth). The amount by whch the second term ncreases relatve to the frst largely depends on the correlaton among the assets. EDOTES Cover [99] shows several stock par examples and generalzes to the concept of a unversal portfolo. Maslov and Zhang [998] use an example of rebalancng between cash and a Russan stock wth negatve growth but wth volatlty hgh enough that rebalancng creates postve portfolo growth. Jamshdan [99] formalzes ths dea. 3 There s a 50% probablty of earnng 50% or losng 5%. The expected growth rate for a large number of trals s 0.5*ln(+0.5)+0.5*ln(-0.5) = For the full nvestment case, 0.5*ln(+.0)+0.5*ln(-0.5) = 0. See Luenberger [998], chap We can also use smulaton to estmate the rebalancng premum for more realstc scenaros. See Sten, emtchnov, and Pttman [008] for an estmate of the rebalancng premum n the emergng markets. 5 See Maclean, Thorpe, and Zemba [0] for a synthess of captal growth theory. 6 Plyakha, Uppal, and Vlkov [0] solate the rebalancng premum from sze, value, and momentum effects. After controllng for these factors, they fnd sgnfcant excess alpha attrbuted to rebalancng. 7 Perold and Sharpe [995] and Wse [996] dscuss dynamc strateges and the ntuton behnd them. 8 Other studes have used dfferent datasets and tme perods and produced smlar results. For example, DMguel, Garlapp, and Uppal [009], Platen and Rendek [00], and Plyakha, Uppal, and Vlkov [0] provde emprcal support to equal-weght and rebalance strateges. 9 ote that the captalzaton-weghted strategy wll have a slghtly dfferent return than the offcal ndex snce t s calculated monthly nstead of daly. For example, the Russell 3000 Index calculated on a daly bass produced a 6.5% return over ths perod. Our monthly calculaton came to 6.47%, whch s close but not equal to the offcal ndex return. 0 Stocks wth data as of January 3, 997, are used. Over tme, as stocks leave the sample, the fnal weght s renvested on ether a market-captalzaton or equal-weghted bass. The smulaton drew one mllon trals of one hundred stocks. VOLATILITY HARVESTIG: WHY DOES DIVERSIFYIG AD REBALACIG CREATE PORTFOLIO GROWTH? FALL 0

11 REFERECES Anderson, R., S. Banch, and L. Goldberg. Wll My Rsk Party Strategy Outperform? Workng paper, Unversty of Calforna Berkeley, 0. Booth, D.G., and E. Fama. Dversfcaton Returns and Asset Contrbutons. Fnancal Analysts Journal, Vol. 48, o. 3 (99), pp Chow, T., J. Hsu, V. Kalesnk, and B. Lttle. A Survey of Alternatve Equty Index Strateges. Fnancal Analysts Journal, Vol. 67, o. 5 (0), pp Cover, T.M. Unversal Portfolos. Mathematcal Fnance, Vol., o. (99), pp. -9. DeMguel, V., L. Garlapp, and R. Uppal. Optmal versus aïve Dversfcaton: How Ineffcent Is the / Portfolo Strategy? Revew of Fnancal Studes, Vol., o. 5 (009), pp Jamshdan, F. Asymptotcally Optmal Portfolos. Mathematcal Fnance, Vol., o. (99), pp Luenberger, D. Investment Scence. ew York, Y: Oxford Unversty Press, 998. Maclean, L., E. Thorpe, and W. Zemba, eds. The Kelly Captal Growth Investment Crteron. World Scentfc Press, Sngapore, 0. Maslov, S., and Y. Zhang. Optmal Investment Strategy for Rsky Assets. Internatonal Journal of Theoretcal and Appled Fnance, Vol., o. 3 (998), pp Perold, A., and W. Sharpe. Dynamc Strateges for Asset Allocaton. Fnancal Analysts Journal, Vol. 5, o. (995), pp Platen, E., and R. Rendek. Approxmatng the umérare Portfolo by aïve Dversfcaton. Quanttatve Fnance Research Center, research paper 8, 00. Plyakha, Y., R. Uppal, and G. Vlkov. Why Does an Equal- Weghted Portfolo Outperform Value- and Prce-Weghted Portfolos? Workng paper, EDHEC, 0. Sten, D.M., V. emtchnov, and S. Pttman. Dversfyng and Rebalancng Emergng Market Countres. The Journal of Wealth Management, Vol., o. 4 (009), pp Wllenbrock, S. Dversfcaton Return, Portfolo Rebalancng, and the Commodty Return Puzzle. Fnancal Analysts Journal, Vol. 67, o. 4 (0), pp Wse, A.J. The Investment Return from a Portfolo wth a Dynamc Rebalancng Polcy. Brtsh Actuaral Journal, Vol., o. 4 (996), pp To order reprnts of ths artcle, please contact Dewey Palmer at dpalmer@journals.com or FALL 0 THE JOURAL OF WEALTH MAAGEMET