3 Portfolio Management

Transcription

1 Mathematcal Modelng Technques 69 3 ortfolo Management If all stock predctons were perfect, portfolo management would amount to the transfer of funds to the commodty that promses the hghest return n the specfed nvestment nterval. Unfortunately, the future s not predctable to that degree of accuracy. onsequently, portfolo management requres a careful dstrbuton of funds n varous stocks so that any one sngle ncorrect predcton does not dramatcally and negatvely affect the performance of the entre portfolo. On the other hand, spreadng the rsk between numerous stocks also mples that a dramatc upsde gans by any one nvestment only helps the portfolo proportonally. Due to ths dynamc, portfolo selecton s dependent on the rsk adversty of the nvestor. Markowtz defned the theoretcal concept of the perfect portfolo, on whch NELION s based [Markowtz 959]. fter analyzng the concepts of return and rsk n ths chapter, I present the parameterzaton of the conflctng goals of hgh return wth low rsk n the optmal portfolo theory.

2 70 ortfolo Management 3. Return The return of a stock n a specfed perod s the percentage ncrease of the value of the nvestment. It s defned as follows: + t t Dt R t = Equaton 3.. t In the equaton above, t s the current prce and t- s the prce at the begnnng of the nterval, whle D t s the dvdend wthn that perod. The dvdend can never be negatve. For perods, whch do not concde wth the fnancal year of the underlyng stock, the dvdend s calculated as a percentage of the total accrued durng the perod. Followng standard nvestment conventon, we assume that the nterval s one year. From Equaton 3.., t s clear that the return R t s postve f t s larger than t-, or the prce of the commodty has ncreased. The return of a portfolo s the weghted sum of the ndvdual stock returns. ( X ) = R t X Rt,, Equaton 3.. In ths equaton X denotes the fracton of the portfolo covered by each nvestment and therefore

3 Rsk 7 n = X = where X 0 Equaton 3..3 Ths requrement does not mpose any restrctons on the allocaton of funds, snce t allows for money kept as cash. The return would then smply be the bank nterest rate, whch may be 0%, dependng on the account type. 3. Rsk Unlke the return of an nvestment, the defnton of rsk s more subjectve. Markowtz assumes a normal dstrbuton of upsde and downsde potental around the return of a commodty, based on ts volatlty ó. Normal Dstrbuton Downsde Rsk Upsde otental 4/6 /6 /6 ì-ó ì ì+ó Fgure 3..: The One-n-Sx Rule In Fgure 3.., the expected return ì defnes the peak of the normal dstrbuton wth ì-σ and ì+σ defnng a /3 margn of

4 7 ortfolo Management return. The downsde potental s /6, hence the name of the rule. It s clear that a small σ reduces potental loss thereby mnmzng the assocated rsk of the equty. We therefore defne the rsk V(X) of an nvestment of value X wth a varance σ as follows. ( X ) X σ V = Equaton 3.. Unlke the return, the rsk can not smply be calculated as the weghted sum of the ndvdual rsks, snce ndvdual stocks can be dependent on smlar external factors. Both Damler- hrysler and Ford are affected negatvely by rsng ol prces, for example, so that a portfolo consstng of these two stocks has a hgher rsk than one wth Damler-hrysler and Mcrosoft, for example, assumng that Mcrosoft and Ford have the same volatlty. onsequently, the systemc rsk of a portfolo ncludes the covarance ρ j of the ndvdual nvestments and j as shown n Equaton 3.. below. V n n ( X ) = X + = ρ X X ρ Equaton 3.. n = j= + j j The frst term represents the nherent rsk of every ndvdual stock, whle the second term captures the rsk assocated wth the correlaton between stocks. Gven a portfolo where the correlaton ρ j between all stocks and j s zero, the rsk V reduces to the smple sum of ndvdual rsks for each stock.

5 The Optmal ortfolo 73 V ( X ) n = X = for ρ j = 0 ρ Equaton The Optmal ortfolo The condtons of Equaton 3..3 are vrtually mpossble to acheve for any n>, but addtonally, ths approach gnored the return of the portfolo. In order to fnd the optmal stock dstrbuton, we look at a sample portfolo wth two stocks wth an equal expected return ì, where ρ j = 0., σ = 0.6 and σ = 0.8. We can plot the rsk of the portfolo as a functon of the nvestment n the frst stock. Rsk of a ortfolo Rsk 0% 0% 40% 60% 80% 00% Stock Dstrbuton Fgure 3.3.: Rsk of a ortfolo If these two stocks were the only avalable choces, an nvestor would deally dstrbute 60% of the avalable captal n stock and the remanng 40% n stock. Ths example

6 74 ortfolo Management shows that the rsk of a portfolo can be mnmzed wthout changng the expected return. In order to calculate ths optmal portfolo, we use Markowtz' approach. He defned the objectve functon, whch assgns a weght between the desre for hgh returns and a low rsk. [ ( R )] V f = E + Equaton 3.3. In ths functon, represents the rsk averson of the nvestor and s dependant on hs nvestment needs. graph of ths functon hghlghts a regon that satsfes the nvestor s requrements for return as well as rsk. The edge of ths regon defnes the portfolos wth the hghest return gven a specfc rsk or conversely, the lowest rsk gve a defned return and s called the Effcent Fronter. Effcent Fronter Excessve expected rsk for a gven expected return Expected Return Effcent Fronter Low expected return for a gven expected rsk Rsk Fgure 3.3.: The Effcent Fronter

7 The Optmal ortfolo 75 In the next step, Markowtz defned the Utlty Functon, whch s also nvestor dependent and descrbes the utlty U(R) of a specfc return R. Ths functon s used to dentfy the desred return when optmzng a portfolo. U ( R ) a + br cr = Equaton 3.3. The coeffcents b and c are not negatve so that the resultng graph wll have a form as shown below. The Utlty Functon Utlty Maxmal Utlty Return Fgure 3.3.3: The Utlty Functon person at the begnnng of hs career can generally afford to take a larger rsk snce he wll generally not depend on the savngs n the near future but would beneft from hgher returns later n lfe. Short-term downward fluctuatons are tolerable to ths group of persons but not for an nvestor who s close to retrement and wll need hs savngs n the near future. Job securty, the plans for a large purchase n the near future

8 76 ortfolo Management or personal rsk averson are other consderatons, whch wll affect these parameters. pplyng the expectaton operator E(.) on Equaton 3.3. we get the followng result. E ( U( R )) + be( R ) ce( R ) = α Equaton Usng the defnton of the varance V ( R ) E( R ) E( R ) [ ] = Equaton we can re-wrte Equaton as follows: E ( U( R )) = a + be( R ) + c E( R ) [ ] cv ( R ) Equaton For a constant expected utlty,, we can solve ths equaton for the expected return E(R ). E ( R ) V ( R ) + = Equaton where a b = + and c ( c) b = Equaton c Ths equaton defnes utlty curves, whch we can add to the graph shown n Fgure 3.3., to arrve at the optmal portfolo as shown below.

9 The Optmal ortfolo 77 The Optmal ortfolo Rsk Expected Return Optmal ortfolo Fgure 3.3.4: The Optmal ortfolo The pont of tangency between the utlty curve and the effcent fronter defnes the optmal portfolo for ths nvestor. Ths pont can be calculated by substtutng Equaton nto Equaton 3.3. and solvng for V(R ) or E(R ). ( ) = f f f R V Equaton ( ) ( ) 4 f R E + + = Equaton Ths expresson unquely specfes the optmal portfolo.

10 78 ortfolo Management The challenge of ths approach s the dentfcaton of the parameters n the utlty and the objectve functons snce they are hghly subjectve and represent relatve weghts and cannot be attached to measurements n the physcal world. 3.4 pplyng the Theory Most tradng systems are extensons of fnancal predcton experments and have the goal of measurng the real-world results that can be assocated wth the forecasts. The smplest form was already mentoned n the experments from Rehkugler and oddg: If an ncrease was predcted, the system purchased one addtonal fcttous unt of the DX, f a decrease was predcted, one was sold. The system dd not permt the ownershp of negatve numbers of the stock, or short postons. Due to ther mathematcal smplcty, tradng strateges based on movng averages are probably the most wdely used techncal rules. These models were promnently used by LeBaron and became the baselne for further comparson [LeBaron 995]. In LeBaron s experment, the sngle movng average ndcator generated buyng (sellng) sgnals when ts value was above (below) the current stock prce. The adjustment of the model requred dentfyng the optmal length of the data wndow. slght mprovement on the basc algorthm could be acheved f tradng sgnals were only generated f the dfference between the movng average and the current prce exceeded a

11 pplyng the Theory 79 specfed band. Ths reduced the number of trades and, by mplcaton, the transacton costs that a real world nvestor has to bear. Movng average oscllators compare a short term and a long term movng average of the stock prce aganst each other. These models frequently use the commonly quoted movng averages of fve, ten, 5, 50 and 00 days for ther comparsons. Buy (sell) sgnals are generated only f the short (long) term movng average rses above that of the long (short) term. gan, frequently the dfference between the two values has to exceed a specfed value n order to trgger a tradng sgnal, so that the number of transactons s kept at bay. Usng both of these movng average tradng systems as a foundaton, Dhardjo and Tan compared artfcal neural network predcton models wth an assocated tradng system to predct proftablty opportuntes n the ustralan Dollar/US Dollar exchange rate [Dhardjo, Tan 999]. Though they found that both systems were proftable n the perod tested, the NN models performed better (annualzed returns between 3% and 9%) than the smple movng average approach (returns of between 8% and 3%). The experments showed that both models were partcularly successful n markets, whch exhbted long term trends. Kumar, Tan and Ghosh used the same ustralan Dollar/US Dollar exchange rate data and bult sophstcated fnancal forecastng models. These ncorporated the chaotc

12 80 ortfolo Management components n numerous ways n an effort to optmze the predctve powers of the models and, by mplcaton, the proftablty of ther system [Kumar et al, 999]. The tradng system worked wth two dfferent rule patterns: attern : If (urrent Forecast revous Forecast) > Delta then Sgnal = Buy Else If (revous Forecast urrent Forecast) > Delta then Sgnal = Sell Else Sgnal = Hold attern : If (urrent Forecast urrent lose) > Delta then Sgnal = Buy Else If (urrent lose urrent Forecast) > Delta then Sgnal = Sell Else Sgnal = Hold The Delta value was used to provde a threshold, whch elmnates excessve trades, snce they were taken nto account wth 0.% of the transacton value n ths experment. Interestngly, though the forecastng models were consderably more complex than the ones used by Dhardjo and Tan, the proftablty ranged between % and 0% annualzed return and thus dd not sgnfcantly help ths goal much.

13 pplyng the Theory 8 Notably mssng from ths lst of tradng strateges s one that addresses the realtes of an ndvdual nvestor, who has to decde not only whch stocks offer good growth opportuntes, but also how to dstrbute hs nvestment between the numerous alternatves. Bookstaber descrbes a smple BSI program that combnes chart analyss wth a smple rsk calculaton algorthm, but does not analyze the success or falure of ths approach gven hstorc data [Bookstaber 985]. rograms wth a smlar focus exst wth nvestment nsttutons or other professonal nvestors who emphasze rsk analyss, however, ths work tends to not get publshed snce t s consdered the strategc advantage of the respectve owner or user communty. Jean Y. Lequarré voced a smlar sentment n the concluson of hs artcle: Ths nablty to dscuss ther fndngs n the open s often frustratng for many of those nvolved n ths actvty and specally the ones who come from academa [Lequarré 993]. Ths thess s an effort to combne the sgnfcant work on fnancal tme seres analyss and predcton wth a coherent tradng strategy that can be adjusted to the preferences and needs of the ndvdual nvestor. The resultng system s desgned to run on common hardware makng t sutable for personal nvestment advce and as a portfolo management tool.