The fallacy of time diversification

Transcription

1 The fallacy of ime diverificaion - a concep ha financial planner do no really underand Wha i he fallacy of ime diverificaion? ü The argumen Financial planner, journali and invemen pundi regularly ae ha a ime increae he andard deviaion of he annualied reurn decreae. While hi i rue (a will be demonraed below) he ue o which i i pu i ofen fundamenally mileading. For inance, he Vanguard group in he US aid in one of i publicaion ha"he volailiy of ock marke reurn diminihe markedly over ime...clearly, over ime, ock marke rik hardly eem exceive even for he mo cauiou inveor" ( Spring 1990 iue of Vanguard' US publicaion "In The Vanguard"). Financial planner and oher argue from he decreae in he andard deviaion of he annualied reurn (an average), ha he uncerainy (which hey equae eiher explicily or implicily wih he andard deviaion of he annualied reurn) of inveing in volaile ae like ock reduce a he ime horizon increae. However, inveor are no inereed in he andard deviaion of he annualied reurn. Raher hey are inereed in heir end invemen value and i i he uncerainy aociaed wih hi ha really maer. In eence hoe who peddle hi argumen are really aering ha he appropriae meaure of he variabiliy of he end invemen i he andard deviaion of he annualied reurn. Unwiingly ome proponen of he argumen may be igning up o he following claim: andard deviaion of he annualied reurn = andard deviaion of he oal reurn. For a ime cale of more han 1 year hi equaion i fale if you ake he average reurn a he annualied reurn. The heoreical underpinning for he mo general cae are quie deailed and have been examined by a number of ubanial wrier uch a Paul Samuelon, Rober Meron, Harry Markowiz and Sanley Ficher, ju o menion he well known people. Significanly he eminal aricle by Meron and Samuelon carrie he ile: "Fallacy of he Log-Normal Approximaion o Porfolio Deciion-Making Over Many Period". They ued expeced uiliy maximiaion principle in heir argumen. A you migh urmie, i i a non-rivial exercie o underand he deailed argumen and ha i why he uperficial ale "argumen" i o aracive o many people - i relieve you of he hard work of underanding he nu and bol. Inveor need o underand he baic poin ha ime doe no wah away all in. I believe i i preciely becaue a lo of ale people have condiioned inveor o believing ha he uncerainy urrounding heir end benefi will decay away o nohingne over ime ha inveor are highly diurbed when hey acually ee large flucuaion in heir porfolio' value. There are ill many good reaon for inveing ome of your money in equiie and a lo of academic reearch ha been done on he ubjec. However, wha i clear from he heoreical andpoin i ha you can' naively believe ha volailiy in your end invemen value will diappear over ime.

2 2 The fallacy of ime diverificaion.nb Underanding he mahemaic We ar wih a implified iuaion o demonrae he hrehold error in he logic decribed above. Suppoe he coninuouly compounded annual rae of reurn on an invemen ha andard deviaion. The invemen horizon i year and r i i he rae of reurn during year i. For compuaional convenience i i aumed ha he r i are independen wih he ame andard deviaion. Thi make he mah nice and clear o he general poin i no obcured. The average reurn i: 1 i=1 r i. The oal reurn i i=1 r i. The developmen of he general argumen i coniderably more deailed han hi and you can ee he Appendix for he main paper on he ubjec. The Meron-Samuelon paper i over 100 page. Now compare he andard deviaion of hee wo reurn. The andard deviaion of he average reurn i: VarJ 1 i=1 r i N = 1 VarI 2 i=1 r i M = = Here "Var" mean he variance and he andard deviaion i he quare roo of he variance. The manipulaion above rely upon he properie of he variance operaor and can be found in any exbook. Clearly a increae he andard deviaion of he average reurn decreae and approache zero in he limi. Thu he andard deviaion of he annualied reurn doe decreae wih increaing ime. The andard deviaion of he oal reurn i: VarI i=1 r i M = 2 = In hi cae he andard deviaion of he oal reurn increae wih ime. For a ime cale of more han 1 year he purpored

3 The fallacy of ime diverificaion.nb 3 In hi cae he andard deviaion of he oal reurn increae wih ime. For a ime cale of more han 1 year he purpored equaion i fale. In fac he andard Brownian moion model ha underpin he enire heoreical edifice involve he aumpion ha reurn follow a geomeric Brownian moion wih andard deviaion and mean (m - 2 ). Thu he underlying model 2 involve increaed reurn variaion wih ime o ha hoe who argumen ha ime reduce uncerainy have o ge over ha analyical fac. While porfolio heory demonrae ha diverificaion of variou or of ae ha are no perfecly correlaed can reul in a decreaed andard deviaion of he expeced annual porfolio reurn, he heoreical bai for ha reul revolve around weighing of ae. Thoe who peddle he ime diverificaion fallacy are eenially ranlaing he logic of weighing of ae o weighing he ime dimenion of he problem. A demonraed above he problem doe no "cale" in hi fahion and he reaoning i fundamenally flawed. Thi i a claic cae of ale people puhing a produc (inve in equiie and i will be alrigh in he long run) wihou having any underanding whaoever of wha i really going on. The deeper heoreical underanding ake much more mahemaical overhead o develop which naurally exclude he va majoriy of he adviory indury. I will come o an overview of ha deeper underanding in an appendix. Making i concree To make he principle concree we can ake a brually imple cae where he rae of reurn in any one year period i eiher +20% or -20%. The wor poible oucome, indeed he caarophic oucome, i where for year he invemen uffer a lo of 20% each year. Wha happen a he ime cale increae? Le V(0) be he value of he invemen a ime = 0. Afer one year he value of he invemen i V(1) = V(0) V(0) = 0.8 V(0). Afer 2 year he value of he invemen i V(2) = V(1) =- 0.2 V(1) = 0.8 V(1) = VH0L. One can proceed inducively o ee ha afer ime he value of he invemen i V() = 0.8 V(0) o ha afer 20 year, ay, he value of he invemen i : V(20) = V(0) = 0.01 V(0). So afer 20 year you have lo 99% of your iniial invemen! By aigning probabiliie o he reurn one can aach probabilie o uch oucome. A widely ued concep of rik i ha of a horfall which occur when he value of he hare porfolio a he horizon dae fall below ome value deermined by a pecified arge rae of reurn. Such a arge rae of reurn i uually a rik free rae. If i acually urn ou ha he mean rae of reurn exceed he rik-free rae of inere, i i rue ha he probabiliy of a horfall decline wih he lengh of he invemen ime horizon. The problem i ha he probabiliy of a horfall i a flawed meaure becaue i compleely ignore how large he poenial horfall may be. A indicaed in he example above, uing a probabiliy of horfall a he meaure of rik, no diincion i made beween a 20% lo and a 99% lo, ince boh are, by definiion, horfall.

4 4 The fallacy of ime diverificaion.nb APPENDIX In porfolio heory he variance of he porfolio' expeced reurn i: Var p = i=1 j=1 w i w j Cov ij where w i and w j are he porfolio weigh of ae i and j and Cov ij i he covariance beween he rae of reurn on ae i and j. In a wo ae world wih weigh w and 1-w he andard deviaion of he porfolio expeced reurn i: p = w H1 - wl wh1 - wl 1 2 Cov 12 w 1 + (1-w) 2 Thu in an equally weighed porfolio (ie 50%/50%) of wo ae wih 1 = 5% and 2 = 10% he porfolio andard deviaion of expeced reurn i le han 50% x 5% + 50% x 10% = 7.5%. Thu diverificaion ha reduced he andard deviaion of he porfolio' expeced annual reurn bu hi i no he ame a demonraing ha a longer ime horizon ha reduced he andard deviaion of he porfolio' expeced oal reurn. The general problem ha been examined exenively: 1. R C Meron and P A Samuelon, "Fallacy of he Log-Normal Approximaion o Porfolio Deciion-Making Over Many Period", Journal of Financial Economic (March 1974), P A Samuelon, "Rik and Uncerainy: A Fallacy of Large Number", Scienia (April - May 1963), P A Samuelon, "The Fallacy of Maximizing he Geomeric Mean in Long Sequence of Inveing or Gambling", Proceeding of he Naional Academy of Science (1971), There are many more recen paper dealing wih he iue. A noed above here i quie a lo of mahemaical overhead in underanding he deailed reul bu i i poible o kech he line of argumen. One ar wih he concep of a horfall which i he amoun by which he value of an equiy porfolio i le han one baed on ome arge rae of reurn uch a he inere rae on a defaul free zero coupon bond. The rik free rae provide a benchmark again which o ae riky equiie. Thi eem o be a reaonable bai upon which o model he relevan behaviour. If i were rue ha equiie were le riky in he long run, hen he co of inuring again earning le han he rik-free rae of inere ough o decline a he lengh of he invemen horizon increae. Bu hi i no he cae - he oppoie i rue.

5 The fallacy of ime diverificaion.nb 5 The argumen run like hi. Suppoe he rik free rae i r and he curren value of he porfolio i S. The coninuouly compounded yearly reurn i aumed o have andard deviaion. Le he fuure value of S afer year a rae r be V. Then V = S r. Nex we can do one of wo hing: le he money ay in he porfolio or pu i all in he bank. If we keep he money in he porfolio he rik we are really meauring i ha of no doing a well a puing he money in he bank. Wha would i co o inure again ending up wih le han V a ime? Such an inurance policy would cover he difference beween he erminal value of he bank invemen and he equiy invemen. A European pu opion on he porfolio wih a rike price V and ime o expiraion can provide he required inurance and he co of hi opion i he price of he inurance we wan. If he value of he required pu opion i P and he correponding call opion i C (having he ame rike price and expiraion dae), he Pu-Call Pariy Theorem ell u ha: P = C + V -r - S = C + (S r ) -r - S = C + S - S = C So he pu and he call have he ame price and you can ue he Black-Schole formula for he value of he call o ge he value of he pu. Uing he andard ymbol eg N(.) i he normal diribuion cumulaive deniy funcion we have: P = C = S N loghsêvl + J r N - V -r N loghsêvl + J r N = S N loghsêvl + J r N - S r -r N loghsêvl + J r N = SANI 1 2 M -N I ME by ymmery conideraion Thi laer expreion give he area under he normal diribuion curve beween - 1 and + 1 and o obviouly 2 2 increae wih increaing. Hence he co of inurance which i our proxy for he rik of he invemen increae wih he ime horizon. Peer Haggrom July 2010