Evaluating SEB Investment Strategy`s recommended Mutual Fund Portfolios

Transcription

1 MÄLARDALEN UNIVERSITY Stockholm, Department of Mathematcs Master Thess n Mathematcs Tutor: Lars Pettersson Evaluatng SEB Investment Strategy`s recommended Mutual Fund Portfolos Alexander Rostam

2 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Abstract Date: Level: D-Thess n Mathematcs, 30p Author: Alexander Mazyar Rostam Ttle: Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Tutor: Lars Pettersson, Department of Mathematcs, Mälardalen Unversty Prevew: SEB Investment Strategy s the functon n SEB that supports busness unts SEB Prvate Bankng and SEB Retal wth nvestment phlosophy and nvestment process. The framework of SEB Investment Strategy encompasses to manage a structured nvestment phlosophy and process to produce a range of nvestment optons and portfolos for dfferent target groups. From January 006 to October 009 forty Proposal for fund portfolos were produced each contanng wrtng on market condton and expectatons plus portfolo recommendatons. Each tme four portfolos consstng of sx mutual funds was recommended, Fund Portfolo 30, 50, 70 and 100. Fund Portfolo 30 (FP30) contaned 30% equty fund and 70% fxed-ncome funds. By same reasonng FP50 contans 50/50 equty- and fxed-ncome funds, FP70, 70% equty funds and 30% fxed-ncome funds and FP100 only equty funds. Purpose: The am of ths work s to evaluate these SEB Investment Strategy recommended portfolos for prvate SEB Retal clents from January 006 to December 009. Evaluaton s done by comparng the performance of recommended portfolos wth portfolos produced by applyng Vascek s Technque and smplfed optmzaton technque. Method: To allow work wth Vascek s Technque n whch we are dependent on a market portfolo, I have created an Index whch ncludes SEB Mutual Funds and ther

3 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos share of the Index s determned from each fund s total assets n relaton to the sum of the total assets under management of all funds nclusve n the Index. Index conssts of 40 mutual funds and 37 mutual funds 008 and 009. The total supply of funds has been reduced to the above numbers by the followng crtera: 1. Clents must be able to nvest n funds through conventonal SEB Fund Account.. No ntaton fees or sales charges. 3. Mnmum hstorcal Net Asset Value prces (NAV-prces) from nd January Daly tradng and at least 300 mllon SEK n assets under management. 5. No Fund-n-Fund products. 6. Only SEB or SEB Choce funds. The closng daly NAV-prces (tme seres) of these funds have been obtaned from seb.se/fonder from nd January 00 to 8 th December 009. Wth prces daly returns are calculated and used for estmaton of hstorcal and average values of varables needed for computng forecasted Alphas and Betas accordng to Vascek s Technque. Mutual funds are then ranked wth respect to excess return over forecasted Beta gven rsk free rate equal to Swedsh government 1 month treasury-bll (SSVX1M) at tme for optmsaton. Top sx ranked funds are ncluded n the optmzaton process. The frst optmzed portfolo gven actual T- bll s then compared to FP100 recommended by SEB Investment Strategy. In order to fnd optmzed solutons to other recommended portfolos premums are added to actual T-bll rate. 3

4 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Table of contents 1. Introducton Theory Markowtz Model (Modern Portfolo Theory) The Sngle-Index Model Vascek s Technque Smplfed Technque for Fndng Optmal Portfolos Method Mutual Funds Purpose Market Portfolo (Index) Obtanng Hstorcal Varables... 3 Hstorcal Alpha, Beta and Varance of Resdulas... 5 Expected Return and Varance of Market Portfolo Obtanng Forecasted Varables Optmzaton Result FP100 vs OP FP70 vs OP FP50 vs OP FP30 vs OP Concluson Recommended Portfolos Optmsed Portfolos References Appendx I: Mutual Funds ncluded n the study Appendx II: Fondportfolos recommended by SEB Investment Strategy Appendx III: Varables needed for the optmsaton process Table 1: Hstorcal Estmates Table : All sub perod data and summary of perods... 6 Table 3: Forcasted varables Appendx IV: Top 10 mutual funds ranked gven forecasts compared to actual outcome

5 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 1. Introducton December 00 I started as part-tme employee at the SEB offce n Vasterås alongsde studes at Mälardalen Unversty. Many tmes I met at the bank customers who needed advce on how best they should allocate ther savngs among the bank's large selecton of funds. I was new and had not the authorty needed to help these customers. They were nstead forwarded to the approprate advsor who n turn found out the purpose of the placement, nvestment horzon, the expected return and the rsk that nvestors were wllng to take. When the advser had the nformaton that was requred, he / she suggested possble locatons. These proposals were based mostly on the prncple of "not puttng all eggs n one basket" and usually went on to spread rsk among dfferent geographcal areas and ndustres. Durng 005 SEB Investment Strategy launched a tool to facltate counsellng. Ths tool s called Placerngs Guden (placement gude) and s avalable to all employees va the SEB Bank's Intranet. Va placement gude, an advser can pck up an nvestment proposal, whch s updated every month wth few exceptons. Four allocaton recommendatons were proposed, Fund Portfolo 30, 50, 70 and 100, each contanng sx mutual funds to facltate for those customers who want an alternatve wth the possblty of monthly savngs whch s only possble n sx funds at a tme va SEB's system. The recommendatons conssted durng most of the perod under test of a fxed-ncome fund, a global fund, a Swedsh fund and three other funds that mostly conssted of a European fund, an emergng market fund, and a North Amercan fund, and at some occasons, Japanese, Eastern European, pharmaceutcals, Nordc and / or natural resources funds. In Aprl 009 after the fnancal crss, these recommendatons are a bt more swtched to nclude more asset classes such as foregn exchange, prvate equty, hedge funds and specal funds. I have for many years wondered for myself s these recommendatons are optmal for our customers. Therefore, I now take on ths project to clarfy and test these proposals by comparng performance of suggested portfolos to optmzed portfolos gven scentfc methods. The results of ths project are presented n Chapter 4 and conclusons are drawn from results and dscussed n Chapter 5. Followng chapter deals wth the theoretcal knowledge necessary to understand the varous steps and methodology (presented n Chapter 3) taken to produce optmzed portfolos. 5

6 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos. Theory Portfolo theory s usually assocated wth models developed by two gentlemen, Harry M. Markowtz and Wllam Sharpe. Markowtz model s based on the thought that nvestors are rsk averse and therefore choose the portfolo wth less rsk gven same return. The optmsaton process for fndng portfolos wth mnmum rsk gven return or maxmum return gven rsk requres calculaton of number of varables gven by below equaton. Where N stands for number of securtes possble to nvest n. For ths paper N s equal to 40 whch means that 380 varables need to be calculated before optmzaton process. N N 1 N Wllam Sharpe smplfed ths work by ntroducng a market portfolo (Index) and relates each stock s performance to ts ndex. Ths reduced the number of varables that must be calculated to 3N +. The smplfed model s called Sngle-Index Model (SIM) n whch each asset s expected return gven ndex return depends on two varables, Alpha and Beta. Alpha s the constant ndependent from ndex return and Beta s a constant both postve and negatve that measures expected change n asset s return gven 1% change n ndex return. The calculaton of Alpha and Beta s based on hstorcal patterns of returns between the asset and ts ndex whch can be done by regresson analyss. Studes made by Blume 1 showed that actual Beta n forecasted perod tended to become closer to 1 than the estmaton of t. Ths resulted n tryng to change the hstorcal Beta for catchng ths dfference. Vascek s Technque corrects for ths tendency by adjustng towards average Beta. Followng two sectons descrbes each model s approach to reach to an optmal soluton. Thrd secton s based on Vascek s Technque and how t s used to change hstorcal values to better ft the future they forecastng and chapter ends by descrbng the smplfed technque used for rankng nvestments and allocatng resources to obtan most optmum soluton. 1 Betas and Ther Regresson Tendences, Journal of Fnance, X, No. 3 (June 1975), pp

7 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos.1 Markowtz Model (Modern Portfolo Theory) Optmsaton can be done by maxmsng return gven rsk, mnmse rsk gven return or by maxmsng Sharpe rato. Gven any of these approaches we frst need to fnd out values for mutual funds returns and rsks. The daly returns were calculated by dvdng the dfference between the NAV-prce day and day 1 wth the NAV-prce of Day 1 as below equaton shows: ( NAV NAV 1) NAV 1 R 1 = Return of Mutual fund day 1. (.1.1) Arthmetc average of returns are then calculated n Excel by buld n functon AVERAGE whch dvdes the sum of all days return for any gven perod wth number of days we have calculated return for. The annual return can be measured by summng daly returns for that partcular year or by summng up for whole perod and dvde by number of years for that perod. The rsks on the other hand are calculated by buld n Excel functon VARP whch sums up the squared yeld dfference from the mean for whole populaton and dvdes t by the number of days n the calculaton mnus one accordng to the followng formula. Takng square root of that value we get standard devaton of mutual fund whch also can be calculated n Excel by STDEVP. T Rj R The Varance of Mutual fund. (.1.) T 1 j 1 Standard Devaton of Mutual Fund. (.1.3) Snce the dea s to optmse the yeld of a dversfed portfolo for mnmum rsk we need to measure the degree of covaraton between funds n the portfolo. Covarances can be calculated n Excel vd short command COVAR whch measures covaraton gven formula (.1.4). Value obtaned s hard to nterpret, therefore we can dvde covarance by the product of standar devatons of funds we have covarance for to get correlaton between funds as 7

8 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos gven by formula.1.5. Correlaton has a value between -1 and 1, f correlaton equals zero then both funds returns are ndependent from each other whereas +1 ndcates perfect postve correlaton and -1 perfect negatve correlaton. T ( Rj R )( Rkj Rk ) k E Covarance between mutual fund and k. (.1.4) j1 T 1 And, k k Correlaton between mutual fund and k. (.1.5) k Where R R ) ( j stands for the devatons of returns from ts mean for securty and by same reasonng ( Rkj Rk ) shows devatons from mean for securty k and k denotes the product of mutual fund and k s Standard Devatons. Gven above formulas we can proceed to portfolo thnkng by examnng equatons for portfolo rsk and return. N w R R p (.1.6) Above formula measures portfolo expected return and shows that ths value equals the sum of weghts w nvested n fund tmes expected return of that nvestment. Formula for N portfolo rsk or varance s the sum of two ndvdual parts, frst w 1 whch s the sum of all ndvdual fund varances tmes weghts nvested n them squared and second, the sum N of w w 1 k1, k k whch measures the product of fund s weght and fund k s weght k tmes covarance between fund and k were never equals k. Thus, the formula that measures portfolo varance s at follows. N N N w w wk k (.1.7) p 1 1 k1, k 8

9 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Now we are back where we started ths chapter, namely to fnd the optmal portfolo through varous approaches. If we for any gven value of rsk fnd the maxmsed level of return by changng weghts we wll fnd an optmsed portfolo gven that rsk. Ths portfolo s then represented by a pont on the graf were we have rsk on the x-axs and return on the y-axs. By repeatng the optmasaton process for other gven rsk values we wll eventually have more ponts on the graf that when combned form the so called effcent fronter. Wth other words we wll maxmse (.1.6) subject to: N w N w w wk k p 1 N 1 N k1, k 3 w 0, = 1,..., N We can also mnmse rsk gven return to acheve the same porpuse. That s, for any gven value of return we mnmze rsk (.1.7) subject to ponts 1 and 3 above. We can also maxmze Sharpe rato, ( R P R F ) P Sharpe Rato (.1.8) RF Rsk free rate gven constrants 1 and 3. Changng the rsk free rate results n dfferent solutons that can be used to plot the effcent fronter. 9

10 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos. The Sngle-Index Model The basc dea of the Sngle-ndex model (SIM) s that when an ndex ncreases securtes n that ndex tend to do the same and when t drops securtes tend to do follow. For ths reason, one can relate securtes return to the return of ts ndex by followng equaton R a R (..1) M Where a s the part of securty s return ndependent of ndex return and that measures expected change n securty s return gven 1% change n ndex return. s the constant In my case ths asserton s not ture, snce we do not have any ndex, nstead we have mutual funds each benchmarked to an Index. Therefore I wll assume that mutual funds are to be consdered as securtes n same ndex and that ther share of ndex s determned by the sze of ther asset under management. Wth ths assumpton we can proceed gven mutual fund ndex M that all dunds more or less correlates wth and that mutual fund s returns can be calculated by equaton..1. Ths equaton can be developed by separatng the ndependent part from the part dependent on ndex performance as follows. a e where s the expected value of a and e the uncertan element or a random varable wth expected value zero. Now equaton can be rewrtten as follows. R a R e (..) M Were e just lke R are both random varables wth E 0, M M e E R M R, and wth e varances e e M M E and R ER R M. By ths reasonng the expected value of mutual funds return can be wrtten as: M R R M (..3) 10

11 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos And the varance, covarance and correlaton as follows: (..4) M e (..5) k k M k k M 1/ k M e k M ek k (..6) Above formulas or equatons are an alternatve approach for fndng return, rsk and covarance between mutual funds. As stated before dong t ths way requres less needed varables then model descrbed n secton (.1). We have also stated that studes made by Blume suggests that Beta n forecasted perod tend to be closer to one then hstorcal Betas and that we need to adjust hstorcal Betas to acheve more accuracy n predctng future behavor of mutual funds. Next chapter descrbes how ths s done accordng to Vascek s technque but before we get there let us have a look at formulas that calculates portfolo return and rsk gven SIM. As n pervous secton, portfolo return s equal to equaton (.1.6), but ths tme we need to replace expected return for each mutual fund wth equaton (..3), whch gves: R p N 1 w N 1 w R M (..7) Replacng expresson for varance and covarance wth (..4) and (..5) n formula (.1.7) results n below formula that calculates portfolo varance when SIM s consdered as model of choce. p N 1 M N 1 N w w w w (..8) k1, k k k M N 1 e 11

12 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos.3 Vascek s Technque Both prevous sectons concerns models that evaluate hstory and assumes that hstory wll repeat t self. But studes lke the one done by Blume shows that ths s not the case and that average Beta tends to be closer to actual Beta then ts hstorcal estmate. Therefore we need a technque that actually changes hstory for better forecast the future. What we are gong to do here s to study how we adjust hstorcal Beta to become forecast Beta and I wll also take t one step further to use a smlar technque that also changes hstorcal Alpha. Hstorcal Alpha, Beta and varance of resduals are all found by regresson analyss accordng to least square method n Excel. Let us begn wth Beta and denote hstorcal Beta for mutual fund n hstorcal perod as and average Beta durng same perod as startng pont as t0 and end of hstorcal perod as T then. If we denote s calculated by regresson analyss durng whole perod and average Beta by weghted sum of all sub perods Beta between t 0 and T. Thus: Regresson( t 0 to T ) (.3.1) And [Regresson( t 0 to t 1 ) + Regresson( t 1 to t ) + + Regresson( t T 1 to T )] / T ( t 1 t T /... ) T (.3.) The am of ths technque s to converge hstorcal Beta towards average Beta n order to get closer to actual Beta n forecasted perod. One way to do t s by addng a half of hstorcal Beta to a half of average Beta. Smlar approach has been wdely used by for example Merrll Lynch. But as t s stated n the book It would be desrable not to adjust all stocks the same amount toward the average but rather to have the adjustment depend on the sze of the uncertanty (samplng error) about Beta. The larger the samplng error, the greater the chance of large dfferences from the average, beng due to samplng error, and the greater the adjustment. Therefore accordng to Vascek s Technque we wll apply formula below that ncorporates what has been stated before: Edwn J. Elton/Martn J. Gruber/Stephen J. Brown/Wllam N. Goetzman, Modern Portfolo Theory and Investment Analyss, Sxth Edton, pp

13 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 1 (.3.3) That s, forecasted Beta s equal to a weghted average of, (.3.4) for average Beta and (.3.5) for hstorcal Beta, denotes varance of resduals from hstorc estmate dvded by varance of the market and denotes varance of Betas nsde bracket of equaton (.3.). Summarsng below: And, (.3.6) e M T tj 1 j1 T 1 (.3.7) That s, large samplng error gves greater value to (.3.6) and therefore more weght (.3.4) to adjust towards average Beta and by same reasonng less adjustment s needed to average Beta n cases wth less uncertanty. 13

14 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos I wll also use smlar but smpler technque to adjust Alpha by summng half of hstorcal estmate of Alpha to half of average estmate of Alpha. Applyng ths gves: 5 1 0,5 0, (.3.8) Replacng and n formula (..7) and (..8) wth and from equatons above gves: And, N N R p w 1 w 1 R M (.3.9) 1 1 p N 1 1 M N 1 N w w w w (.3.10) k 1, k k 1 k1 M N 1 e As n frst secton portfolo rsk (.3.10) can be fxed wth desred level and portfolo return (.3.9) s maxmzed as much as possble n order to fnd the optmsed portfolo. Ths s done gven followng constrants very smlar to those presented on Page 4 exept for pont where we nstead have substtuted wth formula for portfolo varance gven the SIM and Vascek s Technque. N w N 1 w 1 M N 1 N k1, k w w k 1 k1 M N 1 w e p 3 w 0, = 1,..., N And the process can be done by fxng return and mnmzng rsk or by a smplfed optmzaton technque where we nstead use mutual funds excess return over Beta for rankng purposes. Ths alternatve way for fndng optmal portfolos s presented n followng secton. 14

15 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos.4 Smplfed Technque for Fndng Optmal Portfolos 3 Ths s the technque appled for fndng the optmal portfolos that are gong to be compared to portfolos recommended by SEB Investment Strategy. I have chosen ths technque manly because of ts smplcty and ts ablty to predetermne the number of mutual funds nvested n n the fnal optmzed soluton. Why the predetermnaton s mportant wll be explaned later. Recal that one can fnd an optmal portfolo just by maxmzng Sharpe Rato, whch means to fnd the maxmum value of potfolo expected return mnus rsk free rate over portfolo rsk accordng to followng equaton: ( R P R F ) P Sharpe Rato The expresson n the denomnator referes to rsk and denotes n what degree a securty return vares around ts mean. By same reasonng when SIM s accepted and appled the value of Beta denotes the expected change n the rate of return on a securty gven 1% change n the market return. If we accept that Beta denotes rsk we can replace rsk factor n denomnator of above equaton and get the expresson for excess return over Beta: ( R R F ) Excess return over Beta for securty. Ths way we can rank securtes before calculaton of Cut-off Rate C *. The hghest value of C * denote the rate that marks number of securtes to be ncluded n the optmzed portfolo. optmzaton process. In my case I am only nterested n sx mutual funds to nvest n so my Cut-off Rate s the same as the sxth securty C * value. Ths because recommended portfolos by SEB Investment Strategy always consst of maxmum sx nvestments whch s also number of mutual funds that bank clents can nvest n va monthly payments. Let us have a look at an example based on values found when applyng Vascek s technque. Mutual funds n table below are ranked from hghest to lowest value of excess return over Beta gven 3 Edwn J. Elton/Martn J. Gruber/Stephen J. Brown/Wllam N. Goetzman, Modern Portfolo Theory and Investment Analyss, Sxth Edton, pp

16 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos hstorcal data and rsk free rate equal to 1,65% 4. Notce that for smplcty only 30 out 40 mutual funds are ncluded n the table. Table.4.1: Rankng of mutualfunds 006 gven RFR 1,65% Excess Mean Excess Varance Reutrn Mutual Funds Return Return Beta Resduals Over Beta Pennngmarknadsfond SEK -0,08% -1,70% 0,00 0, ,85 Fastghetsfond 19,16% 17,53% 0,04 0, ,14 Svergefond 6,61% 4,98% 0,01 0, ,80 ÖstEurfond Småbo Lux Ack,95% 1,33% 0,07 0, ,85 Nordenfond 10,3% 8,70% 0,03 0, ,76 Lkvdtetsfond SEK -0,45% -,08% -0,01 0, ,06 ÖstEuropafond 8,51% 6,88% 0,13 0, ,03 Realräntefond SEK Lux Utd,83% 1,1% 0,01 0, ,76 Sverge Småbolag C/R 17,3% 15,61% 0,1 0, ,8 ÖstEurfondxRyssland Lux Ack 9,16% 7,53% 0, 0, ,6 Svergefond Småbolag 14,77% 13,15% 0,1 0, ,1 Emergng Markets 11,80% 10,17% 0,1 0, ,85 Asen Små x J Lux Ack 9,76% 8,13% 0,10 0, ,83 Latnamerka 16,36% 14,73% 0,0 0, ,74 Asen x J Lux Ack 10,50% 8,87% 0,1 0, ,74 Japanfond C Lux Ack 6,8% 5,0% 0,09 0, ,59 Aktesparfond,8% 1,0% 0,0 0, ,48 Europa Småbolag 10,67% 9,04% 0,19 0, ,47 Trygg Placerngsfond 3,5% 1,6% 0,04 0, ,38 Trygghetsfond Ekorren 3,56% 1,93% 0,06 0, ,33 Pennngmark Eur Lux Ack,1% 0,50% 0,0 0, ,5 Svergefond C/R 6,63% 5,01% 0, 0, ,3 Schwezfond 5,73% 4,10% 0,18 0, ,3 Japanfond 4,63% 3,00% 0,18 0, ,17 Europa 3,05% 1,43% 0,15 0, ,10 Världenfond,48% 0,85% 0,13 0, ,07 Internetfond 3,96%,33% 0,46 0, ,05 Europa C/R Lux Ack 7,64% 6,0% 1,31 0, ,05 Nordamerka Småbolag,47% 0,85% 0,4 0, ,03 Europa Småbolag Lux Utd,1% 0,59% 0,0 0, ,03 Now, the queston s how many of the top ranked mutual funds that are needed n the optmzed portfolo. In followng table I have calculated the Cut-off Rate for top 30 ranked mutual funds. From table you can see that the value of C ncreases untl t reaches the hghest value 0,1476 for Japanfond. Therefore the Cut-off Rate C * s equal to 0,1476 and denotes that 4 SSVX1M (Swedsh government treasury bll 30 days) 16

17 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos all of the funds up to Japanfond are gong to be ncluded n the optmzed portfolo gven RFR 1,65% and Market varance 0, Table.4.: Calculatons for Determnng Cut-off Rate wth Market Varance = 0, Mutual Funds C Pennngmarknadsfond SEK ,0008 Fastghetsfond ,0149 Svergefond ,015 ÖstEurfond Småbo Lux Ack ,088 Nordenfond ,0305 Lkvdtetsfond SEK ,0345 ÖstEuropafond ,0496 Realräntefond SEK Lux Utd ,050 Sverge Småbolag C/R ,0694 ÖstEurfondxRyssland Lux Ack ,094 Svergefond Småbolag ,1086 Emergng Markets ,1139 Asen Små x J Lux Ack ,1174 Latnamerka ,143 Asen x J Lux Ack ,181 Japanfond C Lux Ack ,197 Aktesparfond ,198 Europa Småbolag ,141 Trygg Placerngsfond ,1434 Trygghetsfond Ekorren ,1443 Pennngmark Eur Lux Ack ,1445 Svergefond C/R ,1460 Schwezfond ,1473 Japanfond ,1476 Europa ,1470 Världenfond ,1458 Internetfond ,1433 Europa C/R Lux Ack ,0506 Nordamerka Småbolag ,0506 Europa Småbolag Lux Utd ,

18 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Before calculatng for weghts to be nvested n each mutual fund let us have a look at formula for Cut-off Rate. Recall that stocks are ranked by excess return over Beta (Rsk) from hghest to lowest. For a portfolo of mutual funds C s gven by C N M 1 ( R 1 N M 1 R F e e ) 1 (.4.1) Now let us calculate some of Cut-off Rate values. We can start by calculatng C for top ranked mutual fund Pennngmarknadsfond SEK. That s, C PennngmarknadsfondSEK 0, *11 0, , , * 1, and for second mutual fund gves C Fastghetsfond 0, * , * 47 0, , ,0149 Proceedng n the same fashon we can fnd all the C `s. Before calculatng for weghts to be nvested n each mutual fund we know that the fnal soluton wll contan 4 mutual funds. That s, all the funds up to Japanfond and the percent nvested n fund s where w 4 Z 1 Z Z R R F C e * (.4.) 18

19 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Applyng above equatons gves followng weghts for all mutual funds ncluded n the portfolo. Table.4.3: Calculatons for Determnng percentage nvested n mutual funds up to C * Mutual Funds e ( R R F ) Weghts Pennngmarknadsfond SEK 6, ,55% Fastghetsfond 46 4, ,% Svergefond 1 3,80 4 0,19% ÖstEurfond Småbo Lux Ack 65, ,90% Nordenfond 8,76 1 1,05% Lkvdtetsfond SEK 7,06 5,63% ÖstEuropafond 103, ,83% Realräntefond SEK Lux Utd 4 1,76 7 0,36% Sverge Småbolag C/R 16 1,8 45 1,43% ÖstEurfondxRyssland Lux Ack 7 1, ,41% Svergefond Småbolag 5 1,1 0 11,14% Emergng Markets 103 0,85 7 3,67% Asen Små x J Lux Ack 71 0,83 48,45% Latnamerka 161 0, ,88% Asen x J Lux Ack 91 0,74 54,73% Japanfond C Lux Ack 50 0,59 1,1% Aktesparfond 5 0,48 0,08% Europa Småbolag 563 0, ,11% Trygg Placerngsfond 85 0, ,98% Trygghetsfond Ekorren 75 0, ,68% Pennngmark Eur Lux Ack 39 0,5 4 0,0% Svergefond C/R 75 0,3 1,13% Schwezfond 57 0,3 0 1,03% Japanfond 00 0,17 5 0,3% Sum ,00% Above mx of mutual funds would be one of the portfolos to be compared wth SEB Investment Strategy recommended portfolo f number of funds n portfolo where not lmted to sx. As you wll see later n next chapter I wll assume that Cut-off Rate C * s always the hghest value of C among top sx ranked mutual funds. Now that we have covered the theoretcal knowledge needed for understandng the steps taken for my study I wll n next chapter step by step explan and show how I been usng all prevous sectons n order to answer questons n problem statement. 19

20 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 3. Method 3.1 Mutual Funds SEB's fund range conssts of approxmately 150 funds, all not taken nto account n ths study because the majorty of them dd not meet the crteras I have set. They are: 1. Clents must be able to nvest n funds through conventonal SEB Fund Account.. No ntaton fees or sales charges. 3. Mnmum hstorcal Net Asset Value prces (NAV-prces) from nd January Daly tradng and at least 300 mllon SEK n assets under management. 5. No Fund-n-Fund products. 6. Only SEB or SEB Choce funds. Crtera 1 to nsure that future allocaton s possble for SEB Retal clents snce some of mutual funds are only avalable to SEB Prvate Bankng clents and some only possble to nvest n va nsurance polces. Crtera because fees for entry or ext complcates calculaton of expected yeld and lmts the possblty for re-allocatons. Crtera 3 to nsurng that data s not dependent on only one knd of market behavor. Frst January 00 was decded to be the startng pont because t was the earlest date that most mutual funds had hstorc data from. Crtera 4 so that mutual fund s share of Index could have a sgnfcant mpact on the market portfolo. Crtera 5 snce fund-n-fund products can most possbly contan funds already ncluded n Index. Crtera 6 because external fund s asset under management are not only based on nflow from SEB s clents and ther magntude can therefore have to much mpact on Index. Due to crtera Index conssted of 40 mutual funds between to and 37 between to All funds along wth ther asset under management 005 to 008 are shown n Appendx I. 0

21 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Mutual funds Fastghetsfond, Asen Små x J Lux Ack and Europa Småbolag Lux Utd had all less then 300 MSEK n assets when measured January 008 and therefore excluded from ndex measurement 008 and Purpose Before we move on to optmsaton steps let us repeat what I want to acheve wth ths work. The objectve s to evaluate the nvestment recommendatons SEB Investment Strategy has proposed January 006 to December 009. Evaluaton could be done by comparng the yeld on the above measurement perod wth the return of an ndex, for example OMX30 or the MSCI World Net Return Index. But then the evaluaton would stop at only one assessment by comparson. What I nstead want to acheve s to evaluate by comparng wth other portfolos generated by applyng the Vascek's Technque. Ths way I can demonstrate f the portfolo allocaton technque results n better nvestments than the proposals gven by SEB Investment Strategy between the years , these proposals are summarzed n Appendx II. Durng those four years new portfolos were recommended on twelve occasons, whch means that at each moment, four dfferent compostons were recommended, FP30, FP50, FP70 and FP100. Each portfolo talored to the nvestor's wllngness to take rsks and expected return. For each portfolo, I wll develop an optmzed soluton usng Vascek's Technque and the smplfed optmzaton process. Ths wll be done on four occasons, the frst four optmzed portfolos OP1, OP, OP3 and OP4 are produced n January 006, ther composton remans unchanged throughout 006 and the process wll be repeated agan n January 007, 008 and 009. For each year one year hstorcal data wll be added to the measurements whch form the bass for the optmzaton process. The dea s to compare OP1 wth FP30, FP50 wth OP, OP3 wth FP70 and fnally OP4 wth FP100. But how wll I be able to produce comparable portfolos? For example, I can measure the rsk of FP30 and optmze the return gven that rsk, and repeat the process for FP50, FP70 and FP100. That way I can take up four optmzed portfolos that ntally have the same rsk as recommend portfolos where returns have been maxmzed. Unfortunately, the results may not only consst of up to sx funds, whch s the requrement for the portfolos f they wll be farly comparable wth recommended portfolos. If I nstead choose to measure the return and mnmze rsk gven that return n the optmzaton process, occurs frst the problem wth a maxmum of sx funds, 1

22 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos and also problems wth negatve returns n some measurement perods. It s because of ths that I have chosen to apply the smplfed optmzaton process where I can pre-determne the number of funds n the portfolo by assumng that C * s equal to the hghest of the sx hghest ranked portfolos Cut-off Rate and by changng rsk-free rate we can create four dfferent portfolos wth dfferent rsk levels. How ths s done we wll return to later n ths chapter when I descrbe how portfolos are developed. The rsk-free nterest rate wll play another mportant role n the optmzaton process. At each optmzaton process, I wll brng the rskfree nterest rate equal to the actual nterest rate on a SSVX1M. The portfolo to be developed wth respect to ths nterest s then compared wth FP100, whch only conssts of equty funds. The reason for ths s that I assumed that, f an nvestor has the opportunty to nvest n the rsk-free rate but chose another opton than he s nterested n a portfolo wth the largest proporton of equty funds. The other three portfolos are optmzed gven rsk-free rate plus % premum each tme. Hgher rsk-free nterest rate contrbutes to lower rsk n the portfolo when the model s weghted more to assets wth less rsk, whch usually happens to be fxedncome funds. By lettng the rsk-free nterest rate to be determned by actual nterest rate on a 30 day treasury bll contrbutes to other benefts. If nterest rates are low, the model wll automatcally be folded more n equty funds for all portfolos snce the base to whch premums are added wll be lower and vce versa when the rsk-free nterest rate s hgher. 3.3 Market Portfolo (Index) As explaned n Chapter secton.4 we need to fnd out values for forecasted Alpha, Beta, varance of resduals, market return and market varance to be able to conduct the optmzaton process. So frst of all we need a market portfolo (Index) to whch we can relate mutual funds return and rsk. Recall that from January 00 to December 007, 40 mutual funds passed the crtera s and therefore were avalable to nvest n when frst optmzaton process started January 006. A lst of all these funds can be seen n Appendx I along wth wealth under management from 005 to 008. Daly tme seres for each of these funds where downloaded from seb.se/fonder from January 00 to December 009 except for three funds whose wealth decreased under 300 MSEK when measured January 008. These three funds Fastghetsfond, Asen Små x J Lux Ack and Europa Småbolag Lux Utd where therefore excluded from Index durng last hstorcal perod 008. Thus Index conssted of 40 funds

23 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos from begnnng of 00 to end of 007. For the frst optmzaton process January 006 each fund s assets as of January 005 where dvded by total assets under management (AUM) at the same tme n order to fnd out t s share of Market portfolo. These proportons were then multpled to ther own prces, then added together for each day to sum up daly tme seres of market portfolo (Index). So f each year conssts of 50 tme seres and we need to fnd out dayl tme seres for Index from January 00 to December 005 then 40 P X P j = 1,, 1000 days Mj 1 j P frst day NAV-prce of Index M1 P frst day NAV-prce of mutual fund 1 X AUM AUM 005 total005 For each subsequent year after 005 new proportons where calculated by updatng AUM for mutual funds and dvde by total AUM gven by volumes n columns of Table n Appendx I. Gven share of Index the procedure where repeaded and dayl tme seres for market portfolo where obtaned for whole perod January 00 to December 008. Gven these prces I could move forward to obtan hstorcal values accordng to steps n theory secton about SIM. 3.4 Obtanng Hstorcal Varables Recal from Chapter.4 Smplfed Technque for Fndng Optmal Portfolos, that mutual funds where ranked wth respect to excess return over Beta, that s where R R F R R M s the component of mutual fund s return that s ndependent of the market s performance. 3

24 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos a constant that measures the expected change n R gven a change n R M and also a measure of rsk. R F = Rsk free rate R M Expected return of the market portfolo All varables except for the rsk free rate are based on hstorcal estmates, but snce Vascek s Technque s the method appled n ths work we are nterested n forecasted values for Alpha and Beta nstead of ther hstorcal estmates. Therefore both Alpha and Beta are replaced by ther forecasted estmates where R 1 R F 1 R ,5 R M 0, 5 1 average Alpha estmated from perodc data average Beta estmated from perodc data e M e M varance of resduals varance of market portfolo T tj 1 j1 T 1 tj sub perod tj Beta From above summary we see that forecasted estmates of Alpha and Beta are dependent on ther hstorcal and average estmates. In the text that follows I wll only descrbe the methods used for measurement of hstorcal data. 4

25 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Hstorcal Alpha, Beta and Varance of Resdulas The reason for groupng above three varables s that all three can be calculated by the use of Excel Regresson analyss tool-pak. Where you choose daly average return of mutual fund as Input Y Range and daly average return of market portfolo as Input X Range. Then, by pressng OK a new Worksheet s produced showng hstorcal Alpha as Intercept, Beta as X Varable and varance of resduals n Resdula row and MS column as shown below for one of the mutual funds Expected Return and Varance of Market Portfolo Average return of market portfolo s smply calculated by the use of Excel command =AVERAGE(number1;[number]; ) where numbers nsde brackets are the daly return data for the hstorcal perod under consderaton. Annualsed expected return can nstead be measured by summng up daly returns for the hstorcal perod needed. Note that ths command calculates the arthmetc mean of arguments nsde brackets. I choosed arthmetc mean nstead of geometrc because the latter only apples to postve numbers whereas data analysed also contan negatve numbers. Market varance s measured followng same steps as above and command VARP whch measures varance wth respect to entre populaton. All hstorcal estmates for perods prevous to forecast year are summarzed n Appendx III, Table 1. 5

26 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 3.5 Obtanng Forecasted Varables Each hstorcal perod was dvded nto sub perods each contanng sx month daly data of return. For examble, the frst hstorcal perod rangng from January 00 to December 005 contaned eght sub perods. For each sub perod average return of market and mutual funds, varance of market, and covarance between Index and mutual funds where computed for calculaton of each sub perod Alpha and Beta. The computaton resulted n eght Alpha and Beta values. Takng average of these values wth Excel command AVERAGE, and VARP of Beta values ( ) we have most of the data needed for computng forecasted Alpha and Beta for year 006 accordng to followng formulas 0, 5 1 0,5 8 j1 j 8 1 j Rj j R Mj j = 1,, 8 jmj j j = 1,, 8 Mj where and 1 8 j1 8 1 j (computed wth hstorcal estmates) e M All subperod data alond summary of perods and forecasted values are summarzed n Appendx III, Table and 3. 6

27 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 3.6 Optmsaton As explaned earler, optmzaton ca be done through varous ways where I have covered four of them. I could measure the rsk value for each recommended portfolo and maxmze return gven that rsk for a soluton comparable to those recommended by SEB Investment Strategy. Dong so could result n portfolos consstng of less or more mutual funds then those sx suggested by SEB Investment Strategy and therefore not comparable snce I should construct portfolos gven same constrants. Measurng return of recommended portfolos and mnmzng rsk gven that return would also result n portfolos wth more or less mutual funds then sx and as I dscovered some perods resulted n negatve return whch are napproprate when computng optmzed portfolos. Ths snce a rskaverse nvestor would never n advance knowng that the portfolo he nvestng n has an expected return whch results n losses. Workng wth Sharpe-Rato through maxmzaton of t gven dfferent rsk-free rates that anables four solutons to compare wth recommended portfolos FP30, FP50, FP70 and FP100 could also result n portfolos not consstng of maxmum sx mutual funds. Due to these matters I fnally decded to apply the smplfed optmzaton technque covered n Chapter 9 of Modern Portfolo Theory and Investment Analyss book 5. The technque was then adjusted to make possble maxmum sx mutual funds to be ncluded n the optmzed soluton. You may argue that ths adjustment may result n portfolos that are not fully optmzed snce as we have seen the soluton wthout adjustment may result n a portfolo wth more than sx mutual funds (example on page 14, Table.4.3). That may be true but outsde the topc of ths paper. For my needs I just needed a technque that wth whch I could predetermne number of mutual funds avalable to nvest n wthout arbtrary thnkng. My next challenge was to determne a way that enables optmzaton that results n four solutons wth dfferent rsk levels. Increasng rsk free rate resulted n portfolos that conssted of hgher degree nterest funds and therefore less rsk. But where to start and how much ncrease? I argued that the lowest rsk free rate and therefore the soluton wth hghest 5 Edwn J. Elton/Martn J. Gruber/Stephen J. Brown/Wllam N. Goetzman, Modern Portfolo Theory and Investment Analyss, Sxth Edton pp

28 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos share of equty funds would equal to actual T-bll at the tme for the optmsaton process. The argument s that gven that rate n the calculaton of mutual funds enables the possblty to only nvest n the rsk free asset but the nvestor s nterested n other nvestments but ncreasng the rate may affect nvestor to adjust to more nterest-bearng alternatves. I decded to ncrease wth % each tme snce lower ncrease dd not have the measurable affect equal to recommended portfolos where share of nterest funds ncreases from 0 % for FP100 to 30 % for FP70, 50 % FP50, and 70 % FP30. An ncrease more than % resulted n portfolos wth to much nterest funds and therefore % seemed best sutable to work wth. As you wll see n next chapter the portfolos optmzed do not consst of equal amount nterent funds to those they are compared wth, some years they are less and some years they are more, and for the case of FP100 wth 100 % equty founds the optmzed soluton always conssts of some degree of nterest funds but that does not necessarrly mean that t s less rsky as t wll be proven n next chapter. 4. Result 4.1 FP100 vs OP1 The frst set of portfolos to be compared s FP100 wth 100% equty funds recommended n January 006 and optmzed portfolo 1 (OP1) resulted gven rsk-free rate 1,65% yearly return and hstorcal data years 00 to 005. The composton of funds n OP1 wll be unchanged throw whole 006 but FP100 wll be reallocated sx tmes accordng to weghts n Appendx II. The comparson wll be contnued n same manner wth new set of optmzed portfolos January 007, 008 and 009. We wll start by examnng each year and fnally summarze for whole perod January 006 to December

29 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos FP100 vs OP1 006 The frst 100% equty fund recommended by SEB Investment Strategy (SEB IS) conssted of funds and weghts accordng to frst weght column n below table. Funds where replaced and weghts updated sx tmes durng 006 whch can be followed n same table. Table 4.1.1: FP100 durng 006 Weghts Mutual Funds Jan Feb-Apr Maj Jun-Aug Sep-Nov Dec Svergefond 50% 50% 50% 50% 50% 50% Globalfond 5% 5% 0% 0% 0% 0% Emergng Markets 10% 10% 10% 10% 10% 10% Östeuropafond 3% 3% 3% 3% 3% 3% Europa 1% 1% 1% 1% Läkemedelsfond 7% 5% Nordenfond 5% 5% Japan C/R Lux Ack 7% 5% BlackRock Global Opp 5% Internetfond 5% By nvestng n frst recommended portfolo and reallocate each day so that allocaton always s the same as latest recommended portfolo an nvestors return was equal to 1,33% durng 006. The optmzed alternatve shown n below table returned durng same perod 4,73%. Table 4.1.: OP1 January 006 Adjusted Mutual Funds Weghts Weghts Pennngmarknadsfond SEK,41% % Fastghetsfond 40,54% 40% Svergefond 0,84% 1% ÖstEurfond Småbo Lux Ack 39,64% 40% Nordenfond 4,68% 5% Lkvdtetsfond SEK 11,89% 1% Summarzng both portfolos below shows that OP1 returned almost dubble the return of FP100 wth almost same level of rsk. Rankng them wth respect to rsk and rsk-free rate 1,65% shows that optmzed soluton rato exceeded recommended portfolos by more than 100%. Note that OP1 contaned 14% nterest-bearng funds compared to FP100 whch contaned none. 9

30 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Table 4.1.3: Summary of data 006 Portfolo Data FP100 OP1 Return 1,33% 4,73% Rsk 9,88% 10,06% Sharpe Rato 1,08,30 FP100 vs OP1 007 Durng 007 recommended portfolo was only reallocated two tmes accordng to weghts n below table wth frst column wghts equal to the weghts for December n Table Table 4.1.4: FP100 durng 007 Weghts Mutual Funds Jan-Apr May-July Aug-Dec Svergefond 50% 50% 50% Globalfond 0% 0% 0% Europa 1% 1% 1% Emergng Markets 10% 10% 10% Östeuropafond 3% 3% JPM Global Natural Resources 5% 5% Internetfond 5% GS BRIC s Portfolo 3% Optmsed alternatve contaned 7% nterest funds compared to 14% when optmzed durng 006 because of hgher rsk-free rate whch ncreased to,985% January 007. Table 4.1.5: OP1 January 007 Adjusted Mutual Funds Weghts Weghts Svergefond 0,61% 1% Pennngmarknadsfond SEK 1,06% 1% Lkvdtetsfond SEK 14,51% 15% ÖstEurfond Småbo Lux Ack 5,% 5% Nordenfond 3,19% 3% Fastghetsfond 44,41% 44% Despte the hgh proporton of fxed-ncome funds OP1 resulted n 13,06% loss durng 007 compared FP100 s 0,09% postve return. 30

31 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Table 4.1.6: Summary of data 007 Portfolo Data FP100 OP1 Return 0,09% -13,06% Rsk 1,3% 11,90% Sharpe Rato neg neg FP100 vs OP1 008 Durng 008 recommended portfolo was only reallocated once n September wth the frst set of portfolo equal to the last one n 007. Table 4.1.7: FP100 durng 008 Weghts Mutual Funds Jan-Aug Sep-Dec Svergefond 50% 50% Globalfond 0% 15% Europa 1% 10% Emergng Markets 10% 7% GS BRIC s Portfolo 3% 3% JPM Global Natural Resources 5% Nordamerka C/R Lux Ack 15% January 008 a Swedsh Government T-bll returned 4,05% whch as expected ncreased the proporton of fxed ncome funds from 7% for prevous year to 48% ths year. Table 4.1.8: OP1 January 008 Adjusted Mutual Funds Weghts Weghts Lkvdtetsfond SEK 17,51% 18% Pennngmarknadsfond SEK 17,4% 17% Nordenfond,0% % Oblfond Flex SEK Lux Ack 13,43% 13% ÖstEurfond Småbo Lux Ack 0,04% 0% ÖstEuropafond 9,58% 30% Both portfolos crashed durng 008, FP100 wth 47,44% and OP1 by 6,7%. Eastern European markets where among hardest-ht markets because of fnancal crss. The adjustment to fxed ncome funds lowered the potental greater loss but stll the hgh degree of 31

32 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos eastern European funds resulted n both hgher volatlty and greater loss compared to recommended portfolo. Table 4.1.9: Summary of data 008 Portfolo Data FP100 OP1 Return -47,44% -6,7% Rsk 16,64%,58% Sharpe Rato neg neg FP100 vs OP1 009 FP100 was reallocated two tmes durng 009 wth frst set of portfolo equal to last one n 008. For the frst tme snce we started the comparson, nvestments n Swedsh equty market decreased from constant 50% to now 0%. Table : FP100 durng 009 Weghts Mutual Funds Jan-Mar Apr-Sep Oct-Dec Svergefond 50% 0% 0% Globalfond 15% 30% 5% Nordamerka C/R Lux Ack 15% 15% 15% Europa 10% 15% 15% Emergng Markets 7% 10% 15% JF Chna Fund 10% GS BRIC s Portfolo 3% Lsted Prvate Equty 10% Rsk-free rate (SSVX1M ) decreased from 4,05% to 1,6% from January 008 to January 009. The,45% decrease resulted n 9% less nvestments n fxed-ncome funds compared to 008. One may argue that even Pennngmark Eur Lux Ack s a nterest-bearng fund. That s true, but for a Swedsh nvestor t s more lkely to be consdered as a FX fund n Euro. 3

33 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos Table : OP1 January 009 Adjusted Mutual Funds Weghts Weghts Lkvdtetsfond SEK 9,5% 9% Pennngmarknadsfond SEK 8,% 8% Pennngmark Eur Lux Ack 16,81% 17% ÖstEuropafond 3,70% 4% Realräntefond SEK Lux Utd,18% % ÖstEurfondxRyssland Lux Ack 39,84% 40% Summary shows that both portfolos returned postve wth almost same level of rsk. OP1 ncreased by 41,61% whch was 8,59% hgher then FP100. Wth only 1,57% more rsk whch resulted n hgher Sharpe-rato. Table 4.1.1: Summary of data 009 Portfolo Data FP100 OP1 Return 33.0% 41,61% Rsk 11,54% 13,11% Sharpe Rato,7 3,05 FP100 vs OP By daly reallocaton and wth portfolos that reflected the last recommended or optmzed portfolo an nvestor where slghtly better of followng SEB Investment Strategy s suggestons. FP100 decreased by % whereas OP1 resulted n 8,99% loss. OP1 was,47% rsker. Data summarzed n followng table and graf. Table : Summary of data Portfolo Data FP100 OP1 Return -,00% -8,99% Rsk 1,96% 15,43% 0,6 0,4 0, 0-0, -0,4-0,6 FP100 OP1 33

34 Evaluatng SEB Investment Strategy s Recommended Mutual Fund Portfolos 4. FP70 vs OP Ths tme we are gong to compare recommended portfolo wth 30% fxed level of nterestbearng funds wth OP optmzed n the same manner as OP1 but wth % added to actual rsk-free rate at the tme for optmzaton process. FP70 vs OP 006 Sx portfolos was recommended by SEB IS durng 006. Oblfond Flex SEK Lux Ack was the fxed ncome fund choosed as a way to lower the rsk n the portfolo for makng t sutable to nvestors wth shorter nvestment horzon or more reluctant to rsk. Table 4..1: FP70 durng 006 Weghts Mutual Funds Jan Feb-Apr Maj Jun-Aug Sep-Nov Dec Oblfond Flex SEK Lux Ack 30% 30% 30% 30% 30% 30% Svergefond 35% 35% 35% 35% 35% 35% Globalfond 19% 19% 15% 15% 15% 15% Emergng Markets 7% 7% 7% 7% 7% 7% Europa 9% 9% 9% 9% Läkemedelsfond 5% 4% Nordenfond 4% 4% Japan C/R Lux Ack 5% 4% BlackRock Global Opp 4% Internetfond 4% The optmzed alternatve gven RFR 1,65% + % contaned nearly same degree of fxedncome to equty funds 8/7 compared to above portfolos whch contaned 30/70. Same mutual funds as OP1 durng 006 where ranked among top sx wth only dfference beng the porton nvested n them. Table 4..: OP January 006 Adjusted Mutual Funds Weghts Weghts Pennngmarknadsfond SEK 5,0% 5% Lkvdtetsfond SEK,48% 3% Fastghetsfond 34,30% 34% ÖstEurfond Småbo Lux Ack 34,9% 34% Svergefond 0,48% 1% Nordenfond 3,43% 3% 34