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Bayesian Kelly Criterion As an Allocation Strategy in Finnish Stock Markets Bayesian Kelly criterion as an allocation strategy in Finnish stock markets Jyväskylä University School of Business and Economics Master’s thesis 2020 Author: Risto Heikkinen Subject: Economics Supervisors: Juha Junttila & Juhani Raatikainen 1 ABSTRACT Author Risto Heikkinen Title Bayesian Kelly criterion as an allocation strategy in Finnish stock markets Subject Type of work Economics Master’s thesis Date Number of pages 10.11.2020 41 The Kelly criterion is an investment strategy, which aims to maximize long-term capital growth rate. There are known limitations in applying it’s original version such as high short-term volatility and uncertainty in estimating future returns. These limitations are traditionally tackled by risking only a fixed fraction of the proposed amount of the capital, but there is no clear consensus in published articles as how to choose this fraction. This thesis combines many earlier presented extensions to the Kelly criterion. These are e.g. estimating the uncertainty of future reaturns with a Bayesian model and a heavy tailed t-distribution and controlling short-term risk with a security constraint based on an investor’s risk tolerance. The end result is an algorithm which optimizes an appropriate allocation between a stock market index and a risk-free rate. In addition, there is a version of the algorithm, where it is possible to allocate the capital to one stock where the investor sees a special potential. In this thesis the potential is measured as the analysts’ target prices and the historical value of this target price information. The developed Bayes-Kelly method is applied to the Finnish stock market and it’s performance is studied during the years 2010 – 2019. The development of the capital which is annually re-allocated to the market index, is compared to the capital which is re-allocated based on the traditional portfolio theory and to a simple strategy where the weight of the risky asset is always 50%. One scope of interest is the portfolio of the Finnish government’s holding company Solidium. It’s performance during the same time period is also a benchmark to Bayes-Kelly strategy. As the last aim of the study, the performed value of adding individual stock to the portfolio based on the analysts’ target prices is investigated. The result was that the Bayes-Kelly strategies’ risk adjusted performance was better than any of the benchmark strategies’ performance based on the Sharpe ratio. Solidium’s wealth accumulation trajectory was similar with the Bayes-Kelly strategy where the accepted annual capital loss was 20%. Because of higher volatility, Solidium had a lower Sharpe ratio. Adding individual stock with the highest potential to the portfolio did not have much effect on the terminal wealth. Adding individual stock based on the analyst target prices reduced the portfolios’ volatility, and hence, improved the Sharpe ratio. Key words Portfolio allocation, Bayesian analysis, Kelly criterion, Stock markets Place of storage Jyväskylä University Library TIIVISTELMÄ Tekijä Risto Heikkinen Työn nimi Bayesläinen Kellyn kriteeri allokointistrategiana Suomen osakemarkkinoilla Oppiaine Työn laji Taloustiede Pro-gradu tutkielma Aika Sivumäärä 10.11.2020 41 Kellyn kriteeriksi kutsutaan sijoitusstrategiaa, jossa tavoitteena on varallisuuden kasvuvauhdin maksimointi pitkällä ajanjaksolla. Sen alkuperäisen version soveltamiseen liittyy heikkouksia kuten suuri varallisuuden vaihtelu lyhyellä ajanjaksolla ja epävarmuus tulevaisuuden tuottojen arvioimisessa. Näitä puutteita on aiemmin paikattu jakamalla riskisijoitusten suuruutta kiinteällä vakiolla, mutta tämän vakion suuruuden valintaan ei ole selvää konsensusta aiemmissa tutkimuksissa. Tässä työssä yhdistetään monta aiemmin ehdotettua laajennusta Kellyn kriteerille. Näitä ovat mm. tulevaisuuden tuottojen epävarmuuden arvioiminen Bayes-mallilla ja paksuhäntäisellä t-jakaumalla sekä lyhyen tähtäimen riskien hallinnointi sijoittajan mieltymyksiin sopivalla turvarajoittella. Lopputuloksena on algoritmi, jonka avulla määritellään sijoittajalle sopiva allokaatio osakemarkkinaindeksin ja riskittömän koron välillä. Lisäksi menetelmästä kehitetään versio, jossa edellisten lisäksi sijoittaja voi allokoida varojaan yksittäiseen osakkeseen, missä näkee erityistä potentiaalia. Työssä potentiaalia mitataan analyytikkojen tavoitehintojen ja tavoitehintojen tuoman historiallisen lisäarvon avulla. Työssä kehitettyä Bayes-Kelly menetelmää sovelletaan Suomen osakemarkkinoille ja tutkitaan sen toimivuutta vuosien 2010 - 2019 aikana. Vuosittain uudelleen markkinaindeksiin allokoidun varallisuuden kehitystä vertaillaan perinteisen portfolioteoriaan pohjautuviin allokointipäätöksiin sekä yksinkertaiseen strategiaan, jossa osakepaino on aina 50%. Mielenkiinnon kohteena on myös Suomen valtion sijoitusyhtiö Solidiumin performanssi kyseisenä ajankohtana ja sen saavuttama tuotto on viimeinen vertailukohta. Lopuksi tutkitaan vielä kuinka paljon lisäarvoa olisi tuottanut yksittäisen osakkeen lisääminen allokaatioon analyytikkojen tavoitehintojen perusteella. Lopputuloksena saatiin, että Bayes-Kelly strategiat tuottivat riskikorjattuna paremmin kuin kaikki vertailumenetelmät Sharpen luvun perusteella. Solidiumin pääoman kehitys vastasi lähes tulkoon Bayes-Kelly strategiaa, jossa sijoittajan hyväksyttävä vuosittainen tappio on 20% varallisuuden arvosta. Suuremman tuottojen vaihtelun takia Solidium kuitenkin hävisi Sharpe- lukujen vertailussa. Eniten potentiaalia omaavan osakkeen lisääminen portfolioon tavoitehintojen perusteella ei vaikuttanut juurikaan loppuvarallisuuseen. Osakkeen lisääminen vähensi tuottojen vaihtelua ja siten paransi hieman Sharpen lukua. Asiasanat Portfolio allokaatio, Bayes analyysi, Kellyn kriteeri, Osakemarkkinat Säilytyspaikka Jyväskylän yliopiston kirjasto Contents 1 Introduction5 2 Theoretical framework7 2.1 Modern portfolio theory.........................7 2.2 Historical review of Kelly criterion....................9 2.3 Kelly growth criterion.......................... 10 2.4 Criticism towards modern portfolio theory and Kelly criterion.... 14 2.4.1 Risk aversion problems...................... 14 2.4.2 Plug-in estimation problems................... 15 2.5 Bayesian statistical analysis....................... 15 3 Data and methology 16 3.1 Model for market index investor..................... 19 3.2 Model for a stock with special potential................. 20 3.3 Bayesian solution and optimization................... 23 4 Results and analysis 24 4.1 Results for the market index investor.................. 25 4.2 Results with individual stock selections................. 27 5 Conclusions 29 A Characteristics of the data on individual stocks 36 2 List of Tables 1 Statistics of monthly changes of stock and index prices, including dividends.................................. 18 2 The performance of market index investors for each strategy...... 27 3 Stock selection for each allocation decision................ 28 4 The performance results for different Bayes-Kelly strategies...... 29 A1 Stock information and parameter estimates at the beginning of March 2010..................................... 36 A2 Stock information and parameter estimates at the beginning of March 2011..................................... 37 A3 Stock information and parameter estimates at the beginning of March 2012..................................... 37 A4 Stock information and parameter estimates at the beginning of March 2013..................................... 38 A5 Stock information and parameter estimates at the beginning of March 2014..................................... 38 A6 Stock information and parameter estimates at the beginning of March 2015..................................... 39 A7 Stock information and parameter estimates at the beginning of March 2016..................................... 39 A8 Stock information and parameter estimates at the beginning of March 2017..................................... 40 A9 Stock information and parameter estimates at the beginning of March 2018..................................... 40 A10 Stock information and parameter estimates at the beginning of March 2019..................................... 41 3 List of Figures 1 Sample of possible portfolios (black points) and the capital allocation line (CAL, as a red line)..........................8 2 Capital growth rate with different bet sizes as a fraction of current wealth.................................... 11 3 Distribution of the monthly aggregate market (OMXHGI) returns during 2000-2019 (black line), it’s normal approximation (red line) and Bayesian predictive posterior distribution based on t-distribution and stochastic parameters (green line).................. 24 4 Weights of the market index with different strategies.......... 25 5 Growth trajectories of $1000 initial capital invested in 2010 with dif- ferent strategies............................... 27 6 Growth trajectories of $1000 initial capital invested in 2010 with dif- ferent Bayes-Kelly strategies........................ 29 4 1 Introduction In financial literature, traditionally optimal asset allocation decisions are based on the mean-variance approach which has roots in modern portfolio theory (MPT) (Markowitz, 1952). In its simplest form it is a myopic strategy that optimizes ex- pected future utility of wealth. A competing approach, as an answer to an argument that investor’s life path is not an ergodic process (Peters, 2019), is based on maxi- mization of investor’s long term capital growth rate. Kelly criterion, which has roots
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