Bayesian Kelly criterion as an allocation strategy in Finnish stock markets

(1)

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

(2)

ABSTRACT

Author

Risto Heikkinen

Title

Bayesian Kelly criterion as an allocation strategy in Finnish stock markets

Subject Economics

Type of work Master’s thesis

Date 10.11.2020

Number of pages 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

(3)

TIIVISTELMÄ

Tekijä

Risto Heikkinen

Työn nimi

Bayesläinen Kellyn kriteeri allokointistrategiana Suomen osakemarkkinoilla

Oppiaine Taloustiede

Työn laji

Pro-gradu tutkielma

Aika 10.11.2020

Sivumäärä 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

(4)

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

(5)

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

(6)

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

(7)

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 expected 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 in the work of John Kelly (Kelly, 1956), is a popular bet sizing strategy amongst the professional sport bettors and blackjack players (Thorp, 2006; Vuoksenmaa et al., 1999). It is also noted as a preferable point of view for risk taking with multiplicative wealth process by Peters (2019).

According to Ziemba (2003) Kelly’s criterion has been successfully used as a stock market investment strategy by investors like Warren Buffet, Edward Thorp and John Maynard Keynes. Also a hedge fund (The Medallion Fund) manager James Simons has been noted as a successful practitioner of Kelly criterion in investing (Ziemba and Ziemba,2007).

The main aim of this thesis is to compare the performance of a developed Kelly criterion investment strategy with the standard version of a mean-variance approach, with a naive benchmark strategy and with the portfolio of the Finnish government’s holding company Solidium. Solidium is stating at its web page that it is a long term investor. In addition, as an institutional investor its investment horizon is not restricted to human life. Solidium’s portfolio is an interesting benchmark for Finnish people because all Finnish taxpayers are implicitly owners of the Solidium equities, too, through the government.

From a practical point of view, this thesis aims to support index fund investors, who believe in the efficient market hypothesis (Fama,1970) or who are not otherwise interested in stock picking. Furthermore, it also serves investors who have some unpriced information of one stock. Useful information could be e.g. analyst ratings or knowledge that is emerging from a day job, a hobby or another special interest as described in the investing bestseller "One Up on Wall Street" (Lynch and Rothchild, 2000).

The original Kelly criterion, as well as the conventional MPT based approaches, have problems such as the uncertainty of parameters that are estimated from historical data or unwanted short term volatility of capital (MacLean et al., 2011a). This work offers a solution for these two problems at the same time by using Bayesian statistical analysis with historical data and adding security constraint which allows investors to control the portfolio’s risk level. Bayesian analysis allows also combin-

(8)

ing information from different types of data, and in our case, analyst target price information is used for stock picking together with historical returns. Most articles about the application of Kelly criterion in investment strategies model stock price fluctuations with geometric Brownian motion assuming normal distribution (Browne and Whitt, 1996; Rotando and Thorp, 1992). It is known that most of the price fluctuations follow a more fat tailed distribution (Mandelbrot and Hudson, 2004;

Taleb et al., 2020). Osorio (2009) has improved the standard Kelly criterion using a fat tailed t-distribution. Even after sixty years since the Kelly criterion was first introduced there is still no clear consensus in published articles as to how it should be applied to the stock markets. The contribution of this thesis is to combine many promising improvements to the original idea. The solution includes parameter uncertainty with Bayesian analysis, security constraint, fat tailed t-distributions and additional information as analysts’ target prices.

The investment strategy in this study is based on allocating capital between the risk-free asset, the Finnish stock market general index and selected individual Finnish stocks. I am going to compare the model’s performance to the performance of the Finnish government’s holding company (Solidium) portfolio and use the same stocks that are the key assets in the Solidium portfolio. Thus, the pool of possible stocks consists of companies that had at least 10% Finnish government ownership in December 2019. The other requirement is that the stock has to have at least 10 years of analyst coverage. Selected stocks present nationally significant companies and their weight in the total market value of the companies in OMX Helsinki index (OMXHPI) was 8% at the end of 2000 and 36% at the end of 2019 (source: Thomson Reuters Datastream).

After applying this approach to real data, I am going to compare its performance against the mean-variance and a naive benchmark strategy, in addition to the performance of Solidium portfolio. As a primary result we are going to see, how well this Bayes-Kelly strategy compares with the competing strategies during the period of the years 2010-2019. As a secondary result we are going to see how much the individual government owned stocks and analyst target prices added value during the same time period compared to investing only in the market index.

As the primary result, during the ten year period we find that all Bayes-Kelly market index strategies performed better than simple mean-variance strategies or a naive benchmark strategy based on Sharpe ratio. The Solidium portfolio achieved similar wealth accumulation trajectory as the constrained Bayes-Kelly market index strategy, where the acceptable yearly loss is 20%. However, because of higher volatily, Solidium’s Sharpe ratio was lower than all Bayes-Kelly and the naive bench-

(9)

mark strategies. As a secondary result, stock selection based on analyst target prices had negligible effect to capital growth but it reduced the volatility of Bayes-Kelly strategies. In addition, the Bayesian model estimated high systematic bias in the mean target prices.

In what follows, I first present the main ideas of modern portfolio theory, Kelly criterion and then the role of Bayesian statistical models in the analysis in Section 2. After that I present the data and models for decision making in Section 3. In Section 4 I show how the competing allocation strategies performed with a real data for investor whose only risky asset is the market index. In addition, I show the performance of stock picking strategies. Finally, Section 5 concludes with the results, discusses about the weaknesses of this approach and offers some ideas how this model could be improved in future studies.

2 Theoretical framework

Here we take a look at the theoretical background of modern portfolio theory and Kelly criterion. We also discuss the criticism towards both of these approaches.

Finally we take a look at Bayesian statistical methods that have been proposed as solution to some of the practical problems in these portfolio allocation approaches.

In mathematical formulations, frictions, like taxes and transaction costs, are ignored.

2.1 Modern portfolio theory

Harry Markowitz introduced the modern portfolio theory (MPT) (Markowitz,1952), where investment portfolio’s performance is measured based on expected return during the holding period and the risk is measured as the standard deviation of annualized returns. The idea is to maximize expected return given the chosen risk level. This methodology is later developed towards single objective decision making problem with utility function that describes investor’s personal risk aversion preferences. Within this framework investor’s objective is to maximize her expected utility (Brandt, 2010).

As a recap and background for the introduction of Kelly criterion, let’s consider a simple case where it is possible to allocate capital to two stocks A and B that have expected returns (r) for one year asE(r_A) = 0.06andE(r_B) = 0.08. Standard deviations of annualized returns are σ_A= 0.15and σ_B = 0.2. An investor allocates proportion w_A of his or her stock portfolio to stock A and proportion w_B = 1−w_A to stock B. In Figure 1 the black points present stock portfolios where w_A has different values between[0,1]. Standard deviations of portfolios are given on X-axis

(10)

end expected profits on Y-axis. According to MPT, efficient portfolios are the ones that have maximum possibleE(r)given standard deviation σ. Thus, for example a decision where w_A = 1 offers an allocation where σ = 0.15 and E(r) = 0.06 and it is not efficient because it is possible to reach higher E(r) with the same risk level.

Instead, decision w_A = 0.4 offers a portfolio where σ ≈0.15 and E(r)≈ 0.072 and it is efficient, as we see from Figure1. In this simple example all the stock portfolios that offerE(r)>0.066 are efficient.

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.00 0.05 0.10 0.15 0.20

Sd(r)

E(r)

Sample of portfolios and capital allocation line

Figure 1: Sample of possible portfolios (black points) and the capital allocation line (CAL, as a red line).

Sharpe (1964) improved MPT by adding a possibility to lend or borrow money with a risk-free rate (r_f). Now it is possible to reach different (E(r), σ) - combina- tions by allocating part of the capital to risk-free asset and part of the capital to risky assets. Efficient portfolios are described with a capital allocation line (CAL) that is tangent to the efficient risky portfolios and crosses the Y-axis at (r_f), i.e.

risk free rate of return. In our example the risk-free rate r_f = 0.01 and CAL is a red line in Figure 1and it hits the risky portfolios at the point where E(r)≈0.069 andσ ≈0.14. This portfolio is called the "tangent portfolio" and it is reached when w_A≈0.55. Mathematical formulation for CAL is

E(r) = r_f +SR·σ, (1) where SR is the Sharpe ratio

E(r_M)−r_f σM

(2) and M is the tangent portfolio. An investor can pick any portfolio from CAL by allocating capital to a risk free asset and a tangent portfolio. Allocations at CAL

(11)

that offer higher expected returns than the tangent portfolio, needs the possibility to borrow money with a rate (r_f).

Final decision for the allocation is based on investor’s utility function U. Because in most cases exact forms of utility function are not known, a practical ways is to decision is to assume concave utility function, approximated (Levy and Markowitz, 1979) by

U(E(r), σ) = E(r)− A

2σ², (3)

where A is a fixed parameter that describes the level of investor’s risk aversion.

Thus, investor chooses the portfolio that maximizes this utility function. Because the mean and the variance of annual returns have such a big role, the traditional Markowitz’s portfolio theory is often called the mean-variance approach.

2.2 Historical review of Kelly criterion

John Larry Kelly was a colleague of a mathematician and the father of the information theory Claude Shannon in Bell Labs. Kelly introduced how Shannon’s ideas in information theory could be capitalized in sport or horse race betting in his seminal paper (Kelly, 1956). Kelly’s idea was developed further byBellman and Kalaba (1957) and Breiman (1961) and they showed that maximizing logarithmic utility function maximizes risk taker’s long run growth of wealth and minimizes the expected time to reach arbitrarily large wealth targets.

Math professor Edward Thorp is a famous practitioner of Kelly criterion. He developed a card counting system for blackjack that he published in Thorp (1966).

During the publication process he met Claude Shannon, who introduced Kelly criterion to Thorp. Shannon and Thorp found common interest in gambling and they developed a wearable computer for tracking roulette ball’s movement and estimating its final position on the wheel. Together they implemented Kelly’s idea to bet sizing in roulette with the help of their computer in Las Vegas during 1960s. Thorp used Kelly criterion also in his investment strategy when he was working as a hedge fund manager since 1970s. He published first time his ideas of Kelly criterion in investing in Thorp (1971). During these years Thorp played card game Bridge and shared investing ideas with a famous investor Warren Buffett. Buffett also included Kelly criterion to his toolbox in decision making. ¹

During 1980s William Ziemba developed and applied horse racetrack strategies that are based on Kelly criterion together with Donald Hausch and Mark Rubinstein

1Background information of John Kelly, Claude Shannon, Edward Thorp and Warren Buffett is based on Thorp’s biographical bookThorp(2017) andMacLeanet al.(2011b).

(12)

(Hausch et al., 1981; Hausch and Ziemba, 1985). Ziemba also has analyzed stock investors’ performance. In addition to Edward Thorp and Warren Buffet, Kelly Criterion has been successfully used as stock investing strategy by John Maynard Keynes and James Simons (Ziemba, 2003; Ziemba and Ziemba, 2007).

Next we are going introduce idea of the Kelly criterion, criticism towards it and some later developments.

2.3 Kelly growth criterion

A long-term risk taker’s typical ultimate object is to maximise average exponential growth of wealth (^V_V^N

0)^N¹, where V₀ is the initial wealth and V_N is wealth after N bets. One bet may refer to a unique bet for a gambler or a portfolio allocation decision for one holding period for a stock investor. For optimization purposes, using a monotonic logarithm function is equivalent to optimizing the original object. As John Larry Kelly originally defined (Kelly,1956), objective for the long-term investor is to maximize

G= lim

N→∞

1

N log(V_N

V₀). (4)

Let us first illustrate Kelly’s idea with a simple gambler’s bet sizing problem. A rich friend offers a gambler an opportunity to toss a fair coin. Her winning would be 2.11 times the wager amount with "heads" and would lose a bet with "tails". The gambler has the chance to repeat the game as many times as she wants. Expected value of profit is now pk = 0.5·2.11 = 1.055, where p = winning probability and k is the decimal odds. How big proportion of her initial wealth the gambler should bet? AfterN repetitions gambler’s wealth in this multiplicative process is

VN = (1 + (k−1)·w)^H(1−w)^TV0, (5) wherewis a bet size as the proportion of current wealth, H is the number of heads and T is the number of tails in N tosses. Following Kelly’s formulation

G = lim

N→∞

H

N log(1 + (k−1)·w) + T

N log(1−w)

(6)

= plog(1 + (k−1)·w) +qlog(1−w), (7) where p = probability of heads and q = probability of tails. Maximizing G with respect towgives a solution for the gambler’s problem generally (Baker and McHale, 2013) as

w^∗ = (k−1)·p−q

k−1 (8)

and in our example, where k= 2.11and p=q= 0.5 w^∗ = 1.11·0.5−0.5

1.11 ≈0.05 (9)

(13)

Thus, if the gambler’s current wealth is $100, she has to place a bet of $5 in order to maximize the long-term capital growth rate.

A critical point w_c is size of the bet making long-term growth equal to zero even though one bet is profitable on average. We need to solve the equation:

plog(1 + (k−1)·w) +qlog(1−w) = 0. (10) In addition to a trivial solutionw= 0, numerically it is possible to find a solution in our example to bew_c≈0.1. In Figure2there is an illustration of the functionG(w) for our example. We can see that, if the gambler bets more than 10% of her wealth, the wealth reduces in time even though the expected profit for one bet is positive.

We can illustrate this phenomenon by a simple example where the gambler tosses a coin two times and gets the expected result: one head and one tail (the order does not matter). We can calculate the end result with three different scenarios applying formula5 as

1. w= 0.025: V₂ = (1 + 1.11·0.025)(1−0.025)·$100≈$100.2 2. w= 0.075: V₂ = (1 + 1.11·0.075)(1−0.075)·$100≈$100.2 3. w= 0.15: V₂ = (1 + 1.11·0.15)(1−0.15)·$100≈$99.2

−0.008

−0.004 0.000

0.00 0.05 0.10 0.15 0.20

w

G

Zones

Danger zone Risky zone Safe zone

Capital growth function G(w)

Figure 2: Capital growth rate with different bet sizes as a fraction of current wealth. Green color refers to safe zone, where positive growth is reached with small risk, yellow color refers to positive growth with high risk. Danger zone, where growth rate is negative, is drawn in red.

(14)

These scenarios are plotted in Figure 2 as the grey vertical lines. In scenario 2 the gambler has profited as much as in scenario 1 but she has been taking more risk. In scenario 3 she has lost money even though expected profit for one bet is positive and she has faced neutral luck in coin toss. Here we see the conflict between short-term gambling and long-term gambling. For one coin toss the gambler maximizes the expected profit by betting all wealth because

E(V₁) = 0.5·(1 + 1.11·w)V₀ + 0.5·(1−w)V₀ (11)

∂E(V₁)

∂w = (1.11−1)·0.5V₀ = 0.055V₀ >0 (12) but this kind of action is fatal for long-term capital growth.

Theorem 1 inThorp(2006) includes a couple of important features based on the findings inBreiman (1961) and Thorp (1969).

1. If g(w)>0, then limN→∞VN =∞almost surely;

2. If g(w)<0, then lim_N→∞V_N = 0 almost surely;

3. Given a strategy Φ^∗ which maximizes G and any other “essentially different” strategy Φ (not necessarily a fixed fractional betting strategy), then limN→∞ VN(Φ^∗)

V_N(Φ) =∞; and

4. The expected time for the current capital V_N to reach any fixed preassigned goal C is, asymptotically, least with a strategy which maximizes G.

In other words, betting less than wc assures that wealth is going to grow in long- term and betting more than w_c assures losses in long-term. Also Kelly criterion is giving higher growth of wealth in infinite time horizon more than any other betting strategy and it also minimizes the expected waiting time to reach some fixed wealth goal C.

One more important feature of Kelly’s criterion is the equivalence with maximizing expected logarithmic utility function,log(V), with any finite investment horizon (Bellman and Kalaba,1957). In the coin toss example this connection is easy to see as

Elog(V₁) = plog(1 + (k−1)·w) +qlog(1−w), (13) which is identical to Formula 7. Because of this feature Kelly’s criterion is often called the "expected log" approach.

Expected log utility as a stock investing criterion has been discussed e.g. by Thorp (2006) and Browne and Whitt (1996). They demonstrated an allocation

(15)

between a single risky asset and a risk-free asset. Within this restriction, wealth at time periodt is

V_t=V_t−1[1 +r_f(1−w_t) +w_ty_t] =V₀

t

Y

i=1

[1 +r_f(1−w_i) +w_iy_i], (14) where y_t is a return of a stock at time period t. If the future stock prices follow geometric Brownian motion, optimal fraction of wealth invested to risky asset is

w^∗ = E(y)−r_f

σ² , (15)

whereE(y)is an expected annual growth rate of a risky asset andσ²is the variance of return. Byrnes and Barnett(2018) introduced a generalized framework for applying the Kelly criterion to stock market. With a multivariate log-normal distribution of future wealth, the solution for a portfolio of two correlated stocks withr_f = 0 is

"

w^∗₁ w^∗₂

#

=M⁻¹b. (16)

NotationM refers to a 2x2 matrix, where

M_ll = 1−2e^µ^l+e^2µ^l^+σ²^l, l ∈ {1,2} (17) and

M₁₂ =M₂₁ =e^µ¹^+µ²^+ρσ¹^σ² −e^µ¹ −e^µ² + 1, (18) whereρ is the correlation between stock returns. Finally

b_l = (e^µ^l−1), l∈ {1,2}. (19) The main difference between the the mean-variance approach (discussed in 2.1) and Kelly approach is that Kelly criterion is derived from infinite investment horizon but mean-variance is derived from a fixed finite investment horizon, and thus, it may lead to a non-optimal solution in long-term. Analogously, the Kelly criterion is optimal as a myopic strategy only if the utility function U = log(V) describes an investor’s single period risk preferences. The optimal weight of risky assets in mean-variance approach is

w_M^∗ = E(rM)−rf

Aσ²_M , (20)

where the relative risk aversion coefficient A = ^−U

00

U⁰ , where U⁰ and U⁰⁰ refer to the first and second derivatives of the utility function respectively. With a logarithmic utility functionA= 1 and thus we end up to a similar result as with Kelly criterion (Formula15) in this special case. More about these connections have been discussed byBaz and Guo (2017).

(16)

2.4 Criticism towards modern portfolio theory and Kelly cri- terion

Traditional versions of modern portfolio theory (mean-variance approach) and Kelly criterion have faced lot of criticism during last decades. DeMiguel et al. (2007) showed that a simple "1/N strategy", that allocates capital evenly to every available asset, has outperformed optimized mean-variance portfolios. Edward Thorp, the early practitioner of Kelly criterion, argued inThorp(1971) that mean-variance may suggest a portfolio that has negative growth in long term. In other words, this means lying in a "danger zone" in Figure2. Markowitz (1976) himself admitted that with a long-term investing strategy investor should filter out portfolio suggestions, where the weight of a risky assets is higher than Kelly criterion suggests. Mean-variance has also been criticized, because it ignores other features of profit distribution than mean, variances and correlation coefficients that are sufficient only with a multinormal distribution. Jondeau and Rockinger (2006) discussed about the effects of a non- normal profit distribution and developed a method that takes into account also skewness and kurtosis of distributions, in addition to the mean and variance. This approach has also limitations because, asTaleb et al.(2020) state, fat tailed return distribution may not have higher moments. Even variance or mean may not have finite values with the most problematic distributions.

2.4.1 Risk aversion problems

Kelly criterion is a very volatile strategy as stated by MacLean et al. (2011a). For short term investing it is optimal only if inverstor’s relative risk aversion coefficient A = 1. According to Gandelman and Hernandez-Murillo (2015) A = 1 is a very good approximation of global average, but it varies wildly between 0.2 - 10. In addition, studies in prospect theory (Kahneman, 2013) claim that normally risk- averse people typically start to seek risks after facing losses. This kind of behavior could be dangerous for long-term capital growth as discussed in Section2.3. To avoid problems with short-term risks, it is commonly recommended to use a "fractional Kelly" strategy, where risky portfolio weights are divided by some fixed fractionf to reduce the risk level. There are multiple suggestions for the selection off. MacLean et al.(1992) used acceptable level of risk measureP(Capital doubles before halves) to derive the value of f and MacLean et al. 2011a based the selection to absolute risk aversion level of power utility function U(V) =δV^δ.

In practise it is not easy to define one’s own risk aversion coefficient. As a practical solution, MacLean et al. (2004) developed Kelly criterion by adding a

(17)

security constraint to the optimization process. Investor can choose an acceptable annual percentage loss, b, for one investment period. This "Kelly growth with security" -strategy maximizes capital growth with a constraintP(V_t+1 <(1−b)V_t)<

α^r), whereα^r is some risk level, for example0.05. This method could also be called as one form of "Value at risk" method.

2.4.2 Plug-in estimation problems

Probably the most critical weakness of both approaches, mean-variance and Kelly criterion, is that stock investors don’t know the real values of expected outcomes, variances and correlations that are needed for optimization. These parameters are usually estimated from historical data. These plug-in solutions, where unknown parameters are replaced with historical values, may be very misleading for the future and thus economically costly to the investor (Brandt, 2010; Kan and Zhou, 2007).

Brandt (2010) illustrates this with a simple example, where the real expected next year’s return is 6% with the real standard deviation 15%. Now if the risk aversion A = 5, the real optimal weight for risky asset w = 53.3% based on mean-variance analysis. If there are 10 years of monthly data, the standard deviation of the estimate sd( ˆw) = 14%and that is calculated in ideal conditions with log-normally distributed returns.

DeMiguel et al. (2007) claim that in reality, there is never enough historical data for mean-variance approach to work in practise as well as in theory. Kelly investor’s basic solution is again "fractional Kelly" to lower the risk that is caused by estimation errors in parameters. However, there is not much literature about choosing the correct fraction f so that it takes into account a possible bias in the parameter estimates and investor’s risk aversion level. Osorio (2009) introduced a risk control solution based on prospect theory. It requires that an investor knows her personal prospect exponent parameter which is not very common.

For the problem of parameter uncertainty, common solution is to use Bayesian statistical analysis. Avramov and Zhou (2010) reviewed different Bayesian versions of the mean-variance approach. In addition, Browne and Whitt (1996) introduced the Bayesian version of Kelly criterion. In the next section we discuss more about the Bayesian analysis.

2.5 Bayesian statistical analysis

In Bayesian scientific philosophy the data are the fact that we observe and the parameters in a model are stochastic variables. Bayesian statistical models are

(18)

based on probability theorem (Gelmanet al., 2013) stated as P(θ |Y) = P(Y |θ)P(θ)

P(Y) , (21)

whereθ represents the vector of unknown parameters (e.g. future mean return and variance) andY the observed data (e.g. historical price fluctuations).

The benefits of using Bayesian models compared to the traditional ones are a possibility to use prior information (P(θ)) that we have before collecting the data, a possibility to combine information from different data sources in one model, a possibility to ask wide range on probability questions, parameter uncertainty is included to predictions, wide range of different models are possible to solve numerically with Markov chain Monte Carlo (MCMC) -methods and usually easier to handle missing values in data as stochastic parameters. Gelmanet al. (2013) summarize: "A prag- matic rationale for the use of Bayesian methods is the inherent flexibility introduced by their incorporation of multiple levels randomness and the resultant ability to combine information from different sources, while incorporating all reasonable sources of uncertainty in inferential summaries. Such methods naturally lead to smoothed estimates in complicated data structures and consequently have ability to obtain better real-world answers."

3 Data and methology

In our empirical analysis we will apply an improved Kelly criterion allocation strategy to Finnish stock markets. We are going to handle the parameter uncertainty by using Bayesian model and control the short-term risk by using the same "value at risk" method as MacLean et al. (2004). We are going to model the future returns as geometric Brownian motion likePraez (1972) where random shocks are assumed to follow a possibly fat tailed Student t-distribution instead of normal distribution.

We are following the idea of Treynor and Black (1973), where the core investment is a market index (OMXHGI in our case) fund but it can be accompanied by an individual stock, where an investor sees extremely high potential. In this work, potential is measured by analyst ratings. The number of individual stocks is limited to only one at the time because modelling a correlation structure of multiple stocks increases model’s complexity and we have only limited amount of data.

The important decision is also the weight of capital allocated to the risk free rate. We are using one year as the re-allocation period and thus the risk free rate is the 12-month Euribor rate. Weight of the risky asset is allowed to be more than one (w > 1) that implies taking a debt with a risk free rate. Furthermore, short

(19)

selling of stock is not allowed, and thus, there is a restriction w ≥ 0. In the case of negative risk-free rate, the investor is allowed to use cash. Capital is going to be re-allocated once in a year. A new period starts at the beginning of March because that is the time of the year, when most of the companies have reported their annual result and the analysts have fresh information in their hands.

First, we are going to allocate capital only between the market index and risk- free rate. After that we apply the strategy of allocating capital between the risk-free rate, market index and one stock at he time with special potential. After applying the strategies to historical data, we are going to compare each one’s performance with

1. a diversified "1/N strategy" (DeMiguel et al., 2007),

2. the mean-variance strategy with relative risk aversion rateA= 1 which is also a simple version of Kelly criterion (see Equation (20))

3. more risk averse mean-variance strategy withA = 5, which is the risk aversion rate, used in Brandt (2010)),

4. more risk seeking mean-variance strategy with A= 0.5 and with

5. Finnish government’s holding company Solidium during the same time period.

We are using as stock candidates the companies where the Finnish government had at least10% ownership at December 2019 because those companies are also the key holdings in the Solidium portfolio. We are using 20 years of monthly historical data from 1.1.2000 until 31.12.2019 from Thomson Reuters Datastream. Decision to restrict data only from this century comes naturally because Finland joined the Euro area in 1999 and since that the Finnish stock market has been working without major changes and individual firms have had a better analyst coverage than in the twen- tieth century. Historical performance of the Solidium portfolio is copied manually from a graph in their web page (https://www.solidium.fi/fi/omistukset/osakesalkun- tuoton-kehitys/), so it’s performance may include some minor tracking error in Sec- tion 4.

We are going to base each investment decision to 10 years of history data and thus companies that have less than 10 years history as listed companies with analyst coverage, have been filtered out. In Table 1 is reported the descriptive statistics of logarithmic returns. Reported statistics are annual mean, standard deviation, number of observations, autocorrelations (lag = 1,2,3,12) and correlation with the OMHX-index-return.

(20)

Table 1: Statistics of monthly changes of stock and index prices, including dividends.

Mean = Mean of annualized returns, Std = monthly standard deviation, N = number of observations, AC1 = autocorrelation of returns with one month lag, AC2 = autocorrelation of returns with two months lag, AC3 = autocorrelation of returns with three months lag, AC12 = autocorrelation of returns with 12 months lag and CorOMXH = correlation between OMHX-index.

Mean Std N AC1 AC2 AC3 AC12 CorOMXH

OMXHGI 0.02 0.07 240 0.19 -0.13 -0.03 0.01 1.00

NESTE 0.13 0.08 176 0.01 -0.06 0.13 0.04 0.55

SAMPO 0.11 0.05 240 -0.13 0.04 0.01 -0.08 0.55

ELISA 0.03 0.08 240 0.08 -0.02 0.04 0.02 0.54

METSO 0.08 0.08 240 0.10 -0.07 0.19 -0.00 0.52

TIETOEVRY -0.02 0.09 240 0.05 -0.16 0.03 -0.01 0.51

STORAENSO 0.01 0.07 240 0.00 -0.13 0.06 0.06 0.46

OUTOKUMPU -0.02 0.08 240 0.12 0.00 0.14 -0.03 0.37

FORTUM 0.12 0.05 240 0.07 0.10 0.02 0.03 0.34

KEMIRA 0.09 0.07 240 -0.07 -0.00 0.08 0.02 0.32

FINNAIR 0.04 0.08 240 0.06 0.07 0.11 -0.01 0.26

Based on autocorrelation with lag = 12, it seems that returns are not autocorre- lated if investment horizon is one year. In addition, we should be concerned about the long term dependency in stock market that is not possible to model with autocorrelations with couple of lags. This common feature of stock markets is explained inMandelbrot and Hudson(2004). The authors measure long term dependency with Hurst exponent H that is 0.5 for independent random walk. For OMXHGI during this periodH = 0.498thus there is no sign of long term dependency. It seems to be safe to model yearly returns as independent of each other. We are not either trying to model the autocorrelation of variance with GARCH model as there is evidence that the volatility autocorrelations in stock markets use to vanish with lags more than one month (Christoffersenet al.,2014;Cont,2005) and our re-allocation period is one year.

One concern is the length of historical data for decision making. The composition of companies and industries change over time, and thus, the historical period cannot be too long. On the other hand, short period does not cover the whole business cycle. A ten year period would be a compromise between these trade-offs. However, ten year historical periods are not identical. The first 10 year period of the data included the burst of the "dot-com bubble" and the financial crisis, during which

(21)

the OMXHGI average yearly return were -0.05. The second 10 year period was quite different part of the business cycle. It included Euro-crisis in 2010-2012 but otherwise that period experienced a long bull-market. Average yearly return was 0.09.

In Bayesian analysis it is possible to reduce over-fitting of the historical data by informative prior values. There are practical examples of over fitting reduction with prior distributions for portfolio selection inFrost and Savarino (1986) and for more general purposes in McElreath (2015). Here we are going to set N(0.06,0.015²) as the prior distribution for the risk premium of OMHXGI. Thus, our model gives more weight to the parameter values that are close 0.06 and reduces the effect of recent business cycle to the future estimates of returns.

In what follows we are going to define the optimization problems for two investors, i.e. a market index investor (together with risk-free rate holdings/debt) and an investor that allocates risky investments between the market index and the stock with the highest analyst ratings (together with risk-free rate holdings/debt).

For both assets, the market index and individual stock, fluctuations in prices and returns are modelled asPraez(1972), so the price of assetiat time pointtis notated asxⁱ_t and logarithmic return rates yⁱ_t= log(_x^xiⁱ^t

t−1) = log(xⁱ_t)−log(xⁱ_t−1). The returns follow geometric Brownian motion

y_tⁱ =µⁱ+σⁱⁱ_t, (22) where ⁱ_t ∼ t_νⁱ(0,1) and νⁱ is a degree of freedom parameter. Hence the original asset prices can be expressed as

xⁱ_t=xⁱ_t−1e^yⁱ^t. (23)

3.1 Model for market index investor

The model for expected values of market index M, µⁱ_t, follows the ideas of Treynor and Black(1973), so

µ^M_t =r_t+RP^M (24)

wherert is the risk free rate and RP^M is the risk premium for the market.

Based on Formula14we have a formula for the capital gain process for an investor with one risky asset, i.e. market index, as

V_t+1 =V_t[1 +r_t+1(1−w_t) +w_t(x^M_t+1

x^M_t −1)]. (25)

(22)

An objective function for Kelly-investor’s decision that is made at time point t= 0 for one time shift is

G₁(w₀) = logV₁ V0

= log [1 +r₁(1−w₀) +w₀(e^y^M¹ −1)]

= log[(1−w₀)(r₁+ 1) +w₀e^y^M¹ ], wherey^M_t is a stochastic variable. For investment horizonT

G_T(w₀) = 1 T

T

X

t=1

log[(1−w₀)(r_t+ 1) +w₀e^y^M^t ]. (26)

We measure the short-term risk with measureφ1 as inMacLeanet al. (2004), so

φ₁ =P(V_T ≥b), (27)

where V_T is the capital after investment horizon T. For example, if an acceptable loss during investment horizon is10% it implies that b= 0.9V₀.

The optimization problem for "Kelly with security" market fund investor is formulated as

maximizeEG_T(w₀) (28a)

subject toP(V_T ≥b)>(1−α^r) (28b)

w₀ ≥0, (28c)

where the "risk-alpha", α^r, is set in the analysis to be 0.05. Very small levels, e.g.

α^r = 0.01, put investor to a situation where she is dependent on the accuracy of modelling the tails of the price distribution. Even though t-distribution is more accurate than the normal distribution with fat tailed return distributions (Praez, 1972; Osorio, 2009; Nkemnole and Abass, 2019), it is better to avoid relying the analysis to small percentiles of the tail. In order to decrease the risk of portfolio, it is possible to increaseb instead of decreasing α^r.

3.2 Model for a stock with special potential

For individual stock i the Treynor and Black (1973) -model is used to modelling expectations based on

µⁱ_t=r_t+βⁱRP^M +αⁱ_t, (29) whereαⁱ_t is an appraisal premium that is modelled based on analysts’ target prices.

We have analysts’ target price information for asset i: mean (m_it), standard deviation (s_it) and number of analysts (n_it). We assume that these analysts represent

(23)

a sample of well informed investors, whose probably biased views about the expected next year’s return follow normal distribution. Thus the target prices follow a log-normal distribution, whereµ^A and σ^A are the average value and the standard deviation of analyst view of the return rate respectively. Values for these parameters are not observed directly but they can be derived from the observed target price information. Based on properties of the log-normal distribution, we derive the distribution for the appraisal premiums as

µ^A_it = log

m²

√s²+m²

−log(xⁱ_t−1) (σ^A_it)² = log

1 + s²

m²

.

We are going to weight the target price information based on uncertainty of the analysts as a group. For this, we use information entropy (Cover and Thomas,2006) as a measure of uncertainty. First, we have to define the maximum and minimum values of entropy as the reference points. A good value for the maximum uncertainty is a view that the future return can be anything between -100% (-10000 basis points) and 100% (10000 basis points). Differential entropy of the uniform distribution U(a, b) is log₂(b −a) bits. Thus, when the returns are measured as basis points and the entropy as bits, the maximum entropy is log₂(10000−(−10000))≈14.29.

The minimum entropy can be considered to be reached, when the uncertainty of the potential is at the level of one basis point. Thus, the minimum entropy islog₂(1) = 0.

We are modelling an analyst’s view with normal distribution and differential entropy of normal distributionh= ¹₂log₂(2πeσ²)(Cover and Thomas,2006). More precisely, we assume that if there is no systematic bias in the analyst ratings,

µ^A_it ∼N

E(µⁱ_t),(σ_it^A)² nit

.

Thus, we define the analyst information weight as a function of analyst standard deviation as basis points and the number of analysts,

v_it= 1− 0.5 log₂(2πe(100σ_it^A)²/n_it)

14.29 , (30)

where the numerator is the entropy of the analyst view and the nominator is the maximum entropy in our case. In other words, we give more weight to the consensus estimateµ^A_it when there is less variation between analysts or the number of analysts is high. Analogously, we give less weight to the analyst ratings when there is more variation between the analysts or the number of analysts is low.

(24)

In addition to the entropy within an analyst group, the whole group of analysts may be biased in their views. Thus, we include one more set of parameters, ψ_i, to the model for the analyst bias. If there is not systematic bias in the views of stock i’s analysts, ψi = 1. In the case ψi = 0, the analyst target prices do not add any useful information to the market price information, and if ψ_i < 0, analyst views could be used as a contraindicator. The parameters are modelled as

ψ_i ∼N(µ_ψ, σ_ψ²), (31) where µ_ψ describes the mean bias of all analysts and σ_ψ the standard deviation between stocks.

Based on Equation (29) we can derive the formula for the appraisal premiums as

µⁱ_t=r_t+βⁱRP^M +αⁱ_t =⇒ αⁱ_t=µⁱ_t−r_t−βⁱRP^M. (32) By replacing the expected value µⁱ_t with the analyst expectations, and adding the weight and bias parameter, we get a model forαⁱ_t

αⁱ_t = (µ^A_it−r_t−βⁱRP^M)v_itψ_i. (33) Finally, we can collect the equations (22), (29), (32) and (31) together for our model for the return of stock i in time period t, yielding

y_tⁱ =r_t+βⁱ(y^M_t −r_t) + (µ^A_it−r_t−βⁱRP^M)v_itψ_i+σⁱⁱ_t (34a)

ⁱ_t ∼t_νⁱ(0,1) (34b)

ψ_i ∼N(µ_ψ, σ_ψ²). (34c)

Parametersβⁱ,RP^M, µ_ψ,σ_ψ,σⁱ and νⁱ are stochastic and prior distributions for them are defined in Section3.3. The optimization problem is the same as in Equation (28), but w₀ is now a vector of two weights; w₀¹ is the weight for the market index fund and w²₀ is the weight of the individual stock with the highest estimated (as posterior mean) appraisal premium α. Stocks that have negative estimated biasˆ parameters,ψˆ_i, are ignored. In a year, when all posterior means of bias parameters are negative, the stock with highest µ^A_i is selected. Returns of this selected stock are notated as y_t. The optimization problem is formulated now as

maximizeEG_T(w¹₀, w²₀) (35a)

subject toP(VT ≥b)>(1−α^r) (35b)

w¹₀ ≥0 (35c)

w²₀ ≥0 (35d)

(25)

and

G_T(w¹₀, w²₀) = 1 T

T

X

t=1

log[(1−w₀¹−w²₀)(r_t+ 1) +w₀¹e^y^t^M +w²₀e^y^t]. (36) Details on solving this problem are next provided.

3.3 Bayesian solution and optimization

A Bayesian model can be solved by using MCMC (Markov chain Monte Carlo) simulation methods. For example, we are interested in a functionf which depends on θ, that is a vector of unknown parameters that have to be estimated based on data x. NowGelman et al.(2013) state that

E(f(θ)|x)≈ 1 S

S

X

s=1

f(θ^s), (37)

where θ^s represents a simulated sample from the posterior probability distribution of θ and S is the total number of simulated samples. In our case, we have data of historical prices on a set of assets x. We have to estimate the posterior probability distribution for future returns of risky assetsy_T and we are optimizing the expected value ofG_T(w,y_T). Thus, Bayesian solution with MCMC simulations to our model is

E[G_T(w,y_T|x)]≈ 1 S

S

X

s=1

G_T(w,y^s_T). (38) We use the probabilistic programming language Stan (Carpenter et al., 2017) and the interface RStan (Stan Development Team, 2020) for solving the model and simulating 60000 (4 chains with 15000 iterations) future scenarios of investment horizon for our portfolio optimization problem. Prior distributions for the unknown parameters in our model are defined as

• νⁱ ∼gamma(2,0.1), i∈0,1,2, . . . , N

(as recommended by Juárez and Steel (2010)),

• σⁱ ∼Half Cauchy(0,0.5), i∈0,1,2, . . . , N (weakly informative distribution),

• RP^M ∼N(0.06,0.015²)as mentioned earlier,

• βⁱ ∼N(1,4), i∈1,2, . . . , N (weakly informative distribution),

• µ_ψ ∼N(1,4)(weakly informative distribution) and

• σ_ψ ∼Half Cauchy(0,0.5) (weakly informative distribution).

(26)

With index (i) the notation 0 refers to market index and N to the number of the analyzed stock market assets.

After solving the Bayesian model, final asset allocations are achieved by constrained optimization where objective function (28a or35a) is replaced with formula 38. Optimization is made numerically using statistical computing environment R (R Core Team, 2020) and package ROI (Hornik et al., 2020) using ALABAMA (Augmented Lagrangian Adaptive Barrier Minimization Algorithm for optimizing smooth nonlinear objective functions with constraints) solver.

4 Results and analysis

First we check how our model estimates the fluctuations of returns. In Figure 3 we plot a distribution of monthly returns of the growth index (OMXHGI) for the whole 20 year period. It is compared to the normal approximation and our model’s predictive posterior distribution for the next month’s return. We see that the actual returns are skewed to left and our symmetric t-distribution with stochastic parameters cannot fit that perfectly. Asymmetric behavior is similar to the diagnostics of US index SP500 (daily observations during 50 years period) inTaleb et al. (2020), where the conclusion was that left tail is considerably thicker than the right one.

However, model’s fit is a huge improvement compared to the normal distribution.

Figure 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).

(27)

4.1 Results for the market index investor

In our main analysis, there are 10 portfolio allocation decisions, one for each year starting in 2010. Seven different investment strategies have been applied for each point in time with 10 years of historical information that were available at each point of allocation decisions.

The first strategy is a risk averse Bayes-Kelly (model that is introduced in Section 3) where the acceptable annual loss is 10%, and hence, security parameterb = 0.9Vt. This strategy is called "BayesKelly10". The second strategy, "BayesKelly20", is a less risk averse version of the same strategy, where the acceptable loss is 20 %.

The third strategy is called "BayesKellyUC", which is an unconstrained version of the same strategy. The fourth strategy, "1/N", is the simple naive strategy where half of the capital is always allocated to market index and the other half to the risk-free asset. In addition, there are also the mean-variance strategies which apply Equation (20) with plug-in estimates from yearly data. The first of these strategies,

"MeanVariance1", has relative risk aversion A = 1. As mentioned earlier, this strategy is equivalent with a the simple Kelly strategy without Bayesian priors or short term risk-control including all the weaknesses that were described in Section 2.4. The second, "MeanVariance0.5", is even more risk seeking strategy with A = 0.5. "MeanVariance5" is more risk averse strategy with A= 5.

Figure 4: Weights of the market index with different strategies. The left panel includes all observations and in the right panel two observations, where w > 4, are censored for easier comparison of different strategies with lower leverage level.

Abbreviations: BK = Bayes-Kelly, MV = mean-variance, UC = unconstrained In Figure 4 we plot the market index weights (w) as a function of time for each strategy. Here we see that the plug-in estimates in the mean-variance strategies

(28)

overfit the historical data. All of these strategies havew= 0 during 2010-2015. But after 2015 there start to be enough evidence for positive returns in 10 year historical period, and the mean-variance strategies take leveraged (debt with risk free rate) positions, where w > 1. The most aggressive strategy, "MeanVariance0.5", takes stock position that is a multiple of 15 times the capital in year 2019.

The Bayes-Kelly -strategies are more stable over time. The constrained strategies hold the stock market weight all the time genuinely positive and do not use leverage (apart from one minor exception). The unconstrained "BayesKellyUC" uses all the time leverage with 1< w <2.5.

The performance results for each strategy are reported in Table2. Strategies are ordered based on the achieved Sharpe ratio that is calculated based on Equation (2). The highest total growth rate of capital was achieved with the least risk averse mean-variance strategy, which ended up to 3.69 times initial capital. However, this result was hugely influenced by one risky and lucky bet in 2019. Even though the

"MeanVariance0.5" strategy had the highest total wealth growth, it had the worst performance as Sharpe ratio because of high volatility (standard deviation of yearly return was 0.354). The "BayesKellyUC" had almost as good capital growth rate as the "MeanVariance0.5" but with lower volatility and it had the best Sharpe ratio 0.721. Based on the Sharpe ratio, the other Bayes-Kelly strategies performed almost as well. Among these Bayes-Kelly strategies, the more risk averse constrain applied, the less there were volatility in performance. The simple "1/N strategy" has lower Sharpe ratio than any of the Bayes-Kelly strategies but it is higher than in any of the mean-variance strategies.

Figure 5shows how a $1000 initial capital invested in 2010 would have accumulated during the 10 year period with these strategies. It shows that the Solidium’s capital growth performance is average among these strategies, almost doubling the initial capital towards the end of the 10 year period. It’s performance follows ap- proximately the "BayesKelly20" strategy. The main difference is that Solidium has higher volatility, and hence, Sharpe ratio is only 0.493 compared to "BayesKelly20"’s 0.703.

(29)

Table 2: The performance of market index investors for each strategy. Average yearly returns (¯y), standard deviations (¯y), total capital growth coefficient during 10 years period (^V_V¹⁰

0 ) and the observed Sharpe ratio.

Strategy y¯ sd(y) ^V_V¹⁰

0 Sharpe 1 BayesKelly (UC) 0.151 0.202 3.54 0.721 2 BayesKelly20 0.078 0.103 2.04 0.703 3 BayesKelly10 0.043 0.052 1.50 0.701

4 1/N 0.050 0.066 1.60 0.664

5 MeanVarianceA5 0.023 0.033 1.25 0.508

6 Solidium 0.080 0.150 1.97 0.493

7 MeanVarianceA1 0.090 0.175 2.15 0.480 8 MeanVarianceA0.5 0.174 0.354 3.69 0.476

Figure 5: Growth trajectories of $1000 initial capital invested in 2010 with different strategies.

4.2 Results with individual stock selections

After comparing the overall risky positions and the performance of different strategies, we take a look at the role of adding an individual stock to the portfolio in terms of improving the performance of Bayes-Kelly strategies. Table3shows the stock selections for each year with stock price together with analysts’ mean target prices and standard deviations at the beginning of the period. In addition, posterior means of appraisal premia, α, and analyst bias parameters, φ, together with the realized annual return premium (dividends included), y−y^M, are presented. Tables A1 - A10in AppendixAshow the same analyst target statistics and parameter estimates for all analyzed stocks. Within the selected stocks, every αˆ ≤ 0.03 indicating low

(30)

expected appraisal premium even though differences between mean target prices and current stock prices indicate higher appraisal premia. Bias estimatesφ <1indicate a high systematic bias in the analyst target prices at least in the investment horizon of one year. This is similar finding withCvitanicet al.(2006), which concluded that analysts’ recommendations has systematic positive bias and they add only a little value to investors.

Table 3: Stock selection for each allocation decision. Starting months of the period, stock names, stock prices (x), target means (m) and target standard deviations (s) are showed together with posterior means of appraisal premium parameter (ˆα) and analyst bias parameter (ψ). The columnˆ y−y^M is the realized annual return premium.

Period start Asset x m s αˆ ψˆ y−y^M

1 March 2010 METSO 17.9 26.2 4.2 -0.018 -0.10 0.27 2 March 2011 METSO 29.4 43.1 5.4 0.030 0.16 0.15 3 March 2012 METSO 28.0 36.1 4.6 0.013 0.12 -0.09 4 March 2013 METSO 25.8 34.3 5.5 0.016 0.14 -0.26 5 March 2014 KEMIRA 11.2 12.1 0.86 0.002 0.10 -0.19 6 March 2015 FINNAIR 3.12 3.60 0.42 0.008 0.13 0.46 7 March 2016 SAMPO 40.9 44.3 3.6 0.010 0.34 -0.11 8 March 2017 SAMPO 42.4 44.6 4.4 0.003 0.28 -0.05 9 March 2018 SAMPO 46.0 48.1 4.5 0.001 0.18 -0.03 10 March 2019 SAMPO 41.7 46.3 3.3 0.012 0.24 -0.07

The total performance results for each Bayes-Kelly strategy are reported in Table 4. The strategies are ordered based on the achieved Sharpe ratio. All strategies performed better when an individual stock was included to the portfolio, compared to a similarly constrained "index fund only" strategy. The individual stocks and analyst ratings helped to decrease portfolios return’s standard deviation but the effects to total growth were negligible or even negative. Figure 6 present visually how $1000 initial capital invested in 2010 would have accumulated during the 10 year period with Bayes-Kelly strategies. Especially the re-allocation in 2015 seem to be critical. On that period market index had negative return (−8%), but Finnair’s 38%return helped to keep strategies, where index fund were joined with individual stock, on a growing wealth trajectory.

Bayesian Kelly criterion as an allocation strategy in Finnish stock markets