This master‘s thesis is focused on creating an application, which would suggest an optimal portfolio of shares from SPAD Stock Market Praha. The application is based on the CAPM model, which is also described in this paper. There is a calculation of securities characteristics and specific portfolio diversification is suggested. The application also allows a user to simulate investments based on his requirements.

The performance of an optimal-weighted portfolio strategy is evaluated when transaction costs are penalized compared to an equal-weighted portfolio strategy. The optimal allocation weights are found by maximizing a modified Sharpe ratio measure each trading day, where modified refers to the expected return of an asset in this context. The leverage of the investment is determined by a conditional expectation estimate of the number of portfolio assets of the next-coming day. A moving window is used to historically measure the transition probabilities of moving from one state to another within this stochastic count process and this is used as an input to the estimator. It is found that the most accurate estimate is the actual trading day’s number of portfolio assets and this is obtained when the size of the moving window is one. Increasing the penalty parameter on transaction costs of selling and buying assets between trading days lowers the aggregated transaction cost and increases the performance of the optimal-weighted portfolio considerably. The best portfolio performance is obtained when at least 50% of the capital is invested equally among the assets when maximizing the modified Sharpe ratio. The optimal-weighted and equal-weighted portfolios are constructed on a daily basis, where the allowed VaR0:05 is €300 000 for each portfolio. This sets the limit on the amount of capital allowed to be invested each trading day, and is determined by empirical VaR0:05 simulations of these two portfolios.

In this thesis, an optimal investment problem is studied for an investor who can only invest in a financial market modelled by an Itô-Lévy process; with one risk free (bond) and one risky (stock) investment possibility. We present the dynamic programming method and the associated Hamilton-Jacobi-Bellman (HJB) equation to explicitly solve this problem. It is shown that with purification and simplification to the standard jump diffusion process, closed form solutions for the optimal investment strategy and for the value function are attainable. It is also shown that, an explicit solution can be obtained via a finite training of a neural network using Stochastic gradient descent (SGD) for a specific case.

The aim of this thesis is to explore the role of bitcoin in an investment portfolio. The paper examines the nature of bitcoin and additionally how bitcoin compares to gold when included in an investment portfolio. This report uses the historical value of bitcoin and investigates with a Mean-Variance model how the risk-adjusted return of an optimized portfolio is affected when bitcoin is a constituent. By comparing Sharpe Ratios from the optimized portfolios, a conclusion can be drawn as to whether bitcoin affects the maximum Sharpe ratio or the global minimum variance point. Our study suggests that including bitcoin in an investment portfolio increases the risk-adjusted return of the portfolio. In addition, portfolios optimized with bitcoin outperform the market. Further, we conclude that bitcoin has a relatively high correlation as compared to gold with the assets in the study. Hence, bitcoin is not the new gold.

This thesis analyzes the mean variance optimization problem with respect to cardinalityconstraints. The aim of this thesis is to figure out how much of an impact transactionchanges has on the profit and risk of a portfolio. We solve the problem by implementingmixed integer programming (MIP) and solving the problem by using the Gurobi solver.In doing this, we create a mathematical model that enforces the amount of transactionchanges from the initial portfolio. Our results is later showed in an Efficient Frontier,to see how the profit and risk are changing depending on the transaction changes.Overall, this thesis demonstrates that the application of MIP is an effective approachto solve the mean variance optimization problem and can lead to improved investmentoutcomes.

Hoarding a too large cash reserve is often unfavourable due to lost investment opportunities. Similarly, an insufficient cash reserve can be detrimental, as one might fail to meet payment obligations. Finding the optimal balance is nothing that is done in the blink of an eye, particularly when the underlying variable is stochastic, e.g., the life span of a human being. Resscapital is a fund manager investing in the secondary and tertiary markets for life insurance policies, also known as the life settlements market. They are currently on a mission to set up a closed-end fund where one of the main challenges is balancing the invested capital and the amount of capital set aside for payment obligations. The stochastic nature of life insurance policies entails the difficulty to foresee future premium payments and face value payouts. Without a model forecasting the cash flows, decisions regarding the cash reserve are based on nothing better than a guesstimate. Thus, with the aim to help determine the minimum cash reserve required to cover the payment obligations, this thesis was initiated. By developing a methodology based on general theory, the objective of this thesis is reached and the purpose fulfilled. The proposed model uses Monte Carlo simulation to generate scenarios that eventually creates a distribution of required cash reserves. Following the inversion principle, the remaining lifetime for each and every individual is simulated from their empirical distribution of survival probabilities, respectively. After simulating the occurrences of demise, an algorithm builds up the cash flows for the entire fund term for that specific scenario based on predetermined parameters. Since cash flows stem from both assets and management, the portfolio must be revalued continuously, demanding a gradual evaluation of the cash flows during the fund term. Repeated a large number of times, the quantile corresponding to any confidence level is attained by using a Value at Risk methodology. Analysis of the results and sensitivity analysis on the parameters provides a deeper understanding of the underlying factors, revealing, among other things, that longevity risk for policies with short life expectancy is a key driver of the required cash reserve. Furthermore, validation of the model shows that the results are sufficient and serve the purpose well.

ESG investing has been a hot topic during several years and there have been numerousstudies examining the relationship between sustainable assets and non-sustainable assetsincluding green bonds, social bonds, environmental bonds, ESG-bonds and ESG indices;conventional bonds, S&P 500, common stocks and non-ESG indices. During negative marketshocks several ESG stocks and indices have been shown to outperform common stocks andindices. Green bonds demonstrated an asymmetric relationship to other assets providinginvestors with an opportunity for diversification. We’ve looked at the relationship andperformance of sustainable assets and non-sustainable assets by using Markowitz portfoliometrics and Engle Rs’ DCC-GARCH. Our findings propose green bonds and treasuries toprovide hedging and diversification opportunities during crises but demonstrate sustainablefixed income assets to underperform non-sustainable fixed income assets during the COVID19 market shock as opposed to previous studies.

Multi-period portfolio optimization (MPO) has gained a lot of interest in modern portfolio theory due to its consideration for inter-temporal trading e effects, especially market impacts and transactions costs, and for its subtle reliability on return predictability. However, because of the heavy computational demand, portfolio policies based on this approach have been sparsely explored. In that regard, a tractable MPO framework proposed by N. Gârleanu & L. H. Pedersen has been investigated. Using the stochastic control framework, the authors provided a closed form expression of the optimal policy. Moreover, they used a specific, yet flexible return predictability model. Excess returns were expressed using a linear factor model, and the predicting factors were modeled as mean reverting processes. Finally, transactions costs and market impacts were incorporated in the problem formulation as a quadratic function. The elaborated methodology considered that the market returns dynamics are governed by fast and slow mean reverting factors, and that the market transactions costs are not necessarily quadratic. By controlling the exposure to the market returns predicting factors, the aim was to uncover the importance of the mean reversion speeds in the performance of the constructed trading strategies, under realistic market costs. Additionally, for the sake of comparison, trading strategies based on a single-period mean variance optimization were considered. The findings suggest an overall superiority in performance for the studied MPO approach even when the market costs are not quadratic. This was accompanied with evidence of better usability of the factors' mean reversion speed, especially fast reverting factors, and robustness in adapting to transactions costs. / Portföljoptimering över era perioder (MPO) har fått stort intresse inom modern portföljteori. Skälet till detta är att MPO tar hänsyn till inter-temporala handelseffekter, särskilt marknadseffekter och transaktionskostnader, plus dess tillförlitlighet på avkastningsförutsägbarhet. På grund av det stora beräkningsbehovet har dock portföljpolitiken baserad på denna metod inte undersökts mycket. I det avseendet, har en underskriven MPO ramverk som föreslagits av N.Gârleanu L. H. Pedersen undersökts. Med hjälp av stokastiska kontrollramen tillhandahöll författarna formuläret för sluten form av den optimala politiken. Dessutom använde de en specifik, men ändå flexibel returförutsägbarhetsmodell. Överskjutande avkastning uttrycktes med hjälp av en linjärfaktormodell och de förutsägande faktorerna modellerades som genomsnittligaåterföringsprocesser. Slutligen inkorporerades transaktionskostnader och marknadseffekter i problemformuleringen som en kvadratisk funktion. Den utarbetade metodiken ansåg att marknadens avkastningsdynamik styrs av snabba och långsammaåterhämtningsfaktorer, och att kostnaderna för marknadstransaktioner inte nödvändigtvis är kvadratiska. Genom att reglera exponeringen mot marknaden återspeglar förutsägande faktorer, var målet att avslöja vikten av de genomsnittliga omkastningshastigheterna i utförandet av de konstruerade handelsstrategierna, under realistiska marknadskostnader. Dessutom, för jämförelses skull, övervägdes handelsstrategier baserade på en enstaka genomsnittlig variansoptimering. Resultaten tyder på en överlägsen överlägsenhet i prestanda för det studerade MPO-tillvägagångssättet, även när marknadsutgifterna inte är kvadratiska. Detta åtföljdes av bevis för bättre användbarhet av faktorernas genomsnittliga återgångshastighet, särskilt snabba återställningsfaktorer och robusthet vid anpassning till transaktionskostnader

The stock market is a non-linear field, but many of the best-known portfolio optimization algorithms are based on linear models. In recent years, the rapid development of machine learning has produced flexible models capable of complex pattern recognition. In this paper, we propose two different methods of portfolio optimization; one based on the development of a multivariate time-dependent neural network,thelongshort-termmemory(LSTM),capable of finding lon gshort-term price trends. The other is the linear Markowitz model, where we add an exponential moving average to the input price data to capture underlying trends. The input data to our neural network are daily prices, volumes and market indicators such as the volatility index (VIX).The output variables are the prices predicted for each asset the following day, which are then further processed to produce metrics such as expected returns, volatilities and prediction error to design a portfolio allocation that optimizes a custom utility function like the Sharpe Ratio. The LSTM model produced a portfolio with a return and risk that was close to the actual market conditions for the date in question, but with a high error value, indicating that our LSTM model is insufficient as a sole forecasting tool. However,the ability to predict upward and downward trends was somewhat better than expected and therefore we conclude that multiple neural network can be used as indicators, each responsible for some specific aspect of what is to be analysed, to draw a conclusion from the result. The findings also suggest that the input data should be more thoroughly considered, as the prediction accuracy is enhanced by the choice of variables and the external information used for training. / Aktiemarknaden är en icke-linjär marknad, men många av de mest kända portföljoptimerings algoritmerna är baserad på linjära modeller. Under de senaste åren har den snabba utvecklingen inom maskininlärning skapat flexibla modeller som kan extrahera information ur komplexa mönster. I det här examensarbetet föreslår vi två sätt att optimera en portfölj, ett där ett neuralt nätverk utvecklas med avseende på multivariata tidsserier och ett annat där vi använder den linjära Markowitz modellen, där vi även lägger ett exponentiellt rörligt medelvärde på prisdatan. Ingångsdatan till vårt neurala nätverk är de dagliga slutpriserna, volymerna och marknadsindikatorer som t.ex. volatilitetsindexet VIX. Utgångsvariablerna kommer vara de predikterade priserna för nästa dag, som sedan bearbetas ytterligare för att producera mätvärden såsom förväntad avkastning, volatilitet och Sharpe ratio. LSTM-modellen producerar en portfölj med avkastning och risk som ligger närmre de verkliga marknadsförhållandena, men däremot gav resultatet ett högt felvärde och det visar att vår LSTM-modell är otillräckligt för att använda som ensamt predikteringssverktyg. Med det sagt så gav det ändå en bättre prediktion när det gäller trender än vad vi antog den skulle göra. Vår slutsats är därför att man bör använda flera neurala nätverk som indikatorer, där var och en är ansvarig för någon specifikt aspekt man vill analysera, och baserat på dessa dra en slutsats. Vårt resultat tyder också på att inmatningsdatan bör övervägas mera noggrant, eftersom predikteringsnoggrannheten.

Optimal portfolios have historically been computed using standard deviation as a risk measure.However, extreme market events have become the rule rather than the exception. To capturetail risk, investors have started to look for alternative risk measures such as Value-at-Risk andConditional Value-at-Risk. This research analyzes the financial model referred to as Markowitz 2.0 and provides historical context and perspective to the model and makes a mathematicalformulation. Moreover, practical implementation is presented and an optimizer that capturesthe risk of non-extreme events is constructed, which meets the needs of more customized investment decisions, based on investment preferences. Optimal portfolios are generated and anefficient frontier is made. The results obtained are then compared with those obtained throughthe mean-variance optimization framework. As concluded from the data, the optimal portfoliowith the optimal weights generated performs better regarding expected portfolio return relativeto the risk level for the investment.