O presente trabalho busca dar início a estudos referentes ao modelo de otimização de portfolios de investimento denominado paridade de risco no cenário brasileiro. Neste trabalho, os índices setoriais da bolsa brasileira (Bovespa) foram utilizados como ativos e com os seus dados foram estimadas carteiras com os modelos de mínima variância, de pesos iguais e de paridade de risco. Verificou-se que no modelo de paridade de risco a forma de obtenção da matriz de covariância exerce pouca influência no resultado final, que é de carteiras com distribuição de pesos e volatilidades intermediárias com relação aos modelos de mínima variância e de pesos iguais. Estes resultados são condizentes com aqueles verificados na literatura que utilizam como base de dados os mercados europeus e americanos. / This paper seeks to initiate studies for the investment portfolios optimization model called risk parity in the Brazilian scene. In this work, the sector indexes of the Brazilian Stock Exchange (Bovespa) were used as assets and their data were used to estimate portfolios with models of minimum variance, of equal weight and of risk parity. It was found that in the risk parity model the form to obtain the covariance matrix has little influence on the final result, that is of a portfolio with weights and distribution of intermediate volatility in relation to the minimum variance models and equal weights. These results are consistent with those found in the literature using as database the European and American markets.

O presente trabalho busca dar início a estudos referentes ao modelo de otimização de portfolios de investimento denominado paridade de risco no cenário brasileiro. Neste trabalho, os índices setoriais da bolsa brasileira (Bovespa) foram utilizados como ativos e com os seus dados foram estimadas carteiras com os modelos de mínima variância, de pesos iguais e de paridade de risco. Verificou-se que no modelo de paridade de risco a forma de obtenção da matriz de covariância exerce pouca influência no resultado final, que é de carteiras com distribuição de pesos e volatilidades intermediárias com relação aos modelos de mínima variância e de pesos iguais. Estes resultados são condizentes com aqueles verificados na literatura que utilizam como base de dados os mercados europeus e americanos. / This paper seeks to initiate studies for the investment portfolios optimization model called risk parity in the Brazilian scene. In this work, the sector indexes of the Brazilian Stock Exchange (Bovespa) were used as assets and their data were used to estimate portfolios with models of minimum variance, of equal weight and of risk parity. It was found that in the risk parity model the form to obtain the covariance matrix has little influence on the final result, that is of a portfolio with weights and distribution of intermediate volatility in relation to the minimum variance models and equal weights. These results are consistent with those found in the literature using as database the European and American markets.

O presente trabalho busca dar início a estudos referentes ao modelo de otimização de portfolios de investimento denominado paridade de risco no cenário brasileiro. Neste trabalho, os índices setoriais da bolsa brasileira (Bovespa) foram utilizados como ativos e com os seus dados foram estimadas carteiras com os modelos de mínima variância, de pesos iguais e de paridade de risco. Verificou-se que no modelo de paridade de risco a forma de obtenção da matriz de covariância exerce pouca influência no resultado final, que é de carteiras com distribuição de pesos e volatilidades intermediárias com relação aos modelos de mínima variância e de pesos iguais. Estes resultados são condizentes com aqueles verificados na literatura que utilizam como base de dados os mercados europeus e americanos. / This paper seeks to initiate studies for the investment portfolios optimization model called risk parity in the Brazilian scene. In this work, the sector indexes of the Brazilian Stock Exchange (Bovespa) were used as assets and their data were used to estimate portfolios with models of minimum variance, of equal weight and of risk parity. It was found that in the risk parity model the form to obtain the covariance matrix has little influence on the final result, that is of a portfolio with weights and distribution of intermediate volatility in relation to the minimum variance models and equal weights. These results are consistent with those found in the literature using as database the European and American markets.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: We consider the problem of portfolio's asset allocation characterised by risk and return. Prior to the 2007-2008 financial crisis, this important problem was tackled using mainly the Markowitz mean-variance framework. However, throughout the past decade of challenging markets, particularly for equities, this framework has exhibited multiple drawbacks. Today many investors approach this problem with a 'safety first' rule that puts risk management at the heart of decision-making. Risk-based strategies have gained a lot of popularity since the recent financial crisis. One of the 'trendiest' of the modern risk-based strategies is the Risk Parity model, which puts diversification in terms of risk, but not in terms of dollar values, at the core of portfolio risk management. Inspired by the works of Maillard et al. (2010), Bruder and Roncalli (2012), and Roncalli and Weisang (2012), we examine the reliability and relationship between the traditional mean-variance framework and risk parity. We emphasise, through multiple examples, the non-diversification of the traditional mean-variance framework. The central focus of this thesis is on examining the main Risk-Parity strategies, i.e. the Inverse Volatility, Equal Risk Contribution and the Risk Budgeting strategies. Lastly, we turn our attention to the problem of maximizing the absolute expected value of the logarithmic portfolio wealth (sometimes called the drift term) introduced by Oderda (2013). The drift term of the portfolio is given by the sum of the expected price logarithmic growth rate, the expected cash flow, and half of its variance. The solution to this problem is a linear combination of three famous risk-based strategies and the high cash flow return portfolio. / AFRIKAANSE OPSOMMING: Ons kyk na die probleem van batetoewysing in portefeuljes wat gekenmerk word deur risiko en wins. Voor die 2007-2008 finansiele krisis, was hierdie belangrike probleem deur die Markowitz gemiddelde-variansie raamwerk aangepak. Gedurende die afgelope dekade van uitdagende markte, veral vir aandele, het hierdie raamwerk verskeie nadele getoon. Vandag, benader baie beleggers hierdie probleem met 'n 'veiligheid eerste' reël wat risikobestuur in die hart van besluitneming plaas. Risiko-gebaseerde strategieë het baie gewild geword sedert die onlangse finansiële krisis. Een van die gewildste van die moderne risiko-gebaseerde strategieë is die Risiko- Gelykheid model wat diversifikasie in die hart van portefeulje risiko bestuur plaas. Geïnspireer deur die werke van Maillard et al. (2010), Bruder and Roncalli (2012), en Roncalli and Weisang (2012), ondersoek ons die betroubaarheid en verhouding tussen die tradisionele gemiddelde-variansie raamwerk en Risiko- Gelykheid. Ons beklemtoon, deur middel van verskeie voorbeelde, die niediversifikasie van die tradisionele gemiddelde-variansie raamwerk. Die sentrale fokus van hierdie tesis is op die behandeling van Risiko-Gelykheid strategieë, naamlik, die Omgekeerde Volatiliteit, Gelyke Risiko-Bydrae en Risiko Begroting strategieë. Ten slotte, fokus ons aandag op die probleem van maksimering van absolute verwagte waarde van die logaritmiese portefeulje welvaart (soms genoem die drif term) bekendgestel deur Oderda (2013). Die drif term van die portefeulje word gegee deur die som van die verwagte prys logaritmiese groeikoers, die verwagte kontantvloei, en die helfte van die variansie. Die oplossing vir hierdie probleem is 'n lineêre kombinasie van drie bekende risiko-gebaseerde strategieë en die hoë kontantvloei wins portefeulje.

Submitted by Joana Vieira (joana_cvieira@hotmail.com) on 2015-10-13T12:58:25Z No. of bitstreams: 1 FINAL REPORT JOANA VEIRA FGV.pdf: 3927213 bytes, checksum: a8f998809220a76b7f10b84fa630e2b0 (MD5) / Rejected by Ana Luiza Holme (ana.holme@fgv.br), reason: Joana, o numero das paginas deve contar a partir da capa, está certo mas só deve aparecer o numero a partir da introdução. fico a disposição. Ana Luiza Holme 3799-3492 on 2015-10-13T16:24:51Z (GMT) / Submitted by Joana Vieira (joana_cvieira@hotmail.com) on 2015-10-13T17:42:24Z No. of bitstreams: 1 FINAL REPORT JOANA VEIRA FGV.pdf: 3951971 bytes, checksum: 667456b57a07dda99ac9aeb852a0c8ee (MD5) / Approved for entry into archive by Ana Luiza Holme (ana.holme@fgv.br) on 2015-10-13T17:44:14Z (GMT) No. of bitstreams: 1 FINAL REPORT JOANA VEIRA FGV.pdf: 3951971 bytes, checksum: 667456b57a07dda99ac9aeb852a0c8ee (MD5) / Made available in DSpace on 2015-10-13T17:49:57Z (GMT). No. of bitstreams: 1 FINAL REPORT JOANA VEIRA FGV.pdf: 3951971 bytes, checksum: 667456b57a07dda99ac9aeb852a0c8ee (MD5) Previous issue date: 2015-09-25 / Taking into account previous research we could assume to be beneficial to diversify investments in emerging economies. We investigate in the paper International Portfolio Diversification: evidence from Emerging Markets if it still holds true, given the assumption of larger world markets integration. Our results suggest a wide spread positive time-varying correlations of emerging and developed markets. However, pair-wise cross-country correlations gave evidence that emerging markets have low integration with developed markets. Consequently, we evaluate out-of-sample performance of a portfolio with emerging equity countries, confirming the initial statement that it has a better a risk-adjusted performance over a purely developed markets portfolio.

Commodity Trading Advisors (CTAs) have gained popularity due to their abilities to generate an absolute return strategy. Little is known about how CTAs work and what variables are important to tune in order to create a profitable strategy. Some investors use CTA-like strategies to leverage their portfolio and create positive returns in times when the spot market is falling. The report is written for Skandinaviska Enskilda Banken and aims to give the bank and readers an understanding on how changes of parameters in a CTA strategy change the outcome of it with focus on three main measurements: Sharpe ratio, drawdown and total return.  The foundation of CTAs is that they rely on signals from some given sets of assets and make investments decisions solely based on them. CTAs can be rule-based with a binomial signal, or they can use a continual signal, like in the report. The thesis aims to recreate a CTA using a continuous momentum signal and with the signal, invest accordingly. Some different variables were tested, most importantly the report focuses on the weights of the assets and investigates if the momentum signal is good as it is or if a risk parity weighting is needed on top of the signal in order to generate a return that matches the expectations of a low drawdown and a high Sharpe ratio.  Beyond the weight allocation, different lookback periods of both the signal and weight were tested. A shorter lookback generated a quicker return that was more sensible to short trends on the market. Which in some cases was profitable but it also lost more of it accumulated return when the trend was "false". The equally weighted signal that only takes the trend into account when allocating the weights of the assets was more volatile it its returns and benefited from a long signal. The CTA results presented can only be seen as an index since it is rebalanced every rebalancing point, the frequency of those points was examined and the strategy was performing well if rebalanced once a week or once a month, every day and once a year did not yield a better result.  As expected, the CTA benefits from trend on the market, no matter the direction of it. The best periods for the CTA were when the market was very volatile, mainly 2008 and 2022. When there is no clear trend, the CTA reacts too slowly and often loses money. One important conclusion is that the CTA never should be used as an investment strategy on its own, rather as a hedging strategy that allocates a fraction of a total long-only portfolio.

In this study within quantitative portfolio optimization, stochastic programming is investigated as an investment decision tool. This research takes the direction of scenario based Mean-Absolute Deviation and is compared with the traditional Mean-Variance model and widely used Risk Parity portfolio. Furthermore, this thesis is done in collaboration with the First Swedish National Pension Fund, AP1, and the implemented multi-asset portfolios are thus tailored to match their investment style. The models are evaluated on two different fund management levels, in order to study if the portfolio performance benefits from a more restricted feasible domain. This research concludes that stochastic programming over the investigated time period is inferior to Risk Parity, but outperforms the Mean-Variance Model. The biggest aw of the model is its poor performance during periods of market stress. However, the model showed superior results during normal market conditions. / I denna studie inom kvantitativ portföljoptimering undersöks stokastisk programmering som ett investeringsbeslutsverktyg. Denna studie tar riktningen för scenariobaserad Mean-Absolute Deviation och jämförs med den traditionella Mean-Variance-modellen samt den utbrett använda Risk Parity-portföljen. Avhandlingen görs i samarbete med Första AP-fonden, och de implementerade portföljerna, med era tillgångsslag, är därför skräddarsydda för att matcha deras investeringsstil. Modellerna utvärderas på två olika fondhanteringsnivåer för att studera om portföljens prestanda drar nytta av en mer restrektiv optimeringsmodell. Den här undersökningen visar att stokastisk programmering under undersökta tidsperioder presterar något sämre än Risk Parity, men överträffar Mean-Variance. Modellens största brist är dess prestanda under perioder av marknadsstress. Modellen visade dock något bättre resultat under normala marknadsförhållanden.

Following the global financial crisis, both risk-based and heuristic portfolio construction methods have received much attention from both academics and practitioners since these methods do not rely on the estimation of expected returns and as such are assumed to be more stable than Markowitz's traditional mean-variance portfolio. In 2016, Lopéz de Prado presented the Hierarchical Risk Parity (HRP), a new approach to portfolio construction which combines hierarchical clustering of assets with a heuristic risk-based allocation strategy in order to increase stability and improve out-of-sample performance. Using Monte Carlo simulations, Lopéz de Prado was able to demonstrate promising results. This thesis attempts to evaluate HRP using walk-forward analysis and historical data from equity index and bond futures, against more realistic benchmark methods and using additional performance measures relevant to practitioners. The main conclusion is that applying hierarchical clustering to risk-based portfolio construction does indeed improve the out-of-sample return and Sharpe ratio. However, the resulting portfolio is also associated with a remarkably high turnover, which may indicate numerical instability and sensitivity to estimation errors. It is also identified that Lopéz de Prado's original HRP approach has an undesirable property and alternative approaches to HRP have consequently been developed. Compared to Lopéz de Prado's original HRP approach, these alternative approaches increase the Sharpe ratio with ~10% and reduce the turnover with 60-65%. However, it should be noted that compared to more mainstream portfolios the turnover is still rather high, indicating that these alternative approaches to HRP are still somewhat unstable and sensitive to estimation errors. / Efter den globala finanskrisen har intresset för riskbaserade och heuristiska metoder för portföljallokering ökat inom såväl akademin som finansindustrin. Det ökade intresset grundar sig i att dessa metoder inte kräver estimering av förväntad avkastning och därför kan antas vara mer stabila än portföljer med grund i Markowitz moderna portföljteori. Lopéz de Prado presenterade 2016 en ny metod för portföljallokering, Hierarchical Risk Parity (HRP), som kombinerar hierarkisk klustring med en heuristisk riskbaserad portföljkonstruktion och vars syfte är att öka stabiliteten och förbättra avkastningen. Baserat på Monte Carlo-simuleringar har Lopéz de Prado lyckats påvisa lovande resultat. Syftet med detta examensarbete är att utvärdera HRP med hjälp av walk-forward-analys och empirisk data från aktieindex- och obligationsterminer. I denna utvärdering jämförs HRP med andra vanliga portföljmetoder med avseende på prestandamått relevanta för portföljförvaltare. Den huvudsakliga slutsatsen är att tillämpning av hierarkisk klustring inom ramen för riskbaserad portföljallokering förbättrar såväl den absoluta avkastningen som Sharpekvoten. Däremot är det tydligt att vikterna i en HRP-portfölj har hög omsättning över tid, vilket kan tyda på numerisk instabilitet och hög känslighet för skattningsfel. Vidare har en oönskad egenskap i Lopéz de Prados ursprungliga HRP-metod identifierats, varför två alternativa HRP-metoder har utvecklats inom ramen för examensarbetet. Jämfört med Lopéz de Prados ursprungliga metod förbättrar de två alternativa metoderna Sharpekvoten med 10% och minskar omsättningen av portföljvikterna med 60-65%. Det bör dock understrykas att även de nya metoderna har en förhållandevis hög omsättning, vilket tyder på att numerisk instabilitet och hög känslighet för skattningsfel till viss del fortfarande kvarstår.