The Nuclear Waste Fund constitutes a financial system that finances future costs of the management of spent nuclear fuel as well as decommissioning of nuclear power plants. The fund invests its capital under strict rules which are stipulated in the investment policy established by the board. The policy stipulates that the fund can only invest according to certain allocation limits, and restricts it to invest solely in nominal and inflation-linked bonds issued by the Swedish state as well as treasury securities. A norm portfolio is built to compare the performance of the NWF’s investments. On average, the NWF has outperformed the norm portfolio on recent years, but it may not always have been optimal. Recent studies suggest that allocation limits should be revised over time as the return and risk parameters may change over time. This study focused on simulating three different portfolios where the allocation limits and investment options were extended to see if these extensions would outperform the norm portfolio while maintaining a set risk limit. Portfolio A consisted of OMRX REAL and OMRX TBOND indexes, Portfolio B consisted of OMRX REAL, OMRX TBOND and S&P Sweden 1+ Year Investment Grade Corporate Bond Indexes, and Portfolio C consisted of OMXR REAL, OMRX TBOND and OMXSPI indexes. The return of each portfolio for different weight distributions of the assets were simulated in MATLAB, and polynomial regression models were built in order to optimize the return as a function of the assets’ weights using a Lagrange Multiplier approach for each portfolio. The results depicted that the maximal returns of Portfolios A, B and C were 4.00%, 4.13% and 7.93% respectively, outperforming the norm portfolio’s average return of 3.69% over the time period 2009-2016.

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.

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.

本篇論文主要是將Black-Litterman模型應用在組合型基金上。從一個組合型基金的基金經理人角度出發，在有限的風險下，如何進行資產配置使其達到報酬極大化的目標？第二章介紹mean-variance模型，以及其模型之缺點。第三章介紹Black-Litterman模型，其不僅可以改善mean-variace模型的缺點，此外允許投資人加入主觀看法，結合數量方法以及投資人的主觀看法是此模型的特色之一。第四章，針對兩個模型的進行比較。最後，我們發現：BLack-Litterman模型不僅符合經濟直覺，進行資產配置時也展現模型的穩定性。 / This paper applies a popular asset allocation model: the Black-Litterman model on a fund of funds. First, an overview is given of the foundations of modern portfolio theory with the mean-variance model. Next, we discuss some improvements that could be made over the mean-variance model. The Black-Litterman model addresses some of these flaws and tries to improve them. Finally, simulation has been performed to compare the performance of the Black-Litterman model to mean-variance optimization. The models have been compared in intuitiveness and stability. The conclusion can be drawn that BL-model improves the mean-variance model, in our simulation, both in intuitiveness and stability.

當產險公司需要同時兼顧競爭力並免於破產時，適當的資產配置就是一項相當重要的決策。然而採用均數-變異數分析(mean‐variance analysis)將受到許多限制，而動態控制理論則是難以實作，因此，我們提出一個新的解決方法。這個方法主要係應用模擬最佳化的演算法，例如基礎的基因演算法(basic genetic algorithm, GA)，多階層演化策略(multi-phase evolutionary strategies, MPES)及多階層基因演算法(multi-phase genetic algorithm, MPGA)等並結合模擬模型，來求解保險公司之資產配置的問題。首先我們建立投資市場及保險業務市場的模擬模型，之後再利用本研究所發展出新的最佳化演算法來搜尋最佳的資產配置。在實務上無法實現的多期投資策略，在我們的研究架構下得以被採用，並且在比較求解結果下，多期投資策略(reallocation strategies)較定額投資策略(re‐balancing strategies)有顯著較佳的績效。在兼顧保險公司投資收益並避免破產的目標函數下，我們所提出的研究方法已證明可以用來協助保險公司建立較佳的資產配置。 / Proper asset allocations are vital for property‐casualty insurers to be competitive and remain solvent. However, popular mean‐variance analysis is limited while dynamic control theory is difficult to implement. We thus propose to apply simulation optimizations such as basic genetic algorithm (GA), multi‐phase evolutionary strategies (MPES) and multi‐phase genetic algorithm (MPGA) to the asset allocation problems of the insurers. We first construct a simulation model of the property‐casualty insurer and then develop simulation optimization techniques to search optimal investment strategies upon the simulation results. The resulted reallocation strategies perform better than re‐balancing strategies used in practice with significant margins. Therefore, our proposal researches can be used to assist insurers to construct better asset allocations.

由於扶養比降低與平均餘命增加，退休理財規劃已被國人所重視。本研究首先介紹退休規劃的流程，依照世界銀行1994年所提出之退休所得三層架構，第一層為強制性社會安全制度的保障，第二層為退休金制度，以及第三層為自願性商業保險儲蓄制度。當退休前的自願儲蓄不足時，即可能產生退休不足度的問題，解決方式為設法提高退休所得；而影響退休所得有三個主要因素，其一為金額之多寡，其二為累積時間之長短，而最為個人能掌握的第三個重要因素為「投資報酬率」之高低，因此如何利用較佳的投資策略與報酬，減少退休所得不足的問題即為一種大課題。本研究主要的目的為考量風險因素後，分析各種投資策略的績效與檢視交易成本對其之影響，並且設法在有無限制風險程度下，找出最大報酬率的策略。分析結果發現固定比例混合法投資策略在各績效衡量指標下與加入交易成本考量皆有較佳的表現，且在限制風險找尋最大報酬的情形下也是如此；但如果是在沒有限制風險找尋最大報酬的情形，固定比例混合法投資策略在以尾端風險為考量之決策目標時，即非最好的策略。 關鍵字：退休規劃、資產配置、投資策略

動態規劃是一種專門用來解決最適化的數學方法，其觀念源自於Bellman (1962)，他提出了動態規劃的最佳原則，然而動態規劃問題不見得有封閉解(closed form solution)，即使其存在，求解過程往往也相當困難且複雜。Vigna & Haberman (2001)用動態規劃方式找出最佳的投資策略並分析確定提撥制(defined contribution)下的財務風險；本研究擬以Vigna & Haberman (2001)的模型為基礎，提出解決動態規劃問題的數值方法。 Vigna & Haberman (2001)推導出確定提撥退休金制度下離散時間的最適投資策略封閉解，透過該模型，我們可以比較本研究所建議的方法與真正封閉解的差異，證實本研究所建議的方法的確可以提供動態規劃問題一個接近且有效率的數值解法。接著根據Yvonne C.(2002、2003)的抽樣方法，希望在進行模擬時，能找出模擬情境的特性並對這些情境進行抽樣，藉此減少情境數以增加電腦運算的效率。最後應用在Vigna & Haberman (2001)的修正模型以及Haberman & Vigna (2002)的模型上，說明了本研究所建議的數值方法也適用在各類型的動態規劃上，包含理論封閉解不存在以及求解非常複雜的問題。

Nos últimos anos, percebeu-se um avanço substancial das metodologias sistemáticas de seleção de ativos em portfólios financeiros, baseadas em técnicas de otimização. A maior pressão por desempenho sobre as gestoras de recursos e a evolução dos softwares e pacotes de otimização foram fatores que contribuíram para esse desenvolvimento. Dentre as técnicas mais reconhecidas utilizadas na gestão de portfólios está a de otimização robusta, cuja aplicação na solução de problemas com dados incertos iniciou-se na década de 1970 e, desde então, vem evoluindo em sofisticação. Partindo de uma extensão recente do método, propõe-se um novo modelo linear que resolve o problema de otimização de um portfólio para múltiplos estágios, com inovações no tratamento da incerteza das estimativas de dispersão dos retornos. Os resultados mostram que o método proposto desempenha muito bem em termos de rentabilidade e de métricas de risco-retorno em momentos de turbulência dos mercados. Por fim, demonstra-se empiricamente que o modelo alcança um desempenho ainda melhor em termos de rentabilidade com a adoção de um estimador eficiente para o valor esperado dos retornos e com a simultânea redução do nível de robustez do modelo. / It has been realized in the last years a remarkable development of the optimization techniques to solve the problem of financial portfolio selection. The pressure on asset management firms to maintain a more stable performance and the evolution of specialized software packages have enabled this positive trend. One of the most recognized approaches applied to the management of investments is the robust optimization, whose use on uncertain portfolio optimization problems has begun in the 1970s and has experienced a substantial growth since then. Building on a recent version of this framework, it is proposed a new linear model of the robust multistage portfolio optimization problem, thereby incorporating uncertainty about dispersion inputs in an innovative way. The results show that this method performs very well during high volatility periods in terms of the terminal wealth and the risk-return tradeoff. Finally, it can be demonstrated empirically that the proposed method outperforms when an efficient return estimator is incorporated to the optimization model and the robustness level is reduced simultaneously.

<div>The contribution of this research work can be divided into two main tasks: 1) implementing this Electronic Warfare Asset Allocation Problem (EWAAP) with the Genetic Algorithm (GA); 2) Comparing performance of Genetic Algorithm to Particle Swarm Optimization (PSO) algorithm. This research problem implemented Genetic Algorithm in C++ and used QT Data Visualization for displaying three-dimensional space, pheromone, and Terrain. The Genetic algorithm implementation maintained and preserved the coding style, data structure, and visualization from the PSO implementation. Although the Genetic Algorithm has higher fitness values and better global solutions for 3 or more receivers, it increases the running time. The Genetic Algorithm is around (15-30%) more accurate for asset counts from 3 to 6 but requires (26-82%) more computational time. When the allocation problem complexity increases by adding 3D space, pheromones and complex terrains, the accuracy of GA is 3.71% better but the speed of GA is 121% slower than PSO. In summary, the Genetic Algorithm gives a better global solution in some cases but the computational time is higher for the Genetic Algorithm with than Particle Swarm Optimization.</div>

Nos últimos anos, percebeu-se um avanço substancial das metodologias sistemáticas de seleção de ativos em portfólios financeiros, baseadas em técnicas de otimização. A maior pressão por desempenho sobre as gestoras de recursos e a evolução dos softwares e pacotes de otimização foram fatores que contribuíram para esse desenvolvimento. Dentre as técnicas mais reconhecidas utilizadas na gestão de portfólios está a de otimização robusta, cuja aplicação na solução de problemas com dados incertos iniciou-se na década de 1970 e, desde então, vem evoluindo em sofisticação. Partindo de uma extensão recente do método, propõe-se um novo modelo linear que resolve o problema de otimização de um portfólio para múltiplos estágios, com inovações no tratamento da incerteza das estimativas de dispersão dos retornos. Os resultados mostram que o método proposto desempenha muito bem em termos de rentabilidade e de métricas de risco-retorno em momentos de turbulência dos mercados. Por fim, demonstra-se empiricamente que o modelo alcança um desempenho ainda melhor em termos de rentabilidade com a adoção de um estimador eficiente para o valor esperado dos retornos e com a simultânea redução do nível de robustez do modelo. / It has been realized in the last years a remarkable development of the optimization techniques to solve the problem of financial portfolio selection. The pressure on asset management firms to maintain a more stable performance and the evolution of specialized software packages have enabled this positive trend. One of the most recognized approaches applied to the management of investments is the robust optimization, whose use on uncertain portfolio optimization problems has begun in the 1970s and has experienced a substantial growth since then. Building on a recent version of this framework, it is proposed a new linear model of the robust multistage portfolio optimization problem, thereby incorporating uncertainty about dispersion inputs in an innovative way. The results show that this method performs very well during high volatility periods in terms of the terminal wealth and the risk-return tradeoff. Finally, it can be demonstrated empirically that the proposed method outperforms when an efficient return estimator is incorporated to the optimization model and the robustness level is reduced simultaneously.