Portfolio diversification has been a subject frequently addressed since the publications of Markowitz in 1952 and 1959. However, the Modern Portfolio Theory and its mean variance framework have been criticized. The critiques refer to the assumptions that return distributions are normally distributed and the symmetric definition of risk. This paper elaborates on these short comings and applies a heuristic downside risk approach to avoid the pitfalls inherent in the mean variance framework. The result of the downside risk approach is compared and contrasted with the result of the mean variance framework. The return data refers to the real estate sector in Sweden and diversification is reached through property type and geographical location. The result reveals that diversification is reached differently between the two approaches. The downside risk measure applied here frequently diversifies successfully with use of fewer proxies. The efficient portfolios derived also reveals that the downside risk approach would have contributed to a historically higher average total return. This paper outlines a framework for portfolio diversification, the result is empirical and further research is needed in order to grasp the potential of the downside risk measures.

This dissertation is comprised of two papers. The first paper (Chapter 1) provides the theoretical foundation for the estimation of systematic downside risk. Using a new approach, I derive a measure of downside systematic risk, downside beta, that is free of the endogeneity problem and thus straightforward to calculate. Since there is no consensus in the literature regarding the appropriate method for the estimation of downside beta, I review the alternative specifications proposed in the past. I explicitly show that the derived formula here is more efficient in capturing downside risk on both theoretical grounds and in terms of empirical results. Using this efficient specification of systematic downside risk, I show that downside beta has increased explanatory power towards the cross-section of equity returns as compared to unconditional beta. In particular, downside beta predicts larger and more significant future premia, insignificant intercepts in portfolio cross-section tests and cannot be subsumed by additional risk factors proposed in the past literature. I attribute this superior performance to the ability of downside risk to capture distress risk and to the fact that it does not penalize (reward) good (bad) events in good states. In the second paper (Chapter 2) that is co-authored with my advisor, Gregory Kadlec, we exploit the notion of downside risk to explain a long-withstanding market anomaly; the long-term stock return reversals. We show that downside betas of past losers are significantly greater than downside betas of past winners, and the inclusion of downside beta in Fama-Macbeth regressions subsumes the reversal effect. / Ph. D.

[pt] O conceito de risco é definido como a distribuição de resultados inesperados devido a alterações nos valores das variáveis que descrevem o mercado. Entretanto, o risco não é uma variável observável e sua quantificação depende do modelo empregado para avaliá-lo. Portanto, o uso de diferentes modelos pode levar a previsões de risco significativamente diferentes.O objetivo principal desta dissertação é realizar um estudo comparativo dos modelos mais amplamente utilizados (medição de variância amostral nos últimos k períodos, modelos de amortecimento exponencial e o GARCH(1,1) de Bollerslev) quanto à capacidade preditiva da volatilidade.Esta dissertação compara os modelos de estimação de volatilidade citados acima quanto à sua capacidade preditiva para carteiras compostas por um conjunto de ações negociadas no mercado brasileiro. As previsões de volatilidade desses modelos serão comparadas com a volatilidade real fora da amostra. Como a volatilidade real não é uma variável observável, usou-se o mesmo procedimento adotado pelo RiskMetrics para o cálculo do fator de decaimento ótimo: assumiu-se a premissa que o retorno médio de cada uma das carteiras de ações estudadas é igual a zero e,como conseqüência disso, a previsão um passo à frente da variância do retorno realizada na data t é igual ao valor esperado do quadrado do retorno na data t.O objetivo final é concluir, por meio de técnicas de backtesting, qual dos modelos de previsão de volatilidade apresentou melhor performance quanto aos critérios de comparação vis-à-vis ao esforço computacional necessário. Dessa forma, pretende-se avaliar qual desses modelos oferece a melhor relação custo-benefício para o mercado acionário brasileiro. / [en] The risk concept is defined as the distribution of the unexpected results from variations in the values of the variables that describe the market. However, the variable risk is not observable and its measurement depends on which model is used in its evaluation. Thus, the application of different models could result in significant different risk forecasts.The goal of this study is to carry out a comparison within the largest used models (sample variance in the last k observations, exponentially smoothing models and the Bollerslev s model GARCH(1,1)). The study compares the models mentioned above regarding its forecast capability of the volatility for portfolios of selected brazilian stocks. The volatility forecasts will be compared to the actual out of sample volatility. As long as the actual volatility is not an observable variable, the same procedure adopted by RiskMetrics in the calculation of the optimum decay factor will be used: it assumes the premise that the average return of which one of the stock portfolios is equal zero and, as the consequence of this fact, the one step variance forecast of the portfolio return carried out on date t is equal to expected value of the squared return of date t.The final objective is to conclude, using backtesting techniques, which of the forecasting volatility models show the best performance regarding the comparison criterions vis-a-vis the demanding computer efforts. By this way, it was aimed to evaluate which of them offer the best cost-benefit relation for the brazilian equity market.

This thesis is focused on the analysis of performance of chosen russian mutual funds on the basis of achieved yield and risk. After short introduction to the russian market of mutual funds, the paper deals with a theoretical background underlying the performance indicators. Risk perception and following construction of indicators are discussed in detail from the perspective of modern and post-modern portfolio theory. The indicators are interpreted and appropriateness of their application is assessed. The analytic part is devoted to the application of discussed methods on 10 open-ended equity mutual funds. Based on the result, the funds are compared with each other and with selected market index.

由於許多資產報酬率的分配呈現厚尾的現象，因此，本文探討將最低報酬要求限制條件加入傳統的平均數╱變異數模型中，考慮在分配已知的情形下，假設資產報酬率的分配為t分配及常態分配，來求取最適的資產配置；在分配未知的情形下，利用古典Bootstrap法、移動區塊Bootstrap法及定態Bootstrap法的抽樣方法來模擬資產報酬率的分配形式，並利用模擬的資產報酬率分配求出最適的資產配置。 同時，本文亦探討資產配置在風險管理上的運用，當分配已知時，若對分配參數的估計正確，則使用的最低要求報酬率就是此資產配置的涉險值，反之，若對參數的估計錯誤時，會對資產配置產生很大的影響及風險管理上的不正確；當分配未知時，利用模擬方法來產生分配，則使用的最低要求報酬率可看成是此資產配置的涉險值。 實證部分選取資料分成本國及全球，研究發現對於何種分配或模擬方法的資產配置績效最好？沒有一定的結論。其原因是各種分配或模擬方法皆必須視資料的性質而定，因此，本論文的貢獻僅在建議使用厚尾分配及利用模擬方法，來符合資產報酬率呈現厚尾的現象，並利用此分配，以期在考慮最低報酬要求限制條件下的資產配置更為精確。 / The distributions of many asset returns tend to be fat-tail. This paper attempts to add the shortfall constraint in Mean-Variance Analysis. When the distribution is known, we find the optimal asset allocation under student-t distribution and normal distribution. On the other hand, we use Classical Bootstrap, Moving Block Bootstrap, and Stationary Bootstrap to stimulate the distribution of asset return, and to obtain the optimal asset allocation. We also examine the risk management of asset allocation. When we use the correct estimators of parameters under the known distribution, the threshold in shortfall constraint is the value-at-risk in asset allocation. Otherwise, if using the wrong estimators, we get the incorrect asset allocation and the improper risk management. When the distribution is unknown, using simulation to generate the distribution, the value-at-risk is the threshold. The empirical study is conducted in two parts, domestic and global asset allocation. The results cannot point out which distributions and simulations are suitable. They depend on the data’s property. The contribution of this paper is to introduce some methods to fit the fat-tail behavior of asset return in asset allocation.

已開發國家中，無論個人或是法人所擁有之財富大多透過金融中介機構管理，因此，財富委由他人管理衍生出現代資本市場中重要的委託關係。委託人與基金管理人產生委任契約時，也必然產生代理問題，即雙方利益不一致所額外增加的成本。為降低代理成本，於委任合約加入對管理人下檔投資風險的要求成為降低代理成本的重要機制。本研究因此探討當基金管理人面對契約存在最低報酬要求時，如何進行最適資產配置決策，並同時分析下檔風險限制改變時對管理人投資行為的影響。研究結果顯示，委任合約增加經理人最低保證收益時，基金管理人傾向增加持股，而經理人風險趨避程度增加時，將減少風險性股票資產，進而持有債券；如果投資目標收益於受委託期間皆不改變，將造成經理人持有債券組合以規避下檔風險，同時卻喪失追求資本利得。 / In most developed countries, financial wealth is not managed directly by the investors, but through a financial intermediary. Hence, the delegated portfolio management is one of the most important principal-agency relationships in the current economy. In addition to that, the principal-agency relationships between the investor and portfolio manager must produce agency cost. In order to reduce these costs, the mandates in the contract become an important factor in reducing the principal-agent problem in a delegated portfolio management framework. In this research, we study how fund managers do asset allocation when they face some guaranteed returns and the relationships between the choices of mandates and the behavior of fund managers. We suppose that the objective of the delegated fund managers is to maximize the expected utility of wealth of the long-term fund at the end of each period and fund managers also have to fulfill some constrains given at the beginning. Finally, we explain how fund managers do optimal asset allocation by our model and some numerical analysis.

現行各國的退休金計畫逐漸地由確定給付制轉變為確定提撥制。這表示投資的風險由原本退休金計畫的發起者(雇主)轉移到了參與者(員工)的身上。為了減少每個確定提撥制計畫參與者的投資風險，本文中採用退休時所得替代率為預估的目標，藉由模擬與最適化的方法找到最適投資策略與最適提撥率。 能反映出時間性的隨機模型在精算科學的領域是日漸重要，本文試著藉由隨機性的變化來估計代替以往精算上各種假設下所求得的負債。本文藉由隨機模擬的方式，得到各種資產在市場上或者是經濟上的價值來建構相關投資標的之報酬率，並利用動態隨機規劃模型去改善財務上避險以及資產負債管理。此外，為了避免模擬分析時間過長的問題，本文採用了情境抽樣的方法去改善電腦模擬分析計算時的效率。 我們主要得到以下結論： (一)確定提撥制下的負債受薪資水準波動的影響，所以此時會持有較 多的指數連結型債券以反應薪資水準及通貨膨脹的影響。整體投 資的結果與Vigna & Haberman (2001) 文中的結果及實務上生命 週期型態(lifestyle)投資方式呈現相同的現象。 (二)考慮每期下跌風險(downside risk)時，期中的投資可能會偏向 於投資風險較高的股票。在每年觀察下跌風險的情況下其投資因 為必須考慮避免每一年的下跌風險，需要比每五年觀察下跌風險 的情況做風險較大的投資，以達到其目標。 (三)在本文的調整投資組合策略下，因為調整次數不多，所以在考慮 交易成本的情況，當交易成本很小時對於整體的最適化資產配置 與最適化提撥率的影響是很小的。在本文的調整投資組合策略 下，交易成本的影響只有在交易成本非常大的情況下才能看得出 來。 (四)均勻抽樣法抽出的400組情境幾乎可以完全的代替4000組情境， 其結果可以看出與未抽樣相同的生命週期型態(lifestyle)投資 方式。而隨機抽樣法的結果雖然也可看出趨勢，但準確性相對於 均勻抽樣法仍稍嫌不足，並不適合用來代替原先的4000組情境。 / A shift from defined-benefit pension plan towards defined-contribution pension plan is currently popular around the world. This means that a serious investment risk transfers from defined-benefit sponsors to the individual members of defined-contribution plans. In order to reduce the risk of individual DC member, we investigate the methodology of finding the optimal contribution rate and asset allocation to reach a certain target of the retirement replacement rate in this paper. Stochastic processes are getting more important to the field of actuarial science. Instead of trying to approximate liabilities by a single deterministic set of actuarial assumption, we seek to take account of market or economic valuation for both assets and liabilities using stochastic simulation. We applied dynamic stochastic programming models to improve financial hedging and asset liability management. Moreover, in order to avoid the problem of time-consuming, we use scenario sampling method to improve the efficiency of computer calculation. We draw four conclusions from our investigations： (1)We will hold more assets in indexed-linked bonds because the pension liability is highly related to the wage- index and inflation rate. The optimal investment strategy is very like the so called "lifestyle" investment strategy. (2)When we consider downside risk, we should hold more risky equities. The investment strategy is more risky when we consider downside risk every year than every 5 years. (3)Under our rebalancing strategy, if the transaction cost is small, the influence on the investment strategy and contribution rate is small. We can see the influence of the transaction cost in a situation that the transaction cost is very big only. (4)There are almost no different between uniform sampling scenarios and original simulation scenarios, so uniform sampling scenarios may replace the original simulation scenarios perfectly. And random sampling method is unsuitable to replace the original simulation scenarios.

Asset allocation is the most influential factor driving investment performance. While researchers have made substantial progress in the field of asset allocation since the introduction of mean-variance framework by Markowitz, there is little agreement about appropriate portfolio choice for multi-period long horizon investors. Nowhere this is more evident than trustees of retirement plans choosing different asset allocation strategies as default investment options for their members. This doctoral dissertation consists of four essays each of which explores either a novel or an unresolved issue in the area of asset allocation for individual retirement plan participants. The goal of the thesis is to provide greater insight into the subject of portfolio choice in retirement plans and advance scholarship in this field. The first study evaluates different constant mix or fixed weight asset allocation strategies and comments on their relative appeal as default investment options. In contrast to past research which deals mostly with theoretical or hypothetical models of asset allocation, we investigate asset allocation strategies that are actually used as default investment options by superannuation funds in Australia. We find that strategies with moderate allocation to stocks are consistently outperformed in terms of upside potential of exceeding the participant’s wealth accumulation target as well as downside risk of falling below that target by very aggressive strategies whose allocation to stocks approach 100%. The risk of extremely adverse wealth outcomes for plan participants does not appear to be very sensitive to asset allocation. Drawing on the evidence of the previous study, the second essay explores possible solutions to the well known problem of gender inequality in retirement investment outcomes. Using non-parametric stochastic simulation, we simulate iv and compare the retirement wealth outcomes for a hypothetical female and male worker under different assumptions about breaks in employment, superannuation contribution rates, and asset allocation strategies. We argue that modest changes in contribution and asset allocation strategy for the female plan participant are necessary to ensure an equitable wealth outcome in retirement. The findings provide strong evidence against gender-neutral default contribution and asset allocation policy currently institutionalized in Australia and other countries. In the third study we examine the efficacy of lifecycle asset allocation models which allocate aggressively to risky asset classes when the employee participants are young and gradually switch to more conservative asset classes as they approach retirement. We show that the conventional lifecycle strategies make a costly mistake by ignoring the change in portfolio size over time as a critical input in the asset allocation decision. Due to this portfolio size effect, which has hitherto remained unexplored in literature, the terminal value of accumulation in retirement account is critically dependent on the asset allocation strategy adopted by the participant in later years relative to early years. The final essay extends the findings of the previous chapter by proposing an alternative approach to lifecycle asset allocation which incorporates performance feedback. We demonstrate that strategies that dynamically alter allocation between growth and conservative asset classes at different points on the investment horizon based on cumulative portfolio performance relative to a set target generally result in superior wealth outcomes compared to those of conventional lifecycle strategies. The dynamic allocation strategy exhibits clear second-degree stochastic dominance over conventional strategies which switch assets in a deterministic manner as well as balanced diversified strategies.

Nesta dissertação, o modelo de seleção de carteiras de Markowitz será estendido com uma análise em tempo discreto e hipóteses mais realísticas. Um produto tensorial finito de densidades Erlang será usado para aproximar a densidade de probabilidade multivariada dos retornos discretos uniperiódicos de ativos dependentes. A Erlang é um caso particular da distribuição Gama. Uma mistura finita pode gerar densidades multimodais não-simétricas e o produto tensorial generaliza este conceito para dimensões maiores. Assumindo que a densidade multivariada foi independente e identicamente distribuída (i.i.d.) no passado, a aproximação pode ser calibrada com dados históricos usando o critério da máxima verossimilhança. Este é um problema de otimização em larga escala, mas com uma estrutura especial. Assumindo que esta densidade multivariada será i.i.d. no futuro, então a densidade dos retornos discretos de uma carteira de ativos com pesos não-negativos será uma mistura finita de densidades Erlang. O risco será calculado com a medida Downside Risk, que é convexa para determinados parâmetros, não é baseada em quantis, não causa a subestimação do risco e torna os problemas de otimização uni e multiperiódico convexos. O retorno discreto é uma variável aleatória multiplicativa ao longo do tempo. A distribuição multiperiódica dos retornos discretos de uma seqüência de T carteiras será uma mistura finita de distribuições Meijer G. Após uma mudança na medida de probabilidade para a composta média, é possível calcular o risco e o retorno, que levará à fronteira eficiente multiperiódica, na qual cada ponto representa uma ou mais seqüências ordenadas de T carteiras. As carteiras de cada seqüência devem ser calculadas do futuro para o presente, mantendo o retorno esperado no nível desejado, o qual pode ser função do tempo. Uma estratégia de alocação dinâmica de ativos é refazer os cálculos a cada período, usando as novas informações disponíveis. Se o horizonte de tempo tender a infinito, então a fronteira eficiente, na medida de probabilidade composta média, tenderá a um único ponto, dado pela carteira de Kelly, qualquer que seja a medida de risco. Para selecionar um dentre vários modelos de otimização de carteira, é necessário comparar seus desempenhos relativos. A fronteira eficiente de cada modelo deve ser traçada em seu respectivo gráfico. Como os pesos dos ativos das carteiras sobre estas curvas são conhecidos, é possível traçar todas as curvas em um mesmo gráfico. Para um dado retorno esperado, as carteiras eficientes dos modelos podem ser calculadas, e os retornos realizados e suas diferenças ao longo de um backtest podem ser comparados. / In this thesis, Markowitz’s portfolio selection model will be extended by means of a discrete time analysis and more realistic hypotheses. A finite tensor product of Erlang densities will be used to approximate the multivariate probability density function of the single-period discrete returns of dependent assets. The Erlang is a particular case of the Gamma distribution. A finite mixture can generate multimodal asymmetric densities and the tensor product generalizes this concept to higher dimensions. Assuming that the multivariate density was independent and identically distributed (i.i.d.) in the past, the approximation can be calibrated with historical data using the maximum likelihood criterion. This is a large-scale optimization problem, but with a special structure. Assuming that this multivariate density will be i.i.d. in the future, then the density of the discrete returns of a portfolio of assets with nonnegative weights will be a finite mixture of Erlang densities. The risk will be calculated with the Downside Risk measure, which is convex for certain parameters, is not based on quantiles, does not cause risk underestimation and makes the single and multiperiod optimization problems convex. The discrete return is a multiplicative random variable along the time. The multiperiod distribution of the discrete returns of a sequence of T portfolios will be a finite mixture of Meijer G distributions. After a change of the distribution to the average compound, it is possible to calculate the risk and the return, which will lead to the multiperiod efficient frontier, where each point represents one or more ordered sequences of T portfolios. The portfolios of each sequence must be calculated from the future to the present, keeping the expected return at the desired level, which can be a function of time. A dynamic asset allocation strategy is to redo the calculations at each period, using new available information. If the time horizon tends to infinite, then the efficient frontier, in the average compound probability measure, will tend to only one point, given by the Kelly’s portfolio, whatever the risk measure is. To select one among several portfolio optimization models, it is necessary to compare their relative performances. The efficient frontier of each model must be plotted in its respective graph. As the weights of the assets of the portfolios on these curves are known, it is possible to plot all curves in the same graph. For a given expected return, the efficient portfolios of the models can be calculated, and the realized returns and their differences along a backtest can be compared.

Nesta dissertação, o modelo de seleção de carteiras de Markowitz será estendido com uma análise em tempo discreto e hipóteses mais realísticas. Um produto tensorial finito de densidades Erlang será usado para aproximar a densidade de probabilidade multivariada dos retornos discretos uniperiódicos de ativos dependentes. A Erlang é um caso particular da distribuição Gama. Uma mistura finita pode gerar densidades multimodais não-simétricas e o produto tensorial generaliza este conceito para dimensões maiores. Assumindo que a densidade multivariada foi independente e identicamente distribuída (i.i.d.) no passado, a aproximação pode ser calibrada com dados históricos usando o critério da máxima verossimilhança. Este é um problema de otimização em larga escala, mas com uma estrutura especial. Assumindo que esta densidade multivariada será i.i.d. no futuro, então a densidade dos retornos discretos de uma carteira de ativos com pesos não-negativos será uma mistura finita de densidades Erlang. O risco será calculado com a medida Downside Risk, que é convexa para determinados parâmetros, não é baseada em quantis, não causa a subestimação do risco e torna os problemas de otimização uni e multiperiódico convexos. O retorno discreto é uma variável aleatória multiplicativa ao longo do tempo. A distribuição multiperiódica dos retornos discretos de uma seqüência de T carteiras será uma mistura finita de distribuições Meijer G. Após uma mudança na medida de probabilidade para a composta média, é possível calcular o risco e o retorno, que levará à fronteira eficiente multiperiódica, na qual cada ponto representa uma ou mais seqüências ordenadas de T carteiras. As carteiras de cada seqüência devem ser calculadas do futuro para o presente, mantendo o retorno esperado no nível desejado, o qual pode ser função do tempo. Uma estratégia de alocação dinâmica de ativos é refazer os cálculos a cada período, usando as novas informações disponíveis. Se o horizonte de tempo tender a infinito, então a fronteira eficiente, na medida de probabilidade composta média, tenderá a um único ponto, dado pela carteira de Kelly, qualquer que seja a medida de risco. Para selecionar um dentre vários modelos de otimização de carteira, é necessário comparar seus desempenhos relativos. A fronteira eficiente de cada modelo deve ser traçada em seu respectivo gráfico. Como os pesos dos ativos das carteiras sobre estas curvas são conhecidos, é possível traçar todas as curvas em um mesmo gráfico. Para um dado retorno esperado, as carteiras eficientes dos modelos podem ser calculadas, e os retornos realizados e suas diferenças ao longo de um backtest podem ser comparados. / In this thesis, Markowitz’s portfolio selection model will be extended by means of a discrete time analysis and more realistic hypotheses. A finite tensor product of Erlang densities will be used to approximate the multivariate probability density function of the single-period discrete returns of dependent assets. The Erlang is a particular case of the Gamma distribution. A finite mixture can generate multimodal asymmetric densities and the tensor product generalizes this concept to higher dimensions. Assuming that the multivariate density was independent and identically distributed (i.i.d.) in the past, the approximation can be calibrated with historical data using the maximum likelihood criterion. This is a large-scale optimization problem, but with a special structure. Assuming that this multivariate density will be i.i.d. in the future, then the density of the discrete returns of a portfolio of assets with nonnegative weights will be a finite mixture of Erlang densities. The risk will be calculated with the Downside Risk measure, which is convex for certain parameters, is not based on quantiles, does not cause risk underestimation and makes the single and multiperiod optimization problems convex. The discrete return is a multiplicative random variable along the time. The multiperiod distribution of the discrete returns of a sequence of T portfolios will be a finite mixture of Meijer G distributions. After a change of the distribution to the average compound, it is possible to calculate the risk and the return, which will lead to the multiperiod efficient frontier, where each point represents one or more ordered sequences of T portfolios. The portfolios of each sequence must be calculated from the future to the present, keeping the expected return at the desired level, which can be a function of time. A dynamic asset allocation strategy is to redo the calculations at each period, using new available information. If the time horizon tends to infinite, then the efficient frontier, in the average compound probability measure, will tend to only one point, given by the Kelly’s portfolio, whatever the risk measure is. To select one among several portfolio optimization models, it is necessary to compare their relative performances. The efficient frontier of each model must be plotted in its respective graph. As the weights of the assets of the portfolios on these curves are known, it is possible to plot all curves in the same graph. For a given expected return, the efficient portfolios of the models can be calculated, and the realized returns and their differences along a backtest can be compared.