Black-Litterman är en allokeringsmodell som gör det möjligt att förena historiska avkastningar med personliga övertygelser om framtida avkastningar från en enskild investerare. Denna studie jämför två kvantitativa metoder i framtagande av felskattningen för framtida prognoser i syfte att kunna minska Black-Littermans subjektivitet. Tidigare litteratur har testat dessa metoder enskilt men aldrig ställt dem mot varandra. De metoder som undersöks använder varianser proportionella mot varianser i marknadsjämvikten, samt varianser från residualer i en faktormodel. Resultatet visar att tillämpandet av varianser framtagna av en GARCH (1,1)-modell är den metod som genererar högst avkastning, samt ger upphov till en fördelning av tillgångar som bidrar till lägst marknadskänslighet. Utifrån denna studie rekommenderas därmed tillämpningen av varianser från residualer i en faktormodel som tillägg för att minska modellens godtycklighet. / The Black-Litterman allocation model unifies historical returns with investor personal views of future returns. The study compares two quantitative methods for the estimation of uncertainty in future views with the goal to mitigate the subjectivity of the Black-Litterman model. Previous literature have investigated and tested these methods independently but a comparison has never been made between them. The two methods consist of using variances in proportion to the variances of market equilibrium and operating the residual variance of a factor model. Results show that the usage of variances estimated by a GARCH (1,1) will generate the highest average returns with an allocation distribution that contributes to least market sensitivity. Furthermore, the study recommends the implementation of variances from residuals with the addition of a factor model to diminish the subjectivity of the Black-Litterman model.

This thesis investigates how neural networks can be used to produce investors' views for the Black-Litterman market model. The study uses two data sets, one with global stock market indexes and one with stock market data from the S&P 500. The task of the neural networks is to produce forecasts for the returns for the next quarter and the following year. The neural network will have to predict whether the market will move up or down and determine if the market movement is less than or equal to one standard deviation, creating four different scenarios. The forecasts are used as input to the Black-Litterman model to generate new portfolios, which are backtested from 2017 until 2022. The index data set was compared to a benchmark portfolio and a portfolio with naive risk diversification, while the S&P 500 data set was compared to market capitalization-weighted and naive portfolios. This resulted in eight different backtests where the neural networks obtained AUC values in the range of 0.56-0.73 and prediction accuracies in the range of 20.9% - 42.1%. The network used for yearly predictions on the index data set was the only network to outperform the benchmark portfolio. It obtained a Sharpe ratio of 1.782, a Sortino ratio of 2.165, and a maximum drawdown of -30.9% compared to the benchmark portfolio, where the corresponding metrics were 1.544, 1.879, and -32.8%. / Detta examensarbete undersöker hur neurala nätverk kan användas för att generera investerares åsikt till Black-Littermans marknadsmodell. Studien använder två dataset, en med globala börsindex och en med börsdata från S6P 500. De neurala nätverkens uppgift är att generera prognoser för avkastning för nästa kvartal samt nästkommande år. Det neurala nätverket måste förutsäga om marknaden kommer att röra sig uppåt eller nedåt, och avgöra om marknadsrörelsen är mindre än eller lika med en standaravvikelse, vilket skapar fyra olika scenarier. Prognoserna användas som input till Black-Litterman-modellen för att generera nya portföljer, som backtestas från 2017 till 2022. Portföljerna som skapades med globala börsindex jämfördes med en benchmarkportfölj och en portfölj med naiv riskspridning. Datasetet med data från S&P 500 jämfördes med marknadsvärdesviktade och naiva portföljer. Detta resulterade i åtta olika simuleringar där de neurala nätverken fick AUC-värden i intervallet 0,56-0,73 och prediktionsnoggrannheter i intervallte 20,9% - 42,1%. Nätverket som användes för årliga prognoser om globala börsindex var det enda nätverket som överträffade jämförelseportföljen. Den fick en Sharpekvot på 1, 782, Sortinokvot på 2,165 och en största kumulativa nedgång på -30,9% jämfört med jämförelseportföljen där motsvarande mätvärden var 1, 544, 1, 879 och -32,8%.

TESIS PARA OPTAR AL GRADO DE MAGISTER EN FINANZAS / Una de las labores más complicadas de un portafolio manager es sin lugar a duda la asignación eficiente de los activos de inversión que tenga bajo su administración. En un mundo que entrega un sin número de instrumentos de inversión, es primordial contar con herramientas que ayuden a tomar las mejores decisiones a la hora de invertir. Dado lo anterior este trabajo se enfocó en el estudio del modelo de Black-Littermam y su aplicación real en el Índice de precios selectivos de acciones (IPSA), dándole especial énfasis al desarrollo de las “views”. Este estudio se desarrolló desde enero del 2010 a junio de 2015. El objetivo de este trabajo fue lograr una metodología con aplicación real, que obtuviese como resultado una rentabilidad mayor al IPSA y a los principales fondos mutuos del país (para ello se consideraron los diez fondos mutuos con mayor patrimonio), cuyo foco de inversión fuera la renta variable nacional. El modelo de Black-Litterman ha tomado gran relevancia desde su creación a principios de los años 90, porque a diferencia del modelo de Markowitz, este permitió incorporar views (visiones) que tienen los inversionistas sobre un activo o sector económico en particular. Según palabras del propio Litterman, el mercado estaría en un permanente estado de equilibrio puntual, en el cual la oferta y la demanda por activos se equiparan; por tanto este equilibrio puntual puede ser comprendido como el “centro de gravedad”, del cual los mercados se desvían en todo instante, según la información que este surgiendo, pero posteriormente el mismo mercado y la información presionaran los precios para que el mercado vuelva a equilibrarse. Este equilibrio de mercado (el cual funciona como base para el modelo), es combinado con el concepto de views, permitiendo adelantarse y tomar ventajas sobre las desviaciones, antes que el mercado vuelva a su equilibrio natural. Para este trabajo se propusieron dos fuentes de información para analizar las views. Una consistente en analizar las visiones que poseen las Administradoras de Fondos de Pensión (AFPs) y otra consiste en analizar las visiones que poseen los analistas listados en bloomberg sobre empresas pertenecientes al IPSA. Los resultados de este trabajo estuvieron acordes a lo esperado, ósea se logró una metodología que pudiese tener una aplicación real y entregar una rentabilidad que estuviese sobre el mercado, además de ser una alternativa real a la estrategia buy and hold aplicada por los fondos mutuos que invierten en acciones locales.

An investor wants to maximize return at the cost of as little risk as possible and theBlack-Litterman model can help see that this condition is met. This thesis willinvestigate whether a portfolio created by using modern portfolio theory can beat thebenchmark index in terms of risk-adjusted return during a five year backtest period(2013-2017). Harry Markowitz provides the mean variance optimization frameworkwhile the practical Black-Litterman model adds the opportunity to tweak performancewith views on stock returns. The method for producing views for the Black-Littermanmodel can vary a lot and is what makes this thesis, for all that we know, unique. Theviews in our model stem from regression on the summed up earnings per share forthe last four quarters multiplied by the corresponding historical price earnings ratioand the historical stock price. The regressions provide data on how over orundervalued the stocks are. Backtesting our modified Black-Litterman model yieldsimpressive results in terms of risk-adjusted return and we encourage other studentsof the financial market to further investigate the performance of our modified portfolio.However most of the results are not statistically significant on a 5% significance leveldue to the need for more data points. This method is purely quantitative and can befully replicated to yield the same results for any interested investor.

[pt] Black e Litterman propuseram um modelo de otimização de portfólio que combina visões do investidor sobre retornos esperados de ativos com o equilíbrio neutro de mercado. No entanto, especificar visões sobre uma carteira de investimentos é uma tarefa difícil, especialmente quando os investidores têm opiniões conflitantes sobre o mesmo ativo. Neste trabalho, é proposto uma nova formulação para otimização de carteiras, que é robusta diferentes à visões do investidor. A nossa abordagem foi testada em dados sintéticos e dados reais disponíveis em uma plataforma do Banco Central do Brasil. Esta plataforma consolida projeções macroeconômicas de mais de uma centena de analistas profissionais e disponibiliza para o mercado numa base semanal. Por fim, é comparado o desempenho desta formulação robusta com o modelo Black-Litterman tradicional frequentemente utilizado na indústria financeira. Os resultados mostram que a metodologia robusta pode providenciar melhor desempenho ajustado ao risco em comparação com o modelo orignial e são menos sensíveis às visões do investor. / [en] Black and Litterman proposed a portfolio optimization model that combines investor s views on future asset s returns with neutral market equilibrium. However, specifying portfolio views is a challenging task, specially when investors have conflicting opinions on the same asset. In this thesis, we suggest a new portfolio optimization formulation that is robust for investor s views. Our approach was tested on synthetic and real data available on a framework developed by Central Bank of Brazil. This online framework collects projections on main macroeconomics variables from more than a hundred professional forecasters and provides public online access on a weekly basis. The performance of this new robust formulation is compared with the traditional Black-Litterman model. The result show that our robust methodology can provide better risk adjusted performance compared to the orignial model and are less sensitive to incorrect inverstor views.

This thesis is an implementation project of a portfolio optimization model, with the purpose of creating a decision support tool. It aims to provide quantitative input to the portfolio construction process at Handelsbanken Fonder, by applying Konno & Yamazaki’s Mean Absolute Deviation method, with a Feinstein & Thapa modification. Additionally, the Black-Litterman model is implemented to approximate the input of expected return. The linear optimization problem was then solved by the Simplex algorithm. The main deliverable is a model that can assist portfolio managers in making investment decisions. Back-testing of the model showed that it did not outperform the benchmark portfolios, which is likely a result of only allowing long positions in the model. Nevertheless, the model provides value by giving the user a second opinion on the efficient frontier, for any given investment decision. / Den här uppsatsen är ettimplementationsprojekt av enportföljoptimerings-modell, med syftet att skapaett beslutsstödjande verktyg. Den strävar efter att ge ett kvantitativt bidragtill portföljallokerings-processen på Handelsbanken Fonder, genom att användaKonno & Yamazaki’s Mean Absolute Deviation-metod med en Feinstein &Thapa-modifiering. Vidare har Black-Littermanmodellen implementerats för attapproximera den förväntade avkastningen. Det linjära optimeringsproblemetlöstes sedan med Simplex-algorithmen. Det huvudsakliga resultatet är en modellsom kan assisterafondförvaltare i investeringsbeslut. Utförda utfallstestvisade att modellen inte överträffade de använda benchmark-fonderna, vilketsannolikt är ett resultat av att modellen enbart tillåterlånga positioner.Likväl, kan modellen vara värdefull genom att erbjuda användaren ett alternativpå den effektiva fronten, för ett givet investeringsbeslut.

[pt] Black e Litterman propuseram um modelo de seleção de portfólio que combina a visão dos investidores acerca de ativos com conceitos de equilíbrio de mercado para construir portfólios ótimos. Entretanto, a eficiência do modelo depende da qualidade da visão futura acerca do retorno dos ativos, o que é desafiador na prática. Com o objetivo de melhorar a aplicação prática do modelo Black-Litterman, o foco desse trabalho é viabilizar novas alocações com base em visões de fatores macroeconômicos que estão fora do universo de alocação. A principal vantagem é que a previsão desses fatores é amplamente fornecida por agentes de mercado. Um estudo de caso baseado nas informações disponibilizadas pelo Banco Central do Brasil é apresentado para validar a estrutura proposta. Os retornos obtidos fora da amostra e ajustados ao risco incorporando a visão de fatores macroeconômicos com a estrutura proposta superaram o modelo de média-variância tradicional e o benchmark local. / [en] Black and Litterman proposed a portfolio selection model that blends investor s views on asset returns with market equilibrium concepts to construct optimal portfolios. However, the model efficiency relies on the performance of investors views regarding tradable assets, which is challenging in practice. Focusing on improving Black-Litterman practical application, this work consists in providing new allocations based upon views on macroeconomic factors, which are largely available but not directly tradable. The main advantage is that predictions on these factors are usually provided by market players. A case study based on the information disclosed by the Brazilian Central Bank is presented to test the proposed framework. The out-of-sample risk-adjusted returns obtained incorporating the players macroeconomic expectations through the use of the proposed framework outperformed the traditional mean-variance model as well as the local benchmark.

Modern portfolio theory has its attractive characteristics of promoting diversification in a portfolio and can be seen as an easy alternative for setting optimal weights for portfolio managers. Furthermore, as portfolio managers try to beat a defined benchmark for their portfolio the Black-Litterman model allows them to include their own prospects on the future return of markets and securities. This thesis examines how the practical use of the Black-Litterman model can affect portfolios' performance. The analysis was done by calculating historical portfolio weights with investor views, without investor views, and with perfect investor views in the Black-Litterman model. Thereafter, calculating historical return and volatility for six multi-asset portfolios between 2017-09-25 and 2021-01-31. This was then compared with benchmark portfolios, which reflect the practical use. These portfolios included historically used investor views and constraints in the mean-variance optimization. The results showed that investor views had a negative effect on total return (lower return) and a positive effect on volatility (lower risk), however, an increased Sharpe ratio. The constraints in the mean-variance optimization used in the benchmark portfolios resulted in a lower total return. In conclusion, the Black-Litterman model showed robustness and did not generate unintuitive or unreasonable portfolios, and it has great potential with increasing accuracy in the investor views. / Modern portföljteori har attraktiva egenskaper vad gäller att främja diversifiering i en portfölj och kan ses som ett enkelt alternativ för att välja optimala vikter för portföljförvaltare. Eftersom portföljförvaltare försöker slå ett definierat benchmark för sin portfölj tillåter dessutom Black-Litterman modellen dem att inkludera sina egna åsikter angående förväntade avkastningar på marknader och värdepapper. Detta examensarbete undersöker hur den praktiska användningen av Black-Litterman modellen kan påverka portföljernas prestation. Analysen gjordes genom att beräkna historiska portföljvikter med Black-Litterman modellen med och utan invetserarens egna åsikter angående förväntade avkastningar, och med perfekta förväntade avkastningar. Därefter beräknades historiska avkastningar och volatiliteter för sex investeringsportföljer mellan 2017-09-25 och 2020-01-31. Detta jämfördes med benchmarkportföljer, vilka återspeglade den praktiska användningen. Dessa portföljer inkluderade historiskt använda förväntade avkastningar och restriktioner i mean-variance optimeringen. Resultaten visade att investerares åsikter angående förväntade avkastningar hade en negativ effekt på avkastningen (lägre avkastning), positiv effekt på volatiliteten (lägre risk), vilket resulterade i en högre Sharpe kvot. Restriktionerna i mean-variance optimeringen som användes i benchmarkportföjerna resulterade i en lägre totalavkastning. Sammanfattningsvis visade Black-Litterman modellen robusthet och genererade inte ointuitiva eller olämpliga portföljer, och modellen har stor potential med ökad träffsäkerhet i investerarens åsikter angående förväntade avkastningar.

Portfolio allocation theories have been studied and used ever since the mid 20th century. Nevertheless, many investors still rely on personal expertise and information gathered from the market when building their investment portfolios. The purpose of this master’s thesis is to examine how personal preferences and expertise perform compared to mathematical portfolio alloca- tion theories and how the risk between these di↵erent strategies di↵er. Using two portfolio allocation theories, the Black-Litterman model and mod- ern portfolio theory (Markowitz), a portfolio managed by the investment firm Placerum Kapitalf ̈orvaltning in Ume ̊a will be compared and challenged to investigate which strategy gives the best risk adjusted return. Using scenario modelling, the portfolios can be compared using both historical data and future forecasted scenarios to analyze the past, present and future of the allocation theories and Placerum’s investment strategy. The first allocation theory, the Black-Litterman model, combines historical information from the market with views and preferences of the investor to select the optimal allocations derived from return and volatility. The second allocation theory, the modern portfolio theory (Markowitz), only uses histori- cal data to derive correlations and returns which are then used to select the optimal allocations. By analysing several risk measures applied on the portfolios historical and forecasted data as well as comparing the performance of the portfolios, it is shown that the investment strategy used at Placerum succeeds with its intentions to achieve relatively high return while reducing the risk. However, the portfolios given using the two allocation theories results in higher potential returns but at the cost of taking on a higher risk. Comparing the two studied allocation theories, it is shown that when using the Black-Litterman model with the assumptions and views defined in this project, modern allocation theory actually beats it in terms of potential return as well as in terms of risk adjusted return, even though its underlying theory is much simpler.

This paper consolidates and compares the applicability and practicality of Black-Litterman model versus traditional Markowitz Mean-Variance model. Although well-known model such as Mean-Variance is academically sound and popular, it is rarely used among asset managers due to its deficiencies. To put the discussion into context we shed light on the improvement made by Fisher Black and Robert Litterman by putting the performance and practicality of both Black- Litterman and Markowitz Mean-Variance models into test. We will illustrate detailed mathematical derivations of how the models are constructed and bring clarity and profound understanding of the intuition behind the models. We generate two different portfolios, composing data from 10-Swedish equities over the course of 10-year period and respectively select 30-days Swedish Treasury Bill as a risk-free rate. The resulting portfolios orientate our discussion towards the better comparison of the performance and applicability of these two models and we will theoretically and geometrically illustrate the differences. Finally, based on extracted results of the performance of both models we demonstrate the superiority and practicality of Black-Litterman model, which in our particular case outperform traditional Mean- Variance model.