Strategy Analysis and Portfolio Allocation1439873/FULLTEXT01.pdfpreferences and expertise perform...

Master Thesis, 30 ECTS Master of Science in Industrial Engineering and Management Department of Mathematics and Mathematical Statistics

Spring 2020

Strategy Analysis and Portfolio Allocation

A study using scenario simulation and allocation theories to investigate risk and

return

Emil Bylund Åberg Johannes Fåhraeus

Copyright © 2020 Emil Bylund Åberg and Johannes Fåhraeus All rights reserved

STRATEGY ANALYSIS AND PORTFOLIO ALLOCATION Submitted in fulfilment of the requirements for the degree Master of Science in Industrial Engineering and Management

Department of Mathematics and Mathematical Statistics UmeåUniversity SE- 901 87 Umea ̊, Sweden

Supervisors: Jun Yu, UmeåUniversity Jens Forsberg, Placerum – Kapitalförvaltning Tomas Tiensuu, Placerum – Kapitalförvaltning

Examiner: Armin Eftekhari, UmeåUniversity

Abstract

Portfolio allocation theories have been studied and used ever since the mid20th century. Nevertheless, many investors still rely on personal expertiseand information gathered from the market when building their investmentportfolios. The purpose of this master’s thesis is to examine how personalpreferences 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 firmPlacerum Kapitalförvaltning in Ume̊a will be compared and challenged toinvestigate which strategy gives the best risk adjusted return. Using scenariomodelling, the portfolios can be compared using both historical data andfuture forecasted scenarios to analyze the past, present and future of theallocation theories and Placerum’s investment strategy.

The first allocation theory, the Black-Litterman model, combines historicalinformation from the market with views and preferences of the investor toselect the optimal allocations derived from return and volatility. The secondallocation theory, the modern portfolio theory (Markowitz), only uses histori-cal data to derive correlations and returns which are then used to select theoptimal allocations.

By analysing several risk measures applied on the portfolios historical andforecasted data as well as comparing the performance of the portfolios, itis shown that the investment strategy used at Placerum succeeds with itsintentions to achieve relatively high return while reducing the risk. However,the portfolios given using the two allocation theories results in higher potentialreturns but at the cost of taking on a higher risk. Comparing the two studiedallocation theories, it is shown that when using the Black-Litterman modelwith the assumptions and views defined in this project, modern allocationtheory actually beats it in terms of potential return as well as in terms of riskadjusted return, even though its underlying theory is much simpler.

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Sammanfattning

Portföljallokering har studerats och använts inom finanssektorn sedan mittenav 1900-talet, trots detta förlitar sig många investerare fortfarande p̊a sitteget omdöme och kunskaper i samspel med vad de kan avläsa fr̊an mark-naden. Syftet med denna examensuppsats är att undersöka hur väl personligapreferenser och expertis presterar jämfört med matematiska allokeringste-orier och hur skiljer sig risken p̊a de portföljer som ges av de olika strategierna?

Genom att applicera och använda tv̊a allokeringsmodeller, Black-Littermanmodellen samt modern portföljteori (Markowitz), kan portföljen Dynamisk,som förvaltas av Placerum Kapitalförvaltning i Ume̊a, jämföras och analyserasi relation till de portföljer givna fr̊an allokeringsmodellerna för att undersökavilken som ger bäst riskjusterad avkastning. Med hjälp av scenariomodeller-ing kan undersökningar p̊a b̊ade historisk data och prognostiserade scenariergenomföras för att analysera den riskjusterade avkastningen p̊a det förflutna,nutiden och framtiden för b̊ade de portföljer givna av allokeringsmodellernasamt Dynamisk.

Den första använda allokeringsmetoden, Black-Litterman, kombinerar his-torisk information fr̊an marknaden med preferenser hos investeraren för attvälja optimala allokeringar härledda fr̊an avkastning och volatilitet. Denandra allokeringsmetoden, modern portföljteori (Markowitz), använder endasthistorisk data för att erh̊alla korrelationer och avkastningar som används föratt välja de optimala allokeringarna.

Fr̊an flertalet riskm̊att applicerade p̊a b̊ade historisk och prognostiserade dataför de olika portföljerna samt genom att undersöka deras prestation visas detatt investeringsstrategin som används av Placerum lyckas med sina avsikter,dvs. att uppn̊a en relativt hög avkastning samtidigt som de minskar risken.Samtidigt visar resultaten att portföljerna givna av de tv̊a allokeringsmod-ellerna har högre potentiell avkastning än Dynamisk men med kostnadenatt de tar p̊a sig en högre risk. Jämförelser mellan Black-Litterman ochMarkowitz visar även att Markowitzportföljen sl̊ar Black-Litterman portföljen(när Black-Litterman används med de antaganden och marknads̊asikter somdefinerats i denna rapport) i s̊aväl potentiell avkastning som riskjusteradavkastning trots att denna underliggande teori är mycket simplare.

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Acknowledgements

There are a couple of people and organizations we want to acknowledge fortheir support and contributions to this master thesis. Without their willing-ness to provide assistance when needed, this project would simply not bepossible.

First and foremost we would like to highlight our deepest gratitude to TomasTienssu and Jens Forsberg at Placerum Kapitalförvaltning for their consistentsupport throughout the entire project work. Despite the current worryingsituation on the financial market, they took their time in providing us withnecessary feedback and insights in every aspect of the project. Their helphave been crucial in achieving credible and desirable results to this thesis.Furthermore, we want to thank Placerum Kapitalförvaltning as an organi-sation, for entrusting us with this cooperation and for their willingness toprovide us with the necessary data needed to complete the project. Theyhave been extremely accommodating and showed a great enthusiasm in ourstay at their company.

We also want to address our appreciation to our university supervisor, Jun Yu,for his important feedback and guidance in this master thesis. His advice andinputs throughout the development of this project have been highly valuablewhich have improved the overall structure and content of this thesis. Moreover,we are highly grateful for the competitive skills and education we have beenprovided from Ume̊a University and the Department of Mathematics andMathematical Statistics. Because of them, we had all of the necessary tools inthe statistical approach to finance and risk management to create this masterthesis.

Lastly, we want to thank our classmates who we have shared this five-yearjourney with, which now has come to an end with the completion of thisthesis. They have been a great source of inspiration and indubitable supportduring this education. We wish you all the best in life and good luck in yourfuture careers.

Johannes F̊ahraeusEmil Bylund ÅbergUme̊a, 27 May, 2020

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Nomenclature

Developed Markets (DM) - Countries that have a stable and large eco-nomic growth with developed capital markets subject to a high level ofregulation and oversight.

Emerging Markets (EM) - Countries that show some of the traits of thedeveloped markets but with a lower level of regulation, market e�ciency andoversight.

MSCI - An American finance company providing a wide range of marketindices from around the world.

Benchmark - Often a market index that an investor choose to compare theirportfolios performance with.

Portfolio - A collection of financial assets built by an investor or investmentfirms.

Return - The value increase or decrease of an investment calculated as thecurrent value divided by the previous value.

Risk - Potential loss of an investment or portfolio, often defined as the stan-dard deviation of returns.

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Contents

Abstract i

Sammanfattning ii

Acknowledgements iii

Nomenclature iv

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . 21.3 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Benchmark Selection . . . . . . . . . . . . . . . . . . . 31.3.2 Portfolio Optimization . . . . . . . . . . . . . . . . . . 41.3.3 Scenario Modeling . . . . . . . . . . . . . . . . . . . . 41.3.4 Stress Tests . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Theory 72.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Expected Shortfall . . . . . . . . . . . . . . . . . . . . 82.1.3 Tracking Error . . . . . . . . . . . . . . . . . . . . . . 82.1.4 Information Ratio . . . . . . . . . . . . . . . . . . . . . 92.1.5 Sharpe Ratio . . . . . . . . . . . . . . . . . . . . . . . 102.1.6 Maximum Drawdown . . . . . . . . . . . . . . . . . . . 10

2.2 Portfolio Theories . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Black-Litterman Model . . . . . . . . . . . . . . . . . . 112.2.2 Modern Portfolio Theory . . . . . . . . . . . . . . . . . 13

2.3 Portfolio Calculations . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 E�cient Frontier . . . . . . . . . . . . . . . . . . . . . 142.3.2 Covariance Estimation . . . . . . . . . . . . . . . . . . 152.3.3 Linear Model . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Constrained Least Squares . . . . . . . . . . . . . . . . 16

2.4 Scenario Modeling . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Vector-Autoregressive Model . . . . . . . . . . . . . . . 162.4.2 Historical Simulation . . . . . . . . . . . . . . . . . . . 18

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3 Method 193.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Benchmark Analysis . . . . . . . . . . . . . . . . . . . . . . . 203.3 Historical Allocation . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.1 The Black-Litterman Model . . . . . . . . . . . . . . . 223.3.2 Modern Portfolio Theory . . . . . . . . . . . . . . . . . 28

3.4 Scenario Modeling & Analysis . . . . . . . . . . . . . . . . . . 283.5 Stress Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Results 344.1 Benchmark Analysis . . . . . . . . . . . . . . . . . . . . . . . 344.2 Historical Allocation . . . . . . . . . . . . . . . . . . . . . . . 384.3 Scenario Modeling & Analysis . . . . . . . . . . . . . . . . . . 404.4 Stress Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Discussion & Conclusions 495.1 Benchmark Analysis . . . . . . . . . . . . . . . . . . . . . . . 495.2 Historical Allocation . . . . . . . . . . . . . . . . . . . . . . . 515.3 Scenario Modeling & Analysis . . . . . . . . . . . . . . . . . . 525.4 Stress Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5 General Discussions & Conclusions . . . . . . . . . . . . . . . 56

6 Final Thoughts 58

References 58

Appendices 61

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1. Introduction

This section will start o↵ with an introduction to the outsourcing company,Placerum Kapitalförvaltning, where their business is described in short as wellas some of the issues they have been facing which lead to the initiation of thisproject. This is followed by a problem description where the specific goals ofthe project are defined followed by the overall purpose of this master’s thesis.The section will end with some delimitations that have been necessary totake and a general outline for the report.

1.1 Background

Placerum Kapitalförvaltning AB is a Swedish investment and insurance firmbased in Ume̊a with an additional o�ce in Örnsköldsvik. They were foundedin 2002 as a franchisee for Länsförsäkringar, but has been an independentcompany as of 2005. Today they have about 20 employees and manage morethan SEK 3 billion for over 3000 private and business customers.[1] The man-agement team at Placerum actively analyses and manages their customers’portfolios depending on di↵erent risk profiles. They manage three portfolios intotal with various amounts of risk which mostly consist of funds from aroundthe world, all three portfolios were started in 2013. The first portfolio, calledFörsiktig, has the lowest risk profile with 0 - 50 percent invested in stockfunds. The second portfolio, called Balanserad, is the middle ground of risklevel with 25 - 75 percent invested in stock funds. The third portfolio, whichhas the highest amount of risk with 50 - 100 percent invested in stock funds,is called Dynamisk. All three portfolios have the remaining capital investedin interest funds. Since Placerum originate their lower risk portfolios fromDynamisk, i.e. that they construct Försiktig and Balanserad with the sameunderlying assets from Dynamisk, while replacing some of the stock funds withless risky interest funds, the project have revolved around analysing Dynamisk.

Placerum’s business is pursuing three main principles: simplicity, commitment,and long-term view. They strive to minimize the risks on their portfolioswithout underperforming relative to the market returns. It is important forPlacerum to build long-lasting relationships with their customers, since acustomer, who commits to a long-term investment plan, is a risk reduction initself. The management team has the overall goal that their portfolios shouldmatch the market returns while obtaining a consistently lower risk. Placerummanages their portfolios using news and analytics provided from outside thecompany in addition with the managers’ own knowledge and predictions.

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Therefore, the managers at Placerum make decisions based on their ownpreferences which might not be supported by investment theories. This wasthe first area identified that could be improved with portfolio theories.

As asset managers, Placerum’s employees are compelled to use a benchmarkindex to compare how their portfolios have performed in relation to the chosenbenchmark. Today, the world index MSCI World converted into SEK is usedas their benchmark. However, since Placerum often invests in a much largerpercentage in Swedish funds/firms than the MSCI World index consists of,it might not be the most suitable benchmark to use. Another problem withthis benchmark is that it is heavily a↵ected by the changes in the USD/SEKcurrency exchange rate since the index is listed in USD while Placerumconverts it into SEK. This implies that the returns in the benchmark canbe deceptive when comparing it to the performance of Placerum’s portfolios.Since the MSCI World index has been positively a↵ected by the USD’sincreased value compared to SEK the last few years, it has become anunfair benchmark comparison.[2] Because of these problems, Placerum findsit di�cult to motivate why they sometimes have a lower return than thebenchmark to their customers. Moreover, since Placerum tries to achieve asignificantly lower risk than the market average, it is not unlikely that theysometimes underperform compared to their benchmark.

1.2 Problem Description

One of Placerum’s current portfolios will be examined in this project. Thechosen portfolio is called Dynamisk and consists of 50-100 percent stock fundswith an average of approximately 90 percent stocks. The remaining part ofthe portfolio consists of short-term rate funds. Since Placerum constructstheir other two portfolios by originating from Dynamisk, it is most reasonableto examine this portfolio. The problem descriptions of this thesis have beendefined as the following three questions.

• Is the current benchmark, MSCI World in SEK, most reasonable to use orshould Placerum change to another benchmark index?

• How would Placerum invest if they implement portfolio allocation the-ories to their investment strategies?

• How does Placerum’s portfolio ”Dynamisk” perform and how risky isit in relation to the portfolios given by portfolio optimization theories?

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1.3 Purpose

The purpose of this master’s thesis is to solve some of the issues that Placerumis facing today revolving their portfolio called Dynamisk. The project hasbeen divided into four main sections. First of all, the current benchmark thatPlacerum is using will be analyzed and evaluated while other alternatives thatmight better replicate their portfolios will be found. Secondly, their portfolioallocations will be compared with portfolios derived from allocation methodssuch as the Black-Litterman model and the modern portfolio theory. Thirdly,using scenario modeling methods like the Vector-autoregressive model, histor-ical data on the portfolio returns as well as future scenarios, will be modelledand analyzed with the purpose of evaluating the constructed portfolios andPlacerum’s current portfolio. Lastly, various stress tests from past eventsmarket downfalls, such as the financial crisis of 2008, will be reconstructed toinvestigate how Placerum’s current portfolio and the portfolios found usingthe portfolio allocation methods would behave and evolve.

1.3.1 Benchmark Selection

As previously stated, the current benchmark that Placerum is comparing theirportfolios with is the MSCI World index. However, Placerum has noticed afew issues with this benchmark. The main problem with the MSCI Worldindex is that it is presented in USD while Placerum’s portfolios are presentedin SEK. This means that Placerum has to convert the index into SEK whenthey want to compare their performance with the index. This might seemlike an insignificant technicality at first glance, but the currency conversionhas a major e↵ect on the index growth performance. Because of the increasedvalue of the USD compared to SEK, it is almost impossible for Placerumto beat their benchmark in terms of returns, which has been di�cult toexplain to their customers. Therefore, the first task of this project is to finda more suitable benchmark that better represents their portfolio for a fairercomparison of returns.

The goal here is to create a new benchmark that better replicates Dynamisk.To do this, other benchmarks will be created, tested and analysed by usingdi↵erent underlying components and various proportions of those compo-nents. By calculating the tracking error on the previous benchmark and theconstructed benchmarks on historical data and future simulated scenarios, acomparison can be made of how well they are replicating Placerum’s portfolio.

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In addition to the tracking error, one can also calculate how often the port-folio is over- and underperforming compared to the benchmarks. The mostpreferable result would be to find a benchmark that has a low tracking errorwhile also underperforming compared to Dynamisk on average.

1.3.2 Portfolio Optimization

Since Placerum manages their portfolios using news and analytics providedfrom outside the company in addition with the managers’ own knowledgeand predictions, their investment strategies might not be supported by con-ventional portfolio allocation theories. As an example, since Placerum is aSwedish investment firm, they usually tend to build their portfolios with alarger weight in Swedish and North American markets than other parts ofthe world. For this reason, a number of allocation methods will be used toshow Placerum how they could optimize their portfolio returns according tomathematical theories. After thorough discussions with Placerum’s invest-ment team, they have requested a method that allows them to implementtheir own preferences and views of the market in the model.

To optimize the portfolio allocations while still meeting Placerum’s specificrequests, the Black-Litterman model will be the main choice of methodbecause of its ability to combine the markets’ views and the investors’ viewsto receive optimal allocation weights. The goal is to use the Modern Portfoliotheory in addition to the Black-Litterman model to produce a more diverseand credible result. It will also act as a benchmark to the Black-Littermanmodel to investigate if the investors’ views increase or decrease the portfolioreturns. The results from these models will be compared with Placerum’scurrent portfolio on historical and forecasted data so a conclusion can bedrawn regarding their performance.

1.3.3 Scenario Modeling

Scenario modeling is the process of simulating potential outcomes of a portfo-lio for a given time period. Both likely events and unlikely worst case scenarioscan be simulated depending on what the analyst wants to know. Using statis-tical and mathematical principles such as the Vector-Autoregressive model,scenario modeling can estimate shifts in the portfolio value in potential futureevents. Important to note is that the accuracy of the models is determinedby the input data and the assumptions that the analyst make.

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Since a major part of this project is about the study of how Placerum’sportfolios and the portfolios derived from allocation theories would behave infuture and past events, generating computer simulated scenarios will be a keypart of the project work. The goal is to predict future behavior of returnsbased on the markets’ past behaviour by assuming that future will resemblethe past.

1.3.4 Stress Tests

One of Placerum’s specific request was to simulate a financial crisis as well asreplicating past financial downfalls with the purpose of evaluating how theirportfolio would perform in such scenarios. To reconstruct a past financialcrisis, like the financial crisis of 2007-2008, one could simply recreate the samereturns from that time by collecting historical data and implementing thosereturns on the underlying assets of the portfolio. This method will show whatwould happen to Placerum’s portfolio Dynamisk if the same, or similar event,would occur again. To create a new hypothetical financial crisis, one can usedi↵erent scenario modelling methods to create future downfalls of the market,which are in a reasonable possibility of occurring.

When the past and potential future financial downfalls have been constructed,the scenarios will be implemented on Placerum’s portfolio Dynamisk, theportfolios constructed from the portfolio allocation theories as well as themarket. The purpose of doing this is to evaluate how well the portfolioscan resist such events compared to the market downfall by analysing thedevelopment of returns. By choosing a desired risk exposure, conclusions ofwhich portfolio is preferable in a financial crisis can then be made by theportfolio managers.

1.4 Delimitations

A delimitation that arised at the early stages of this project was that theamount of data that will be implemented in our methods and calculations hadto be limited. A decision was made to only use weekly historical data on theunderlying indices in the historical allocation methods since daily data is notpublicly available for all indices further back in time. Moreover, the amountof data that would have to be handled if daily data were to be used in allaspects of the project, would be too time consuming and wouldn’t contributeenough advantages or insights to the project to motivate the di�culties ofusing daily data.

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Therefore, weekly data was deemed to be su�cient to achieve credible resultssince we analyse the behaviour of our allocation methods over a long period oftime. Another delimitation worth mentioning is that the underlying assets ofall portfolios in this project is constructed with indices. This means that theyhave been built with the intention of replicating the behaviour of portfoliosthat have been constructed with stocks as closely as possible. However, theevolution of Placerum’s portfolio Dynamisk, will not be an exact copy of theactual portfolio since it has been constructed with correlating indices in thismaster thesis.

1.5 Outline

The overall structure of this master’s thesis will follow a chronological orderof the project work with the purpose of making the report easily understoodregardless of what kind of background the reader has. The goal is to make thisthesis a rewarding read-through regardless if you are a professional, studentor finance enthusiast.

Section 2 presents all of the relevant theories that have been used in theproject work. All of the mathematical and statistical methods will be de-scribed in a chronological order so that the reader has all of the necessaryknowledge before continuing with the report.

Section 3 goes further in to detail of how the theoretical methods have beenused in practice. The purpose of this section is to inform the reader of how thescenario modelling and the portfolio allocation methods have been conducted.This will make the results more credible and add to the overall transparencyof the thesis.

Section 4 contains the results from all of the methods and modelling discussedin the previous section. The outcome of the benchmark analysis, portfoliooptimizations, scenario modelling and stress tests will be presented in itsbasic numerical form without any conclusions drawn from the results.

Section 5 presents our discussions and conclusions from the obtained resultsand what these results imply if the methods and models were to be im-plemented in a practical use. It will also involve our recommendations forPlacerum on how they should interpret the outcome of the project work. Thefinal section contains our final thoughts regarding this master’s thesis andsome personal notes that we wanted to share with the reader.

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2. Theory

The theoretical and mathematical foundations that will be used in this reportwill be presented in this section with the intent to give the reader a deeperunderstanding of the methods and results that are going to be discussedfurther into the report. The theory section contains mathematical formulasand background information in areas such as financial risk measures, portfolioallocation theories, scenario modeling and other calculations that are relevantto this master’s thesis.

2.1 Risk Measures

2.1.1 Value-at-Risk

Value-at-risk (VaR) is one of the most common risk measure used in financialinstitutions and is quite simple to understand and straightforwardly calculated.It is a statistical measure that quantifies the financial risk for a portfolio overa given time frame. For investors, it is an indication on how much value anasset or portfolio can lose in a worst-case scenario as well as the occurrenceratio of that loss. A drawback of Value-at-risk is that it does not give anyinformation about the severity of losses which might occur below the givenconfidence level. The VaR of a portfolio at the confidence level ↵ is given bythe smallest number l such that the probability that the loss L exceeds l isno larger than (1� ↵). Formally, [3, pp. 37-39]

V aR↵ = inf{l 2 R : P (L > l) 1� ↵} = inf {l 2 R : FL(l) � ↵}

where FL(l) = P (L l) is the distribution function of the loss L. Typicalvales for the the confidence level are ↵ = 0.95 or ↵ = 0.99.

Using scenario modelling of financial portfolios, VaR can be derived fromselecting the ↵-percentile of losses from that portfolio. Assuming that wehave N scenarios with losses of a portfolio over the time horizon T, the VaRat the confidence level ↵ is then the value of the ↵-percentile of the worstlosses from these N scenario values.

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2.1.2 Expected Shortfall

Expected shortfall (ES) is another common risk measurement used in financialinstitutions and attempts to address the shortcomings of the VaR measure. Itis derived from calculating a weighted average of the worst-case scenarios usedin the value-at-risk calculations. Simpler put, it tells an investor how largethe expected loss of a portfolio or an asset is, given that we find ourselves inthe given quantile, which is usually 95 or 99 percent. This risk measure ismore practical to use for volatile investments than VaR and usually leads toa more conservative approach when choosing risk exposure. The value of theexpected shortfall is calculated by the following formula, [3, pp. 43-47]

ES =1

1� ↵ ⇤Z 1

↵

qu(FL)du

where ↵ is the cut-o↵ point on the distribution where the analyst sets the VaRbreakpoint and qu(FL) is the quantile function of the distribution functionFL.

Expected shortfall is thus related to VaR by,

ES =1

1� ↵ ⇤Z 1

↵

V aRu(L)du (2.1)

As one can see, expected shortfall calculates the average of all losses largerthan the decided confidence level and is therefore looking further down thetail of the loss distribution than VaR. It is intuitive that |ES| � |V aR| for agiven confidence level since ES depends on the same loss distribution as VaRwhile averaging the losses behind the given confidence level. To implementexpected shortfall in our simulated future returns of the portfolios, one simplytakes the worst 5 or 1 percent of the simulated returns, calculates the averagevalue of those returns and multiply the result with the value of the portfolio.

2.1.3 Tracking Error

When asset managers want to analyze the performance of their portfolios, theyoften use a benchmark that has a similar behaviour as the portfolio. Trackingerror is a commonly used metric to understand how well the portfolio isperforming compared to the benchmark by calculating the di↵erences betweenthe returns of the portfolio and the returns of the benchmark. More precisely,it can be described as the mean squared error of the return di↵erences betweena portfolio and a benchmark. Tracking error is calculated by the followingformula,[4 pp. 78-79]

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TrackingError =

sPNn=1(Pn � Bn)2

N � 1 (2.2)

where Pn is the return of the portfolio in period n, Bn is the return of thebenchmark in period n and N is the number of return periods used. Ahigh tracking error indicates that the benchmark returns are far from theportfolio returns, i.e. that the benchmark and the portfolio is not behavingin a similar way. However, tracking error does not tell us anything regardingif the portfolio is overperforming or underperforming in terms of returnscompared to the benchmark.[5]

2.1.4 Information Ratio

The information ratio (IR) is often used to measure an asset managers abilityto generate excess returns compared to the returns of a benchmark. It mea-sures how much the portfolio is overperforming relative to the benchmarkwhile incorporating tracking error to analyze the consistency of that perfor-mance. Information ratio is calculated by subtracting the portfolio returnsover a given time period with the benchmark returns which is divided by thetracking error.

IR =Rp �Rb

TrackingError(2.3)

Where Rp is the return of the portfolio over a period, Rb is the return ofthe benchmark over the same period and TrackingError the tracking error(Section 2.1.3) between the two during the period.

If we have a high positive information ratio the value comes either from a lowtracking error or big di↵erence between the portfolio and benchmark return.A low tracking error while having positive numerator means that the managerhave achieved a better return than the benchmark, while keeping the portfolioclose to the benchmark which shows an ability to generate excess returns.A high value of the numerator would also indicate that the manger has anability to generate excess returns, but would also result in a higher trackingerror in most cases.

Negative values of information ratio means that the benchmark had higherreturns over the period than the portfolio and indicates that the managerhave have failed to achieve excess returns in relation to the benchmark.[6 pp.433]

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2.1.5 Sharpe Ratio

The Sharpe ratio is a measurement tool that helps investors to evaluate thereturn of a portfolio compared to the risk of the portfolio. Developed byWilliam F. Sharpe in 1966, it has become one of the most commonly usedmethods to calculate the risk-adjusted return of an investment. It is derivedfrom subtracting the risk-free rate from the mean return of a portfolio oran asset for a given time period and dividing the result with the standarddeviation of the portfolio,

SharpeRatio =Rp �Rf

�p(2.4)

where Rp is the return of the portfolio, Rf is the risk free rate and �p isthe standard deviation of the portfolio. The risk free rate is assumed tobe 0 in this master’s thesis. The Sharpe ratio gives information regardingif the portfolio returns is due to smart investment strategies or due to ahigh-risk taking. A high Sharpe ratio on a portfolio implies a more attractiveinvestment for an investor since this means that the ratio between return andrisk is high which suggests that one have achieved a relatively high returnfrom the risk taken.[6 pp. 433]

2.1.6 Maximum Drawdown

Maximum Drawdown (MDD) measures the maximum observed loss for aninvestment over a given period. By investigating the maximum drawdown aninvestor can get information of the risk and the volatility of the investment,even if positive results have been achieved at the end of the time period.Maximum drawdown is derived from the following equation,

MDD = min1

2.2 Portfolio Theories

2.2.1 Black-Litterman Model

The Black-Litterman model is an analytical portfolio allocation method thatoptimises the asset allocation of a portfolio. The model combines historicaldata of asset returns with the views of the asset manager to create an optimalportfolio that lies within the investors’ risk tolerance. The model is thereforeenabling portfolio managers to combine their unique views of the market withthe market equilibrium which leads to diversified and intuitive portfolios. Themodel was introduced by Fisher Black and Robert Litterman in 1990 withthe purpose of solving some of the problems with earlier portfolio allocationmethods like the Modern portfolio theory, which often produce unintuitiveand highly-concentrated portfolios which becomes problematic for practicaluse.[8 pp. 1-2]

First of all, the combined excepted returns need to be calculated which canbe derived from the following formula, [9]

µBL =⇥(⌧⌃)�1 + V T⌦�1V

⇤�1 ⇥(⌧⌃)�1⇧+ V T⌦�1Q

⇤(2.6)

and the co-variance matrix of the returns becomes: [9]

⌃BL = (1 + ⌧)⌃� ⌧ 2⌃V T�⌧V ⌃V T + ⌦

��1V ⌃ (2.7)

where⌃ is the co-variance matrix of historical returns,⌧ is the uncertainty scalar,⇧ is the Implied Equilibrium Return Vector,V is a matrix that identifies the assets involved in the views,⌦ is a diagonal co-variance matrix of error terms from the expressed viewsrepresenting the uncertainty in each view,Q is the View Vector.

Implied Equilibrium Return

The Implied Equilibrium Return is given by the following formula,

⇧ = �⌃wmkt (2.8)

where wmkt is the market weights of the assets (given using Constrained LeastSquares, Section 2.3.4) and � is a risk aversion parameter, representing therisk tolerance which is given by:

11

� =SharpeRatio

�mkt, �mkt =

qwTmkt⌃wmkt

The uncertainty scalar

The uncertainty scalar (⌧) is intended to regulate the uncertainty of eachview and the estimated equilibrium returns. There exists a number of waysfor estimating this parameter but in this report it will be estimated as: [9]

⌧ =1

Nwhere N is the number of time steps used to estimate the co-variance matrixfrom historical returns.

Views

The V -matrix in Equation (2.6 and 2.7) defines which assets that is involvedin each view, Q represent the excess returns of the views and ⌦ is the vari-ance/error of the views.

Each view can either be relative or absolute, if the view is relative the re-spective row in the V -matrix sums to 0 and if the view is absolute it sums to 1.

Each view, Vi, in the V -matrix can be expressed as:

Vi = Qi + ⌦i

Therefore, the variance/error of view i can be calculated as:

⌦i = ⌧vi⌃vTi

where vi is the i:th row in the V matrix, representing view i.

12

2.2.2 Modern Portfolio Theory

The Modern portfolio theory (MPT) is a method for portfolio managersto maximize the expected returns based on a given a level of risk. It wasintroduced in 1952 when Harry Markowitz published this pioneering portfoliotheory in his paper ”Portfolio Selection” in The Journal of Finance, which hewas later awarded a Nobel prize for.[10] Markowitz argued that a portfolio’srisk and return should not be observed as two separate characteristics but astwo closely related portfolio measurements. MPT can be used to construct aportfolio of assets with an allocation that either gives a maximized returngiven a level of risk or a minimized risk given a level of expected return. Thetheory states that the higher risk that the investor is willing to take, thehigher return the investor can expect to achieve.[11]

In a scenario where there are two portfolios available o↵ering the sameexpected return, MPT assumes that an investor is risk averse, i.e. that theinvestor will choose the less risky portfolio. This means that an investor willonly accept a riskier portfolio, where risk is expressed as the volatility of theportfolio, if this is compensated with a higher expected return. The trade-o↵between risk and return will di↵er for each investor, but the implication isthat a rational investor will always choose a portfolio that have the mostfavourable risk adjusted return. To make use of the MPT, one has to calculatethe expected return and the volatility of the portfolio. Expected return iscalculated by the following formula,[12 pp. 62]

E[Rp] =nX

i

wiE(Ri) (2.9)

where Rp is the portfolio return, wi is the weighting of asset i, Ri is theestimated return of asset i and n is the number of assets in the portfolio. Thevariance of the portfolio can then be expressed with the following formula,

V ar[Rp] = wt⌃w (2.10)

where ⌃ is the estimated covariance matrix of estimated returns.

By combining the concepts above with the e�cient frontier (presented insection 2.3.1 below), it is possible to find the optimal portfolio allocationsamong a set of underlying assets so that the given portfolio have the highestexpected return for a defined level of risk, or vice versa.

13

To further develop the MPT it is possible to use it together with the conceptof robust optimization, which tries to take uncertainties of estimated values(return and covariance matrix) into account. Since financial portfolios exist ina much larger universe than the chosen set of assets studied, there will alwaysexist some uncertainties within the estimations. However, the concept ofrobust optimization will not be used in this master’s thesis but is an importanttopic to study further to evolve the results given in this report.

2.3 Portfolio Calculations

2.3.1 E�cient Frontier

The e�cient frontier is the set of optimal portfolios that o↵er the highestexpected return for a defined level of risk, i.e the portfolio that solves thefollowing minimization problem for a given target return µ0.

minw

1

2wT⌃w

s.t. wTµ = µ0

wT~1 = 1

w � 0where µ is the estimated return of the underlying assets and w is the vectorof asset weights. The second constraint makes sure that the total investmentweights is equal to 1, i.e that all capital is invested. The last constraintensures that the weights of each asset is equal or larger than zero, i.e thatthere is no shorting allowed.

The e�cient frontier starts with the portfolio that minimizes the risk (volatil-ity), i.e the portfolio that solves the above problem without the (wTµ = µ0)constraint. For portfolios with no shorting allowed the e�cient frontier endswith investing 100% in the underlying asset with the highest estimated return.

14

2.3.2 Covariance Estimation

To estimate the covariance matrix ⌃ between several financial assets, a movingaverage procedure is used. The purpose of using moving average is to smoothout short term fluctuations while highlighting long term trends or cycles. Thefollowing formula estimates the covariance matrix ⌃ of several assets usinghistorical data. [3. pp. 64-65]

⌃ =1

T

TX

i=1

RtRTt (2.11)

where Rt is a vector of demeaned returns of the assets in time t and T is thelength of the time horizon.

2.3.3 Linear Model

Let X1, X2, . . . , Xp�1 be (p� 1) explanatory variables (predictors) and Y bea response variable from an observed sample of size n, i.e.

(y1, x11, x12, . . . , x1(p�1))

(y2, x21, x22, . . . , x2)(p�1))

...

(yn, xn1, xn2, . . . , xn(p�1))

where Y = {y1, y2, . . . , yn} andX1 = {x11, x21, ..., xn1}, ..., Xp�1 = {x1(p�1), x2(p�1), . . . , xn(p�1)}.The response variable Y may be modelled in terms of the predictorsX1, X2, . . . , Xp�1.The general form of the model then becomes:

Y = f(X1, X2, . . . , Xp�1) + "

where f is a smooth, continuous function and " is the error in this repre-sentation. The function f(·) is assumed to have a restricted linear form,i.e.:

Y = �0 + �1X1 + · · ·+ �(p�1)X(p�1) + " (2.12)

where �i, i = 0, 1, . . . , (p� 1) are p unknown parameters.

15

2.3.4 Constrained Least Squares

Constrained least squares is a linear least squares problem with constraintson the solution. Constrained least squares are often used to solve constrainedlinear regression and other problems on the following form.

minw

||wX � Y ||2

s.t. Aw aBw = b

(2.13)

whereX is a input matrix of size m⇥ p,w is the weights for the input matrix X,Y is a vector of size m with response variables,A & B are matrices where each row represent one inequality or equalityconstraint.

(The weights w can also be called parameters � as in Linear Model (Section2.3.3))

2.4 Scenario Modeling

2.4.1 Vector-Autoregressive Model

When analysing multivariate time series, the Vector-Autoregressive model(VAR) is one of the most successful and easy to use model for portfoliomanagers. It has proven to be quite accurate on replicating the behaviourand movement of financial time series for historical and forecasted portfolioreturns. The purpose of the VAR-model is to predict future returns of anasset based on its past performance by assuming that the future will resemblethe past. However, this assumption can lead to inaccurate predictions ifextreme changes in the underlying market would occur such as a technologicaltransformation of an industry.[13]

A general VAR(p)-model is defined by the following formula.

Yt = A0 + A1Yt�1 + ...+ ApYt�p + ✏t (2.14)

whereYt is a vector of returns in time t,A0 is a vector of intercepts,

16

A1,...,Ap are matrices of parameters,✏t is a vector of errors with E[✏t] = 0, without serial correlation and havingcovariance matrix ⌃.

The equation for a VAR(1)-model then becomes:

Yt = A0 + A1Yt�1 + ✏t

To estimate the VAR(1) parameters A0 & A1 we use historical log-returns forall k assets in the investment universe,

yt,1 = a0,1 + a1,1yt�1,1 + · · ·+ a1,kyt�1,k + ✏t,1yt,k = a0,k + ak,1yt�1,1 + · · ·+ ak,kyt�1,k + ✏t,k

where yt,k is the return of asset k in time t, and ak,k is the A1 parameter forasset k used to calculate yt,k.

Since none of the parameters in A0 or A1 occurs in more than one equation,the parameters of each equation can be estimated separately.

We then make use of the least square method to estimate the parameters,and for asset 1 we get:

0

BBB@

1 yT�1,1 yT�1,2 . . . yT�1,K1 yT�2,1 yT�2,2 . . . yT�2,K...

......

. . ....

1 y1,1 y1,2 . . . y1,K

1

CCCA⇤

0

BBB@

a0,1a1,1...

ak,1

1

CCCA=

0

BBB@

yT,1yT�1,1

...y2,1

1

CCCA

= M = a1 = V1

The least-squares solution of a1 is given by the following equation.

a1 = (MTM)�1MTV1

If we let V = (V1, V2, . . . , Vk) where Vi = (yT,i, yT�1,i, . . . , y2,i), then all pa-rameters can be expressed as:

B = (A0|AT1 )

which is given by:

B = (MTM)�1MTV ) (2.15)

17

2.4.2 Historical Simulation

Historical simulation is a scenario modeling method where future scenar-ios of length T are created by randomly selecting historical returns of theasset and using these to create a simulated evolution of the asset in thefuture. The method uses no model parameters except the choice of histori-cal time window length from which the historical returns will be selected from.

Method:

1. Choose a window length of historical returns.

2. Randomly select T returns, with replacement, from the historical windowand use these to create a scenario of length T.

3. Repeat to create several scenarios.

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3. Method

The practical and methodological work used throughout the thesis projectwill be described in this section. The section is divided into five parts:”Data Collection”, ”Benchmark Analysis”, ”Portfolio Allocation”, ”ScenarioModeling & Analysis” and ”Stress Test”, where the first part describes howthe relevant data have been collected and the four later parts focuses on howthe goals of this project have been solved. The purpose of this section is togive the reader a more in-depth view on how the practical work have beencarried out and what kind of challenges have appeared during the projectwork.

3.1 Data Collection

The first practical focus point of the thesis was to collect and preprocess thedata needed to solve the problems stated for this thesis. From the outsourcer,i.e Placerum, a number of Excel files were provided containing price data oftheir portfolios and di↵erent model portfolios with their underlying assetsthroughout the years. This data was then complemented with historicaldata from a number of stock indices and assets from around the world. Thehistorical data are daily prices from 2000-01-01 to 2019-12-31 as long as theyare publicly available.

Listed below is a short description of the most frequently used assets duringthis project, the full list of assets can be found in the appendix.

• Dynamisk - Placerum’s high risk portfolio, consisting of approximately90% stock funds and 10% rate funds.

• MSCI World - Large and Mid cap index representing 23 DevelopedMarkets (USA, Japan, UK, Germany, etc).

• MSCI EM - Large and Mid cap index representing 26 EmergingMarkets (China, Korea, Brazil, etc).

• OMXS 30 - Stock market index for the Swedish stock market consistingof the 30 most traded stocks.

• S&P 500 - Stock market index that measures the stock performanceof 500 large companies listed on stock exchanges in the United States.

• Euro Stoxx 50 - Stock market index containing fifty of the largestand most liquid stocks in the European zone.

19

• OMX T-Bill - Rate index designed to show the value growth trendfor a certain type of interest-bearing Swedish security.

When the data had been collected, it was thoroughly inspected to look formissing, strange or other extreme values by plotting the assets and visuallylook for abnormalities. Each value determined to be incorrect was replacedby the mean value of the day before and the day after the incorrect value.Besides inspecting the data, it was rearranged in such a way that it would beeasily accessible in further methods and simulations.

3.2 Benchmark Analysis

To find a better alternative to Placerum’s current benchmark, which is theMSCI World index converted into SEK, a total of eight di↵erent benchmarkalternatives have been examined. The composition and weighting of eachbenchmark have been decided in cooperation with Placerum so that thebenchmarks will somewhat represent their portfolios underlying assets. Themost reasonable choices of indices were agreed to be OMXS 30, MSCI World,MSCI Emerging Markets and OMX T-Bill since they best represent the areaswhere Placerum usually invest. These indices were then distributed in variousweights where some were converted in to SEK and some were kept in theiroriginal currency. A decision was made to not mix currencies within thesame benchmark since this would not be practically possible to implementfor Placerum.

Listed below, the benchmarks that have been tested and analysed as well asthe benchmark that Placerum is using today can be found.

• Current Benchmark - 90% MSCI World (SEK) + 10% T-Bill (SEK)

• Benchmark 1 - 90% OMXS (SEK) + 10% T-Bill (SEK)

• Benchmark 2 - 90% MSCI World (USD) + 10% T-Bill (USD)

• Benchmark 3 - 90% OMXS (USD) + 10% T-Bill (USD)

• Benchmark 4 - 40% OMXS (SEK) + 30% MSCI World (SEK) + 20%MSCI EM (SEK) + 10% T-BILL (SEK)

• Benchmark 5 - 40% OMXS (SEK) + 40% MSCI World (SEK) + 10%MSCI EM (SEK) + 10% T-BILL (SEK)

20

• Benchmark 6 - 45% OMXS (SEK) + 45% MSCI World (SEK) + 10%T-BILL (SEK)

• Benchmark 7 - 22,5% OMXS (SEK) + 45% MSCI World (SEK) +22,5% MSCI EM (SEK) + 10% T-BILL (SEK)

• Benchmark 8 - 22,5% OMXS (USD) + 45% MSCI World (USD) +22,5% MSCI EM (USD) + 10% T-BILL (USD)

Each benchmark was tested on how well it performs in relation to Placerum’sportfolio called Dynamisk by calculating the tracking error (Equation 2.2)from historical returns between 2015 and 2019 as well as how often Dynamiskover- and underperform compared to each benchmark during that time period.The results from these tests will then show which benchmark that wouldhave represented Placerum’s portfolio most accurate historically in termsof returns. Moreover, the benchmarks were also analysed on how well theyreplicate Dynamisk for simulated future scenarios. To do this, 2000 scenariosfor a five year time period were simulated using the Vector-Autoregressivemodel (Section 2.4.1). 10 years of historical returns for Dynamisk, USD/SEKcurrency changes and each of the benchmarks underlying indices were usedin equation (2.15) to estimate the VAR parameters which were then used tosimulate the future scenarios.

For each simulated scenario, the benchmark returns are determined and thetracking error (Equation 2.2) between each benchmarks and Dynamisk iscalculated. How often Dynamisk is over- and underperforming compared toeach benchmark in every scenario is also calculated in the same way as forthe historical data. When the tracking error has been calculated for eachscenario, the mean value for each benchmark is derived to be able to comparethe benchmark performances against each other.

The results from the benchmark analysis on the historical data and the simu-lated future scenarios are then presented in separate plots. Which benchmarkthat best replicates the portfolios in terms of tracking error and performancewas decided in cooperation with Placerum’s management team.

In order to see patterns and analyse the results, the historical evolution onthe assets is plotted in both SEK and USD. For the scenarios, the evolutionof each asset is plotted for two randomly selected scenarios. For an easiercomparison, the assets starting value is all set to 100 in the beginning of thehistorical data (i.e February 2015).

21

3.3 Historical Allocation

The general method to find the historical returns of each portfolio allocationmodel is conducted by starting in the first week of 2013 and using the past twoyears of the underlying indices historical data, converted to SEK, to find thestarting weights. All indices are converted to SEK since Placerum’s portfolioDynamisk are sold in SEK and to best compare the allocation theories tothis portfolio it was decided that all the underlying assets should also beconverted in to SEK. After four weeks, the portfolio is rebalanced with a newtwo-year historical return window prior to that point in time which becomesthe new portfolio weights for the four coming weeks. Each model starts inthe first week of 2013 and ends in the last week of 2019.

The portfolio allocation methods have 13 underlying assets to choose from.The assets are OMXS 30, OMX T-Bill, OMXS T-Bond, OMXS Small Cap,S&P 500, Russell 2000 (US Small Cap), Euro Stoxx 50, DAX 30, Hang Seng,MSCI EM Asia, Nikkei 225, MSCI EM Latin America, MSCI EM Europeand Middle East. The allocation methods also consider four currencies inrelation to SEK which are USD, EUR, HKD and JPY since these are thecurrencies that the underlying indices are traded in.

3.3.1 The Black-Litterman Model

To make use of the Black-Litterman model, the Black-Litterman combinedexpected returns is calculated by inserting the inverse co-variance matrix ofhistorical returns, an uncertainty scalar ⌧ , the ⇧-vector which identifies theimplied equilibrium market return vector, the view vector and the uncertaintymatrix in equation (2.6). The Black-Litterman co-variance matrix of returnsis then derived by inserting the same variables in equation (2.7).

Given the Black-Litterman return vector and the co-variance matrix, theoptimal allocation is derived using the e�cient frontier (Section 2.3.1) andthe allocation that has the most similar Sharpe ratio to what Dynamisk hadduring the same two year historical window is chosen.

The benchmarks used in the Black-Litterman model when calculating the im-plied equilibrium market return will be the current benchmark that Placerumis using today, (i.e 90% MSCI World in SEK + 10% OMX T-Bill) and Bench-mark 6 (Section 3.2). All of the methods described in the Black-Littermansection will be conducted on both of these benchmarks.

22

Market PortfolioTo calculate the market weights (i.e the portfolio that the market suggests oneshould choose to match the given benchmark) and the implied equilibriummarket returns of the Black-Litterman model, a linear regression with amoving estimation window of two years is used. This means that the marketportfolio weights are found by using actual historical returns of the underlyingindices from two years prior to the given time step in a constrained leastsquares minimization problem (Section 2.3.4), and by doing so finding theoptimal portfolio weights.

To avoid unrealistic weights in the market portfolio, a couple of constraintshave been implemented on the model which gives a maximum and minimuminvestment in each market. Listed below is every constraint on the Black-Litterman market weights where USA represents S&P 500 and Russel 2000,Europe represents Euro Stoxx 50 and DAX 30, Sweden represents OMXS30and OMXS Small Cap, Rates represents OMX T-Bill and OMX T-Bond, Asiarepresents Hang Seng, Nikkei 225 and Emerging Markets represents MSCIEM Asia, MSCI EM Latin America and MSCI EM Europe & Middle East.

• Constraint 1 - Sweden 10-50%

• Constraint 2 - USA 10-60%

• Constraint 3 - Europe 10-40%

• Constraint 4 - Rates

To find the A-matrix and the a-vector, equation (2.13) is used where Y isan array of returns for the benchmark, X is an matrix of returns of theunderlying indices in SEK during the given time period and w is the vectorof allocation weights wmk. Since the constraints defined above are inequalityconstraints, the A-matrix and the a-vector become as seen below, where eachcolumn represents an underlying index and each row represents a constraint.

A =

0

BBBBBBBBBBBB@

�1 0 0 �1 0 0 0 0 0 0 0 0 01 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 �1 �1 0 0 0 0 0 0 00 0 0 0 1 1 0 0 0 0 0 0 00 0 0 0 0 0 �1 �1 0 0 0 0 00 0 0 0 0 0 1 1 0 0 0 0 00 1 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 1 1 0 00 0 0 0 0 0 0 0 0 1 0 1 1

1

CCCCCCCCCCCCA

aT =�0.1 0.5 0.1 0.6 0.1 0.5 0.2 0.3 0.3

�

Besides the inequality constraints above, one equality constraint exists whichis that the total weights is equal to one, i.e that no shorting is allowed andthat all investment capital is invested (i.e no asset have negative weights).

Given the market weights, the implied equilibrium market return vector ⇧is calculated as in Equation (2.8). To do this, the risk aversion parameter �is needed, which is calculated by dividing the one year Sharpe ratio of thebenchmark with the volatility of the market. The Sharpe ratio is calculatedas stated in Section 2.1.5 where Rp is the one year return of the last year,Rf is assumed to be zero and �p is the volatility of the benchmark which isestimated as in Section 2.3.2.

Black-Litterman ViewsTo implement the preferences and investment strategies of Placerum’s invest-ment team, a number of market views have been defined that represents theirspecific opinions about the future market developments. These views willmake predictions about the indices returns in such a way that the portfolioweights will redirect towards Placerum’s investment preferences. The viewswill therefore either accelerate or decelerate the weighting of certain indicesin the Black-Litterman portfolio.

24

Listed below is every view of the market that has been implemented in theBlack-Litterman model. Note that each percentage is stated as a percentageincrease or decrease of the return from the previous year, not in percentagepoints. The annual mean is derived from the average return of the previoustwo years.

• View 1 - Sweden: If the annual mean return of OMXS30 is above 15%,the return of the next year will decrease with 15%; if the annual meanreturn is between -5% and 15% it will increase with 10%; and if theannual mean return is below -5% it will increase with 15%.

• View 2 - USA: If the annual mean return of S&P 500 is above 15%,the return of the next year will decrease with 15%; if the annual meanreturn is between -5% and 15% it will increase with 10%; and if theannual mean return is below -5% it will increase with 15%.

• View 3 - Europe: If the annual mean return of Euro Stoxx 50 is above15%, the return of the next year will decrease with 15%; if the annualmean return is between -5% and 15% it will increase with 10%; and ifthe annual mean return is below -5% it will increase with 15%.

• View 4 - Sweden: If the annual mean return of OMXS30 and OMXSSmall Cap is larger than the yearly mean return of emerging markets,the di↵erence will increase with 10% the following year.

• View 5 - Sweden: If the mean annual return of OMXS30 is larger thanthe mean annual return of S&P 500 and Euro Stoxx 50, the di↵erencewill increase with 5% the following year.

• View 6 - Emerging Markets: If the annual mean return of EM is largerthan the annual mean return of DM, the return of the next year willdecrease with 20%, otherwise it will increase with 5%.

• View 7 - DM Asia: If the mean return of Hang Sen and Nikkei 225 islarger than the annual mean return of DM World (OMXS 30, S&P 500& Euro Stoxx), the return of the next year for Hang Sen and Nikkei225 will decrease with %10.

• View 8 - Sweden: If the annual mean return of OMXS Small Cap isabove 20%, the return of the next year will decrease with 25%; if theannual mean return of OMXS Small Cap is positive and the annualreturn of OMXS30 is negative, the return of OMXS Small Cap the nextyear will decrease with 10% and otherwise it will increase with 20%.

25

• View 9 - USA: If the annual mean return of Russel 2000 is above 20%,the return of the next year will decrease with 25%; if the annual meanreturn of Russel 2000 is positive and the annual return of S&P 500 isnegative, the return of Russel 2000 the next year will decrease with 10%and otherwise it will increase with 20%.

In general, the views implemented into the Black-Litterman model, specifiedin cooperation with Placerum, are optimistic about the Swedish market.Moreover, the views are also optimistic about USA and Europe comparedto the the emerging markets, which means that there exists a scepticism to-wards the emerging markets. These views correspond closely with Placerum’sinvestment strategies, since their portfolios tend to have a much larger weightinvested in developed markets, especially in Sweden and USA. (Section 1.3.2)

Given the views defined above, the V-matrix and the ⌦-matrix is formulatedas below, where each row represent one view and each column represents oneunderlying asset.

V =

0

BBBBBBBBBBBBBBBBBB@

1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 0 0 00.5 0 0 0.5 0 0 0 0 0 �1/3 0 �1/3 �1/31 0 0 0 �0.5 0 �0.5 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 1 0 00 0 0 1 0 0 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0

1

CCCCCCCCCCCCCCCCCCA

⌦ =

0

BBB@

⌧v1⌃vT1 0 . . . 0

0 ⌧v2⌃vT2. . .

......

. . . . . . 00 . . . 0 ⌧v12⌃vT12

1

CCCA

26

Row 4 and 5 in the V-matrix , representing view 4 and 5, is the only tworelative views used which can be seen in the matrix since the row sums to 0and not 1. Row 6, 7 & 8 all represent view 6 since each asset will decrease or in-crease by di↵erent values, the same goes for row 9 & 10 which represent view 7.

In the error matrix ⌦, the value of the parameter ⌧ is defined as ⌧ = 1N(Section 3.3.1), where N are the number of observations used, i.e two years ofweekly data or N = 2 ⇤ 52 = 104.

The view vector Q of the excess return has equally amount of rows as theV-matrix, where each row represent the views excess return, this return is,as previously stated, based on the mean annual return during the two yearhistorical window used at each rebalancing period.

Combined distribution and AllocationThe purpose of the Black-Litterman model is to combine the market portfolio,i.e. how the market suggests one should allocate their portfolio to match agiven benchmark, with the investors own views and preferences of the market.To do this, the calculated market portfolio and implied equilibrium market re-turns are combined with the views to calculate the expected Black-Littermanreturn and co-variance matrix as in Equations (2.6) & (2.7).

Given the Black-Litterman return and co-variance matrix, the e�cient frontieris calculated as described in Section 2.3.1 for no shorting portfolios. Fromthe e�cient frontier, the Sharpe ratio of each portfolio on the frontier iscalculated as in Section 2.1.5. Di↵erent investment strategies were testedsuch as choosing the portfolio with a maximized Sharpe ratio or the portfoliothat is maximizing the return. However, choosing the portfolio with the mostsimilar Sharpe ratio as Dynamisk during the two year historical window wasdecided to be the most fair comparison of the two portfolios since this meansthat they both have the same risk and return trade o↵.

When the Black-Litterman portfolio has been found, the development of theportfolio is calculated using the actual returns combined with the currencychanges in the given time step. As stated above, the portfolio is rebalancedafter four weeks by finding the new Black-Litterman weights, which is doneby repeating the entire process above. This means that after four weeks, themarket portfolio is recalculated with the new estimation window, the viewsare implemented to find the new Black-Litterman return- and covariance-matrices, the e�cient frontier is updated and the Black-Litterman portfoliowith the closest Sharpe ratio to Dynamisk is chosen.

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When the evolution of the two Black-Litterman portfolios have been calculated,the average annual Sharpe ratio and the total seven year Sharpe ratio for bothportfolios as well as for Dynamisk is calculated. The mean portfolio weightsof the two Black-Litterman portfolios are also calculated and presented.

3.3.2 Modern Portfolio Theory

In addition to using the Black-Litterman method, the modern portfolio theory(i.e Markowitz) is also tested as an allocation method for the same time periodas the Black-Litterman portfolios. Just as for the Black-Litterman portfolio(Section 3.3.1), the e�cient frontier for the Markowitz portfolio is calculated,but now the expected return is estimated by the mean weekly return of eachasset during the two year historical window while the covariance matrix isestimated as in Section 2.3.2 on the assets returns during the time window.

Furthermore, the Sharpe ratio for each Markowitz portfolio on the new ef-ficient frontier is calculated (Section 2.1.5) and the portfolio with the mostsimilar Sharpe ratio as Dynamisk during the 2 year time window is chosen.

When the Markowitz portfolio has been found, the evolution of the portfolio isthen calculated using the actual returns of the underlying assets, converted into SEK, in the same way as the Black-Litterman portfolio, where the portfoliois rebalanced every four weeks just like the previous model. As for the Black-Litterman portfolios and Dynamisk, the average annual Sharpe ratio and thetotal, seven year, Sharpe ratio for the Markowitz portfolio is calculated. Themean portfolio weight of the Markowitz portfolio is also derived and presentedtogether with the portfolio weights of the Black-Litterman portfolios.

3.4 Scenario Modeling & Analysis

The general method to find the portfolio weights for each allocation methodin future scenarios is designed in the same way as the historical time period(Section 3.3). The starting point is the first week of 2020 where, once again,two years of prior data is used to find the market weights and to calculatethe views. The portfolio is rebalanced every four weeks with the new, twoyear, time window which becomes the investment strategy for the coming fourweeks. This method is then repeated for the entire period of the generatedscenarios which has been set to five years.

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Scenario Creation2000 scenarios are generated using the Vector-Autoregressive (VAR) model(Section 2.4.1), where historical log-returns are used to estimate the interceptand the A-parameters in Equation (2.14). To decide the size of the lagparameter p, historical autocorrelation with lag one to ten of all underlyingassets as well as Dynamisk were studied. No clear pattern could be seen forthe di↵erent autocorrelations and the lag that gave the highest correlationvaried for each asset. However, more then half of the assets had highestautocorrelations for lag 1. Therefore, it was decided to use a VAR(1)-modelbecause of the complexity and computation time of VAR(p)-models usinghigher lag p and since VAR(1)-models are more convenient for analyticalderivations.

Since a VAR(1)-model is used, the VAR-parameters A0 and A1 are esti-mated using Equation (2.15) in which M and V are a log-return matrixand vector of historical prices in SEK from the past 10 years, respectively.The parameters are then used together with the current day’s log-returnsand normal distributed, randomly chosen, errors to calculate the next day’slog-returns. Since the errors are normal distributed with mean 0 and co-variance ⌃, these are given by extracting random values from this distribution.

When generating the scenarios for Dynamisk, the returns of the portfolioare assumed to follow a similar pattern as the historical returns. The futurereturns of Dynamisk have been derived from the actual historical values ofthe portfolio while the individual return of its underlying assets have notbeen considered. This means that Dynamisk has been viewed as a singleasset when simulating its future returns. The other assets used in the Vector-Autoregressive model are the same ones used in the Black-Litterman modelin addition with the MSCI World Index, which is used for the two benchmarks.

To simulate the evolution of the four currencies, historical simulation (Section2.4.2) is used, where historical returns of the simulated currency are randomlychosen to create the 2000 scenarios with the same time period as the portfoliosunderlying assets. These simulated scenarios of the currency evolution arethen used to convert all underlying assets to SEK.

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Portfolio AllocationFor each scenario, the Black-Litterman allocation model is used as in Section3.3.1. Using 2 years of weekly returns prior to the allocation period, theimplied equilibrium market returns and the Black-Litterman views are usedto calculate the expected Black-Litterman returns and the co-variance matrixas described in Equation (2.6) & (2.7). The Black-Litterman returns and theco-variance matrix is then used to find the Black-Litterman portfolios. Thesame two benchmarks used in the historical allocation are used in this part.Besides the Black-Litterman allocation, modern portfolio theory (i.e Markowitz)has also been implemented on the generated scenarios in the same way as insection 3.3.2.

AnalysisWhen the allocations of the two Black-Litterman portfolios and the Markowitzportfolio have been implemented on the scenarios, a number of risk measuresand performance measures are calculated to analyse the performance of theportfolios against Dynamisk.

Firstly, the one- and five-year value at risk for each investment is calculatedas defined in Section 2.1.1 using both 95% and 99% as the confidence level.Taking the 95% and 99% worst one- and five-year return among the 2000scenarios for each investment (Black-Litterman, Markowitz and Dynamisk)gives an indication of their risk level.

Secondly, the one- and five-year expected shortfall of the four investmentsare also calculated (Section 2.1.2) using confidence level of 95% and 99% bytaking the mean value of the worst 5% and 1% of the investment scenarios.

The one and five year percentile values of the scenarios for each investmentare then extracted, where the percentiles extracted are 5, 10, 25, 50, 75, 90and 95 for both one- and five-year returns.

Moreover, the one- and five-year maximum drawdowns are calculated as pre-sented in Section 2.1.6 for each of the four portfolios. For each scenario and foreach portfolio, the one- and five-year maximum drawdowns are calculated, forwhich the mean value of the 2000 scenarios for each investment is calculatedand presented.

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The tracking error between Dynamisk and both studied benchmarks, the Black-Litterman investments and their respective benchmark and the Markowitzinvestment and both studied benchmarks are calculated as presented in Sec-tion 2.1.3. The tracking error is calculated using both the one-year and thefive-year returns in Equation (2.2), and the mean value of the scenarios foreach investment is then presented.

The information ratio for both the one- and five-year return is then calculatedfor the same combination of portfolios and benchmark as for the tracking error.(Section 2.1.4) As for the tracking error, the information ratio is calculated foreach scenario and investment strategy while the mean value of all scenarios ispresented.

Lastly, the Sharpe ratio for each of the four portfolios (Dynamisk, the twoBlack-Litterman portfolios and Markowitz) is calculated using both the one-and five-year return. (Section 2.1.5) The Sharpe ratio is calculated for each sce-nario and investment strategy and the mean value of all scenarios is presented.

When all risk and performance measures have been calculated, a number ofplots are created to visualize the results.Moreover, for each investment, the5, 50 and 95 percentile return values are plotted in separate plots, and theone-year 95% value at risk is plotted against the one-year mean (50 percentile)return of each investment.

3.5 Stress Test

To compare the risk and stability of the portfolios in extraordinary situations,so-called stress test can be used. In these stress tests, one uses historicalperiods of extreme losses or gains which are then implemented on the studiedportfolios to see how they would behave in such scenarios. Alternatively, onecan create new stress test where the analyst decides how large the marketdownfall should be to see how the portfolio would behave in a potential futurefinancial crisis. A total of six stress tests has been implemented on the threeallocation portfolios (the two Black-Litterman portfolios and the Markowitzportfolio) as well as on Dynamisk. The six stressed scenarios used are:

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1. The financial crisis of 2008, where the scenario starts with the bankruptcyof Lehman Brothers, 15 September 2008, and ends at the last day of2008.

2. The first week after the bankruptcy of Lehman Brothers, 15 - 20September 2008.

3. The month with the largest market increase after the financial crisis2008, 2 March - 2 April 2009.

4. The Chinese stock market turbulence of 2015, August 10th - September4th 2015.

5. The Covid-19 stock market crisis of 2020, February 21st - March 27th2020.

6. The worst week of the Covid-19 stock market crisis of 2020, Week 11(9-13 March).

To create these stressed scenarios, actual historical returns from the relevantperiods were gathered for the indices used in this report. To be able to usethese scenarios on Dynamisk, an example portfolio replicating Dynamiskbuilt from the indices used in the allocation methods had to be created. Indiscussion with the managers at Placerum, it was decided that the exampleportfolio which best represents their investment strategies was to be weightedas follows.

• 28% in OMXS 30

• 12% in OMXS Small Cap

• 10% in OMX T-Bill

• 14% in S&P 500

• 6% in Russell 2000

• 20% in Euro Stoxx

• 10% in MSCI EM

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In order to compare the di↵erent portfolios, a decision was made to use abuy & hold strategy during all stress test. As previously stated, all under-lying indices used were transformed into SEK since Dynamisk is sold andbought in SEK so that the stress test becomes as true to the reality as possible.

First of all, the portfolio weights for the two Black-Litterman portfolios andthe Markowitz portfolio were found by assuming that we stand in the end of2019 and use the previous two years of historical data of the indices in a simi-lar way as described in Section 3.3. For the Black-Litterman portfolios, theimplied equilibrium market returns are calculated and the views are definedin order to get the Black-Litterman returns and co-variance matrix. Usingthe returns and the co-variance matrix, the e�cient frontier is calculated fromwhich the portfolio weights for the portfolio which best match the Sharperatio of Dynamisk are given. For the Markowitz portfolio, the mean weeklyreturn during the previous two years of historical returns for each index isfound which, together with the estimation of the co-variance matrix, is usedto calculate the e�cient frontier. The portfolio weights can then be derivedin the same way as for the Black-Litterman method.

When all portfolio weights have been found, the returns of the stressedscenarios are implemented on the portfolios where an initial investment of100 is used. Since a Buy & Hold strategy is used, no rebalancing of theportfolio weights are made. When the stress tests have been applied on theportfolios one by one, the evolution of the portfolio values is presented inseparate figures as well as a table with the portfolio percentage increase ordecrease for each stress test and each portfolio.

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4. Results

This section will present all of the acquired results from the methods thatwere discussed in the previous section. The results will be demonstrated in achronological order of the methodical work in numerical values and variousdescriptive graphs. First of all, the result from the benchmark analyses will beshown, followed by the portfolio allocations and performance derived from theportfolio theories used. Moreover, the outcome from the scenario modelingon future scenarios will be presented as well as the results from the stresstests and the risk measure calculations.

4.1 Benchmark Analysis

In Figures 4.1, 4.2 and 4.3 the evolution of the underlying asset values for thehistorical data and two, randomly chosen, simulated scenarios can be studied.

Figure 4.1: Historical evolution of assets, using starting values of 100.

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Figure 4.2: Evolution of assets for a random future scenario, where the initialvalue is the end value in Figure 4.1.

Figure 4.3: Evolution of assets for a random future scenario, where the initialvalue is the end value in Figure 4.1.

The results of the benchmark analysis can be seen in Figures 4.4 and 4.5presented below, where the x-axis represents the tracking error of the bench-marks and the y-axis represents the percentage of weeks where Dynamisk hada higher weekly return than the compared benchmark.

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Figure 4.4: Tracking error and performance comparison between Dynamiskand the benchmarks for historical data.

In Figure 4.4, which presents the result using historical data from 2015 to2020, one can see that Placerum’s current benchmark is the one which Dy-namisk is outperforming the least. However, the current benchmark is one ofthe benchmarks which has the lowest tracking error. One can also see thatbenchmarks 4,5 & 6 is grouped together with a relatively low tracking errorand also underperforming compared to Dynamisk in most cases. The samecan not be said of benchmarks 1 and 3 which had the largest tracking error.

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Figure 4.5: Tracking error and performance comparison between Dynamiskand the benchmarks for forecasted scenarios.

In Figure 4.5 the result of the 2000 generated future scenarios are presented.This figure show a similar result as the historical data. Placerum’s currentbenchmark is among the benchmarks with the lowest tracking error and isthe benchmark which Dynamisk outperforms the least. Benchmarks 4,5 & 6are once again grouped together where all three of them are outperformedby Dynamisk more then 50% of the time while also obtaining a low trackingerror. The similarities between the historical data and the generated futurescenarios continue for Benchmarks 1 & 3 which is outperformed by Dynamiskthe most while also obtaining the largest tracking error.

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4.2 Historical Allocation

In Figure 4.6 the evolution of the four portfolios between the beginning of2013 and the end of 2019 is presented. Each portfolio uses a starting value of100 for easier comparisons.

Figure 4.6: Historical evolution of the portfolios, using starting values of 100.

Table 4.1: The portfolio value increase of the historical evolution.

Dynamisk Black-Litterman 1 Black-Litterman 2 MarkowitzPortfolio Increase 115.17% 118.69% 109.46% 145.53%

When studying the graph (Figure 4.6) and Table 4.1, the clear winner at theend of 2019 is the Markowitz portfolio which had an increase of its portfoliovalue with 145.53% during the seven year time period. The other threeportfolios have a similar end result as each other, where the Black-Littermanportfolio that tries to replicate Placerum’s current benchmark ends just abovethe other two portfolios with an increase of 118.69%, followed by Dynamiskwhich grew 115.17% and then the second Black-Litterman portfolio that triesto replicate Benchmark 6 which had a portfolio value increase of 109.46%.

The average portfolio weights for both Black-Litterman portfolios and theMarkowitz portfolio are presented in Table 4.2. The sum of the underlyingasset weights equals to 1 since the entire portfolio is always invested.

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Table 4.2: The average portfolio allocation weights for every index during thehistorical evolution.

Weights Black-Litterman 1 Black-Litterman 2 MarkowitzOMXS 30 0.0680 0.148 7.46e-06T-Bill 0.0835 0.0925 4.37e-05T-Bond 0.0449 0.0462 0.0805

OMXS Small Cap 0.347 0.312 0.254S&P 500 0.141 0.104 0.00029

Russel 2000 0.0695 0.0785 0.0040Euro Stoxx 50 0.0206 0.0171 3.96e-06

DAX 30 0.0074 0.0081 0.362Hang Seng 0.0385 0.0279 3.84e-05

MSCI EM Asia 0.0758 0.0780 0.194Nikkei 225 0.0471 0.0357 0.0020

MSCI EM EU & ME 0.0246 0.0214 0.102MSCI EM LA 0.0321 0.0316 8.78e-05

The average annual Sharpe ratio for the four portfolios is presented in Table4.3 and the total seven-year Sharpe ratio for the entire historical time windowof the portfolios is presented in Table 4.4. Once again, the clear winner amongthe four portfolios is the Markowitz portfolio on both the annual averageand on the total seven year period. The runner up is Dynamisk, followed byBlack-Litterman 1 and lastly Black-Litterman 2.

Table 4.3: The mean annual Sharpe Ratio for each portfolio during thehistorical time window.

Dynamisk Black-Litterman 1 Black-Litterman 2 MarkowitzAnnual SR 1.366 1.245 1.216 1.542

Table 4.4: The total seven year Sharpe Ratio for each portfolio during theentire historical time window.

Dynamisk Black-Litterman 1 Black-Litterman 2 MarkowitzSeven year SR 4.594 4.272 3.998 5.564

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4.3 Scenario Modeling & Analysis

In Figures 4.7-4.9 the 5, 50 and 95 percentile evolution of all portfolios arepresented, respectively.

Figure 4.7: Forecasted evolution of the portfolios for the 5:th percentile ofthe 2000 scenarios.


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The one and five year percentile values for each portfolio is presented in Table4.5 and 4.6 respectively. The values represent the percentage increase ordecrease of the portfolio value. Together with the figures above, one can seethat Dynamisk has the smallest decrease and increase for both the one andfive year period, and that the Markowitz portfolio have the highest potentialreturn (i.e largest values on the high percentiles.)

Table 4.5: The average 1-year percentile values of each portfolio for the future,generated scenarios.

1 Year 5% 10% 25% 50% 75% 90% 95%Dynamisk -4.66% -1.95% 3.47% 9.81% 16.42% 23.34% 27.63%

Black-Litterman 1 -7.78% -3.84% 2.67% 11.12% 20.36% 29.64% 34.73%Black-Litterman 2 -8.23% -4.61% 1.83% 9.82% 19.19% 27.95% 32.59%

Markowitz -8.16% -4.09% 2.71% 10.94% 20.35% 30.20% 35.95%

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Table 4.6: The average 5-year percentile values of each portfolio for the future,generated scenarios.

5 Year 5% 10% 25% 50% 75% 90% 95%Dynamisk 18.70% 27.21% 44.35% 66.22% 88.80% 111.66% 137.60%

Black-Litterman 1 7.97% 20.17% 40.52% 67.85% 98.35% 133.92% 160.05%Black-Litterman 2 6.00% 19.99% 37.81% 65.23% 92.83% 127.92% 154.76%

Markowitz 12.76% 22.47% 42.59% 68.96% 100.35% 139.22% 161.91%

Risk MeasuresThe first risk measure, presented in Table 4.7, is the one and five year valueat risk (VaR) for each portfolio on a 95% and 99% confidence level. Thevalues represent the percentage increase or decrease of the portfolio value.

Table 4.7: The 1-year and 5-year Value at Risk for each portfolio at a 95%and 99% confidence level.

VaR Dynamisk Black-Litterman 1 Black-Litterman 2 Markowitz1-year 95% -4.66% -7.78% -8.23% -8.16%1-year 99% -9.32% -13.27% -13.92% -15.08%5-year 95% 18.70% 7.97% 6.00% 12.76%5-year 99% 4.81% -7.52% -9.66% -15.08%

Secondly, the one- and five-year expected shortfall (ES) for each portfolio onboth the 95% and 99% confidence levels are presented in Table 4.8 below. Thevalues represent the percentage increase or decrease of the portfolio value.

Table 4.8: The 1-year and 5-year Expected Shortfall for each portfolio at a95% and 99% confidence level.

ES Dynamisk Black-Litterman 1 Black-Litterman 2 Markowitz1-year 95% -7.59% -11.41% -11.18% -12.03%1-year 99% -11.90% -15.83% -14.56% -17.88%5-year 95% 10.24% -1.99% -1.41% 0.06%5-year 99% -1.20% -13.96% -13.00% -21.30%

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The VaR and ES values in the two tables above show that Dynamisk hasleast negative values for all time horizons and confidence levels and that theMarkowitz portfolio have the most negative values, especially for the 99%confidence level. Both Black-Litterman portfolios have similar values for bothVaR and ES.

In Table 4.9, the one and five year maximum drawdown for each of the fourportfolios can be found. The values represent the percentage decrease of theportfolio value.

Table 4.9: The 1-year and 5-year Maximum Drawdown for each portfolio.

MD Dynamisk Black-Litterman 1 Black-Litterman 2 Markowitz1-year -6.31% -8.15% -8.23% -8.43%5-year -10.81% -14.13% -14.26% -13.84%

In Figure 4.10, the one- and five-year tracking error of Dynamisk, the firstBlack-Litterman portfolio, the Markowitz portfolio compared with the currentbenchmark (90% MSCI World + 10% T-Bill) are presented. In Figure 4.11,the one and five year tracking error of Dynamisk, the second Black-Littermanportfolio, the Markowitz portfolio compared with Benchmark 6 (45% MSCIWorld + 45% OMXS 30 + 10% T-Bill) are presented.

Table 4.10: The 1-year and 5-year Tracking Error for each portfolio comparedto the Current Benchmark.

TE Current Benchmark Dynamisk Black-Litterman 1 Markowitz1-year 0.0201 0.0209 0.02025-year 0.0204 0.0215 0.0215

Table 4.11: The 1-year and 5-year Tracking Error for each portfolio comparedto Benchmark 6.

TE Benchmark 6 Dynamisk Black-Litterman 2 Markowitz1-year 0.0161 0.0096 0.01135-year 0.0165 0.0107 0.0120

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The one and five year information ratio of Dynamisk, the first Black-Littermanportfolio, the Markowitz portfolio compared with the current benchmark arepresented in Figure 4.12 and the one and five year information ratio betweenDynamisk, the second Black-Litterman portfolio, the Markowitz portfoliocompared with benchmark 6 can be seen in Figure 4.13.

Table 4.12: The 1-year and 5-year Information Ratio for each portfoliocompared to the Current Benchmark.

IR Current Benchmark Dynamisk Black-Litterman 1 Markowitz1-year -0.6671 -0.1527 -0.17475-year -9.8215 -7.7581 -6.9624

Table 4.13: The 1-year and 5-year Information Ratio for each portfoliocompared to Benchmark 6.

IR Benchmark 6 Dynamisk Black-Litterman 2 Markowitz1-year 1.0447 1.9781 2.71385-year 3.4954 5.6956 9.7575

The last risk measure presented is the one and five year Sharpe ratio for eachof the four portfolios which can be seen in Table 4.14. The result show thatDynamisk and the Markowitz portfolios have the highest/best ratio for boththe one- and five-year case and that the second Black-Litterman portfoliohave the lowest/worst.

Table 4.14: The 1-year and 5-year Sharpe Ratio for each portfolio.

SR Dynamisk Black-Litterman 1 Black-Litterman 2 Markowitz1-year 1.1611 1.1803 1.0781 1.17255-year 3.3126 3.2224 3.0746 3.3895

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4.4 Stress Test

The evolution of the four studied portfolios in the first two stress test, thefinancial crisis of 2008 and the week after the bankruptcy of Lehman Brothers,can be seen in Figures 4.10 & 4.11 respectively, where the portfolios havebeen given starting values of 100 for a easier comparison. The evolution ofthe portfolios during the third stress test, i.e. the market increase after thefinancial crisis of 2008 is presented in figure 4.12.

Figure 4.10: Portfolio Evolution - Stress Test 1

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Figure 4.11: Portfol

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