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 Umeå University
SE- 901 87 Umeå, Sweden Supervisors:
Jun Yu, Umeå University
Jens Forsberg, Placerum – Kapitalförvaltning Tomas Tiensuu, Placerum – Kapitalförvaltning Examiner:
Abstract
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.
Sammanfattning
Portf¨oljallokering har studerats och anv¨ants inom finanssektorn sedan mitten av 1900-talet, trots detta f¨orlitar sig m˚anga investerare fortfarande p˚a sitt eget omd¨ome och kunskaper i samspel med vad de kan avl¨asa fr˚an mark- naden. Syftet med denna examensuppsats ¨ar att unders¨oka hur v¨al personliga preferenser och expertis presterar j¨amf¨ort med matematiska allokeringste- orier och hur skiljer sig risken p˚a de portf¨oljer som ges av de olika strategierna?
Genom att applicera och anv¨anda tv˚a allokeringsmodeller, Black-Litterman modellen samt modern portf¨oljteori (Markowitz), kan portf¨oljen Dynamisk, som f¨orvaltas av Placerum Kapitalf¨orvaltning i Ume˚a, j¨amf¨oras och analyseras i relation till de portf¨oljer givna fr˚an allokeringsmodellerna f¨or att unders¨oka vilken som ger b¨ast riskjusterad avkastning. Med hj¨alp av scenariomodeller- ing kan unders¨okningar p˚a b˚ade historisk data och prognostiserade scenarier genomf¨oras f¨or att analysera den riskjusterade avkastningen p˚a det f¨orflutna, nutiden och framtiden f¨or b˚ade de portf¨oljer givna av allokeringsmodellerna samt Dynamisk.
Den f¨orsta anv¨anda allokeringsmetoden, Black-Litterman, kombinerar his- torisk information fr˚an marknaden med preferenser hos investeraren f¨or att v¨alja optimala allokeringar h¨arledda fr˚an avkastning och volatilitet. Den andra allokeringsmetoden, modern portf¨oljteori (Markowitz), anv¨ander endast historisk data f¨or att erh˚alla korrelationer och avkastningar som anv¨ands f¨or att v¨alja de optimala allokeringarna.
Fr˚an flertalet riskm˚att applicerade p˚a b˚ade historisk och prognostiserade data f¨or de olika portf¨oljerna samt genom att unders¨oka deras prestation visas det att investeringsstrategin som anv¨ands av Placerum lyckas med sina avsikter, dvs. att uppn˚a en relativt h¨og avkastning samtidigt som de minskar risken.
Samtidigt visar resultaten att portf¨oljerna givna av de tv˚a allokeringsmod- ellerna har h¨ogre potentiell avkastning ¨an Dynamisk men med kostnaden att de tar p˚a sig en h¨ogre risk. J¨amf¨orelser mellan Black-Litterman och Markowitz visar ¨aven att Markowitzportf¨oljen sl˚ar Black-Litterman portf¨oljen (n¨ar Black-Litterman anv¨ands med de antaganden och marknads˚asikter som definerats i denna rapport) i s˚av¨al potentiell avkastning som riskjusterad avkastning trots att denna underliggande teori ¨ar mycket simplare.
Acknowledgements
There are a couple of people and organizations we want to acknowledge for their support and contributions to this master thesis. Without their willing- ness to provide assistance when needed, this project would simply not be possible.
First and foremost we would like to highlight our deepest gratitude to Tomas Tienssu and Jens Forsberg at Placerum Kapitalf¨orvaltning for their consistent support throughout the entire project work. Despite the current worrying situation on the financial market, they took their time in providing us with necessary feedback and insights in every aspect of the project. Their help have been crucial in achieving credible and desirable results to this thesis.
Furthermore, we want to thank Placerum Kapitalf¨orvaltning as an organi- sation, for entrusting us with this cooperation and for their willingness to provide us with the necessary data needed to complete the project. They have been extremely accommodating and showed a great enthusiasm in our stay 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 and inputs throughout the development of this project have been highly valuable which have improved the overall structure and content of this thesis. Moreover, we are highly grateful for the competitive skills and education we have been provided from Ume˚a University and the Department of Mathematics and Mathematical Statistics. Because of them, we had all of the necessary tools in the statistical approach to finance and risk management to create this master thesis.
Lastly, we want to thank our classmates who we have shared this five-year journey with, which now has come to an end with the completion of this thesis. They have been a great source of inspiration and indubitable support during this education. We wish you all the best in life and good luck in your future careers.
Johannes F˚ahraeus Emil Bylund ˚Aberg Ume˚a, 27 May, 2020
Nomenclature
Developed Markets (DM) - Countries that have a stable and large eco- nomic growth with developed capital markets subject to a high level of regulation and oversight.
Emerging Markets (EM) - Countries that show some of the traits of the developed markets but with a lower level of regulation, market efficiency and oversight.
MSCI - An American finance company providing a wide range of market indices from around the world.
Benchmark - Often a market index that an investor choose to compare their portfolios performance with.
Portfolio - A collection of financial assets built by an investor or investment firms.
Return - The value increase or decrease of an investment calculated as the current value divided by the previous value.
Risk - Potential loss of an investment or portfolio, often defined as the stan- dard deviation of returns.
Contents
Abstract i
Sammanfattning ii
Acknowledgements iii
Nomenclature iv
1 Introduction 1
1.1 Background . . . 1
1.2 Problem Description . . . 2
1.3 Purpose . . . 3
1.3.1 Benchmark Selection . . . 3
1.3.2 Portfolio Optimization . . . 4
1.3.3 Scenario Modeling . . . 4
1.3.4 Stress Tests . . . 5
1.4 Delimitations . . . 5
1.5 Outline . . . 6
2 Theory 7 2.1 Risk Measures . . . 7
2.1.1 Value-at-Risk . . . 7
2.1.2 Expected Shortfall . . . 8
2.1.3 Tracking Error . . . 8
2.1.4 Information Ratio . . . 9
2.1.5 Sharpe Ratio . . . 10
2.1.6 Maximum Drawdown . . . 10
2.2 Portfolio Theories . . . 11
2.2.1 Black-Litterman Model . . . 11
2.2.2 Modern Portfolio Theory . . . 13
2.3 Portfolio Calculations . . . 14
2.3.1 Efficient Frontier . . . 14
2.3.2 Covariance Estimation . . . 15
2.3.3 Linear Model . . . 15
2.3.4 Constrained Least Squares . . . 16
2.4 Scenario Modeling . . . 16
2.4.1 Vector-Autoregressive Model . . . 16
2.4.2 Historical Simulation . . . 18
3 Method 19
3.1 Data Collection . . . 19
3.2 Benchmark Analysis . . . 20
3.3 Historical Allocation . . . 22
3.3.1 The Black-Litterman Model . . . 22
3.3.2 Modern Portfolio Theory . . . 28
3.4 Scenario Modeling & Analysis . . . 28
3.5 Stress Test . . . 31
4 Results 34 4.1 Benchmark Analysis . . . 34
4.2 Historical Allocation . . . 38
4.3 Scenario Modeling & Analysis . . . 40
4.4 Stress Test . . . 45
5 Discussion & Conclusions 49 5.1 Benchmark Analysis . . . 49
5.2 Historical Allocation . . . 51
5.3 Scenario Modeling & Analysis . . . 52
5.4 Stress Test . . . 55
5.5 General Discussions & Conclusions . . . 56
6 Final Thoughts 58
References 58
Appendices 61
1. Introduction
This section will start o↵ with an introduction to the outsourcing company, Placerum Kapitalf¨orvaltning, where their business is described in short as well as some of the issues they have been facing which lead to the initiation of this project. This is followed by a problem description where the specific goals of the project are defined followed by the overall purpose of this master’s thesis.
The section will end with some delimitations that have been necessary to take and a general outline for the report.
1.1 Background
Placerum Kapitalf¨orvaltning AB is a Swedish investment and insurance firm based in Ume˚a with an additional office in ¨Ornsk¨oldsvik. They were founded in 2002 as a franchisee for L¨ansf¨ors¨akringar, but has been an independent company as of 2005. Today they have about 20 employees and manage more than 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 in total with various amounts of risk which mostly consist of funds from around the world, all three portfolios were started in 2013. The first portfolio, called F¨orsiktig, has the lowest risk profile with 0 - 50 percent invested in stock funds. The second portfolio, called Balanserad, is the middle ground of risk level with 25 - 75 percent invested in stock funds. The third portfolio, which has the highest amount of risk with 50 - 100 percent invested in stock funds, is called Dynamisk. All three portfolios have the remaining capital invested in interest funds. Since Placerum originate their lower risk portfolios from Dynamisk, i.e. that they construct F¨orsiktig and Balanserad with the same underlying assets from Dynamisk, while replacing some of the stock funds with less 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 portfolios without underperforming relative to the market returns. It is important for Placerum to build long-lasting relationships with their customers, since a customer, who commits to a long-term investment plan, is a risk reduction in itself. The management team has the overall goal that their portfolios should match the market returns while obtaining a consistently lower risk. Placerum manages their portfolios using news and analytics provided from outside the company in addition with the managers’ own knowledge and predictions.
Therefore, the managers at Placerum make decisions based on their own preferences which might not be supported by investment theories. This was the first area identified that could be improved with portfolio theories.
As asset managers, Placerum’s employees are compelled to use a benchmark index to compare how their portfolios have performed in relation to the chosen benchmark. Today, the world index MSCI World converted into SEK is used as their benchmark. However, since Placerum often invests in a much larger percentage in Swedish funds/firms than the MSCI World index consists of, it might not be the most suitable benchmark to use. Another problem with this benchmark is that it is heavily a↵ected by the changes in the USD/SEK currency exchange rate since the index is listed in USD while Placerum converts it into SEK. This implies that the returns in the benchmark can be deceptive when comparing it to the performance of Placerum’s portfolios.
Since the MSCI World index has been positively a↵ected by the USD’s increased value compared to SEK the last few years, it has become an unfair benchmark comparison.[2] Because of these problems, Placerum finds it difficult to motivate why they sometimes have a lower return than the benchmark to their customers. Moreover, since Placerum tries to achieve a significantly lower risk than the market average, it is not unlikely that they sometimes underperform compared to their benchmark.
1.2 Problem Description
One of Placerum’s current portfolios will be examined in this project. The chosen portfolio is called Dynamisk and consists of 50-100 percent stock funds with an average of approximately 90 percent stocks. The remaining part of the portfolio consists of short-term rate funds. Since Placerum constructs their other two portfolios by originating from Dynamisk, it is most reasonable to examine this portfolio. The problem descriptions of this thesis have been defined as the following three questions.
• Is the current benchmark, MSCI World in SEK, most reasonable to use or should 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 is it in relation to the portfolios given by portfolio optimization theories?
1.3 Purpose
The purpose of this master’s thesis is to solve some of the issues that Placerum is facing today revolving their portfolio called Dynamisk. The project has been divided into four main sections. First of all, the current benchmark that Placerum is using will be analyzed and evaluated while other alternatives that might better replicate their portfolios will be found. Secondly, their portfolio allocations will be compared with portfolios derived from allocation methods such 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 modelled and analyzed with the purpose of evaluating the constructed portfolios and Placerum’s current portfolio. Lastly, various stress tests from past events market downfalls, such as the financial crisis of 2008, will be reconstructed to investigate how Placerum’s current portfolio and the portfolios found using the portfolio allocation methods would behave and evolve.
1.3.1 Benchmark Selection
As previously stated, the current benchmark that Placerum is comparing their portfolios with is the MSCI World index. However, Placerum has noticed a few issues with this benchmark. The main problem with the MSCI World index is that it is presented in USD while Placerum’s portfolios are presented in SEK. This means that Placerum has to convert the index into SEK when they want to compare their performance with the index. This might seem like an insignificant technicality at first glance, but the currency conversion has a major e↵ect on the index growth performance. Because of the increased value of the USD compared to SEK, it is almost impossible for Placerum to beat their benchmark in terms of returns, which has been difficult to explain to their customers. Therefore, the first task of this project is to find a more suitable benchmark that better represents their portfolio for a fairer comparison 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 using di↵erent underlying components and various proportions of those compo- nents. By calculating the tracking error on the previous benchmark and the constructed benchmarks on historical data and future simulated scenarios, a comparison can be made of how well they are replicating Placerum’s portfolio.
In addition to the tracking error, one can also calculate how often the port- folio is over- and underperforming compared to the benchmarks. The most preferable result would be to find a benchmark that has a low tracking error while also underperforming compared to Dynamisk on average.
1.3.2 Portfolio Optimization
Since Placerum manages their portfolios using news and analytics provided from outside the company in addition with the managers’ own knowledge and predictions, their investment strategies might not be supported by con- ventional portfolio allocation theories. As an example, since Placerum is a Swedish investment firm, they usually tend to build their portfolios with a larger weight in Swedish and North American markets than other parts of the world. For this reason, a number of allocation methods will be used to show Placerum how they could optimize their portfolio returns according to mathematical theories. After thorough discussions with Placerum’s invest- ment team, they have requested a method that allows them to implement their own preferences and views of the market in the model.
To optimize the portfolio allocations while still meeting Placerum’s specific requests, the Black-Litterman model will be the main choice of method because of its ability to combine the markets’ views and the investors’ views to receive optimal allocation weights. The goal is to use the Modern Portfolio theory in addition to the Black-Litterman model to produce a more diverse and credible result. It will also act as a benchmark to the Black-Litterman model to investigate if the investors’ views increase or decrease the portfolio returns. The results from these models will be compared with Placerum’s current portfolio on historical and forecasted data so a conclusion can be drawn 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 scenarios can 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 future events. Important to note is that the accuracy of the models is determined by the input data and the assumptions that the analyst make.
Since a major part of this project is about the study of how Placerum’s portfolios and the portfolios derived from allocation theories would behave in future and past events, generating computer simulated scenarios will be a key part of the project work. The goal is to predict future behavior of returns based on the markets’ past behaviour by assuming that future will resemble the past.
1.3.4 Stress Tests
One of Placerum’s specific request was to simulate a financial crisis as well as replicating past financial downfalls with the purpose of evaluating how their portfolio would perform in such scenarios. To reconstruct a past financial crisis, like the financial crisis of 2007-2008, one could simply recreate the same returns from that time by collecting historical data and implementing those returns on the underlying assets of the portfolio. This method will show what would happen to Placerum’s portfolio Dynamisk if the same, or similar event, would occur again. To create a new hypothetical financial crisis, one can use di↵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, the portfolios constructed from the portfolio allocation theories as well as the market. The purpose of doing this is to evaluate how well the portfolios can resist such events compared to the market downfall by analysing the development of returns. By choosing a desired risk exposure, conclusions of which portfolio is preferable in a financial crisis can then be made by the portfolio managers.
1.4 Delimitations
A delimitation that arised at the early stages of this project was that the amount of data that will be implemented in our methods and calculations had to be limited. A decision was made to only use weekly historical data on the underlying indices in the historical allocation methods since daily data is not publicly available for all indices further back in time. Moreover, the amount of data that would have to be handled if daily data were to be used in all aspects of the project, would be too time consuming and wouldn’t contribute enough advantages or insights to the project to motivate the difficulties of using daily data.
Therefore, weekly data was deemed to be sufficient to achieve credible results since we analyse the behaviour of our allocation methods over a long period of time. Another delimitation worth mentioning is that the underlying assets of all portfolios in this project is constructed with indices. This means that they have been built with the intention of replicating the behaviour of portfolios that have been constructed with stocks as closely as possible. However, the evolution of Placerum’s portfolio Dynamisk, will not be an exact copy of the actual portfolio since it has been constructed with correlating indices in this master thesis.
1.5 Outline
The overall structure of this master’s thesis will follow a chronological order of the project work with the purpose of making the report easily understood regardless of what kind of background the reader has. The goal is to make this thesis a rewarding read-through regardless if you are a professional, student or finance enthusiast.
Section 2 presents all of the relevant theories that have been used in the project work. All of the mathematical and statistical methods will be de- scribed in a chronological order so that the reader has all of the necessary knowledge before continuing with the report.
Section 3 goes further in to detail of how the theoretical methods have been used in practice. The purpose of this section is to inform the reader of how the scenario modelling and the portfolio allocation methods have been conducted.
This will make the results more credible and add to the overall transparency of the thesis.
Section 4 contains the results from all of the methods and modelling discussed in the previous section. The outcome of the benchmark analysis, portfolio optimizations, scenario modelling and stress tests will be presented in its basic numerical form without any conclusions drawn from the results.
Section 5 presents our discussions and conclusions from the obtained results and what these results imply if the methods and models were to be im- plemented in a practical use. It will also involve our recommendations for Placerum on how they should interpret the outcome of the project work. The final section contains our final thoughts regarding this master’s thesis and some personal notes that we wanted to share with the reader.
2. Theory
The theoretical and mathematical foundations that will be used in this report will be presented in this section with the intent to give the reader a deeper understanding of the methods and results that are going to be discussed further into the report. The theory section contains mathematical formulas and background information in areas such as financial risk measures, portfolio allocation theories, scenario modeling and other calculations that are relevant to 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 financial institutions and is quite simple to understand and straightforwardly calculated.
It is a statistical measure that quantifies the financial risk for a portfolio over a given time frame. For investors, it is an indication on how much value an asset or portfolio can lose in a worst-case scenario as well as the occurrence ratio of that loss. A drawback of Value-at-risk is that it does not give any information about the severity of losses which might occur below the given confidence level. The VaR of a portfolio at the confidence level ↵ is given by the smallest number l such that the probability that the loss L exceeds l is no 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. Typical vales for the the confidence level are ↵ = 0.95 or ↵ = 0.99.
Using scenario modelling of financial portfolios, VaR can be derived from selecting the ↵-percentile of losses from that portfolio. Assuming that we have N scenarios with losses of a portfolio over the time horizon T, the VaR at the confidence level ↵ is then the value of the ↵-percentile of the worst losses from these N scenario values.
2.1.2 Expected Shortfall
Expected shortfall (ES) is another common risk measurement used in financial institutions and attempts to address the shortcomings of the VaR measure. It is derived from calculating a weighted average of the worst-case scenarios used in the value-at-risk calculations. Simpler put, it tells an investor how large the expected loss of a portfolio or an asset is, given that we find ourselves in the given quantile, which is usually 95 or 99 percent. This risk measure is more practical to use for volatile investments than VaR and usually leads to a more conservative approach when choosing risk exposure. The value of the expected 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 VaR breakpoint and qu(FL) is the quantile function of the distribution function FL.
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 larger than the decided confidence level and is therefore looking further down the tail of the loss distribution than VaR. It is intuitive that |ES| |V aR| for a given confidence level since ES depends on the same loss distribution as VaR while averaging the losses behind the given confidence level. To implement expected shortfall in our simulated future returns of the portfolios, one simply takes the worst 5 or 1 percent of the simulated returns, calculates the average value 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, they often use a benchmark that has a similar behaviour as the portfolio. Tracking error is a commonly used metric to understand how well the portfolio is performing compared to the benchmark by calculating the di↵erences between the 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 between a portfolio and a benchmark. Tracking error is calculated by the following formula,[4 pp. 78-79]
T rackingError =
sPN
n=1(Pn Bn)2
N 1 (2.2)
where Pn is the return of the portfolio in period n, Bn is the return of the benchmark in period n and N is the number of return periods used. A high tracking error indicates that the benchmark returns are far from the portfolio returns, i.e. that the benchmark and the portfolio is not behaving in a similar way. However, tracking error does not tell us anything regarding if the portfolio is overperforming or underperforming in terms of returns compared to the benchmark.[5]
2.1.4 Information Ratio
The information ratio (IR) is often used to measure an asset managers ability to generate excess returns compared to the returns of a benchmark. It mea- sures how much the portfolio is overperforming relative to the benchmark while incorporating tracking error to analyze the consistency of that perfor- mance. Information ratio is calculated by subtracting the portfolio returns over a given time period with the benchmark returns which is divided by the tracking error.
IR = Rp Rb
T rackingError (2.3)
Where Rp is the return of the portfolio over a period, Rb is the return of the benchmark over the same period and T rackingError 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 low tracking error or big di↵erence between the portfolio and benchmark return.
A low tracking error while having positive numerator means that the manager have achieved a better return than the benchmark, while keeping the portfolio close to the benchmark which shows an ability to generate excess returns.
A high value of the numerator would also indicate that the manger has an ability to generate excess returns, but would also result in a higher tracking error in most cases.
Negative values of information ratio means that the benchmark had higher returns over the period than the portfolio and indicates that the manager have have failed to achieve excess returns in relation to the benchmark.[6 pp.
433]
2.1.5 Sharpe Ratio
The Sharpe ratio is a measurement tool that helps investors to evaluate the return of a portfolio compared to the risk of the portfolio. Developed by William F. Sharpe in 1966, it has become one of the most commonly used methods to calculate the risk-adjusted return of an investment. It is derived from subtracting the risk-free rate from the mean return of a portfolio or an asset for a given time period and dividing the result with the standard deviation 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 is the standard deviation of the portfolio. The risk free rate is assumed to be 0 in this master’s thesis. The Sharpe ratio gives information regarding if the portfolio returns is due to smart investment strategies or due to a high-risk taking. A high Sharpe ratio on a portfolio implies a more attractive investment for an investor since this means that the ratio between return and risk is high which suggests that one have achieved a relatively high return from the risk taken.[6 pp. 433]
2.1.6 Maximum Drawdown
Maximum Drawdown (MDD) measures the maximum observed loss for an investment over a given period. By investigating the maximum drawdown an investor 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, M DD = min1<j<k<T
(Rk Rj) Rj
(2.5) where Riis the return of the investment in time i and j & k are two time points during the window of the size T. The equation searches for the minimum value since the maximum loss is equal to the minimum return. To exemplify maximum drawdown one can imagine an investment with a time horizon of one year where the yearly return resulted in +10%. At first glance, this would seem like a stable and secure investment, but it doesn’t tell the investor anything regarding what happened during that time period. A large value on maximum drawdown implies that the underlying asset had a major downfall during that year, which could mean that the investment is volatile, while a small value on maximum drawdown implies a more stable investment.[7]
2.2 Portfolio Theories
2.2.1 Black-Litterman Model
The Black-Litterman model is an analytical portfolio allocation method that optimises the asset allocation of a portfolio. The model combines historical data of asset returns with the views of the asset manager to create an optimal portfolio that lies within the investors’ risk tolerance. The model is therefore enabling portfolio managers to combine their unique views of the market with the market equilibrium which leads to diversified and intuitive portfolios. The model was introduced by Fisher Black and Robert Litterman in 1990 with the purpose of solving some of the problems with earlier portfolio allocation methods like the Modern portfolio theory, which often produce unintuitive and highly-concentrated portfolios which becomes problematic for practical use.[8 pp. 1-2]
First of all, the combined excepted returns need to be calculated which can be derived from the following formula, [9]
µBL=⇥
(⌧ ⌃) 1+ VT⌦ 1V⇤ 1⇥
(⌧ ⌃) 1⇧ + VT⌦ 1Q⇤
(2.6) and the co-variance matrix of the returns becomes: [9]
⌃BL = (1 + ⌧ )⌃ ⌧2⌃VT ⌧ V ⌃VT + ⌦ 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 views representing 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 Least Squares, Section 2.3.4) and is a risk aversion parameter, representing the risk tolerance which is given by:
= SharpeRatio
mkt
, mkt =
q
wTmkt⌃wmkt
The uncertainty scalar
The uncertainty scalar (⌧ ) is intended to regulate the uncertainty of each view and the estimated equilibrium returns. There exists a number of ways for estimating this parameter but in this report it will be estimated as: [9]
⌧ = 1 N
where N is the number of time steps used to estimate the co-variance matrix from historical returns.
Views
The V -matrix in Equation (2.6 and 2.7) defines which assets that is involved in 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⌃viT
where vi is the i:th row in the V matrix, representing view i.
2.2.2 Modern Portfolio Theory
The Modern portfolio theory (MPT) is a method for portfolio managers to maximize the expected returns based on a given a level of risk. It was introduced in 1952 when Harry Markowitz published this pioneering portfolio theory in his paper ”Portfolio Selection” in The Journal of Finance, which he was later awarded a Nobel prize for.[10] Markowitz argued that a portfolio’s risk and return should not be observed as two separate characteristics but as two closely related portfolio measurements. MPT can be used to construct a portfolio of assets with an allocation that either gives a maximized return given a level of risk or a minimized risk given a level of expected return. The theory states that the higher risk that the investor is willing to take, the higher return the investor can expect to achieve.[11]
In a scenario where there are two portfolios available o↵ering the same expected return, MPT assumes that an investor is risk averse, i.e. that the investor will choose the less risky portfolio. This means that an investor will only accept a riskier portfolio, where risk is expressed as the volatility of the portfolio, 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 is that a rational investor will always choose a portfolio that have the most favourable risk adjusted return. To make use of the MPT, one has to calculate the expected return and the volatility of the portfolio. Expected return is calculated by the following formula,[12 pp. 62]
E[Rp] = Xn
i
wiE(Ri) (2.9)
where Rp is the portfolio return, wi is the weighting of asset i, Ri is the estimated return of asset i and n is the number of assets in the portfolio. The variance 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 efficient frontier (presented in section 2.3.1 below), it is possible to find the optimal portfolio allocations among a set of underlying assets so that the given portfolio have the highest expected return for a defined level of risk, or vice versa.
To further develop the MPT it is possible to use it together with the concept of robust optimization, which tries to take uncertainties of estimated values (return and covariance matrix) into account. Since financial portfolios exist in a much larger universe than the chosen set of assets studied, there will always exist some uncertainties within the estimations. However, the concept of robust optimization will not be used in this master’s thesis but is an important topic to study further to evolve the results given in this report.
2.3 Portfolio Calculations
2.3.1 Efficient Frontier
The efficient frontier is the set of optimal portfolios that o↵er the highest expected return for a defined level of risk, i.e the portfolio that solves the following minimization problem for a given target return µ0.
minw
1 2wT⌃w s.t. wTµ = µ0
wT~1 = 1
w 0
where µ is the estimated return of the underlying assets and w is the vector of asset weights. The second constraint makes sure that the total investment weights is equal to 1, i.e that all capital is invested. The last constraint ensures that the weights of each asset is equal or larger than zero, i.e that there is no shorting allowed.
The efficient 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 efficient frontier ends with investing 100% in the underlying asset with the highest estimated return.
2.3.2 Covariance Estimation
To estimate the covariance matrix ⌃ between several financial assets, a moving average procedure is used. The purpose of using moving average is to smooth out short term fluctuations while highlighting long term trends or cycles. The following formula estimates the covariance matrix ⌃ of several assets using historical data. [3. pp. 64-65]
⌃ = 1 T
XT i=1
RtRTt (2.11)
where Rt is a vector of demeaned returns of the assets in time t and T is the length of the time horizon.
2.3.3 Linear Model
Let X1, X2, . . . , Xp 1 be (p 1) explanatory variables (predictors) and Y be a 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} and
X1 ={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 predictors X1, 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.
2.3.4 Constrained Least Squares
Constrained least squares is a linear least squares problem with constraints on the solution. Constrained least squares are often used to solve constrained linear regression and other problems on the following form.
minw ||wX Y||2 s.t. Aw a
Bw = b
(2.13)
where
X 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 equality constraint.
(The weights w can also be called parameters as in Linear Model (Section 2.3.3))
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 portfolio managers. It has proven to be quite accurate on replicating the behaviour and movement of financial time series for historical and forecasted portfolio returns. The purpose of the VAR-model is to predict future returns of an asset based on its past performance by assuming that the future will resemble the past. However, this assumption can lead to inaccurate predictions if extreme changes in the underlying market would occur such as a technological transformation of an industry.[13]
A general VAR(p)-model is defined by the following formula.
Yt = A0 + A1Yt 1+ ... + ApYt p+ ✏t (2.14) where
Yt is a vector of returns in time t, A0 is a vector of intercepts,
A1,...,Ap are matrices of parameters,
✏t is a vector of errors with E[✏t] = 0, without serial correlation and having covariance 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 for all k assets in the investment universe,
yt,1 = a0,1+ a1,1yt 1,1+· · · + a1,kyt 1,k+ ✏t,1
yt,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 for asset 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 BB B@
1 yT 1,1 yT 1,2 . . . yT 1,K 1 yT 2,1 yT 2,2 . . . yT 2,K
... ... ... . .. ...
1 y1,1 y1,2 . . . y1,K
1 CC CA⇤
0 BB B@
a0,1 a1,1
...
ak,1
1 CC CA=
0 BB B@
yT,1 yT 1,1
...
y2,1
1 CC CA
= 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)
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 the asset and using these to create a simulated evolution of the asset in the future. 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 window and use these to create a scenario of length T.
3. Repeat to create several scenarios.
3. Method
The practical and methodological work used throughout the thesis project will be described in this section. The section is divided into five parts:
”Data Collection”, ”Benchmark Analysis”, ”Portfolio Allocation”, ”Scenario Modeling & Analysis” and ”Stress Test”, where the first part describes how the relevant data have been collected and the four later parts focuses on how the goals of this project have been solved. The purpose of this section is to give the reader a more in-depth view on how the practical work have been carried out and what kind of challenges have appeared during the project work.
3.1 Data Collection
The first practical focus point of the thesis was to collect and preprocess the data 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 of their portfolios and di↵erent model portfolios with their underlying assets throughout the years. This data was then complemented with historical data from a number of stock indices and assets from around the world. The historical data are daily prices from 2000-01-01 to 2019-12-31 as long as they are publicly available.
Listed below is a short description of the most frequently used assets during this project, the full list of assets can be found in the appendix.
• Dynamisk - Placerum’s high risk portfolio, consisting of approximately 90% stock funds and 10% rate funds.
• MSCI World - Large and Mid cap index representing 23 Developed Markets (USA, Japan, UK, Germany, etc).
• MSCI EM - Large and Mid cap index representing 26 Emerging Markets (China, Korea, Brazil, etc).
• OMXS 30 - Stock market index for the Swedish stock market consisting of the 30 most traded stocks.
• S&P 500 - Stock market index that measures the stock performance of 500 large companies listed on stock exchanges in the United States.
• Euro Stoxx 50 - Stock market index containing fifty of the largest and most liquid stocks in the European zone.
• OMX T-Bill - Rate index designed to show the value growth trend for a certain type of interest-bearing Swedish security.
When the data had been collected, it was thoroughly inspected to look for missing, strange or other extreme values by plotting the assets and visually look for abnormalities. Each value determined to be incorrect was replaced by 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 be easily accessible in further methods and simulations.
3.2 Benchmark Analysis
To find a better alternative to Placerum’s current benchmark, which is the MSCI World index converted into SEK, a total of eight di↵erent benchmark alternatives have been examined. The composition and weighting of each benchmark have been decided in cooperation with Placerum so that the benchmarks will somewhat represent their portfolios underlying assets. The most reasonable choices of indices were agreed to be OMXS 30, MSCI World, MSCI Emerging Markets and OMX T-Bill since they best represent the areas where Placerum usually invest. These indices were then distributed in various weights where some were converted in to SEK and some were kept in their original currency. A decision was made to not mix currencies within the same benchmark since this would not be practically possible to implement for Placerum.
Listed below, the benchmarks that have been tested and analysed as well as the 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)
• 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’s portfolio called Dynamisk by calculating the tracking error (Equation 2.2) from historical returns between 2015 and 2019 as well as how often Dynamisk over- and underperform compared to each benchmark during that time period.
The results from these tests will then show which benchmark that would have represented Placerum’s portfolio most accurate historically in terms of returns. Moreover, the benchmarks were also analysed on how well they replicate Dynamisk for simulated future scenarios. To do this, 2000 scenarios for a five year time period were simulated using the Vector-Autoregressive model (Section 2.4.1). 10 years of historical returns for Dynamisk, USD/SEK currency changes and each of the benchmarks underlying indices were used in equation (2.15) to estimate the VAR parameters which were then used to simulate the future scenarios.
For each simulated scenario, the benchmark returns are determined and the tracking error (Equation 2.2) between each benchmarks and Dynamisk is calculated. How often Dynamisk is over- and underperforming compared to each benchmark in every scenario is also calculated in the same way as for the historical data. When the tracking error has been calculated for each scenario, the mean value for each benchmark is derived to be able to compare the 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 benchmark that best replicates the portfolios in terms of tracking error and performance was decided in cooperation with Placerum’s management team.
In order to see patterns and analyse the results, the historical evolution on the assets is plotted in both SEK and USD. For the scenarios, the evolution of each asset is plotted for two randomly selected scenarios. For an easier comparison, the assets starting value is all set to 100 in the beginning of the historical data (i.e February 2015).
3.3 Historical Allocation
The general method to find the historical returns of each portfolio allocation model is conducted by starting in the first week of 2013 and using the past two years of the underlying indices historical data, converted to SEK, to find the starting weights. All indices are converted to SEK since Placerum’s portfolio Dynamisk are sold in SEK and to best compare the allocation theories to this portfolio it was decided that all the underlying assets should also be converted in to SEK. After four weeks, the portfolio is rebalanced with a new two-year historical return window prior to that point in time which becomes the new portfolio weights for the four coming weeks. Each model starts in the 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 Europe and Middle East. The allocation methods also consider four currencies in relation to SEK which are USD, EUR, HKD and JPY since these are the currencies 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 combined expected returns is calculated by inserting the inverse co-variance matrix of historical returns, an uncertainty scalar ⌧ , the ⇧-vector which identifies the implied equilibrium market return vector, the view vector and the uncertainty matrix in equation (2.6). The Black-Litterman co-variance matrix of returns is then derived by inserting the same variables in equation (2.7).
Given the Black-Litterman return vector and the co-variance matrix, the optimal allocation is derived using the efficient frontier (Section 2.3.1) and the allocation that has the most similar Sharpe ratio to what Dynamisk had during 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 Placerum is 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-Litterman section will be conducted on both of these benchmarks.
Market Portfolio
To calculate the market weights (i.e the portfolio that the market suggests one should choose to match the given benchmark) and the implied equilibrium market returns of the Black-Litterman model, a linear regression with a moving estimation window of two years is used. This means that the market portfolio weights are found by using actual historical returns of the underlying indices from two years prior to the given time step in a constrained least squares minimization problem (Section 2.3.4), and by doing so finding the optimal portfolio weights.
To avoid unrealistic weights in the market portfolio, a couple of constraints have been implemented on the model which gives a maximum and minimum investment 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 OMXS30 and OMXS Small Cap, Rates represents OMX T-Bill and OMX T-Bond, Asia represents Hang Seng, Nikkei 225 and Emerging Markets represents MSCI EM 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 <20%
• Constraint 5 - Asia <30%
• Constraint 6 - Emerging Markets <30%
The constraints have been specified in cooperation with Placerum so that the market portfolio will represent their general investment pattern. In general, the constraints will limit the weighting of rates, Asian markets and emerging markets in the market portfolio, while the Swedish, North American and European markets are less restricted.
Using the stated constraints, a constrained least squares problem (Section 2.3.4) is solved to find the allocation wights wmk which match the used bench- mark the best. In other words, the linear model (Equation 2.12) is used where Y is the return of the benchmark, 0 = 0 and Xi is the return of underlying asset i in SEK.
To find the A-matrix and the a-vector, equation (2.13) is used where Y is an array of returns for the benchmark, X is an matrix of returns of the underlying indices in SEK during the given time period and w is the vector of allocation weights wmk. Since the constraints defined above are inequality constraints, the A-matrix and the a-vector become as seen below, where each column represents an underlying index and each row represents a constraint.
A = 0 BB BB BB BB BB BB
@
1 0 0 1 0 0 0 0 0 0 0 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0
0 0 0 0 1 1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0
0 0 0 0 0 0 1 1 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 1 1 0 0
0 0 0 0 0 0 0 0 0 1 0 1 1
1 CC CC CC CC CC CC A
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 which is that the total weights is equal to one, i.e that no shorting is allowed and that 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 the benchmark with the volatility of the market. The Sharpe ratio is calculated as 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 is estimated as in Section 2.3.2.
Black-Litterman Views
To implement the preferences and investment strategies of Placerum’s invest- ment team, a number of market views have been defined that represents their specific opinions about the future market developments. These views will make predictions about the indices returns in such a way that the portfolio weights will redirect towards Placerum’s investment preferences. The views will therefore either accelerate or decelerate the weighting of certain indices in the Black-Litterman portfolio.
Listed below is every view of the market that has been implemented in the Black-Litterman model. Note that each percentage is stated as a percentage increase or decrease of the return from the previous year, not in percentage points. The annual mean is derived from the average return of the previous two 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 mean return is between -5% and 15% it will increase with 10%; and if the annual 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 mean return is between -5% and 15% it will increase with 10%; and if the annual mean return is below -5% it will increase with 15%.
• View 3 - Europe: If the annual mean return of Euro Stoxx 50 is above 15%, the return of the next year will decrease with 15%; if the annual mean return is between -5% and 15% it will increase with 10%; and if the annual mean return is below -5% it will increase with 15%.
• View 4 - Sweden: If the annual mean return of OMXS30 and OMXS Small 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 than the mean annual return of S&P 500 and Euro Stoxx 50, the di↵erence will increase with 5% the following year.
• View 6 - Emerging Markets: If the annual mean return of EM is larger than the annual mean return of DM, the return of the next year will decrease with 20%, otherwise it will increase with 5%.
• View 7 - DM Asia: If the mean return of Hang Sen and Nikkei 225 is larger 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 Nikkei 225 will decrease with %10.
• View 8 - Sweden: If the annual mean return of OMXS Small Cap is above 20%, the return of the next year will decrease with 25%; if the annual mean return of OMXS Small Cap is positive and the annual return of OMXS30 is negative, the return of OMXS Small Cap the next year will decrease with 10% and otherwise it will increase with 20%.
• 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 mean return of Russel 2000 is positive and the annual return of S&P 500 is negative, 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, specified in cooperation with Placerum, are optimistic about the Swedish market.
Moreover, the views are also optimistic about USA and Europe compared to the the emerging markets, which means that there exists a scepticism to- wards the emerging markets. These views correspond closely with Placerum’s investment strategies, since their portfolios tend to have a much larger weight invested in developed markets, especially in Sweden and USA. (Section 1.3.2) Given the views defined above, the V-matrix and the ⌦-matrix is formulated as below, where each row represent one view and each column represents one underlying asset.
V = 0 BB BB BB BB BB BB BB BB BB
@
1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0
0.5 0 0 0.5 0 0 0 0 0 1/3 0 1/3 1/3
1 0 0 0 0.5 0 0.5 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0
1 CC CC CC CC CC CC CC CC CC A
⌦ = 0 BB B@
⌧ v1⌃v1T 0 . . . 0 0 ⌧ v2⌃v2T . .. ...
... . .. . .. 0
0 . . . 0 ⌧ v12⌃vT12 1 CC CA
Row 4 and 5 in the V-matrix , representing view 4 and 5, is the only two relative views used which can be seen in the matrix since the row sums to 0 and 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 ⌧ = N1 (Section 3.3.1), where N are the number of observations used, i.e two years of
weekly data or N = 2⇤ 52 = 104.
The view vector Q of the excess return has equally amount of rows as the V-matrix, where each row represent the views excess return, this return is, as previously stated, based on the mean annual return during the two year historical window used at each rebalancing period.
Combined distribution and Allocation
The 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 a given 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-Litterman return and co-variance matrix as in Equations (2.6) & (2.7).
Given the Black-Litterman return and co-variance matrix, the efficient frontier is calculated as described in Section 2.3.1 for no shorting portfolios. From the efficient frontier, the Sharpe ratio of each portfolio on the frontier is calculated as in Section 2.1.5. Di↵erent investment strategies were tested such as choosing the portfolio with a maximized Sharpe ratio or the portfolio that is maximizing the return. However, choosing the portfolio with the most similar Sharpe ratio as Dynamisk during the two year historical window was decided to be the most fair comparison of the two portfolios since this means that they both have the same risk and return trade o↵.
When the Black-Litterman portfolio has been found, the development of the portfolio is calculated using the actual returns combined with the currency changes in the given time step. As stated above, the portfolio is rebalanced after four weeks by finding the new Black-Litterman weights, which is done by repeating the entire process above. This means that after four weeks, the market portfolio is recalculated with the new estimation window, the views are implemented to find the new Black-Litterman return- and covariance- matrices, the efficient frontier is updated and the Black-Litterman portfolio with the closest Sharpe ratio to Dynamisk is chosen.
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 both portfolios as well as for Dynamisk is calculated. The mean portfolio weights of 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 period as the Black-Litterman portfolios. Just as for the Black-Litterman portfolio (Section 3.3.1), the efficient frontier for the Markowitz portfolio is calculated, but now the expected return is estimated by the mean weekly return of each asset during the two year historical window while the covariance matrix is estimated 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 most similar Sharpe ratio as Dynamisk during the 2 year time window is chosen.
When the Markowitz portfolio has been found, the evolution of the portfolio is then calculated using the actual returns of the underlying assets, converted in to SEK, in the same way as the Black-Litterman portfolio, where the portfolio is rebalanced every four weeks just like the previous model. As for the Black- Litterman portfolios and Dynamisk, the average annual Sharpe ratio and the total, seven year, Sharpe ratio for the Markowitz portfolio is calculated. The mean portfolio weight of the Markowitz portfolio is also derived and presented together 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 method in 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 calculate the views. The portfolio is rebalanced every four weeks with the new, two year, time window which becomes the investment strategy for the coming four weeks. This method is then repeated for the entire period of the generated scenarios which has been set to five years.
