Optimizing the Nuclear Waste Fund’s Profit André Ramirez Alvarez & Zakaria Kazi-Tani

Stockholm Business School

Bachelor’s Degree Thesis 15 HE Credits

Subject: Business Administration

Autumn semester 2017

Supervisor: Christian Lundström

Abstract

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

Acknowledgements

First and foremost, we would like to express our sincere gratitude and thank-fulness to our supervisor, Christian Lundstrom, PhLic., for his assistanceand dedicated involvement, and without whom this paper would have neverbeen accomplished. Thank you for your patience and support over this pastsemester.

Finally, we would like to thank our families, for their love, support andunderstanding.

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem discussion . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Research Question . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Boundaries and limitations . . . . . . . . . . . . . . . . . . . 5

2 Literature review 62.1 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 The Markowitz Portfolio Theory . . . . . . . . . . . . 62.1.2 Strategic and tactical asset allocation . . . . . . . . . 82.1.3 The Nuclear Waste Fund . . . . . . . . . . . . . . . . 10

2.2 Previous Research . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 The First to Fourth AP-Funds . . . . . . . . . . . . . 12

2.3 Summary of Literature review . . . . . . . . . . . . . . . . . . 13

3 Research Design 153.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Research Strategy . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Two security categories . . . . . . . . . . . . . . . . . 173.2.2 Three security categories . . . . . . . . . . . . . . . . 183.2.3 Several Inequality Constraints Optimization . . . . . . 183.2.4 Data Collection and Sample Size . . . . . . . . . . . . 213.2.5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 213.2.6 Method Criticism . . . . . . . . . . . . . . . . . . . . . 21

3.3 Epistemological and ontological assumptions . . . . . . . . . . 223.4 Research ethical reflection . . . . . . . . . . . . . . . . . . . . 23

4 Analysis and Findings 244.1 Empirical presentation and Results . . . . . . . . . . . . . . . 24

4.1.1 Portfolio A . . . . . . . . . . . . . . . . . . . . . . . . 264.1.2 Portfolio B . . . . . . . . . . . . . . . . . . . . . . . . 274.1.3 Portfolio C . . . . . . . . . . . . . . . . . . . . . . . . 30

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4.1.4 Summary of results . . . . . . . . . . . . . . . . . . . . 324.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Discussion and critical reflection 35

6 Conclusion 37

7 Limitations of Research 39

A Matlab script of Portfolio A 43

B Matlab script of Portfolio B 46

C Matlab script of Portfolio C 52

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Nomenclature

β The beta coefficient in the regression model

ε The error term of the regression model

L The Lagrangian

λ The Lagrangian Multiplier

ρ The correlation coefficient

σi,j The covariance between asset i and asset j

σP The standard deviation of the norm portfolio

σp The standard deviation of a portfolio

Pi The index closing price year i [SEK]

Pi−1 The index closing price year i-1 [SEK]

Pt−1 The value of a portfolio a time t-1 [SEK]

Pt The value of a portfolio at a given time t [SEK]

Ri The expected return of asset i [%]

Rp The portfolio return [%]

Rt The return percentage of a portfolio at a given time t[%]

wi The weight allocated in asset i

BLUE Best Linear Unbiased Estimator

E[Rp] The expected return of a portfolio [%]

FFAPF First to fourth AP-funds

MPT Markowitz Portfolio Theory

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NWF Nuclear Waste Fund

OLS Ordinary Least Squares

OMRX REAL OMRX Real Return Bond Index

OMRX TBOND OMRX Treasury Bond Index

OMXSPI OMXPI Stockholm Stock Index

S&P Corp S&P Sweden Investment Grade Corporate Bond Index

SAA Strategic Asset Allocation

TAA Tactical Asset Allocation

VAR[Rp] The variance of a portfolio return

W A matrix containing the securities’ allocation weights

x The independent variable in the regression models

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Chapter 1

Introduction

1.1 Background

When it comes to investments, the risk and return of assets changes overtime. Static risk and allocation limits are factors that can have large impacton the outcome of the investment. Therefore, it is important that the in-vestment is subject to recurrent revisions. The exercise of such revisions isessential for funds with fundamental functions in society, such as pension ornuclear waste funds. As these funds manage capital that without which fun-damental social functions in society would cease to work, they must handlethe capital with responsibility. In other words, the investments cannot bejeopardized by taking too large risks. Therefore, risk and allocation limitsare applied to the investment policies of the funds.

Recently, there have been revisions in the Swedish pension system onbehalf of the Ministry of Finance (Finansdepartementet). The AP-funds,Allmanna Pensionsfonder or National Public Pension Funds in English,which constitute one of the three components for the Swedish pension system(McKinsey & Company, 2013), are in the core of this amendments. SinceJanuary 1 2001, the First to fourth AP-funds, FFAPF, have been used asbuffer funds (McKinsey & Company, 2013). Since the introduction of thebuffer system, the buffer funds have had an annual mean return of 5.6%,outperforming the annual mean income index which for the same period was3.0% (Skr. 2016/17:130, 2017). This reflects the advantage and strategicdecision of introducing such system.

The number of diverse assets have contributed to satisfying returns andis also expected to contribute to cushion negative effects from declines in thestock market (Skr. 2016/17:130, 2017). However, the allocation has beenmanaged according to strict allocation rules (Skr. 2016/17:130, 2017), whichlimits the possibilities of alternative asset investments. Due to these limita-tions, the Ministry of Finance has made a proposal to revise and change theallocation rules, and is expecting to implement the changes by 2018 (Skr.

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2016/17:130, 2017).There is a number of similar funds which have fundamental functions in

the Swedish society. One of these is the Nuclear Waste Fund, which haveexisted since the 1980’s. The Nuclear Waste Fund constitutes a financialsystem, and was established by the Swedish government to finance futurecosts of the management of spent nuclear fuel as well as decommissioningof nuclear power plants (Karnavfallsfonden, 2016a). By collaborating withthe Swedish Radiation Safety Authority as well as the Swedish NationalDebt Office, the NWF strive to ensure that the fund’s capital is sufficientto achieve its purpose (Karnavfallsfonden, 2016c).

Like the FFAPF, the NWF undergoes strict investment rules which aredetermined by the board. Regarding the investment possibilities, the fundis allowed to invest only in nominal and inflation-linked bonds issued by theSwedish state as well as treasury securities (Karnavfallsfonden, 2017c). Anorm portfolio is defined to indicate the long-term allocation for the Nu-clear Waste Fund. The allocation of the norm portfolio consists of 70%investments with nominal return and 30% investments with real return(Karnavfallsfonden, 2017c). Although the NWF must follow the allocationrules, there is still room for tactical allocation. In other words, the NWFmanages its investments actively. The investments can thus deviate fromthe norm portfolio within the intervals specified in Table 1.1

Table 1.1: The Nuclear Waste Fund’s allocation limits, according to itsinvestment policy (Karnavfallsfonden, 2017c).

Investment class Minimal [%] Maximal [%]

Nominal 50 90

Real 10 50

In 2009, the restriction of solely investing in bonds issued by the Swedishstate was abolished Karnavfallsfonden (2017c), as a proposal of investing incovered bonds was approved by the Swedish government. The approval wasbased on the fact that covered bonds would generate a higher nominal re-turn than bonds issued by the Swedish state for a small increase in risk(Karnavfallsfonden, 2017c). Covered bonds have ever since been an invest-ment option for the NWF portfolio. For comparison purposes, the indexesOMRX Real Return Bond Index (OMRX REAL) and OMRX TreasuryBond Index (OMRX TBOND) are used (Karnavfallsfonden, 2016b). OMRXREAL is used as comparison index for the portfolio part with index-linkedinvestments, whereas OMRX TBOND is used for the portfolio part withfixed-income investments. Both of these indexes consists of bonds issued bythe Swedish state and have benchmark status (Karnavfallsfonden, 2016b).A composite index is built out of these two indexes, set up as presented

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in Table 1.2. These benchmark indexes have been used since 2009-01-01,when the investment policy was changed with respect to the investment op-tions. However, between 2009-01-01 and 2014-01-01, there have been slightvariations in the choice of benchmark indexes Karnavfallsfonden (2016c).

Table 1.2: Composition of norm portfolio (Karnavfallsfonden, 2016c).

Index Weight [%]

OMRX REAL 30

OMRX TBOND 70

The Legal, Financial and Administrative Services Agency has set a goalfor its asset management “. . . that the Fund’s nominal return over a five-yearperiod should exceed the standard portfolio’s comparison index by an aver-age of 0.5 percentage points per annum.”(Karnavfallsfonden, 2017c). TheNuclear Waste Fund’s returns compared to the norm portfolios return arepresented in Table 1.3. As can be seen, the active investment managementhas paid off, as the norm portfolio had a lower mean return between the 2012to 2016 compared to the NWF’s nominal return. This can be explained bythe fact that the fund invests in bonds with a long time to maturity whilstthe bond yield has been continuously decreasing (Karnavfallsfonden, 2016c).The returns have however been insufficient due to the NWF having a deficitin capital, which prevents the fund from fulfilling its purpose. The deficit isa direct consequence of restricting the investment options that characterizesthe fund (Karnavfallsfonden, 2016c).

Table 1.3: Nuclear Waste Fund’s annual nominal returns, norm portfolio’sannual return and excess return. All values give in percent.

Year Nominal return Norm portfolio return Difference

2012 4.60 2.50 2.10

2013 -0.57 -1.36 0.79

2014 10.69 10.72 -0.03

2015 -0.52 0.39 -0.91

2016 5.76 5.01 0.75

Mean return 3.91 3.37 0.54

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1.2 Problem discussion

As mentioned in the previous section, the Nuclear Waste Funds manages itsinvestments actively and has outperformed the norm portfolio on averagein recent years, which can be observed in Table 1.3. However, this doesnot necessarily mean that the portfolio is optimal, at least not all the time.Furthermore, the investment policy stipulated by the government limits thepossibilities of the portfolio, as the fund mainly is allowed to only investin nominal and inflation-linked bonds. Extending the number of allowedassets in which the fund can invest, or by changing the risk and allocationlimits, could therefore increase the average return of the fund and thereforedecrease the current capital deficit.

It should be mentioned that the difference in returns in Table 1.3 arisesdue to two actively tactical strategies; the allocation of assets and the se-lection of financial instruments. The asset allocation refers to how the in-vestments are distributed among various instruments, whereas the selectionof financial instruments refers to, as the name implies, the selection of theindividual instruments. More on this topic is discussed in subsection 2.1.2.

1.3 Research Question

Due to the Nuclear Waste Fund’s constraining investment rules, the averagereturn of the fund may have not always been optimal. For that reason, thestudy conducted in this thesis aims to answer the following question:

• What is the profit of changing the allocation limits and extending theinvestment options of the Nuclear Waste Fund?

1.4 Purpose

Due to its potentially dangerous nature, nuclear waste must be handled withgreat responsibility. This requires large investments in the nuclear wastemanagement and the disposal of the nuclear waste. Thus, the Nuclear WasteFund’s function is a public matter of great importance, as the managementof spent nuclear fuel and decommissioning of nuclear is vital for the publicsafety and the environment. In order to improve the conditions for futurehandling of nuclear waste, it is of interest to invest the fund’s capital toobtain as large returns as possible while considering the risks that are beingtaken. This leads us to the purpose of the thesis, which is to investigate ifand how the portfolio could have historically performed better. This studyis of an experimental character and will hence use a quantitative researchapproach.

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1.5 Boundaries and limitations

When investigating the optimal portfolio, the risk boundaries will be set toallow only portfolio compositions which do not exceed 1.05 times standarddeviation of the norm portfolio returns during the specified period. Also,there will be no selection of instruments, but only the asset allocationswill be considered. This means that only the effects from the allocationsand not from the instruments themselves will be studied. Furthermore, theresearch is delimited to only consider Swedish corporate bonds and Swedishstocks with respect to the extension of the investment possibilities. Theindex that will be used for the corporate bonds is the S&P Sweden 1+Year Investment Grade Corporate Bond Index, and OMXSPI Index for thestocks. In addition, management fee’s are neglected in the calculations.Due to the variations in choice of benchmark indexes, the report will onlyconsider the composition of the two indexes OMRX REAL and OMRXTBOND with their corresponding weights shown in Table 1.2. Finally, asthe restrictions of covered bonds were introduced in 2009, the time seriesthat will be processed will range between 2009-12-28 − 2016-12-30.

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Chapter 2

Literature review

The Nuclear Waste Fund invests under strict allocation rules, and is re-stricted to invest solely in nominal and inflation linked bonds. Furthermore,the fund has set allocation rules, which allows the NWF to invest a cer-tain amount of capital in respective bonds, defined as the norm portfolio,as introduced in Table 1.2, section 1.1. The purpose of this study is toevaluate what the profit could have been if the Nuclear Waste Fund wouldhave changed the constraining rules. Hence the purpose converges into thefollowing research question:

• What is the profit of changing the allocation limits and extending theinvestment options for the Nuclear Waste Fund?

In this chapter, a literature study will be conducted, where theories,strategies and models relevant to the study will be introduced. These in-volve the fundamentals of the Markowitz Portfolio Theory, and Strategic andTactical asset allocation, which are strategies used for managing the NWF.Section 2.2 will consist of previous research conducted in the same domainas this thesis, and will introduce a memorandum written by the Ministryof Finance, that investigates the opportunities of optimizing the First toFourth AP funds returns by changing their allocation rules. Finally, section2.1.3 will consist of background information about the NWF and the fund’sfuture challenges that might give incentive to invest in assets beyond theexisting ones.

2.1 Theoretical Framework

2.1.1 The Markowitz Portfolio Theory

The Markowitz Portfolio Theory, MPT, was pioneered and established byHarry Markowitz in his seminal article “Portfolio Selection” at the year of1952 (Markowitz, 1952). Markowitz claimed that the outcome of a dynamic

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market is reflected by uncertain investors that strive after risk-averse invest-ments (Markowitz, 1952). According to Mangram (2013), diversification ofa certain number of securities in a portfolio is a fundamental concept in theMarkowitz model in order to obtain a risk-averse portfolio and maximize theexpected return. Furthermore, Markowitz (1952) emphasizes that the lowerthe correlation is between the securities in a portfolio, the more risk averseit will be. The MPT is hence used to create and select a portfolio based onmaximizing the portfolio’s expected return and minimizing the investmentrisk.

Markowitz (1952) demonstrates that by choosing a proper weighted col-lection of investment assets, they can together exhibit a lower investmentrisk. An uncertain investor might as well set an intact risk limit for a port-folio in order to maximize the portfolios’ expected return for that given risk,by testing different weight combinations and finding the one combinationthat would generate the highest expected return (Markowitz, 1952). How-ever, Markowitz had to make certain foundational assumptions in order toconstruct the MPT. Those assumptions are as follows (Markowitz, 1952):

1. The investors are considered rational.

2. Investors will only invest with a higher risk profile if it is compensatedwith an increase in expected return.

3. The investors will receive all relevant information related to their in-vestment decision in time or frequently.

4. Investors have the option to borrow or lend an unlimited sum of capitalat a risk free interest rate.

5. Markets are perfectly efficient and exclude taxes and transaction costs.

6. The investor has the option to choose securities, whose performancesare independent of other portfolio investments.

7. The standard deviation σ is an adequate measure of risk.

The seven assumptions mentioned above have been criticized in multiplearticles, claiming they might not reflect reality and are hence not always ap-plicable. McClure (2017) claims that investors do not always act rationally.This can be seen in the herd behavior, a common phenomenon that occurswhen investors act as a collective with no centralized direction (McClure,2017). Furthermore, it may not always be correct that investors only investwith a higher risk profile if it is compensated with an increase in expected re-turn. In fact, McClure (2017), claims that an investor can purchase a riskyasset such as a derivative, to decrease the total risk of a portfolio. Thisstrategy will thus not be compensated with an increased expected return.

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Mangram (2013) emphasizes that the fact that investors frequently re-ceive the relevant information in time does not reflect the reality. Mangram(2013) claims that the information flow is considered asymmetric betweenthe outside investor and a firm, since the firm will always have superiorinformation, which is symbolized by example given insider trading. Thefact that investors have unlimited access to capital is also an incorrect as-sumption to make, since every investor has limited credits (Mangram, 2013).Subsequently, the MPT is based on the assumption that markets are per-fectly efficient. However, the assumption does not consider potential marketfailures such as information asymmetry, according to Mangram (2013). His-torical events such as bubbles and busts clearly demonstrate that marketsare not perfectly efficient (Mangram, 2013). The fact that the investor hasthe option to choose independent securities is also an assumption that hasbeen criticized. It has been demonstrated that no such instruments exist.During intensive periods that characterize uncertainty, seemingly indepen-dent investments do exhibit characteristics of correlation (McClure, 2017).

2.1.2 Strategic and tactical asset allocation

The purpose of using a Strategic Asset Allocation, SAA, is mainly to cre-ate a strategic mix of assets that will give the portfolio a satisfying balancebetween the expected return and risk for a long term investment horizon(Eychenne et al., 2013). A fund may for instance use an SAA approach todetermine the proportion of the portfolio to be invested in nominal bonds,real bonds, stocks and other securities over a long time horizon within aspecific risk interval. Due to the assets’ performance, the portfolio alloca-tions will be periodically rebalanced back to the initial settings wheneverthey deviate significantly (Eychenne et al., 2013). However, applying a longterm investment strategy on a portfolio requires in practice that the futurevariables such as the volatility of each asset, the correlation between theassets, and other relevant variables will have to be determined in order tocalculate the future risk and expected return (Eychenne et al., 2013). Thefuture values are often determined based on their historical values, whichindicates that the historical trend will remain active until the end of theinvestment horizon. However, Eychenne et al. (2013) states that relying onhistorical figures may not be an accurate method since it disregards the factthat the market is dynamic and can thus undergo changes and experienceeconomic shocks, which in turn might impact the investment portfolio.

According to Baird (2009), strategic asset allocation might be an appro-priate investment strategy when the market is stable. Baird (2009) howeverclaims that during periods where the market undergoes crises and shocks,the correlation between the assets may converge towards 1.0, which indi-cates that the benefits coming from diversifying the portfolio will be brokendown. A strategic approach may hence fail to provide adequate protection

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to investors during dynamic periods. Baird (2009) further emphasizes thatduring bear-markets, that are usually characterized by increasing marketinterest rates, the attractiveness of solely using SAA and investing in eitherbonds issued by the state or index securities is diminished given that theinvestor will receive a minimal rate of return (Baird, 2009).

Using tactical asset allocation, TAA, and SSA as complementary compo-nents within a comprehensive investment framework is believed to give theinvestor more flexibility to hedge a predicted future increase in risk or de-crease in return (Swedbank, 2015). In fact, multiple funds apply both TAAand SAA in order to make the portfolio more efficient and maximize thereturn. A tactical asset allocation is described as a short term investmentstrategy that uses active management and can be designed to either hedge ordraw benefits from the momentum coming from dynamic movements in thefinancial market, compared to the SAA that is a more passive and long termportfolio management strategy (Hyman, 2009). The investment horizon ofa TAA can differ between several months, applying a trading strategy, tothree years applying a thematic strategy (Hyman, 2009).

According to Brennan et al. (1997), in order to use a TAA, it is crucialthat the investor receives the correct information in order to anticipate possi-ble declining or increasing trends in due time. Furthermore, not all decisionsusing a TAA will reflect a beneficial outcome since the excess return mightbe negative due to bad investment decisions (Brennan et al., 1997). Baird(2009) states that the tactical allocation interval is essential for the risk inthe trading. A “go-anywhere” approach might significantly increase the risk,especially in volatile markets (Baird, 2009). By investing at the right time,the investor can prevent potential losses and also achieve potential gains.However, a wrong-timed investment might result in under-performance andeven significant losses. It is hence essential to set a consequent interval forthe tactical allocation given that in example a fund has set risk boundaries(Baird, 2009). Baird (2009) further emphasizes that an investor applyingTAA might experience periodic dip falls, and it is thus essential for the in-vestor to stay committed to its investment strategy throughout that periodin order for the strategy to pay off.

Baird (2009) also mentions that a fund’s overall profit can be affected bythe costs and fees that may arise by applying a TAA approach. The mostcommon fee is the management fee, which is an inevitable fee that the ownerof the portfolio will have to pay in order for the portfolio to be managed(Baird, 2009). The fee is generally higher for a portfolio that uses TAA,mostly due to the strategy being active and thus time demanding. The taxcosts have however the largest impact on the portfolio, and increase in linewith the active trading (Baird, 2009). In other words, the more transactionsthat are made in the portfolio, the larger the tax costs will be. It is henceof big importance to consider these costs before applying a TAA. Thus,Baird (2009) draws the conclusion that all investors should apply TAA to a

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certain grade, but also taking into account the potential under-performanceand increased tax liabilities if the portfolio is not managed efficiently.

2.1.3 The Nuclear Waste Fund

In the early 1980’s, the Swedish government decided to implement a financialsystem with the purpose of handling nuclear fuel waste and decommissionnuclear power plants. This resulted in the founding of the Nuclear WasteFund at the year of 1996 (Karnavfallsfonden, 2017b). The fund’s capitalcomes from a fee paid by holders of a license to own, or operate nuclearfacilities (Karnavfallsfonden, 2017b). The fee itself is managed by the NWFand is determined depending on how much kWh electricity that is deliveredby the nuclear facility (Karnavfallsfonden, 2017b). If the license holder can-not pay the demanded fee, and the capital of the fund is insufficient in orderfor the NWF to fulfill its purpose, the state, and thus the tax payers, willbe responsible to provide the missing capital and make up for the deficit(Karnavfallsfonden, 2017b).

The fund has until the year of 2016 achieved their target of an averagenominal expected return per annum of 0.5 percentage points above the com-parison index between 2012-2016 (Karnavfallsfonden, 2017a). Since the fundwas initiated in 1996, the average nominal return per annum was estimatedto 0.7 percentage points above the comparison index (Karnavfallsfonden,2017a). The NWF illustrates this by making an assumption that the fundreceived a 100 SEK payment in 1996, whereas the nominal return of the100 SEK is shown in Figure 2.1 (Karnavfallsfonden, 2017a). Figure 2.1 alsodepicts the return of the 100 SEK following the comparison index, as well asthe inflation that has to be deducted from the nominal return of the NWFin order to determine the real return (Karnavfallsfonden, 2017a). Hence,the NWF concludes that (Karnavfallsfonden, 2017a):

• the nominal return of the 100 SEK payment that was made to theNWF is estimated to 356 SEK in 2016. By deducting the inflation,equivalent to 25 SEK, the real return is evaluated to 331 SEK.

• if 100 SEK were invested following the comparison index, the capitalwould be equivalent to 304 SEK in 2016.

The success of the NWF in terms of nominal return can be explained bythe fact that the bond yields have been in a decreasing trend in line with thedecreasing interest rate in the financial market (Karnavfallsfonden, 2016c).The NWF has therefore been able to hedge the decrease by investing inlong-term bonds, that is bonds with a long date to maturity. The increasednominal return comes thus from the revaluation profits on the bond holdings(Karnavfallsfonden, 2016c). The NWF have however predicted that thetrend of the decreasing bond yield will most likely be broken, which will

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Figure 2.1: The nominal return of the NWF vs the return of comparisonindex over the time span of 1996-2016, given that 100 SEK is to be managed(Karnavfallsfonden, 2017a).

reflect on the nominal return of the portfolio (Karnavfallsfonden, 2016c). Inaddition, even though the NWF achieved their set goal of annual nominalreturn, the generated capital has not been sufficient to decommission nuclearpowerplants, which is one of the main reasons why the NWF was initiated(Lundin, 2015). The chairman Dan Barr Lundin (2015) claims that thedeficit is approximately 10.7 BSEK in the NWF, which in other words is thecapital needed for the system to be in balance. Barr further emphasizes thatan increased interest rate will impact the NWF’s nominal return negatively(Lundin, 2015). Precautions will thus have to be taken in order to keepthe nominal return stable. The NWF have in fact proposed to the Swedishgovernment to allow placements in other types of securities other than theones that are permitted, and refer primarily to corporate bonds and stocks(Karnavfallsfonden, 2016c). The NWF’s assessment states that this couldbe done with a moderate increase in the overall risk level in the portfolio.The government chose however to reject the proposal due to the risk increase(Karnavfallsfonden, 2016c).

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2.2 Previous Research

2.2.1 The First to Fourth AP-Funds

The First-Fourth AP-Funds’ main role is to manage the buffer in the incomepension system, a platform where the surplus from the pension payments istransfered to, as well as capital is taken from when there are deficits in thepension payouts (AP-fonderna, 2017). The surplus occur when the pensionpayments are larger than the pension payouts, while the deficits correspondto the opposite, that is when the payouts are larger than the payments (AP-fonderna, 2017). The fund is managed by strictly following the allocationrules that were set in order to create a stability in the income pension systemand achieve their goal to maximize the return on a long-term basis relativeto the risk (AP-fonderna, 2017). Some of the current allocation rules of theFFAPF are listed below (Finansdepartementet, 2017):

1. A maximal of 5% of the fund capital may be allocated in stocks orshares in venture capital companies.

2. At least 30% of the fund capital is to be allocated in interest-bearingsecurities with low credit and liquidity risk.

3. Capital may be invested in derivatives, excluding derivatives with com-modities as underlying assets.

Since the allocation rules were implemented in the 1990’s, the financialmarket has been constantly evolving and growing in line with the develop-ment of technology (Finansdepartementet, 2017). Fundamental tools usedfor research have been invented and innovated. The access to informationhas been exponentially increasing, which has resulted in creating, developingand innovating a number of economic theories and models. Because of this,Finansdepartementet (2017) highlights in a memorandum that the alloca-tion rules need to be reformed in order to improve the funds circumstancesand find more efficient allocation solutions, especially considering the lowinterest rates in today’s financial market. This should be done by followingthe Prudent Man Rule, which refers to the rule in which the investment ad-visers only invest in securities that are believed to generate a positive returngiven that the risk limit is maintained (Finansdepartementet, 2017). TheMinistry of Finance’s proposition of the new allocation rules mainly involvethat:

• Up to 40% of the real value of the FFAPFs’ assets are to be illiquid(Finansdepartementet, 2017). Illiquid assets are all assets that are notconsidered to be easily exchanged into cash, such as for instance realestate, large packages, antiques and certain types of debt instruments.The maximal limit of 5% for investments in unlisted shares will alsobe eliminated (Finansdepartementet, 2017).

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• At least 20% of the market value of the assets that each of the FFAPFpossess is to be placed in interest-bearing securities with low creditand liquidity risk (Finansdepartementet, 2017). The reform involvesin other words a decrease in the current allocation percentage by 10percentage points from 30% to 20%.

According to Finansdepartementet (2017), this change is considered togenerate a higher expected return in a long term perspective for the FFAPF.By investing in illiquid assets, the funds can also benefit from the liquiditypremiums in the financial markets (Finansdepartementet, 2017). In addi-tion to generating a higher expected return, illiquid assets will contributeto a good diversification of the porfolio making it more risk averse (Finans-departementet, 2017). An investigation made by McKinsey & Company,presented in Skr. 2016/17:130 (2017), clearly illustrates that investing in al-ternative assets have in fact been efficient, generating an annual mean returnof 12.3% during the time-span of 2001-2016 compared to the annual meanreturns of 6.9% as well as 5.1% generated by stock and interest-investments.Investing in alternative assets may even be beneficial for a portfolio in amarket that is characterized by low interest rates, which reflects diminishedreturns from interest funds (Skr. 2016/17:130, 2017).

In the early 2000’s, the market rate was in a decreasing trend (Finans-departementet, 2017). Purchasing a bond today with a time to maturityof 10 years would generate an annual return of 0.57%, while a bond with atime to maturity of two years would generate a negative annual return of-0.64% (Finansdepartementet, 2017). By complying with the current rule ofa minimum amount of 30% being invested in interest-bearing securities withlow credit and liquidity, the low generated expected returns would have tobe compensated by investments with a higher risk in order to achieve theset return targets (Finansdepartementet, 2017).

The reform proposals mentioned above have both drawn good and badattention. Riple (2017) claims that the reform proposals raised by the gov-ernment addresses the opposite of the measures that should actually betaken. According to Riple (2017), the time is inappropriate to switch fromlow to high risk investments, since it would result in a more aggressive port-folio seen from a risk perspective. Riple (2017) further explains that thereturn that has to be sacrificed in order to reduce the risk may have neverbeen this low, drawing the conclusion that it would be a strategically betteroption to ”...keep the least risky assets until they are reasonably paying toswitch to investments with a higher risk”.

2.3 Summary of Literature review

The Markowitz portfolio theory, MPT, refers to optimizing a portfolio’sreturn based on a given risk. An uncertain investor striving to create a low-

13

risk portfolio can center the portfolio around an intact risk, and further testdifferent asset allocations in order to identify which allocation that wouldgenerate the highest return for that given risk. However, the theory isconstructed in accordance with certain assumptions that might not alwaysbe generalized in the real market.

The MPT is fundamental in this study, which purpose is to find theoptimal asset allocation of the NWF that would maximize the return fora given risk limit. The MPT is consistent with strategic and tactical allo-cation approaches. The reason behind using a strategic asset allocation isto create an appropriate composition of assets that would give the portfolioa satisfying balance between the expected return and risk for a long terminvestment horizon. The assets performances will however lead to a devia-tion from the initial settings, and the allocations of the portfolio will haveto be periodically rebalanced. In order to hedge potential future risks, atactical asset allocation approach can be applied. Tactical asset allocationrefers to an active short term allocation management that benefits fromthe momentum coming from dynamic movements in the financial market.A memorandum written by the Swedish Ministry of Finance emphasizesthat the current strategic and tactical asset allocation limits of the First toFourth AP Funds, FFAPF, are outdated and have to be reformed in orderto maximize the return of the funds. The Ministry of Finance has submittedproposals for new strategic and tactical allocation limits, in which, accord-ing to an analysis made by Mckinsey & Company, would generate a higherreturn, given that the prudent man’s rule is followed.

The NWF may face a similar challenge as the FFAPF. Currently, theNWF insufficient funds to deploy nuclear power plants despite achieving theset return target, will need to change their strategic and tactical allocationas an expected interest rate increase will take place. The effects of an in-creasing interest rate is in turn reflected in decreasing bond yields. Sincethis study aims to invest in securities other than the ones that are allowed,while maintaining the risk limit, this study will, similar to the FFAPF, fol-low by the prudent man’s rule. Lundin emphasizes that the Nuclear WasteFund’s allocation policy will need to change, and that securities such ascorporate bonds and stocks should be allowed to be invested in, to hedgethe upcoming increasing interest rate. During the year of 2016, the NWFmade an attempt to convince the Swedish government to allow placementsin types of bonds other than what is currently permitted, referring primar-ily to corporate bonds and foreign bonds. The government chose howevernot to proceed with the proposal. One of the main purposes in this thesisis to test whether changing the allocation limits and extending the invest-ment options would have generated a higher average return. This wouldin turn lead to a change in the NWF strategic and tactical allocation sinceadditional assets would have to be considered in the NWF portfolio.

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Chapter 3

Research Design

3.1 General Approach

A quantitative study is pointed out by Bahari (2010) as a research methodin which the one performing the research applies post positivist claims inorder to develop the knowledge . This research method involves interpreta-tion of a vast amount of statistical data coming from experiments and otherpredetermined instruments that produces statistical data (Bahari, 2010).This type of research approach is thus considered appropriate for the thesisstudy, that will use historical and statistical data to optimize the historicalNWF portfolio. Furthermore, the study conducted in this thesis is of anexperimental character. According to Yanoff (2017), the difference betweenan observational study and an experimental study lies in the manipulationof the real variables of the study. An experimental study changes the realvariables of the study that are believed to have an impact on the outcome ofthe study (Yanoff, 2017). An observational study in turn does not manipu-late the real variables, and involves mainly observations of a set of featuresover a specific period (Yanoff, 2017).

Qualitative and quantitative research approaches are two different typesof research methods that can be incorporated in scientific studies (Bahari,2010). According to Bahari (2010), a qualitative research is defined as“. . . one in which the researcher usually makes knowledge claims based onconstructivist perspectives”, meaning that the research method mainly in-volves narratives, phenomenologies, grounded theory studies or case studies.In other words, a qualitative research method emphasizes words rather thanquantification. A qualitative research method is hence not considered anappropriate method for the study performed in this thesis.

The selected approach for the study is consistent with the deductiveresearch approach. So far, a theoretical framework has been presented,consisting of academic articles, electronic sources and technical reports. Thepurpose is to introduce the reader to the theory and concepts that concerns

15

the subject. On this basis, relevant data will be gathered and processedto obtain findings, which are expected to answer the research question. Asmentioned in the introductory section, the purpose is to find out how theNuclear Waste Fund’s portfolio could have been composed to give a higherreturn.

3.2 Research Strategy

The research strategy will consist of two chosen methods. The main methodapplied to the research is Markowitz Portfolio Theory, mentioned in sec-tion 2.1.1. MPT is selected due to its external validity and relevance for thethesis’ purpose. The general method of computing the return percentage ofa portfolio Rt at a given time t is defined by Equation (3.1).

Rt =Pt

Pt−1− 1 (3.1)

where Pt is the value of the portfolio at a given time t and Pt−1 the valueof the portfolio at time t− 1.

The MPT considers statistical measures such as the expected return,correlation and variance of each individual asset (Markowitz, 1952). Theexpected return of a portfolio E[Rp] containing n assets is thus computedby summarizing the product of wi, the relative amount invested in asset i,with its expected return defined as Ri in Equation (3.3). As each weightingfactor is a proportion of the whole portfolio, the following condition applies:

n∑i=1

wi = 1 (3.2)

Now let σ2p define the portfolio variance, and σi,j the covariance betweenRi and the expected return of asset j, Rj . The variance of the portfolio isthen computed according to Equation (3.4), and the standard deviation σpis defined by Equation (3.5).

E[Rp] =n∑

i=1

wiRi (3.3)

σ2p = V AR[Rp] =n∑

i=1

n∑j=1

σi,jwiwj (3.4)

σp =√V AR[Rp] (3.5)

However, MPT could only answer the research question to a certainextent, and needed thus to be completed with an additional method. The

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second method applied to the study was a polynomial regression model,which is a form of regression analysis used for assessing the return as afunction of the assets’ weight. The Ordinary Least Squares (OLS) method,was used since it is in terms of efficiency, consistency and unbiasedness, thebest linear unbiased estimator, BLUE, for linear regression models (Brooks,2008). By minimizing the sum of the squared residuals using OLS, theclosest fit between the function and the data points was obtained (Brooks,2008). A first order polynomial was first applied, and subsequently, a secondorder polynomial. The second order polynomial was clearly the best fit tothe data. A third degree order was not used, with the motivation that theuncertainty would increase and the second degree polynomial was accurateenough for the purpose of the study.

3.2.1 Two security categories

Consider the dependent variable Yi, the independent variable xi and thecoefficients β1, β2,...,βk+1. Then, in one variable, the polynomial regressionmodel can be expressed as follows (Jorgensen, 1993):

Yi = β1 + β2xi + β3x2i + ...+ βk+1x

ki + ε (3.6)

where ε is the error term of the model. Rewriting the equation as a systemof equations, expressing β as b, we obtain the following expressions:

−→Y =

Y1Y2...Yn

,Xn,k =

1 x1 · · · xk11 x2 · · · xk2...

.... . .

...1 xn · · · xkn

,−→b =

b1b2...bn

,−→ε =

ε1ε2...εn

As Equation (3.6) is valid for the population, the sample regression functioncan be expressed as:

Y = Xb (3.7)

The polynomial regression method was chosen because it is appropri-ate for complex non-linear models which can be described by polynomialfunctions (Jorgensen, 1993). When applied to this study, the polynomialfunction will illustrate the portfolio return as a (polynomial) function of theweight of security x.

The above model is useful for models of one independent variable. Thus,it cannot be directly applied to a model where both nominal and real bondsare accounted for (that is, two variables). However, Equation (3.2) states aconstraint that is valid for in this case. Thus, the model is reduced to onevariable, and becomes:

Rp = b1 + wib2 + w2i b3 + ...+ wk+1b

ki (3.8)

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where wi is the weight of either securities. The sample regression functioncan then be expressed as:

−→Rp = W

−→b (3.9)

where−→Rp is the array of portfolio returns and W is a matrix containing the

securities’ weights. The coefficients are then determined by solving for−→b :

−→b = (WTW)−1WT−→Rp

Having the coefficients, the polynomial regression model can be determined.The degree that will be applied will depend on the complexity of the datapoints’ trend.

The return values are simulated for different weight values in MATLABand stored in an array, with their corresponding weight values. Also, theconstraint that no portfolio exceeding 1.05 times the norm portfolio’s stan-dard deviation is applied. From that point, the coefficients are determinedfor the polynomial regression model.

3.2.2 Three security categories

When introducing a third security, it is not possible to use the polynomialregression model in one variable, even when using the constraint in Equa-tion (3.2) which reduces it to two variables. Hence, a polynomial regressionmodel in two variables is introduced. For simplicity, a polynomial of seconddegree will be used, and is expected to be sufficient for the polynomial fit.It can be expressed by the following equation:

Rp = b1 + b2w1 + b3w2 + b22w21 + b33w

22 + b23w1w2 (3.10)

Analogous to the two security categories method, the portfolio returns forthe three security categories were simulated in MATLAB and subsequentlyused to determine the polynomial regression model.

3.2.3 Several Inequality Constraints Optimization

The Lagrange Multiplier is a method utilized to find the extremas of amulti-variable function subject to constraints (Feldman, 2011). For the caseof optimizing a function f(x) with more than one inequality constraint andn variables of x, the function is subject to the following constraints (Nijhoff,2005):

g1(x) ≤ b1....gm(x) ≤ bmi = 1,m,

the Lagrangian is subsequently expressed as

18

L = f(x)− λ1g1(x)− ....− λmgm(x) (3.11)

whereas the n first order equality conditions are to be solved according tobelow (Nijhoff, 2005)

∂L∂x1

= 0, ....,∂L∂xn

= 0

Adding the m complementary slackness conditions, following expression isobtained (Nijhoff, 2005):

λ1(g1(x)− b1) = 0, ...., λm(gm(x)− bm) = 0

These equations are n + m equations for the n + m unknowns x1, ...., xn,λ1, ....λm. Solely solutions which fullfill the 2k inequalities according tobelow are admissible (Nijhoff, 2005).

λ1 ≥ 0, ...., λm ≥ 0, g1(x) ≤ b1, ...., gm(x) ≤ bmIf these are not fullfilled, it can either be explained by the fact that theextrema of the first order equalities either is outside the constraint domain,or that the Lagrange multiplier is of a negative profile. If there is howevera solution of the first order conditions, the next step would be to examinewhich of the constraints that are binding and which are not. Solely con-straints with λi > 0 are binding. In order do determine if the stationarypoint is a maxima, we will consider the bordered Hessian that is built fromthe Lagrangian and the binding constraints. Constraints that are not bind-ing are not considered when determining the character of the stationarypoint. (Nijhoff, 2005)

After the regression model of the return has been calculated, the La-grangian will be utilized in this study to optimize the polynomial regressionmodels subject to constraints due to the risk and allocation limits. Hence,the weight allocations in respective securities that generates the highest re-turns will be determined by using a Lagrangian Multiplier approach. Theconstraints for the two security regression model are expressed below:

w1 ≤ 1 (3.12)

w21σ

21 + (1− w1)

2σ22 + 2w1w2σ1,2 ≤ (1.05σP )2 (3.13)

where σP is the standard deviation of the norm portfolio. Using the two-security polynomial regression model as the function that will be optimized,The Lagrangian will be expressed as follows:

L(w1, λ1, λ2) = b1 + w1b2 + w21b3 − λ1(w1 − 1)

− λ2(w21σ

21 + (1− w1)

2σ22 + 2w1w2σ1,2 − (1.05σP )2) = 0(3.14)

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As for the three security polynomial equation expressed in Equation (3.10),the following four constraints are considered:

w1 + w2 ≤ 1 (3.15)

− w1 ≤ 0 (3.16)

− w2 ≤ 0 (3.17)

w21(σ21 − 2σ1,3 + σ23) + w2

2(σ21 − 2σ2,3 + σ23)

+ 2w1w2(σ1,2 − σ1,3 − σ2,3 + σ23) + 2w1(σ1,3 − σ23)

+ 2w2(σ2,3 − σ23) + σ23 ≤ (1.05σp)2

(3.18)

The constraints will be formulated as functions in the Lagrangian function,according to:

g1(w1, w2) = w1 + w2 − 1 (3.19)

g2(w1) = −w1 (3.20)

g3(w2) = −w2 (3.21)

g4(w1, w2) = w21(σ21 − 2σ1,3 + σ23) + w2

2(σ21 − 2σ2,3 + σ23)

+ 2w1w2(σ1,2 − σ1,3 − σ2,3 + σ23) + 2w1(σ1,3 − σ23)

+ 2w2(σ2,3 − σ23) + σ23 − (1.05σp)2

(3.22)

Now, using the three security polynomial equation as the function f(w1, w2)that is to be optimized, the complete Langrangian equation subject to thefour constraints is, according to Equation (3.11) expressed as:

L(w1, w2, λ1, λ2, λ3, λ4) = b1 + b2w1 + b3w2 + b22w21 + b33w

22

+ b23w1w2 − λ1(w1 + w2 − 1)− λ2(−w1)− λ3(−w2)− λ4(w21(σ21

− 2σ1,3 + σ23) + w22(σ21 − 2σ2,3 + σ23) + 2w1w2(σ1,2 − σ1,3 − σ2,3

+ σ23) + 2w1(σ1,3 − σ23) + 2w2(σ2,3 − σ23) + σ23 − (1.05σp)2) = 0

(3.23)

The optimization of the two and three securities polynomial regressions willbe performed using Wolfram Mathematica.

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3.2.4 Data Collection and Sample Size

Due to the nature of the research question, the research will be performed us-ing a quantitative data collection method. In order to calculate the NuclearWaste Funds historical returns, data of the OMRX TBOND and OMRXREAL bond indexes’ prices is collected. The data was collected from thefinance portal OnVista Group (OnVista, 2017). OnVista Group is part ofOnVista media GmbH, a subsidiary of the bank Comdirect, who in turn is asubsidiary of well known Commerzbank. It is one of the most visited financeportals in Germany (OnVista Media, 2017).

As for the S&P Sweden Investment Grade Corporate Bond Index, thedata for the index prices will be collected from the S&P Dow Jones Indicesdata base (S&P Dow Jones Indices, 2017b). The S&P Dow Jones Indeceshas received various international awards, some entitling it ”the best indexprovider” (S&P Dow Jones Indices, 2017a). As all data is available to thepublic, it can be accessed without the need of any special permission.

Finally, data regarding the OMXSPI stock index was collected from theNasdaq Nordic data base, found on the Nasdaq Inc official site (Nasdaq Inc,2017)

3.2.5 Data Analysis

Once the index prices were gathered, the annual returns could be calculatedusing Equation (3.1). The annual return was calculated for each year be-tween 2009 − 2016 (9 years). Here, Ri is the annual return year i, Pi theindex closing price year i and Pi−1 the index closing price year i− 1. Threedifferent portfolios consisting of different categories of securities were builtin order to simulate different outcomes for different compositions.

3.2.6 Method Criticism

Reliability

The reliability of a study is according to Denscombe (2016) dependent onwhether the study would generate the same results at a different occasion.The reliability can thus be analyzed based by looking at following aspects(Denscombe, 2016).

1. The study is performed under the same conditions, but at a lateroccasion. The results from both occasions are subsequently comparedto control whether they deviate or not.

2. The researcher should identify whether other researchers would sharethe same conclusions given that they were observing the same event.

In this study, by using the collected sample data, the regression modelsthat were built and optimized generated in-sample results. Hence, it is not

21

of importance whether the study is performed on a later occasion, since astudy with the same sample data would generate the same in-sample results.A study similar to the one in this thesis but with different sample data mightgenerate different results since the data would differ. Hence, the reliabilityof this study is considered overall high.

Validity

The validity of a study is an occurrence of a systematical weakness in theresearch design, and assesses the quality, accuracy and utility of the exper-imental design (Druckman et al., 2011). Validity is generally divided intotwo sub-groups; internal validity and external validity. The internal validitystudies the weakness of the tools, models or instruments used in the studywhile the external validity regards whether the results can be generalized insituations other than the ones in this study (Druckman et al., 2011).

One significant weakness has been identified. The study in this thesiswill be performed using the Markowitz theory. It is presumed that theMarkowitz portfolio theory is applicable given the assumptions mentionedin section 2.1.1. These assumptions have however been criticized since theydo not always correspond to the real market, hence it slightly reduces thedegree of validity. The other tools and models used in this study are howeverconsidered accurate and reliable.

3.3 Epistemological and ontological assumptions

Epistemology is defined as the theory of knowledge, and comprises of whatknowledge should be considered acceptable and how that knowledge is tobe obtained (Bahari, 2010). According to Bahari (2010), epistemological as-sumptions can be regarded as associated with the nature of knowledge as wellas the methods used to acquire that knowledge. Positivism is an epistemo-logical assumption that is based on the view that only information obtainedthrough observation is trustworthy (Bahari, 2010). The knowledge can inother words only be significant if it was acquired through observations. Ba-hari (2010) further claims that a researcher that performs a positivism studymust be independent, and its properties should be measured and obtainedusing objective methods such as gathering data and perform experimentalmethods. Using statistical analysis to get quantifiable findings is thus verycommon in positivism approaches (Bahari, 2010). The study in this thesiswill be performed experimentally, and will hence be of a positivist character.

Ontology studies the nature of reality, that is the theory of its existence,the theory of what exists in the world and how these entities interrelatewhich one other (Bahari, 2010). According to Saunders et al. (2007), theassumptions made by researchers are based on the way the world operatesas well as the commitment held to particular views. Saunders et al. (2007)

22

further mentions two aspects of ontology that are common within the fieldof business and management; objectivism and subjectivism. Objectivism isbased on the assumptions that social entities exist in reality, independentof social actors (Saunders et al., 2007). Subjectivism on the hand is basedon assumptions that “. . . social phenomena are created from the perceptionsand consequent actions of social actors” (Saunders et al., 2007).

3.4 Research ethical reflection

Ethics refers to building, stimulating and maintaining an awareness anda discussion about how to act as an individual (Vetenskapsradet, 2017).Although certain issues related to ethics must be regulated formally, ethicsdeals not only with laws and regulations (Vetenskapsradet, 2017). Ethicalaspects are particularly important in research since it has a major long termimpact on society (Vetenskapsradet, 2017). Thus, the ultimate responsibilitylies within the researcher, to ensure that the research is of good quality andis morally acceptable (Codex, 2017).

It is thus of great importance not to leave any room for any types ofmisinterpretations. The assumptions made in this study are clearly statedin the report. Furthermore, it should be emphasized that this study doesnot aim to explain how the fund should invest in the future. The studyhas rather given the NWF the basis and perspective of how its historicalinvestments could have been made, which could further be of help for futureinvestments.

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Chapter 4

Analysis and Findings

4.1 Empirical presentation and Results

The three portfolios that were composed for the simulations and analysisare listed in Table 4.1. The notation will be used henceforth to refer to therespective portfolio.

Table 4.1: Compositions of the studied portfolios.Notation Securities

A OMRX REAL OMRX TBOND -

B OMRX REAL OMRX TBOND S&P Corp

C OMRX REAL OMRX TBOND OMXS PI

Since portfolio A consists of solely 2 securities, a polynomial regressionconsisting of two securities was performed. The regression model is intro-duced in subsection 4.1.1. The other portfolios, that is portfolio B and C,consist of three securities, and the regression model will therefore consist ofthree securities. The regression model for Portfolio B and C are presentedin subsection 4.1.2 nad subsection 4.1.3, respectively.

In Table 4.2, the closing prices of the last trading day each year between2008 and 2016 of each security are listed. The reason behind choosing thelast trading day every year is for the consistency in the calculations. Fur-thermore, this data was used to calculate the annual returns of each securityindex according to Equation (3.1). The results are listed in Table 4.3.

24

Table 4.2: Years and closing prices of the last trading day each year between2008 and 2016 in respective indexes [SEK].

Year OMRX REAL OMRX TBOND S&P CORP OMXSPI

2008 4865.73 5359.92 171 204.22

2009 5032.71 5309.63 182.59 299.5

2010 5271,02 5465.69 186.89 368.54

2011 5939.89 6191.04 197.81 307.04

2012 5988.64 6288.16 208.9 343.94

2013 5679.85 6081 214.75 423.66

2014 6098.84 6820.72 229.77 473.89

2015 6209.74 6805.52 229.62 505.13

2016 6657.91 7082.77 236.46 534.56

Table 4.3: Dates and returns each year between 2009 and 2016 in respectiveindexes [%].

Date OMRX REAL OMRX TBOND S&P CORP OMXSPI

2009 3.43 -0.94 6.78 4.66

2010 4.73 2.94 2.36 23.05

2011 12.69 13.27 5.84 -16.69

2012 0.82 1.57 5.6 12.01

2013 -5.15 -3.3 2.80 23.18

2014 7.38 12.16 6.99 11.85

2015 1.81 -0.22 -0.07 6.60

2016 7.22 4.07 2.98 5.82

Mean 4.00 3.54 4.13 8.12

Table 4.4 demonstrates the correlation matrix, which investigates thedependence between the indexes’ returns in the NWF portfolio. The closerthe correlation is to -1 and 1 respectively, the higher the dependence isbetween the returns of the indexes. Now utilizing the Evans’ correlationscale for the absolute value of the correlation ρ, following conditions apply(Evans, 1996):

• 0.0 ≤ ρ ≤ 0.19 - The correlation is very weak

• 0.20 ≤ ρ ≤ 0.39 - The correlation is weak

• 0.40 ≤ ρ ≤ 0.59 - The correlation is moderate

25

• 0.60 ≤ ρ ≤ 0.79 - The correlation is strong

• 0.80 ≤ ρ ≤ 1.0 - The correlation is very strong

As can be seen, the REAL and TBOND indexes have a very strong correla-tion, followed by REAL and OMXSPI that have a strong correlation. Thecorrelation is moderate between TBOND and S&P Corp as well as TBONDand OMXSPI, and weak between S&P CORP and OMXSPI as well as S&PCorp and REAL.

Table 4.4: Correlation matrix of the returns of the indexes between 2009and 2016.

REAL TBOND S&P CORP OMXSPI

REAL 1.00

TBOND 0.87 1.00

S&P CORP 0.37 0.49 1.00

OMXSPI -0.74 -0.58 -0.33 1.00

4.1.1 Portfolio A

Figure 4.1: 2D plot of simulated data points of Portfolio A’s returns andcorresponding weight-% of OMRX REAL

26

In Figure 4.1, the obtained data points from the simulation in MATLAB arepresented in a 2D plot. Applying the polynomial regression method to thedata points, according to Equation (3.8), a second degree polynomial curvewas fitted to the data points. It was concluded that a second degree curvewould be sufficient to fit the data points, as the curve was almost linear.This is also depicted by the polynomial regression equation:

Rp,A = −3.7 · 10−6w21 + 0.049w1 + 3.5 (4.1)

where w1 is denoted as the weight invested in OMRX REAL.When optimizing the polynomial equation to obtain the maximal return,

it must be done according to the constraints as in Equation (3.12) andEquation (3.13). The following constraints are considered:

g1,A(w1) = w1 − 1 (4.2)

g2,A(w1) = 3.6 · 103(w1 − 1.0)2 + 2.8 · 10−3w21

− 5.6 · 10−3w1(w1 − 1.0)− 3.5 · 10−3(4.3)

These constraints, along with Equation (4.1), gives the Lagrange equation:

L(w1, g1,A, g2,A) = 0.49w1 − λ1(w1 − 1)− λ2(0.0036(w1 − 1)2 + 0.0028w21

− 0.0056w1(w1 − 1)− 0.0035)− 0.037w21 + 3.5

(4.4)

Finally, the optimization is run on Wolfram Mathematica, giving the result:

Rp,1 ≈ 4.00

forw1 = 1.00

4.1.2 Portfolio B

In MATLAB, all the data points of portfolio B’s returns were simulated fordifferent compositions with respect to the assets’ weights. These were after-wards plotted in 3-dimensional graphs, as showed in Figure 4.2, Figure 4.3and Figure 4.4. The color shades simulates a fourth dimension in order todepict another property. A total of 500856 data points were simulated andused for the plots.

After inserting the data points for the returns of the indexes that arepart of Portfolio B, the plot in Figure 4.2 was fitted using the polynomialfit model described in Equation (3.10). The third security in this portfolio,

27

Figure 4.2: 3D plot of simulated data points of Portfolio B’s return andcorresponding weight-% of OMRX REAL and OMRX TBOND.

that is, S&P CORP, will be denoted as w3. In order to evaluate the weightcombinations that generate the maximal return of Portfolio B, the followingfunction will be optimized using the Lagrange Multiplier method:

Rp,B(w1, w2) = 0.041− 3.20 · 10−4w1 − 0.0047w2

− 0.0010w21 − 0.0018w2

2 − 0.0011w1w2

(4.5)

The polynomial function in Equation (4.5) is subject to the following con-straints:

g1,B(w1, w2) = w1 + w2 − 1 (4.6)

g2,B(w1, w2) = 0.0025w21 + 0.0043w1w2 − 2.7 · 10−4w1

+ 0.0028w22 + 2.2 · 10−4w2 − 0.003

(4.7)

g3,B(w1) = −w1 (4.8)

g4,B(w2) = −w2 (4.9)

28

Figure 4.3: 3D plot of simulated data points of Portfolio B’s assets’ weight-%and standard deviation.

Inserting the polynomial function and the constraints in the Lagrangiangives:

L(w1, w2, λ1, λ2, λ3, λ4) = 0.041− 3.20 · 10−4w1 − 0.0047w2

− 0.0010w21 − 0.0018w2

2 − 0.0011w1w2 − λ1(w1 + w2 − 1)

− λ2(0.0025w21 + 0.0043w1w2 − 2.7 · 10−4w1 + 0.0028w2

2

+ 2.2 · 10−4w2 − 0.003)− λ3(−w1)− λ4(−w2) = 0

(4.10)

Subsequently, Equation (4.10) was optimized using Wolfram Mathematica,which generated the following results:

Rp,B ≈ 4.13%

forw3 = 1.00

29

Figure 4.4: 3D plot of simulated data points of Portfolio B’s assets’ weight-%and return.

4.1.3 Portfolio C

Analogous to Portfolio B, the data points of Portfolio C’s returns were sim-ulated in MATLAB. However, for convenience regarding the analysis, twoplots were generated; a 3-dimensional graph and a 2-dimensional graph, pre-sented in Figure 4.5 and Figure 4.6, respectively. In the same manner asportfolio B, a total of 500856 data points were simulated.

In the regression model used in this study, OMXSPI is denoted asw3. The polynomial regression model of Figure 4.5 is according to Equa-tion (3.10) expressed as:

Rp,C(w1, w2) = 0.13− 0.069w1 − 0.072w2 − 0.00019w21

− 0.00020w1w2 − 0.00039w22

(4.11)

When optimizing the polynomial equation to determine the maximal returnof portfolio C, the following two constraints in which the equation is subjectto, have to be considered:

g1,C(w1, w2) = w1 + w2 − 1 (4.12)

30

Figure 4.5: 3D plot of simulated data points of Portfolio B’s return andcorresponding weight-% of OMRX REAL and OMRX TBOND.

g2,C(w1, w2) = 0.046w21 + 0.096w1w2 − 0.077w1 + 0.051w2

2

− 0.08w2 + 0.029(4.13)

g3,C(w1) = −w1 (4.14)

g4,C(w2) = −w2 (4.15)

Which leads to the following Lagrange equation:

L(w1, w2, λ1, λ2, λ3, λ4) = 0.13− 0.069w1 − 0.072w2 − 0.00019w21

− 0.020w1w2 − 0.039w22 − λ1(w1 + w2 − 1)− λ2(0.046w2

1

+ 0.097w1w2 − 0.077w1 + 0.051w22 − 0.08w2 + 0.029)− λ3(−w1)

− λ4(−w2)

(4.16)

Equation (4.16) was optimized using Wolfram Mathematica, which gener-ated the following results:

Rp,C ≈ 7.93%

31

Figure 4.6: 2D plot of portfolio C with the weight-% of OMRX REAL andOMRX TBOND.

forw1 = 42.7, w2 = 17.1, w3 = 40.2

4.1.4 Summary of results

Table 4.5: Optimal weight compositions of portfolios. Values of portfolioA-C are taken from the simulation [%].

Portfolio OMRX REAL OMRX TBOND S&P Corp OMXS PI

Norm 30 70 - -

A 100 0 - -

B 0 0 100 -

C 42.7 17.1 - 40.2

32

Table 4.6: Comparison of portfolio annual returns and mean return (calcu-lated as the geometric mean of years 2009-2016) [%]. The ”active return”indicates the mean return above benchmark for the stated time period. TheNuclear Waste Fund’s actual return is also presented.

Year A B C NFW return Norm portfolio

2009 3.43 6.78 19.58 2.5 0.37

2010 4.74 2.36 11.60 3.0 3.48

2011 12.69 5.84 0.76 9.37 13.10

2012 0.82 5.61 5.75 4.60 1.34

2013 -5.16 2.80 7.32 -0.57 -3.85

2014 7.38 6.99 11.14 10.69 10.73

2015 1.82 -0.07 2.98 -0.52 0.39

2016 7.22 2.98 5.39 5.76 5.02

Mean return 4.00 4.13 7.93 4.28 3.69

Active return 0.31 0.44 4.24 0.59 -

4.2 Analysis

According to the findings, Portfolio A, which consists of two types of assets,has a maximal mean return equal to 4.00% for the time period 2009-2016.The value of the return exceeds that of the norm portfolio by 0.31 percentagepoints. As this corresponds to a weight w1 = 1, it means that investing 100%in the OMRX REAL asset would give the maximal return. This can also beseen in Figure 4.1. This means an extension of the investment limit equalto 70 percentage points from that of the norm portfolio (see Table 1.2),and a 50 percentage points extension from the allocation limit stated inthe Nuclear Waste Fund’s investment policy (see Table 1.1). In order tounderstand why this is the case, the constraints must be examined. Oneof them, which is expressed in Equation (4.2), tells nothing else than thata maximal of 100% can be invested, which is trivial. The next constraint,expressed in Equation (4.3), means that a maximal standard deviation equalto 1.05 times the standard deviation of the norm portfolio’s return willbe tolerated. Therefore, as a 100% investment in OMRX REAL does notviolate the risk limit according to the calculations, the maximal return ofthe portfolio is simply equivalent to that of the index itself, as presented inTable 4.3. Would a 100% investment in OMRX REAL have meant a higherstandard deviation than the limit, the results would have obviously beendifferent. Furthermore, as can been seen in Figure 4.1, the second degreepolynomial regression is accurate.

Regarding Portfolio B, the maximal mean return was found to be 4.13%

33

during 2009-2016. This value exceeds that of the benchmark return by0.44 percentage points. The corresponding allocation distribution were com-prised weights of w1 = w2 = 0 and thus w3 = 1. Therefore, a 100% allocationin S&P Swedish Corporate Bonds would generate the maximal return, un-der the defined circumstances. This is depicted in Figure 4.1 as well. Therisk limit is not either violated. In fact, Figure 4.3 and Figure 4.4 illustratesthat solely investing in S&P Swedish Corporate Bonds generates the lowestrisk. In contrast to the results of Portfolio A, this would not only meanan extension to the allocation limits from that of the norm portfolio and tothat of the Nuclear Waste Fund’s investment policy, but it would also implyan extension of the investment options, as corporate bonds do not form apart of the investment options according to the policy. However, the resultsprove that such an extension would imply a higher return on average forthe defined time period. The results would also follow the same explanationas Portfolio A; the constraints in subsection 4.1.2 give rise to the obtainedresults. This means that a total investment in OMXSPI does not violatethe risk limit of 1.05 times the standard deviation of the norm portfolio’sreturn.

As for Portfolio C, the maximal mean return for the period 2009-2016 wasfound to be 7.93%, outperforming the benchmark return by 4.24 percent-age points. The composition of this portfolio was calculated to the weightsw1 = 42.7%, w2 = 17.1% and w3 = 40.2%, meaning that 42.7% is investedin OMRX REAL, 17.1% in OMRX TBOND and 40.2% in OMXSPI. How-ever, this result is rather vague, and not too precise. In Figure 4.6, it canbe observed that there is a linearity between the OMRX TBOND weightand the OMRX REAL weight. This linearity indicates that the OMXSPIweight remains constant. The return peaks whenever the weight sums onthe bottom line as well as the top line of the OMRX TBOND and OMRXREAL are 60% and 100%.

34

Chapter 5

Discussion and criticalreflection

As introduced in chapter 4, the three portfolios that were built outperformedthe norm portfolio during the time period 2009-2016. Considering the find-ings and analysis, this suggests that an extension of the allocation limits aswell as an extension of the investment options would have significantly im-proved the average return compared to the norm portfolio, without increas-ing the risk. It should be pointed out that this does not indicate that suchcompositions would necessarily outperform the norm portfolio in the future,as only historical data have been studied. The results are in this contextin-sample results. Additionally, in spite the fact that all three portfolios out-performed the norm portfolio, only Portfolio C managed to outperform theactual NWF portfolio. In other words, the NWF’s portfolio outperformedboth Portfolio A and B. However, the NWF portfolio is subject to not onlytactical asset allocation but also the selection of instruments, whereas thebuilt portfolio’s only consider the asset allocation.

The MPT emphasizes that risk aversion can be obtained by reducing thecorrelation between the returns of the selected securities in a portfolio. Bydoing this, the securities will more likely not move in the same direction.As presented in Table 4.4, the correlation between the securities in portfolioA is very strong, which according to Markowitz, implies that the risk inPortfolio A is generally high. Looking at portfolio B, that has the lowestcorrelation of all three portfolios, the optimal weight allocation giving themaximal return also generates the lowest risk. This portfolio constitutes agood example for risk-averse investors. Now given that the maximal returnof Portfolio A is the lowest of all three portfolios while generating a highrisk, it should according to the MPT be excluded as an investment option.In fact, one of the seven assumptions mentioned in subsection 2.1.1 regardsthat an investor will only invest with a higher risk if the outcome of theinvestment is a higher expected return. Such is the case in Portfolio C, that

35

has the second highest correlation of all three portfolios, but generates thehighest expected return of 7.93%.

However, the assumptions in which the MPT is based on, do not reflectthe real market, as mentioned in subsection 2.1.1. One of the assumptionssaying that taxes and transaction costs are excluded, is one that cannotbe applied for the NWF since using a tactical asset allocation approachrequires multiple transactions, followed by transaction fees. Subsequently,Eychenne et al. (2013), emphasizes that relying on historical figures whencomputing the expected return is not sufficient since the market is dynamicand undergoes changes, the same historical trend is not necessarily alwaysactive. The future expected return can therefore not solely rely on the resultsin this study.

As for the methods applied in this study, a polynomial regression wasperformed, whereas the polynomial functions of each portfolio were opti-mized using a number of constraints. The constraints were taken from theMPT and are derived in section 3.2. Given that an optimization of the NWFhistorical return was to be made, cross-fertilizing a polynomial regressionanalysis with the constraints from the MPT yielded satisfying results, andwas a rather important process in order to perform this study and fulfill itspurpose. Furthermore, solely applying a strategic asset allocation approachmay as mentioned in subsection 2.1.2 fail to provide adequate protectionduring periods where the market undergoes crises and shocks. This is dueto the correlation between the assets converging towards 1.0. The optimalweight compositions of portfolios A to C that were computed in chapter 4,correspond to the strategic allocation that would yield the highest return.However, in future investments, the fund must apply a tactical allocation tohedge potential market fluctuations.

The knowledge contribution provided by this study is suitable for fundsthat, among other things, seek to optimize their returns while maintainingthe risk limit. We also show that managers can benefit from reviewing theirallocation limits over time. The Nuclear Waste Fund, whose agenda, from asocietal perspective, is to prevent and handle potential environmental dam-ages caused by nuclear power plants, can use this study to gain perspectiveof how its historical investments could have been made in order to increasethe historical return. This can further help the fund and provide betterconditions for its future investments.

36

Chapter 6

Conclusion

The purpose of the Nuclear Waste Fund is to manage nuclear waste and de-commission nuclear power plants. The fund, which is subject to strict rulesregarding allocation and securities selection, has achieved its average returntarget. Nevertheless, the fund’s capital is still insufficient to fulfill its pur-pose as it has a capital deficit. In addition, the interest rate that is expectedto increase, might lead to a decreasing bond yield followed by a lower nom-inal return by investing only in bonds carried out by the state. The NWFhas therefore proposed to extend the investment options by investing in cor-porate bonds and stocks. The proposal was however rejected by the Swedishgovernment. The purpose of this study is to investigate how and if the port-folio could perform better by changing the allocation limits, and extendingthe investment options, while maintaining the risk limit. Therefore, threeportfolios were built; Portfolios A, B and C. Portfolio A, containing OMRXREAL and OMRX TBOND indexes, generated an average return of 4% be-tween 2009-2016. Portfolio B, containing OMRX REAL, OMRX TBONDand S&P Corp indexes, generated an average return of 4.13%, and finally,Portfolio C, consisting of OMRX REAL, OMRX TBOND and OMXSPIindexes, generated an average return of 7.93%. The three portfolios outper-formed the NWF norm portfolio, and it is thus concluded that by extendingthe investment options and changing the allocation limits, the profit couldincrease by 0.31, 0.44 and 4.24 percentage points respectively, without vi-olating the set risk limit, between the time period 2009-2016. The resultstherefore suggests that revisions of the allocation limits over time couldbenefit the Nuclear Waste Fund’s profit.

This study was conducted on the basis of a number of auxiliary assump-tions, whereas one of the main assumption to consider was that the specifiedrisk limit of the NWF was the most optimal one in order for the NWF tofulfill its objectives. This might however not be the case. Therefore, theprofit that comes from changing the asset allocation rules might changegiven that the risk limit is different. A potential continuation study to this

37

thesis would be to determine whether the NWF set risk limits are optimalin order for the NWF to achieve its objective of yielding a sufficient averagereturn to fulfill its objectives. Furthermore, the thesis does not take intoaccount the effects of investing in other alternative securities such as foreignbonds, to exemplify. The effects of including foreign bonds in the NWFwould rather be an interesting approach to observe, and may be a potentialstudy to conduct in future research.

38

Chapter 7

Limitations of Research

Due to mainly time limitations, the research in the study was limited to onlystudy three different portfolios. A more thorough study could be made wereportfolios contain larger variations of asset classes. Furthermore, the port-folio compositions were simulated under the constraint regarding the risklimit. Once more, the study could have covered the portfolios disregardingthe risk constraint had there been more time.

Additionally, when performing the simulations, a finite number of datapoints were generated. Had this number of data points been extended, moreaccurate results could probably have been obtained. However, due to limitedprocessing capacity in the available computer systems, it was not reasonableto use extend the number of data points, as the computation time wouldhave been unreasonably large.

39

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42

clc; clear all; close all

Rr= [0.034317564 ...

0.047352222 0.126895743 0.008207223 -0.051562625 ...

0.073767793 0.018183786 0.072172104];

Rn= [-0.009382603 ...

0.029391879 0.132709685 0.015687187 -0.032944454 ...

0.121644466 -0.002228504 0.040738988];

%% Standard deviation for 70/30 portfolio

rptest=0.3*Rr+0.7*Rn;

sd=std(rptest);

%% Algorithm

w=[0:0.01:1];

% onev=ones(length(Rn),1)

i=1;

y=1;

for i=i:length(w)

Rpi= w(i)*Rr+(1-w(i))*Rn;

sdi=std(Rpi);

stdev2(i)=std(Rpi);

if sdi<=1.05*sd

Appendix A

Matlab script of Portfolio A

43

Rpi=Rpi+1;

Rp(y)=geomean(Rpi)-1;

wr(y)=w(i);

wn(y)=1-wr(y);

stdev(y)=std(Rpi);

y=y+1;

end

i=i+1;

end

% scatter3(wr*100,wn*100,Rp*100,'.')

% xlabel('OMRXREAL weight-%')

% ylabel('OMRXTBOND weight-%')

% zlabel('Portfolio return (%)')

figure()

plot(wr*100,Rp*100,'.','MarkerSize',10)

xlabel('OMRXREAL weight-%')

ylabel('Portfolio return (%)')

title('Portfolio A')

a=wr';

a2=ones(length(wr), 1);

X=[a2, a, a.^2];

Y=Rp';

sol=X\Y

b1 = sol(1); b2 =sol(2); b3 = sol(3);

syms w1 lambda1 lambda2

RP= 100*(b1 + b2*w1 + b3*w1^2);

% -3.6764e-6*x^2 + 0.0048912*x + 3.5453

%% Max returns

[MAXRETURN, position] = max(Rp);

w1formax = wr(position);

w2formax = 1-w1formax;

annualreturns = (w1formax*Rr+w2formax*Rn)';

44

meanreturn = (geomean(annualreturns+1)-1)*100

format bank

annualreturns*100

format long

%%

var1 = var(Rr);

var2 = var(Rn);

cov12 = cov(Rr,Rn);

cov12 = cov12(1,2);

constraint1 = w1-1;

constraint2= w1^2*var1+(1-w1)^2*var2 + 2*w1*(1-w1)*cov12

- 1.05^2*sd^2;

Lagrange = RP - lambda1 * constraint1 - lambda2*

constraint2

45

clc; clear all; close all; format long

Rr= [0.034317564 0.047352222 0.126895743 0.008207223 -

0.051562625 ...

0.073767793 0.018183786 0.072172104];

Rn= [-0.009382603 0.029391879 0.132709685 0.015687187 -

0.032944454 ...

0.121644466 -0.002228504 0.040738988];

Rc = [0.067777778 0.02355003 0.058430093 0.0560639

0.02800383...

0.069941793 -0.000652827 0.029788346];

rptest1=0.246*Rr+0.3586*Rn+(1-0.246-0.3586)*Rc;

%% Standard deviation for 70/30 portfolio

rptest=0.3*Rr+0.7*Rn;

sd=std(rptest);

%% Algorithm

w=[0:0.001:1];

i=1;

y=1;

for i=i:length(w)

Appendix B

Matlab script of Portfolio B

46

w1=w(i);

alpha = 1-w1;

w2_temp =[0:0.001:alpha];

j=1;

for j=j:length(w2_temp)

w2=w2_temp(j);

w3=alpha-w2;

Rpi=w1*Rr+w2*Rn+w3*Rc;

sdi=std(Rpi);

if sdi<=1.05*sd

Rpi=Rpi+1;

Rp(y)=geomean(Rpi)-1;

wr(y)=w1;

wn(y)=w2;

stdev(y)=std(Rpi);

y=y+1;

end

end

i=i+1;

end

% cftool(wr*100,wn*100,Rp*100, stdev)

x1=wr';

x2=wn';

onearray=ones(length(wr), 1);

X=[onearray, x1, x2, x1.^2, x2.^2, x1.*x2];

Y=Rp';

coeffs=X\Y;

b1=coeffs(1); b2=(coeffs(2));

b3=coeffs(3);b22=coeffs(4);b23=coeffs(5);

b33=coeffs(6); % polynomial regression coefficients

47

%% Max returns

[MAXRETURN, position] = max(Rp);

w1formax = wr(position);

w2formax = wn(position);

w3formax = 1-w1formax-w2formax;

annualreturns = (w1formax*Rr+w2formax*Rn+w3formax*Rc)';

meanreturn = (geomean(annualreturns+1)-1)*100

format bank

annualreturns*100

format long

%% Plots

figure()

markerSize = 10;

scatter3(wr*100,wn*100,Rp*100,markerSize, stdev,'.')

title('Portfolio B')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('Portfolio return (%)')

c = colorbar;

c.Label.String = 'Standard deviation';

ylim([0 100])

figure()

pointsize = 10;

scatter(wr, wn, pointsize, stdev)

colorbar

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

h = colorbar;

ylabel(h, 'Standard deviation')

title('Portfolio B')

markerSize = 10;

scatter3(wr*100,wn*100,stdev,markerSize, 100*Rp,'.')

title('Portfolio B')

48

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('Standard deviation')

c = colorbar;

c.Label.String = 'Portfolio return (%)';

figure()

title('Portfolio B')

pointsize = 10;

scatter(wr, wn, pointsize, Rp*100)

colorbar

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

h = colorbar;

ylabel(h, 'Return (%)')

figure()

ws=1-wn-wr;

markerSize = 10;

scatter3(wr*100,wn*100,ws*100,markerSize, Rp*100,'.')

title('Portfolio B')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('S&P Corp weight-%')

c = colorbar;

c.Label.String = 'Return (%)';

ylim([0 100])

figure()

ws=1-wn-wr;

markerSize = 10;

scatter3(wr*100,wn*100,ws*100,markerSize, stdev,'.')

title('Portfolio B')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

49

zlabel('S&P Corp weight-%')

c = colorbar;

c.Label.String = 'Standard deviation';

ylim([0 100])

%% Part 2 - Optimization

var1 = var(Rr);

var2 = var(Rn);

var3 = var(Rc);

cov12 = cov(Rr,Rn);

cov12 = cov12(1,2);

cov13 = cov(Rr,Rc);

cov13 = cov13(1,2);

cov23 = cov(Rn,Rc);

cov23 = cov23(1,2);

syms w1 w2 lambda1 lambda2 lambda3 lambda4

Rp = b1+b2*w1+b3*w2+b22*w1^2+b23*w1*w2+b33*w2^2;

g1 = w1+w2-1;

g2 = (w1^2*(var1 - 2*cov13 + var3) +w2^2*(var2 - 2*cov23

+ var3)+...

+2*w1*w2*(cov12-cov13-cov23+var3)+2*w1*(cov13-

var3)+...

+2*w2*(cov23-var3)+var3) - 1.05^2* sd^2;

%0.0464*x^2 + 0.0965*x*y - 0.0765*x + 0.051*y^2 -

0.0802*y + 0.0329<=0.003199741654098

g3 = -w1;

g4 = -w2;

Lagrange = Rp - lambda1 * g1 - lambda2 *g2 - lambda3 *

g3 - lambda4 * g4;

% close

% Optimality conditions

dLdw1 = vpa(diff( Lagrange, w1),3) ==0; %%%%% condition

1

50

dLdw2 = vpa(diff( Lagrange, w2),3) ==0; %%%%% condition

2

condition3 = lambda1 * g1 == 0; %%% condition 3

condition4 = lambda2 * g2 == 0 ; %%%% condition 4

condition5 = lambda3 * g3 == 0 ; %% condition 5

condition6 = lambda4 * g4 == 0; %% condition 6

51

clc; clear all; close all; format long

Rr= [0.034317564 0.047352222 0.126895743 0.008207223 -

0.051562625 ...

0.073767793 0.018183786 0.072172104];

Rn= [-0.009382603 0.029391879 0.132709685 0.015687187 -

0.032944454 ...

0.121644466 -0.002228504 0.040738988];

Rs = [0.466555675 0.230517529 -0.166874695

0.120179781...

0.231784614 0.118562054 0.065922471 0.05826223];

rptest1=0.246*Rr+0.3586*Rn+(1-0.246-0.3586)*Rs;

%% Standard deviation for 70/30 portfolio

rptest=0.3*Rr+0.7*Rn;

sd=std(rptest);

%% Algorithm

w=[0:0.001:1];

i=1;

y=1;

for i=i:length(w)

Appendix C

Matlab script of Portfolio C

52

w1=w(i);

alpha = 1-w1;

w2_temp =[0:0.001:alpha];

j=1;

for j=j:length(w2_temp)

w2=w2_temp(j);

w3=alpha-w2;

Rpi=w1*Rr+w2*Rn+w3*Rs;

sdi=std(Rpi);

if sdi<=1.05*sd

Rpi=Rpi+1;

Rp(y)=geomean(Rpi)-1;

wr(y)=w1;

wn(y)=w2;

stdev(y)=std(Rpi);

y=y+1;

end

end

i=i+1;

end

% cftool(wr*100,wn*100,Rp*100, stdev)

x1=wr';

x2=wn';

onearray=ones(length(wr), 1);

X=[onearray, x1, x2, x1.^2, x2.^2, x1.*x2];

Y=Rp';

coeffs=X\Y;

b1=coeffs(1); b2=(coeffs(2));

b3=coeffs(3);b22=coeffs(4);b23=coeffs(5);

b33=coeffs(6);

53

%% Max returns

[MAXRETURN, position] = max(Rp);

w1formax = wr(position);

w2formax = wn(position);

w3formax = 1-w1formax-w2formax;

annualreturns = (w1formax*Rr+w2formax*Rn+w3formax*Rs)';

meanreturn = (geomean(annualreturns+1)-1)*100

format bank

annualreturns*100

format long

%% Plots

figure()

markerSize = 10;

scatter3(wr*100,wn*100,Rp*100,markerSize, stdev,'.')

title('Portfolio C')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('Portfolio return (%)')

c = colorbar;

c.Label.String = 'Standard deviation';

ylim([0 100])

figure()

pointsize = 10;

scatter(wr, wn, pointsize, stdev)

colorbar

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

h = colorbar;

ylabel(h, 'Standard deviation')

title('Portfolio C')

markerSize = 10;

scatter3(wr*100,wn*100,stdev,markerSize, 100*Rp,'.')

title('Portfolio C')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

54

zlabel('Standard deviation')

c = colorbar;

c.Label.String = 'Portfolio return (%)';

figure()

pointsize = 10;

scatter(wr, wn, pointsize, Rp*100)

title('Portfolio C')

colorbar

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

h = colorbar;

ylabel(h, 'Return (%)')

figure()

ws=1-wn-wr

markerSize = 10;

scatter3(wr*100,wn*100,ws*100,markerSize, Rp*100,'.')

title('Portfolio C')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('OMXSPI weight-%')

c = colorbar;

c.Label.String = 'Return (%)';

ylim([0 100])

figure()

ws=1-wn-wr

markerSize = 10;

scatter3(wr*100,wn*100,ws*100,markerSize, stdev,'.')

title('Portfolio C')

xlabel('OMRXREAL weight-%')

ylabel('OMRXTBOND weight-%')

zlabel('OMXSPI weight-%')

c = colorbar;

55

c.Label.String = 'Standard deviation';

ylim([0 100])

%% Part 2 - Optimization

var1 = var(Rr);

var2 = var(Rn);

var3 = var(Rs);

cov12 = cov(Rr,Rn);

cov12 = cov12(1,2);

cov13 = cov(Rr,Rs);

cov13 = cov13(1,2);

cov23 = cov(Rn,Rs);

cov23 = cov23(1,2);

syms w1 w2 lambda1 lambda2 lambda3 lambda4

Rp = b1+b2*w1+b3*w2+b22*w1^2+b23*w1*w2+b33*w2^2;

g1 = w1+w2-1;

g2 = (w1^2*(var1 - 2*cov13 + var3) +w2^2*(var2 - 2*cov23

+ var3)+...

+2*w1*w2*(cov12-cov13-cov23+var3)+2*w1*(cov13-

var3)+...

+2*w2*(cov23-var3)+var3) - 1.05^2* sd^2;

%0.0464*x^2 + 0.0965*x*y - 0.0765*x + 0.051*y^2 -

0.0802*y + 0.0329<=0.003199741654098

g3 = -w1;

g4 = -w2;

Lagrange = Rp - lambda1 * g1 - lambda2 *g2 - lambda3 *

g3 - lambda4 * g4;

% close

% Optimality conditions

dLdw1 = vpa(diff( Lagrange, w1),3) ==0; %%%%% condition

1

dLdw2 = vpa(diff( Lagrange, w2),3) ==0; %%%%% condition

2

condition3 = lambda1 * g1 == 0; %%% condition 3

condition4 = lambda2 * g2 == 0 ; %%%% condition 4

56

condition5 = lambda3 * g3 == 0 ; %% condition 5

condition6 = lambda4 * g4 == 0; %% condition 6

57

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