!

A Quantitative Study of OptimalAsset Allocation in a Mean-CVaR

& Mean-Variance FrameworkAuthor:

Marcus Nilsson

Supervisors:

Lisa Hed, Umeå UniversitetJörgen Olsén, Sirius International

Marcus Nilsson25th June 2014Examensarbete, 30hpCivilingenjörsprogrammet i industriell ekonomi, risk management, 300 hp

Abstract

Optimal portfolio selection has been an area of great focus ever sincethe inception of modern portfolio theory as proposed by Harry Markowitz.This project has applied Markowitz modern portfolio theory to an invest-ment universe created from the output of an economic scenario genera-tor. This project is a collaboration between the author and the Swedishreinsurer Sirius International. The investment universe stems from Sir-ius International’s own modeling of assets. In this universe, a frame-work for extracting optimal portfolios is developed using the risk mea-sures Conditional-Value-at-Risk (CVaR) and the variance. The projectanalyses how a set of efficient portfolios perform in terms of risk versusreward, creating both a mean-CVaR and a mean-variance framwork. Theproject also analyses the cost of constraints enforced in the optimizationof efficient portfolios.

Its main findings conclude that the mean-CVaR framework is prefer-able for an insurance company for several reasons. The mean-CVaR frame-work is more reliable than a mean-variance framework because it relieson the modeling of the tails of the loss distribution, whereas the mean-variance optimization will only capture the variance of a loss distribution.The mean-variance framework will as such only reflect the risks in theirentirety for an underlying normal distribution. Furthermore, the mean-CVaR framework offers more stability, making it favorable for a dynamicinvestor who may want to adjust their portfolio holdings according toshifting market conditions. The report also concludes that the additionalconstraints enforced in the optimization can be quantified in terms of alimited return tradeoff.

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Sammanfattning

Det teoretiska området för optimala portföljval har fått väldigt my-cket uppmärksamhet ända sedan Harry Markowitz först presenterade det.Det här projektet har tillämpat Markowitz moderna portföljteori på ettinvesterings universum som är skapat från data från en ekonomisk sce-nario generator. Projektet är ett sammarbete mellan författaren och detsvenska återförsäkringsföretaget Sirius International. Investeringsuniver-sumet bygger på Sirius International’s egen modellering av tillgångar. Idetta universum så skapas ett ramverk där man kan extrahera optimalaportföljer utifrån riskmåtten förväntat bortfall och variansen. Projek-tet analyserar hur olika effektiva portföljer presterar i termer av risk ochavkastning, och skapar både ramverket avkastning-förväntat-bortfall samtramverket avkastning-varians. Projektet analyserar också de kostnadersom är förenliga med begränsningar i optimeringen av de effektiva port-följerna.

De främsta slutsatserna är att ett ramverk för avkastning-förväntat-bortfall är att föredra för ett försäkringsbolag av flera anledningar. Ramver-ket för avkastning-förväntat-bortfall är mer tillförlitligt än ett ramverk föravkastning-varians eftersom ramverket förlitar sig på en underliggandefördelnings svans, medan ett avkastning-varians ramverk enbart kom-mer att fånga variansen hos en underliggande fördelning. Rameverketför avkastning-varians kommer således bara att fånga hela risken hosen underliggande normalfördelning. Ramverket för avkastning-förväntat-bortfall är även stabilare, vilket gör det fördelaktigt för en investeraresom vill kunna justera sin portfölj efter marknadens rörelser. Rapportenkommer också fram till att de begränsningar som tillkom i optimeringenkan kvantifieras i termer av en begränsad avkastningsavvägning.

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AcknowledgementsI would like to express my deep gratitude to all the people who have con-

tributed to this report, and to Sirius International for starting this collaborationwith me in the first place. Everyone at Sirius International has been very helpfuland provided me with plenty of fruitful discussions and invaluable insights. Aspecial thanks to my supervisor Jörgen Olsén, Sirius International, for aidingme in this project and continuously pushing me towards new challenges andexploring new interesting areas. I also want to thank my supervisor Lisa Hed,Umeå Universitet, for guiding me through this project and providing me withcontinuous feedback and academic guidance.

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Contents

1 Introduction 61.1 Sirius International . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background 62.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Motivation to using modern portfolio theory . . . . . . . . . . . . 7

3 Theory 83.1 Market risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Risk measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Coherent risk measures . . . . . . . . . . . . . . . . . . . 93.2.3 Value-at-risk . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.4 Conditional-Value-at-Risk . . . . . . . . . . . . . . . . . 10

3.3 Modern portfolio theory . . . . . . . . . . . . . . . . . . . . . . . 113.3.1 Positive semidefinite covariance matrix . . . . . . . . . . . 12

3.4 Post-modern portfolio theory . . . . . . . . . . . . . . . . . . . . 133.4.1 Optimization of conditional value-at-risk . . . . . . . . . 13

3.5 Bond pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5.1 Corporate and government bonds . . . . . . . . . . . . . . 163.5.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Economic scenario generator models . . . . . . . . . . . . . . . . 163.6.1 Interest rates . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.2 Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.3 Cash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.6.4 Mortgage-backed securities . . . . . . . . . . . . . . . . . 173.6.5 Exchange rates . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Policyholders funds and shareholders funds . . . . . . . . . . . . 18

4 Method 194.1 Grouping of asset types . . . . . . . . . . . . . . . . . . . . . . . 204.2 Exchange rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Equity returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 Cash returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Mortgage-backed security returns . . . . . . . . . . . . . . . . . . 224.6 Corporate and government bond returns . . . . . . . . . . . . . . 22

4.6.1 Short term bonds . . . . . . . . . . . . . . . . . . . . . . 234.7 Modified conditional-value-at-risk . . . . . . . . . . . . . . . . . 234.8 Theoretical portfolio . . . . . . . . . . . . . . . . . . . . . . . . . 234.9 Positive semidefinite covariance matrix . . . . . . . . . . . . . . . 244.10 Constraints for the optimization problem . . . . . . . . . . . . . 24

4.10.1 Asset type constraints . . . . . . . . . . . . . . . . . . . . 254.10.2 Currency constraints . . . . . . . . . . . . . . . . . . . . . 25

4

4.11 Estimating an efficient frontier . . . . . . . . . . . . . . . . . . . 26

5 Results 265.1 Investment universe . . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Efficient Frontier using CVaR . . . . . . . . . . . . . . . . . . . . 29

5.2.1 Representation of portfolios on the efficient frontiers . . . 315.3 Efficient frontier using standard deviation . . . . . . . . . . . . . 37

5.3.1 Investment universe using the standard deviation . . . . . 375.3.2 Efficient Frontier using the standard deviation . . . . . . 38

5.4 Analysis of CVaR versus standard deviation . . . . . . . . . . . . 41

6 Conclusion 42

7 References 44

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1 Introduction

This project will model and analyse Sirius International’s own asset allocationfrom a risk versus reward perspective. A framework for the risk versus rewardanalysis will be created to easily analyse future portfolios as well as a theoret-ical portfolio of assets using primarily two different risk measures. In additionto creating this framework, this project will analyse the various optimizationconstraints that are imposed on Sirius International.

1.1 Sirius InternationalSirius International is a reinsurance company, which means that they offer in-surance to other insurance companies, which are clients of Sirius International.Sirius International underwrites insurance mainly for health, travel, contingency,aviation and property. Their funds are invested in a wide range of assets for op-timal diversification but also because they operate internationally, underwritingmany different currencies and need to mitigate their currency risk.

2 Background

For a long time, the notion of risk has been used to reflect the uncertainty of anoutcome. This resulted in the risk measure being equally affected by positiveas well as negative outcomes. This notion has since been developed to bettercapture the negative tails of the distribution, thus focusing on the probabilityof default. Several risk measures has been developed for this purpose, such asthe Value-at-Risk (VaR, see 3.2.3) and the Conditional-Value-at-Risk (CVaR,see 3.2.4), the latter being a modification of VaR.

Market risk encompasses the risk of large losses occurring in the market.Market risk is a collective term of four different types of risk, namely equity risk,interest rate risk, currency risk and lastly commodity risk. Sirius Internationalis primarily exposed to equity risk, interest rate risk and currency risk and thesewill therefore be the focus of this project.

A natural step when analyzing market risk would be to investigate how onesportfolio allocation should look like to minimize the market risk, while receivingoptimal return. This was indeed the question posed by Harry Markowitz whenhe first introduced his paper on modern portfolio theory in 19521. Back thenthe risk measure of choice was the variance, but today VaR and CVaR arestandardized measures that are widely accepted and used as a new way toquantify risks.

As a reinsurance company, Sirius International is required to monitor andcontrol their exposure to market risk. Sirius International has developed aninternal model for this purpose, and can use this internal model to produceestimates of the market risk

1The Journal of Finance, Vol. 7, No. 1, p. 77-91

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There is a regulatory framework for companies operating within the insur-ance industry called Solvency I, which enforces rules that the insurance industrymust oblige to. The regulations also control how much solvency capital a com-pany must have in reserve to ensure that the company does not go insolvent.These regulations are being revised and updated with the new Solvency II regu-lations, that are scheduled to be enforced 1 January 2016. The new regulationswill be stricter in terms of risk management disclosure, as well as changingthe formula for calculating the solvency capital requirements along with numer-ous other changes that affect the insurance industry. These changes places anincreasingly large focus on risk management. For this reason, the risk manage-ment of optimal asset allocation is ever more important, and why this projectwas initiated.

2.1 ObjectiveThe objective of this project is to produce a framework for risk versus rewardanalysis for Sirius International’s asset holdings. The analysis will have two pri-mary objectives. The first objective is to analyse how different asset allocations,including a theoretical portfolio developed by Sirius International’s finance de-partment, performs in terms of risk versus reward. The second objective is toanalyse the cost of constraints imposed on Sirius International with regard totheir investment guidelines.

2.2 Delimitations• Will only look at a theoretical portfolio of specified holdings (see 4.8) and

not a complete investment universe, although the framework can be scaledto include other assets than those used in this project.

• Will lump together asset types based on how the economic scenario gen-erator models them, for example, investments in affiliates and convertiblebonds are both treated as equity holdings.

• Will only investigate the optimal portfolio with regards to optimizing riskand return for the asset side, with no consideration for matching cashflows of the assets and liabilities.

• Will not include currency hedges in the optimization.

2.3 Motivation to using modern portfolio theoryMarkowitz created the foundation of modern portfolio theory as we know it.The model is still considered true by many, but new risk measures have sincebeen developed, which may prove superior to the standard deviation, whichis the risk measure originally proposed by Markowitz. His framework can bescaled and modified in many ways, to better match the projects needs, and istherefore a good starting point for this project.

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3 Theory

In this section, relevant theory will be introduced and defined in order to geta better understanding of the theoretical foundation that this project buildsupon. Central theory for this project includes various risk measures, relevantoptimization theory and the modeling of assets in the economic scenario gener-ator.

3.1 Market riskMarket risk is often divided into four subcategories: equity risk, interest raterisk, currency risk and commodity risk. For Sirius International, the commodityrisk is not a major risk, because of their few investments in this area. Marketrisk can be diversified to a large extent, but in times of an economical crisis,the entire market may move highly correlated, thus making the diversificationuseless. This type of risk is called systematic risk and is impossible to completelyavoid. Because the diversification of risks is one of the central aspects of thisproject, it is important to acknowledge that systematic risk exists and maycreate scenarios that is difficult to forecast for this project.

3.2 Risk measuresWhen choosing a risk measure one must first understand when and why it isused. For this project we are both interested in risk management in the tail of theloss distribution, as well as the volatility of returns and will choose risk measuresbased on these needs. We will now examine three of the most commonly usedrisk measures.

3.2.1 Variance 2

For risk management, the objective is to minimize the potential risk of losingmoney. A common measure of risk has long been the variance, or standarddeviation (volatility). This risk measure has both good properties and badproperties. Among the good properties is the fact that it is easy to communicateand easily used for various statistical purposes. Because variance registers bothpositive and negative events equally, it is not ideal when only interested in thebad outcomes. The variance risk measure also has the downside of not beingcoherent (see 3.2.2), which is an attractive trait for a risk measure.

Variance is defined as:‌

V ar(y) = �

2 = E[(y � E[y])2]

‌where y is a random variable.

2Luenberger, David G., Investment Science, p.143

8

The standard deviation, or volatility, is simply the square root of this ex-pression:

‌

� =pE[(y � E[y])2]

‌A covariance measure indicates how two variables are related to each other.The covariance is defined as:‌

Cov(x, y) = E[(x� E[x])(y � E[y])]

‌As can be seen in the definition of covariance, a variables covariance with

itself is simply the variance of the variable. This covariance measure is one ofthe central pillars in modern portfolio theory (see 3.3).

3.2.2 Coherent risk measures 3

In the world of risk management, a risk measure can be classified as coherent.The coherency trait is often sought after in a risk measure because it grantscertain mathematical properties. In order for a risk measure to be classified ascoherent, four mathematical properties must hold:

‌

• Monotonicity: If Z1 and Z2 is portfolios of assets, for all future out-comes, the value of Z1 > value of Z2, then the risk of holding Z1 must belower than the risk of holding Z2.

• Sub-additive: The risk of a portfolio of assets should not be greater thanthe risks of holding each asset separately.

• Positive homogeneity: By doubling the value invested in each asset ofa portfolio, the risk will also be doubled.

• Translation invariance: If the loss distribution, L, is increased by anamount alpha, then the risk is increased by the same amount alpha.

3.2.3 Value-at-risk4

Value-at-risk (VaR) is one of the ‘newer’ risk measures, emerging in the early90’s. In the upcoming Solvency II regulations, submitting a VaR is a require-ment. It has many good qualities such as it’s only affected by the loss distri-bution far out in the tail, as well as simplicity of stating the risk as a certainmonetary amount. Because Value-at-risk is not sub-additive, it can be mislead-ing and is subsequently not a coherent risk measure.

3McNeil et al, Quantitative Risk Management, p. 238-2484McNeil et al, Quantitative Risk Management p. 38-41

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Before defining VaR, we can look at an illustrative example of what VaReffectively means:

If a risk manager states that the 99.5% one year VaR is 100 million, thatmeans that the company has a 0.5% chance of loosing at least 100 million in thecoming year, or conversely; the company is expected to lose at least 100 millionevery 200 years.

Note that VaR does not specify how big losses that can occur if the lossesexceed 100 million, it simply states the loss associated with a specific percentileof the distribution.

‌DefinitionIf L is the loss of a portfolio, and ↵ 2 (0, 1) then‌

V aR↵(L) = inf {l 2 R : FL(l) � ↵}

‌where FL is the distribution of losses and ↵ is a percentile.As an example of VaR not being sub-additive, suppose there are two bonds

in a portfolio, each having 4% default rate. Holding these bonds separately willresult in a V aR0.95 of zero respectively, however holding them in a portfolio willyield a risk of at least one bond defaulting at 7.84% giving the portfolio a clearpotential for loss at the V aR0.95 level.

3.2.4 Conditional-Value-at-Risk5

Conditional-Value-at-Risk (CVaR), often called Tail-VaR(TVaR) or expectedshortfall, evaluates what the expected loss will be, conditional on being beyondthe quantile ↵. It accounts for the whole tail of the loss distribution, and notjust the quantile value.

The CVaR is effectively the weighted average of the losses that exceed acertain quantile.

‌Definition‌

CV aR↵ =1

1� ↵

ˆ 1

↵

V aR�(L)d�

‌Because CVaR is the weighted average of the losses that exceed a quantile

the following relationship holds for every quantile ↵:‌

V aR↵(L) CV aR↵(L)

5McNeil et al, Quantitative Risk Management p. 44-46

10

‌It is worth noting that two companies with the same VaR can still have

drastically different risk profiles, which a difference in CVaR would reflect.While CVaR does indeed seem like a more attractive risk measure compared

to VaR and the variance, it is not always preferable. CVaR is often estimatedusing either historical data, or simulated data. Because CVaR calculates theweighted average of a certain percentile, far out in the tail of the distribution,it is often based on very few observations. While variance uses the whole distri-bution as a basis for estimating the variance, and the VaR only reflects the lossassociated with a certain percentile, CVaR is heavily affected and reliant on asufficiently large data set.

An example of this would be comparing the risk for a simulated distribution,and noting how the risk changes as the number of simulations included in thedistribution increases. Two distributions are based on the same simulation al-gorithm, one holds 1000 simulations, another holds 10 000 simulations. In thiscase the variance would probably change relatively little, because it is based onmany data points in either case. The V aR0.99 and CV aR0.99 however, wouldprobably change a lot, given that the CVaR is estimated from 100 values insteadof 10.

3.3 Modern portfolio theory6

This modern portfolio theory (MPT) is based on the popular portfolio the-ory proposed by H. Markowitz in 1952. The theory is based on a number ofassumptions, the most noteworthy being:

• There are no frictions such as taxes and transaction costs.

• It is possible to invest in very small fractions of all different assets.

• Investors are only concerned about expected return and risk.

• The returns on the assets are individually different and cannot be ex-pressed as linear combinations of other assets.

• There must be at least two different assets in the investment universe.

Return is measured as a percentage gain or loss, in the MPT and throughoutthis project. Modern portfolio theory aims to find the minimal variance portfoliofor any given expected return. The idea behind his theory is that all possibleportfolios that can be formed using a predetermined number of assets, whichwill form a feasible set of portfolios. Of all these feasible portfolios, there is onlyone portfolio per expected return that has the lowest variance. In order to findthese portfolios, Markowitz formulates the problem:

‌6Luenberger, David G, Investment Science, p. 157-161

11

minimize

1

2

nX

j=1

xixj�ij

‌The 1

2 in front of the expression is for convenience only. This will minimizethe variance of the portfolio.

‌

subject to

nX

i=1

xir̄i = r̄

‌where the left hand side is the expected return of the portfolio.‌

nX

i=1

xi = 1

‌where r̄i is the mean return of each asset i, xi represents the weight of each

asset i, in every portfolio, and �ij is the correlation between assets i and j. Thereturns are assumed to follow a normal distribution in this optimization. Theoptimization will only reflect the entire risk appropriately if the returns followa normal distribution.

An additional constraint is often used to constraint the optimization, onlyallowing for positive investments in assets:

‌

xi � 0

for i = 1, . . . n.‌By solving this optimization problem for any number of different expected

returns, the optimization will result in an efficient frontier, representing everyportfolio which has the highest expected return among all portfolios with thesame level of risk. The efficient frontier stretches from the minimum risk port-folio, up to the maximum return portfolio.

3.3.1 Positive semidefinite covariance matrix7

The covariance matrix that is used in the optimization of the efficient frontiermust be symmetric and positive semidefinite (PSD) in order to be a viablecovariance matrix. The PSD trait for our purposes means that the portfoliovariance must be zero or positive. A matrix ⌃ is PSD if and only if:

‌7Glasserman, Paul. Monte Carlo Methods in Financial Engineering, p. 64

12

x

T⌃x � 0 8x2 R

‌where the left hand side is the variance of a portfolio with investment weights

x. This is equivalent to the requirement that all eigenvalues of ⌃ being nonneg-ative.

3.4 Post-modern portfolio theoryThe (post-) modern portfolio theory (PMPT) model is in many ways based onthe same principles as ordinary modern portfolio theory. The changes has todo with the way risk is measured. By swapping out the variance for a moredownside oriented risk measure such as the CVaR, it can yield an interestinganalysis that is complementary to standard MPT.

3.4.1 Optimization of conditional value-at-risk 8

This theory will closely follow an article by Rockafellar and Uryasev, publishedin Journal of Risk in 2000. The main idea presented in the article is that CVaRcan be used to optimize portfolios much the same way as standard MPT. Thefollowing theory is used in the MATLAB built in class PortfolioCVaR, whichwill be used in this project. The relationship between CVaR and VaR (see 3.2.4)is one of the central pillars in the optimization theory, because a low CVaR willguarantee a low VaR as well. The task is to create a minimization problem thatwill minimize CVaR using linear optimization. The theory has its foundationin MPT and as such inherits many constraints from it.

The expected return of a portfolio is the weighted expected return of allassets that the portfolio is invested in.

‌

rp =NX

n=1

xn ⇤ rn (1)

‌Where rp is the target expected return, N is the number of assets, x repre-

sents the weights in the portfolio. Only portfolios with an expected return of atleast rp will be admitted, the optimization will run over all different return tar-gets, while only allowing feasible portfolios having at least this return. Becauseshorting of assets is not allowed, all portfolio weights must be zero or positive:

‌

xj � 0 (2)

‌8Rockafellar and Uryasev, Optimization of Conditional Value-at-Risk, Journal of Risk

13

All portfolios must be completely invested in the market, which is formulatedas:

‌nX

j=1

xj = 1 (3)

‌Although CVaR is usually defined in terms of monetary value, because this

project uses return distributions, it is defined here as percentage returns. Thetask of minimizing CV aR� (expressed as ��(x)) can be stated as:

‌

minimize ��(x) over x✏X (4)

‌Where x represents a portfolio in the feasible set of portfolios X that still

adheres to the constraints imposed on feasible portfolios (1), (2) and (3). Theminimization of ��(x) is the same as the minimization of the performance func-tion F� , which is formulated as:

‌

F�(x,↵) = ↵+1

1� �

ˆy✏R

[f(x,y)� ↵]+p(y)dy

‌which can be approximated with the function F̃� :‌

F̃�(x,↵) = ↵+1

q(1� �)

qX

k=1

[�x

Tyk � ↵]+

‌where‌

[t]+ =

(t when t > 0

0 when t 0

‌where x is a portfolio in the feasible set of portfolios, ↵ is the threshold of

VaR, yk is a random sample of returns from each of our asset classes, q is thenumber of scenarios for each asset class and � is any specified probability levelin (0,1). A more in depth explanation of why the preceding equations hold canbe found in the referenced article.

The minimization of F̃� can be reduced to convex programming using auxil-iary real variables, uk, for k = 1, . . . , q, it is equivalent to minimizing the linearexpression

‌

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↵+1

q(1� �)

qX

k=1

uk

‌subject to the linear constraints of‌

uk � 0

‌and‌

x

Tyk + ↵+ uk � 0

‌for k = 1, . . . , q.It is worth noting that for the reduction to linear programing, yk can have

any distribution, and is not reliant on being normally distributed. This is oneof the key differentiators between PMPT and standard MPT, the latter requirethe returns to be normally distributed to accurately describe the complete dis-tribution, while PMPT does not.

The optimization will result in an efficient frontier, representing every port-folio which has the highest expected return among all portfolios with the samelevel of CVaR.

3.5 Bond pricingFixed income securities are very important assets for any insurance company,including Sirius International, because of their predetermined payoff. Becauseinsurance companies have liabilities such as insurance claims, it is important toalways have enough money to pay for these claims, in order to stay solvent. Be-cause the income from fixed income securities is predetermined, most insurancecompanies portfolios contain a significant portion of this type of asset.

The price of the bond today (at t0) is the present value of all future cashflows:

‌

P0 =NX

t=1

C

(1 + Y TMt)t+

FV

(1 + Y TMN )N

‌Where Y TMt is the observed yield to maturity for a certain time to maturity

t. Yield to maturity is the anticipated return of a bond if held until maturity.C is the annual (for our purpose) coupon payment and FV is the face value (orprincipal value). The face value is the value that is paid out at the end of thebonds life cycle. N is the number of coupon payments that are paid out untilmaturity. The bond is said to reach maturity when its life cycle ends. Because

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the bond price is dependent on an interest rate that fluctuates, the expectedreturn of a bond is calculated as:

‌E[rBond] =

P1

P0� 1

‌Where P0 is simply the bonds present value and P1 is the bonds expected

value at time 1, that for this project means one year from today.

3.5.1 Corporate and government bonds

This project will price and analyse both corporate bonds as well as governmentbonds. The pricing of both are as described in section 3.5. The difference be-tween the two types of bonds lie in which interest rate that is applied to thebond. The interest rates are modeled according to the different ratings by Stan-dard & Poor’s: BBB, A, AA, AAA and GVT for this project. For each of theseratings it is modelled for maturities 0.5, 1, 2, 3 . . . , 30 years. The governmentbonds are all priced using the GVT interest rates, while the corporate bondsare priced using the other ratings, depending on how risky the corporate bondinvestment is deemed to be.

3.5.2 Duration

A bonds duration is a measure for the sensitivity of the bond price to a changein interest rates. The duration states what the weighted average income time isfrom the fixed income security. For a zero coupon bond the duration is simplythe same as the bonds maturity, since there are no payments before that. For astandard coupon bearing bond, the duration is lower than the time of maturity,how much lower depends on the size of the coupon payments. A high couponrate means larger coupon payments, which would decrease the weighted averagetime of the payments (the duration) to some degree.

3.6 Economic scenario generator models9

The main idea for this project is that the risk and return can be estimatedusing Monte Carlo techniques on simulated data. After obtaining the estimatedrisk and return, an optimization algorithm with be used to obtain an efficientfrontier that holds all optimal portfolios.

The Economic scenario generator (ESG) that Sirius currently licenses is onefrom Barrie & Hibbert (B&H). They have been working on and improving theirESG for a long time and is considered industry leading by many. The ESGuses Monte Carlo simulation paths for the joint behavior of financial marketrisk factors. They employ a number of tools to stay relevant to companies suchas delivering quarterly updates to the ESG parameters, to better match currentmarket conditions. While their simulation algorithms are finely tuned in a very

9Based on Sirius International internal ESG documentation

16

complex way and not disclosed in detail, the algorithms can be described in ageneral way.

3.6.1 Interest rates

The modeling of interest rates is essential to many of the ESG models and willtherefore be covered first. The nominal interest rates are modeled using anextended Two-Factor Black-Karasinsky (2BK) model. Nominal interest ratesare the simplest interest rate before taking inflation into account. It is modeledas follows:

‌

dlnr = a1[lnm(t)� lnr(t)]dt+ �1(dZ1(t) + �dt)

‌

dlnm = a2[µ0 � lnm(t)]dt+ �2(dZ2(t) + �dt)

‌Where r is the short rate, m is the mean reversion level, � is the market

price of risk, µ0 is the average mean reversion level, the a parameters control thespeed of mean reversion and Z1 and Z2 are independent Brownian motions. The2BK model is very similar to a Two-Factor Vasicek model, with the difference ofusing the logarithm of the short rate, in order to ensure positive interest rates.Without the logarithm of the short rates, interest rates may be negative forshort maturities, a trait that is not sought after for our purposes.

3.6.2 Equity

Models excess equity return, that is in excess of the short rate. It is based onseven variables where five of them uses a deterministic volatility and describesthe exposure of a local currency to a global factor that has an effect on alllocal currencies. A sixth variables is added to introduce stochastic volatility toeach of the economies. In order to account for any remaining volatility, anothervariable is added which relies on a stochastic volatility jump diffusion (SVJD)equity model.

3.6.3 Cash

The cash return from the ESG output is estimated using a short term interestrate and extending it in order to get the yearly return on cash investments.

3.6.4 Mortgage-backed securities

Mortgage-backed securities (MBS) are a type of asset-backed security wherethe cash flows are backed by the interest and principal payments on a set ofmortgage loans. The ESG modeling of these MBS is very complex, however,the MBS can be described in a general way. Mortgage holders have the option to

17

refinance (at a cost) their mortgage. When interest rates fall, mortgage holdersmay compare the value of refinancing the mortgage to a lower rate with the costof refinancing the mortgage.

It is presumed, in general, that mortgage holders do not have the neces-sary information or knowledge to choose to refinance their mortgage optimally.Because of this, a laggard distribution is used to account for this suboptimalbehavior of refinancing. If a mortgage holder chooses to refinance, or to payback the mortgage entirely ahead of schedule, the return on the MBS invest-ment will decline. Because the return is based upon mortgage holders payingoff the mortgage for a long time, earning interest on the loan, paying back themortgage ahead of schedule will have a negative effect on the MBS investment’sreturn.

3.6.5 Exchange rates

Exchange rate (or FX Rate) output from the ESG is based upon the assumptionof Purchasing Power Parity (PPP). The PPP assumption states that an adjust-ment factor is needed on the exchange rate between two currencies in order forthe exchange to be equivalent to each currency’s purchasing power.

Let " be the real exchange rate, then:‌

dln(") = ↵(µ� ln("(t)))dt+ �"dZ"(t)

‌where as usual ↵ describes the speed of mean reversion and the term �"dZ"(t)

represents the shock term that creates short term fluctuations in the real ex-change rate with proportional volatility �".

3.7 Policyholders funds and shareholders fundsAs an insurance company, Sirius International is required to manage their as-sets and liabilities to stay solvent. Sirius International makes the importantdistinction of separating investments of policyholders funds from investmentsof owners funds. Liabilities are referred to as technical provisions (TP) andthe assets that correspond to these technical provisions is roughly translatedto the policyholders funds. The assets in excess of the policyholders funds isthe shareholders funds, which correspond roughly to Sirius Internationals ownfunds. Both policyholders funds as well as shareholders funds are assets thatare available for investing in various assets.

It is common to strive for immunization of the portfolio, that is, to strivefor a similar duration of cash flows from the assets to that of the technicalprovisions. This is of course because the claims made by the policyholders,should coincide with an income from the investment of the policyholders funds.This immunization is realised through having a similar duration of the portfolio,to that of the technical provisions. Simply having a similar duration of the cashflows and the technical provisions will however not guarantee that the cash flows

18

are matched completely. Duration is an interesting measure that can be includedin an optimal asset allocation, but it is beyond the scope of this project.

When optimizing Sirius International’s assets, both the shareholders funds aswell as the policyholders funds must be taken into consideration. It is likely thatthe policyholders and the shareholders have very different preferences for targetreturns and the risks that accompany them. The policyholders main concernis that Sirius International stay solvent, and is therefore mainly interested inthe risks far out in the tail of the distribution. Shareholders are of course alsovery keen on keeping Sirius International a solvent company, but it is not theironly concern. Most shareholders invest in a company with the expectation ofhaving a certain return on their investment. Because of this, the shareholdersmight want to look at not only risk measures such as the tail-oriented CVaR,but also other risk measures such as the variance, which captures the standarddeviations of the return. The variance would capture the fluctuations of thereturn and could reflect how often the return would fall short of the expectedreturn which would be of interest to the shareholders.

4 Method

The data used in this risk versus reward optimization is an excel file with atheoretical portfolio of assets that is characteristic for a reinsurance companysuch as Sirius International. The excel file containing a theoretical portfolio(see 4.8) has been developed together with the finance department at SiriusInternational, in order to get a good representation of what a plausible portfoliocould look like. A MATLAB file containing 100 000 simulations from the B&HESG is also used, with each scenario containing a 1 year prediction into thefuture.

Each simulation contains the following data, separate for each of the threecurrencies EUR, SEK and USD:

• Equity capital gain and Equity dividend yield

• Cash returns

• Mortgage-backed security return

• Interest rate curves for maturities 0.5 to 30 years for the ratings BBB, A,AA, AAA and government interest rates

• FX returns with USD as a base currency.

• The starting interest rate values that the ESG uses for simulation of newinterest rates is also included in this simulation file.

Because the investment universe consists of more assets than our theoreticalportfolio currently holds, each asset type will be modeled for all three currenciesEUR, SEK and USD, regardless of the theoretical portfolio’s current position.

19

In order to create a CVaR framework for risk and return, both the risk andreturn need to be estimated for all different assets. The ESG output need notonly be converted into return distributions, it also needs to be consolidated ina base currency for easy comparison between assets.

For this thesis, the consolidation currency of choice is the US dollar, USD.This means that all returns will take the FX Rate into account and consolidatein USD when comparing return and risk.

4.1 Grouping of asset typesThe B&H ESG models assets according to these different asset classes

• Equity, which is a collective term that includes common stock, convertiblebonds, hedge funds, investments in affiliates and private equity partner-ships.

• Cash, which includes only cash investments.

• Mortgage backed securities (MBS), which only holds these MBS.

• Corporate Bonds, which includes corporate bonds as well as commercialmortgage backed securities (CMBS) and asset backed securities (ABS)

• Government Bonds, which includes agency and foreign government bonds.

Note that these assets are aggregated in a very coarse way. Equity in particularholds very many different kinds of investments, and it is important to keep thisin mind when judging investments into this asset class. While equity may bedeemed a risky investment, an investment in affiliates may not be associatedwith the same risk, and might even bear a required investment. This is becauseinvestments in affiliates may not shift from day to day, but must be kept forlonger periods of time.

4.2 Exchange ratesAs mentioned previously, this project has chosen the USD as a base currency,meaning that for all purposes, the project will consolidate in USD in order tocompare returns for different assets classes.

The FX Rate is usually quoted as CurrencyBaseCurrency , which for our purposes

translates to CurrencyUSD . Whenever a currency translation is needed that means

that the inverse of the equation above must be used:‌

1Currency

USD

=USD

Currency

‌The calculation of the return distributions are done for all three currencies

for all asset classes.

20

An example of the consolidation in USD:To translate a 100 SEK bond value into a comparable USD bond value, the

formula above can be used:100SEK ⇤ USD

SEK ⇡ 15.5 USD using today’s FX Rate (Beginning of year, USDSEK

is 6.4339 with the current ESG calibration).

4.3 Equity returnsAll currencies are treated in the same way, with FXReturns being the returnon the chosen currency per dollar. Because of the consolidation in USD, theFXReturns for USD equity holdings is zero. FXRate0 is the present FXRate

for the chosen currency stated in CurrencyUSD and the FXRate0 for USD holdings

is therefore 1.‌

EquityV aluet = (1+CapitalGain+DividentY ield)⇤ 1

FXRate0 ⇤ (1 + FXReturns)

‌

EquityV alue0 =1

FXRate0

‌

EquityReturn =EquityV aluet

EquityV alue0� 1

‌

4.4 Cash returnsCash returns are treated the same way as equity, treating the FXRates thesame way as described previously.

‌

CashV aluet = (1 + CashReturn) ⇤ 1

FXRate0 ⇤ (1 + FXReturns)

‌

CashV alue0 =1

FXRate0

‌

CashReturns =CashV aluet

CashV alue0� 1

‌

21

4.5 Mortgage-backed security returnsMBS returns are treated the same way as equity and cash returns, treating theFXRates the same way as described previously.

‌

MBSV aluet = (1 +MBSReturn) ⇤ 1

FXRate0 ⇤ (1 + FXReturns)

‌

MBSV alue0 =1

FXRate0

‌

MBSReturns =MBSV aluet

MBSV alue0� 1

‌

4.6 Corporate and government bond returnsIn order to calculate a return distribution for fixed income securities, severalfactors has to be taken into account. As described in section 3.5, in order toprice a bond you need its coupon value, face value, years to maturity and finallythe interest rate for each of those years until maturity. The interest rate thatshould be used for each year until maturity should correspond to the bondsrating, currency and years to maturity.

For example, the pricing at t0 for a USD corporate bond with 3 years tomaturity, AA rating, a coupon of 5% and a face value of 1000 USD would be:

‌

P0 =50

(1 + USDAA1Y ear)+

50

(1 + USDAA2Y ear)2+

50 + 1000

(1 + USDAA3Y ear)3

‌For pricing of bonds at t0 the interest rate used in calculations is observed

on the market and therefore deterministic.The same formula is also used when pricing the bond one year from today,

at t1, but is using interest rate scenarios for this pricing instead.In order to calculate the return distribution the following formula holds‌

BondReturn =P1 ⇤ 1

FXRate0⇤(1+FXReturns)

P0 ⇤ 1FXRate0

� 1

‌While the different bonds are simplified using a limited number of different

maturities and ratings, the framework still models them with an approach that

22

is more granular than for the other asset classes. The different bonds are rep-resented in the model by having 73 different asset classes, compared to the 9different asset classes that represent equity, cash and MBS (one asset class percurrency). This increased granularity for bonds will allow the asset allocation alot more freedom to invest into various bonds, than for other asset classes. Thehigh granularity may affect the optimal portfolios so that it invests more intocertain bonds than it would have done if the bonds had the same granularity asthe other assets.

4.6.1 Short term bonds

The bonds with one year to maturity is a special case because the only stochasticpart in bond pricing is the interest rates, which for one year bonds is determin-istic. The interest rates used to price the bond today (at t0) are observed onthe market and the price at t1 is simply the facevalue of the bond plus the an-nual coupon payment. For EUR and SEK bonds, there is a currency risk thatfollows with the consolidation in USD, which adds fluctuations to the return ofthe bond. However, USD bonds are not affected by currency risk in this project,and the USD bonds with one year to maturity hence has a deterministic valueand no risk allotted to them.

4.7 Modified conditional-value-at-riskWhile this project does put a lot of confidence in the way the ESG operates,sometimes it is beneficial to have a conservative view of an asset risk. Forthis reason a Modified Conditional-Value-at-Risk (MCVaR) has been developed.This MCVaR will assume that the distribution of returns outputted from theESG is correct, but shifts the center of the distribution to zero instead of beingcentered around the expected value. This means that the company view is thatfalling short of the expected return is also included in the risk. The definitionof MCVaR is as follows:

‌

MCV aR = CV aR+ E[r]

‌This MCVaR may not differ much from the CVaR if the distribution has

relatively long tails. For short tails however, the choice between MCVaR andCVaR matters a lot, and ultimately comes down to how conservative the com-pany’s risk appetite is. MCVaR is most often used when insurance companieswant to estimate their capital requirements with a conservative level of risk.

4.8 Theoretical portfolioAs previously mentioned, the theoretical portfolio used in this project has beendeveloped by the Sirius International’s finance department in order to get a port-folio that is characteristic to a reinsurance company such as Sirius International.

23

This means that the portfolio has a high emphasis on fixed income securities,which spans a large range from relatively risky bonds with short maturities, tosafe investments with long maturities.

After consolidating the portfolio in USD, the theoretical portfolio has ap-proximately the following portfolio weights:

‌

Asset class Portfolio weightEquity 19%Cash 11%MBS 2%

Corporate bonds 31%Government bonds 37%

‌While the model will include a much more granular version of the portfolio,

this table is simplified in order to get a good overview of how the portfolio relatesto the efficient frontier (see Figure 4). The theoretical portfolio has investmentsin almost all assets that make up our investment universe (see 5.1).

4.9 Positive semidefinite covariance matrixIn order to perform the optimization of the efficient frontier using the stan-dard deviation, the covariance matrix that is used must be positive semidefinite(PSD). Because of numerical precision errors, the covariance matrix estimatedfrom the different asset return series is not PSD. In order to clean up the co-variance matrix and make it PSD, any negative eigenvalues must be identifiedand removed. For this project, this is done by identifying the smallest negativeeigenvalue, and adding the absolute value of this to the diagonal of the covari-ance matrix. This is not an entirely accurate and correct methodology, butconsidering that the eigenvalue is negative at the roughly 18th decimal point,and that our financial data has nowhere near this precision, it should not affectthe result in any meaningful way.

4.10 Constraints for the optimization problemIn order to control the optimization problem and make the asset allocation morerealistic, a set of constraints are enforced to limit the investments into variousasset classes and currencies.

Two groups of constraints have been chosen that will constrain the possibleportfolios that make up the efficient frontier. The first group focuses on assetclass constraints and is only enforced in the optimization of the policyholdersfunds. The reason for only applying asset class constraints for the policyholdersfunds is because Sirius International enforces stricter investment guidelines forthese funds than for their shareholders funds.

24

4.10.1 Asset type constraints10

Sirius International enforces investment guidelines for the investment of theirpolicyholders funds. The optimization of the policyholders funds is subject tothe following asset type constraints:

‌

0.95 Bonds+ Cash 1

‌

0 BondsHighRisk 0.2

‌which means that Bond+Cash investments range between 95% and 100%

of each portfolio. This leaves up to 5% to be invested into other types ofassets. BondsHighRisk refers to all corporate bonds that are deemed to be belowinvestment grade. These constraints forces each portfolio of policyholders fundsto have a lower risk than a portfolio that can more freely choose its mix of assetclasses. These constraints also allows for flexibility regarding the duration ofthe investments, which is favorable when matching cash flows and liabilities.Sirius International currently has a duration of interest bearing investments of2.2 years.

4.10.2 Currency constraints

The optimization will simultaneously enforce a set of currency constraints. Thecurrencies chosen are the same as those modeled by the ESG. Constraints areplaced on both EUR and USD to various degrees in order to increase the realismof the optimization. The constraints are chosen with care for the sole purpose ofincreasing the realism of the model. The chosen constraints can be formulatedas:

‌

0 EUR 0.2

‌

0 USD 0.5

‌

0 EUR+ USD 0.67

‌which means that EUR and USD have quite significant restrictions while

SEK investments have no restrictions imposed on them. Because Sirius Inter-national is a Swedish company, and therefore may not regard SEK as a currency

10Sirius International Insurance Corporation Annual Report 2013, Note 2

25

exposure, these constraints are realistic. These currency constraints are enforcedfor both the shareholders funds as well as the policyholders funds and is there-fore also enforced in the resulting combined frontier. In reality, there are a lotmore types of constraints than those that appear in this optimization, but it isdifficult to account for all possible constraints that should limit the asset allo-cation. The constraints are easily modified in order to view efficient frontiersthat are subject to stricter or more flexible constraints.

4.11 Estimating an efficient frontierThe MATLAB built-in class PortfolioCVaR is used together with methods andcommands that are available in Matlab in order to set constraints in the op-timization. The optimization theory that is the foundation of this MATLABclass is the theory described in the theory section (see 3.4.1). For this CVaRoptimization, the 99% percentile has been chosen and is used for all CVaR calcu-lations for easy comparison of different assets and portfolios. The optimizationis applied to both the shareholders funds frontier and the policyholders fundsfrontier, using different constraints for each frontier.

As for the standard MPT optimization, the built-in class Portfolio is usedtogether with appropriate functions that are used to set covariance betweenassets, various constraints and plot the different frontiers. The theory that thisoptimization is built upon can be found in section 3.3.

5 Results

In this section the results from this project will be presented. The results canbe divided into four different parts. The first part will present the investmentuniverse that is the foundation of this project. The second and third part willpresent the results from the optimal asset allocation using CVaR and standarddeviation respectively as risk measures in a MPT framework. The fourth andfinal part of the section will include a comparative analysis of the two differentoptimizations.

5.1 Investment universeTogether with Sirius International’s finance department, a total of 82 differentasset classes has been chosen, spanning three different currencies and five differ-ent groups of asset classes. These assets are modeled by the ESG and togetherspans our investment universe. It is important to acknowledge that there ofcourse exists far more asset types and currencies than those that are includedin this project.

‌

26

Figure 1: Investment universe by asset class using CVaR

‌

Figure 2: Investment universe by currency using CVaR

27

‌In Figure 1 and Figure 2 it is possible to identify grouping behaviour both

regarding asset classes and currency. A few assets stand out from the crowdby having an exceptional high return

risk ratio. A high ratio is the consequence ofbeing further to the north west than other assets in Figure 1 or 2. The equityassets clearly has a very high return, but their risk is also very high, giving thema fairly common ratio.

The grouping behaviour is more easily identifiable in Figure 2, which groupsthe assets by currency. This is because the SEK and EUR currencies have aexchange rate component that models how they will appreciate or depreciateversus the USD currency. The ESG forecasts a strengthening of both EUR andSEK versus the USD, which will increase the return of said currencies whenconsolidating in USD. The SEK is expected to strengthen by 101 basis pointsagainst the USD, while the EUR is expected to strengthen by 83 basis pointsagainst the USD. This exchange rate component not only increases the returnbut also the currency risk of the asset classes it is applied to.

The USD bonds with one year left until maturity are considered risk free,since they are priced using current interest rates instead of forecasted rates.Because of this, their value is deterministic. SEK and EUR bonds with oneyear left until maturity are of course also priced using current interest rates,but because of the consolidation in USD, these bonds have some currency riskapplied to them.

It is important to consider that those assets that do have a high returnrisk

ratio using the CVaR risk measure, might not be quite as favourable using theMCVaR measure (see 4.7). A high ratio may indicate that it is a very short taileddistribution, and that the risk may therefore be substantially higher using theMCVaR risk measure. An example of this would be one of the USD corporatebonds, which has a return

risk ratio of 0.45, meaning that its return is almost halfas large as its risk. Using the MCVaR risk measure would in this case result ina risk that is almost 50% greater than the risk calculated using the CVaR riskmeasure.

‌

28

Figure 3: Least square regression of the assets in the investment universe

‌In Figure 3, linear regression has been used to separate the assets by per-

formance. Placement above the line indicates which assets that perform aboveaverage, while placement below the line indicates performance that is belowaverage. From this regression at least two assets perform notably well, one ofthem is the SEK equity, which has the highest mean return of all the assetsrepresented in this investment universe. The other asset is a SEK corporatebond with BBB rating and two years until maturity, this asset is the highestreturning corporate bond in Figure 1.

5.2 Efficient Frontier using CVaRThe efficient frontier holds 40 optimal portfolios for varying return targets, whileminimizing the risk for each return target. Three frontiers are created for anal-ysis:

• Efficient frontier for shareholders funds

• Efficient frontier for policyholders funds

• Efficient frontier combining both shareholders funds and policyholdersfunds.

The three frontiers are based on the same simulated scenario data. The differ-entiating factor in each frontier is the individual optimization constraints that

29

are enforced for the frontier (see 4.10). The purple ⇤ represents the investmentuniverse that is available to each efficient portfolio that is created (see 5.1 for areview of the investment universe).

Figure 4 is the resulting efficient frontier that is estimated using CVaR as arisk measure. Because CVaR is calculated from return distributions, the CVaRis consequently stated as a percentage loss of the total portfolio value.

‌

Figure 4: Efficient frontier using CVaR

‌The combined frontier is a combination of the shareholders frontier and the

policyholders frontier, approximately weighted to match Sirius Internationalsown ratio between policyholders funds and shareholders funds.

The theoretical portfolio clearly falls short of the optimal combined frontier,this is primarily because of two reasons. The first reason is that the theoreticalportfolio stems from a potential real world portfolio, containing all the differ-ent constraints that applies to real world investments, many of which are notaccounted for in this optimization. The second reason that the theoretical port-folio under-performs is because it contains an investment in 80 different assetclasses (two of the modeled 82 assets are not included in this portfolio), eventhose that has a poor return

risk ratio. While diversification in itself is surely ben-

30

eficial, the theoretical portfolio may not be optimal in this sense. The feasiblereturn from the combined frontier is 2.16% E[r] 6.94%, which means thatthe minimum risk portfolio has a return of 2.16%.

5.2.1 Representation of portfolios on the efficient frontiers

The 40 optimized portfolios on the efficient frontier can be illustrated by graphsthat show how the optimal portfolio weights change when moving along theefficient frontier. This means that the portfolios to the left have a low risk andlow return, and the portfolios on the right have a high risk and a high return.These portfolios are the result of asset allocation optimization that will yield thehighest return of all portfolios with the same level of risk. The risk of portfolioj is higher than the risk of portfolio i, if j > i.

Figures 5, 7 and 9 has colour mapping of each of the 82 different assetclasses that ranges from equity (1-3, dark red), cash (4-6, dark red), MBS (7-9,dark red), corporate bonds (10-69, orange to light blue) and government bonds(70-82, dark blue)

‌Shareholders funds efficient frontierThe shareholders frontier has the following weights for each of the 40 port-

folios as a cross section for each portfolio on the x-axis.‌

Figure 5: Areaplot illustrating the 40 efficient portfolios for the shareholdersfunds.

31

‌Because the first few portfolios has a low return target, there is little to

no investments in equity, which has a high return and a high risk, and a largeportion of bonds and other low return asset types. Because equity has a highreturn and a high risk, the latter portfolios on the frontier tend to increase theinvestment into equity until the whole portfolio consists of only equity (the darkred colour). Because there is no constraint for investing in Sirius International’slocal currency, SEK, it is possible to have a portfolio consisting solely of SEKequity which is the case here. It is also worth noting that there are no invest-ments into government bonds. Because this is a representation of the efficientshareholders frontier, the CVaR in portfolio 1 is roughly 8% and the CVaR inportfolio 40 is roughly 53%.

‌

Figure 6: 3D-plot including asset classes and illustrating the 40 efficient port-folios for shareholders funds

‌Figure 6 is another representation of the area plot, where the different asset

classes are more easily deciphered. Note that the asset classes have a dedi-cated axis and is no longer represented by colours. As is very notable in thegraph, two assets are heavily favored. One is a SEK bond below investmentgrade that has a fairly high risk accompanied by a fairly high return. Thisasset is too risky for the safest portfolios, which is why it is complemented byassets having a very low risk in these portfolios. When the expected returnincreases towards the middle of the frontier, at the twentieth portfolio, the in-

32

vestments in the SEK bond tend to decline while shifting the resources intoSEK equity. This happens because SEK equity has the highest return out ofall assets that are modeled, and is therefore the only asset that can exist at theright end of the frontier given that there is no restrictions on SEK investments.

Policyholders funds efficient frontierThe policyholders frontier has the following weights for each of the 40 port-

folios as a cross section for each portfolio on the x-axis.‌

Figure 7: Areaplot illustrating the 40 efficient portfolios for the policyholdersfunds

‌The areaplot depicting the changes in the portfolios covering the policyhold-

ers funds look a lot more diversified than the portfolios covering the shareholdersfunds. This is because of the added constraints on the asset class compositionthat is imposed on all portfolios on the efficient policyholders frontier. Becausethis is a representation of the efficient policyholders frontier, the CVaR in portfo-lio 1 is roughly 8% and the CVaR in portfolio 40 is roughly 20%. The constraintsthat restrict this particular frontier has a big impact on performance. While norestriction is placed on SEK investments, the two top performing assets (whichare SEK, see section 5.1) are both deemed as risky assets which are very lim-ited for this frontier. The top performing SEK corporate bond with a ratingbelow investment grade is capped at 20% of portfolio value, and this investmentis almost maxed out throughout the whole frontier. The capped investments

33

are very notable in Figure 7, where some investments are almost constant formost of the portfolios. The capped investments besides the top performing SEKcorporate bond is the SEK equity, capped at 5%, and a EUR corporate bond,capped at 20%.

‌

Figure 8: 3D-plot including asset classes and illustrating the 40 efficient port-folios for shareholders funds

‌It is noteworthy that both the shareholders funds portfolios and the poli-

cyholders funds portfolios only choose to invest in roughly ten different assets.

Combined frontierThe combined frontier weights the shareholders frontier and the policyhold-

ers frontier to get an approximation of an optimal efficient frontier for SiriusInternational’s complete asset allocation. This combined frontier still adheresto the individual constraints imposed on the shareholders frontier and policy-holders frontier respectively.

The results from combining the shareholders funds frontier with the policy-holders frontier can be seen in Figure 9 and 10.

‌‌

34

Figure 9: Areaplot illustrating the 40 efficient portfolios for shareholders andpolicyholders funds combined

Figure 10: 3D-plot including asset classes and illustrating the 40 efficient port-folios for shareholders and policyholders funds combined

35

‌When examining the asset weights more closely a few notable patterns arise.

For low return targets, the opimization has a preference for the USD bonds withone year until maturity, these bonds provide a relatively low return for zero risk(see 4.6.1). This is unsurprising considering that it is basically free return, albeita low return. When the return target increases, these bonds will no longer sufficeand their investment will be shifted into assets with a higher return. The middleof the frontier (portfolios 15-25) has a relatively diverse allocation, even thoughSEK is favored with roughly 60%-70% of the portfolio. Much of this SEKinvestment is allotted to a single SEK corporate bond, the highest performingcorporate bond seen in Figure 1. Towards the end of the frontier (portfolios 26-40), the SEK currency dominates the other currencies, having huge investmentsboth in the previously mentioned corporate bond, but also in SEK equity. Thisis unsurprising because it is reasonable that the highest returning portfolio hasinvestments predominantly in the highest returning assets.

‌Currency weights for the combined frontier‌

Figure 11: Currency weights for the 40 efficient portfolios for the combinedfrontier

36

‌Figure 11 illustrates the changes in currency weights over the 40 optimal

portfolios on the combined frontier. For low risk portfolios, a good diversificationof different currencies is preferred, while the more risky portfolios strive toincrease its investments into SEK, which holds the highest returning assets.

5.3 Efficient frontier using standard deviationAs stated previously, although CVaR is a preferred risk measure, the sharehold-ers may very well have an interest in the variance of the asset returns.

5.3.1 Investment universe using the standard deviation

Figure 12 and 13 depicts the investment universe using the standard deviationas a risk measure

‌

Figure 12: Investment universe by asset class using standard deviation

‌

37

Figure 13: Investment universe by currency using standard deviation

‌In comparison to Figure 1 and 2, this investment universe has an even more

clear grouping of currencies, separating the EUR and SEK from the USD. Eventhough there are differences between the two universes, clear similarities canalso be identified. It is likely that the assets with a high CVaR will also have ahigh standard deviation, which these similarities confirm. The top performingassets using CVaR are also top performing assets with regards to the standarddeviation.

5.3.2 Efficient Frontier using the standard deviation

The following efficient frontier enforces the same constraints as the CVaR opti-mization, but is using the standard deviation as a measure of risk.

‌

38

Figure 14: Efficient frontier using standard deviation

39

‌As is evident in Figure 14, the theoretical portfolio performs a bit better

when using standard deviation, placing itself closer to the efficient frontier com-pared to the CVaR optimized efficient frontier. The feasible return from thecombined frontier is 1.06% E[r] 6.94%, which means that the minimumvariance portfolio has a return of 1.06%.

‌

Figure 15: Areaplot illustrating the 40 efficient portfolios for shareholders andpolicyholders funds combined using the standard deviation

‌Comparing Figure 9 to Figure 15, the second half of the combined efficient

frontier looks quite similar. However, the first half of the frontier looks dras-tically different. While the combined frontier using CVaR looks quite stablefor these portfolios, the standard deviation counterpart makes large shifts inportfolio weights between portfolios, resulting in a very jagged area plot.

‌

40

Figure 16: 3D-plot including asset classes and illustrating the 40 efficient port-folios on the combined frontier using standard deviation

‌

5.4 Analysis of CVaR versus standard deviationAn analysis of the different optimizations is possible due to the fact that theyhave the same constraints imposed on them. When analysing the different CVaRand standard deviation optimizations, it is important to consider that the CVaRoptimization is a modified version of the regular MPT. Because of this, bothoptimizations carries many of the same assumptions. The biggest drawback ofthe standard deviation optimization is that it assumes that the scenarios followa normal distribution, which is violated in this project since most assets returndistribution is more complex than a standard normal distribution. Since theCVaR optimization does not make assumptions of the different asset distribu-tions, it may prove more reliable because of this.

An interesting discovery when comparing the different efficient frontiers isthat the minimum risk portfolio on each combined frontier is associated withvery different portfolio returns. While the least risky CVaR portfolio has areturn of 2.16%, the least risky standard deviation portfolio has a return of1.06%. It is difficult to compare two portfolios on a risk basis when they areusing different risk measures. How do you compare a CVaR of 8.05% to avolatility of 4.07%?

The first twenty portfolios on either combined efficient frontier differ a lot

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from each other. This is mainly due to the fact that there are so many feasibleportfolios in the lower target return region, typically below 4% return. Thejaggedness of the standard deviation portfolios for these first twenty portfoliosis not present in the CVaR optimization. Above 4% return, the number offeasible portfolios decrease quickly until there is only one feasible portfolio thathas the highest possible return 6.94%.

For the cost of constraints, when comparing the maximum return portfoliofor the policyholders funds, with the same level of risk for the shareholders funds,the difference in return is very similar for both CVaR and the standard deviation.The difference is roughly 1 percentage point in return for both optimizationsmeaning that the asset type constraints that are imposed on the policyholdersfunds cost at most roughly 1 percentage point in return. When investigatingthe minimum risk portfolio, it is important to acknowledge that the minimumvariance portfolio has a lower return than the minimum CVaR portfolio. Theminimum risk portfolios can be seen in the table below:

‌

Asset class Theoretical Portfolio CVaR VarianceEquity 19% 1.7% 0%Cash 11% 5.8% 50%MBS 2% 0% 2%

Corporate bonds 31% 92.5% 0%Government bonds 37% 0% 48%

‌While the asset allocation differs a lot for the minimum risk portfolios, so

does the expected return for these portfolios.

6 Conclusion

For this project, a framework for analysing risk versus reward has been devel-oped and implemented according to Sirius International’s own way of modelingand estimating risk and reward. This framework presents the investment uni-verse and allows for extracting efficient portfolios from this investment universe.

One of the objectives presented at the outset of this report, was to investigatehow a set of portfolios perform in terms of risk versus reward. This objectivehas been fulfilled by analysing optimal portfolios on two different efficient fron-tiers. The analysis of respective frontier also showed that while no transactioncosts exists for either optimization, the CVaR optimized frontier is more sta-ble than the optimization with regard to the standard deviation. This meansthat for a real world investor, following the recommendations of a mean-CVaRoptimized frontier would be both easier and cheaper than following the recom-mendations of a mean-variance optimized frontier. This of course only holds fora dynamic investor, who may want to adjust the target return or target risk oftheir portfolio when market conditions change.

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The second objective was to analyse the cost of constraints that the opti-mization is dependent on. Both efficient frontiers point towards a cost of at most1% return, for the asset type constraints that are enforced for the policyholdersfunds.

The CVaR optimization is more appropriate for an insurance company suchas Sirius International for primarily two reasons. The first reason is that CVaRis a better risk measure than the standard deviation, and is more suited for aninsurance company that puts a large emphasis on the modeling of the tails ofthe distributions. The VaR and CVaR is already a very integral part of the riskmanagement for insurance companies and would therefore require little transi-tional effort. The second reason that the CVaR is better for optimizing assetallocation is that it makes no assumptions for the underlying distributions of theassets. Some assets have a distribution that differs a lot from the normal dis-tribution, and the use of CVaR optimization could include these assets withoutcompromising the reliability of the results.

Criticism of the selection of assets included in the investment universe andthe granularity of the individual asset classes is justified to some degree, butis alleviated for a number of reasons. The investment universe is only meantto represent a suitible investment universe for an insurance company such asSirius International, and with an expanded investment universe comes the addedcomplexy and time consuming optimization that it would entail.

The criticism that could be pointed towards a skewed granularity favoringfixed income securities is also alleviated by the fact that this is an insurancecompany which relies heavily on fixed income securities to match their durationof cash income to the duration of their liabilities. The skewed granularity wouldas such only increase the realism of the model, as long as it is used by aninsurance company and not by a company whose only concern is risk and return.However, many of the portfolios do favor a particular bond that has a remarkablyhigh return, if the fixed income granularity would be decreased, it would havea big impact the optimal asset allocation.

This project places a lot of confidence in the way the ESG forecasts sce-narios, but some scepticism could be raised regarding the ESG’s influence bycurrent market conditions. While the USD equity mean return of 4.5% may bereasonable, it is but a forecast that B&H supports, other investors may havedifferent expectations that this project does not consider.

The framework could be developed further in many different ways, suchas building special constraints regarding the duration of each portfolio. Theduration constraint could be combined with a larger asset and liability framworkthat could match each portfolio’s duration to the duration of the liabilities.Another extension of the project could be to include capital requirements foreach of the efficient portfolios. Because Solvency II places different capitalrequirement weights on different types of assets, it could be an integral part ofthe optimized asset allocation. Expanding the framework established here isleft for further studies in this field.

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7 References

Glasserman, Paul. Monte Carlo Methods in Financial Engineering, New York:Springer, 2003‌Luenberger, David G., Investment Science, Oxford University Press, 1998‌Markowitz, Harry. Portfolio Selection. The Journal of Finance, Vol. 7, No. 1.March (1952).‌McNeil, Alexander J., Frey, Rudiger and Embrechts, Paul. Quantitative RiskManagement, Princeton University Press, 2005‌Rockafellar, R. Tyrell and Uryasev, Stanislav. Optimization of ConditionalValue-at-Risk, Journal of Risk, Vol. 2, No. 3, (2000).‌Sirius International Insurance Corporation, Annual Report, 2013

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