INTEREST RATE RISK: A comparative study aimed at finding the most crucial shift in interest rate curves for a life insurance company

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Master thesis, 30 credits

M.Sc. in Management & Industrial Engineering, Risk Management 300 credits Department of Mathematics and Mathematical Statistics Spring 2019

INTEREST RATE RISK

A comparative study aimed at finding the most crucial shift in interest rate curves for a life

insurance company

Felix Gyllenberg, Leonard Åström

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INTEREST RATE RISK

A comparative study aimed at finding the most crucial shift in interest rate curves for a life insurance company

Department of Mathematics and Mathematical Statistics Umeå University

Umeå, Sweden Supervisors:

Markus Ådahl, Umeå University Alexander Rufelt, Skandia Examiner:

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Abstract

Risk management is applied in many financial institutions under regulatory su- pervision. Life insurance companies face many challenges to ensure policy holders of future payouts. The inverted balance sheet of life insurance companies imply that the policy holder pay premiums in advance to the insurance company to later receive payouts at the age of retirement. This means a great responsibility for the life insurance company to be able to meet future liabilities. Due to this, one of the largest risks facing a life insurance company is the interest rate risk. Future liabilities depend on the interest rates and the difference in duration in assets and liabilities creates an imperfect negative correlation between the movements in assets and liabilities when the interest rate change.

The bond market holds different types of bonds such as government bonds, housing bonds and corporate bonds with different maturities within each subgroup.

The relationship between these subgroups and maturities within these subgroups are interesting to investigate in a forecasting point of view. This relationship is usually referred to as the term structure of interest rates and changes in the term structure are referred to as shifts.

This thesis aims to find which of the three shifts, level, slope and curvature, that is most important to capture in interest rate models. This is investigated using three different simulation techniques and the results show that the first shift repre- senting a level shift of the whole term structure has the largest effect on Skandia’s balance sheet.

Sammanfattning

Riskhantering tillämpas i många finansiella institut under tillsynsövervakning.

Livförsäkringsbolagen står inför många utmaningar i arbetet att garantera kun- der framtida utbetalningar. Försäkringsbolagens inverterade balansräkning in- nebär att försäkringstagaren betalar premier i förväg till försäkringsbolaget för att senare få utbetalningar vid pensionsåldern. Detta innebär ett stort ansvar för livförsäkringsbolaget för att kunna möta framtida skulder. På grund av detta är ränterisken en av de största riskerna för ett livförsäkringsbolag. Framtida skulder beror på räntorna och skillnaden i varaktighet i tillgångar och skulder skapar en icke perfekt negativ korrelation mellan rörelser i tillgångar och skulder när räntan ändras.

Obligationsmarknaden innehåller olika typer av obligationer så som statsobliga- tioner, bostadsobligationer och företagsobligationer med olika löptider inom re- spektive undergrupp. Förhållandet mellan dessa undergrupper och löptider inom

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dessa undergrupper är intressant att undersöka. Detta förhållande brukar kallas räntekurva och förändring i räntekruvan kallas för skift.

Examensarbetet syftar till att hitta vilken av de tre skift, nivå, sluttning och krökning som är viktigast att fånga i räntemodeller. Detta undersöks med hjälp av tre olika simuleringstekniker och resultaten visar att det första skiftet som rep- resenterar en nivåförskjutning av hela räntekurvan har störst påverkan på Skandias balansräkning.

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Acknowledgement

We would like to thank Alexander Rufelt at Skandia for guidance and help throughout the whole process of compiling this thesis. Alexander has given us insight in the industry of life insurance companies to help us understand the dynamics of the processes. Also we would like to thank our supervisor at Umeå university Markus Ådahl for introducing us to the subject and his guidance in the choice of subject and models used in this thesis.

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Acronyms

AIC Akaike Information Criteria ALM Asset and Liability Management

BE Best Estimate

BOF Basic Own Funds

EIOPA European Insurance and Occupational Pensions Authority ERM Enterprise Risk Management

ESG Economic Scenario Generator OGARCH Orthogonal GARCH

PC Principal Component

PCA Principal Component Analysis

RM Risk Margin

TP Technical Provision YTM Yield To Maturity

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List of Figures

1.1 Standard Formula for the Solvency Capital Requirements . . . 7

1.2 Plausible future scenario for Skandia . . . 9

2.1 The first three principal components . . . 12

2.2 Principal Component Analysis - Two dimensional . . . 14

2.3 Balance Sheet Structure . . . 15

2.4 The relation between maturity of a zero coupon bond and yield. . . 17

2.5 The inverse relationship between a bonds value and interest rate. . 18

2.6 The relationship between bond prices and changes in yield. . . 19

2.7 Flows of fixed and floating rates between two companies and the swap dealer. . . 21

2.8 Nelson and Siegel parameters . . . 23

2.9 Volatility cluster . . . 26

2.10 Autocorrelation on simple returns . . . 27

2.11 Autocorrelation on simple squared returns . . . 27

4.1 Historical data . . . 38

4.2 Skandia’s expected cash flows . . . 39

4.3 Log-likelihood estimation value with Akaike information criteria. . . 40

4.4 Shifts in yield curve . . . 41

4.5 Variance explained by Principal Components . . . 43

4.6 Historical trend . . . 44

4.7 Emerging trend . . . 45

4.8 No trend . . . 45

4.9 Distribution of BOFs . . . 47

4.10 Historical trend . . . 48

4.11 Emerging trend . . . 48

4.12 No trend . . . 49

4.13 Distribution of BOFs . . . 50

6.1 The evolution and simple returns of Swedish bonds. The Swedish bonds are shifted down. . . 54

6.2 Volatility comparison between the same Swedish bonds in different time periods. . . 55 6.3 Normalized evolution of Swedish bonds and the Swedish index OMXS30 56

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List of Tables

4.1 Initial Solvency Balance Sheet (millions) . . . 36

4.2 Data used in the models . . . 37

4.3 Asset portfolio . . . 40

4.4 Effects on Balance Sheet for shifts in β₀ (in million SEK) . . . 42

4.5 Effects on Balance Sheet for shifts in β₁ (in million SEK) . . . 42

4.6 Effects on Balance Sheet for shifts in β2 (in million SEK) . . . 42

4.7 Effects on Balance Sheet for shifts in β₀ (in million SEK) . . . 42

4.8 Effects on Balance Sheet for shifts in β₁ (in million SEK) . . . 42

4.9 Effects on Balance Sheet for shifts in β2 (in million SEK) . . . 43

4.10 Correlation between assets . . . 44

4.11 Probability of loss with certain level and time horizon - Historical trend . . . 46

4.12 Probability of loss with certain level and time horizon - Emerging trend . . . 46

4.13 Probability of loss with certain level and time horizon - No trend . . 46

4.14 Probability of loss with certain level and time horizon - Historical trend . . . 49

4.15 Probability of loss with certain level and time horizon - Emerging trend . . . 49

4.16 Probability of loss with certain level and time horizon - No trend . . 49

6.1 Correlation between the index OMXS30 and Swedish bonds . . . . 56

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Introduction

1.1 Background

Risk management is an essential part of all financial institutions and they are su- pervised by authorities and regulators thoroughly. For a life insurance company specifically the key regulations are provided by Solvency II and the supervising authority in Sweden is Finansinspektionen. Furthermore the balance sheet of an insurance company differs from other industries. The liabilities are calculated by discounting expected cash flows in the future with certain expectations on the yield curve which is set by the European Insurance and Occupational Pensions Authority (EIOPA) which is part of the European Commission. By offering insurance policies the company is obliged to payments in the future. The size of these payments are uncertain and estimated based on mortality data and transfers of policy holders to other insurance companies to mention some parameters. In order for the company to be solvent, the value of the assets at a specific day needs to be higher than the value of the liabilities.

To ensure policy holders’ safety regarding the choice of insurance company the European Commission introduced Solvency II as an upgrade of the previous regulations, Solvency I. The new regulations demand a robust risk management and transparency in reporting key numbers to authorities and consumers in every company offering insurance policies. The update in regulations came as an effect of the financial crisis in 2008 (EIOPA, 2016).

1.1.1 Skandia

Skandia is a Swedish pension company founded in 1855 offering services for financial needs and security. Skandia is a mutual company which means that surpluses are rewarded to the policy holders. In the group are also the subsidaries Skan- dia Fonder, Skandiabanken and Skandia Investment Management among others.

However this thesis will focus on the life insurance company (Skandia, 2019).

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1.2 Problem description

The task to match the assets with the liabilities can be problematic from several points of view. The liabilities are estimated and often based on historical data which can lead to wrong assumptions and calculations of future cash flows. Also, the availability of liquid assets is sometimes low. For example, the cash flows for a pension policy reaches many years into the future and the insurance company has to include these cash flows in the current balance sheet. However, to find bonds with these maturity times is difficult and not always available at all. Furthermore, it might not be the best strategy to lock in capital in long maturity bonds depending on the current interest rate level and expectations of future interest rates.

1.3 Purpose

The purpose of this thesis is to find the movements in the yield curve that has the largest effect on Skandia’s balance sheet. An asset portfolio matching Skandia’s true duration will be replicated and compared to a made-up portfolio of higher duration to examine the sensitivity to shifts in the yield curve.

1.4 Delimitations

An insurance company faces many different risks such as market risk, operation risk and credit risk. This study will only focus on interest rate risk which is a part of the market risk since a large part of the assets in Skandia are allocated in interest bearing assets. The liabilities are also affected by changes in interest rates as they are discounted with a yield curve based on market prices (Skandia, 2017).

This thesis will evaluate the balance sheet for market scenarios affecting Skandia and project the balance sheet over a time horizon of five years. The asset side consists of different types of assets that are treated as zero coupon bonds with different maturities for simplification but the asset portfolio duration will be matched to Skandia’s real duration and the liabilities in the model used will approximately have the same duration as Skandia’s real liability duration. The liability cashflows reach for 60 years into the future. As the thesis focuses on interest rate risk noth- ing will be allocated in stocks. The true amount allocated in stocks will be held fixed in this thesis and thereby included in the balance sheet but not affected by the market scenarios. Due to limited data on future expected cash flows beyond

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60 years, this thesis assumes that the liabilities are constant. This means that the same liabilities will be used in every time step of simulations to represent the liability side of Skandias liabilities.

1.5 Life Insurance

Life insurance companies offer policy holder to pay premiums in advance to later, after the age of retirement, receive money back. The money paid to the insurer are invested collectively in a portfolio consisting of stocks, bonds, real estate, private equity, derivatives and commodities. The reserves are usually divided up in one guaranteed part and one bonus part. The guaranteed part ensures the policyholder of a specific amount of money in the future, regardless of the economical development. It is usually called the first order reserve and is calculated based on premiums paid by the policyholder growing with a guaranteed rate. This requires the insurance company to invest the premiums in a safe but still value increas- ing way. Policyholders have the chance of earning extra money if the insurance company’s portfolio performs well. This part of the contract is referred to as the second reserve and depends on current actual value of the assets invested in. The policyholder is always guaranteed the value of the first order reserve but if the second order reserve has a higher value than the first, future payouts to the policy holder will be based on the second order reserve’s value.

Other types of models are unit-linked insurance where the premiums are invested in funds and the risk of losses are held by the policyholder without guaranteed payouts.

To limit risk of bankruptcy, insurance companies calculate future expected cash flow based on a group of policyholders with similar characteristics. Contracts are formed to limit risk and still bring value to the policyholders. When a person with a contract dies, multiple things can happen. Either the relatives of the dead receive a one time payment, a monthly payment or no payment at all depending on the contract.

1.6 Solvency II

Solvency II is a regulatory framework in European Union law and has been ap- proved by the European Parliament in 2014 and implemented in 2016. The regu-

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lations are designed to handle risk management issues within the insurance industry. More precisely, it determines how data should be used to calculate different risk measurements such as the Solvency Capital Requirement (SCR) and how to communicate this information to the supervising authority as well as insurees. In- surance companies are obligated to follow Solvency II but can individually apply the regulations on their businesses in the form of internal models (EIOPA, 2014).

SCR is the minimum capital an insurance company under Solvency II regulation must hold. SCR is calculated using Value at Risk (VaR) on a 99,5% confidence level on a one year horizon. Insurance companies can either come up with their own model or use a standardized approach to calculate SCR. The standard SCR formula divides total SCR into sub-risks and a capital requirement for each sub- risk is calculated. Capital requirement from each sub-risk is then consolidated using a correlation matrix to sum up to the total SCR. The correlation matrix represents how the different sub-risks correlate with each other to produce a fair total SCR.

Figure 1.1 – Standard Formula for the Solvency Capital Requirements

The reason for Solvency II was the shortcomings in the previous regulatory framework, Solvency I. The old framework had been prevailing for over 40 years and was simplistic in many cases. Models did not result in accurate estimations for the insurance companies which lead to doubtful allocations of capital.

Similarly to the regulatory framework applied within the banking industry, the Third Basel Accord, Solvency II is divided into three “pillars”. The first one sets out the rules regarding quantitative requirements and calculations of assets and technical provisions (TP). Technical provisions are the insurance companies’ liabilities and a lot of companies have liabilities stretching out for many years ahead.

Key metrics that are to be revealed are Solvency Capital Requirement, Minimum

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these companies to be solvent they should match their assets to the liabilities. To do that each company has to discount the cash flows by a yield curve recommended by the European Commission.

The second “pillar” sets out directives about governance and ensures that risk management is implemented and understood by the entire organization. This means that the company needs to conduct internal audits and controls. Also, the actuar- ial function within the organization plays an important role verifying models with respect to risk taking.

The third and last “pillar” sets out guidelines about transparency. The financial institutions that are obliged to follow Solvency II need to publish reports to authorities about the financial statement of the institution. This includes reporting the risk profile of the firm and several key numbers such as the ability to meet the minimum SCR.

One of the main purposes of Solvency II is to create incentives for large insurance companies to match the duration of their assets and liabilities. By doing that the amount of capital required can be reduced and free for the company to invest in the market (European Commission, 2015).

1.7 Asset-Liability Management

Enterprise Risk Management (ERM) is an important part of any financial institution. It sets out guidelines for how the company should assess different types of risks and how to manage these. One part of ERM is called Asset-Liability Man- agement (ALM) and involves actions a business takes that affects the assets’ and liabilities’ cash flows. The objective of ALM can differ between different companies and also within a company. For instance it can be used to examine various strategies or to calculate capital requirements for financial institutions. The Society of Actuaries (2003) defines ALM as:

“ALM can be defined as the ongoing process of formulating, implement- ing, monitoring and revising strategies related to assets and liabilities to achieve an organization’s financial objectives, given the organization’s risk tolerances and other constraints."

On the asset side of a firm, investments are to be conducted with the objective of maximizing profits. The assets of an insurance company commonly consists of

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stock, real-estate, private equity, bonds, etcetera. However, financial institutions have additional objectives of matching the liability cash flows to ensure solvency and liquidity in order to fulfill the obligations towards insurance holders as well as Solvency II (Skandia, 2017).

Matching large asset allocations with respect to complex liability cash flows can be problematic due to changes in mortality, longevity and transfers to other insurance companies. ALM is a tool to capture these risks and to manage them. The ALM model is typically based on either stress tests from historical data or on stochastic simulated data. In the latter case all components affecting the balance sheet can be simulated such as market development and policy holder behaviour (Gerstner et al., 2008).

One of the most tangible risk that insurance companies deal with is the interest rate risk since it has effect on both the asset side as well as the liability side of the company. This results from the fact of long-term policy contracts resulting in high duration of the liabilities. Matching the characteristics of the liabilities and assets with each other can be problematic from several points of view that have been discussed in earlier sections (Deelstra and Janssen, 2001).

An example of how the results from the ALM model can be projected on the balance sheet is shown in Figure 1.2. The figure shows market value of the assets (blue line), the estimation of future liabilities (red line) and the surplus (green line).

Figure 1.2 – Plausible future scenario for Skandia

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1.8 Economic Scenario Generator

An Economic Scenario Generator (ESG) is a mathematical model designed to generate future market scenarios used by financial institutions to estimate risk and effects on the firms balance sheet. The ESG is the foundation of the ALM model and can be designed in various forms to fit the current state of the market and perform likely market scenarios. The ESG in this thesis will primarily be based on three methods, namely Nelson-Siegel, Historical Simulation of principal components and a time dependent volatility model using GARCH simulation. The Nelson-Siegel method is not based on historical market data but on the yield curve of a distinct day. To examine the interest rate risk of Skandia, market scenarios will be generated as direct shocks on the level, slope and curvature of the yield curve in one time step. This is not likely to happen in an actual market but it captures the sensitivity of the ALM model on interest rate changes.

The Historical Simulation of principal components will only be able to simulate returns that has occurred in the history which limits the outcome to some extent.

The time varying model simulates volatilities that depend on the volatility in the previous time step and can create more extreme scenarios. The model will simulate monthly scenarios on a time horizon of five years. It should also be noted that these models depend on the historical data and will take on the same trend that has been observed in the history in its original form. Scenarios will also be simulated with regards to expectations of the interest rate evolvement in the future.

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Theory

The following part of the thesis will present relevant theory regarding the subject and problem description.

2.1 Principal Component Analysis

Principal component analysis (PCA) is a method of reducing the dimensionality of a data set consisting of correlated variables without losing vital information.

In other words, PCA describes the variation in the data set with less number of uncorrelated variables. The new variables that are created are called the principal components (PCs). These are ordered by how much information about the variance they describe from most to least so the first component is the direction in which the data varies the most. In many cases the vast majority of the variance can be described from just a few PCs as is the case for interest rates with different maturities. The interest rates are highly correlated since there would be arbitrage opportunities otherwise. The method of PCA can be seen as that the PC vectors form lines through the origin that minimizes the distance from each data point to the line itself.

Definition (Correlation). The correlation matrix of a financial time series of returns X with n time steps is defined as ρ(X). By introducing the standardized vector Y such that Y_i = √ ^Xⁱ

V ar(Xi) the correlation matrix, ρ, can be defined as ρ(X) = COV (Y). The elements of the matrix ρ are given by:

ρ_i,j = COV (X_i, Y_i) pV AR(X_i)V AR(Y_i)

Figure 2.1 shows the first three PCs that often are used to explain the variance in bond yields with different maturities. The first PC refers to the level of the interest rates and usually accounts for about 90% of the total variance. The second PC refers to the slope in interest rates and the third PC refers to the curvature.

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Figure 2.1 – The first three principal components

Suppose a data set consisting of m observations measuring p variables. That is, a p-dimensional data set X₁, X₂, . . . , X_p. In this thesis the observations consist of simple returns of bond prices. As explained in the previous section, all of these p dimensions are not important when describing the data set and therefore it is preferable to reduce the number of dimensions in order to more easily visualize the data. PCA looks for a small number of dimensions where the observations varies the most along each dimension. The dimensions that are found form a linear combination of the p variables.

Definition (Simple Returns). Let P_t be the price of an assets at time t. The simple return over the last time period t − 1 to t is defined as:

Y_t= P_t− P_t−1

The first PC, Z₁, denoted in equation 2.1 is a normalized linear combination of X₁, X₂, . . . , X_p with the largest variance followed by the second PC, Z₂ which also is a normalized linear combination of X₁, X₂, . . . , X_p but with the second highest variance and so on.

Z₁ = φ₁₁X₁+ φ₁₂X₂+ . . . + φ_p1X_p (2.1)

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The elements, φ₁₁, . . . , φ_p1 refer to the loadings of the first PC and it follows that the sum of the loadings squared add up to one since the PC is normalized. The squared factor loading represents what percentage of the variance in an original variable is explained by the factor, i.e the loading describes the weight that the variables from the original data gets in the first PC. The loadings correspond to the correlation between the original data points and the newly created points in the principal component vector (Hastie et al., 2013)

According to Carol (2008) principal components explain a data set with less variables and can be transformed back to the original data as a result of orthogonality in the matrix of eigenvectors. The PCs are defined by:

Definition (PCA). Consider a T × n matrix denoted by X consisting of n number of returns and T observations and let V denote its correlation matrix.

The principal components are the columns of the T × n matrix, P defined by:

P = XW

where W is the n × n orthogonal matrix of eigenvectors of V. The original data has now been transformed into orthogonal returns in P. Equation 2.1 can be transformed back to the original variables since W is orthogonal, W⁻¹ = W⁰ by :

X = P W⁰

To receive the principal components for a two-dimensional data set, the data points are projected orthogonally onto a line and then the sum of all the distances, d₁, . . . , d_m from the projected data points to the origin squared are calculated, see Figure 2.2. The next step is to rotate the line and repeat the same steps.

When this has been repeated for all possible solutions the sums of the squared distances are compared. The sums are also known as the eigenvalues of each PC.

The line that has the largest eigenvalue corresponds to the first PC. To maximize the distance from the projected data point to the origin is equal to minimize the distance from the original data point to the projected point as shown in Figure

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slope larger than 1 states that the variable on the Y-axis has more variance than the variable on the X-axis.

Figure 2.2 – Principal Component Analysis - Two dimensional

The PC vector that runs through the origin can be scaled into a unit vector and is then referred to as the eigenvector for the PC in question. When calculating the second PC in this two-dimensional problem the steps are similar as for the first PC.

However, for the second PC, the line running through the origin is perpendicular to the first PC. Lastly, by taking the eigenvalue of the first PC divided by the sum of both eigenvalues one receives information about how much of the total variance is represented by the first PC.

2.2 Balance Sheet Model

The balance sheet of a life insurance company contains assets and liabilities just as any other company. The difference from "ordinary" companies is that the liabilities consist of future payments to the policy holders. The complexity of the liabilities occurs since they are dependent on unknown but probable happenings in

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the future. Payments to policy holders are not known in advance and arise when accidents or pension dates occur.

The liabilities mainly consist of the Best Estimates (BE) calculated by discounting future expected cash flows using interest rates from the generated market scenarios. The balance sheet object Risk Margin (RM) is defined as the costs for another insurance company to take over the unhedgeable risks if the insurance company in question faces bankruptcy. Another word for the BE plus the RM is technical provisions. In this thesis the Basic Own Funds is calculated as the difference of the market value of the assets and the BE in every time step. The insurance company will be considered bankrupt if the BE exceeds the market value of the assets at any point in time. The structure of the balance sheet is shown in Figure 2.3 (EIOPA, 2015).

Figure 2.3 – Balance Sheet Structure

The balance sheet in Figure 2.3 visualizes one risk measure in the framework Sol- vency II, namely the Solvency Capital Ratio. The SCR in this thesis is calculated by simulating different market scenarios and calculating the BOF at the time horizon of one year in each simulation. Then the Value at Risk at the confidence interval of 99.5% denotes the SCR. This corresponds to a risk of being insolvent once in 200 years. The approach of calculating the SCR is decided in accordance with Skandia.

Balance Sheet

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The balance sheet used in this thesis is simplified to some extent and is modelled using discrete time steps. The start of the simulation is denoted by t = 0 and ends at t = T where ∆t is expressed as one month. The asset side of the balance sheet consists of the market value of Skandia’s portfolio and is projected at every time step t_k. The liabilities are built up by the 1st order reserve based on premiums paid by customers and the allocated bonus based on market scenario (Gerstner et al., 2008). The liabilities in this thesis are the real guaranteed cash flows stated in Skandia’s solvency report and does not state the amount corresponding to 1st order reserve or allocated bonus.

Investment strategies

Life insurance companies have long duration of liabilities as a result of future payouts and to ensure that the capital requirement is met the company can evaluate the investment strategy to get as good returns as possible and also match the future liabilities. Fluctuations in assets value causes the portfolio value to change and in order to reach goals the portfolio might have to be rebalanced depending on the investment strategy. For example the company might want to change the amount allocated in fixed income securities depending on market evolution. The most simple investment method is the so called Buy-and-hold method. An initial mix of different underlyings is bought and then held without rebalancing the portfolio. Short for a version of Buy-and-hold strategy is BH(60/40) which means that 60 percent of total capital is invested in for example stocks and 40 percent of capital invested in for example government bonds. The simple method is also easy to analyze and creates a portfolio downward protection since part of the investment is in almost risk-free government bonds (Perold and William F., 1988).

2.2.1 Bonds

A bond is a financial asset and can be issued by governments, states or corpo- rations. Issuing a bond is a way of borrowing money for a predetermined time and cost. There are both coupon paying and zero coupon bonds meaning that the holder of the bond either receives payment with specified interval during the life of the bond or not. The holder of a coupon paying bond usually receives payments annually or semi-annually throughout the life of the bond with predetermined size expressed in relation to the end-payment at maturity. The contract between a bond holder and issuer is called bond indenture and it specifies the type of bond, the value of the bond, maturity date, coupon rate and the face value of the bond.

The yield of a bond is the return an investor can expect on average if the bond is purchased for asked price and held until maturity. By average means that the

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investor sometimes will receive more, sometimes less. This is mainly due to the fact of default (Bodie et al., 2011).

The bond value is the current price of the bond today, the face value is the amount the bondholder will receive from the issuer at maturity date and the coupon rate is the size of the coupon payment in relation to the face value. Zero coupon bonds have coupon rates equal to zero which means that the holder will only receive face value at maturity.

Bonds are issued with different time to maturity. The relationship between maturity time and yield to maturity can be graphically illustrated with a yield as in Figure 2.4. The shape of this curve can vary a lot depending on the market expectations. In the normal case, as shown in Figure 2.4, one can expect a higher yield for a bond with longer holding period.

Figure 2.4 – The relation between maturity of a zero coupon bond and yield.

Bonds issued by the government are usually considered risk-free due to the low risk of default. The risk of default is also called credit risk and government bonds are most of the times rated AAA by bond-rating agencies such as Moody’s or Standard

& Poor’s. The holder of a government bond will receive a lower yield compared to corporate bonds with the same maturity due to the risk-free investment. The yield of government bonds are often used as discount rates for future premiums or payment obligations.

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Companies can also issue bonds to borrow money. Most companies have a higher credit risk than the government which implies that they have to issue bonds with higher yields (Bodie et al., 2011).

2.2.2 Bond price

The value of a bond at a specific point in time depends on its future cash flows and the change in value of the currency due to inflation. The present value calculation depends on the interest rate r. The sum of the risk free rate, inflation rate and a premium to compensate for the risk of the bond equals the interest rate or discount rate. Trivially, this means that a more secure bond with higher rating will have a higher price than an equal bond but with lower rating (ibid.).

Bond Value =

T

X

t=1

Coupon

(1 + r)^t +Face Value

(1 + r)^T (2.2)

The relationship between the bond value and interest rate is illustrated in Figure 2.5 and shows how the bond value changes depending on the interest rate. When interest rates decrease, the bond values increase and the opposite for the case when interest rate increase (ibid.).

Figure 2.5 – The inverse relationship between a bonds value and interest rate.

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The term yield refers to the earning generated by the bond to the investor. The earning can come from coupon payments or the face value. A higher interest rate will make the investor demand a higher yield for bonds in general.

2.2.3 Duration

An inverse relationship exists between bond prices and yields. This affect investors especially when interest rates are volatile and Duration is a measurement for how much the investor is affected. In practice, it states how much the market value of a fixed income security changes after a 1% change in the yield. Another way of looking at duration is to define it as when the bond will pay off. A coupon bond will generally have a lower duration than a zero coupon bond since there are payments prior to the maturity date. Figure 2.6 shows the relationship between bonds with different maturities and different coupon rates (Bodie et al., 2011).

The change in bond value for a specific percentage change in yield to maturity depends on the Duration of the bond. Figure 2.6 shows how the value of a 5 and 30 year zero coupon changes relative to changes in yield to maturity. The 5 year bond has an YTM of 3% and the 30 year bond has an YTM of 4.5% in Figure 2.6.

Lower Duration bonds are less sensitive in price towards changes in the interest rate or yield compared to a bond with higher duration. This is a fact because of the longer time to payment.

Figure 2.6 – The relationship between bond prices and changes in yield.

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Definition (Duration). Duration is a sensitivity measure based on weighted average maturity of cash flows. Consider n present values P V₁, P V₂, . . . P V_n with time in years to payment denoted by t1, t2, . . . tn:

V =

n

X

i=1

P V_i

Duration = Pn

i=1t_iP V_i Pn

i=1P Vi

= Pn

i=1t_iP V_i

V =

n

X

i=1

t_iP V_i V where ^{P V}_Vⁱ represents the proportion of cash flow i.

Insurance companies aim to match the Duration of the assets with the Duration of the liabilities. This is strived for to hedge the company and its portfolios from interest rate risk. By matching Duration the company assures that the assets and liabilities are affected in the same way and size in case of interest rate changes.

The assets in this thesis will only consist of zero coupon bonds and they have a Duration of their time to maturity measured in years. i.e a zero coupon bond with maturity time of five years has a Duration of five years. The Duration of a bond portfolio is calculated in 2.3.

D = w₁D₁+ w₂D₂+ · · · + w_kD_k (2.3)

where w₁, w₂, . . . , w_k are proportions of the total portfolio value invested in each asset. The Duration of the liabilities is more complex and is calculated in 2.4 (McCaulay, 2013).

D = Pn

k=1tkcfke^−r^k^t^k Pn

k=1cf_ke^−r^k^t^k (2.4)

where t_k is the time period of the future cash flow, cf_k and r_k is discount rate.

The benefits of matching the Duration of the two sides of the balance sheet is that the market value of the assets will change just as much as the liabilities in case of a change in the yield curve. A mismatch in duration can cause the liabilities to be greater than the assets and make the company insolvent.

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2.2.4 Swaps

Swaps are a set of future contracts with multiple alike agreements instead of one.

Interest rate swaps allow two parties to exchange cash flows in order to speculate or hedge against interest rate risk. These swaps do not trade on exchange so the two parties need to customize the contract to fit both parties’ needs. The interest rate swaps can be used by a bank to hedge their floating interest rate risk by changing parts to fixed-rate basis. The bank would then have to enter a swap contract with another party to receive floating rate matching the interest rate and instead pay a fixed rate. The swap dealer is the party that arranges the swap contracts between different parties. The dealer who has taken one side of the swap wants to find another trader with the opposite rate type to neutralize the position (Bodie et al., 2011).

Figure 2.7 – Flows of fixed and floating rates between two companies and the swap dealer.

2.3 Liability model

Term structure estimation

The term structure reflects expectations of market evolvement about changes in interest rates and is usually measured by mean of the spot rate (or yield curve) on zero coupon bonds. One issue with the term structure is that the market does not usually offer bonds with all maturities that is of interest (Diebold and Canlin,

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2006).

The solution to this is to create a term structure model to estimate interest rates between and beyond the true bond yields. With either spot rate curves, forward rate curves or discounting curve the entire term structure can be represented and interest rates with other maturities can be found. This is of essence for all companies working in the financial sector. Future contracts, costs and obligations need to be discounted to the same time to be comparable. Since it is often hard to find zero coupon bonds with long maturity, the companies have to rely on the term structure model result to be able to discount obligations further away. This is a common problem, especially for pension funds since they can receive payments 50-60 years before they need to pay out the money (Diebold and Canlin, 2006).

2.3.1 Inter- and Extrapolation Methods

Life insurance companies are obligated to use a risk free interest rate curve to discount future debt cash flows. Since the market does not offer interest rates for all maturities, inter- and extrapolation has to be done to receive rates with the desired maturities. Two known methods to achieve this are the Nelson and Siegel and the Smith-Wilson method.

Nelson and Siegel

The Nelson and Siegel model is a three-factor method estimated on known bonds yields at a specific point in time to extrapolate future maturities. It can also be used to interpolate bond yields between liquid maturities.

y_t(τ ) = β_0t+ β_1t(1 − e^−λ^t^τ

λ_tτ ) + β_2t(1 − e^−λ^t^τ

λ_tτ − e^−λ^t^τ) (2.5)

β₀, β₁ and β₂ are the three factors that are estimated to fit a ploynomial line to the known yield curve. As Equation 2.5 shows the three -parameters are adjusted for each time step t. Their factor loadings each represent different movements that affect the yield curve as seen in Figure 2.8. The Loading on β₀ is equal to 1, preventing the yield curve to decay to zero. The second loading for β₁ is ^1−e_λ^−λtτ

tτ

which is designed to start at 1 and quickly decay towards zero and represents the short term movement in the term structure. The third loading is ^1−e_λ^−λtτ

tτ − e^−λ^t^τ which starts at zero to increase relatively fast before decaying towards zero. This third loading shows the movement in the middle of the term structure. The form of the three β-factors can be related to the principal components used to describe

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the variation in interest rates corresponding to level, slope and curvature (Diebold and Canlin, 2006).

Figure 2.8 – Nelson and Siegel parameters

Smith-Wilson

The Smith-Wilson inter- and extrapolation method adapts to today’s known rates of bonds with different maturities and an ultimate forward rate (UFR) of a bond with high maturity. Extrapolation is executed beyond the last liquidity point to create the entire term structure of risk free rates to be used when discounting future expected cash flows.

The term structure pricing function for N known zero coupon bonds with maturities u₁, u₂ . . . u_N.

P (t) = e^{−U F R∗t}+

N

X

j=1

ζ_jW (t, u_j) (2.6)

m_i = P (u_i) (2.7)

where the symmetric Wilson W (t, u_j) is defined as:

W (t, u_j) = e−U F R (t+u_j)α min(t, uj)−0.5e^{−α max(t,u}^j⁾(0.5e^{α min(t,u}^j⁾−0.5e^{−α min(t,u}^j⁾)

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where:

- N = The number of zero coupon bonds with known price function - m_i = The market price of zero coupon bonds

- u_i = The maturities of the zero coupon bonds in m_i - t = Time to maturity

- α = The mean reversion speed of convergence towards UFR

To calculate the unknown parameters ζ₁, ζ₂ . . . ζ_N a linear combination of N func- tions is solved. The system of equations is the following:

m₁ = P (u₁) = e^{−U F R∗u}¹ +PN

j=1ζ_jW (u₁, u_j) m₂ = P (u₂) = e^{−U F R∗u}² +PN

j=1ζ_jW (u₂, u_j) ...

m_N = P (u_N) = e^{−U F R∗u}^N +PN

j=1ζ_jW (u_N, u_j) (2.9)

The equation system 2.9 becomes:

m = p = µ + W ζ (2.10)

where:

m = (m₁, m2, . . . , mN)^T

p = (P (u₁), P (u₂), . . . , P (u_N))^T

µ = (e^{−U F R∗u}¹, e^{−U F R∗u}², . . . , e^{−U F R∗u}^N)^T ζ = (ζ₁, ζ2, . . . , ζN)^T

And W is the N × N matrix from Equation 2.8:

W =







w(u₁, u₁) w(u₁, u₂) . . . w(u₁, u_N) w(u₂, u₁) w(u₂, u₂) . . . w(u₂, u_N)

... ... ... ... w(u_N, u₁) w(u_N, u₂) . . . w(u_N, u_N)







The solution to Equation 2.10, ζ, is the difference between p and µ and the inverse of the Wilson matrix W .

ζ = W⁻¹(p − µ) = W⁻¹(m − µ) (2.11)

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From Equation 2.11 and Equation 2.6 the price of illiquid zero coupon bonds can be calculated to create the entire interest rate curve used to discount future expected cash flows (Finanstilsynet, 2010).

2.3.2 Discounting Cash flows

The cash flows that arise during the policyholders’ lifetime are discounted to present value using the interest rates generated by the ESG model and extrapo- lated to maturities of 60 years. The discounting is conducted according to equation 2.12 in every timestep, t.

P V_t= CF_te^−r^t^t (2.12)

where P V_t denotes the present value of cash flow, CF_t.

2.4 Scenario Generation Techniques

Autocorrelation

Time series of financial instruments often show signs of volatility clustering as shown in Figure 2.9. Periods of highly volatile returns are followed by periods of returns with low volatility. This plays an important part in order to simulate realistic returns and not receive uniformly distributed returns.

Definition (Autocorrelation). Given the assets y₁, y₂, . . . y_n observed at time periods t₁, t₂, . . . t_n and the lag k, the autocorrelation function is defined as:

ρ_k= Pn−k

i=1(r_i− ¯r)(r_i+1− ¯r) Pn

i=1(ri− ¯r)²

Where r_i is the market return for time period i and ¯r denotes the mean return.

Autocorrelation is a measure that describes patterns in time series. The method of calculating autocorrelation is to compare a time series with lagged version of

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Figure 2.9 – Volatility cluster

the same time series. A strictly positive- or negative autocorrelation indicates that the time series is predictable to some extent. As Figures 2.10 and 2.11 show, the autocorrelation is strictly positive for the squared returns of the first principal component of the data used in the thesis but not for the regular returns. This states that there is a pattern for high- and low periods of volatility but no pattern in the returns themselves indicating uncertainty about a negative or positive return. The evidence of volatility clustering motivates the GARCH simulation method (Embrecht et al., 2005).

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Figure 2.10 – Autocorrelation on simple returns

Figure 2.11 – Autocorrelation on simple squared returns

2.4.1 Historical simulation

Simulating future market outcomes using historical returns is called historical simulation. The method is simple and easy to adapt to many different cases and assumes that history repeats itself. The workset is to randomly draw historical returns from historical data and based on the previous time period value generate expected value of the assets in the next period. The drawback with historical sim-

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ulation is that it will not consider volatility clustering nor expect more extreme returns than history has shown (Danielsson, 2011).

Example

Consider returns on a period of P days and n zero coupon bonds with maturity 1, 2,. . . , n. Denote the value of the bonds today by X_i, i = 1, . . . , n.

To simulate the value of every zero coupon bond the next period;

1. Generate a random number Z between 1 and P.

2. Draw the Z:th return from the historical returns

3. Add the drawn return from step 2 to the latest observed value of the bonds.

4. Repeat step 1 to 4

2.4.2 GARCH simulation

The GARCH model is designed to forecast volatility of the next time period depending on past time periods log-return and volatility. The general GARCH(p,q) can consider data from multiple time periods back in time:

σ²_t = ω +

p

X

i=1

α_iY_t−i² +

q

X

j=1

β_jσ_t−j² (2.13)

Y_t= σ_tZ_t (2.14)

where σ is the standard deviation, Y denotes daily returns, ω is a constant ensuring that the variance can never become zero. α is the reaction coefficient explaining how much the previous return affects the future volatility. β is called the persis- tence coefficient explaining how much the previous volatility affects the volatility in the next time step. GARCH(1,1) only consider data from the last time period t-1:

σ²_t = ω + αY_t−1² + βσ²_t−1 (2.15)

Yt= σtZt (2.16)

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GARCH(p,q) models future conditional variance σ_t² to estimate movements in the market. Eventually, these shocks will slowly move towards the unconditional variance ¯σ_t² specified in Equation 2.17.

¯

σ² = ω

1 − (α + β) (2.17)

There are multiple extensions to the original GARCH(p,q) to handle some of the weaknesses. One issue with multivariate GARCH is the amount of parameters that are needed to predict the correlation matrix in the next period for all assets.

The Orthogonal GARCH or OGARCH is one way to reduce the size of parameter estimation. It uses the orthogonal uncorrelated coefficients from principal component analysis to create a covariance matrix which is then used to simulate next periods volatility of each asset respectively (Alexander, 2001).

Akaike Information Criteria

GARCH(p,q) models require parameter estimation which can be extensive with a high number of assets. There is always a risk of over fitting the model to historical data, which can cause faults in future estimations. The chosen number of lags in the models mentioned above will result in different numbers of parameters to estimate and to ensure that the correct number of lags are used an evaluation criteria such as Akikie Information Criteria can be used. This criteria is based on the Log-likelihood function and attempts to find the number of lags that will not

"under-fit" nor "over-fit" the model. A model with a high number of lags has a tendency to "over-fit" the model by replicating historical patterns in an unnatural way. By assuming that the fitting error is normally distributed the AIC can be expressed as in Equation 2.18 (Nakamura et al., 2006)

AIC(p) = n lne^Te

n + 2p (2.18)

where n is the number of observations, e is the fitting error and p is the number of parameters. The value of p that minimizes the function is the optimal (ibid.).

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Maximum Likelihood Estimator

The parameters used in a GARCH(1,1) model are estimated using the Maximum Likelihood method. The method determines the parameters ω, α, β that maximizes the likelihood of observed returns in the market, yt, . . . , yT recursively. The log- likelihood function to maximize is stated as the following:

L(ω, α, β) = −T − 1

2 log(2π) − 1 2

T

X

t=2

{log(σ_t²) + y²_t

σ_t²} (2.19)

where the term −^{T −1}₂ log(2π) can be removed since it is a constant and will not affect the maximization and σ²_t = ω + αy_t−1² + βσ_t−1² from 2.15 (Danielsson, 2011).

2.4.3 Monte Carlo

In finance, many decisions has to be made relying on assumptions about the future.

One way to project future outcomes is to use the so called Monte Carlo simulations. The purpose of the method is to project many different outcomes based on random variables and draw conclusions about the probability of a specific outcome.

Applying Monte Carlo simulation requires assumptions about the distribution of data or the expected outcome which to some extents is a limitation of the Monte Carlo method. However, the Monte Carlo method is a powerful tool that can be applied to many different problem that can not easily be solved analytically. In a stock market a dependency exist to some level between the single stocks in the market. The problem of evaluating a certain portfolio with correlation can be solved by using Monte Carlo simulation. The random variables generated to simulate future returns for every time period are assumed to have the same correlation as the historical returns over a specified period (Platon and Constantinescu, 2014).

Definition (Monte Carlo). Consider the integral I =

Z

Ω

f (x)µ(dx)

where f is an arbitrary function and µ is a measure over Ω with the numerical

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approximation

I =ˆ 1 N

n

X

i=1

f (xⁱ)

where {X}ⁿ_i=1 is n random numbers with distribution µ. According to the Strong Law of Large Numbers, the approximation ˆI converges to I as n goes towards infinity.

n→∞lim I −ˆ→ I

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Method

3.1 ALM

The ALM model created in this thesis aims to capture different market scenarios and project the result in the simplified solvency balance sheet. To do this, certain assumptions have been made. The cash outflows are assumed to be constant in every time step which can be interpreted as the liabilities being constant over time.

The liability side is constructed by the Best Estimates. i.e the discounted value of the future cash flows. The cash flows are determined by the size of the premiums paid by the customer and the future payments from the company to the customer.

After setting up the framework of the model different market scenarios are generated to evaluate future solvency of the company. These scenarios are constructed with different methods to capture the interest rate risk. Principal Component Analysis will be used to capture the likelihood of different events based on historical scenarios. Another method based on the Nelson-Siegel framework will be used in order to capture direct shocks to the yield curve without respect to the likelihood of the event.

By using these scenarios, the allocation strategy Buy and Hold is evaluated in its original form but also by matching the duration of the assets with the duration of the liabilities. The simplified asset portfolio containing zero-coupon bonds aims to replicate the duration of Skandia’s actual portfolio likewise the duration of the liabilities. In every time step in each scenario the balance sheet is calculated.

Algorithm ALM

1. The first step in the model is to extract the expected cash flows.

2. Calculate BE for every time step by discounting expected future cash flows for each time period.

3. The value of the assets depends on the change in bond price. This value is calculated by multiplying the total amount invested in each asset with the specific period return of the asset.

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4. The Basic Own Fund is calculated by subtracting the value of Best Estimate from the value of the Assets in all of the 60 time steps.

5. This is repeated until desired number of simulations is met.

6. The SCR is calculated using Value at Risk at 99.5% on the simulated BOF with one year time horizon.

3.2 Scenarios

3.2.1 Nelson and Siegel Scenario

We are interested in finding which of the first three principal components that is most important to capture in a short rate model. According to Diebold and Canlin (2006) the shifts level, slope and curvature are represented by both the principal components and the parameters in the Nelson and Siegel model.

To do this, we want to find which type of shift that affects the balance sheet in a negative direction the most. To ensure that we expose our ALM-model to the specified shifts, we use the Nelson-Siegel model and adopt it to the current interest rates and interpolate and extrapolate to recieve data for all maturities. Then we vary the parameters β₀, β₁ and β₂ within an interval of +/- 1.00 percentage point individually to create different scenarios corresponding to the three shifts described by principal components. These scenarios are run through our ALM- model to determine the effect on the entire balance sheet.

The purpose of generating these scenarios is to show how the different shifts affect the balance sheet. However, this method does not consider how possible a certain scenario is since it is manufactured.

Algorithm Nelson-Siegel

1. Adopt the β-parameters in the Nelson Siegel model to current bond rates for different maturities.

2. Create an interval to change β -parameters +/- 1.00 percentage unit.

3. Create one scenario for every β-parameter individually

4. Calculate rates for all bonds using the Nelson-Siegel model using manipulated β-parameters.

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3.2.2 Historical Simulation of Principal Components

Another way of generating market scenarios is using the principal components based on historical data. By applying historical simulation a perspective of likelihood is added when random principal components are pulled from the historical distribution.

Algorithm Principal Component Simulation

1. Denote X as a time series of returns from highly correlated fixed income securities with different maturities.

2. Subtract the mean, ¯X from each time series in order to standardize the data.

3. Calculate the covariance matrix, Σ for the matrix of standardized returns 4. Calculate the Eigenvectors, W and Eigenvalues, λ of the covariance matrix 5. The principal components, P are then calculated as P = XW

6. Determine the desired amount of variance the principal components are to explain and use the subset of principal components.

7. Randomly select historical principal components, Phist as simulated values.

8. Retrieve the simulated return thanks to the inverse relationship W⁻¹ = W⁰. Data points can be projected as X = P W^T and by pulling random principal components from historical data the simulated returns are expressed as X_sim = P_histW^T

9. Proceed by adding the simulated returns to the last observation of the asset to create a path.

10. Repeat step 7 through 9 until desired number of simulations is met.

3.2.3 OGARCH Simulation

To calculate future volatility and return of the portfolio, the OGARCH(1,1) can be used. GARCH is a univariate method but can be applied to the independent principal components individually to create a multivariate model.

Algorithm OGARCH Simulation

1. Calculate the score Y_t−1 and volatility σ_t−1 for a principal component of the current period.

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2. Estimate the parameters ω, α and β using for example Maximum Likelihood.

3. Calculate σ²_t by using equation 2.15

4. Generate a standard normal random number Z_t with standard deviation σ and mean µ.

5. Calculate the simulated score using equation 2.14 6. Update σ_t−1 with σ_t and Y_t−1 with Y_t

7. Repeat step 1 to 6 for the next principal component.

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Results

This section shows the results on the solvency balance sheet when creating stressed scenarios using the three techniques: Nelson Siegel, Historical Simulation and GARCH(1,1) simulation. Three different types of scenarios have been created using the historical simulation and GARCH techniques. The first type of scenario is based on the historical trend in the market from 2007. The second type of scenario is created by adding a drift to the simulated scenarios. The drift is set to be the opposite of the historical trend in the market. That means if the interest rates have a negative trend historically the trend in these simulations will be positive. This reflects an expectation of rising interest rates in the future. The third type of scenario is set to have no drift. This means that the 50th percentile of the scenarios will remain at the current interest rate level. The three types of simulations will go under the names Historical trend, Emerging trend and No trend. The initial values for the solvency balance sheet is found in Table 4.1.

Assets BE BOF

429,130 238,210 190,920

Table 4.1 – Initial Solvency Balance Sheet (millions)

4.1 Data

The ESG model created in this thesis is based on daily historical data from the time period 2007-06-19 to 2019-02-08 of Swedish and American government bonds and Swedish housing bonds. Swedish swap rates are also simulated in order to discount the liability cash flows. The bonds are used to construct the company asset portfolio and the swap rates are simulated in order to construct the discount rate based on criterias from EIOPA. A list describing the data used in the thesis is found in Table 4.2.

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Assets Swap Rates SE GVB 2Y OMX SEK SWAP 2Y SE GVB 5Y OMX SEK SWAP 3Y SE GVB 7Y OMX SEK SWAP 4Y SE GVB 10Y OMX SEK SWAP 5Y US GVB 5Y OMX SEK SWAP 6Y US GVB 10Y OMX SEK SWAP 7Y US GVB 15Y OMX SEK SWAP 8Y SE BoObl 2Y OMX SEK SWAP 9Y SE BoObl 5Y OMX SEK SWAP 10Y

Table 4.2 – Data used in the models

As a result of the current- and historical state of interest rates in the market, simple returns have been used in the construction of the ESG-model. This makes the occurrence of negative rates possible. Figure 4.1 shows the historical evolution of the interest rates used in the thesis. The data is highly correlated as expected which motivates the use of PCA. The differences in the level of the interest rate is mainly a result from the differences in the Swedish and American bond market.

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(a) American bonds (b) Swedish bonds

(c) Swedish housing bonds (d) Swedish swap rates Figure 4.1 – Historical data

Cash flows

The annually expected future cash flows calculated by Skandia are shown i Figure 4.2. In this thesis they are uniformly divided into monthly cash flows. The positive cash flows arise from maturing bonds and premiums. The negative cash flows correspond to future pay outs. The figure visualizes the mismatch in duration between assets and liabilities.

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Figure 4.2 – Skandia’s expected cash flows

4.1.1 Akaike Information Criteria

The Akaike Information Criteria was used to determine the number of lags in the GARCH(p,q)-model. Figure 4.3 shows the log likelihood estimation results for P=1 to P=4. This means that simulations using GARCH will depend only directly on the last time periods return and volatility.

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Figure 4.3 – Log-likelihood estimation value with Akaike information criteria.

The number of lags was chosen to one for the GARCH(p,q) model based on the data used in this thesis.

4.2 Assets

The portfolio of assets used in the model is constructed to replicate Skandia’s asset duration of approximately three years. However, in line with Skandia’s request only 44% of the total assets will be allocated. The rest of the assets will be held fixed.

The allocation in percentage of total investment in bonds can be found in Table 4.3. The amount invested in US and Housing bonds was chosen together with Skandia to match their actual portfolio.

Asset Allocation SE GVB 2Y 56.0%

SE GVB 5Y 7.0%

SE GVB 7Y 3.5%

SE GVB 10Y 3.5%

US GVB 5Y 9.0%

US GVB 10Y 4.5%

US GVB 15Y 1.5%

SE BoObl 2Y 10.5%

SE BoObl 5Y 4.5%

Table 4.3 – Asset portfolio

INTEREST RATE RISK: A comparative study aimed at finding the most crucial shift in interest rate curves for a life insurance company