University of Groningen Master’s Thesis Econometrics Interest rate risk modeling using the Dynamic Nelson-Siegel model

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University of Groningen

Master’s Thesis Econometrics

Interest rate risk modeling using the Dynamic Nelson-Siegel

model

Gašper Močnik s2818612

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Master’s Thesis Econometrics Supervisor: Prof. Dr. P. A. Bekker

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Interest rate risk modeling using the Dynamic Nelson-Siegel

model

Gašper Močnik

Abstract

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1 Introduction

In the beginning of 2016, the European insurance regulation changed, as new directive called Solvency II became applicable. The new regulatory regime changed the way insurance compa-nies calculate their required amount of capital. Solvency II imposes a so called total balance sheet approach, where risks stemming from the assets and liabilities have to be taken into account when calculating the required amount of capital. One of the risks that insurance companies face is interest rate risk which affects the company’s assets as well as liabilities. Solvency II prescribes the way of calculating capital requirement stemming from interest rate risk, but insurance companies can develop internal models . The purpose of this thesis is therefore to try to develop an alternative way of calculating interest rate risk, which could potentially be used as an internal model.

In this thesis we will make use of the popular dynamic Nelson-Siegel model for term structure of interest rates . The model was originally proposed by Nelson and Siegel (1987) and then later presented in the dynamic environment by Diebold (2006). They have shown that the model fits the cross-section of yields well and that it produces better out-of-sample forecasts than various other term structure models. The model also has many variations and extensions. Diebold, Rudebusch, and Aruoba (2006) and De Pooter, Ravazzolo, and Van Dijk (2010) include macroeconomic variables into the model whereas Diebold, Li, and Yue (2008) and Modugno and Nikolaou (2009) extend the model to allow for term structures of multiple countries.

Majority of applications of dynamic Nelson-Siegel model explore forecasting performance, but there is little written about using it for simulation of future yields. This is what we will explore in this thesis. We will estimate dynamic Nelson-Siegel model for term structures of two countries, namely the US and Germany. The estimated model will then be used to simulate future yield curves. These will in turn be used to calculate interest rate risk of a portfolio of Sava Reinsurance company. Results will be compared to the standard approach described in Solvency II.

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2 European insurance regulation

2.1 Solvency II

Solvency II is the new European regulatory regime for insurance and reinsurance companies. Framework of Solvency II is set up in Directive 2009/138/EC (Solvency II) which was adopted in November 2009 and was later amended by directive 2014/51/EU (Omnibus II). It became applicable on 1 January 2016, replacing previous regulatory regime called Solvency I. Solvency II has a three pillar structure. First pillar defines the capital requirements for insurance and reinsurance companies, second pillar deals with governance and third pillar deals with compliance and reporting. Below we briefly describe the calculation of capital requirements within the Solvency II regime.

Solvency II introduces two boundaries for the amount of capital of insurance or reinsurance companies. First one is Solvency Capital Requirement, hereafter abbreviated SCR, and the second one is Minimum Capital Requirement, hereafter abbreviated MCR. SCR is defined as

the Value-at-Risk of the basic own funds1 _{of an insurance or reinsurance undertaking subject}

to a confidence level of 99.5 % over a one-year period. The amount of capital in the size of SCR should provide insurance companies with enough capital to withstand loss corresponding to a 1 in 200 year event. It has to cover at least the following risk modules

• non-life underwriting risk, • life underwriting risk, • health underwriting risk, • market risk,

• credit risk, • operational risk.

Insurance and reinsurance companies have two options available for calculation of SCR. First one is to calculate SCR according to standard formula and the second one is to use full or partial internal model. Standard formula is briefly described in Directive 2009/138/EC and in more detail in Commission delegated regulation supporting the Directive 2009/138/EC. Figure 1 presents risk modules and sub-modules of SCR according to standard formula approach.

1

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Figure 1: Structure of SCR standard formula modules

The main component of SCR is Basic Solvency Capital Requirement, hereafter abbreviated BSCR. It is divided into main risks modules that represent most important risks that insurance or reinsurance company faces. These are non-life underwriting risk, life underwriting risk, health underwriting risk, market risk and counterparty default risk. All of these modules, with the exception of counterparty default risk module, are then further divided into sub-modules. Calculation of SCR is done from bottom to top, meaning that the first step consists of calculating partial SCR for those sub-modules of SCR that are at lowest level in the tree structure shown in Figure 1. For each of these sub-modules, the standard formula prescribes the method for calculating its partial SCR. These usually come in shocks to either directly the value of asset or liability, or to some parameters that are used in valuation. These shocks were calibrated using the Value-at-Risk measure with a 99.5 % confidence over one year period. Next, we move one level up in the tree and calculate partial SCR for each of the modules in that level. This is done by aggregating bottom level partial SCR using correlations given

in Directive 2009/138/EC. For example, partial SCR for market risk, denoted SCRmarket, is

calculated as SCRmarket = s X i X j Corrij · SCRi· SCRj, (1)

where the the i and j go through all sub-modules of market risk and Corrij is the correlation

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way, i.e. by aggregating the partial SCR from modules directly below it. It is calculated as BSCR = s X i X j Corrij· SCRi· SCRj+ SCRintangibles, (2)

where i and j now go through all of the main sub-modules of BSCR directly below it in the tree structure, with the exception of partial SCR for intangible asset risk, which is just added. Finally, the SCR of an insurance company is calculated as

SCR = BSCR + SCRoperational+ Adj, (3)

where SCRoperational is partial SCR for operational risk and Adj is adjustment for loss

ab-sorbing capacity of technical provisions and deferred taxes.

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3 Yield curve modeling

3.1 Yield curve basics

Before we describe dynamic Nelson-Siegel model we will describe some of the fundamental

concepts used in yield curve modeling. Let Pt(τ )denote the price at time t of discount bond,

i.e. a bond that pays $1 at maturity, which matures at time t + τ. Assuming continuous compounding, the price of such bond is given by

Pt(τ ) = e−τ yt(τ ), (4)

where yt(τ )is yield to maturity of discount bond with maturity τ, also known as zero-coupon

yield or spot rate. Equation (4) is called a discount curve, it represents the price of $1 receivable as a function of maturity τ. We can of course invert (4) to get zero coupon yield curve or spot curve as

yt(τ ) = −

1

τ log(Pt(τ )). (5)

Another important concept is forward rate. Forward rate is interest rate set at time t for

period between t + τ1 and t + τ2, where we assume that τ2> τ1. We then have the following

relation

eyt(τ1)τ1_eft(τ1,τ2)(τ2−τ1)_{= e}yt(τ2)τ2_. (6)

Solving (6) for ft(τ1, τ2)gives

ft(τ1, τ2) = −

log(Pt(τ2)) − log(Pt(τ1))

τ2− τ1

. (7)

Taking the limit of (7) as τ2− τ1 ↓ 0yields the instantaneous forward rate

ft(τ ) = −

∂ log(Pt(τ ))

∂τ = −P

0

t(τ )/Pt(τ ), (8)

where ft(τ ) is instantaneous forward rate. It can be interpreted as return for infinitesimal

period after t + τ. Using (5) and (8) we get the following relation for zero coupon yields

yt(τ ) = 1 τ Z τ 0 ft(u)du. (9)

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3.2 Dynamic Nelson-Siegel model

In their paper, Diebold (2006) use the functional form proposed by Nelson and Siegel (1987). In particular, Nelson and Siegel (1987) use the following form for the instantaneous forward rate curve

ft(τ ) = β1t+ β2te−λtτ+ β3tλtτ e−λtτ. (10)

Using (9), we have the following form for the zero coupon yield

yt(τ ) = β1t+ β2t 1 − e−λtτ λtτ + β3t 1 − e−λtτ λtτ − e−λtτ . (11)

Equation (11) is the so called dynamic Nelson-Siegel model as proposed in Diebold (2006).

Parameters β1t, β2t and β3t of (11) are latent factors called level, slope and curvature,

respec-tively. Fourth parameter λt affects the exponential decay of factor loadings. Factor loadings

of β1t, β2t and β3t are 1, 1 − e−λtτ /λtτ and 1 − e−λtτ /λtτ − e−λtτ, respectively. To

understand why parameters are interpreted as level slope and curvature, it is useful to inspect each factor loading as a function of maturity, τ. This is shown in Figure 2.

Figure 2: Nelson-Siegel factor loadings as functions of maturity in months. Blue line represents loading

on β1t, red curve represents loading on β2t and black curve represents loading on β3t.

Loading on β1t is a constant 1, so β1t can be interpreted as a long term factor since,

unlike the other two factors, constant factor loading ensures it will have an effect on yields with long maturities. Because it affects yields of all maturities with the same magnitude it

is also called level, it determines the overall level of yield curve. Loading on β2t starts at 1

and then decays to 0 as maturity increases. Parameter β2t thus has a greater effect on yields

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the slope of the yield curve. Parameter β2t has therefore been interpreted as slope. Loading

on β3t starts at 0, rises, and then decays to 0 as maturity increases. It mostly affects yields

at medium maturities, so β3t can be thought of as a medium term factor. It is also called

curvature, since it affects yields with medium maturity the most and therefore changes the curvature of the yield curve.

By taking the limit of (11) when τ approaches infinity, we get lim

τ →∞yt(τ ) = β1t. (12)

The level factor can therefore be interpreted as yield of infinite maturity. By taking the limit of (11) when τ approaches zero, we get

lim

τ →0yt(τ ) = β1t+ β2t. (13)

This means that the short rate is the sum of β1t and β2t.

The key insight of dynamic Nelson-Siegel model is that the variation of yield curve is explained by the variation of just three latent factors.

3.3 Dynamic Nelson-Siegel model for multiple countries

Since we are interested in modelling interest rate risk of a portfolio of assets and liabilities, we want to be able to jointly model dynamics of yield curves of multiple countries at the same time. Insurance companies usually have investments and liabilities that are denominated in multiple currencies. To calculate interest rate risk we need a model that can capture joint movement of yields of multiple countries. We therefore move to the global environment, one with multiple countries, each of them having their own term structure of interest rates. We assume that yields are correlated across countries and we are interested in how to jointly model their evolution in time. We can easily extend the framework proposed in Diebold (2006) to include multiple countries. One could in principle estimate equation (11) separately for each country and then jointly model the dynamics of latent factors. We can also combine equation (11) for all countries in the model and estimate it with OLS in one step. We describe this below.

For clarity, let us change the notation for latent factors. We denote β1t , β2t and β3t

as lt, st and ct, respectively, since we interpret them as latent level, slope and curvature

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where i = 1, . . . , N and t = 1, . . . , T .

Now suppose that we observe zero coupon yields at Ji different maturities for country i

so τ = τ1, τ2, . . . , τJi . By stacking yields for all observed maturities for country i at time t,

we can rewrite (14) as

yit= Λifit+ εit, (15)

where yit = (yit(τ1), . . . , yit(τJi))

0 _{is J}

i× 1 of zero coupon yields, fit = (lit, sit, cit)0 is 3 × 1

vector of latent factors , εit= (it(τ1), . . . , it(τJi))

0 _{is J}

i× 1vector of error terms and

Λi =        1 1−e_λ−λitτ1 itτ1 1−e−λitτ1 λitτ1 − e −λitτ1 1 1−e_λ−λitτ2 itτ2 1−e−λitτ2 λitτ2 − e −λitτ2 . . . . 1 1−e_λ−λitτJi itτ_Ji 1−e−λitτJi λitτ_Ji − e −λitτ_Ji       

is Ji× 3matrix of factor loadings.

By stacking equation (15) for all countries we get

yt= Λft+ εt, εt∼ (0, H) (16)

where yt = (y01t, y2t0 , . . . , yN t0 )

0

is (J1+ J2+ . . . + JN) × 1 vector of zero coupon yields,

ft = (f1t0 , f2t0 , . . . , fN t0 )

0 _{is a (3N) × 1 vector of latent factors, ε}

t = (ε01t, ε02t, . . . , ε0N t)

0

is (J1+ J2+ . . . + JN) × 1 vector of errors and factor loadings matrix Λ is of dimension

(J1+ J2+ . . . + JN) × (3N )and is defined as Λ =        Λ1 0 . . . 0 0 Λ2 . . . 0 . . . . 0 0 . . . ΛN        .

We assume error terms εt are independent over time and distributed with mean 0 and

(J1+ J2+ . . . + JN) × (J1+ J2+ . . . + JN) covariance matrix, which we denote H.

Next, we need to specify the dynamics of ft. We assume it follows first order vector

autoregressive process

ft= c + Φft−1+ ηt, ηt∼ (0, Q) (17)

where c is 3N × 1 vector of constants, Φ is 3N × 3N matrix. Error terms ηt are assumed to

be independent over time and distributed with mean 0 and 3N ×3N covariance matrix which

we denote Q. Additionally, we assume that εt and ηtare independent of each other and that

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The model (17) could also be represented differently. By subtracting ft−1from both sides

of (17) we get

∆ft= c + (Φ − I) ft−1+ ηt (18)

= c + ˜Φft−1+ ηt. (19)

Note that the model (17) does not introduce any bounds to the factors. Since we will use DNS model to simulate future interest rates by simulating factors from estimated model (17), it is desirable that the level factors remain positive. Level factor represents yield of infinite maturity in DNS model and since yields of highest maturities are usually positive, we want to bound the level factor to be positive. We do this by estimating the model with transformed factors, where level factors of each country are transformed as

f_it? = (log (lit) , sit, cit)0.

We assume that transformed factors follow first order vector autoregressive process so we have

f_t? = c2+ Φ2ft−1? + ut, ut∼ (0, Q2) (20)

where f?

t = (f1t?, f2t?, . . . , fN t? )is (3N) × 1 vector of transformed factors, c2 is (3N) × 1 vector

of constant terms Φ2 is a 3N × 3N matrix of parameters and ut is (3N) × 1 vector of error

terms. We assume error terms utare distributed with mean 0 and 3N ×3N covariance matrix

which we denote Q2.

By subtracting f?

t−1 from both sides of (20) we can rewrite the model as

∆f_t? = c2+ (Φ2− I) ft−1? + ut (21)

= c2+ ˜Φ2ft−1? + ut. (22)

We assume that ut and εt are independent.

Diebold (2006) propose estimation of DNS model in two steps. The first step consists of estimation of model (16). They use the fact that model (16) can be estimated by OLS if one

is willing to fix the parameter λtat some predetermined value. As noted in Nelson and Siegel

(1987), λt affects the decay of the factor loadings. Large values of λt mean that the factor

loadings will decay to zero quickly and model (16) will provide a good fit at short maturities

at the expense of a poorer fit at longer maturities. Small values of λtcorrespond to the slow

decay of factor loadings which will be able to provide a good fit for longer maturities at the

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curvature factor achieves its maximum. Diebold (2006) calibrate λt so that curvature factor

loading is maximized at precisely 30 months. They get that λt= λ = 0.0609per month. With

parameter λ now fixed, the model (16) can be estimated by OLS at each t. This produces time

series of estimated vector of factors ˆft. Alternatively, one could also estimate parameter λtby

estimating the model (16) with NLS for each t or by treating λ as time invariant parameter, but rather than calibrating it beforehand, it could be estimated so that the overall sum of squared residuals (SSR) from model (16) is minimized. The second step consists of estimation

of model (17), where latent factors ft is replaced by its estimate ˆft which was estimated in

the first step. Hence, this estimation approach is called the two-step method in Diebold et al. (2006) and Diebold and Rudebusch (2013). Note that we measure τ in months when fitting the Nelson-Siegel model.

Note that if one uses two-step estimation approach, no explicit restrictions on covariance matrices of error terms H and Q are necessary. These matrices may contain large number of parameters and if the parameters are estimated using maximum likelihood estimation, the large number of parameters might make the optimization of the likelihood function infeasible. When estimating the model with two-step method, estimation of covariance matrices H and

Q does not introduce any additional computational burden and we therefore chose to leave

them unrestricted.

Diebold et al. (2006) present an alternative estimation approach they call one-step DNS. They use the fact that DNS model can be treated as dynamic factor model and then use a Kalman filter for factor extraction. The parameters are then estimated by maximum likeli-hood. Dynamic factor models are useful in situations where a high dimensional data set is driven by a low dimensional set of latent factors, which is the case for modeling yield curves. They can be elegantly written in state space form. State space system consists of observation or measurement equation and state or transition equation. The observation equation relates observations to the latent factors and the state equation describes the dynamics of latent fac-tors. The observation equation is given by (16) and state equation is given by (17). Diebold et al. (2006) assume covariance matrix H is diagonal which implies that error terms in (16) are uncorrelated. This restriction is done to reduce the number of parameters in the model since covariance matrix of measurement equation can be quite large. They also assume unre-stricted covariance matrix Q, which allows error terms in (17) to be correlated. Additionally

they assume that error terms εt and ηt are independent of each other and of initial state

f0. For any given parameter configuration Kalman filter gives optimal filtered estimates of

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step method is in principle superior to the two-step method but that little is lost when using

two-step method since there is usually enough cross-sectional variation that ft is estimated

very precisely at each t.

Diebold (2006) examine out-of-sample forecasting performance of DNS model estimated with two-step method. They estimate two variations of the model, one where latent factors are modeled as VAR(1) process and one where each latent factor is modeled as AR(1) process. Forecasting performance of the two variations of DNS model is compared to selection of other models with random walk model being the benchmark. Comparison is done for yields of various maturities and for various forecast horizons. They use monthly US treasury yields for period from January 1985 through December 2000. Out-of-sample forecasting is done for the period from January 1994 through December 2000. Their results show that DNS with AR(1) factor dynamics produces out-of-sample forecasts with lower RMSE than random walk benchmark. The difference is negligible at forecast horizon of 1 month, but as forecast horizons becomes longer the differences between DNS and random walk increase in favour of the former.

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4 Interest rate risk modeling

4.1 Interest rate risk modeling with Dynamic Nelson-Siegel model

We now turn to modeling interest rate risk. Interest rate risk is the risk of adverse changes in value of portfolio due to the fluctuations in the term structure of interest rates. Insurance companies have interest rate sensitive investments as well as liabilities, so changes in interest rates affect both sides of company’s balance sheet. Therefore, interest rate risk has to be calculated on the basis of net asset value, hereafter abbreviated NAV, which is the difference in value between assets and liabilities. For the purpose of modelling interest rate risk, we take into account only those assets and liabilities that are sensitive to interest rate risk movement. Assets that are sensitive to interest rate risk movement belong to the class of so called fixed income investments. Fixed income investments have the property that the issuer of such a security has obligation to pay interest and principal according to predefined fixed schedule. To find the value of such an investment, each future cash flow stemming from it can be discounted with appropriate interest rate. The value of fixed income investment is then the sum of discounted cash flows. Since the majority of fixed income investments are not risk-free, one can not simply discount the future cash flows with risk-free interest rates but has to take into account the risk associated with the investment into account, basically meaning that that the spread should be added to risk-free interest rate when discounting the cash flows. The market value of fixed income security, assuming continuous compounding, can therefore be written as

M VF I =

n

X

i=1

CFF I,ie−τa,i(y(τa,i)+s), (23)

where MV is market value of fixed income security, CFF I,iis the cash flow from fixed income

investment with maturity τa,iyears, y(τa,i)is the spot rate (risk-free) with maturity τa,iand s

is spread that is added to spot rate which is constant for all maturities. In practice however, the value of majority of fixed income investments is known and formed on the market. At

given time of valuation, MV , CFiand y(τa,i)are known so we can solve (23) for s numerically.

The spread over risk-free term structure s is also called the Z-spread. Note that because zero coupon yields are represented on annualized basis, the time to maturity must be measured in years which is represented by the subscript a.

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M Vliabilities= n

X

i=1

CFliabilitiesie

−τa,iy(τa,i)_, (24)

where CFliabilitiesi is expected cash flow from liabilities with maturity τa,iyears. We can then

define NAV as

N AV = M Vassets− M Vliabilities, (25)

where MVassets is the market value of the whole fixed income portfolio.

Under the Solvency II standard formula approach, the modules of BSCR are calibrated using Value-at-Risk measure with a 99.5 % confidence level over one year period. We therefore choose to use the same measure and confidence level in our analysis. The following definition of VaR is taken from McNeil, Frey, and Embrechts (2015). Let L denote a loss and α ∈ (0, 1) dentote a cofidence level. VaR at confidence level α is then given by the smallest number l such that the probability that L exceeds l is smaller or equal to (1 − α). It is formally defined as

V ARα= inf {l ∈ R : P (L > l) ≤ 1 − α} = inf {l ∈ R : FL(l)) ≥ α} . (26)

Note that L is a positive number. To calculate interest rate risk we therefore need the distribution of losses in NAV that stem from interest rate risk movement. Suppose we are at

valuation date t. The first step is to calculate NAV at time t, which we denote NAVt. Then,

we calculate Z-spread for every fixed income investment in the portfolio. Note that we use fitted yields from Nelson-Siegel model when discounting cashflows and not actually observed yields, since the former are available for any maturity. We will be modeling interest rate risk that arises due to fluctuations of risk-free interest rates and we will assume that Z-spread is constant through time.

The second step consists of simulating large number S of one-year ahead zero-coupon yield curves via simulation of Nelson-Siegel factors from the estimated model (17). Suppose we are at time t. The h-day ahead vector of Nelson-Siegel factors is given by

ft+h=  I + h−1 X j=1 Φj  c + Φhft+ h X j=1 Φh−jηt+i (27) = h−1 X j=0 Φjc + Φhft+ h X j=1 Φh−jηt+i, (28)

where Φ0 _{= I}_{. The conditional expectation of f}

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Since we assumed the ηtare independent over time, we have the following expression for the

conditional variance of ft+h at time t

Σf(h) =vart(ft+h) = h X j=1 Φh−jQ Φh−j 0 . (30)

If we make an additional assumption that error terms ηtare normally distributed, the

condi-tional distribution of ft+h given ft is also normal. We have

ft+h|t ∼N (µf,t(h), Σf(h)) , (31)

where µf,t(h)is given in (29) and Σf(h)is given in (30).

Now suppose that we have estimated the DNS model. Let ˆc and ˆΦbe the estimated vector

of constants and matrix of autoregressive coefficients from model (17), respectively, and ˆQ

be the estimated covariance matrix of error terms from the model (17). Also, let ˆfT be the

estimated vector of factors at the last observation date T . We could then estimate µf,T(h)

and Σf(h)using estimated parameters as

ˆ µf,T(h) = h−1 X j=0 ˆ Φjc + ˆˆ ΦhfˆT, (32) and ˆ Σf(h) = h X j=1 ˆ Φh−jQˆ ˆΦh−j 0 . (33)

One way of simulating the h - period ahead factors is therefore to simulate them from

multi-variate normal distribution with mean ˆµf(h) and covariance matrix ˆΣf(h).

If we do not assume normality we do not know the exact h - period ahead distribution and so we have to do the simulation differently. In this case we simulate the factors iteratively, one period at the time, using estimated parameters and error terms that are drawn from empirical

distribution of residuals from model (17) which we denote ˆηt. Concretely, we have

˜

fT +1 = ˆc + ˆΦ ˆfT + ˜ηT +1, (34)

where ˜fT +1 denote a simulated factors in period T + 1 and ˜ηT +1 denotes a simulated vector

of errors which were drawn from empirical distribution of ˆηt. By iterating (34) h times gives

a simulated path of factors from T to T + h. This is then repeated large number of times to get the distribution of factors at time T + h.The simulation is performed in the same way for model (20).

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simulation we then revalue the portfolio of assets and liabilities using the simulated one-year ahead yields, keeping the spread s constant. We get simulated one-year ahead loss in NAV as Lsim,T +250= N AVt− N AVsim,T +250, where sim = 1, . . . , S. This way we get the distribution

of one-year ahead losses in NAV. The final step is to calculate VaR of simulated distribution of Lsim,T +250.

4.2 Interest rate risk under Solvency II

Interest rate risk is defined in Article 105 of Directive 2009/138/EC as sensitivity of the values of assets, liabilities and financial instruments to the changes in term structure of interest rates or in the volatility of interest rates. A detailed description on interest rate risk capital requirement calculation according to the standard formula is given in Articles 165, 166 and 167 of Commission delegated regulation supplementing the Directive 2009/138/EC. Capital requirement for interest rate risk is defined as the larger of the following:

1. the sum, over all currencies, of the capital requirements for the risk of an increase in the term structure of interest rates;

2. the sum, over all currencies, of the capital requirements for the risk of a decrease in the term structure of interest rates.

The capital requirement for the risk of an increase in the term structure of interest rates for given currency is defined as the loss in the basic own funds (NAV), that would result from an instantaneous increase in basic risk-free interest rates for that currency and vice versa for the risk of decrease of term structure of interest rates.

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Maturity (in years) Increase Decrease 1 70 % 75 % 2 70 % 65 % 2 64 % 56 % 4 59 % 50 % 5 55 % 46 % 6 52 % 42 % 7 49 % 39 % 8 42 % 36 % 9 44 % 33 % 10 42 % 31 % 11 39 % 30 % 12 37 % 29 % 13 35 % 28 % 14 34 % 28 % 15 33 % 27 % 16 31 % 28 % 17 30 % 28 % 18 29 % 28 % 19 27 % 29 % 20 26 % 29 % 90 20 % 20 %

Table 1: Increase and decrease of risk free term structure at valuation date t according to Solvency II standard formula.

For maturities other than those in Table 1, the increase and decrease should be linearly interpolated. Interest rates of maturities shorter than 1 year should increase or decrease by 70% and 75%, respectively, and interest rates of maturities greater than 90 years should increase or decrease by 20 %. Increase of basic risk free interest rates at any maturity should be at least 1 percentage point, even if upward shocks in Table 1 do not result in increase of such magnitude. If the basic risk free interest rates are negative, the decrease should be nil.

The first step in calculation of interest rate risk according to Solvency II standard formula

is to calculate NAVt and calculate Z-spread for every fixed income security in the portfolio.

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5 Description of the data

5.1 Yield data

We will estimate the dynamic Nelson Siegel model for zero coupon yields of two countries, the US and Germany. US zero coupon yields are based on daily US Treasury constant maturity yields, which are available on FRED website and are updated daily. The yields are of 11 different maturities, namely 1, 3, 6, 12, 24, 36, 60, 84, 120, 240 and 360 months. Constant maturity yields are available from early 80’s onwards (precise year depends on maturity), but yields are not available at all maturities all the time. We therefore chose to restrict our sample to the period when yields are available for all above specified maturities. After this restriction, we are left with period spanning from July 31 2001 until December 31 2015.

Daily German zero coupon yields are available at Bundesbank’s web page.2 _{These yields}

are fitted yields from Nelson-Siegel-Svensson model. Bundesbank reports yields of maturities from 6 months to 30 years with yearly interval. One and three-month zero coupon yields are not available and we therefore choose to include four and eight-year yields instead of one and three-month yields in order to have the same number of maturities in German data as we do in US data.

We also observe that the time series of US zero coupon yields contains less observations than the time series of German zero coupon yields for the period from July 31 2001 until December 31 2015. US yields are observed at 3607 different dates whereas German yields are observed at 3666 different dates. In order to be working with time series observed at the same time we find the intersection of dates of both datasets and subset both datasets accordingly. In the end we are left with 3547 observation dates.

We present descriptive statistics of US zero coupon yields in Table 2 and descriptive statistics of German zero coupon yields in Table 3. We also present descriptive statistics of empirical level, slope and curvature. For US term structure we define empirical level as thirty-year zero coupon yield, empirical slope as the difference between thirty-year zero and one-month zero coupon yield coupon yield and empirical curvature as two times two-year yield minus the sum of thirty-year yield and one-month yield. Empirical factors for German term structure are defined analogously, with the exception that instead of 1-month zero coupon yield we use 6-month zero coupon yield since we do not have the data for the very shortest maturities.

2

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Maturity in months Mean Standard deviation Minimum Maximum 1 1.31 1.63 0.00 5.26 3 1.36 1.65 0.00 5.16 6 1.46 1.67 0.02 5.26 12 1.58 1.63 0.08 5.23 24 1.85 1.54 0.16 5.22 36 2.13 1.45 0.28 5.19 60 2.69 1.30 0.56 5.16 84 3.13 1.17 0.92 5.34 120 3.56 1.04 1.47 5.46 240 4.33 1.05 2.09 6.31 360 4.41 0.85 2.34 6.01 Empirical level 4.41 0.85 2.34 6.01 Empirical slope 3.10 1.46 -0.76 5.05 Empirical curvature -2.02 1.23 -4.07 0.73

Table 2: Descriptive statistics of US zero coupon yield curve in p.p.

Maturity in months Mean Standard deviation Minimum Maximum

6 1.651 1.516 -0.430 4.500 12 1.721 1.526 -0.440 4.710 24 1.884 1.528 -0.450 4.770 36 2.069 1.522 -0.400 4.750 48 2.259 1.508 -0.320 4.880 60 2.446 1.486 -0.210 5.030 84 2.786 1.427 -0.080 5.230 96 2.936 1.396 -0.030 5.300 120 3.193 1.338 0.070 5.430 240 3.801 1.216 0.410 5.940 360 3.860 1.280 0.540 6.270 Empirical level 3.860 1.280 0.540 6.270 Empirical slope 2.762 1.104 -0.136 4.731 Empirical curvature -1.743 0.739 -3.150 0.190

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US zero coupon yield curve was on average increasing, with the shorter end of the yield curve being more volatile than the long end. German zero coupon yields curve was on average increasing as well, with shorter end being more volatile than the long end, but the difference in volatility between the short and long end of the curve is smaller. Comparing the minimum values of yields reveals that German zero coupon yields have fallen below zero, whereas the US yields have remained positive. The longest maturity for which German yields were negative is eight years. We present plot of time series of one, five, ten and thirty-year yields for the US and Germany in Figure 3.

2005 2010 2015 0 2 4 6 Time Yield in p .p 2005 2010 2015 0 2 4 6 Time Yield in p .p .

Figure 3: Time series of selected yields in p.p. for US (left) and Germany (right). Black lines show one-year yields, blue lines show five-year yields, red lines show ten-year yields and green lines show thirty-year yields.

We see that before the financial crisis short term yields rose substantially and came close or even surpassed the level of long term yields. This period of flat yield curves lasted from August 2005 until December 2007 in US and from September 2006 until December 2008 for Germany. We plot a set of yield curves for the US and Germany in Figure 4.

0 50 100 150 200 250 300 350 0 2 4 6 Maturity in months Yield in % 0 50 100 150 200 250 300 350 0 2 4 6 Maturity in months Yield in %

Figure 4: Selected yield curves for US (left) and Germany (right). Yield curves on July 31 2001 are presented in red and the ones on December 31 2015 are presented blue. Yellow curves represent randomly selected dates.

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Addi-tionally, we plot yield curves at 20 randomly selected dates. We see that yield curves vary substantially through time. Overall level of yield curve decreased significantly from the first observation to the last, but the overall shape remained similar.

We present the correlation matrix of the set of US and German yields in Table 26 in Appendix E. First consider the correlation between yields of the same country. We see that the yields of similar maturities exhibit very strong correlation, this fact holds for yields of both countries. As the difference between maturities of yields increases, the correlation among them gets less strong. This fact is more apparent for US yields. Comparing cross-country correlations we see that the US and German yields are highly correlated as well. This fact seems to justify the need for joint modelling of term structures of different countries.

5.2 Assets and Liabilities data

The assets and liabilities data were provided to us by Sava Reinsurance company. Unfor-tunately we are not allowed to disclose more information about portfolio than reported in yearly report, since the detailed data about the portfolio is confidential. Table 4 presents the structure of investment portfolio of Sava Reinsurance company by the type of investment as reported on December 31 2015.

Type of investment Value on reporting date % of total value

Cash and cash equivalents 285,950.17 0.06

Corporate bonds 112,016,284.28 24.82 Deposits 4,923,272.59 1.09 Deposits at cedents 5,784,421.48 1.28 Government bonds 102,191,733.73 22.64 Investment property 113,702.10 0.03 Investments in subsidiaries 208,240,720.50 46.14 Loans given 2,834,952.65 0.63 Mutual funds 4,075,691.35 0.90 Stocks 10,892,491.15 2.41

Table 4: Value of investment portfolio of Sava Reinsurance company by type of investment in EUR.

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Another important information is the currency in which the investments are denominated. It is crucial since it will determine which term structure will be used for discounting the cash flows stemming from assets and liabilities. The information about the currency structure is not reported at investments level but rather at the the level of all the assets, but it can serve as an approximation of the currency structure of investments. We present it in Table 5.

Currency Ammount % of total value

Euro 458,352,974.00 80.29 US Dollar 43,593,750.00 7.64 Korean Won 18,370,624.00 3.22 Chinese Yuan 8,876,770.00 1.55 Indian Rupee 6,507,058.00 1.14 Bangladeshi Taka 2,403,782.00 0.42 Rest 32,761,753.00 5.74

Table 5: Currency structure of assets of Sava Reinsurance company. All the values are in EUR.

As much as around 80 % of the assets are denominated in Euros making it by far the most important currency of the portfolio. It is followed by the US Dollar and the two currencies together represent almost 88 % of total assets.

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6 Model estimation

We estimate the DNS model using two-step method as proposed in Diebold (2006). The first step consists of estimating model (16). In our two-country setup, we model a set of yields of

11 different maturities in both countries. Therefore, the vector of yields yt and the vector of

error terms εt are of dimensions 22 × 1, loadings matrix Λ is of dimension 22 × 6 and the

vector of latent factors ftis of dimension 6 × 1. Covariance matrix of εt, H, is of dimension

22 × 22 and, since it is left unrestricted, it contains 253 parameters. If parameter λ is fixed

(16) can be estimated with OLS at each time t as ˆ

ft= Λ0Λ

−1

Λ0yt. (35)

The covariance matrix H is estimated as ˆ H = 1 T T X t=1 ˆ εtεˆt0, (36) where ˆεt= yt− Λ ˆft.

As noted before, parameter λ could also be treated as time invariant and estimated so that overall sum of squared residuals of the model (16) is minimized. We present this approach below. The sum of squared residuals at time t is

SSRt(λ) = yt− Λ Λ0Λ −1 Λ0yt 0 yt− Λ Λ0Λ −1 Λ0yt (37) = y_t0yt− yt0Λ Λ 0 Λ−1Λ0yt. (38)

Sum of squared residuals over all t is therefore SSR(λ) = T X t=1 y_t0yt− T X t=1 y0_tΛ Λ0Λ−1 Λ0yt. (39)

Note that λ affects SSR through Λ since it affects its elements. Once the parameter λ has been estimated, the factors can be estimated as shown in (35).

The second step in the estimation procedure is to estimate dynamics of Nelson-Siegel factors estimated in the first step. For the purpose of estimating model (17), we treat the

estimated factors ˆft as observations of ft . Model (17) contains a lot of parameters that

need to be estimated. There are 6 parameters in constant vector c, 36 parameters in matrix

Φ and 21 parameters in covariance matrix Q. We have assumed that covariance matrix Q

is unrestricted, so it allows the error terms of (17) to be correlated. In total we have 63 parameters to estimate. The model (17) can be rewritten as

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where A = (c Φ) is 6 × 7 matrix of parameters and zt−1 is 7 × 1 vector defined as zt−1 =

1 f_t−10 0

. By stacking (40) for all observations we get

F = AZ + U , (41)

where F = (f2, . . . , fT) is 6 × (T − 1) matrix, Z = (z1, . . . , zT −1) is 7 × (T − 1) matrix and

U = (η2, . . . , ηT) is 6 × (T − 1) matrix. Parameter matrix A is then estimated by OLS as

ˆ

A = F Z0 ZZ0−1

. (42)

The covariance matrix Q is estimated as ˆ

Q = 1

T − 8U ˆˆU

0

, (43)

where ˆU = F − ˆAZ. We divide by T − 8 since T − 1 observations are used and there are

7 parameters in each equation of the VAR model (17). The estimate of covariance matrix of parameters is given by

c

varvec( ˆA)= ZZ0−1

⊗ ˆQ, (44)

where ⊗ denotes Kronecker product.

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7 Results

7.1 Estimation of Dynamic Nelson-Siegel

Below we present the estimation results of DNS model for US and Germany. We begin by presenting the results of the first step of the two-step method. We find that the overall

SSR given in (39) is minimized at λ = 0.0387.3 _{At this value of λ, loading on curvature}

factor is maximized at 46.36 months. We see that λ that minimizes the overall SSR is significantly lower than the proposed in Diebold (2006). Descriptive statistics of estimated

factors ˆft=ˆlU S,t, ˆsU S,t, ˆcU S,t, ˆlGER,t, ˆsGER,t, ˆcGER,t are given in Table 6.

Mean Standard deviation Minimum Maximum Augmented Dicky-Fuller statistic

ˆ_l_{U S} _4.94 _0.94 _2.59 _6.81 _-1.65 ˆ sU S -3.62 1.74 -6.06 0.57 -1.42 ˆ cU S -2.87 1.95 -7.50 1.74 -2.48 ˆ lGER 4.31 1.21 0.68 6.77 -0.88 ˆ sGER -2.65 1.23 -5.16 0.00 -1.69 ˆ cGER -2.82 1.53 -5.75 1.12 -2.80

Table 6: Descriptive statistics of estimated Nelson-Siegel factors for the US and Germany. Factors were estimated using λ = 0.0387.

Comparison of the level factors of the two countries reveals that it was higher for US term structure compared to German term structure, indicating that the yields of longest maturities were higher in US. Mean of slope factor was negative for both countries, indicating that the yields curves were on average upward sloping. It was higher for German term structure, indicating that German yield curves were flatter on average. Average curvature factor is similar in both countries. Level factor was found to be the least volatile, followed by slope and curvature, respectively. The last column presents test statistic from Augmented Dicky-Fuller test. Augmented Dicky-Dicky-Fuller test is used to test for the presence of unit root in time series, i.e. to test if the time series are stationary or not. Under the null hypothesis the series contains a unit root and is therefore not stationary, and under alternative the unit root is not present. If the test statistic is lower than the critical value, we can reject the null hypothesis. For this sample size, the critical value for the test statistic at α = 0.05 is −2.86. The presence of unit root therefore can not be rejected for any of the estimated factors.

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To assess the in-sample fit of DNS model, we present descriptive statistics of residuals from DNS model for the US and Germany in Table 7 and Table 8, respectively.

Maturity in months Mean Standard deviation Minimum Maximum RMSE

1 -0.029 0.122 -0.712 0.281 0.125 3 -0.014 0.046 -0.413 0.174 0.048 6 0.037 0.085 -0.164 0.716 0.093 12 0.022 0.103 -0.282 0.551 0.106 24 0.005 0.083 -0.182 0.213 0.083 36 -0.019 0.052 -0.350 0.170 0.055 60 -0.009 0.072 -0.228 0.211 0.073 84 0.002 0.096 -0.244 0.232 0.096 120 -0.027 0.073 -0.226 0.308 0.078 240 0.090 0.138 -0.170 0.501 0.165 360 -0.060 0.129 -0.561 0.218 0.142

Table 7: Descriptive statistics of residuals in p.p. from DNS model for US term structure. DNS model was estimated using λ = 0.0387.

We see that for US terms structure mean residuals are low at all maturities. The last column presents RMSE which we use as a measure of goodness of fit. The model appears to fit the worst at maturities of twenty and thirty years as well as at one month maturity.

Maturity in months Mean Standard deviation Minimum Maximum RMSE

6 -0.016 0.051 -0.347 0.197 0.053 12 0.012 0.028 -0.219 0.296 0.030 24 0.020 0.051 -0.147 0.247 0.055 36 0.007 0.042 -0.149 0.109 0.042 48 -0.009 0.029 -0.105 0.056 0.031 60 -0.019 0.025 -0.103 0.051 0.031 84 -0.018 0.037 -0.119 0.112 0.042 96 -0.009 0.048 -0.117 0.169 0.048 120 0.019 0.070 -0.127 0.247 0.073 240 0.076 0.081 -0.051 0.236 0.111 360 -0.061 0.111 -0.356 0.158 0.127

Table 8: Descriptive statistics of residuals in p.p. from DNS model for German term structure. DNS model was estimated using λ = 0.0387.

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indicate that the model fits the worst at twenty and thirty year maturities. For US, the overall RMSE of the model with estimated λ is 0.103 p.p. and for Germany the overall RMSE equals 0.066 p.p. Overall, the DNS model fits the German yield curves better. This might be due to the fact that data for German yields are fitted yields of Svensson model, which is an extension of Nelson-Siegel model, and as such therefore fit the Nelson-Siegel functional form better. The cross-sectional fit of the model at four different dates for both countries is presented in Figure 7 and Figure 8 in Appendix A. At the chosen dates, the model appears to fit German term structure better.

Time series of the estimated factors for both countries are presented in Figure 5. There is apparent co-movement of factors of the same type. The correlation coefficients between

the estimated factors of the same type are ρˆlU S, ˆlGER

= 0.92, ρ (ˆsU S, ˆsGER) = 0.75 and ρ (ˆcU S, ˆcGER) = 0.73. 2005 2010 2015 3 4 5 6 Estimated level Time Le vel in p .p . 2005 2010 2015 1 2 3 4 5 6 7 Estimated level Time Le vel in p .p . 2005 2010 2015 −6 −5 −4 −3 −2 −1 0 Estimated slope Time Slope in p .p . 2005 2010 2015 −5 −4 −3 −2 −1 0 Estimated slope Time Slope in p .p 2005 2010 2015 −6 −4 −2 0 2 Estimated curvature Time Cur vature in p .p . 2005 2010 2015 −6 −5 −4 −3 −2 −1 0 1 Estimated curvature Time Cur vature in p .p .

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Table 9 presents the estimation results of model (18). We chose to present the estimation results of this variation of the model due to the fact that all the estimated factors are highly

persistent and the estimated diagonal elements of ˆΦ were all near one with extremely high

t-statistics.

constant lU S,t−1 sU S,t−1 cU S,t−1 lGER,t−1 sGER,t−1 cGER,t−1

∆lU S,t 0.033 -0.013 -0.000 -0.001 0.007 -0.000 0.003 (3.422) (-3.660) (-0.280) (-1.309) (2.983) (-0.264) (2.235) ∆sU S,t -0.024 0.013 0.000 0.004 -0.011 -0.004 -0.002 (-2.002) (3.102) (0.127) (4.047) (-3.614) (-2.128) (-1.461) ∆cU S,t -0.068 0.017 0.006 -0.004 -0.007 0.001 -0.008 (-2.290) (1.619) (1.595) (-1.666) (-0.894) (0.113) (-2.332) ∆lGER,t -0.000 0.008 -0.001 -0.001 -0.008 0.003 0.003 (-0.056) (2.630) (-0.840) (-1.201) (-3.552) (1.902) (2.474) ∆s_GER,t -0.006 0.002 0.005 -0.000 -0.001 -0.007 -0.000 (-0.546) (0.525) (3.513) (-0.257) (-0.552) (-4.136) (-0.201) ∆cGER,t -0.089 0.016 0.002 0.012 -0.002 -0.002 -0.021 (-3.321) (1.636) (0.636) (5.233) (-0.335) (-0.458) (-6.384)

Table 9: Table of estimated coefficients and their respective t-statistics from model (18). First column

represents ˆc and the rest of the columns correspond toΦ.ˆ˜

The process is mean reverting if the coefficients on the diagonal of ˆ˜Φ are less than zero.

This is indeed the case for all of the factors except sU S,t and cU S,t where the coefficients of

their lagged values are not statistically different from zero. The cross-factor coefficients are low and do not seem important. The estimation results of model (21) are presented in Table 16 in Appendix B. The results are similar to the ones of model (18).

The residuals ˆηt = (ˆη1t, . . . , ˆη6t) from the model (18) were tested for normality to asses

if it is reasonable to assume that distribution of ηt is normal. The skewness and kurtosis of

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5.99and therefore the null hypothesis of normality is rejected for all of the series of residuals.

Similar results are found for residuals ût= (û1t, . . . , û6t) from model (21), which we present

in Table 17 of Appendix B. Exception is kurtosis coefficient of ˆu4t, which is extremely high.

ˆ

η1t ηˆ2t ηˆ3t ηˆ4t ηˆ5t ηˆ6t

Skewness -0.053 -0.208 0.178 0.134 -0.446 0.000 Kurtosis 5.710 9.274 9.309 7.121 8.605 9.009 Jarque-Bera test statistic 1089.448 5850.883 5908.808 2524.955 4766.844 5343.379

Table 10: Sample skewness and kurtosis coefficients and values of the Jarque-Bera test statistic of residuals from model (18).

7.2 Simulation of Dynamic Nelson-Siegel

Having estimated the models (17) and (20), we performed simulation of the factors using the estimated parameters. We have set the number of simulations to 10000 and simulation horizon was set to 250 days to approximate one-year horizon. As described before, we do the simula-tion in two ways. First, we simulated the 250-days ahead factors under the assumpsimula-tion that

error terms ηt and ut are multivariate normal. The mean and covariance matrix of one-year

ahead simulated factors from the model (17) are given in (32) and in (33), respectively, where

his replaced by 250. The h-day ahead mean and covariance amtrix of the transformed factors

are not given explicitly, but they can be estimated analogously as those of the model (17). We denote the models (17) and (20) with normal errors DNS-N and DNS2-N, respectively. Simulated one-year ahead transformed level factors of both countries from model DNS2-N were transformed back to the original representation by exponentiating.

The second way of simulating the factors or transformed factors is by iteratively simulating one-day ahead at the time, where error terms are drawn from empirical distribution of residuals

ˆ

ηtand ˆut. We will denote models (18) and (21) with errors drawn from empirical distribution

as DNS-E and DNS2-E, respectively. As before, one-year ahead level factors from model DNS2-E were transformed back to the original representation by exponentiating. As a starting value for both simulations we chose the estimated factors on the last observation date T , i.e. December 31 2015, which equal

ˆ

fT = (3.32, −3.20, −0.84, 2.05, −2.21, −4.06)0.

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Mean Standard deviation Minimum Maximum lU S 3.47 0.65 1.24 6.06 sU S -1.81 1.08 -6.10 2.40 cU S -2.15 1.71 -8.39 4.45 lGER 2.28 0.70 -0.38 5.27 sGER -1.75 0.74 -4.87 1.49 cGER -3.18 1.29 -8.91 1.78

Table 11: Descriptive statistics of simulated one-year ahead Nelson-Siegel factors in p.p. for the US and Germany from model DNS-E.

Mean Standard deviation Minimum Maximum

lU S 3.47 0.65 0.92 5.73 sU S -1.81 1.06 -6.02 1.91 cU S -2.15 1.72 -8.07 4.76 lGER 2.29 0.71 -0.31 4.90 sGER -1.76 0.74 -4.71 0.95 cGER -3.18 1.30 -8.19 2.38

Table 12: Descriptive statistics of simulated one-year ahead Nelson-Siegel factors in p.p. for the US and Germany from the model DNS-N.

Mean Standard deviation Minimum Maximum

lU S 3.65 0.58 2.08 6.67 sU S -2.15 1.10 -6.33 1.78 cU S -2.05 1.71 -8.49 4.70 lGER 2.34 0.57 0.89 6.03 sGER -1.89 0.74 -4.92 1.14 cGER -3.14 1.30 -8.94 1.71

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Mean Standard deviation Minimum Maximum lU S 3.66 0.58 1.94 6.51 sU S -2.15 1.09 -6.51 2.11 cU S -2.04 1.72 -8.25 4.55 lGER 2.35 0.58 0.95 6.03 sGER -1.89 0.74 -4.68 0.81 cGER -3.14 1.30 -8.01 1.43

Table 14: Descriptive statistics of simulated one-year ahead Nelson-Siegel factors in p.p. for the US and Germany from the model DNS2-N.

Let us first consider the results of models DNS-E and DNS-N. We see that simulated factors are on average almost the same for both models. Comparing average values to the starting values we observe that both level and curvature factor have drifted towards their sample mean, which holds for US as well as Germany. Slope factor of both countries have, on the other hand, moved further away from the sample mean. Standard deviations from both models are almost the same as well. Looking at the extremes we observe that both models give negative values for the German level factor. Recall that under the Nelson-Siegel functional form, level factor represents the yield of infinitely long maturity, which effectively means that the yields of the longest maturities will be below zero for some simulations. Such a situation seems implausible, even though we observe negative yields even for ten-year German government bonds nowadays. Note that simulation of autoregressive process depends on the starting values. Since we initialized the simulations using estimated of factors from the last observation date when German level factor was very low, the simulated values fall below zero for some simulations. The simulated factors from models DNS2-N and DNS2-E are similar to the ones simulated from DNS-N and DNS-E, with the exception that both level factors are now bounded above zero.

We present descriptive statistics of one-year ahead simulated yields from all four models for Germany in Appendix C and for the US in Appendix D. Table 18, Table 19, Table 20 and Table 21 of Appendix C present simulated one-year ahead yields from models DNS-E, DNS-N, DNS2-E and DNS2-N for Germany, respectively. Table 22, Table 23, Table 24 and Table 25 of Appendix D present the same for the US. The yields are presented at the same maturities as the original data.

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in-dicating that the yields have on average drifted upwards. Comparing standard deviations reveals that the yields of medium length maturities are the most volatile. Models DNS-E and DNS-N produce negative yields for some simulations. In fact, negative yields appeared for all observed maturities, which is an obvious drawback of simulation using DNS model with unbounded level factor. This problem is circumvented by bounding the level factor to be pos-itive, which is visible from results of models DNS2-E and DNS2-N. The fact that simulated level factor is bounded in these two models has resulted in positive simulated yields for twenty and thirty-year maturity which is more consistent with the historical data.

For the US the results are similar. Average simulated one-year ahead yields from models DNS-N and DNS-2 are higher than those observed on the last observation date. Standard deviations reveal that the short end of the curve is most volatile, which is consistent with historical data. Looking at the minimum values we see that simulations from both models give negative yields. The highest maturity that still exhibits negative simulated yields is seven years for DNS-E model and ten years for DNS-N model. The fact that the simulated US level factor did not drop below zero for any of the simulations ensured that twenty and thirty-year yields stayed positive for all simulations. Results of simulations with models DNS2-E and DNS2-N are similar with the difference that negative yields occur only for maturities up to five years. Although negative yields have not been observed in the US during the sample period, we conclude that they are certainly plausible, as is seen nowadays is Europe.

Even though models with transformed factors ensured that the simulated one-year ahead yields of highest maturities remained positive, some simulations still result in yields of short and medium maturities being much lower than those observed in the sample.

7.3 Results of intererst rate risk modeling

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histogram of simulated losses in NAV.

Mean Standard deviation Minimum Maximum V aR0.95 V aR0.99 V aR0.995 DNS-E 2,222,986.27 2,258,351.76 -6,632,230.49 11,175,935.73 5,903,699.30 7,392,251.19 8,024,957.61 DNS-N 2,221,305.60 2,276,857.53 -6,983,826.50 10,953,194.87 5,953,351.90 7,368,944.17 8,057,080.47 DNS2-E 2,254,315.74 1,998,523.35 -5,886,827.49 11,709,646.78 5,579,723.29 7,170,008.09 7,799,342.62 DNS2-N 2,274,399.99 1,981,008.83 -4,386,921.85 11,709,646.78 5,580,039.39 7,107,644.28 7,703,313.96

Table 15: Summary of simulated ∆NAV in EUR

Histogram of ∆ NAV from model DNS−E.

∆ NAV in EUR

Frequency

−5e+06 0e+00 5e+06 1e+07

0

500

1000

1500

Histogram of ∆ NAV from model DNS−N.

∆ NAV in EUR

Frequency

−5e+06 0e+00 5e+06 1e+07

0

500

1000

1500

Histogram of ∆ NAV from model DNS2−E.

∆ NAV in EUR

Frequency

−5e+06 0e+00 5e+06 1e+07

0

500

1000

1500

2000

Histogram of ∆ NAV from model DNS2−N.

∆ NAV in EUR

Frequency

−5e+06 0e+00 5e+06 1e+07

0

500

1000

1500

2000

Figure 6: Histogram of simulated changes in NAV

Simulated distributions of losses from all four models have mean around 2.2 million EUR. This is an indication that portfolio of Sava Reinsurance is adversely affected by the rise of interest rates since the simulated one-year ahead yields are on average higher than the yields on December 31 2015. Comparing VaR of the four models we see that models DNS2-E and DNS2-N produce slightly lower estimates of interest rate risk at all the presented confidence levels. The difference between models where errors were assumed to be normally distributed and the models where no such assumptions were made is small, both versions give nearly identical estimates of interest rate risk.

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Comparing the two approaches reveals that simulation using DNS model gives much higher estimate of interest rate risk. An obvious advantage of DNS approach is that it gives the whole distribution of one-year ahead losses in NAV whereas Solvency II standard formula approach

only gives one number that should correspond to V aR0.995. Insurance companies could be

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8 Conclusion

The goal of this thesis was to try to build a model for the calculation of interest rate risk for portfolio of insurance or reinsurance company, that would provide an alternative to the interest rate risk calculation according to the standard approach described in Solvency II framework. We chose to model zero-coupon yields with the popular dynamic Nelson-Siegel model, which was estimated for the US and German data.

We then proceeded with the simulation of one-year ahead yields by jointly simulating one-year ahead Nelson-Siegel factors of both countries using the estimated DNS model. An appealing feature of this approach is the fact that Nelson-Siegel functional form gives the zero-coupon yield of any maturity, thereby eliminating the need for interpolation between yields. However, the model also has its problems. It turned out that simulations based on DNS model result in one-year ahead German level factor being negative for some simulations. This result is not in line with theory, since level factor represents yield of infinite maturity under Nelson-Siegel functional form, and it seems implausible to imagine the yields of the highest maturities being negative. This problem arises due to the fact the distribution of one-year ahead simulated factors depends on the starting values, and because German level factor was low at the end of our sample, it is not surprising that it falls below zero for some simulation. We have managed to circumvent this problem by simulating the transformed factors, where both level factors have been transformed by taking the natural logarithm. Even with this adjustment, the simulated one-year ahead yields for short and medium maturities were still considerably lower than the lowest values observed in the sample for some simulations. We thus think that DNS model is not the best model for simulating yield curves in low yield environment.

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References

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Diebold, Francis X., Canlin Li, and Vivian Z. Yue (2008). Global yield curve dynamics and interactions: a dynamic nelson–siegel approach. Journal of Econometrics 146 (2), 351–363. Diebold, Francis X. and Glenn D. Rudebusch (2013). Yield Curve Modeling and Forecasting:

The Dynamic Nelson-Siegel Approach. Princeton University Press.

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EU (2009). Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Rein-surance (Solvency II). Official Journal of the European Union.

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Appendices

A

Fitted yields from dynamic Nelson-Siegel model

● ●●● ● ● ● ● ● ● ● 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 7

Observed and fitted yields on July 31 2001

Maturity in months Y ield in p .p . ● ● ●● ●● ● ● ● ● ● 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 7

Observed and fitted yields on August 24 2005

Maturity in months Y ield in p .p ●●●●● ● ● ● ● ● ● 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 7

Observed and fitted yields on September 15 2009

Maturity in months Y ield in p .p ● ● ● ● ● ● ● ● ● ● ● 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 7

Observed and fitted yields on December 31 2015

Maturity in months

Y

ield in p

.p

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● ●●●●● ●● ● ● ● 0 50 100 150 200 250 300 350 0 2 4 6

Observed and fitted yields on July 31 2001

Maturity in months Y ield in p .p ● ●●●●● ●● ● ● ● 0 50 100 150 200 250 300 350 0 2 4 6

Observed and fitted yields on August 24 2005

Maturity in months Y ield in p .p ● ● ● ● ● ● ● ● ● ● ● 0 50 100 150 200 250 300 350 0 2 4 6

Observed and fitted yields on September 15 2009

Maturity in months Y ield in p .p ● ●●●●● ●● ● ● ● 0 50 100 150 200 250 300 350 0 2 4 6

Observed and fitted yields on December 31 2015

Maturity in months

Y

ield in p

.p

Figure 8: Observed yields versus fitted yields for German term structure. Dots represent actual yields and blue lines correspond to Nelson-Siegel fitted yields from the model with λ = 0.0387.

B

Estimation results of model (21)

constant log lU S,t−1 sU S,t−1 cU S,t−1 log lGER,t−1 sGER,t−1 cGER,t−1

∆ log lU S,t 0.011 -0.009 -0.000 -0.000 0.003 -0.000 0.001 (3.094) (-2.767) (-0.142) (-0.903) (1.826) (-0.360) (2.209) ∆sU S,t -0.031 0.033 -0.000 0.003 -0.019 -0.004 -0.002 (-1.611) (1.814) (-0.021) (3.466) (-2.070) (-1.910) (-1.234) ∆c_{U S,t} -0.107 0.084 0.007 -0.004 -0.027 -0.000 -0.008 (-2.255) (1.836) (1.638) (-1.797) (-1.146) (-0.018) (-2.329) ∆ log lGER,t -0.008 0.015 0.000 -0.000 -0.009 0.000 0.001 (-1.849) (3.376) (0.282) (-1.791) (-4.174) (0.784) (2.113) ∆sGER,t -0.005 0.005 0.005 -0.000 -0.003 -0.007 -0.000 (-0.324) (0.289) (3.354) (-0.376) (-0.363) (-3.994) (-0.079) ∆cGER,t -0.113 0.058 0.001 0.012 0.000 -0.001 -0.020 (-2.632) (1.396) (0.315) (5.332) (0.001) (-0.267) (-6.317)

Table 16: Table of estimated coefficients and their respective t-statistics from model (21). First column

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ˆ

u1t uˆ2t ˆu3t uˆ4t uˆ5t uˆ6t

Skewness -0.054 -0.256 0.181 0.146 -0.452 -0.013 Kurtosis 6.687 9.269 9.310 25.107 8.615 9.019 Jarque-Bera test statistic 2014.372 5855.104 5911.927 72315.572 4787.669 5361.056

Table 17: Sample skewness and kurtosis coefficients and values of the the Jarque-Bera test statistic of residuals from model (21).

C

Descriptive statistics one-year ahead simulated German yields

6 0.399 0.579 -1.867 2.436 12 0.333 0.606 -2.070 2.593 24 0.324 0.661 -2.206 2.865 36 0.409 0.696 -2.269 3.061 48 0.536 0.712 -2.209 3.365 60 0.677 0.716 -2.157 3.612 84 0.946 0.711 -1.949 3.978 96 1.063 0.706 -1.834 4.114 120 1.259 0.697 -1.622 4.323 240 1.750 0.684 -1.034 4.790 360 1.926 0.686 -0.815 4.952

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Maturity in months Mean Standard deviation Minimum Maximum 6 0.400 0.583 -1.516 2.486 12 0.335 0.610 -1.820 2.599 24 0.325 0.667 -2.098 2.925 36 0.411 0.703 -2.198 3.102 48 0.538 0.720 -2.172 3.190 60 0.679 0.725 -2.075 3.257 84 0.948 0.720 -1.812 3.556 96 1.066 0.715 -1.679 3.664 120 1.262 0.706 -1.442 3.828 240 1.754 0.693 -0.809 4.262 360 1.931 0.694 -0.576 4.476

Table 19: Descriptive statistics of German one-year ahead simulated zero coupon yields from model DNS-N.

6 0.339 0.538 -1.778 2.553 12 0.288 0.550 -1.830 2.783 24 0.302 0.589 -1.818 3.201 36 0.404 0.615 -1.927 3.619 48 0.543 0.624 -1.859 3.955 60 0.692 0.623 -1.694 4.226 84 0.972 0.609 -1.272 4.626 96 1.093 0.600 -1.064 4.775 120 1.295 0.586 -0.697 5.002 240 1.796 0.561 0.258 5.507 360 1.976 0.560 0.479 5.682

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Maturity in months Mean Standard deviation Minimum Maximum 6 0.345 0.533 -1.927 2.835 12 0.294 0.543 -1.949 3.051 24 0.308 0.583 -1.816 3.440 36 0.410 0.610 -1.578 3.773 48 0.549 0.621 -1.397 4.055 60 0.698 0.622 -1.150 4.292 84 0.978 0.609 -0.718 4.656 96 1.100 0.601 -0.579 4.795 120 1.302 0.588 -0.334 5.013 240 1.804 0.565 0.317 5.509 360 1.984 0.564 0.556 5.682

Table 21: Descriptive statistics of German one-year ahead simulated zero coupon yields from model DNS2-N.

D

Descriptive statistics one-year ahead simulated US yields

1 1.654 0.913 -1.847 5.872 3 1.646 0.906 -1.890 5.787 6 1.641 0.900 -1.930 5.667 12 1.654 0.894 -1.942 5.451 24 1.741 0.885 -1.776 5.282 36 1.868 0.862 -1.477 5.524 60 2.144 0.794 -0.782 5.683 84 2.383 0.728 -0.241 5.668 120 2.647 0.664 0.237 5.572 240 3.044 0.612 0.976 5.446 360 3.186 0.613 1.136 5.603

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Maturity in months Mean Standard deviation Minimum Maximum 1 1.647 0.906 -1.453 5.181 3 1.639 0.900 -1.501 5.169 6 1.634 0.894 -1.585 5.146 12 1.648 0.890 -1.670 5.091 24 1.735 0.883 -1.611 5.150 36 1.863 0.861 -1.381 5.277 60 2.139 0.794 -0.826 5.305 84 2.379 0.728 -0.541 5.226 120 2.643 0.663 -0.202 5.084 240 3.041 0.611 0.335 5.154 360 3.183 0.612 0.529 5.334

Table 23: Descriptive statistics of US one-year ahead simulated zero coupon yields from model DNS-N.

1 1.506 0.901 -2.148 5.758 3 1.513 0.891 -2.190 5.710 6 1.530 0.881 -2.222 5.641 12 1.582 0.871 -2.201 5.516 24 1.727 0.856 -1.928 5.305 36 1.895 0.830 -1.495 5.226 60 2.221 0.756 -0.531 5.554 84 2.489 0.684 0.103 5.645 120 2.775 0.611 0.795 5.850 240 3.198 0.547 1.623 6.206 360 3.349 0.545 1.891 6.324

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Maturity in months Mean Standard deviation Minimum Maximum 1 1.508 0.893 -2.203 5.406 3 1.516 0.883 -2.152 5.399 6 1.533 0.873 -2.065 5.386 12 1.586 0.863 -1.863 5.354 24 1.732 0.849 -1.402 5.276 36 1.902 0.824 -0.985 5.257 60 2.229 0.750 -0.432 5.111 84 2.497 0.679 0.115 5.035 120 2.783 0.608 0.723 5.200 240 3.206 0.546 1.420 5.721 360 3.356 0.545 1.673 5.924

Table 25: Descriptive statistics of US one-year ahead simulated zero coupon yields from model DNS2-N.

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University of Groningen Master’s Thesis Econometrics Interest rate risk modeling using the Dynamic Nelson-Siegel model