A benchmark for interest rate risk using a Markowitz approach

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A Benchmark for Interest Rate Risk

using a Markowitz Approach

J.V. Verheijen

Amsterdam, October 29, 2014

A thesis in partial fulfillment of the requirements for the degree of Master of Science in Financial Econometrics

Department of Quantitative Economics University of Amsterdam

Principal Advisor: Dr. H.P. Boswijk Second Advisor: Dr. S.A. Broda

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Preface

This study was conducted at request of ABN AMRO, further referred to as ”the bank”. I would like to extend my sincerest thanks and appreciation to Wilson Jan Kansil and the Balance Sheet Analysis team, for providing the opportunity and their support. Further I would like to recognize Martijn van Rooijen, from ABN AMRO, and dr. Peter Boswijk, from the University of Amsterdam, for their guidance and input on the subject. Finally I would like to emphasize that the views expressed in this thesis are those of the author. No responsibility for them should be attributed to the ABN AMRO.

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1. Introduction

The objective of this thesis is to calculate a benchmark to measure the performance of steering transactions on the interest rate mismatch. The mismatch naturally follows from the balance sheet of a bank, because it consists mostly of short-term liabilities and long-term assets. Since the yield curve is in general a monotonically increasing and concave function the long-term yields are higher than the yields for short maturities and therefore the mismatch (usually) generates positive results. However, because of this difference in interest rates and the mismatch in duration, the bank is exposed to interest rate risk.

Interest rate risk is the bank’s exposure to adverse movements in the interest rates. There are four sources of interest rate risk, which are basis -, optionality -, repricing - and yield curve risk. Accepting interest risk is normal for banks and can be an important source of profitability and shareholder value. Nevertheless, excessive risk taking can significantly threaten the bank’s earnings and its capital. Therefore, effective risk management is needed to secure the safety and soundness of banks. To maintain effective risk management the bank for international set-tlements 1 (BIS) requires banks to have standards for Performance Measurement. Here lies the relevance of this thesis, which tries to obtain a benchmark for the steering transactions for the duration mismatch.

From a management perspective the mismatch, which follows from the

bal-1_{The Basel Committee on Banking Supervision is a Committee of banking supervisory} au-thorities which was established by the central bank Governors of the Group of Ten countries in 1975. It consists of senior representatives of bank supervisory authorities and central banks from Belgium, Canada, France, Germany, Italy, Japan, Luxembourg, Netherlands, Spain, Sweden, Switzerland, United Kingdom and the United States.

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ance sheet, is steered via duration. Duration is the most commonly used measure of risk in bond investing. The duration mismatch is steered with swap transactions, which can be used to make the balance sheet more (or less) sensitive to changes in the interest rates. These transactions have an impact on the Net Interest Income (NII) and the development of the Market Value of Equity (MVE) of the bank. When the bank receives floating rates from a swap transaction increasing rates lead to a higher NII. The MVE is also affected by changes in the interest rates, an increase (decrease) of the yield curve leads to a decrease (increase) in MVE, since the MVE is the discounted value of all future cash flows that are discounted with lower (higher) yields. The benchmark needs to take the changes in NII and MVE into account.

This thesis tries to obtain a benchmark for the NII and MVE by investigating the return and market value of an optimal bond portfolio, constructed by the Markowitz approach. This mean-variance approach aligns with the objective of the bank, which is maximizing its returns given a prespecified level of risk. Hence, when comparing the NII and the market value of the steering portfolio with the benchmark, the magnitude and structure of risk of the latter must correspond to the risk of the steering transactions.

To perform a mean variance optimization the expected return and (co-) vari-ances of the available bonds are needed. The expected return and variance of (fixed-income) bonds follow from forecasts of the yield curve, hence a yield curve model is needed. The yield curve models used in this thesis are the Dynamic Nel-son Siegel model (DNS) and Affine Arbitrage Free NelNel-son Siegel model (AFNS). The empirical results of these models are compared with those obtained from the Random Walk model and a Principal Component Analysis.

The benchmark for the return generated by the duration mismatch is ob-tained in four steps. First, different models are used to obtain the yield curve. Secondly, the zero coupon fixed-income yield curve is bootstrapped from the swap curve. Thirdly, the bond prices, and hence the period holding returns, are obtained using the latter yield curve. Finally, the bond portfolio returns are optimized with respect to their variance.

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This thesis is divided into five chapters. The second chapter explains the framework used to obtain the benchmark. The third chapter elaborates on the different models in detail. The fourth chapter describes the empirical results of the followed framework. Finally in Chapter five the conclusions are given.

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2. Literature and background

2.1 Mismatch results

As discussed in the introduction this thesis tries to obtain a benchmark for the results generated from the duration mismatch1. This mismatch arises primarily from the fact that the repricing period of the assets typically exceeds the repricing period of the liabilities. To understand the concept of mismatch results it is important to understand the basics of interest rate risk and the balance sheet. This section gives a short introduction to both concepts. The remainder of the chapter introduces the models needed for the benchmark portfolio.

2.1.1 Interest rate risk

Interest rate risk is the exposure of a bank’s financial condition to adverse move-ments in interest rates. Accepting this risk is a normal part of banking and can be an important source of profitability and shareholder value (Basel Committee on Banking Supervision, 2004). Banks are typically exposed to four sources of interest rate risk, which include basis -, optionality -, repricing - and yield curve risk. The interest rate risk that follows from the duration mismatch are repricing - and yield curve risk, hence a short introduction might be helpful.

Repricing risk arises from the timing difference in maturity (for fixed-rate) and repricing (for floating-rate) of bank assets, liabilities, and off balance sheet positions. For instance a bank that funded a long-term fixed-rate loan with a

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short-term deposit could face declining net interest income (NII) if interest rates increase. This decline follows from the fixed, and therefore unchanged (long-term) income together with the increased (variable) funding costs. The second source of interest rate risk, yield curve risk, follows from the same timing differences but arises from non parallel changes of the yield curve. For instance, the value of a position in 10-year bonds hedged by a position in 5-year bonds could decline if the yield curve steepens. In this case, the present value of the 10-year position decreases, because it is discounted at higher rates, which is not offset by the value change of the hedged positions because the corresponding rates did not change or changed less. Therefore the total position decreases in value when the interest rate curve steepens. These two sources of interest rate risk, repricing - and yield curve risk, affect the balance sheet of bank and the interest income. The next section elaborates the balance sheet to examine the exposure of the bank towards these two sources of interest rate risk.

2.1.2 Balance sheet

The bank fulfils a maturity transformation role by financing long term assets with short term liabilities. Under normal conditions this ensures a positive NII, since the interest income generated by assets (long-term) exceeds the interest expenses paid for liabilities (short-term). To connect the balance sheet of the bank to interest rate risk a duration gap analysis is often used. A duration gap analysis examines the sensitivity of the market value of the financial institutions net worth to changes of the interest rates. This analysis is based on modified duration, a modified version of the Macaulay model.

Modified duration = Macaulay duration

1 + Y T M_n . (2.1) Here is n the number of coupon payments per year, Y T M the yield to maturity and Macaulay duration is given by

Macaulay duration = Pn t=1 t·C (1+y)t n·M (1+y)n

Current bond price, (2.2)

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The modified duration, further referred to as duration, of an instrument is an important measure for investors to consider, as bonds with higher durations carry more risk and have a higher price volatility than bonds with lower durations. For zero coupon bonds the duration equals the time to maturity, for plain vanilla bonds, which offer coupon payments, the duration is shorter than time to maturity. An important fact of duration is that it is an additive measure, which implies that the duration of a portfolio is the weighted average duration of all individual assets. A positive interest mismatch is ensured when the duration of assets is higher than the duration of the liabilities. This implies that the liabilities are repriced more frequently than the assets on the balance sheet. Hence with a positive in-terest mismatch and increasing inin-terest rates both the inin-terest mismatch and the market value of equity decreases. The interest mismatch decreases since liabilities are repriced earlier than assets and the interest expenses are elevated because of increased rates. The market value of equity decreases since the market value of assets decreases more than the market value of liabilities. The asset value changes more because the duration of assets is higher, which implies has a higher sensitivity to changes in the interest rates. To make the concept of duration and duration gap more tangible, a fictive balance sheet is considered in Appendix A.1. The current approach of the bank is based on this gap analysis and is elaborated in the next section.

2.2 Current approach

The duration gap is managed by taking receiver - and payer positions in swaps. A net receiver position means that the bank receives fixed and pays floating rates. An increase of the interest rates would increase the rates payed (while not affecting the rates received) and therefore decreases the market value of the position. A net payer position means that the banks pays fixed and receives floating rates. In this case an increase of the interest rates would increase the market value of the position. With these positions it is possible to make the balance sheet less -, or more sensitive to changes of the interest rates. Net payer positions can be used to

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decrease the duration of equity and therefore make the balance sheet less sensitive to changes in the yield curve.

The interest mismatch is managed by the Asset & Liability Committee (ALCO) by means of duration, market value of equity-at-Risk (MVE-at-Risk) and Net interest income (NII). When the bank reduces the duration of the balance sheet, net payer swaps are needed for hedging. Given the level of duration the bank has a certain NII and MVE, which are both determined by developments of the interest rates. It is important to measure the NII and MVE given this level of duration and the steering actions. Especially the NII is affected by the steering transactions. The proposed benchmark tries to provide a performance measure for these two statistics.

The calculation of the duration mismatch only contains an interest rate risk component, this should be reflected in the benchmark. Therefore, the benchmark should be a portfolio containing only an interest rate risk component, which can be done with a portfolio of bonds. Note, that the assumption is made that bonds are default free. This portfolio will result in coupon payments, which is comparable with the NII of the bank. Further the portfolio has a changing market value, the present value of all the future cash flows, which can be compared with the market value of the balance sheet. Secondly, the objective of the bank, generate the best results possible given the risk that is taken, should be reflected in performance measurement and therefore in the benchmark. This characteristic of the high-est return given the risk that is taken leads to a mean variance optimized bond portfolio as our benchmark. The next section will introduce the mean-variance optimization used for the benchmark.

2.3 Mean-variance optimization

The previous section elaborated on the need for a mean-variance optimized portfo-lio as the benchmark (portfoportfo-lio). When performing a mean-variance optimization both the return and variance of available bonds are needed. Hence, this section gives a short introduction of the mean-variance approach.

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2.3.1 Mean-variance optimization

The benchmark is an optimized fixed income portfolio using the mean-variance ap-proach proposed by Markowitz (1952), to be specific a Sharpe ratio optimization. The mean-variance optimization is widely used by managers for portfolio con-struction and to develop quantitative asset allocation strategies. However, these strategies are often restricted to equity portfolios. For the selection of fixed income portfolios managers often use duration.

There are two reasons that explain why mean-variance optimization is rarely used in fixed income portfolio selection. The first argument is the relative stable behavior and low historic variability of bonds, which discouraged the use of ad-vanced techniques to exploit the risk-return trade-off. However the variability in bond markets has increased a lot since the crisis, even in markets with low default probabilities, see Korn and Koziol (2006). This increase in volatilities encourages the use of more sophisticated methods like a Sharpe optimization for bond portfolio selection. Secondly, difficulties in obtaining the expected returns and covariances of the fixed income portfolios has restrained the use of the mean-variance ap-proach in fixed-income portfolios. Fabozzi and Fong (1994) argued that if returns and covariances were easily available fixed income portfolio optimization would be equivalent to that of equity portfolios. Factor models like the DNS model have greatly simplified the computation of the expected return and covariance matrix. Together, the increase in volatility of the bond markets and the introduction of factor models encourage the use of mean-variance optimization techniques for fixed income portfolios.

The mean-variance approach states how investors can maximize their returns and minimize their risks. The mean-variance optimization provides analytical solutions in a large class of models, restricting the investment constraints to be affine. The market considered consist of N bonds, with prices Ptat generic time t.

The optimization provides an N -dimensional vector ω∗, the most suitable portfolio allocation for a given investor, which follows from the investor preferences and the information on the market.

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Collecting information on the investor and the market

The information on the investor consist of knowledge of the investors’ current situation and the objective of the investor. The investors’ current situation can be summarized in a portfolio ω which corresponds to his wealth at the time the decision is made:

Wt = Pt0x0. (2.3)

Note that at time t, the moment of optimizing the portfolio, the prices Pt, the

initial amount of assets x0 and therefore the wealth of the investor are known.

The objective of the investor is determined by his preferences. When using the mean-variance approach only the first two portfolio moment are considered in the optimization. This approach is justified if the individuals expected utility depend only on the mean and variance of the portfolio return. If the utility function is quadratic, which implies that all derivatives of order three and higher are equal to zero, it follows from a Taylor expansion that the expected utility given by

E[Rp(ω)] = U (E[Rp(ω)]) + 1 2E[(Rp(ω) − E[Rp(ω)]) 2_]U00 (E[Rp(ω)]) = U (E[Rp(ω)]) + 1 2V[Rp(ω)]U 00 (E[Rp(ω)]). (2.4)

Therefore a quadratic utility function leads to an expected utility, which only de-pends on the first two portfolio moments, independent of the distribution of the portfolio returns. Recall that a power utility is assumed, which implies constant relative risk aversion. Levy and Markowitz (1979) showed that the mean-variance analysis can be regarded as an second order Taylor-series approximation of the standard utility functions, such as the power - and exponential utility function. Hence assuming the power utility justifies the use of the mean-variance optimiza-tion.

In general the investor has multiple objectives, which depend on the alloca-tion ωtfrom (2.3). The assumption is made that the main objective of the investor

is the return of the bond portfolio. Any objective of the investor is a linear func-tion of the allocafunc-tions and of the market vector. In case of the main objective, the portfolio return is a linear combination of the returns of the N available bonds

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over the investment horizon T of the investor.

R(ω) = R0_Tω, (2.5)

where RT is an N -dimensional vector of returns of the available bonds.

It is further assumed that the investors evaluates his net returns in terms of their value at risk, which means that the investor obtains a higher utility if the variance of his investment is smaller. Therefore the secondary objective of the investor is minimizing the variance of the portfolio, that is

S(ω) = −V ar(R0_Tω). (2.6)

The investors’ current portfolio, investment horizon, main objective and util-ity function are all information on the investor needed. To complete the informa-tion needed for the mean-variance optimizainforma-tion, informainforma-tion on the market is needed.

The information needed from the market are the current prices of the N bonds and the future prices at the investment horizon T . The current prices are deterministic and known. The future prices PT are random variables and follow

from the yield curve models. Combining the information from the investor and the market, allows to obtain the most suitable portfolio allocation, which is described in the next section.

Computing the optimal portfolio allocation

With the information on the investors and the market, the most suitable portfolio allocation can be found. Combining the the primary and secondary objective of the investor, the expected utility of each portfolio allocation can be approximated by

U (Rp(ω)) = F (E[Rp(ω)], V[Rp(ω)]) (2.7)

The optimal portfolio allocation approach suggested by Markowitz is defined as: ω(v) = argmax

V[Rp(ω)]=v

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where v ≥ 0. The solution of the optimization in (2.8) is the (mean-variance) efficient frontier, which are all portfolio allocations that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return.

Given the efficient frontier the investor can obtain the most suitable portfolio allocation by selecting the allocation such that:

ω∗ = ω(v∗) = argmax

v≥0

U (ω(v)). (2.9)

This section described a two step procedure to obtain the most suitable portfolio allocation for a given investor. Note that only risky bonds were available, when there is also a risk free bond available additional allocations are available. The next section elaborates on the Sharpe ratio, which takes these additional allocations into account.

2.3.2 Sharpe Ratio optimization

The mean-variance approach in the previous section was a two step calculation. The first step was the computation of the efficient frontier and the second step was optimizing the investors’ utility given the efficient frontier. Note, that in the optimization only risky bonds were available. When considering a market with a risk free asset the set with possible portfolio allocations increases. The Sharpe Ratio optimization takes these additional allocations into account, which results in the capital market line. The allocations on the capital market line are a linear combination between the risk free rate and the optimal portfolio, which is the tangent of the capital market line and the efficient frontier.

Adding a risk free bond shifts the objective from obtaining the efficient fron-tier to finding the optimal portfolio. This can be done by obtaining the tangent to the efficient frontier, that is finding the linear combination between the risk free rate and the set of feasible allocations with the highest slope. Following this procedure results in the Sharpe optimal portfolio

max

ω S(ω) =

E[ eR(ω)]

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where eR(ω) is the excess return relative to the risk-free (short) rate.

The two components for portfolio selection are the expected return of the investment and the variance. For the returns of the bonds within the portfolio the simple holding-period returns are used that is the return of buying a (zero-) coupon bond with maturity τ at time t and selling this bond at time T , where T ≤ τ . Hence the holding-period return of a bond is given by

h(t, τ, T ) = P (T, τ − (T − t)) − P (t, τ )

P (t, τ ) , (2.11)

which shows that at time t, the moment of optimizing the portfolio allocation, P (t, τ ) is deterministic term and P (T, τ − (T − t)) is a stochastic term. Therefore the expected return of a single bond is defined as

E[h(t, τ, T )|Ft] = E P (T, τ − (T − t)) − P (t, τ ) P (t, τ ) Ft = E[P (T , τ − (T − t))|Ft] − P (t, τ ) P (t, τ ) . (2.12)

The variance of the bond return is given by V[h(t, τ, T )|Ft] = V P (T, τ − (T − t)) − P (t, τ ) P (t, τ ) Ft = V[P (T , τ − (T − t))|Ft] (P (t, τ ))2 . (2.13)

Both (2.12) and (2.13) show that the distribution of the future price P (T, τ − (T − t)) is needed, which follows from a yield curve model.

This section shortly introduced the mean-variance and Sharpe ratio optimiza-tion. To perform a Sharpe optimization both the expected return and covariances are needed, which follow from the forecast of the yield curve. The yield curve fore-cast depends on the yield curve model that is used. The next section introduces the DNS model, the yield curve model used in this thesis.

2.4 Yield curve model

To obtain a mean-variance optimized bond portfolio the expected bond returns and covariances are needed. The DNS model is used to fit and model the yield

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curve. With these future yields (and their distribution) it is possible to obtain the distribution of the future prices and returns, needed for the mean variance optimization.

The DNS model tries to explain the yield curve using three factors, which are level, slope and curvature. Once these factors are estimated based on historical rates, an auto-regressive (AR) structure is added to forecast them. Finally, a Monte Carlo simulation is used to obtain the distribution of the yield curve. The expected bond returns and covariances follow from the future yields and their distribution, obtained from the Monte Carlo simulation. The next chapter gives a detailed overview of bond pricing and the DNS - and AFNS model.

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3. Theory

The proposed benchmark is a mean-variance optimized bond portfolio. To apply a mean-variance optimization the expected returns and covariances of the available bonds are needed, which follow from the distribution of the future yield curves. The distribution is obtained using the DNS- and AFNS model. In this thesis the empirically relevant assumption is made that the expectation hypothesis does not hold. The expectation hypothesis states that the expected holding-period returns on bonds of different maturities should be equal. However Engle et al. (1987) show that risk premia change systematically with the perceived uncertainty which lead deviations. Consequently, price dynamics under the risk neutral measure Q are different from price dynamics under the real measure P, which requires to know how to change the probability measure. This section elaborates on bond pricing, changing the probability measure and the yield curve models used in this thesis.

3.1 Pricing Bonds in continuous time

The affine term structure models following the work from Duffie and Kan (1996) all have closed form expressions for the price of zero coupon bonds. Zero coupon bonds pay a terminal notional at maturity date, often normalized to one, without intermediate coupon payments and disregarding default risk. A zero-coupon bond with maturity τ is currently traded at P (t, τ ). Buying the bond at time t, holding it and selling it at time T the n-period holding-period return is given by

h(t, τ, T ) = P (T, τ − (T − t)) − P (t, τ )

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where n ≤ τ and T = t+n. When selling the zero-coupon bond before the maturity date the holding period excess return is usually random, because it depends on the unknown P (T, τ − (T − t)). At maturity the return of the bond is known. It follows that the price of a zero-coupon bond, and therefore the holding-period return, is a random variable until maturity, and is a deterministic quantity at maturity. This implies that the statistical properties of the price and return of bonds depend on their time to maturity. Therefore, the bond price and return are non-ergodic processes1_{and traditional statistical techniques do not apply (Meucci,}

2009, p.110).

A pricing problem involves conditioning on the current market data, through the fundamental theorem of asset pricing it is possible to price under an equivalent martingale measure or “risk-neutral-probability measure” Q. The fundamental theorem of asset pricing states that for a stochastic process, the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage (Delbaen and Schachermayer, 1994). This allows pricing without knowing the exact risk preferences of the investors, in case of complete markets. Hence prices are future expected payoffs discounted at the risk free rate, where expectations are computed using the risk neutral measure Q. Usually the face value of a zero-coupon bond is normalized to one, hence the price is given by

P (t, τ ) = EQ t h e−Rtt+τrudu i , (3.2)

where rt is the instantaneous spot rate.

Under the risk neutral probability measure the expected return on bonds is the risk free rate, which implies that the expected excess return is zero. However, in the case of interest rate risk prices are needed under the restriction of a certain exposure to the term structure of interest rates. This implies that the investors’ risk attitudes need to be considered, which requires the price dynamics under the probability measure P. Hence obtaining the expected bond prices consist of two steps. The first step is changing the probability measure P to Q. The second step

1_{A stochastic system is called ergodic if it tends in probability to a limiting form that is} independent of the initial conditions, (Horst, 2007). Hence, non-ergodic implies path dependency, in our case time to maturity.

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is determining the dynamics of the short rate r, which we will do with a factor model. That is, making r a function of a state vector x, and factor loadings. The assumption is made that the state vector x is a Markov process under Q. Doing so, it is possible to rewrite (3.2) as a function of these state vector and time to maturity, which leads to

P (t, τ ) = f (xt, τ ). (3.3)

When obtaining the evolution of the bond prices by rewriting (3.2), assumptions about the dynamics under the P- and Q measure are needed, which will be inves-tigated in the next section.

3.1.1 Change the probability measure for bond pricing

In this thesis the future bond prices are modelled with a DNS and an AFNS model. The DNS models the price dynamics directly under the P-measure but the AFNS model models these dynamics under the risk neutral measure Q. Since the empirically relevant assumption is made that the local expectation hypothesis2

does not hold, modelling with the AFNS model requires the change of probability measure. An additional advantage of pricing under the probability measure P, is that we have an intuition of the parameters, which we do not have under the risk neutral measure. This section elaborates on changing the probability measure required for the AFNS model.

It is important to realize that under the risk-neutral measure the expected returns are always equal to the riskless rate that is

EQ t [h(t, τ, T )] = µ ∗ f(x, τ ) = e −r(t,T ) (3.4) where r(t, T ) is the yield at time t for maturity T and a function of the state variables x. The AFNS model assumes that, under the risk neutral measure Q, the state vector x solves

dxt= µ∗x(xt)dt + σx∗(xt)dzt∗ (3.5)

2

The Local Expectation Hypothesis states that the data generating measure P and the risk neutral measure Q coincide (Piazzesi, 2009).

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where z_t∗ _{is a standard vector Brownian motion under the risk neutral measure Q.} Now we can change the probability measure using Girsanov’s theorem3_{. Girsanov’s}

theorem states that, for a Brownian motion, an absolutely continuous change of measure is equivalent to change of drift. Note that changes of probability measure do not affect the variance on innovations of the state vector x.

The dynamics under the P measure are obtained in four steps. First, (3.2) states that at maturity the bond price equals the payoff, which implies that f (x, 0) = 1 ∀x. Secondly, the exponential function within the expectation (3.2) implies a strictly positive price. Thirdly, Itˆo’s lemma4 _{implies that f (x, τ ) is also}

an Itˆo process, hence

df (xt, τ )

f (xt, τ )

= µ∗_f(xt, τ )dt + σf∗(xt, τ )dzt∗ (3.6)

with an instantaneous expected bond return µ∗_f(xt, τ ) = − ˙ fτ(x, τ ) f (x, τ ) + f0(x, τ )> f (x, τ ) µ ∗ x(x) + 1 2tr σ_x∗(x)σ_x∗(x)>f 00_{(x, τ )} f (x, τ ) , (3.7) where ˙fτ(x, τ ) = ∂f (x,τ )_∂τ , fτ0(x, τ ) = ∂f (x,τ ) ∂x , f 00 τ(x, τ ) = ∂2_{f (x,τ )} ∂x∂x and tr denotes

trace. The drift µ∗_x(x) and volatility σ∗_x(x) of the state vectors are still under the risk neutral measure. The fourth step, changing the measure, captures the risk adjustment of the future prices. This change of measure involves a strictly positive martingale ξ, which is a martingale if Novikov’s condition5 _{is satisfied and starts}

at ξ0 = 1. The differential equation is given by

dξt

ξt

= −σξt(xt)

>

dt (3.8)

Again applying Girsanov’s theorem, we see that z∗_t is a Brownian motion under Q, hence

dz_t∗ = dzt+ σξ(xt)>dt (3.9)

3_{Girsanov’s Theorem is stated in C.1} 4_Itˆ_{o’s lemma is stated in C.2}

5

Novikov’s condition: E[e12 RT

0 σ ∗

ξ(xu)σ∗ξ(xu)>du_{] < ∞, a more detailed overview is given by}

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Substituting this definition of z_t∗ 3.9 into 3.5 we find

dxt= (µ∗x(xt) + σx∗(xt)σ>ξ(xt))dt + σx∗(xt)dzt (3.10)

When looking at 3.10, we see that the volatility is unaffected and only the drift changes by the change in risk measure. This is known as the diffusion invariance principle.

This section showed how to change the probability measure, which is needed for the AFNS model. Both the DNS- and the AFNS will be introduced in the remainder of this chapter.

3.2 Modelling yield curves and stylized facts

Portfolio selection with respect to interest rate risk involves measuring the expo-sure of one’s portfolio to adverse movement in the term structure of interest rates. Because the yield curves are not observed in practice, we have to estimate these from the (historical) bond prices. There are two different classes of term structure models to model these curves. The first class are affine term-structure models by building on the work of Vasicek (1977) and Cox et al. (1985). This class of mod-els works with the restriction that arbitrage opportunities are eliminated. These restrictions are appealing since bonds are traded in well-organized, highly liquid markets. These models have the advantage that the possess good tractability and a good economic foundation, however these models have difficulty capturing devia-tions from the expectation theory, see Bolder (2006). The second class, introduced by Diebold and Li (2003), works directly under the probability measure P. These models are basically a time-series description of the term structure and provides a better forecast than the affine models. A disadvantage of this approach is the lack of the theoretical model foundation. However recent work from Christensen et al. (2009) improved the theoretical foundation by imposing the arbitrage free restriction, which led to the AFNS model.

A good model for the yield curve should be able to capture at least some of five stylized facts. First, the average yield curve is increasing and concave over

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time. Secondly, the yield curve can take on a variety of shapes, for example up-and downward sloping, humped up-and S-shapes. Thirdly, yield dynamics are (very) persistent, which means that there are high correlations, in particular on short term. Further the short end of the curve is more volatile then the long end. And finally, yields for different maturities have high cross-correlations.

The next section elaborates on the basis for the DNS and the AFNS model. Bolder (2006) gives a thorough derivation of the Nelson Siegel models, however for completeness this will partly be repeated in this thesis.

3.3 Nelson Siegel Term Structure Models

Recall that in the last section five stylized facts of the yield curve were stated. From these five stylized facts follows a typical yield curve, which Nelson and Siegel (1987) associated with solutions to differential or difference equations. This section introduces the Nelson-Siegel model and follows the work of Diebold and Li (2003) and Christensen et al. (2011) that result in the DNS - and AFNS model. The starting point in the Nelson Siegel models is the instantaneous forward rate, given as

f (t, τ ) = lim

T →τf (t, T, τ ), (3.11)

where f (t, T, τ ) is the continuously compounded forward interest rate that is f (t, T, τ ) = 1 T − τ ln P (t, τ ) P (t, T ) (3.12) Substituting the continuous forward interest rate into 3.11 the instantaneous

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for-ward rate can be obtained f (t, τ ) = lim T →τ 1 T − τ ln P (t, τ ) P (t, T ) = lim T →τ ln P (t, τ ) − ln P (t, T ) T − τ = lim T →τ ln P (t, τ ) − ln P (t, T ) T − τ = lim T →τ ∂ ∂T(ln P (t, τ ) − ln P (t, T )) ∂ ∂T(T − τ ) = lim T →τ P0(t,T ) P (t,T ) 1 = −P 0_{(t, τ )} P (t, τ ) (3.13)

The fourth equation is obtained using L’Hˆopital’s rule6_{. The instantaneous forward}

can be seen as the overnight interest rate, therefore it is possible to derive the yield curve as a function of the instantaneous forward curve

−P 0_{(t, τ )} P (t, τ ) = f (t, τ ) − ∂ ∂τ(ln P (t, τ )) = f (t, τ ) − ∂ ∂τ(ln e −y(t,τ )(τ −t) ) = f (t, τ ) Z τ t ∂ ∂s(y(t, s)(s − t))ds = Z τ t f (t, s)ds y(t, τ )(τ − t) − y(t, t)(t − t) = Z τ t f (t, s)ds y(t, τ ) = 1 τ − t Z τ t f (t, s)ds (3.14)

It follows that the the zero-coupon yield is an equally-weighted average of forward rates. Nelson and Siegel (1987) proposed a functional form for f (t, τ ) which results in a parsimonious representation of the yield curve. This form will be introduced in the next section.

6_L’Hˆ_{opital’s rule: if lim} x→c f (x) g(x) = 0, +∞ or −∞, limx→c f0(x) g0_(x) exists and g0(x) 6= 0 then limx→c f (x) g(x) = limx→c f0_(x) g0_(x)

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3.3.1 Nelson-Siegel model

The model originally suggested by Nelson and Siegel (1987) was a functional form for f (t, τ ), which is given by

f (t, τ ) = x0+ x1e−λtτ + x2λtτ e−λtτ (3.15)

This is a parsimonious representation of a yield curve and does not depend on the expectations theory of term structure. Further it does not enforce the theoretically appealing condition of no arbitrage. From this functional form of the forward rate we can obtain a closed form solution of the corresponding yield curve by substituting this functional form into (3.14).

y(t, τ ) = x0,t+ 1 − e−λtτ λtτ x1,t+ 1 − e−λtτ λtτ − e−λtτ x2,t (3.16)

Where y(t, τ ) is the zero coupon yield curve with τ denoting the time to maturity, and x0,t, x1,t, x2,t and λt are model parameters. Diebold and Li (2003) build on

this model and made two important adjustments, which led to the DNS model. This model will be elaborated on in the next section.

3.3.2 Dynamic Nelson-Siegel model

The first adjustment that Diebold and Li (2003) made, was a clear interpretation of the factors. This interpretation can be derived from Figure 3.1. When investigating the zero-coupon yield curve in (3.16), one can see that the terms affect different tenors on the yield curve. The first loading on x0 is one, a constant, which does

not decay to zero when t goes to τ . Diebold and Li (2006) interpret this loading as a long term factor. The loading on x1 is 1 − e−λtτ /λtτ , a function that starts at

one but quickly monotonically decays to zero, hence Diebold and Li interpret this as a short term factor. Finally, the last loading on x2, 1 − e−λtτ /(λtτ ) − e−λtτ,

is function that starts at zero (therefore not short-term), increases and decays to zero again (therefore not a long-term); hence Diebold and Li interpret this as a medium-term factor. Diebold and Li re-interpreted this Nelson-Siegel curve as

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a dynamic model that reduces the dimensionality via a factor structure. They interpreted these time-varying long-, short- and medium-term factors respectively as level, slope and curvature factor.

Figure 3.1: Factor Loadings of the Dynamic Nelson Siegel

The second important adjustment that Diebold and Li (2003) made, was to make the coefficients, i.e. the weights on level, slope and curvature, vary over time. We can define the time-varying coefficients as

Xt=     x0,t x1,t x2,t     , F (t, τ ) =     f0(t, τ ) f1(t, τ ) f2(t, τ )     =     1 1−e−λtτ λtτ 1−e−λtτ λtτ − e −λtτ     .

Using these time-varying coefficients and substituting these into the yield curve defined by Nelson-Siegel that is (3.16) we obtain the Dynamic Nelson-Siegel yield curve y(t, τ ) = x0,t + 1 − e−λtτ λtτ x1,t+ 1 − e−λtτ λtτ − e−λtτ x2,t. (3.17)

Under the assumption that λt can be treated as a constant value 7, it is possible

to write the bond price at time t as the function

P (t, T ) = e−F (t,T )>Xt_, _(3.18)

7_{Treating λ}

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where is F (t, T ) a deterministic process of factor loadings. The stochastic process is induced by X, the usual choice for {Xt, t ≥ 0} is given by

dXt = κ(θ − Xt)dt + C>ΣdWt, (3.19)

where κ, C, Σ ∈ R3x3 _{and X}

t, θ, dWt ∈ R3x1 where Σ is a diagonal matrix, C is

a Cholesky composition of the instantaneous correlation matrix, and {Wt, t ≥ 0}

is a standard Brownian Motion under the P-measure. Note that from (3.19) we can see that this model imposes a vector-auto regressive structure on the factor coefficients. Xt− Xt−1≈ κ(θ − Xt−1) + εt Xt≈ κθ + (1 − κ)Xt−1+ εt Xt≈ α + βXt−1+ εt, (3.20) where εt∼ N (0, Ω)).

Note that Ω is defined as C>ΣΣC. However this model does not enforce the theoretical appealing no-arbitrage condition. In fact, Filipovi´c (1999) showed that independent of the stochastic process, it is impossible to enforce the no arbitrage condition at the bond prices resulting from Nelson Siegel yield curve. Christensen et al. (2011) suggest a model which is theoretically rigorous that simultaneously displays empirical tractability, good fit and good forecasting performance. This model will be elaborated on in the next section.

3.3.3 The arbitrage-free Nelson-Siegel model

Christensen et al. (2011) came up with a solution to overcome this theoretical weak-ness. Their derivation starts from the standard continuous-time affine structure of Duffie and Kan (1996). In the first proposition Christensen et al. (2011) assume that the the instantaneous risk-free rate is rt = X0,t+ X1,t. This follows from the

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factor loadings of the Nelson Siegel models, the instantaneous short rate follows from y(t, 0) = X0,t+ X1,t. The state variables are given by Xt = (X0,t, X1,t, X2,t)

and described by a system of stochastic differential equations defined by

    dX0,t dX1,t dX2,t     =     0 0 0 0 λt −λt 0 0 λt             θ₀Q θ₁Q θ₂Q     −     X0,t X1,t X2,t         dt + Σ     dW_0,tQ dW_1,tQ dW_2,tQ    

Then the zero-coupon bonds are P (T − t) = EQ_t he−RtTr

0 udu

i

= eF0(T −t)X0,t+F1(T −t)X1,t+F2(T −t)X2,t+A(t,τ )_, _(3.21)

where F0(T − t), F1(T − t), F2(T − t) and A(t, τ ) are solutions to the system of the

ordinary differential equation     dF0(T −t) dt dF1(T −t) dt dF2(T −t) dt     =     1 1 0     +     0 0 0 0 λt −λt 0 0 λt         F0(T − t) F1(T − t) F2(T − t)     and dA(t, τ ) dt = −F (T − t) >_KQ_θQ₋1 2 3 X j=1 (Σ>F (T − t)F (T − t)>Σ)jj, (3.22)

with boundary conditions F0(T − t) = F1(T − t) = F2(T − t) = A(t, τ ) = 0. The

solution to the system of ordinary differential equations is given by F0(T − t) = −(T − t) F1(T − t) = − 1 − e−λt(T −t) λt F2(T − t) = (T − t)e−λt(T −t)− 1 − e−λt(T −t) λt A(t, τ ) = (KQθQ)2 Z T t F1(s, T )ds + (KQθQ)3 Z T t F2(s, T )ds +1 2 3 X j=1 Z T t (Σ>F (s, T )F (s, T )>Σ)jjds (3.23)

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Finally, Christensen et al. (2011) show that the yields are given by y(t, τ ) = X_t1+1 − e λtτ λtτ X_t2+ 1 − e −λtτ λtτ − e−λtτ X_t3− A(t, τ ) τ , (3.24)

where again τ = T − t. Here is −A(t,τ )_{T −t} an unavoidable “yield adjustment term”, which only depends on the maturity of the bond T not on the time. In the next section this “yield adjustment term” is elaborated more in detail.

“The Yield Adjustment Term”

The DNS-model does not state the choice of P-dynamics, the choice of P-dynamics is irrelevant for the yield curve. However in the AFNS-model the volatility Σ affects both the P-dynamics and the yield curve through this yield adjustment term. Christensen et al. (2011) show that the adjustment term is identified when the drift term θQ _{= 0. Christensen et al. (2011) made two conclusions based on}

this result. First, the fact that AFNS-zero coupon yields are given by an analytical formula greatly facilitates empirical implementation of the AFNS models. Second, only the six terms ¯A, ¯B, ¯C, ¯D, ¯E and ¯F are identified. λt the maximally flexible

AFNS specification that can be identified has the triangular volatility matrix

Σ =     σ11 0 0 σ21 σ22 0 σ31 σ32 σ33     .

3.4 Forecasting

The ability to forecast is a crucial element of the model, since the expectation hy-pothesis does not hold empirically for the term structure of interest rates. There-fore it is important that the model can capture these deviations and produce good forecasts.

In order to forecast with the DNS model the time evolution of the factors needs to be specified. We have chosen for an auto-regressive (AR) process of order one given by

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ˆ

xi,t+1= ˆφ0+ ˆφ1xi,t. (3.25)

Following the work of Diebold and Li (2006) the vector autoregressive struc-ture (VAR) is disregarded and the autoregressive strucstruc-ture of order one is chosen. Diebold and Li (2006) argue that the inferiority of the VAR-model is caused by at least two reasons. First, VARs tend to produce poor forecasts of economic variables. Secondly, the factors display little cross-variation and are not highly correlated so that the appropriate multivariate model is close to a stacked set of univariate models.

To evaluate the forecasting performance of the different models, one can calculate the root-mean-square-error (RMSE), given by

RM SEmodel(τ ) = v u u t 1 T − t0 T X t=t0 (ˆyt(τ ) − yt(τ ))2, (3.26)

where ˆyt(τ ) is the yield forcasted by the model, yt(τ ) is the observed yield and the

forecast interval is given by [t0, T ]. When evaluating the forecasting performance

with the RMSE, a smaller value of the RMSE corresponds to a better forecast. The random walk is taken as the benchmark, since this is a simple no change forecast, hence a minimum standard for the accuracy of the forecast.

The last section elaborated the different yield curve models used to obtain the price forecasts of the different bonds. These forecasts are then used the obtain the expected returns needed for the mean variance optimization for the bench-mark portfolio. The yield curve models are compared with a principal component analysis to investigate the number of factors used and the fit of the curve, which will be introduced in the next section.

3.5 Models for comparison

This thesis focuses on the DNS - and the AFNS model, which are compared with a principal component analysis (PCA) and the Random Walk model, therefore

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these models are only elaborated as an introduction.8

3.5.1 Random Walk model

The Random Walk (RW) model is used as comparison for the other models. The RW is often reported as being difficult to beat in out-of sample forecast perfor-mance, and is given by

yt+h(τ ) = yt(τ ) + t(τ ), (3.27)

where t(τ ) is a White Noise process. A White noise process is a serially

uncorre-lated, zero-mean, constant and finite variance process. Hence the forecast of this models is given by

ˆ

yt+h(τ ) = yt(τ ) (3.28)

Assuming a random walk model for interest rates implies a simple ”no change” forecast for the individual yields. Note that the Nelson Siegel models and the PCA reduces the dimensionality of the data set where the RW models does not. In this model the h-months ahead prediction of a bond yield is simply given by the at t known yield.

3.5.2 Principal Component Analysis

The principal component analysis is used to test two aspects of the Nelson Siegel models. First, it is tested whether three factors are necessary. Secondly, the coefficients of a PCA with three components is compared with the factor of the Nelson Siegel models, to investigate whether factors can explain the variance.

PCA provides an approximation of the data in terms of the product of the principal components and the corresponding coefficients, see Wold et al. (1987). These two matrices try to capture the important patterns within the data set. The

8_{A more detailed description of the Random Walk model is given by Durrett (2010). An} overview of the Principal Component is given by Wold et al. (1987).

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principal components are linear transformations of the original data set and the corresponding coefficients of these principal components are calculated such that the first principal component contains the maximum variance of the data set, the second coefficient tries to capture the maximum (remaining) variance. An import characteristic of the principal components is the fact that they are uncorrelated so that they explain different patterns in the data.

For the PCA to work properly the mean must be subtracted from each of the dimensions, which produces a data set whose mean is zero. Consider a data set X, with n observations of m varianbles. The mean of m variables can be constructed as an m dimensional vector given by

µ = 1

n(x1 + ... + xn), (3.29)

where xi ∈ Rmx1 and n is the number of observations. Using the mean of the data

set given by (3.29) it is possible to recenter the data set

H = [x1− µ| ... |xn− µ] (3.30)

Note that H has the same dimensions as the original data set but has mean zero. The second step is to calculate the covariance matrix of this re-centered data set H. The covariance matrix can be defined as

S = 1

n − 1BB

>

(3.31) Thirdly, the eigenvalues and eigenvectors of the covariance matrix must be calcu-lated. Since S is an covariance matrix, this matrix is symmetric, therefore it can be orthogonally diagonalized by the Spectral Theorem9_{, which results in}

Svi = λivi, (3.32)

where λi is a scalar called the eigenvalue of S, and viis a m-dimensional eigenvector

of S, which are the principal components of the data set. These are important

9_{Spectral Theorem: If A is symmetric, then A is ortogonally diagonalizalbe and has only real} eigenvalues. In other words, there exists real numbers λ1, ..., λm(the eigenvalues) and orthogonal, non-zero real vectors v1, ..., vm(the eigenvectors) such that for each i = 1, ..., n: Avi = λivi, see Halmos (1963).

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since they provide important information about the data. The eigenvector who results in the best fit of the data is the first principal component of the data set since it explains the most variation.

This section introduced the principal component analysis and the random walk used as a comparison to the factor models. The next section describes the Monte Carlo simulation used to obtain the covariances of the bonds, which are needed for the mean variance optimization.

3.6 Monte Carlo simulation

With the DNS and the AFNS the yield curve is explained with only three factors. Fitting the yield curve with these models results in three series of beta’s from which we can obtain the historical (co-)variances. With these variances it is possible to simulate future yield curves with the last observed curves and the an normally distributed error with the historical variance around zero. Normally an AR(1) structure is used to describe the time-varying process of the factors, however in this case this led to a high level of negative yields hence for the Monte Carlo simulation is chosen for a Random Walk model behind the factors. The choice for the RW does not solve the problem of negative rates completely but leads to a reduction of the number of negative yields. The process to simulate the factors of the Nelson Siegel models is given by

xi,t+1= xi,t+ ·ηi,t

where,

ηi,t ∼ N (0, Ωi,i))

(3.33)

Here is xi,t the i-th fitted factor of the DNS and AFNS model and Ωi,i the

corre-sponding historical variance. I use equation 3.33 to simulate the distribution of the different factors at time t + 1 given time t.

With the Monte Carlo simulation a 100.000 possible curves are simulated, based on the last observed yield curve and historical variance. This results in a

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distribution at each tenor of the yield curve. This section elaborated on bond pricing, the yield curve models and the simulation techniques used to obtain the distribution of the bond returns and their forecasts. The next section will discuss the results based on this framework.

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4. Empirical Results

This chapter elaborates on the data that are used and the results from modelling the Euro Swap Curve with the models described in Chapter 3. First, the data are described extensively with a visual approach and descriptive statistics. Secondly, the results from the models are given.

4.1 Data description

Recall that this paper tries to obtain a benchmark for results generated from the mismatch. This is done by constructing an optimal bond portfolio with the Markowitz approach. In order to use this approach the return and variances of the available bonds need to be obtained. Starting with the price the term structure needs to be obtained, which are modelled with two models based on Nelson-Siegel. Estimating the yield curves with the DNS - and AFNS model requires data. The data used to obtain the model parameters are end-of-month observations of the Euro Swap Curve, in the time-period January, 1999 to April 2014 extracted from Bloomberg. This period leads to 184 observations for each maturity. For each time-point the maturities that are observed are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 20, 30 and 50 years, leading to 15 points on the yield curve, which we can fit with our three factor models. Given the data for different maturities at different time points results in a panel data structure.

When looking at the data plot in Figure 4.1, one first notices the cut off of a small number of curves. This results from the fact that the 50 year yield was not available for all time-points. It can be seen that the yield curve most often is

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normally upward-sloping, but also downward-sloping (“inverted”), hump shaped and trough shaped (“inverted humped shaped”) are observed. Further one could see that the short-term yields vary more than the long-term yields, but that small changes in the short-term yields highly affect the long-term yields.

Figure 4.1: Shapes Euro Swap Curve

In order to get a better understanding of the evolution of the term structure of time one could look at Figure 4.2. Again it is highly apparent that the typical yield curve is upward sloping. Further one could notice the significant decrease of the yield curve around 2007/2008, the beginning of the financial crisis. This decrease is followed by an increase of the slope of the yield curve. This steepening of the curve could be explained by an increase in demand in short-term high quality bonds, which pushes the prices downwards. Finally one could see that in comparison with the last 15 year the yield curve is very low at the moment.

Table 4.1 shows the descriptive statistics for the Euro Swap Curve. The mean, standard variation, minimum and maximum are shown for each maturity. These descriptive confirm the observations made from Figure 4.2 and 4.1, for example the increasing mean and decreasing variance with longer maturities.

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Figure 4.2: Evolution Euro Swap Curve

Maturity τ Mean Standard Deviation Minimum Maximum

1 2.66 1.46 0.33 5.38 2 2.83 1.43 0.38 5.52 3 3.01 1.40 0.47 5.59 4 3.19 1.16 0.60 5.65 5 3.35 1.32 0.75 5.70 6 3.49 1.28 0.91 5.76 7 3.62 1.25 1.06 5.82 Years 8 3.73 1.22 1.21 5.86 9 3.83 1.20 1.35 5.89 10 3.91 1.18 1.47 5.95 12 4.04 1.15 1.69 6.06 15 4.19 1.12 1.90 6.16 20 4.30 1.12 1.92 6.22 30 4.30 1.15 1.86 6.22

Table 4.1: Evolution Euro Swap Curve

This section described the used data to model the yield curve. The remainder of this chapter elaborates on the performance of the models, the next section starts with investigating the need for three factor models.

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4.2 Specification model

The models used in this thesis are three factor models based on the Nelson Siegel curve. It is interesting to investigate whether three factors are needed to correctly model the yield curve. This sections elaborates on a empirical analysis based on a PCA model, to investigate whether three factors are needed.

(a) The first three principal components (b) Factor loadings Figure 4.3: The first three principal components of the data set

When investigating whether three factor are necessary a PCA is very useful. Calculating the explained variance of each principal component, it follows that the first principal component explains nearly ninety-seven percent of the variance in the data set. This coincides with the conclusion of Figure 4.3a, which shows that the level has the highest variation. When explaining the variance over time the first principal component would be sufficient, however the added value of the remaining two principal component follow from fitting the yield curve at a specific moment in time, which can be seen from (4.8).

From the principal component analysis also follows that the first three prin-cipal components could be interpreted as level, slope and curvature. This can be seen from figure 4.3b. It shows that the first principal component affects all matu-rities equally which could be interpreted as a level factor, which was explained in 3.3.2. The second principal component affects mostly the short end of the curve,

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while the third factor mainly affects the middle of the curve, and can therefore be interpreted as slope and curvature.

This section showed the advantage of the two additional factors, slope and curvature and concludes that the three factor models results in a better forecast of the yield curve. The next section elaborates on the performance of these three factor models.

4.3 Empirical Results

A logical question when modelling the yield curve is “when is the yield curve model good enough?”. The answer of this question starts with defining the purpose of the yield curve model. When using the yield curve model to price bonds, one must evaluate on how well the model matches bond prices and the movements of the interest rate. Since this thesis tries to obtain a recurring benchmark, which is an optimal investment strategy, another important aspect of the yield curve model is its sensitivity to dynamics in the yield movements. Evaluating the proposed models one must investigate along three different aspects of the model. The first aspect is the pricing performance of the model, which can be evaluated with the root mean squared error. Secondly, one must investigate whether the model does capture the time evolution correctly, which can be done by checking the residuals of the model. Finally, one must investigate the stability of the model parameters. The outline of this section is according these three aspects. Both the DNS and AFNS model are evaluated with respect to each aspect and compared with the Principal Component Analysis and Random Walk.

4.3.1 Forecasting Performance

Forecasting of the bond prices is the main purpose of the used models and there-fore an important aspect. When obtaining a benchmark portfolio the return and variances of these returns need to be accurate. In this section the forecasting per-formance of each model is evaluated both in- and out-of-sample fit. The first eighty percent of the data is used to estimate the parameters. The remaining twenty

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per-cent is used to evaluate the forecasting performance of the model. To make the structure of the data set and the use of it more clear this is shown in Figure 4.4. Note that it is impossible to give an absolute measure of pricing performance of the different models. However, to investigate the performance of the DNS - and the AFNS model, these models are compared with a PCA and the Random Walk model, which results in a relative performance measure.

Figure 4.4: Structure and use of the data set

Fit of the Euro swap curve

When following the procedure introduced by Diebold and Li (2006), a reasonable fit for the Euro Swap Curve is found. The results for four specific dates are stated in Figure 4.6. When looking at the different graphs we see that the model can produce different shapes. The best fit is found for the yield curve on 30-Apr-2014 and 29-Apr 2005. For 29-Apr-2011 and 30-Apr-2008 the curvature is larger and as expected the DNS has difficulties capturing this higher curvature.

The random walk model states that the best prediction of the swap curve is the last observed swap curve. An important consequence is that when the swap curve remains constant over time this model gives the best forecast. This can be seen for the swap curve on 30-Apr-2014, 29-Apr-2011 and 29-Apr-2005 in Figure 4.7. However when the swap curves change over time the random walk model results in estimation errors, which can be seen at 30 of April 2008, when the swap

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curve increased with respect to the last observed period. Comparing the DNS model and the random walk for 30 of April 2008, the DNS model gives a slightly better fit. An explanation for this finding is that the increase of the swap curve was observed in the observed time point prior to 30 April 2008. When looking at the data it follows that the increase of the swap curve on 30 April 2008 was the third increase in a row, where each observation is the end of the month.

Looking at the fit of the third model, the principal component analysis, which is shown in Figure 4.8, it can be seen that the model only captures the level of the curve. This is hardly surprising since only one principal component is included. When again comparing this model with the DNS model the advantages of the two additional factors become apparent, since a one factor model gives a poor fit. Interesting is the shape of the curve fitted by the first principal component, which is especially evident in the yield curve fit of April 2008. This shape can be explained when looking more closely at the loading of the first principal component from Figure 4.3b and is shown in Figure 4.5. The resulting yield curve follows the shape of factor loading of the first principal component.

Figures (4.6) and (4.8) show that the two additional factors of the DNS-model highly improve the forecast of the yield curve. This increased performance in forecasting justifies the additional factors. When comparing the DNS with the RW and the AFNS it is not obvious which model performs best, a more detailed investigation of the two models is needed. Because the main objective of the model is to estimate the yield curve in order to obtain bond prices, it is interesting to investigate how the estimation errors of the swap curve influence the yield curve.

Figures 4.6 and 4.7 encourage the use of Nelson Siegel models because of the good fit. However it is interesting to investigate when these models results in a good fit or not. The in-sample forecasts from 29 April 2011 until 31 October 2014 can be compared with the realised rates. This is done by comparing the factors level, slope and curvature of the Nelson Siegel models with the Random Walk model. When doing so one can distinguish two different period within the sample. The first period is from 29th of April until the end 2012, the second period is the remaining of the sample.

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Figure 4.5: Factor loading of the first principal component

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Figure 4.7: Forecasting the swap curve with the Random Walk model

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In the first period, 29th of April until the end 2012, the DNS model often shows a comparable or better fit for the level of the yield curve when compared with the Random Walk. The slope factor mostly coincide to that of the random walk, that is in ten out of the 12 cases the slope coincides. However the curvature factor shows a remarkable worse fit than the random walk. In this period the yield curve is flattening over time or even decreasing, and the DNS model results in an underestimation of the curvature in most cases. In this case the AFNS would result in a better fit since the yield adjustment term mainly affects the curvature factor, because the effect increases with maturity.

In the second period, 31th of January 2013 until the 31th of April 2014, the DNS model and Random Walk show coincide in most cases. In this period the yield curve is fairly constant over time. This shows that within a stable yield curve environment, without large shocks in the three factors, the Nelson Siegel models result in a good forecast of the yield curve. In this case the AFNS would overestimate the curvature factor because the adjustment term is subtracted from the DNS forecasted curve, which was a good fit. An explanation for the overestimation of the curvature factor is that the variance of the factors within this period is lower than of the sample variation. Therefore the adjustment term, which is increasing with the variance matrix Σ, is too high for the low variance period. The long term tenors are especially affected because the adjustment term is monotonically increasing in time to maturity, which is shown in Figure 4.9. Hence it is possible to conclude that the AFNS model results in a good forecast when the estimated variation of the period is a good representation of the realized volatility, even for an autoregressive structure of order 1.

Fit of the zero coupon fixed-income yield curve

In the last sub-section the fit of the Euro swap curve was elaborated, however the corresponding errors with respect to the zero coupon fixed-income yield curve are more important, since the objective is zero coupon bond pricing. When investi-gating the fit of the zero coupon fixed income-yield curve one can distinguish two different cases, namely in-sample and out-of-sample fit. However, Duffee (2002),

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Figure 4.9: Structure and use of the data set

and Diebold and Li (2003) argue that the ability of the model to fit the data is a poor measure of the capability of capturing the interest rate dynamics. Instead of the fit of the data, one need to consider the ability to forecast the zero coupon fixed-income yield curve as a measure to capture the interest rate dynamics. As mentioned earlier, the forecasting is done by using 80 percent of the observations to estimate the model parameters and use the remaining data to compare the observed yields with the forecasts produced by the different models. Recall that an AR(1) is used for the evolution of the state variables. Further the forecast performance is measured with the RMSE (3.26).

The conclusions drawn from Table 4.2 coincide with the conclusion made in the last subsection. That is, the Random Walk and the DNS perform comparable when forecasting the swap curve and the corresponding zero curve (zero coupon curve). The principal component analysis performs worse then the latter models. This section described the performance of point estimation of the Euro swap curve and corresponding zero coupon fixed-income yield curve, which is needed for bond pricing. Hence it is interesting to know how these fitting errors of the zero coupon fixed-income yield curve affects the bond prices. The next subsection tries to investigate this effect on bond prices.

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Tenor RW DNS AFNS PCA 1 0.0970 0.1340 0.1335 0.9631 2 0.1315 0.1533 0.1525 0.9584 3 0.1577 0.1969 0.1942 0.8562 4 0.1774 0.2226 0.2167 0.6971 5 0.1872 0.2325 0.2232 0.5348 10 0.2000 0.1941 0.1868 0.3937 12 0.2039 0.2050 0.2078 0.5778 15 0.2089 0.2035 0.2897 0.7819

Table 4.2: Root Mean Square Errors for zero coupon fixed-income yield curve.

Bond Prices

The last two sections described the fit of the Euro swap curve and the correspond-ing zero coupon fixed-income yield curve of the different models. These fittcorrespond-ing errors lead to different prices for the zero coupon bonds. This section compares the forcasted prices that result from the different models and the observed swap rates. These pricing errors are measured for each tenor, again by using the root mean squared error as measure. The fit of bond prices of the different models will not differ from the fit of the yield curve, nevertheless this section is added to show that the fitting error are magnified for the longer maturities due to the higher sensitivity with respect to the interest rates.

The conclusions that can be drawn from Table 4.3 are in line with the Figures 4.6 - 4.8 and Table 4.2. The principal component model shows larger pricing errors in comparison with the random walk and the DNS model. Further, one could notice that the pricing errors increase with the tenors. The intuition behind the latter is that the swap rates contain coupon payments. When for example the 30 year swap rate is estimated incorrectly, the estimation error affects the cash flow of 30 years, whereas the estimation error of the 1 year swap rate only affects the one year cash flow. Therefore the estimation errors in the swap rates have a larger effect on the bond prices for longer tenors.

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Tenor RW DNS AFNS PCA 1 0.0974 0.1348 0.1342 0.9475 2 0.2578 0.3015 0.3191 1.8693 3 0.4590 0.5725 0.5651 2.4762 4 0.6786 0.8495 0.8280 2.6463 5 1.0469 1.0902 1.1474 2.4213 10 1.6519 1.5925 1.5377 3.2392 12 1.9070 1.8194 1.8492 5.5067 15 2.2338 2.3000 3.1871 8.6599

Table 4.3: Root Mean Square Errors for bond prices.

From this section the conclusion can be made that adding additional factors to the one factor model significantly improves the fit of the Euro swap curve. This better fit is reflected in the fit corresponding zero coupon fixed-income yield curve and the bond prices. However this section only elaborated the time point pricing performance of each model and has not investigated the ability to capture the interest rate dynamics of time and the stability of the parameters. In the next section this ability to capture the interest rate dynamics is elaborated.

4.3.2 Time Evolution

In the last section the forecasting performance of each model was given. Since the benchmark is based on optimal investment strategies one could be interested in the sensitivity over time of this benchmark and if the model captures the evolution of the yield curve over time. Hence this section elaborates on the ability of each model to capture time evolution of the Euro swap and zero coupon fixed-income yield curve. This is done by evaluating the residuals of each model to look whether there is any autoregressive structure left in the residuals.

Figure 4.10 shows the autocorrelations and residual autocorrelations of the three estimated factors of the DNS model. The residual autocorrelations result from the AR(1) model fit to the DNS factors. From the figure one can conclude

A benchmark for interest rate risk using a Markowitz approach