Comparison of yield curves' models based on in-sample fit

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Master’s Thesis

Comparison of Yield Curves’ Models

based on In-Sample Fit

Yijia Lu

Student number: 11371587 Date of final version: August 4, 2017 Master’s programme: Econometrics

Specialisation: Financial Econometrics Supervisor: Prof. dr. H. P. Boswijk Second reader: Dr. S. A. Broda

Faculty of Economics and Business

Amsterdam School of Economics

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If preferred you can change the number and order of the sections (but the order you use should be logical) and the heading of the sections. You have a free choice how to list your references but be consistent. References in the text should contain the names of the authors and the year of publication. E.g. Heckman and McFadden (2013). In the case of three or more authors: list all names and year of publication in case of the rst reference and use the rst name and et al and year of publication for the other references. Provide page numbers.

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Statement of Originality

This document is written by Yijia Lu who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Introduction

The yield curve plays a remarkable role in macro-regulation and finance, which is universally recognized by researchers and monetary policy makers. It is a basic tool for forecasting interest rates, pricing financial assets, allocating investment portfolios, and implementing monetary policy. Therefore, it is necessary to understand the evolution of yield curves.

Many previous researchers have developed various modeling approaches to fit yield curves. Nelson and Siegel (1986) firstly introduced a parsimonious model of yield curves (NS model). This model has a few parameters which has the attractive feature, that is simple and flexible enough to represent the many different shapes of yield curves. Many countries’ central banks use this model to construct and release yield curves. Since this static models cannot capture the dynamics of yield curves, researchers transfer this model into the dynamic version. Litterman and Scheinkman (1991) firstly proposed to use factors to analyze yield curves. According to their characteristics, these factors can be classified as a level factor, a slope factor, and a curvature factor. Diebold and Li (2006) proposed the Dynamic Nelson-Siegel (DNS) model. In this paper, they introduced a new yield curve modeling framework which is named as the DL model. They found that three of the unknown parameters in the NS model can be interpreted as three latent factors of yield curves. Due to the good performance of the DL model in empirical analysis, this model can also be generalized to other models with more factors.

Svensson (1994) improved the Nelson-Siegel model (NSS model) by adding more parameters to better fit yield curves. The dynamic version of the Svensson model is presented by De Pooter (2007) based on the idea proposed by Diebold and Li (2002). The dynamic Nelson-Siegel-Svensson (DNSS) model includes the second curvature factor that can better fit the medium maturity bonds.

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CHAPTER 1. INTRODUCTION 2

The DNSS model can enhance the ability of representing complex forms of yield curves, such as a bumped shape, or a V shape. Similarly, the NSS model also can be extended to the generalized Nelson-Siegel (GNS) model which includes one more term. The dynamic version of this model which includes the second slope factor is referred as the dynamic generalized Nelson-Siegel (DGNS) model. This extension was firstly introduced by Chistensen, Diebold, and Rudebusch (2011) since the DNS, and DNSS model cannot achieve arbitrage-free consistency.

Adding affine structure and arbitrage-free conditions into the DNS, DNSS, DGNS models can make them tractable and theoretically rigorous. Duffie and Kan (1996) proved that the single factor models proposed by Merton (1973), Vasicek (1977), and Cox, Ingersoll Jr, and Ross (1985) are affine models. The yields can be expressed as a linear function of latent factors after introducing affine models. Unfortunately, Duffee (2002) focused on completely affine models using quasi-likelihood es-timation method, he concluded he cannot forecast the future yield curves well. This failure is caused by too many local maxima of quasi-maximum likelihood function. The Arbitrage-Free principle is the fundamental assumption of financial markets. This principle is widely used in financial asset pricing since the fair price means there is no arbitrage opportunity in financial markets. Because bonds are traded in mature and liquid markets, the attraction of imposing theoretical restrictions to eliminate the arbitrage opportunities is huge. Ho and Lee (1986) firstly used arbitrage-free pric-ing theory to derive a term structure model of interest rates. In recent years, combinpric-ing the affine and arbitrage-free conditions, Duffee (2002)’s problem finally was overcome by Christensen, Diebold, and Rudebusch (2007). They introduced a new class of Gaussian affine arbitrage-free models on the basis of the DNS model. This class of models are named as the arbitrage-free dynamic Nelson-Siegel (AFDNS) model. According to the similar approach, Christensen, Diebold, and Rudebusch (2011) extended the five-factor DGNS model to the arbitrage-free dynamic generalized Nelson-Siegel (AFDGNS) model.

All models introduced above are theoretical dynamic term structure models, it is necessary for researchers to concentrate on the best fitting empirical dynamic term structure models. Relying on the finding of Christensen, Diebold, and Rudebusch (2011), we can restrict the volatility matrix Σ in the AFDNS model to be diagonal. Then we can set the less significant parameters in the mean-reverting matrix K to zero, which can decrease the number of unknown parameters in K, and boost the efficiency of the AFDNS model. The best specification of the AFDNS model is provided by

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Christensen, and Rudebusch (2012). This most efficient model is named as the CR model. However, this best-fitting empirical dynamic term structure model has encountered problems when yields are close to the zero lower bound. In 2008, the tremendous financial crisis resulted by the sub-prime mortgage crisis which arose in United States, then, resulted in a financial crisis over the world. In order to deal with the recession, the United States Federal Reserve (Fed) lowered the interest rates, reaching and crossing the zero lower bound since December, 2008. This policy has been conducted until December, 2015. During this period, the United States Treasury had the yields which are close to the zero lower bound. Though the previous arbitrage-free models can capture the dynamics of yields which are near the zero lower bound, these models fail to fit the yield curves in two aspects. Firstly, the short rate prediction is unrealistic, which the positive probability for negative short rate forecast. Christensen and Rudebusch (2013) confirmed this by plotting the graph of the conditional probability of negative short rate prediction from the CR model. They found that before December, 2008, the condition probability was approximately zero, while after that, the significant probability cannot be ignored. Secondly, these models neglect the facts that the short rate was close to the zero lower bound, and the long-term rates became relatively flat during this period. Hence, in this paper, they proposed the shadow rate B-CR model to find the solution to this problem. Christensen and Rudebusch (2013) also compared the performance between the CR model and the shadow rate B-CR model, and made the conclusion that the B-CR model fits the yields at the zero lower bound with high efficiency.

Two estimation methods are widely applied in yield curve modeling. Diebold and Li (2002) argued instead of utilizing nonlinear least squares to estimate the DNS model, a more simple and convenient two-step ordinary least squares estimation method can be used. They fixed λ in the DNS model to obtain the loadings of factors, then used ordinary least squares to estimate. After that, the factors’ dynamics can be estimated by a V AR(1) model. This estimation method is regarded as the two-step estimation method. For continuity with the existing literature about arbitrage-free version of models, Diebold and Li (2002) have shown that the DNS model can be transformed to a dynamic version with latent factor model. More specifically, Diebold, Redebusch, and Aruoba (2006) used the transition matrix to control the dynamics of the latent factors. This factor model representation can be the independent-factor type, or the correlated-factor type. In that paper, using the Kalman filter for the maximum likelihood estimation for factor models was introduced. Henceforth, the Kalman filter is commonly used in yield curve modeling. This method is the second approach to do yield

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curve modeling. For instance, Christensen, Diebold, and Rudebusch (2011) employed the Kalman filter to estimate arbitrage-free versions of factor models. For the shadow-rate model, since the tran-sition equation is not linear, Christensen and Rudebusch (2013) used the extended Kalman filter to estimate the B-CR model.

Based on all models and methods I mentioned above, the in-sample fit of these models can be compared. Diebold, Redebusch, and Aruoba (2006) compared the DNS model using two-step es-timation along with the state-space representation with the maximum likelihood eses-timation, they concluded that the maximum likelihood estimation provides a better in-sample fit. However, Bolder and Liu (2007) attained a contradictory result that the two-step estimation provides a fit with more precision. In this paper, empirical analysis will be applied to the Nelson-Siegel, the Nelson-Siegel-Svensson, and the generalized Nelson-Siegel models to compare the two-step estimation and the maximum likelihood estimation. Further, Christensen, Diebold, and Rudebusch (2007) compared the in-sample fit between the independent-factor DNS model, the correlated-factor DNS model, the independent-factor AFDNS model, and the correlated-factor AFDNS model, they found that the correlated-factor DNS model provides little competitive edge in fitting the yield curves. They also found that the independent-factor AFDNS model outperforms the the correlated-factor AFDNS model. In this paper, I will extend the comparison to the generalized Nelson-Siegel model. Chris-tensen and Rudebusch (2013) constructed the shadow rate B-CR model depending on the AFDNS framework, then he compared the B-CR model with the CR model. They concluded the B-CR model provides a similar fit before the end of 2008, when yields are near the zero lower bound, the B-CR model has an evidently better fit compared with the CR model. In this paper, these findings stated in previous papers will be verified by using more data in the zero lower bound state.

The research questions which will be investigated in this paper are:

1. For the dynamic factor models, which estimation methods are better? The two-step estimation method or the Kalman filter maximum likelihood estimation method?

2. After adding more factors into the DNS model, does the in-sample fit becomes better?

3. After imposing the arbitrage free restriction, what happens to the in-sample fit for the dynamic three-factor, four-factor, and five-factor model?

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The paper is organized as follow: In Chapter 2, some basic concepts which will be further used in the modeling part will be presented. In this chapter, I will first introduce the basic formulas for computing yield curves. Then, the arbitrage-free pricing principle and the affine factor models will be introduced which is the foundation for the arbitrage-free versions of models. Finally, some concepts of the option-based shadow-rate models will be given which will be further used in the B-CR model. Chapter 3 will derive and present all term structure models studied in this paper. After that, descriptive statistics will be shown in Chapter 4. Chapter 4 will also illustrate the estimation methods for different specific models in detail. In Chapter 5, all empirical results will be presented, and finally in Chapter 6, conclusions and further implications of this thesis will be given.

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Chapter 2

Modeling the Term Structure of Interest

Rates: Basic Concepts

The term structure of interest rates depicts the relation between the yields on different bonds and their time to maturity, at a specific time, so the yield curves can be used to represent the term structure of interest rates. Theoretically, the term structure of interest rates provides mathematical expression of yield curves on a zero-coupon bond. Hence, we can start the yield curves modeling with the zero-coupon bond.

2.1 Interest Rates

In this section, I will derive the basic formula used by Nelson-Siegel (1987) to model the yield curves (formula 2.4). Consider a zero-coupon bond with nominal value of $1. Let P (t, T ) be the price, at time t, of this zero-coupon bond with maturity T. This bond is also named as the T-bond. We assume:

P (T, T ) = 1 f or all T P (t, T ) ≤ 1 f or all t ≤ T

The equations above can be explained by the discount effect. Denote y(t, T ) be the continuously compounded yield to maturity. The price of zero-coupon bond at time t can be discounted as:

P (t, T ) = 1 × e−y(t,T )(T −t)

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CHAPTER 2. MODELING THE TERM STRUCTURE OF INTEREST RATES: BASIC CONCEPTS 7

This leads to the expression of continuously compounded yield to maturity:

y(t, T ) = −log P (t, T )

(T − t) (2.1)

To introduce forward rates, we consider a replicating strategy which involves 3 time points: the current time t, the expiry time T , and the maturity time S of a forward contract. The relation among these 3 time points is: t < T < S. This replicating strategy has the following cash flows:

At time t: Short one T-bond, and long P (t,T )_{P (t,S)} S-bonds. At time T : pay one dollar

At time S: obtain P (t,T )_{P (t,S)} dollar Table 2.1: Cash Flows of Forward Replicating Strategy

At time t, we have the cash flow: 1 × P (t, T ) − P (t,T )_{P (t,S)} × P (t, S) = 0, Hence, in this strategy, $1 will be invested at time T , and P (t,T )_{P (t,S)} will be received at time S. Hence, we can derive several interest rates depended on this strategy. The simple forward rate for [T, S] prevailing at time t is:

P (t, T ) P (t, S) = 1 + (S − T )F (t; T, S) =⇒ F (t; T, S) = 1 (S − T ) P (t, T ) P (t, S) − 1

The continuously compounded forward rate for [T, S] prevailing at time t is: P (t, T )

P (t, S) = e

y(t,T ,S)(S−T )

=⇒ y(t, T, S) = log P (t, T ) − log P (t, S) S − T

Taking S → T , the instantaneous forward rate can be obtained:

f (t, T ) = lim

S→Ty(t, T, S) = −

∂ log P (t, T )

∂T (2.2)

The instantaneous short rate or risk-free rate at time t can be derived from (2.2):

r(t) = f (t, t) = lim

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Using (2.2) and the assumption P (T, T ) = 1 for all T , we can derive the relation between the bond price and the forward rate:

P (t, T ) = exp − Z T t f (t, s)ds

Using the expression between the bond price and the forward rate and Equation (2.1), we can obtain the relation between yield to maturity and forward rate, that is:

y(t, T ) = 1 (T − t)

Z T

t

f (t, s)ds (2.4)

2.2 Arbitrage-Free Pricing Principle and Affine Factor Models

Following Duffie and Kan (1996), I introduce the factor model for yield curves. Assume a complete probability space (Ω, F , Q), with the augmented filtration {Ft}t≥0generated by a standard Brownian motion WQ _{∈ R}n_{, also assume the stochastic process {X}

t} on some open set D ⊂ Rn. Define the market price of a zero-coupon bond at time t maturing at T be P (Xt, T ). According to the definition of instantaneous short rate or risk-free rate at t in (2.3) and the relation between the bond price and the forward rate, the risk-free interest rate equals:

r(Xt) = lim T →t−

log P (Xt, T ) (T − t)

Risk neutral pricing which discounts the cash flows by the risk-free interest rate r can be used to price a zero-coupon bond. The risk neutral measure, also known as the Q measure, is closely related to risk neutral pricing. Usually, we refer the real world probability measure as the P measure. The P measure is equivalent to the Q measure, which means P(A) = 0 ⇐⇒ Q(A) = 0. Starting from this section, I denote τ = T − t for simplicity. The price of zero-coupon bond can be written on the basis of risk neutral pricing:

P (Xt, τ ) = EQ exp − Z t+τ t r(Xs)ds Xt (2.5)

where EQ _{is the conditional expectation under the Q measure.}

Suppose that the Markov process {Xt}t≥0 satisfies the following stochastic differential function:

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where µQ_X(Xt) : D → Rn is the drift term, and σ Q

X(Xt) : D → Rn×n is the diffusion term (volatility term). An important characteristic for continous stochastic differential equations is that the volatil-ity is the same under the P measure and Q measure, that is σXP(Xt) = σQ_X(Xt). This characteristic can be proved using the Cameron-Martin-Girsanov Theorem.

Let the price of zero-coupon bond P (Xt, τ ) follows a Ito process, that is:

dP (Xt, τ ) = µ Q P(Xt, τ )P (Xt, τ )dt + σ Q P(Xt, τ )P (Xt, τ )dW Q t

Applying Ito’s lemma, we have:

µQ_P(Xt, τ ) = − Pτ(Xt, τ ) P (Xt, τ ) + PX(Xt, τ ) 0 P (Xt, τ ) µQ_X(Xt) + 1 2tr σ_XQ(Xt)σ Q X(Xt)0 PXX(Xt, τ )0 P (Xt, τ ) )

where PX, Pτ, PXX are partial derivatives of P . The transformation from Q to P based on Girsanov’s theorem is characterized by:

dQ dP = ξ where ξ is defined as:

ξt= exp − Z t 0 σξ(Xs)dWt− 1 2 Z t 0 σξ(Xs)σξ(Xs)0ds

where σξ : D → R1×N is the diffusion term (volatility) of ξ, with ξ0 = 1. Define dWtQ = dWtP + σξ(Xt)0dt, substitute this into (6), and combine the fact that σXP(Xt) = σQX(Xt), obtain:

dXt = (µPX(Xt) + σXQ(Xt)σξ(Xt)0)dt + σXP(Xt)dWtP

Hence, for the augmented filtration {Ft}t≥0 generated by a standard Brownian motion WP ∈ Rn, the Markov process {Xt}t≥0 satisfies the following equation:

dXt= νXP(Xt)dt + σXP(Xt)dWtP (2.7)

where νP

X(Xt) : D → Rn is the drift term, and σXP(Xt) : D → Rn×n is the diffusion term (volatility term). In conclusion, the information above tells for the stochastic process {Xt} can be analyzed under both P and Q.

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Harrison and Pliska (1981) argued that in complete markets, the existence of a Q measure is equivalent to the absence of arbitrage opportunities. Hence, in this paper, the probability measure Q can be used in the AFDNS model and AFDGNS model. Then, the arbitrage-free models will be extended to the real-world dynamics, that is the arbitrage-free models under P.

For asset pricing, the instantaneous risk-free rate can be defined as an affine function of Xt:

r(Xt) = ρ0(t) + ρ1(t)0Xt

where ρ0(t) ∈ R, ρ1(t) ∈ RN, ρ1(t) depends on the dimension of Xt, or number of factors of Xt. Suppose {Xt} satisfies the following process under risk neutrality:

dXt= KQ(t)(θQ(t) − Xt)dt + ΣQ(t)DQ(Xt)dWtQ (2.8)

where D is a diagonal matrix:

D(Xt) =           √ α1+ β1Xt · · · 0 .. . . .. ... 0 · · · √αn+ βnXt          

, and (2.8) is (2.6) with specific drift and diffusion term.

µQ_X(Xt) = KQ(t)(θQ(t) − Xt)

σ_XQ(Xt) = ΣQ(t)DQ(Xt) Since σQ_X(Xt) = σXP(Xt), (2.8) can be rewritten as:

dXt = KQ(t)(θQ(t) − Xt)dt + Σ(t)D(Xt)dWtQ

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differential equation under Q to stochastic differential equation under P:

dXt= KP(t)(θP(t) − Xt)dt + Σ(t)D(Xt)dWtP (2.9)

Equation (2.9) shows the structure of the transition equation which will be useful for the later mod-eling part.

Now, we start to introduce some basic concepts of the measurement equation which is necessary for our modeling. Duffie and Kan (1996) found the risk-neutral price of zero-coupon bond (2.5) can be expressed using the exponential affine function under risk neutral valuation:

P (Xt, τ ) = exp(A(τ ) + B(τ )0Xt) (2.10)

According to Feynman–Kac formula, A(τ ) and B(τ ) satisfy the following ordinary differential equa-tions: dA(τ ) dτ = ρ0− B(τ ) 0 KQθQ− 1 2 n X j=1 (Σ0B(τ )B(τ )0Σ)jjαj (2.11) dB(τ ) dτ = ρ1+ (K Q₎0 B(τ ) − 1 2 n X j=1 (Σ0B(τ )B(τ )0Σ)jjβj (2.12)

The boundary conditions are: A(0) = 0 and B(0) = 0. According to (2.1), the continuously com-pounded yield to maturity can be rewritten as:

y(t, τ ) = −log P (Xt, τ )

τ (2.13)

Subsititute (2.10) into (2.13), the yield to matruity becomes:

y(t, τ ) = −A(τ )

τ −

B(τ )0

τ Xt (2.14)

Equation (2.14) shows the the structure of measurement equation for arbitrage-free version of model which will be used in the later modeling part.

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2.3 Option-Based Shadow-Rate Models

In this section, I will continue to use the notation in Section 2.1. Black (1995) first proposed the concept of a shadow short rate which is used to solve the zero lower bound problem. Define the shadow short rate as st, then the instantaneous risk-free interest rate which is restricted by the zero lower bound takes the form of:

r(t) = max{0, st}

so I can refer this interest rate with the underline as the zero lower bound interest rate. However, Black’s model is difficult to implement due to the curse of dimensionality. In order to overcome this problem, Krippner (2012) presented a new approach to approximate Black’s model using an option-based method. He assumed the physical currency exists and can be held without cost. Without the currency, the zero-coupon bond price (shadow-rate zero-coupon price) can be traded above par, with the existence of currency, this price will be restricted to be traded below par, so the negative short rates will not be observed. This implies, with the currency, the zero lower bound zero-coupon bond price equals the shadow-rate zero-coupon bond price maturing at t + τ minus an American call option price on the same shadow-rate zero-coupon bond with strike price 1$. This implication can be written as:

P (t, τ ) = PS(t, τ ) − CS,A(t, τ, τ ; 1)

Due to the availability of early exercise for American call options, the payoff of this American call option is difficult to calculate. Krippner (2012) introduced an auxiliary zero-coupon bond price which is based on the corresponding European call option to approximate the zero lower bound zero-coupon bond price of the American call option. The auxiliary zero-coupon bond price equals the shadow-rate zero-coupon bond price maturing at t + τ + δ minus a European call option price on a shadow-rate zero-coupon bond with strike price 1$ maturing at t + τ , that is:

Pa(t, τ + δ) = PS(t, τ + δ) − CS,E(t, τ, τ + δ; 1)

By setting δ → 0, the zero lower bound zero-coupon bond instantaneous forward rate is defined as:

f (t, τ ) = lim δ→0 − d dδlog Pa(t, τ + δ)

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Krippner (2012) derived the zero lower bound zero-coupon bond instantaneous forward rate to be:

f (t, τ ) = fS(t, τ ) + z(t, τ ),

where fS(t, T ) is the shadow-rate zero-coupon bond instantaneous forward rate, and z(t, τ ) is the option effect which has the expression:

z(t, τ ) = lim δ→0 − d dδ CS,E_{(t, τ, τ + δ; 1)} PS_{(t, τ + δ)}

For Gaussian affine term structure models, Krippner (2012) derived the expression for the zero lower bound zero-coupon bond instantaneous forward rate:

f (t, τ ) = fS(t, τ )N f S_{(t, τ )} ω(t, τ ) + ω(t, τ )√1 2πexp − 1 2 fS_{(t, τ )} ω(t, τ ) 2! , (2.15)

where N (·) is the cumulative normal distribution function, and

ω(t, τ )2 = 1 2limδ→0 ∂2_{v(t, τ, τ + δ)} ∂δ2 .

Note that v(t, τ, τ + δ) is the conditional variance in the shadow-rate zero-coupon bond option. Equation (2.15) will be used later for construction of the yield-to-maturity for the shadow-rate B-CR model in the later modeling part.

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Chapter 3

Modeling the Term Structure of Interest

Rates: Models

In this section, by using the basic concepts presented in the previous section, all yield curves models can be introduced. Firstly, all dynamic factor models will be presented, then, all arbitrage-free dynamic models will be gave, finally, based on the arbitrage-free framework, two empirical models, the CR model and the shadow-rate B-CR model will be explained.

3.1 Dynamic Nelson-Siegel Model

Nelson and Siegel (1987) first proposed a parsimonious model for yield curves. This model in-volves relative few parameters and these parameters have evident economic meanings. They described the relation between instantaneous forward rates and time to maturity τ as:

f (t, τ ) = β1+ β2· exp(−λτ ) + β3λτ · exp(−λτ )

Using the notation of τ , Equation (2.4) becomes:

y(t, τ ) = 1 τ Z t+τ t f (t, s)ds (3.1) 14

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CHAPTER 3. MODELING THE TERM STRUCTURE OF INTEREST RATES: MODELS 15

Applying the formula of instantaneous forward rates, the yield to maturity has the following expres-sion: y(t, τ ) = 1 τ Z t+τ t (β1+ β2· exp(−λs + β3λs · exp(−λs)) ds = β1+ β2 1 − e−λτ λτ + β3 1 − e−λτ λτ − e −λτ

where β1, β2, β3, λ are all parameters.

Based on this basic model, Diebold and Li (2006) introduced the dynamic framework, and trans-formed β1, β2, β3 into time varying parameters Lt, St, Ct which represents a level factor, a slope factor, a curvature factor, respectively.

y(t, τ ) = Lt+ St 1 − e−λτ λτ + Ct 1 − e−λτ λτ − e −λτ (3.2)

Model (3.2) is named as the dynamic Nelson-Siegel model, briefly, the DNS model.

These factors contain strong mathematical implications in practice. The factor loading for Lt is F L1(τ ) = 1 which has a constant effect on yield to maturity. Since this loading of the factor is independent of time to maturity τ , naturally, a change of Lt will have the same effect for all yields of different maturity, so, regarding Lt as level factor is reasonable.

The factor loading for Stis F L2(τ ) = 1−e

−λτ

λτ . The trend of F L2(τ ) can be analyzed by calculating the limit values and taking the first order derivative. Applying de l’ Hopital’s rule to calculate the limit values: lim τ →0F L2(τ ) = limτ →0 1 − e−λτ λτ = lim τ →0 (1 − e−λτ)0 (λτ )0 = lim τ →0 λe−λτ λ = 1 lim τ →∞F L2(τ ) = limτ →∞ 1 − e−λτ λτ = 0

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The first order derivative w.r.t τ is:

dF L2(τ )

dτ =

e−λτ(λτ + 1) − 1 λτ2

The Taylor expansion ex =P∞ i=0 xi i! = 1 + x + x 2 + O(x 3_{) leads to:} dF L2(τ ) dτ < 0

Hence, these imply F L2(τ ) is a decreasing function in the domain (0, ∞).

The factor loading for Ct is F L3(τ ) = 1−e

−λτ

λτ − e

−λτ_{, based on the same approach:}

lim τ →0F L3(τ ) = limτ →0 1 − e−λτ λτ − e −λτ = 0 lim τ →∞F L3(τ ) = limτ →∞ 1 − e−λτ λτ − e −λτ = 0 The sign of the first order derivative is:

dF L3(τ ) dτ = e−λτ(1 + λτ + λ2τ2) − 1 λτ2 > 0 f or τ < τ ∗ dF L3(τ ) dτ = e−λτ(1 + λτ + λ2_τ2_{) − 1} λτ2 < 0 f or τ > τ ∗

Hence, these imply F L3(τ ) firstly increases from 0, reaching maximum at τ ∗, then decreases to 0 when τ goes to infinity.

According to the trends of F L1(τ ), F L2(τ ), F L3(τ ), we can derive:

|y(t, ∞) − y(t, 0)| = St

Thereby, St can describe the absolute difference between long term and short term yields. Hence, regarding St as the slop factor is reasonable. Finally, Ct is closely related to the yield curvature, if Ct > 0, yield curves are peak-shape, if Ct< 0, yield curves are U-shape.

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(a) Nelson-Siegel Model (b) Nelson-Siegel-Svensson Model (c) Generalized Nelson-Siegel Model

Figure 3.1: Factor Loadings for Siegel, Siegel-Svensson, Generalized Nelson-Siegel

Panel(a) shows the factor loadings for Lt, St, and Ct with λ = 0.5978. Panel (b) shows the factor loadings

for L_t, S_t, C_t1, and C_t2 with λ₁ = 1.0549 λ2 = 0.2329. Panel (c) shows the factor loadings for Lt, St1, St2, Ct1,

and C_t2 with λ₁ = 1.0549 λ2= 0.2329.

information. Since y(t, ∞) = Lt, the level factor Lt also can be used to represent the long-term factor. Since the rapidly decreasing of F L2(τ ) with time to maturity, the slope factor St also can be interpreted as the short-term factor. At the same time, F L3(τ )’s variation is evident for neither short-term nor long-term yields. The change of F L3(τ ) is most prominent for medium-term yields. Hence, the curvature factor Ct also can be interpreted as the medium-term factor. The economic implications of these factors can be confirmed by Figure 3.1(a).

3.1.1 Independent-factor DNS model

The basic dynamic Nelson-Siegel model presented by Diebold and Li (2006) is quite limited for further investigation for yield curves. Diebold, Rudebusch, and Aruoba (2006) introduced the factor model representation for the DNS model. The factor model representation always consists of state transition equation and measurement equation. If factors Lt, St, Ct follow independent autoregressive process with first order. Assume Ft = (Lt, St, Ct)0, µ = (µL, µS, µC)0, then the state transition equation can be written as:

Ft− µ = K(Ft−1− µ) + ηt (3.3)

where K is a 3 × 3 diagonal matrix, that is: K =      κ11 0 0 0 κ22 0 0 0 κ33      .

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The i.i.d white noise disturbance ηt has a covariance matrix which is given by:

Q =      q₁₁2 0 0 0 q2 22 0 0 0 q2 33     

Assume y(t, τ ) = (y(t, τ1), · · · , y(t, τN))0, the measurement equation consists of the yields for N different bonds with different maturities:

y(t, τ ) = HFt+ t (3.4)

Where H is a N × 3 matrix, that is:

H =         1 1−e_λτ−λτ1 1 1−e−λτ1 λτ1 − e −λτ1 1 1−e_λτ−λτ2 2 1−e−λτ1 λτ2 − e −λτ2 .. . ... ... 1 1−e_λτ−λτN N 1−e−λτN λτN − e λτN        

The i.i.d. white noise disturbance t has a covariance matrix which is a 3 × 3 diagonal matrix with non-negative diagonal elements, that is:

V =      ν₁₁2 0 0 0 ν2 22 0 0 0 ν₃₃2     

For further investigation based on Kalman Filter, the white noise disturbances of state transition equation (3.3) and measurement equation (3.4) are assumed to be orthogonal to each other, that is:

  ηt t  ∼ W N     0 0  ,   Q 0 0 V     (3.5) 3.1.2 Correlated-factor DNS model

The correlated-factor DNS model assumes factors Lt, St, Ctfollow a autoregressive process with first order which allows the factors to be correlated. Christensen, Diebold, and Rudebusch (2007) extended the independent-factor DNS model to more general case. The state transition equation is

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still (3.3), but with matrix K being a unrestricted matrix, rather than a diagonal matrix.

In the correlated-factor DNS model, the disturbances ηt(L), ηt(S), ηt(C) are also allowed to be correlated. Hence, the covariance matrix of the disturbances is given by:

Q = qq0 =      q11 0 0 q21 q22 0 q31 q32 q33           q11 q21 q31 0 q22 q32 0 0 q33      =      q2 11 q11q21 q11q31 q21q11 q221+ q222 q21q31+ q22q32 q31q11 q31q21+ q32q22 q312 + q232+ q233     

The measurement equation for the correlated-factor DNS model is the same as the independent-factor DNS model.

3.2 Dynamic Nelson-Siegel-Svensson Model

The limit of Nelson-Siegel model is insufficient in-sample fit for short-term and medium-term yields. At the same time, it is difficult for the Nelson-Siegel model to describe complex forms of yields. To improve the in-sample fit and enhance the availability of the Nelson-Siegel model, Svensson (1995) included a fourth term, the second curvature term β4.

y(t, τ ) = β1+ β2 1 − e−λ1τ λ1τ + β3 1 − e−λ1τ λ1τ − e−λ1τ + β4 1 − e−λ2τ λ2τ − e−λ2τ

Similarly, employing the DL model, the dynamic Nelson-Siegel-Svensson model can be obtained by replacing βs by dynamic factors Lt, St, Ct1, Ct2. The DNSS model is given by:

y(t, τ ) = Lt+ St 1 − e−λ1τ λ1τ + C_t1 1 − e −λ1τ λ1τ − e−λ1τ + C_t2 1 − e −λ2τ λ2τ − e−λ2τ (3.6)

The factor loadings of the DNSS model are presented in Figure 3.1(b).

3.2.1 Independent-factor DNSS model

The independent-factor DNSS model has the same structure as the independent-factor DNS model, but with the second curvature term. So assume Ft = (Lt, St, Ct1, Ct2)

0

, µ = (µL, µS, µ1C, µ2C) 0

. For the independent-factor DNSS model, the state transition equation is still (3.3), but with K being

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a 4 × 4 matrix, that is: K =        κ11 0 0 0 0 κ22 0 0 0 0 κ33 0 0 0 0 κ44        .

The i.i.d. white noise disturbance ηthas a covariance matrix which is given by: Q = diag(q112 , q222 , q332 , q442 ). The measurement equation is (3.4) with a N × 4 matrix H.

H =         1 1−e_λ−λ1τ1 1τ1 1−e−λ1τ1 λ1τ1 − e −λ1τ1 1−e−λ2τ1 λ2τ1 − e −λ2τ1 1 1−e_λ−λ1τ2 1τ2 1−e−λ1τ2 λ1τ2 − e −λ1τ2 1−e−λ2τ2 λ2τ2 − e −λ2τ2 .. . ... ... ... 1 1−e_λτ−λτN N 1−e−λ1τN λ1τN − e λ1τN 1−e−λ2τN λ2τN − e λ2τN        

where the disturbance t for measurement equation is i.i.d. white noise with a covariance matrix V = diag(ν2

11, ν222 , ν332 , ν442 ).

3.2.2 Correlated-factor DNSS model

The correlated-factor DNSS model is similar to the previous models. The state transition equa-tion is (3.3) with K being a 4 × 4 unrestricted matrix. The matrix K is given by:

K =        κ11 κ12 κ13 κ14 κ21 κ22 κ23 κ24 κ31 κ32 κ33 κ34 κ41 κ42 κ43 κ44       

The covariance matrix for disturbances ηt(L), ηt(S), ηt(C1), ηt(C2) is Q = qq0 where

q =        q11 0 0 0 q21 q22 0 0 q31 q32 q33 0 q41 q42 q43 q44       

The measurement equation for correlated-factor DNSS is the same as the independent-factor DNSS model.

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3.3 Dynamic Generalized Nelson-Siegel Model

The five-factor model is firstly proposed by Christensen, Diebold, and Rudebusch (2011) in order to solve the arbitrage-free consistency problem. By introducing the second slope, and the second curvature factor, this model can boost the flexibility obviously and can achieve arbitrage-free consistency. We name this five-factor model as the Dynamic generalized Nelson-Siegel model, briefly, the DGNS model. The yield to maturity can be rewritten using more factors:

y(t, τ ) = Lt+ St1 1 − e−λ1τ λ1τ + S_t2 1 − e −λ2τ λ2τ + C_t1 1 − e −λ1τ λ1τ − e−λ1τ +C_t2 1 − e −λ2τ λ2τ − e−λ2τ (3.7)

The factor loadings of the DGNS model are shown in Figure 3.1(c).

3.3.1 Independent-factor DGNS model

Assume Ft = (Lt, St1, St2, Ct1, Ct2)0, µ = (µL, µ1S, µ2S, µ1C, µ2C)0. Similarly, for the independent-factor DGNS model, the state transition equation is (3.3) with a 5 × 5 matrix K. The matrix K is given by: K =           κ11 0 0 0 0 0 κ22 0 0 0 0 0 κ33 0 0 0 0 0 κ44 0 0 0 0 0 κ55          

The disturbance ηt follows an i.i.d. white noise process with a covariance matrix Q = diag(q2

11, q222 , q233, q244, q255). The measurement equation for the DGNS model is (3.4) with a N × 4 matrix H. H =         1 1−e_λ−λ1τ1 1τ1 1−e−λ2τ1 λ2τ1 1−e−λ1τ1 λ1τ1 − e −λ1τ1 1−e−λ2τ1 λ2τ1 − e −λ2τ1 1 1−e_λ−λ1τ2 1τ2 1−e−λ2τ2 λ2τ2 1−e−λ1τ2 λ1τ2 − e −λ1τ2 1−e−λ2τ2 λ2τ2 − e −λ2τ2 .. . ... ... ... ... 1 1−e_λ−λ1τN 1τN 1−e−λ2τN λ2τN 1−e−λ1τN λ1τN − e −λ1τN 1−e−λ2τN λ2τN − e −λ2τN        

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3.3.2 Correlated-factor DGNS model

The state transition equation for the correlated-factor DGNS model is (3.3) with K being a 5 × 5 unrestricted matrix rather than a diagonal matrix, that is:

K =           κ11 κ12 κ13 κ14 κ15 κ21 κ22 κ23 κ24 κ25 κ31 κ32 κ33 κ34 κ35 κ41 κ42 κ43 κ44 κ45 κ51 κ52 κ53 κ54 κ55           .

The covariance matrix for disturbances is given by: Q = qq0 where

q =           q11 0 0 0 0 q21 q22 0 0 0 q31 q32 q33 0 0 q41 q42 q43 q44 0 q51 q52 q53 q54 q55          

The measurement equation for the correlated-factor DGNS model is the same as the independent-factor DGNS model.

3.4 Arbitrage-Free Dynamic Nelson-Siegel Model

For a three-factor arbitrage-free affine model with a random variable Xt = (Xt1, Xt2, Xt3), Since changes of long-term and short-term interest rates can affect bond prices in different ways, the risk-free rate is defined as: r(Xt) = Xt1 + Xt2, the system of stochastic differential equation under Q measure is given by:

     dX_t1 dX2 t dX_t3      =      0 0 0 0 λ −λ 0 0 λ                θQ₁ θQ₂ θQ₃      −      X_t1 X2 t X_t3           dt + Σ      dW_tQ,1 dW_tQ,2 dW_tQ,3      , λ > 0

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For the three-factor model, (2.10) can be rewritten as:

P (Xt, τ ) = exp(A(τ ) + B1(τ )Xt1+ B 2

(τ )X_t2+ B3(τ )X_t3) (3.8)

B1(τ ), B2(τ ), and B3(τ ) in (3.8) are the solutions of the following differential function,

dB(τ ) dτ =      1 1 0      +      0 0 0 0 λ 0 0 −λ λ      B(τ ) (3.9) and dA(τ ) dτ = −B(τ ) 0 KQθQ− 1 2 3 X j=1 (Σ0B(τ )B(τ )0Σ)jj (3.10)

The solutions for B1_{(τ ), B}2_{(τ ), and B}3_{(τ ) are:}

B1(τ ) = −τ B2(τ ) = − 1 − e −λτ λτ B3(τ ) = − 1 − e −λτ λτ − e −λτ

and according to Singleton (2006), we can fix the mean of factors θ_iQ = 0, i = 1, 2, 3 for simplicity, the solution for A(τ ) takes form of:

A(τ ) = 1 2 3 X j=1 Z t+τ t (Σ0B(s, τ )B(s, τ )0Σ)jjds

Hence, the yield to maturity under the arbitrage-free affine model has the following expression:

y(t, τ ) = −A(τ ) τ + X 1 t + X 2 t 1 − e−λτ λτ + X_t3 1 − e −λτ λτ − e −λτ (3.11)

Comparing (3.2) and (3.11), we find the arbitrage-free affine model contains additional term −A(τ )_τ which is called yield-adjustment term. Hence, for arbitrage-free affine model, X_t1, X_t2, X_t3 are re-garded as the level factor, the slope factor, the curvature factor, respectively. The exact expression

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of yield adjustment term is presented in Appendix.

Up to now, all valuations are based on risk neutral dynamics (Q measure). In the real world, the arbitrage opportunities do not exist, and all dynamics is measured by the physical probability (P measure). Recall the Cameron-Martin-Girsanov theorem, we have:

dW_tQ = dW_tP + Φtdt (3.12)

To transfer the model under Q measure to the model under P measure, Duffee (2002) defined: Φt= φ0+ φ1Xt

where φ0 is (3 × 1) matrix, and φ1 is (3 × 3) matrix. Hence, under physical probability P, the stochastic differential function is given by:

dXt= KP(t)(θP(t) − Xt)dt + ΣdWtP (3.13)

where dW_tP = dW_tQ− Φtdt

Then, I will present the independent-factor AFDNS model and correlated-factor AFDNS model. Assume dXt = (dXt1, dXt2, dXt3)

0

, θP = (θP

1, θ2P, θP3) 0

. Since KP(t) and θP_{(t) are time constant} matrix, (3.13) can be rewritten as the state transition equation for the independent-factor AFDNS model, that is:

dXt= KP(θP − Xt)dt + ΣdWtP (3.14)

where KP _{is a 3 × 3 diagonal matrix and Σ is a 3 × 3 diagonal matrix. The matrix K}P _{and Σ are:}

KP =      κP 11 0 0 0 κP₂₂ 0 0 0 κP 33      Σ =      σ11 0 0 0 σ22 0 0 0 σ33     

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The correlated-factor AFDNS model allows the correlations between X_t1, X_t2, and X_t3, the transition equation is the same as the independent-factor AFDNS model but with different KP _{and Σ. The} matrix KP _{and Σ are:}

KP =      κP 11 κP12 κP13 κP₂₁ κP₂₂ κP₂₃ κP 31 κP32 κP33      Σ =      σ11 0 0 σ21 σ22 0 σ31 σ32 σ33     

According to (3.11), the measurement equation for both independent-factor and correlated-factor model takes form of:

y(t, τ ) = −A(τ )

τ + HFt+ t (3.15)

where t for measurement equation is i.i.d. white noise.

The matrix H in the above equation is the same as the matrix H of the DNS model.

3.5 Arbitrage-Free Dynamic Generalized Nelson-Siegel Model

Christensen et al. (2011) argued it is impossible to generate factor loading structure based on affine arbitrage-free model for only one slope factor, two curvature factors. Hence, the arbitrage-free dynamic Nelson-Siegel-Svensson model does not exist.

For five-factor model, we define the random variable: Xt= (Xt1, Xt2, Xt3, Xt4, Xt5), which is similar to the three-factor AFDNS model, assume the risk-free rate is:

r(Xt) = Xt1+ X 2 t + X

3 t

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The system of stochastic differential equation under Q measure is given by:           dX1 t dX_t2 dX3 t dX4 t dX_t5           =           0 0 0 0 0 0 λ1 0 −λ1 0 0 0 λ2 0 −λ2 0 0 0 λ1 0 0 0 0 0 λ2                               θ₁Q θ₂Q θ₃Q θ₄Q θ₅Q           −           X1 t X_t2 X3 t X4 t X_t5                     dt + Σ           dW_tQ,1 dW_tQ,2 dW_tQ,3 dW_tQ,4 dW_tQ,5           , λ1 > λ2 > 0

For five-factor model, (2.10) can be rewritten as:

P (Xt, τ ) = exp(AG(τ ) + B1(τ )Xt1+ B 2

(τ )X_t2+ B3(τ )X_t3+ B4(τ )X_t4+ B5(τ )X_t5) (3.16)

B1(τ ), B2(τ ), B3(τ ), B4(τ ), and B5(τ ) in (3.16) are the solutions of:

dB(τ ) dτ =           1 1 1 0 0           +           0 0 0 0 0 0 λ1 0 0 0 0 0 λ2 0 0 0 −λ1 0 λ1 0 0 0 −λ2 0 λ2           B(τ ) (3.17) and dAG(τ ) dτ = −B(τ ) 0 KQθQ− 1 2 5 X j=1 (Σ0B(τ )B(τ )0Σ)jj (3.18)

The solutions of (3.17) and (3.18) are:

B1(τ ) = −τ B2(τ ) = − 1 − e −λ1τ λ1τ B3(τ ) = − 1 − e −λ2τ λ2τ B4(τ ) = − 1 − e −λ1τ λ1τ − e−λ1τ B5(τ ) = − 1 − e −λ2τ λ2τ − e−λ2τ

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For simplicity, the solution for AG(τ ) is given by:

AG(τ ) = 1 2 5 X j=1 Z t+τ t (Σ0B(s, τ )B(s, τ )0Σ)jjds

Hence, the yield to maturity under the arbitrage-free affine model takes form of:

y(t, τ ) = −A G_{(τ )} τ + X 1 t+X 2 t 1 − e−λ1τ λ1τ + X_t3 1 − e −λ2τ λ2τ + X_t4 1 − e −λ1τ λ1τ − e−λ1τ + X_t5 1 − e −λ2τ λ2τ − e−λ2τ (3.19)

The expression for −AG_τ(τ ) is similar to the expression for −A(τ )_τ . Assume dXt = (dXt1, dXt2, dXt3, dXt4, dXt5)

0

, θP = (θP

1, θP2, θP3, θP4, θP5) 0

. Similarly to the AFDNS model, for the independent-factor AFDGNS model, the transition equation still takes the form of (3.14), but with a 5 × 5 diagonal matrix of KP _{and Σ. The matrix K}P _{and Σ are given by:}

KP =           κP₁₁ 0 0 0 0 0 κP 22 0 0 0 0 0 κP₃₃ 0 0 0 0 0 κP 44 0 0 0 0 0 κP 55           Σ =           σ11 0 0 0 0 0 σ22 0 0 0 0 0 σ33 0 0 0 0 0 σ44 0 0 0 0 0 σ55          

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different matrix KP and Σ. The matrix KP and Σ are given by:

KP =           κP 11 κP12 κP13 κP14 κP15 κP₁₂ κP₂₂ κP₂₃ κP₂₄ κP₂₅ κP 31 κP32 κP33 κP34 κP35 κP 41 κP42 κP43 κP44 κP45 κP₅₁ κP₅₂ κP₅₃ κP₅₄ κP₅₅                     σ11 0 0 0 0 σ21 σ22 0 0 0 σ31 σ32 σ33 0 0 σ41 σ42 σ43 σ44 0 σ51 σ52 σ53 σ54 σ55          

The measurement equation for both independent-factor AFDGNS model and correlated-factor AFDGNS model is similar to the AFDNS model, that is:

y(t, τ ) = −A G_{(τ )}

τ + HFt+ t (3.20)

where t for measurement equation is i.i.d. white noise.

The matrix H in the above equation is the same as the matrix H of the DGNS model.

3.6 Empirical Dynamic Term Structure Models

3.6.1 CR Model

Christensen and Rudebusch (2012) determined the best fitting specification of transition equation for the AFDNS model is given by:

     dX1 t dX_t2 dX3 t      =      10−7 0 0 κP₂₁ κP₂₂ κP₂₃ 0 0 κP 33                0 θ₂P θP 3      −      X1 t X_t2 X3 t           dt +      σ11 0 0 0 σ22 0 0 0 σ33           dW_tP,1 dW_tP,2 dW_tP,3      (3.21)

This model is referred as the CR model. The measurement equation for the CR model is still equation (3.15).

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3.6.2 Shadow-Rate B-CR Model

With the development of Gaussian affine models, Christensen and Rudebusch (2013) added the the conception of shadow-rate model into the AFDNS model to create the shadow-rate AFDNS model. An important assumption proposed by Krippner (2013) states that the zero lower bound risk free rate r(Xt) can be written as:

r(Xt) = max{0, Xt1+ X 2 t}

The yield-to-maturity on shadow-rate zero coupon bonds has the same structure as the AFDNS model, according to Equation (2.1) and (2.2), the shadow-rate zero-coupon bond instantaneous forward rate can be calculated as:

fS(t, τ ) = ∂ (τ y(t, τ )) ∂τ This implies:

fS(t, τ ) = X_t1+ X_t2e−λτ + X_t3λτ e−λτ + Af(τ ) (3.22)

Where Af_{(τ ) is given in Appendix. The yield-to-maturity is derived from the zero lower bound} zero-coupon bond instantaneous forward rate which takes form of:

y(t, τ ) = 1 τ Z t+τ t f (t, s)ds (3.23) Where f (t, s) = fS(t, s)NfS_ω(s)(t,s)+ ω(s)√1 2πexp −1 2 _fS_(t,s) ω(t,s) 2

, and for the B-AFDNS model, ω(s) takes the form:

ω(s)2 = σ₁₁2 s + (σ2₂₁+ σ2₂₂)1 − e −2λs 2λ + (σ₃₁2 + σ₃₂2 + σ₃₃2 ) 1 − e −2λs 4λ − 1 2se −2λs₋ 1 2λs 2_e−2λs + 2σ11σ21 1 − e−λs λ + 2σ11σ31 −se−lambdas+1 − e −λs λ + (σ21σ31+ σ22σ32) −se−2λs+1 − e −2λs 2λ (3.24)

Hence, (3.23) is the measurement equation for the shadow-rate B-CR model with the diagonal matrix Σ, and the transition equation for the shadow-rate B-CR model is the same as the CR model which

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Chapter 4

Data Description and Estimation Methods

The data used in this paper are historical weekly U.S. Treasury Constant Maturity rates with 10 different maturities, from August 3rd, 2001 to November 27th, 2015 provided by Federal Reserve Bank. The data includes 1-month, 3-month, 6-month, 1-year, 2-year, 3-year, 5-year, 7-year, 10-year, and 20-year of treasury yields. During the 2008 financial crisis, on December 16th, 2008, the Federal Open Market Committee (FOMC) decided to target range for federal funds rate of 0 to 0.25 percent. Hence, we define the time period before December 19th, 2008 as the normal state (since weekly data used), it includes 385 observations, and from this time on, define as the zero-bound state, it includes 363 observations.

The yield curves presented in Figure 4.1 shows that the long-term U.S. treasury bonds have relatively higher yields, and the short-term U.S. treasury bonds exhibit relatively lower yields. This pattern is violated before the 2008 financial crisis, more precisely, from the beginning of 2006 to the beginning of 2008. During this special period, the yields of short-term U.S. treasury bonds increased dramatically, while the yield rates of long-term bonds did not present evident changes. From Figure 4.2(a), we can find from the beginning of 2006 to the beginning of 2008, the short-term and long-term bonds had similar yields. Then, the yield rates for all treasury bonds fell violently, especially the short-term treasury bonds. In September 2008, the short-term treasury bonds nearly hit the zero-bound. After that, since the policy of conducted by FOMC, the yield rates for treasury bonds maintained relatively steady and low until November, 2015 (zero-bound state).

Descriptive statistics are presented in Table 4.1. The means of treasury bond yields with different maturities coincide with pattern mentioned above. The standard deviations for the normal state and the zero-bound state exhibit a totally different trend. For the normal state, the standard deviations

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CHAPTER 4. DATA DESCRIPTION AND ESTIMATION METHODS 32

(a) Normal State (b) Zero-bound State

Figure 4.1: U.S. Treasury Bond Yields

Panel(a) shows the time-series plots of the U.S. Treasury Bond Yields with maturities 1-month, 3-month, 6-month, 12-6-month, 24-6-month, 36-6-month, 60-6-month, 84-6-month, 120-6-month, and 240-month during the normal state. Panel(b) shows the time-series plots of the U.S. Treasury Bond Yields with maturities 1-month, 3-month, 6-3-month, 12-year, 24-3-month, 36-3-month, 60-3-month, 84-3-month, 120-3-month, and 240-month during the zero-bound state.

for the treasury bond yields decrease with the time-to-maturity increases, while for the zero-bound state, the standard deviations for the treasury bond yields increase with time-to-maturity increases.

4.1 Two-Step Estimation for Diebold-Li Models

Naturely, the three-factor model (3.2) can be estimated by nonlinear least squares regression. For simplicity and numerical convenience, Diebold and Li (2006) have introduced a new approach which only requires the ordinary least squares. This approach is called the Two-Step method. According to this paper, we need to first pre-specified λ, then we can calculate F L2(τ ) and F L3(τ ) based on the λ. After that, the ordinary least squares can be used to estimate the factors Lt, St, and Ct. In the second step, the factors’ dynamics can be estimated using the V AR(1) specification, that is:

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CHAPTER 4. DATA DESCRIPTION AND ESTIMATION METHODS 33 Maturity in Months Normal State

Mean(×100) Std Skewness Kurtosis 1 2.4951 1.5040 0.4920 1.8670 3 2.5779 1.5130 0.4681 1.7757 6 2.7362 1.5118 0.4393 1.6896 12 2.8643 1.3936 0.3800 1.6977 24 3.1165 1.1930 0.1566 1.7193 36 3.3490 1.0367 0.0143 1.7973 60 3.7884 0.7669 -0.1788 2.0957 84 4.0870 0.6194 -0.2732 2.5246 120 4.3671 0.4767 -0.2248 2.9677 240 4.9532 0.4115 0.1109 3.4519 Maturity in Months Zero-bound State

Mean(×100) Std Skewness Kurtosis 1 0.0602 0.0499 1.0188 3.3051 3 0.0797 0.0585 0.9561 3.6966 6 0.1427 0.0879 1.3632 4.9977 12 0.2420 0.1342 1.1055 3.6267 24 0.5450 0.2631 0.6096 2.3329 36 0.8714 0.3933 0.2979 2.1666 60 1.5325 0.5507 0.1454 2.2787 84 2.0804 0.6298 0.2238 2.2507 120 2.5852 0.6331 0.3250 2.0268 240 3.2971 0.7111 0.2800 1.7700 Table 4.1: Descriptive Statistics for U.S. Treasury Bond Yields

The top panel shows the descriptive statistics for the normal state, and the bottom panel shows the descriptive statistics for the zero-bound state.

In this equation, the ˆA is the estimated coefficient matrix for the V AR(1) model, the vector ˆc is the constant for the V AR(1) model.

The most significant problem for this method is to choose appropriate value of λ.

The U.S. Treasury bonds can be classified into three types. Bonds with maturity less than one year are called short-term bonds, for bonds with maturity longer than ten years are named as long-term bonds. Other bonds are medium-long-term bonds. Recall the economic information about factors mentioned in Section 3.1. The appropriate λ can maximize the factor loading on medium-term factor or curvature factor, that is:

max λ 1 − e−λτ λτ − e −λτ

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the longest maturity in my data is 240 months instead of 120 months in Diebold and Li (2006), I choose 36 months in the maximization problem. This results in λ = 0.5978.

For the four-factor model (3.7), we have two values λ1 and λ2 which need to be pre-specified. since the second curvature factor is introduced, additional restriction is imposed on λ1, and λ2, that is, λ2 can maximize the factor loading on the second curvature factor for a maturity which should be as least 1 year shorter than the first curvature term. Hence, I finally choose 39 months and 18 months. This leads to λ1 = 1.0549, and λ2 = 0.2329. For the five-factor model (3.11), the same λ1 and λ2 can be used.

After obtaining λ = 0.5978, λ1 = 1.0549, and λ2 = 0.2329, the ordinary least square estimation can be employed to estimate {Lt, St, Ct} for the three-factor model, {Lt, St, Ct1, Ct2} for the four-factor model, {Lt, St1, St2, Ct1, Ct2} for the five-factor model.

4.2 Kalman Filter MLE for Diebold-Li Models

Diebold et al. (2006) used the Kalman filter for maximum likelihood estimation in order to estimate yield curves that incorporate macroeconomic factors. Kalman filter is an efficient way to estimate the DNS, DNSS, and DGNS models. For the AFDNS and AFDGNS model, Christensen et al. (2009) showed the Kalman filter is also a feasible approach. In this section, I will firstly introduce the structure of the Kalman filter, then present the specific algorithm for different models.

The framework of the Kalman filter requires a state-space representation which includes a state transition equation, and a measurement equation. Define Xt ∈ Rnbe the unobservable state variable at time t, assume Xt follows the following process:

Xt= T Xt−1+ CIt−1+ ηt, ηt∼ N (0, Q) (4.1)

Equation (4.1) is referred as the state transition equation, where T is called state transition matrix which can describe the dynamics of the state variable Xt. C is the control matrix which can control the input It∈ Rn, and ηtis the process noise which has a constant covariance matrix Q. The relation between the unobservable state variable Xt and the observed state variable yt is described by the

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measurement equation. The measurement equation takes form of:

yt = HXt+ t t∼ N (0, V ) (4.2)

where H maps the unobservable state Xt to observable vector yt, and t is the measurement noise which follows a normal distribution with a constant covariance matrix V . The random variable ηt and t are assumed to be independent which leads to the joint distribution as follow:

  ηt t  ∼ N     0 0  ,   Q 0 0 V    

Consider the prediction of the unobservable state variable Xtwhich base on the information up to time t − 1, let Gt−1 be the observable information at time t − 1. More precisely, Gt−1 = {y1, y2, · · · , yt−1}. Hence, the prediction of Xt based on Gt−1 is given by:

ˆ

Xt|t−1 = E[Xt|Gt−1]

= E[T Xt−1+ CIt−1+ ηt|Gt−1]

= T E[Xt−1|Gt−1] + CIt−1+ E[ηt|Gt−1]

= T ˆXt−1|t−1+ CIt−1 (4.3)

Pt|t−1 = E[(Xt− ˆXt|t−1)(Xt− ˆXt|t−1)0|Gt−1]

= T E[(Xt−1− ˆXt−1|t−1)(Xt−1− ˆXt−1|t−1)0|Gt−1]T0 + E[ηtη0t]

= T Pt−1|t−1T0+ Q (4.4)

Since ˆXt|t−1 and Pt|t−1 only base on the information up to time t − 1, we call these as the priori (predicted) estimates. Then, we can use new information yt to make the update.

Let ˆtbe the measurement residual, and assume that the measurement residual has the covariance matrix St. According to equation (4,2) and (4.3), the measurement residual is given by:

ˆ

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Equation (4.2) implies:

yt ∼ N (HXt, V ) This leads to:

St = V[ˆt] = V[yt− H ˆXt|t−1] = V[yt] + HV[ ˆXt|t−1]H0

= V + HPt|t−1H0 (4.6)

The optimal Kalman gain can be obtained by minimizing E Xt− ˆXt|t 2

. Hence, the optimal Kalman gain is:

Kt= Pt|t−1H0St−1 (4.7)

Based on the optimal Kalman gain and the priori estimates, we can obtain the posteriori (updated) estimates. The posteriori estimate of the unobservable state variable Xt is given by:

ˆ

Xt|t= ˆXt|t−1+ Ktˆt (4.8)

And the posteriori estimate of its covariance matrix is given by:

Pt|t= (I − KtH)Pt|t−1 (4.9)

According to all Kalman filter estimation components mentioned above, the Maximum likelihood estimation can be applied to estimate the unknown parameters. The Kalman filter iteration starts with initial state variable X0, and its initial covariance matrix P0, then we can obtain the optimal state variable, covariance matrix and parameters recursively. Let ψ = (H, T, I, Q, V ) be the set of unknown parameters. The joint probability density is given by:

f (ψ; y1, y2, · · · , yT) = T Y t=1 1 (2π)N/2_|S t|−1/2 e−12(ˆ 0 tS −1 t ˆt) (4.10)

Then the log-likelihood function is:

`(ψ; y1, y2, · · · , yT) = T X t=1 −N 2log(2π) − 1 2log|St| − 1 2ˆ 0 tS −1 t ˆt (4.11)

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where T is the total number of observations in this paper. The maximum-likelihood estimation problem is:

max

ψ `(ψ; y1, y2, · · · , yT) This is the same as:

−min ψ ( N T 2 log(2π) + T X t=1 1 2log|St| + 1 2ˆ 0 tS −1 t ˆt ) (4.12)

This minimization problem can be solved by Nelder-Mead algorithm which is implemented by Date and Wang (2009).

Now, we can apply the above Kalman filter framework for the DNS, DNSS, DGNS, AFDNS, AFDGNS, CR model, and shadow-rate B-CR model.

4.2.1 Kalman Filter for Dynamic Nelson-Siegel Model

For the Dynamic Nelson-Siegel model, the factors Lt, St, and Ct are treated as the unobservable state variables Xt under Kalman filter framework, i.e. XtDN S = (Lt, St, Ct)0. Equation (3.3) can be rewritten as:

Ft= KFt−1+ (I3− K)µ + ηt (4.13)

According to the structure of (4.3), assume X_tDN S = Ft, for the independent-factor DNS model, the state transition equation takes form of:

X_tDN S = TIDN SX_t−1DN S+ (I3− TDN S)IDN S+ ηt ηt∼ N (0, QIDN S) (4.14)

where TIDN S _{is a 3 × 3 diagonal matrix, the input vector is I}DN S _{= µ, and the covariance matrix} QIDN S _{= qq}0 _{with q being a 3 × 3 diagonal matrix. The measurement equation is:}

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CHAPTER 4. DATA DESCRIPTION AND ESTIMATION METHODS 38 Where HDN S =         1 1−e−λτ1 λτ1 1−e−λτ1 λτ1 − e −λτ1 1 1−e_λτ−λτ2 2 1−e−λτ1 λτ2 − e −λτ2 .. . ... ... 1 1−e_λτ−λτN N 1−e−λτN λτN − e λτN        

Similarly, for the correlated-factor DNS model, the state transition equation is given by:

X_tDN S = TCDN SX_t−1DN S+ (I3− TDN S)IDN S+ ηt ηt∼ N (0, QCDN S) (4.16)

where TCDN S _{is an unrestricted matrix, and Q}CDN S _{= qq}0_{, with q =}      q11 0 0 q21 q22 0 q31 q32 q33      Other

components are the same as the independent-factor DNS model.

4.2.2 Kalman Filter for Dynamic Nelson-Siegel-Svensson Model

Similar to the dynamic Nelson-Siegel model, the dynamic Nelson-Siegel-Svensson model adds the second curvature factor. Hence, the unobservable state variable is given by: XDN SS

t = (Lt, St, Ct1, Ct2) 0_. According to the structure of (4.3), assume XDN SS

t = Ft, for the independent-factor DNSS model, the state transition equation is given by:

X_tDN SS = TIDN SSX_t−1DN SS+ (I4− TDN SS)IDN SS + ηt ηt∼ N (0, QIDN SS) (4.17)

where TIDN SS is a 4 × 4 diagonal matrix, the input matrix is IDN SS = µ, and the covariance matrix QIDN SS _{= qq}0 _{with q being a 4 × 4 diagonal matrix. The measurement equation for the} independent-factor DNSS model is given by:

yt= HDN SSXtDN SS+ t, t ∼ N (0, VDN SS) (4.18) Where HDN SS =         1 1−e_λ−λ1τ1 1τ1 1−e−λ1τ1 λ1τ1 − e −λ1τ1 1−e−λ2τ1 λ2τ1 − e −λ2τ1 1 1−e_λ−λ1τ2 1τ2 1−e−λ1τ2 λ1τ2 − e −λ1τ2 1−e−λ2τ2 λ2τ2 − e −λ2τ2 .. . ... ... ... 1 1−e_λ−λ1τN 1τN 1−e−λ1τN λ1τN − e λ1τN 1−e−λ2τN λ2τN − e λ2τN        

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The correlated-factor DNSS model is easy to obtain by changing the state transition equation (4.17):

X_tDN SS = TCDN SSX_t−1DN SS+ (I4− TDN SS)IDN SS + ηt ηt∼ N (0, QCDN SS) (4.19)

where TCDN SS is a 4 × 4 unrestricted matrix, and QCDN SS = qq0, with q =        q11 0 0 0 q21 q22 0 0 q31 q32 q33 0 q41 q42 q43 q44        .

4.2.3 Kalman Filter for Dynamic Generalized-Nelson-Siegel Model

The dynamic generalized-Nelson-Siegel model can be obtained by adding the second slope and the second curvature factors. So, the state variable is XDGN S

t = (Lt, St1, St2, Ct1, Ct2)

0_{. According to} the structure of (4.3), assume X_tDGN S = Ft, the state transition equation for the independent-factor DGNS model takes form of:

X_tDGN S = TIDGN SX_t−1DGN S+ (I5− TDGN S)IDGN S + ηt ηt∼ N (0, QIDGN S) (4.20)

where TIDN SS _{is a 5 × 5 diagonal matrix, the input matrix is I}DN SS _{= µ, and the covariance matrix} QIDN SS _{= qq}0

with q being a 5×5 diagonal matrix. The measurement equation for independent-factor DGNS model is given by:

yt= HDGN SXtDGN S+ t, t∼ N (0, VDGN S) (4.21) Where HDGN S =         1 1−e_λ−λ1τ1 1τ1 1−e−λ2τ1 λ2τ1 1−e−λ1τ1 λ1τ1 − e −λ1τ1 1−e−λ2τ1 λ2τ1 − e −λ2τ1 1 1−e_λ−λ1τ2 1τ2 1−e−λ2τ2 λ2τ2 1−e−λ1τ2 λ1τ2 − e −λ1τ2 1−e−λ2τ2 λ2τ2 − e −λ2τ2 .. . ... ... ... 1 1−e_λ−λ1τN 1τN 1−e−λ2τN λ2τN 1−e−λ1τN λ1τN − e λ1τN 1−e−λ2τN λ2τN − e λ2τN        

For the correlated-factor DGNS model, the state transition equation is:

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where TCDGN S is a 5 × 5 unrestricted matrix,

and QCDGN S = qq0, with q =           q11 0 0 0 0 q21 q22 0 0 0 q31 q32 q33 0 0 q41 q42 q43 q44 0 q51 q52 q53 q54 q55           .

4.3 Kalman Filter MLE for Gaussian Affine Models

The state transition equation for the Gaussian affine models can be written as:

Xt = (I − exp(−KP∆t))θP + exp(−KP∆t)Xt−1+ ηt (4.23)

The measurement equation for the Gaussian affine models takes form of:

yt= A + HXt+ t (4.24)

Assume the error terms follow:   ηt t  ∼ N     0 0  ,   Q 0 0 V    

Where V is a diagonal matrix which is defined in section 3, and unlike the non-Gaussian models, Q can be calculated by:

Qt= Z ∆t

0

e−KPsΣΣ0e−(KP)0sds (4.25)

The prediction of Xt based on Gt−1 is given by: ˆ

Xt|t−1 = Φ0+ Φ1Xˆt−1|t−1 (4.26)

For the Gaussian affine models, the covariance matrix is given by Σ, the prediction of Σt based on Gt−1 is:

Comparison of yield curves' models based on in-sample fit

Master’s Thesis