Empirical estimation of the Ultimate Forward Rate using the Kalman Filter and Smoother

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Forward Rate

Using the Kalman Filter and Smoother

Ryan Tjin

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Ryan Tjin Student nr: 5967562

Email: jrtjin@gmail.com Date: July 31, 2015

Supervisor: dhr. prof. dr. ir. M.H. (Michel) Vellekoop Second reader: dhr. dr. T.J. (Tim) Boonen

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Abstract

Since the introduction of the Ultimate Forward Rate (UFR) there has been an ongoing debate amongst stakeholders in the pension and finan-cial sector. The discussion mostly revolves around the chosen model and their input parameters. The proposed UFR method is the para-metric Smith-Wilson extrapolation function with a fixed UFR value and Last Liquid Point (LLP). Therefore the question arises if there is an empirical estimation method for the UFR, which is objective, transparent and dynamic?. In this thesis a model is constructed which incorporates all market data and returns parameters from which an UFR value is computed. This is done by applying a Kalman Filter to the Vasicek short rate model. It was clearly shown that the Kalman Filter and Smoother can successfully filter noise from observed yields if the underlying state structure is known. Thus providing a possible so-lution for the LLP debate. However, the model coped with the known limitations of the Vasicek model and therefore needs to be researched further in order to actually serve as an alternative to the proposed solution by EIOPA.

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Thank you Mr. Vellekoop for the necessary structure, knowledgeable guidance and excellent communication during the development of this thesis. The speed and accuracy of your replies, the assessments of my work etc. were really great. I could not have wished for better supervisor, once again thank you!

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Introduction

Since the introduction of the Ultimate Forward Rate (UFR) there has been an ongoing debate amongst stakeholders in the pension and financial sector. Pension funds are obliged to be constantly aware of the value of their future obligations in order to maintain sufficient provisions. The UFR method was created in order to stabilize the yield curve at high maturities by extrapolating the yield curve from a predetermined Last Liquid Point (LLP) to a set (Ultimate) forward rate of 4.2%. The use of the UFR is motivated by the notion that at high maturities, financial products are illiquid resulting in highly volatile long term yields when values are determined in a market consistent manner. Stabilizing the long term yield is desirable, since this directly affects the volatility of the funding ratio which in turn determines strategical decisions for pension funds.

The UFR discussion mostly revolves around the chosen model and its inputs since studies have shown that the input parameters can significantly influence model outcomes (Jiang, 2014). There are three main concerns, the height of the UFR, the determination of the Last Liquid Point and the extrapolation method. Especially the UFR value and LLP are subject to heavy debate, since these are static fixed numbers which are prede-termined by EIOPA. Therefore the main research question of this thesis is: Is there an empirical estimation method for the UFR, which is objective, transparent and dynamic? In this thesis a model is constructed which uses weighted market data from which an UFR value follows, instead of a fixed LLP and a fixed UFR. The model insures that all market information is taken into account whenever there is sufficient data, thus staying as close as possible to market consistent valuation. The height of the UFR follows from the models parameters, which can be updated dynamically whenever new information becomes available. These parameters follow a transparent computation method, there-fore mitigating uncertainties regarding possible changes in UFR policies by EIOPA.

As a starting point, the well known Vasicek short rate model is used. The limitations of the model are well known, but in combination with the Kalman Filter and Smoother it produces new insights for the UFR discussion.

In Chapter 2 the background and drawbacks of the current UFR model are discussed with the basics of term structure modelling in theory and in practice. Chapter 3 de-scribes the Vasicek model, Kalman filter and the mathematical derivation for our model. Chapter 4 discusses model findings and the implications for the current UFR method-ology. Chapter 5 summarizes the most noteworthy findings and features suggestions for further research.

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Chapter 2

Ultimate Forward Rate

This chapter covers interest rate models in general, the motivation for the current Ul-timate Forward Rate, the current computation methodology and the motivation for a more objective model.

2.1 Motivation for the UFR

The value of Dutch Pension Assets exceeds 1000bn euro1. Pension funds have the task to hold and invest these assets in order to meet the retirement entitlements of their members. This section will discuss the relevant mechanics involved in pension asset and liability management and the historical requirements regarding liability valuation. Together they illustrate the motivation for the Ultimate Forward Rate.

2.1.1 Economic context

It is well known that the average life expectancy increases, and politicians are slowly catching up by increasing the retirement age. As a result, people save longer for their retirement, which could also be followed by longer retirements. Therefore the time hori-zons of pension funds liabilities in the Netherlands increase even further. This calls for investments with very long maturities if one were to match the duration of these liabilities.

The ratio between assets and liabilities is called the funding ratio or solvency ratio. Based on the funding ratio, pension rights are indexed for inflation or even decreased when the funding ratio is low. Besides internal decision making, the Dutch government has ordered a minimal funding ratio of 105%. Funds that do not meet this, must submit a recovery plan to the Dutch regulator. Suffice to say the valuation of liabilities plays a crucial role in pension fund decision making.

To complicate things, one cannot decouple liabilities from assets. In an ideal world, the pension fund matches their liabilities exactly with assets that have the same duration and payout. When interest rate fluctuations occur and for instance the interest rates drop, both the value of liabilities and fixed-income securities like government bonds increase. When they are duration matched this doesn’t really matter since expected income and payout stay the same. In reality, however, there is an asset and liability mismatch. The total liabilities of a pension fund exist of more then just the pension rights, therefore they invest in multiple asset classes in order to earn returns on invest-ments that exceed these pension right liabilities. With these extra returns the cost of overhead can be covered and the ambition to raise pension benefits with inflation and wages can be honored. By investing in other non fixed-income products these funds are exposed to interest rate risk.

1

Global Pension Asset Study by Towers Watson values Dutch Pension Assets at 1359bn USD as of January 2014.

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A RBS Insurance and ALM advisory team (2013) say that the current market con-ditions with decreasing interest rates put extra pressure on businesses with this type of asset liability mismatch. This pressure results in an increased demand for hedging which puts ’downward pressure on rates’ and in turn also has a negative effect on funding and solvency ratios.

The introduction of an other valuation mechanism would decouple the valuation of liabilities with these kind of market movements, and perhaps provide for a more stable valuation of the liabilities. On the other hand it will also introduce new risks and thus induce other hedging behaviour, which will be discussed in the following sections.

2.2 Interest rates

In the valuation of pension rights the risk free term structure of interest is used to discount these liabilities. Suffice to say, modelling the term structure is of value to institutions with these kind of obligations. Although no perfect model exist, this section covers the basics of interest rate modelling and explains the choices made for the model of this thesis.

2.2.1 Zero-coupon rate

The risk free interest rate is the return rate that one would get of an investment without financial risk. Risk free investments are usually associated with investing in government bonds. Although in practice there is no such thing as a risk free investment, certain countries are considered highly unlikely to default on their interest payments and there-fore their bonds are considered risk free. Examples of such investments are the U.S. Treasury Bills and German government bonds.

The interest gained per year with an n-year risk free bond without coupon payments is also referred to as the n-year zero-coupon rate or the n-year spot rate. These interest rates have an inverse relation with the discount factors for their respective maturities, which can be used to compute the net present value of future cash flows. zero-coupon rates or yields in this thesis are denoted by yt,T where t represents the time and T the

time to maturity.

2.2.2 The forward rate

Given the zero-coupon rates, one can compute the forward rate. The forward rate can be seen as a spot rate which is defined in the future. Thus instead of defining an expected interest rate for some defined time period from now on, the forward rate defines an expected interest rate for some future time period. For example, this could be the expected interest rate for a 5 year period 3 years from now. The forward rate is depicted as ft,t+n.

However it is more common, to speak about the 1-year forward rate, which is also called the short rate, depicted as st. Thus when one knows the term structure of

in-terest, one can define the forward rates and vice versa. Higher forward rates result in higher expected interest rates. The mathematical relations between the zero-coupon rate, forward rate and short rate are shown in the following formulas:

(1 + y0,t)t= t−1 X n=0 (1 + sn) (1 + ft,t+n)t= t+n−1 X n=t (1 + sn) (1 + y0,t)t(1 + ft,t+n)n= (1 + y0,t+n)t+n

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4 Ryan Tjin — Empirical estimation of the UFR

The Ultimate Forward Rate depicts the ’last’ forward rate, which in practice is the forward rate for some time in the far future.

2.2.3 Term structure models

Models describing the zero-coupon interest rates over time are also referred to as term structure models. Stochastic term structure models can be classified as equilibrium mod-els or no-arbitrage modmod-els. Equilibrium modmod-els use economic theories and assumptions to construct their model and fit model parameters to historical data like prices of swaps and bonds. On the other hand no-arbitrage models construct term structures that repli-cate historical data, but might stray away from economic theories. Thus equilibrium models provide an estimate for today’s term structure as an output while no-arbitrage models use today’s term structure as an input (Hull 1993).

In 1977, Oldrich Vasicek introduced one of the first models that made an explicit characterization of the term structure. The Vasicek model describes the process for the short rate over time and can be classified as an equilibrium model. The first of three assumptions on which it builds is that the short rate follows a continuous Markov process with no jumps. Secondly he assumes that bond prices are solely driven by the short rates over the term of the bond, as a result yields and bond prices for all maturities thus depend on one short rate process. Lastly, the assumption is made that markets are efficient, thus transparent and rational. The first assumption determines the model form, which is described in chapter 3, that short rates are driven by a single stochastic element (one factor) and are pulled back to a certain long-term mean (mean reversion) with a force that is determined by the deviation from the mean. The latter assumption ensures that the short rate does not diffuse to infinite values. Thus, the Vasicek model models the short rate as a function of the long-term expected short rate, mean reversion speed, and a random error term. Since short rates are not directly observed in the market, the observed data is transformed and model parameters can be derived using a maximum likelihood estimation or least squares estimation.

After Vasicek, other one-factor short rate models were introduced. Rendleman and Bartter (1980) constructed a model that had a constant rate of change for the short rate, instead of a mean reverting short rate. Their model’s objective was to model the short rate like stocks. After that, the Cox, Ingersoll and Ross (CIR) model (1985) was introduced. This model, has the same mean reverting property as the Vasicek model, but has a volatility which depends on the short rate itself. This modification results in positive model parameters and avoids negative short rates.

As said before, equilibrium models have a hard time fitting the actual data, therefore they were extended which lead to no-arbitrage models. Examples are the Ho-Lee model (1986) and the Hull-White model (1990). Both have an expected short rate which is variable through time. Ultimately models arose that had a varying volatility parameter (Black, Derman, and Toy model) and eventually Hull White (1990) created a model where all parameters are time varying called the Extended Vasicek model.

Besides stochastic models, there are models which do not necessarily consider un-derlying drivers of the term structure of interest. Instead, they characterize the shape of the yield curve by factors like level, curvature, and slope. These models, referred to as Nelson-Siegel Models (NSM), can be extended to capture macroeconomic factors. The factors are captured using latent variables, an example of this is Diebold, Rudebusch, and Auroba (2006) whom model the yield as a function of annual price inflation, the federal funds rate and others. NSM-type models have one property that is similar to the Smith-Wilson method which is used in the proposed UFR method (which will be discussed later in this chapter), namely both models are static in terms of UFR mod-elling. In other words, NSM-models and the Smith-Wilson method have predetermined fixed shape parameters which are therefore subject to debate. Since this thesis focuses on the forward rate as the sole stochastic driver for the term structure and a dynamic

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Figure 2.1: Long-term interest rate defined by ECB as 10 yr to maturity2 model with no a priori shape parameters is preferred for transparency reasons, NSM type models are not considered in the estimation process. However,

The Vasicek model is still widely used today because it was one of the first interest models and more so because it is easy to implement and estimate. The CIR model is a bit harder to estimate since the short rates are not normally distributed. Since negative interest was a phenomenon which was regarded as impossible, CIR was (and probably still is) considered superior to the Vasicek model. Now (June 2015) that short maturity interest rates are actually below zero, this argument holds less value. Also considering the main objective of this thesis, determining an alternative objective UFR by applying the Kalman Filter, we are more interested in the Filter’s dynamics. Therefore a ’simple’ Vasicek model is implemented, hopefully stimulating further researchers whom also want to explore different interest rate model dynamics in combination with the Kalman Filter.

2.2.4 Dutch Term structure

Until 2007 pension funds were allowed to use a fixed discount rate of 4% to value their future liabilities. Thus, they did not use an actual term structure to discount pension rights. At the time, it was believed that interest rates would stay high and investment returns, on average, far exceeded 4%. When the actual market rates were higher than the discount rate, pension funds would value their liabilities higher than they actually are. Thus one could regard this as a prudent measure. But times have changed and so have interest rates and investment returns. The long-term interest rates are shown in figure 2.1. It is visible that in the late 90’s the long-term interest rate dropped below 4% and has dropped even further since the crisis. Thus, pension funds were systematically undervaluing their liabilities, which called for a new valuation system.

The Financieel Toetsings Kader (FTK), the Dutch regulatory framework for pension funds, was introduced in 2007 and states that liabilities should be discounted using in-terest rates which are consistent with rates that are observed in financial markets. The

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values of discount rates are constructed from the values of swaps, which are financial products that exchange fixed interest rates for ’floating’ interest rates. Swaps are avail-able in a wide range of maturities and swap markets are considered to be very liquid. Since these rates come from financial markets and institutions are willing to trade on these rates, these are considered to be more in line with reality than a fixed discount rate.

The DNB interest rate term structure was constructed using Euribor swap rates, without smoothing the forward rates, with maturities 1 - 10, 12, 15, 20, 25, 30, 40 en 50 years. Forward rates were considered constant between unknown maturities and the 49-year forward rate was used for all forward rates with maturity 50 and higher. This market consistent valuation resulted in a increase of the discount rates which in turn resulted in a decrease in funding ratios. An additional effect was increased volatility in the liabilities with high maturities since they are discounted using a variable interest rate instead of a fixed one. This increase in volatility could be countered by a hedge using the same swaps on which the new term structure was based.

The financial crisis of 2008 and 2009 ultimately lead to another version of the term structure. The Dutch regulator deemed the valuation of the risk-free rate uncertain, given the unlikely market circumstances and low liquidity in the long end of the swap market. Instead of the current observed market rates, a 3-month averaging mechanism was imposed in December of 2011. This resulted in lower discount rates thus higher funding ratios.

However, this new term structure method did not end the discussion surrounding the liquidity of financial products with long maturities. This motivated the introduction of the Ultimate Forward Rate in 2012 which provides a solution for this issue. The details and dynamics are discussed in the following section.

2.3 UFR Specifications & Drawbacks

The Ultimate Forward Rate methodology thus deals with illiquid or non-available mar-ket data. This section provides the technical specifications set by EIOPA and the draw-backs of their suggested Ultimate Forward Rate method.

2.3.1 Smith-Wilson Extrapolation Method

The method used to extrapolate market rates to the UFR, is the Smith-Wilson method. It allows one to fit a curve to observed financial products whilst defining a convergence speed parameter and an UFR level, which are discussed later on. It is a generally accepted model and considered to be easily implementable3.

In the QIS5 specifications, EIOPA compared the Smith-Wilson method to a linear extrapolation method4 which was used in QIS4. The Smith-Wilson method was chosen over the QIS4 method for a number of reasons. The main difference between the two techniques is the input parameters. Smith-Wilson has an undefined convergence period but a defined convergence rate, while the linear method has a predefined convergence period and an undefined convergence rate. In the QIS4 specifications, the linear method is simplified further, by assuming constant forward rates after a certain maturity, there-fore having no convergence period. Other motivations in favor of the Smith-Wilson are that the output is a smooth differentiable forward curve and it can be used for both interpolation and extrapolation.

Another characteristic of the Smith Wilson function is a perfect fit to the input yields. In general, this is considered to be a good thing since it is perfectly in-line with market data, however in a research paper by Barrie+Hibbert (2010) it is also said that

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CEIOPS demonstrates its simplicity through a downloadable Excel file.

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this could be its biggest weakness. When fitting noisy bond prices, due to tax or liquidity effects, for example, an exact fit will lead to a bumpy curve. This could be compensated using the convergence parameter, but this will obviously affect the extrapolated part of the curve as well.

As mentioned before, the Smith Wilson function has a convergence parameter, alpha, which is set to 0.1 by EIOPA. This value stems from a research paper by Thomas et al. (2008). However, this research paper calibrated various extrapolation methods for the South African term structure and EIOPA does not clarify why the same convergence speed is also justified for the UFR method in general. If the resulting UFR curve does not reach the UFR in a predefined period, alpha is re-calibrated. This essentially takes away one of the main differences between the Linear and Smith-Wilson methods, imposing a ’fixed’ convergence period. The financial supervisor of Sweden (2010) also stresses the fact that there is ”a lot of work” to be done in setting objective criteria in (re-)calibrating the alpha.

It is clear that there is a need to objectively determine model parameters and perhaps for a more objective model. This is further emphasized in the following sections where the other input parameters of the Smith Wilson model, the height of the UFR and the LLP, are discussed in more detail.

2.3.2 UFR Value

The Ultimate Forward Rate is a fixed value to which the 1-year forward rate converges, thus being the ’last’ 1-year forward rate. The value of 4.2% is set by EIOPA5 and consists of expected real interest rate of 2.2% (for Eurozone) and an expected inflation of 2% (for the Eurozone). These values differ over countries and region.

This expected real interest rate is based on research by Dimson et al. (2012), who shows that the average real interest rate was 2.3% in the second half of the 20th century. The Dutch UFR Committee (2013), instated by the Dutch regulator in order to evaluate the EIOPA UFR method and alternatives, says this is mostly due to post world war two growth in order to reduce inflation, and is not representative for current projections. Dimson et al. (2012) also shows that interest rates can substantially diverge from the long-term average for longer periods of time.

Expected long-term inflation of 2% is set by EIOPA in line with inflation targets of central banks. The European Central Bank aims at ”inflation rates below but close to 2%” and the Federal Reserve states it aims at 2%.

In general, the Actuarieel Genootschap (Dutch Actuarial Society) and the CPB (Dutch Bureau for Economic Policy Analysis) are in agreement that a more realistic UFR would be below 4%. A study among numerous scientists, as a comment on the technical specifications by EIOPA, does not lead to a specific value, but they say that 4.2% is ”far from market consistent”. A UFR level which is higher than realized interest rates will result in the undervaluation of future liabilities and overestimation of the current funding ratio. The latter makes it less likely that current pension rights are cut and premiums are increased, which will result in big generation effects in favor of current pensioners says CPB (2012).

Both the scientists and the UFR Committee suggest a UFR level which is not fixed. The first suggests a liquidity and maturity weighted average market yield of 10 to 30 years instruments while the latter suggests a ten year average of the 20-year forward rate. The latter computation method results in a UFR of 3.9% (as of July 2013). These solutions also increase transparency for financial institutions, as the UFR level is defined as functions of known data. Otherwise, these institutions have to anticipate or insure themselves against changes in UFR policies by EIOPA.

The debate on the height of the UFR is still ongoing, but given the reasons above, 5_{Details can be found in Solvency II Calibration Paper, 2010}

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a non-predefined UFR might be the better option. Therefore, a possible transparent solution to determine the UFR is described in this thesis by deriving the UFR with the Kalman Filter algorithm.

2.3.3 Last Liquid Point

For really long maturities, financial products are less liquid or none existing. Therefore a Last Liquid Point (LLP) is defined after which market data is deemed unreliable. Solvency II regulations set the LLP at 20 years.

The Dutch UFR Committee points out that numerous experts are in agreement regarding good liquidity in the market of products with maturities till 10 years and that liquidity issues arise after the 30-year point. Some parties also say that liquidity for 20-year and 30-year products does not differ much. Kocken et al. (2012) supports this notion and say that there is no reason to assume illiquidity before a maturity of 30 years. Either way, there is no consensus on what the exact Last Liquid Point should be. Also, there is no objective measure for liquidity, because of for instance over-the-counter deals that happen outside the observable market.

Both the UFR Committee and Kocken point out that there is a high-interest rate sensitivity around the LLP. The shape of the Smith Wilson curve relies heavily on the last forward rate, which is the last interpolated forward rate between bonds with a maturity of 20 years and the bond with the highest maturity up to 20. The Committee, therefore, suspects that demand for products around the LLP might increase and, as a result, demand for higher maturities will decrease, making the liquidity argument of the LLP a self-fulfilling prophecy. The study by Kocken et al. (2012) shows that sensitivity around the LLP is increased six-fold essentially turning the Last Liquid Point in the Most Illiquid Point, if all institutions would hedge the UFR curve.

The proposed solutions all suggest a non-fixed LLP. This ensures that market data is not disregarded after the LLP but gradually weighted down, removing the sensitivity around the LLP. In practice, the Dutch regulator already uses this for pension funds, after the LLP the ’market’ forward rates are weighted with the Smith Wilson forward rate. Therefore, they speak about a first smoothing point at the 20-year mark. Other suggestions include Kocken et al. (2012) who proposes a modified version of the Smith Wilson model and the UFR Committee (2012) who propose forward rates which are computed with what they say is a simplified Kalman Filter. The latter method extrap-olates the forward rates to a weighted average of the forward rates with maturity 25, 30, 40 and 50, which is in turn averaged with the result from the day before. This thesis will discuss the use of an actual Kalman filter in light of the UFR.

2.4 The Kalman Filter

Named after Rudolf K´alm´an, the Kalman Filter is a recursive linear filtering algorithm. This section describes origins of the filter and how it can be used in the debate illustrated in the previous section.

2.4.1 Origins and background

Although the Kalman Filter carries only Kalman’s name, it was developed between 1958 and 1961 and credit is given to Swerling (1958), Kalman (1960) and Kalman and Bucy (1961) by numerous sources. The algorithm returns estimates of system states when given noisy observations of a linear system. Thus, as the name suggests, it takes in observations, filters out the noise and gives the best estimation of an un-observable (state) variable. It is a recursive algorithm, so each time a new observation comes in a new estimate is given that minimizes the variance of the estimation error. Since it is recursive, the algorithm can run in real time which increases its practical applications.

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It was originally developed in order to assist in spacecraft navigation, where it was used to estimate the exact location of spacecrafts given ’noisy’ location measures. But soon it was picked up in engineering, computer science, econometrics and many more fields and nowadays can be found in computer games and smartphones. In this thesis, the filter is applied to term structure modeling to provide a less objective alternative in determining an Ultimate Forward Rate.

2.4.2 Kalman Filter, Vasicek model and the UFR

The term structure is modeled after bond and swap prices which are observed in the market. Forward rates, however, can not be observed directly. It is also shown that there is a direct mathematical relationship between the term structure and forward rates. Thus, one could say that the Kalman filter could provide us with an optimal estimate of the forward rate (system state estimates) when provided with noisy measurements of the term structure. This could lead to new insights in the discussion surrounding the term structure extrapolation method, the UFR level, and the last liquid point.

One of the characteristics of the Smith Wilson function is that it disregards market data after a certain point. The last liquid point is the last point where market data is taken into account, and it is shown that various opinions exist regarding the maturity of this point. If the term structure is modeled using a Kalman Filter, all data is used and it is up to the filter to determine the weight given to the observation and to filter out the noise that might exist due to liquidity issues.

When combined with the Vasicek model for the short rate, the Kalman Filter can provide us with estimates for the expected mean, a mean reversion parameter, and the volatility. Using the parameters the Vasicek model has a closed-form solution for the forward rate when it tends to infinity. This is explained in the model specifications chapter and will be used to compare to the UFR level standards.

Naturally this is subject to the limitations of the Vasicek model and one can hold the same model choice discussion, but it does provide a more transparent way in determining the Ultimate Forward Rate. The method and results can be applied in two ways, first the Kalman Filter and Vasicek model can replace the Smith Wilson function altogether, however, this is not ideal since the Vasicek model can not replicate market prices exactly. The second application is to use the model outcome as input for other extrapolation methods, where the UFR is given a priori, as is the case with Smith Wilson. In the next chapter, we will discuss the Kalman Filter used to estimate the (Vasicek) short rate process (and thus the UFR).

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Chapter 3

Model Specifications

As discussed in the previous chapter, using data after the LLP could result in over-or underestimation due to various reasons. In this chapter, techniques are proposed in order to derive information using this data even if it contains noise. As a starting point, yields are modeled using a Vasicek process and information is ’extracted’ using principles of the Kalman Filter.

3.1 Vasicek Short Rate Model

As stated in the first chapter, the Vasicek Model proposed by Vasicek (1977), is a mean-reverting single-factor model for the short rate. It is assumed that short rates converge to a long-term mean level with a certain speed and the process is driven by one stochastic process. The short rate model consists of a drift term and a diffusion term.

The uni-variate stochastic differential equation of the Vasicek Short Rate Model follows an Ornstein-Uhlenbeck process and is of the form:

dst = k[θ − st]dt + σdwt (3.1)

where

st = short rate,

θ = long-term short rate level, k = rate of convergence, k[θ − st]dt = drift term,

σ = volatility of the short rate, wt = standard Brownian Motion,

dwt = diffusion term

The drift term is responsible for the long-term convergence, if k is larger the model will reach the long-term mean faster. The diffusion term is responsible for the variance in the model, a larger σ will result in a more volatile process.

Vasicek (1977) shows that given formula (3.1) bond prices p(t, T ) can be shown to satisfy the following equation:

p(t, T ) = eA(t,T )−B(t,T )st _(3.2) where B(t, T ) = 1 − e −k(T −t) k , (3.3) A(t, T ) = (B(t, T ) − T + t)(k 2_{θ − σ}2_/2) k2 − σ2B(t, T )2 4k (3.4) 10

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Terms B(t, T ) and A(t, T ) are deterministic, thus the log-bond price p(t, T ) is a linear function of the short rate or which we will later refer to as the state variable. Therefore, the Vasicek Model is regarded as an Affine Term-Structure Model, a classification of interest rate models that was introduced by Duffie and Kan (1996).

Given the bond price and assuming no arbitrage, we can also derive the zero coupon yield ytin affine form:

p(t, T )e(T −t)y(t,T ) = 1,

y(t, T ) = −ln p(t, T ) T − t

(3.5)

Substituting (3.2) in (3.5) it follows that: y(t, T ) = −A(t, T )

T − t +

B(t, T )st

T − t , discretized (3.6) From equation (3.6) the limit of T to infinity can be derived:

lim T →∞ B(t, T ) T − t = limT →∞ 1−e−k(T −t) k T − t = 0, lim T →∞ A(t, T ) T − t = limT →∞ (B(t,T )−T +t)(k2_θ−σ2_/2) k2 T − t − σ2B(t,T )2 4k T − t = k 2_{θ − σ}2_/2 k2 _{T →∞}lim B(t, T ) − T + t T − t − σ2 4kT →∞lim B(t, T )2 T − t = θ − σ 2 2k2 lim T →∞ B(t, T ) −T + t + −T + t T − t −σ 2 4k B(t, T )2 T − t = θ − σ 2 2k2 (0 − 1) − 0 = σ 2 2k2 − θ, lim T →∞y(t, T ) = limT →∞ −A(t, T ) T − t + B(t, T )st T − t = θ − σ 2 2κ2

Vasicek (1972) shows that the same formula holds for the long-term short rate, this equation will be used as the approximation of the Ultimate Forward Rate given Vasicek parameters:

s∞ = θ − σ 2

2κ2

The yield of a very long bond (without market price for risk) is equal to s∞ given

above.

The Vasicek differential equation (formula 3.1) can be discretized using Ito’s Lemma: st+∆t= θ(1 − e−k∆t) + e−k∆tst+ wt+∆t (3.7) where wt ∼ N 0,σ 2 2k(1 − e 2k∆t₎ , i.i.d. st+∆t|st ∼ N θ(1 − e−k∆t) + e−k∆tst, σ2 2k(1 − e 2k∆t₎

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3.2 Kalman Filter for Vasicek Short Rate model

Each iteration, when a new observation is taken into account, the weight of the obser-vation is determined and the system state is updated. The weight of the obserobser-vation is determined using a measurement equation. This equation represents the observed vari-able in an affine model form and is used to forecast the observed value. The forecast of the previous time period is compared to the actual observation and the error determines the weight of the observation. After that, the state variables in the transition equation are updated using the observation and weight. Combined, the transition equation and measurement equation are called the state space form of a model. In the first section, the general Kalman filter formulas are given and applied to the Vasicek short rate model. The second section describes the iterative process to determine the performance of the filter. The last section shows the outcome.

3.2.1 Kalman Filter and Kalman Smoother equations

In order to apply the Kalman Filter to the Vasicek Short Rate Model, the state space form of the Vasicek Model has to be derived. After that, equations for the Kalman filter can be construed and the Kalman smoother can be derived.

Kalman Filter

Koopmans et al. (2012) shows that mean-reverting models, like the Vasicek Model, have the following measurement equation:

~

yt= ~Ztαt+ ~dt+ ~εt, εt∼ N (0, ~HT) (3.8)

Where ~yt, ~Zt, ~dt and ~ε are vectors over maturities T. Given formula (3.5) we can

specify the measurement equation: ~

Zt =

B(t, T ) T − t ,

αt = st (the state variable),

~

dt = −

A(t, T ) T − t , ~

HT = Var[~εt] a T x T variance matrix

Koopmans et al. (2012) shows the that the transition equation is of the form: αt+1= Ttαt+ ct+ Rtηt, ηt∼ N (0, Qt)

α1∼ N (a1, P1)

(3.9) Using (3.7) this leads to the following specifications of the transition equation:

Tt = e−k∆t,

αt = st (the state variable),

ct = θ(1 − e−k∆t), Rt = Identity Matrix, Qt = σ2 2k(1 − e −2k∆t_), a1 = θ, at = The estimate of αt, P1 = σ2 2k

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The resulting Kalman Filter is of the form: vt= yt− Ztat− dt, Ft= ZtPtZt0+ Ht, a_t|t= at+ PtZt0Ft−1vt, Pt|t= Pt− PtZt0Ft−1ZtPt, at+1 = Ttat|t+ ct, Pt+1= TtPt|tTt0+ RtQtR0t (3.10) where Yt = past observations y1, ..., yt−1,

vt = the forecasted error or innovation term,

at|t = E [αt|Yt],

P_t|t = Var[αt|Yt] ,

at+1 = E [αt+1|Yt],

Pt+1 = Var[αt+1|Yt] ,

Ft = Var[vt|Yt−1]

The equation for Pt+1 can be rewritten as follow:

Pt+1= TtPt|tTt0+ RtQtR0t,

Pt+1= TtPt(Tt− KtZt)0t+ RtQtRt0,

(3.11)

Similarly equation for at+1 can be rewritten as follow:

at+1= Ttat|t+ ct,

at+1= Ttat+ TtPtZt0Ft−1vt+ ct,

at+1= Ttat+ Ktvt+ ct,

(3.12)

where

Kt = TtPtZt0Ft−1, the Kalman Gain Matrix

Together, equations (3.11) and (3.12) are also known as the prediction step of the Kalman Filter.

Kalman Smoother

Given the output of the Kalman Filter, it is possible to calculate a smoothed conditional mean and variance of the states. This takes into account all observed values and not just the ones until time t. The smoother consists of a backwards recursion and has the following form: rt−1 = Zt0F −1 t vt+ L0trt, Nt−1 = Zt0Ft−1Zt+ L0tNtLt, ˆ αt = at+ Ptrt−1, Vt = Pt− PtNt−1Pt, (3.13) where

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n = of observations, ˆ

αt = E [αt|Yt−1, vt, vt+1...vn],

Vt = Var [αt|Yt−1, vt, vt+1...vn] ,

rt = the weighted sum of innovations,

rn = 0,

Nt = Var[rt],

Nn = 0,

Lt = Tt− KtZt,

This form is equal to the state space form without mean adjustment. The smoother relies on the term vt, which is also called the innovation term, and is equal to vt in the

non-mean adjustment state space form. Log Likelihood

The log likelihood when initial states are known is of the following form (see Koopmans):

log L(Yn) = − np 2 log2π − 1 2 n X t=1 log|Ft| + vt0F −1 t vt (3.14) 3.2.2 Iteration Process

The Kalman filter and smoother are used to derive information on the (Vasicek) short rate, which is assumed to be the underlying driver for the observed yields. Using the Filter and Smoother equations derived in the previous section, the iteration process can be described.

Parameter initialization

Before the recursion is started, initial values of the expected state and its variance are constructed. Formula (3.9) states these as:

E [α1] = a1 = θ

Var [α1] = P1 =

σ2 2k

Now that we have our initial values we can start the recursion. Update measurement equation

Every step we have an expected state and expected variance. The expected state can be used to compute an expected yield, using the measurement equation.

Using formula (3.8) we can state that:

E [yt|Yt−1] = ZtE [αt|Yt−1] + dt (3.15)

The predicted error follows from taking the observed value of yt and deduct its

expec-tation (3.13) from it.

vt = yt− E [yt|Yt−1],

vt = yt− Ztat− dt, from (3.10)

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Compute Kalman Gain

The prediction error in (3.15) is used to update expectations for the state variable. First we compute the conditional variance of the prediction error:

Var [vt|Yt−1] = ZtVar [αt|Yt−1]Zt0+ Ht

Ft = ZtPtZt0+ Ht, from (3.10)

(3.17) Using the conditional variance we can compute the Kalman Gain Matrix:

Kt = TtVar [αt|Yt−1]Zt0Var [vt|Yt−1]−1

Kt = TtPtZt0Ft−1, as stated above

(3.18)

Forecast state

Now that all variables for the recursion have been computed, we can ’forecast’ our state variable for the next time step.

E [αt+1|Yt] = TtE [αt|Yt−1] + Ktvt+ ct

at+1 = Ttat+ Ktvt+ ct, from (3.12)

(3.19) The resulting state variance follows:

Var[αt+1|Yt] = TtVar[αt|Yt−1] (Tt− KtZt0) + RtQtR0t

Pt+1 = TtPt(Tt− KtZt)0t+ RtQtR0t, from 3.11

(3.20)

Log Likelihood

In order to compute the log likelihood in the end (or over an arbitrary time period), each time step the addition to the log likelihood is computed with:

LLt= − p 2log2π − 1 2log|Ft| − 1 2v 0 tF −1 t vt (3.21)

In the next chapter we will show the results of applying this algorithm to our data set.

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Chapter 4

Model Testing / Comparison

In this chapter, the performance of the Kalman is measured by applying it to different data sets. Short rates are not observed in the market, thus no ’real’ short rate data exists. Therefore, short rate data is simulated for different variance structures, used as input for the Filter and eventually compared to the Kalman Filter estimates. After establishing its performance on simulated data, the Kalman algorithm is used on market data and its results and implications are discussed.

4.1 Simulated Data

A number of steps are taken in order to measure the performance of the Kalman Filter. We begin by simulating short rates using the Vasicek process, these are converted to yields and then noise is added. The resulting yields including noise are used as input values for the Kalman filter and Kalman smoother. The outputs of the Kalman filter and smoother, Kalman estimates of the original short rate, can then be compared to the original rates. This section describes the steps taken.

4.1.1 Simulate Vasicek short rates and yields

We simulate n = 50 short rates using equation (3.7) which are converted to yields by using equation (3.6) for each maturity. The input parameters for the Vasicek model are:

θ = 0.05 k = 0.3 σ = 0.01 ∆t = 0.1 T = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30] p = length(T ) s1 = 0.01 which yield:

st = n × 1 short rate vector

After the short rates are computed, yields can be computed using the state space form in equation (3.6). This results in n (Vasicek) yields per maturity. Noise is added to these yields with the same variance matrix as in equation (3.8). The εtis constructed by

multiplying an n × p matrix filled with i.i.d. N (0, 1) random values with the Cholesky decomposition of HT the p × p variance matrix of εt. In the next sections, different

variance structures are simulated. This step results in the following variables: 16

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dt = p × 1 constant vector

Zt = p × 1 coefficient vector

Yvasicek,t = n × p ’original’ yield matrix

εt = n × p noise matrix

Yt = Yvasicek,t+ εt, n × p yield + noise matrix

Now that the noisy yields are constructed, the Kalman Filter is used in order to find the underlying Vasicek parameters. In order to do so, an algorithm searches for the set of Vasicek parameters which maximise the likelihood function which is defined in 3.14. Thus each time noisy yields are fed to the algorithm it gives a set of estimates and the corresponding log likelihood:

ˆ

θ = Estimated long term mean ˆ

k = Estimated convergence speed ˆ

σ = Estimated short-rate volatility ˆ

Ht = Estimated variance of the noise on yield

LLt = n × 1 partial log likelihood vector

4.1.2 Performance results with singular yield variance

Without initial short rate estimation

In order to determine a general idea of how the Kalman Filter performs, the simulation of yields and parameter estimation is repeated 100 times with three different parameter estimation algorithms. The yields are simulated using the Vasicek process and noise variance HT is considered constant for all t and all maturities T and set to 0.0001.

The table below shows the average and standard deviation of the estimated parameters compared to the actual parameters for each algorithm.

Active-Set Interior-Point SQP Actual Mean Std. dev. Mean Std. dev. Mean Std. dev. theta 0.05 0.0500 0.0007 0.0501 0.0008 0.0500 0.0008 kappa 0.3 0.3451 0.0120 0.3470 0.0094 0.3463 0.0102 sigma 0.01 0.0168 0.0016 0.0169 0.0016 0.0167 0.0017 HT 0.0001 0.0001 3.6e-06 0.0001 3.4e-06 0.0001 3.8e-06

All the algorithms seem to do well on estimating the long term mean theta and the Yield Noise. Thus, an accurate estimation of the long term short rate can be made given the short rate process and thus yield process are driven by the Vasicek models. However the same does not hold for the mean reversion speed and the short rate noise given their means and standard deviations. There is an overestimation in both these parameters, this will be addressed later on. In later computations, it became clear that the Interior-Point algorithm is preferred above the others. This is due to the fact that the other algorithms have a tendency to ignore the parameter estimation bounds. When this happens, the algorithms resort to imaginary numbers and the process can take a long time or might even freeze.

For illustration purposes, the simulated Yield of the Vasicek process with added noise and the predicted yield with maturity 1 are shown in figure 4.1. It is clear to see that the shape of the predicted yield by the Kalman Filter is a lot more like the original Vasicek Yields compared to the noisy yields. This seems to happen for all maturities, which are shown in figure 4.3. In the second graph (4.2) the Vasicek short rates are shown together with the filtered state vector and the smoothed state vector. The Kalman filtered short

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rate and smoothed short rate are following the actual short rate quite well. However the smoothed short rate, as the name suggests, is smoother and seems to be less volatile. This could be due to the backwards calculation of the smoothed short rate. When all the information is known, the prediction should be better. The initial starting value of the short rate is equal to theta thus one sees the discrepancy for time 0, but this will be addressed in the next section as well.

Figure 4.1: Yields, Yields with added noise and Kalman Filtered Yield, Maturity 1 yr

Figure 4.2: Vasicek short rate, Kalman Filtered short rate and Kalman Smoothed short rate

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With initial short rate estimation

As mentioned before, the kappa and sigma are overestimated in the previous model. This appears to be due to the starting value of the short rate. The Kalman process from the previous section assumes a starting value equal to theta. When the starting value is calibrated like the other parameters, this generates different results. In the table below the parameter results are given after 100 simulations for each optimization algorithm. It can be seen that the estimated parameter values, kappa and sigma, are all within one standard deviation from the actual means, for each calibration method. Theta and Ht

have similar results as in the previous section and the estimator for s1 is really close to

the actual value.

Active-Set Interior-Point SQP Actual Mean Std. dev. Mean Std. dev. Mean Std. dev. theta 0.05 0.0497 0.0008 0.0498 0.0009 0.0500 0.0010 kappa 0.3 0.3211 0.0664 0.3112 0.0466 0.3139 0.0491 sigma 0.01 0.0098 0.0020 0.0098 0.0024 0.0096 0.0024 Ht 0.0001 0.0001 3.7e-03 0.0001 4.0e-03 0.0001 3.3e-06

s1 0.01 0.0094 0.0037 0.0097 0.0042 0.0102 0.0040

As a result of calibrating the starting short rate, the initial jumps in expected yield and estimated short rate, which were seen in the previous section, disappear. The short rate now starts at the ’correct’ starting value, which is seen in figure 4.5. This also has an effect on the Kalman Measurement Prediction of the yield which is illustrated in figure 4.4. Apart from the initial jump, there does not seem to be a lot of differences between this estimation method and the one of the previous section.

Thus given initial short rate estimation and equal noise variance for all maturities, it is clear to see that the Kalman filter is able to filter out the observed noise on the yields. Combined with the simulation results, it can also be said that the starting value should be included as an estimation parameter in order to get accurate results. Hence, the models in the next sections will build upon this model.

Figure 4.4: Yields, Yields with added noise and Kalman Filtered Yield, with initial short rate calibration, Maturity 1 yr

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Figure 4.5: Vasicek short rate, Kalman Filtered short rate and Kalman Smoothed short rate, with initial short rate calibration

4.1.3 Performance results with multiple yield variance

One of the motivations for an UFR is that observed market prices for swaps and bonds with long maturities are not representative, due to a lack of liquidity. Implying an inac-curate market value. Thus, one could say that observed market prices for long maturity products contain more noise. This section, therefore, adds variable noise variance to the simulated yields for each maturity.

Linear increasing yield noise variance

Using the Active-Set algorithm, 50 simulations are run using a linearly increasing yield noise variance given by formula:

HT(i, i) = 0.0001 + 0.000001T (i),

HT(i, j) = 0

The estimated Vasicek parameters theta. kappa and sigma all fall within one stan-dard deviation of the actual values. More importantly this also holds for the noise variance. It also appears that the standard deviation increases as the noise increases, which is expected.

Actual Mean Std. dev. theta 0.05 0.0497 0.0012 kappa 0.3 0.3268 0.0624 sigma 0.01 0.0099 0.0024 s1 0.01 0.0097 0.0043

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Fixed Yield Noise Variance Maturity 30 Linear Increasing Yield Noise Variance Maturity 30

HT, T(i)= Actual Estimate Std. dev. HT, T(i)= Actual Estimate Std. dev.

1 1.1e-04 1.09e-04 0.13e-04 7 1.7e-04 1.67e-04 0.27e-04 2 1.2e-04 1.19e-04 0.16e-04 8 1.8e-04 1.85e-04 0.26e-04 4 1.4e-04 1.35e-04 0.17e-04 10 2.e-04 2.04e-04 0.28e-04 3 1.3e-04 1.35e-04 0.15e-04 15 2.5e-04 2.58e-04 0.35e-04 5 1.5e-04 1.50e-04 0.20e-04 20 3e-04 2.98e-04 0.38e-04 6 1.6e-04 1.58e-04 0.23e-04 30 4e-04 3.93e-04 0.49e-04 For illustrative purposes, we zoom in on the yield for maturity 30, for the previous model and this model. Thus comparing the situation with yield noise variance 0.0001 and 0.0004. In this figure, the difference in noise is visible, as the fact that increased noise influences the Kalman prediction (increased standard deviation of the estimate).

4.1.4 Performance results with correlated yield variance

Using the Active-Set algorithm, 30 simulations are run using an correlated yield noise variance structure. This structure is of the general heterogeneous auto regressive 1 (ARH1) form given by formula:

HT(i, i) = 0.0001 + 0.000001T (i)

HT(i, j) = HT(i, i) ∗ HT(j, j) ∗ ρT (i)−T (j)

ρ = correlation factor

Thus, it is still assumed that noise variance increases in maturity but also that covariance increases when maturities are closer to each other. Dependence between yield over maturities would be logical if market noise were totally independent this could lead to very diverging yields over maturity and in turn to huge arbitrage possibilities. Using the covariance matrix form above, yields are constructed and the Kalman Filter and Smoother are applied to find the Vasicek parameters. Plots of the fits can be found in Appendix A.

Actual Mean Std. dev. theta 0.05 0.0502 0.0014 kappa 0.3 0.2988 0.0624 sigma 0.01 0.0097 0.0024 s1 0.01 0.0103 0.0043

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HT, T(i) = Actual Estimate Std. dev. HT, T(i) = Actual Estimate Std. dev.

1 1.1e-04 1.10e-04 0.17e-04 7 1.7e-04 1.71-04 0.26e-04 2 1.2e-04 1.21e-04 0.14e-04 8 1.8e-04 1.79e-04 0.25e-04 4 1.4e-04 1.30e-04 0.21e-04 10 2.e-04 1.96e-04 0.24e-04 3 1.3e-04 1.43e-04 0.21e-04 15 2.5e-04 2.51e-04 0.29e-04 5 1.5e-04 1.48e-04 0.20e-04 20 3e-04 3.00e-04 0.41e-04 6 1.6e-04 1.57e-04 0.22e-04 30 4e-04 4.00e-04 0.65e-04 The results show that the Vasicek parameters for theta, kappa, sigma, and s1 can

be found by the algorithm. Also the actual parameter for the yield variance fall well within one standard deviation. The estimation for the correlation parameter ρ is off by quite a bit and the standard deviation is very big in comparison to the mean, thus not excluding 0. Looking at the detailed results one can see that the Kalman estimation ’hits’ the upper bound value 1 for ρ quite a lot. Therefore, it can be concluded that the Kalman Filter and Smoother, given correlated yield data, is unable to model the covariance structure correctly.

4.2 Historical data

4.2.1 Dutch Yield Data

Vasicek Estimation

In order to provide a context for the Kalman Filter and Smoother estimations, the reg-ular Vasicek parameter estimation is applied to Dutch historical yield data. ’Snapshots’, thus data with fixed time stamps, are used to estimate the Vasicek parameters present in formula 3.1. In order to do so, Formula 3.2 and 3.5 are used in sequence to generate yields given arbitrary Vasicek parameters. This is repeated until the difference between the actual data and generated yields converge to a minimum.

In order to filter out any irregularities caused by the yield models and UFR, which are described in chapter 2, data used in this section is taken from 12-2003 until 31-12-2005. The data contains Dutch term structures as published by the Dutch regulator DNB on a monthly basis. For each month, the yields with time to maturity 1, 2, 3, 4, 5, 6, 7, 8, 10, 15, 20 and 30 are taken as observations. The snapshots illustrated in figure 4.6 are from dates 31-12-2003 and 30-06-2005 respectively. A decrease in yields can be seen over the year and a half time span. In earlier experiments with varying starting values, the calibration either fitted well or overestimated theta with a very low mean reversion parameter (details can be found in Appendix A). However, when using starting values that are close to what is expected, more realistic results can be produced which are shown in the table below.

30-12-2003 31-03-2004 30-06-2004 30-09-2004 31-12-2014 31-03-2005 31-06-2005 theta 0.0572 0.0583 0.0564 0.0547 0.0511 0.0483 0.0456 kappa 0.3172 0.2389 0.3272 0.2503 0.2016 0.2365 0.1762 sigma 0.16e-04 0.29e-12 0.26e-09 0.62e-17 0.62e-18 0.51e-11 0.16-e11

s1 0.0173 0.0152 0.0192 0.0195 0.0212 0.0204 0.0175

MSE 0.053e-05 0.264e-05 0.048e-05 0.081e-05 0.077e-05 0.092e-05 0.350e-05 UFR 0.0572 0.0583 0.0564 0.0547 0.0511 0.0483 0.0456

The implied UFR is almost the same as theta since the modeled volatility is very low. Using these snapshots, model parameters can be plotted over time. Where it can be seen that the expected short rate decreases and the mean reversion fluctuates but also has a decreasing trend.

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Figure 4.6: Vasicek parameters for snapshot 31-12-2003 and 30-06-2005

Figure 4.7: Vasicek parameters for time 31-12-2003 to 30-06-2005

Kalman Estimation

The Kalman Filter and Smoother are also applied to Dutch historical yield data, which is published by the Dutch regulator. For each month, the yields with time to maturity 1, 2, 3, 4, 5, 6, 7, 8, 10, 15, 20 and 30 are taken as observations. Also, all yields are assumed to have their ’own’ noise, thus there is a noise variance for each time to maturity. These error terms are assumed to be independent and normally distributed.

As above, the short rate is assumed to follow a Vasicek process. The table below shows the estimated Vasicek parameters, implied UFR and the yield noises.

Estimate theta 0.0599 kappa 0.1521 sigma 0.0072 s1 0.0282 UFR 0.05878

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HT, T(i) = Estimate HT, T(i) = Estimate HT, T(i) = Estimate

1 2.23e-05 5 1.00e-09 10 5.13e-06

2 1.02e-05 6 4.26e-07 15 1.073e-05

3 3.65e-06 7 1.50e-06 20 1.67e-05

4 6.83e-07 8 2.72e-06 30 3.16e-05

Figure 4.8: Dutch Yield data vs Kalman Prediction

In figure 4.8 the Kalman prediction for the yields is shown together with the DNB yield data. For time to maturity range 5yr till 10 yr, the shape of the predicted yield seems to follow the actual yield. Whereas for the short time to maturities, 1-5 yr, and the long time to maturities, 10+ yr, the predicted form seem skewed. This might be due to the assumption that all yields are driven by the same short rate. And in order to minimize log likelihood, the Kalman filter ’chooses’ the middle maturity which results in minimal prediction errors.

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These results, also compared with the ’plain’ Vasicek estimations from the previous section, seem plausible. But the mean short rate is a bit higher than in the snapshots. The implied UFR of 5.88% is also at the high end of the spectrum of the snapshots, this could imply that the Kalman Smoother is ’slow’ in catching short rate movements.

4.2.2 Euribor Swap Data

Vasicek Estimation

It is common practice to use swaps to construct yields for higher maturity. Data used in the previous section is partially derived from swaps as well. Therefore, historical Euribor swap data is fitted in the same manner as before. For comparison purposes, (daily) swap data is taken for the same period as the Dutch Yield data. The data quotes swaps with maturity 1 - 10, 12, 15, 20, 25 and 30. Before the data is fitted, quotes are interpolated linearly for missing maturities. Afterwards, the zero coupon rates are bootstrapped using the standard formula:

zj = 1 − ij j−1 P k=1 zk 1 + ij

zi = zero coupon bond price

ij = swap price

Using the estimates of the DNB snapshots as starting points yields the results shown in the table below. Original estimates are in Appendix A, where the same overestima-tion of theta can occur if parameters have different starting values. The range for the estimated long term short rate is 4.5% - 5.8%, and mean reversion rate kappa in the range 0.17 - 0.31.

30-12-2003 31-03-2004 30-06-2004 30-09-2004 31-12-2014 31-03-2005 31-06-2005 theta 0.0570 0.0581 0.0564 0.0546 0.0509 0.0482 0.0454 kappa 0.2947 0.2312 0.3050 0.2384 0.1986 0.2260 0.1714 sigma 0.82-e08 0.49e-14 0.30e-08 0.999e-12 0.22e-17 0.83e-13 0.30-e18

s1 0.0175 0.0118 0.0184 0.0180 0.0188 0.0216 0.0139

MSE 0.12e-05 0.37e-05 0.16e-05 0.16e-05 0.14e-05 0.16e-05 0.44e-05 UFR 0.0570 0.0581 0.0564 0.0546 0.0509 0.0482 0.0454

These estimates are very similar to that of the DNB snapshots which are due to the fact that the DNB yield curve was probably constructed partially using Euribor swap data.

Kalman Estimation

As before, maturities 1 - 8, 10, 15, 20 and 30 are taken as observations. The table below shows the results, these are similar to the results with DNB yield data. The long term short rate is a little bit higher, but the mean reversion speed is lower.

In Appendix A the same dynamics as in the previous section can be seen, the Kalman filter ’matches’ maturity 6. One could argue that given the yield term structure, which is usually increasing in maturity, the Filter’s behaviour is to choose the ’middle’ maturity in order to maximize the log likelihood.

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1 2.97e-05 5 5.16e-07 10 2.55e-06

2 1.47e-05 6 1.00e-07 15 7.81e-05

3 6.59e-06 7 3.45e-07 20 1.42e-05

4 2.27e-06 8 1.00e-06 30 3.67e-06

4.3 Present Data

4.3.1 Euribor Swap Data

Although the DNB publishes yield data each month, this data can not be used as it already has the (DNB) UFR methodology applied to it. In the previous section it was shown that for historical data, from before the introduction of other yield curve methods and the UFR, Euribor data produced similar results as DNB. Therefore more recent Euribor data will be used as the only source for present data in this section. The number of input dates will stay the same, however, thus the data used is of 30-04-2012 to 30-04-2014.

Vasicek Estimation

In order to set expectations for the parameters, snapshots are taken just like in previous sections, however, this time all the known maturities will be taken into account. The known maturities are: years 1-10, 12, 15, 20, 25, 30, 40, 501.

s1 14.5e-009 27.8e-009 123.2e-009 37.8e-009 1.7e-012 8.36e-012 123e-009

MSE 46.0e-006 24.2e-006 26.2e-006 13.2e-006 19.6e-006 25.9e-006 32.3e-006 UFR 0.0288 0.0275 0.0298 0.0295 0.0285 0.0302 0.0326

It is clear to see that times have changed. Yields have dropped significantly and therefore the expected short rate and UFR have also dropped. The implied UFR is increasing but far below 4.2%.

Kalman Estimation with multiple yield variance

If historical patterns are applicable to more recent data, the Kalman theta should be higher than what is seen in the snapshots, but signify a drop in comparison to the historical data. The Filter and Smoother are first applied assuming independent yield variance between maturities. In the table below the estimates are shown using maturities 1-10, 12, 15, 20, 25, 30, 40 and 50.

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28 Ryan Tjin — Empirical estimation of the UFR Estimate theta 0.0376 kappa 0.3217 sigma 0.0538 s1 0.0282 UFR 0.0236

HT, T(i) = Estimate HT, T(i) = Estimate HT, T(i) = Estimate HT, T(i) = Estimate

1 2.790e-06 6 3.608e-06 12 6.742e-06 40 6.0465e-06

2 3.651e-06 7 3.967e-06 15 7.681e-06 50 1.1240e-05

3 66.68e-06 8 6.146e-06 20 6.557e-06

4 3.627e-06 9 4.239e-06 25 4.980e-06

5 3.413e-07 10 4.380e-06 30 4.364e-05

The estimated long term mean for the Euribor data is higher than what is expected from the previous section. The mean reversion rate seems reasonable but combined with the estimated volatility of around 5%, the implied UFR value of 2.3% is a lot lower than the proposed UFR value of 4.2%. But looking at an excerpt of the fit plots, for maturities 1, 9 and 50, it is seen that the fit of maturity 50 is really poor because the estimated yield does not intersect with the actual yield. This is caused by the way the Kalman Filter works, which is illustrated further by varying the input data for different maturities.

Figure 4.9: Euribor data maturities (1-10, 12, 15, 20, 25, 30, 40, 50) vs Kalman Prediction In the table below, Kalman estimations are shown based on Euribor data with maturities 1-10, 12, 15, 20, 25 and 30. Estimate theta 0.0437 kappa 0.1643 sigma 0.0191 s1 0.0050 UFR 0.369

HT, T(i) = Estimate HT, T(i) = Estimate HT, T(i) = Estimate HT, T(i) = Estimate

1 6.57e-06 5 3.13e-07 9 1.64e-07 20 7.11e-06

2 4.64e-06 6 5.82e-08 10 3.31e-07 25 2.05e-05

3 2.15e-06 7 1.06e-13 12 8.40e-07 30 3.77e-05

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An implied UFR of 3.7% is higher than the snapshots in the previous section but lower than the proposed 4.2% by the UFR methodology and the 3.9% by the Dutch UFR committee. Also, one can see that the fit is still focused around maturities 4-12. As expected, excluding high maturities results in poor fits for high maturities. In figure 4.10 shows the Kalman estimate for maturity 30 has no intersection with the actual yield at all. Full fit can be found in the appendix.

Figure 4.10: Euribor data maturities (1-10, 12, 15, 20, 30) vs Kalman Prediction When higher maturity series are included at the cost of excluding lower maturity series, a whole different picture is drawn. The results in the table below show the Kalman estimations on data with maturities 1-6, 10, 15, 20, 30, 40 and 50.

1 2.54e-06 5 1.50-07 20 7.01e-06

2 1.64-06 6 5.56-07 30 3.94e-06

3 2.65-07 10 4.38e-06 40 4.42e-06

4 6.04-13 15 1.02e-05 50 4.54e-06

The estimated Vasicek parameters are a lot closer to what one would expect given the snapshots in the previous section. An UFR of 2.6% is a lot lower than the UFR of 4.2%. Using the same maturities as in figure 4.10, it is seen in figure 4.11 that the fit for maturity 30 is much better. The down side is also shown, the fits for maturity 1 and 10 seem to be worse. A full plot can be seen in Appendix.

Concluding, the Kalman Filter and Smoother try to fit most series by maximizing the log likelihood. In the case of more lower maturity inputs, it becomes harder for the model to fit higher maturities. In those cases, the higher maturity are given less weight (property of the Kalman filter) therefore resulting in a higher overall expected short rate and UFR. The opposite also holds, when adding higher maturity series in favor of lower maturities, the ’focus’ of the filter shifts to a higher (’middle’) maturity. Therefore fitting higher maturities better, which, in this case, results in a lower expected short rate. Thus, the combination of the Vasicek model and the Kalman Filter is very

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Figure 4.11: Euribor data maturities (1-6, 10, 15, 20, 30, 40, 50) vs Kalman Prediction

sensitive to the inputted values, which is probably due to the limitations of the Vasicek model. Using a model which captures the differences between maturities better should Therefore be investigated further in order to avoid a new discussion regarding which maturities to feed the Kalman Filter.

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Conclusion

Using the well known Vasicek model in combination with the practical Kalman Fil-ter, a model was constructed to provide new insights to further the discussion regarding the Ultimate Forward Rate. The model purpose was to replace the a priori UFR value of 4.2% with a less subjective (and political) method. While taking into account data for all maturities but letting the Kalman Filter correct for uncertainties regarding the higher maturities.

First the model was tested using simulated data, this allowed us to compare the actual values with the model estimates. It can be concluded that the model, given a ’Vasicek term structure’ with independent yield variance, estimates the mean short rate, mean reversion level noise and yield noise well within one standard deviation. However when the yield variance is correlated, the model was unable to estimate the correlation coefficient correctly. Therefore, correlated variance structure was not pursued further and not applied to real data.

After determining that the model worked on simulated data, it was applied to monthly historical data for the time period 31-12-2003 till 30-06-2005. This time period was chosen because it was before the Dutch regulator started to apply UFR methods and it is well before the financial crisis of 2008. Using snapshots and plain Vasicek estimation, it was shown that the Kalman estimations for the mean short rate were consistently higher, resulting in an implied UFR of 5.9% and 6.3% for DNB yield data and Euribor swap data respectively. Where the model fitted yield noise almost perfectly with simulated data, estimations of the historical data showed different results. The model minimized noise at the middle maturities in order to maximize the log likelihood. One can only conclude that the yield process is inconsistent with the Vasicek structure and should be investigated further.

Seeing that the Euribor curve with its plain Vasicek and Kalman estimates were close to those of the Dutch yield data, Euribor swap data was used as a current data set and probably can be used as a proxy for the unpolluted’ Dutch yield curve. The current data consisted of monthly Euribor swap data from time period 30-04-2012 to 31-10-2014. The current data also included the 40 and 50-year maturities. When added as inputs to the known series, one could see that there was a poor fit for the higher maturities. Therefore, the input maturities were varied besides using all maturities. This emphasized the previously observed model behaviour even further. The resulting UFR estimates varied from 2.7% to 4.5%, given different maturities as input. Although most researchers are more inclined to agree with the former value, this sparks a new ’input discussion’. The Kalman Filter and Vasicek model as a solution therefore just replaces one subjective matter with another and thus can not be directly used as a replacement for the proposed UFR method.

It was clearly shown that the Kalman Filter and Smoother can successfully filter noise from observed yields if the underlying state structure is known, thus solving for

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noisy observed market prices for high maturity products. Given the age of the Vasicek model and the extensive research in term structure modeling afterwards, one could ex-pand upon the research of this paper by applying the Kalman Filter and Smoother to extensions of the Vasicek model in order to capture yield behaviour better. This ulti-mately would have the added benefit of fitting actual data better and perhaps replacing the Smith-Wilson approach in total. Thus, it is not possible to construct a model which is completely objective, with the Kalman Filter and a short rate model. But, by using these techniques it is possible to obtain a more transparent (the computation method is known and clear) and dynamic (the UFR value is not a fixed state) model.

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Vasicek estimation, DNB data

The Vasicek estimation parameters for multiple dates are shown in the table below. The results vary across dates. Some come close to what might be expected (t=31-12-2003), other seem to overvalue the theta with a very low mean reversion (t=31-03-2005). Running multiple times, with different starting values also yields variable results. Thus it can be concluded that plain Vasicek parameter estimation does not produce consistent results. This can be resolved using constrained solution values.

30-12-2003 31-03-2004 30-06-2004 30-09-2004 31-12-2014 31-03-2005 31-06-2005 theta 0.0572 0.1530 0.1077 0.0547 0.1206 0.0483 0.0456 kappa 0.3172 0.0650 0.1278 0.2503 0.0533 0.2365 0.1762 sigma 0.34e-04 0.0306 0.0416 0.24e-13 0.0219 0.22e-10 0.73-e07

s1 0.0173 0.0152 0.0192 0.0195 0.0212 0.0204 0.0175

However this anomaly can be corrected using different starting values for the opti-mization algorithm.

Vasicek estimation, Euribor data

s1 0.0175 0.0118 0.0184 0.0180 0.0188 0.0216 0.0139

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Empirical estimation of the Ultimate Forward Rate using the Kalman Filter and Smoother

Forward Rate

Using the Kalman Filter and Smoother

Ryan Tjin

Contents

Introduction

Chapter 2

Ultimate Forward Rate

2.1

Motivation for the UFR

2.2

Interest rates

2.3

UFR Specifications & Drawbacks

2.4

The Kalman Filter

Chapter 3

Model Specifications

3.1

Vasicek Short Rate Model

3.2

Kalman Filter for Vasicek Short Rate model

Chapter 4

Model Testing / Comparison

4.1

Simulated Data

4.2

Historical data

4.3

Present Data

Conclusion

Vasicek estimation, DNB data

Vasicek estimation, Euribor data

Euribor Kalman Fits

Performance results with correlated yield variance