Modeling Interest Rate Shocks: an Empirical Comparison on Hungarian Government Rates Balazs Toth

Transcription

1 Modeling Interest Rate Shocks: an Empirical Comparison on Hungarian Government Rates Balazs Toth Abstract This paper compares the performance of different models in predicting loss events due to interest rate movements. The goal is to find a shocked yield curve, so that only 0.5% of the cases would give higher loss on a predetermined portfolio. Three approaches are investigated, a one-factor model proposed by Cox-Ingersoll-Ross, a Multifactor framework, and Principal Component Analysis (PCA). By calibrating them on Hungarian historical data, one can use the parameters in a Monte Carlo simulation to generate future term structures. We calculate the Value at Risk (VaR) of four different portfolios according to each framework, and compare them to the empirically occurred losses. We find that although the no-arbitrage one factor model and the PCA projects similar VaR-s, in case of three portfolios they highly overestimate, while for one they underestimate the possible losses. The Multifactor approach provides the closest match with the history, only in case of a 30 years annuity it is not conservative enough. Keywords: Factor Model; PCA; Shock Event; Worst Case Scenario; Term Structure of Interest Rates Master Thesis 2008 Department of Econometrics & Operations Research Tilburg University B. Toth s b.toth@uvt.nl

2 Abstract Introduction No arbitrage one-factor approach Risk-neutral world The market price of risk Real versus risk-neutral world Introduction to one factor no-arbitrage models Vasicek model CIR model Estimation by GMM Generating future r t -s Multi-factor approach The HJM framework The volatility structure Univariate AR modeling Merging the volatility structure and the AR together Estimating kappa Estimating the system of equations in (3.) How to use the estimates for forecasts PCA approach The common factors Mathematical background Determining the shocked curves Adjusting for autocorrelation, calculating phi Data Data input Inverse term structure Evaluated portfolios Empirical comparison Empirical results of the no arbitrage modeling Empirical result of Multifactor model Empirical results of PCA Historical losses Conclusion Appendix A Appendix B Acknowledgement References Tables Figures... 54

3 . Introduction According to some special rules, given by the regulator of financial markets in a particular economy, insurance companies and pension funds have to reserve an amount of capital to protect themselves against possible losses. These losses could be due to operational mistakes, change in demographical effects, i.e., change in mortality rate, or average life expectancy, or simply originating from financial market events. An example of the latter is the variation of interest rates. The market of pension and insurance products is highly competitive, so the company has to sell them close to the fair price. What is the actual fair price? For sure, it includes the price of the risk, which is due to a possible interest rate shock. An interest rate shock is a movement of the yield curve in such a way, that the discounted value of a portfolio changes. Depending on the direction of the movement, the value of the portfolio will increase/decrease imposing a gain, or a loss for the holder of the portfolio. Therefore, for risk management it is essential to have a joint model for the term structure movements in the relevant countries, where the company has operations [Driessen, Melenberg, Nijman (2003)]. As it was mentioned, an insurance company has to prepare itself for such a loss event, which occurs only once in 200 years. Translating it to the language of statistics, the 99.5 percentile worst case movement of the yield curve has to be estimated. The goal is to find a shocked yield curve, so that only 0.5% of the cases would give a higher loss on a predetermined portfolio. This paper investigates three different approaches to forecast the term structure movements of the future. One of them is the no arbitrage one factor approach. The short term riskless rate is one of the most fundamental and important prices determined on the financial markets [Chan, Karolyi, Longstaff, Sanders (992)]. A variety of models have been built to exploit, and explain its behavior, including the ones by Vasicek (977), Cox-Ingersoll-Ross (985), Ho-Lee (986), Longstaff (989), Hull-White (990), Black- Derman-Toy (990), and Black-Karasinski (99). This paper analyzes two of them, namely the one proposed by Vasicek and Cox-Ingersoll-Ross. The model parameters are estimated on historical data, and used in a Monte Carlo simulation to generate future values of the short rate. In particular, according to each simulation the entire term structure is calculated to be able to evaluate any kind of portfolio. Insurance companies have to hold project capital based on once in a 200 years shock. 2

4 Modeling the interest rate evolution through the short rate has some advantages, mostly the large flexibility one has in choosing the related dynamics [Brigo, Mercurio 200]. However, these models have also some clear drawbacks. For example, the exact calibration to the initial curve of discount factors is in most of the cases not possible to achieve. They also have the undesirable property that all bond returns are perfectly correlated, and thus these models do not accurately characterize the observed term structure of interest rates and its changing shape over time [Kuo, Paxson (2006)]. To develop the fit of the calibration, more factors need to be used. As a further approach, one can apply two-factor short rate models. One, introduced by Hull and White [Hull, White (994)] is widely used by pension funds in their hedge models. Although having more flexibility, beside a factor, which drives the stochastic mean reverting process, it still uses only the short rate as an input to determine the yield curve for the entire horizon. There is also a wide range of literature for three and four factor models; [Jarrow (996)], [Frye (996)], [Hull (2000)], [James, Webber (2000)], [Wilmott (2000)]. By increasing the number of factors, on the one hand we allow for more flexible shapes of the predicted yields to fit with the observed ones, on the other hand the formula to calculate the zero coupon bond prices becomes more complex, if analytically expressible at all. As a second approach, a multifactor model is applied in accordance with the general framework developed by Heath-Jarrow-Morton 2 [Heath, Jarrow, Morton (992)], but with an important simplification. In this paper ex ante, stressed security prices are not postulated to be free of arbitrage opportunities. Therefore, the no arbitrage constraint is not a crucial point of modeling the interest rate movements. Hence, instead of the bounded drift structure derived by HJM, a univariate autoregressive drift structure is used. By assumption, the volatility of any point of the yield curve will be dependent on the variation of other points of the yield curve. After exploiting the strength of interdependence, a system of equations with some key maturities is estimated. The results are used in a Monte Carlo simulation to obtain diverse outcomes of the term structure. Such multifactor models are extremely complex systems, thus their application may require enormous computational sources. Most interest rate variations are parallel shifts, when the interest rates for all the maturities increase/decrease with a couple of basis points. However, the steepness and the curvature change of the curve also play an important role. Some papers introduce that the three before mentioned factors could describe the movements of the yield curve with a reasonable accuracy. Principal Component Analysis (PCA) is a statistical method, which 2 Referred later as HJM. 3

5 can be applied to identify the above mentioned most common features of the yield curve variation. Frye (997) applied PCA to determine the VaR of a portfolio. Driessen, Melenberg and Nijman (2003) analyzed, how a multi-country version of the model can be used for risk management purposes, as well as for pricing cross currency interest rate derivatives. The paper applies PCA to identify the three common factors, driving the movements of the term structure. By using extreme values of the factors, the worst case movement of the entire yield curve is expressible. In the empirical analysis, the models are calibrated on historical data. We use weekly observations of some key maturities (3, 6 month,,3,5,0 and 5 year) of the Hungarian yield curve, starting form December 2000 to July In case of each approach, the entire time horizon is used as an input, in order to dispose with comparable results. Finally, the Value at Risk of four different portfolios are investigated, and compared to each other according to the three different models and the history. In case of the one-factor models, the 3-month rate was treated as the short rate. The Vasicek framework is not able to fit with the Hungarian term structure at all, it even shows negative values of interest rates at certain maturities. The level dependent volatility of the CIR model is an essential feature to match with the Hungarian yields. Therefore, the simulation was applied only based on the CIR approach. We find that for three portfolios it gives significantly higher possible losses than the history, while for the fourth, it predicts a value less than the half of the empirical one. The Multifactor approach uses all the maturities as input. Therefore, it can be interpreted as a seven-factor model. It turns out that the predicted losses are closer to the occurred ones. However, in case of a 30 years annuity the result is not conservative enough. The three most common factors of the PCA explain 95% of the yield curve movements. The first, second, third factor represent the parallel shift, the slope change, and the curvature movements, respectively. It shows similar results to the one-factor approach, the difference between the predicted losses and the historical ones are rather high. Although according to the comparison presented in this paper, the seven factor model shows the most common features with the history, the shortage of disposable historical data also has to be a matter of concern. Apparently, we do not have 200 years of historical data in any economy. Therefore, the comparison between any model and the history has to be treated with doubtful attention. Risk managers also have to consider their computational capability and recourses. The remainder is organized as follows. The next three sections (2, 3 and 4) describe in details the above mentioned 3 different approaches (no-arbitrage one factor, multifactor 4

6 and PCA respectively), their estimation methodologies, and how the results can be used to predict worst case movements of the term structure. In section 5 the data is described, while section 6 shows the empirical results of the different approaches. Section 7 concludes. 2. No arbitrage one-factor approach No arbitrage one-factor term structure models usually targets to suit a process for the short rate r. The final aim is to explore what the process r implies about bond prices. The short rate at time t, r t generally referred as the instantaneous spot rate, applies to an interest rate for an infinitesimally short period of time t, at period t. According to these models, bond, option and other derivative prices only depend on the process followed by r [Hull (2002)]. It is important to note that with an adjustment between the two, they try to describe the evolution of r in both the risk-neutral world, and in the real world. 2.. Risk-neutral world Since the seminal papers of Black and Scholes 3 on the pricing of options, the term riskneutral valuation has widely been spread out in the financial communities. The basic idea behind the risk-neutral valuation is the theory of arbitrage free financial markets. Consider an investor in the fictive risk-neutral world, meaning that he is assumed to be risk-neutral. In that case, the only decisive feature in an investment decision (i.e. investing in risky assets: stock or option; or risk free assets: bond) is the expected return on a particular asset. If there was a difference between the returns of the different assets by going short in the one with a low return and taking a long position in the one with a higher return would result in an arbitrage profit. To put this in another way, by investing a cumulative amount of 0, the investor could achieve positive profit, with a positive probability. Nevertheless the most powerful statement of financial theory is the absence of arbitrage. It means excluding the opportunity to gain positive profits with positive probability, by zero investment. Absence of arbitrage exists in the risk neutral world only if the expected return on every asset were equal. We know, that the risk free bond has the same return both in the risk-neutral, and the real world; it is called the risk free rate, f r. As a 3 [Black, Scholes 973] 5

7 consequence, in the risk neutral world all the expected asset returns must be equal to the risk free rate. Certainly, investors are typically risk-averse in the real world. For higher risk, they require higher return. Accordingly, the asset returns differ, depending on how much uncertainty they capture. The more volatile the stock or the derivative in the real world is, the higher the expected return. The following chapter aims to exploit the correspondence between the volatility, and the expected return The market price of risk As mentioned before, to be able to calculate bond prices, first the evolution of r has to be derived. Consider two risky investment assets, A and B, and keep the notation for their prices A and B. Assume, that there is only one source of uncertainty W (Wiener process), and that their evolution can be written by the Stochastic Differential Equations (SDEs) as follows: dat A t dbt B t = µ dt+ σ dw A B A = µ dt+ σ dw da and db refer to the change in the price, B (2.) µ A and µ B are the expected returns respectively, σ A and σ B represent the volatility (the riskiness) of the assets. Since both SDEs are driven by the same uncertainty, by a linear combination of them we can create an instantaneously riskless portfolio, Π (i.e., consisting of σ B amount of A). B Π= ( σ BB) A ( σ A ) A B σ pieces of asset B and The change in the portfolio value can be written as the linear combination of the changes in asset prices. By substituting (2.) into (2.2): dπ= σ BdA σ AdB (2.2) B A ( µ σ µ σ ) dπ= AB dt A B B A Since the portfolio is riskless, as a consequence of no arbitrage possibilities it has to earn the risk free rate. dπ = f r dt = Π ( µ Aσ B µ Bσ A) ( σ σ ) B A dt A A 6

8 r σ r σ = µ σ µ σ f f B A A B B A r σ f µ A µ B B f r = λ (2.3) σ The numerators in (2.3) are the expected returns over the risk free return, usually referred to as excess expected return. The before derived analysis shows that the quotients of the excess return, and the volatility for any asset driven by the same uncertainty are equal. The quotient λ is usually called the market price of risk. B As mentioned before, in the real world investors expect a higher return/excess return on a more volatile asset. In brief, we derived a theoretical amount λ, that is able to express the before mentioned context in one number. Moreover, this number is the same for every asset, driven by the same uncertainty, W (Wiener process). Once we dispose by the market price of risk, by multiplying it with the volatility of the asset, we can calculate the difference between the return of the asset and the risk free rate: r = (2.4) f µ λσ Equation (2.4) helps to be able to switch between the real, and the risk neutral world Real versus risk-neutral world Once being familiar with the concept of the market price of risk, the transition between the real and the risk neutral world becomes easy to manage. Assuming an asset A, whose price follows a stochastic process under the real world measure depicts as follows: Here da = µ A dt+ σ A dw t A t A t t µ A is the drift coefficient, while σ A is the volatility parameter. dw is the change in the value of the Wiener process, the uncertainty factor in the model. According to a before mentioned statement, under the risk neutral measure the asset has to have a return equal to the risk free rate. Using (2.4) one can write: f ( λσ ) da = r + A dt+ σ A dw (2.5) t A t A t t f da = r A dt+ σ A dw t (2.6) t t A t Here, dw is a Wiener process under the risk neutral measure, corresponding to dw under the real measure. t dw = λdt+ dw (2.7) Since we possess the knowledge how to transform processes from the real world to the risk neutral and back, we can revert to one factor no-arbitrage interest rate models and the determination of bond prices. t 7

9 2.4. Introduction to one factor no-arbitrage models Despite the severity of one factor no-arbitrage models (see section ), most of them can be described in a common framework. A general Ito-process for the one factor short rate models can be written as: (, ) (, ) dr= m r t dt+ s r t dw (2.8) Here, r is the instantaneous short rate, and both the mean m, and the standard deviation s of the process can be a function of the level of r, and time t. Depending on how m() and s() are defined, we can obtain one of the models mentioned in section. This paper will focus on two short rate no-arbitrage models, namely the Vasicek, and the Cox-Ingersoll-Ross (CIR). Both of them have a mean reverting process included in the drift term, and the feature that none of the parameters either in the drift term, or the volatility is time dependent. These two latter are essential when one would like to estimate the coefficients of the models on historical data Vasicek model In the Vasicek model the risk neutral process for r can be written as follows: ( ) t = ɶ 4 t +σ t dr a b r dt dw To draw a comparison with the general form in equation (2.8), m( r, t) = a( b rt ) s( r, t) (2.9) ɶ and = σ. b ɶ stands for the long-run mean, while a shows how fast the mean reverting property of the process r is. Although from historical data, only the real world evolution of the process r is known, from equation (2.5)-(2.7) we are already familiar with how to transfer parameters across the real world and the risk neutral measure. Analogously to (2.5)-(2.6) one can obtain the following process of the Vasicek model under the real world measure: ( ) dr = a b r dt+ σ dw (2.0) t t t Notice that, the parameters a and σ are considered to be equal in (2.9) and (2.0). The comparison between b ɶ and b can be written as: 4 indicates a value in the risk neutral world. 8

10 bɶ σλ = b (2.) a Assume, that the model has the same features also in discrete time. dt will stand for a time step in the dataset. Once we have historical data on r, the coefficients of the above written equation (2.0) can be estimated by the following regression: t u dr = β + β r + u t 0 t t = σε ε N t t t ( 0,) (2.2) dr is the absolute change in the level of the short rate, dr = r+ r. Obviously, β = a, t t t while the constant term, β 0 will be equal to an aggregation of parameters, namely β 0 = ab. σ is simply assumed to be equal to the standard deviation of the error term, t 2 ( 0, ) u N σ. To estimate the coefficients β 0, β and σ, GMM was used 5. The reader might raise the question, still how will we determine the four parameters ( a, b, λ, σ ) out of three (,, ) β β σ. Naturally, it is needed to introduce a further 0 restriction. As a first step, the market price of risk will be set to zero, by assuming bɶ = b. Using this assumption the projection of the zero-coupon bond prices will be calculated. The most advantageous feature of the whole Vasicek framework is the simplicity of the calculation of zero-coupon bond prices after knowing the parameters a, b λ and σ. There exists an explicit formula for doing so. Let P( t, T ) represent the zero-coupon bond price at time t, maturing at T. Where ( t) ( ) ( ) (, ) t P t, T = A t, T e B t T r (2.3) a T e B( t, T) = a A( t, T) = exp 2 a 2 ( B( t, T) T+ t 2 )( a b aσλ σ 2) σ B( t, T) 2 2 4a 6 (2.4) By adjusting the market price of risk, we will try to converge with the projection to the last estimated spot curve (27 th July 2007). The difference between them is measured as the sum of the square distances at each corresponding point on the entire horizon (0-360 month). Basically the following minimization problem has to be solved: 5 To have an insight in why the given method is adequate, and how the process works in details, see section 2.7 and Appendix A. 6 For further details on the derivation of the formulas see [Vasicek (977)], or [F. Jamshidia (989)]. 9

11 T ( r( t" last" T) P( t" last" T) ) min,, (2.5) λ T= Here, r( t T ) is the spot curve on the 27 th July 2007, P( t ) " last", " last", T is the zero coupon bond price calculated by formulas (2.3) and (2.4) for the same date. T indicates the time to maturity in months. Equations (2.3) and (2.4) are derived on continuously compounded interest rates. Since the arising difference between using annually versus continuously compounded interest rates is negligible regarding other weaknesses of the framework (i.e. using 3 month spot rates instead of instantaneous ones, or continuous framework vs. discrete data), we can dispense with modifications. The results are used in a particular simulation method. This procedure will be introduced in chapter CIR model The CIR model is extremely similar to the already introduced Vasicek framework. The only difference is that the variance of the process is dependent on the level of the short rate. It will turn out to be an essential freedom of the model in terms of fitting the actual yields of empirical data. To put this in another way, in the process describing SDE the change of the Wiener process will be multiplied by the square root of r t, beside σ. The instantaneous interest rate r, under the risk neutral measure will look like as follows: dr = a bɶ r dt+σ r dw (2.6) ( ) t t t t The comparison with (2.8) can be written as m( r, t) = a( b rt ) ɶ, and s ( r t ), σ rt =. For similar reasons as it was mentioned in the previous chapter, (2.6) has to be transferred to represent the evolution of r t under the real-world measure. ( ) dr = a b r dt+ σ r dw (2.7) t t t t To compare b ɶ in (2.6) and b in (2.7), the formula (2.) is still applicable. The corresponding regression is similar to the one in equation (2.2), therefore the estimation procedure is also similar. The only difference is the assumption on the standard deviation of u t. Under the CIR framework the regression equation works out as follows: dr = β + β r + u t 0 t t u = σ rε ε N t t t t ( 0,) (2.8) 0

12 It means that the volatility is proportional to the actual yield level. β 0, β and σ will be estimated by GMM. The corresponding moment restrictions are introduced in chapter 2.7. Also the way, how the predicted bond prices and the actual one will converge remains the same, namely by adjusting the parameter λ, the market price of risk. However, one has to notice, that in this way the values of the estimated market price of risk in the Vasicek and the CIR framework are not directly comparable 7. On the other hand, the formulas to calculate the price of the zero coupon bonds become different. Fortunately, there still exists an analytical formula, but by involving a further term r t in the model the expression becomes more complex. (, ) B t T (, ) A t T = ( ) ( ) (, ) t P t, T = A t, T e B t T r (2.9) 2 2 a + 2σ ( T t) ( e ) 2 ( ) 2 2 a + 2σ ( T t) ( σ ) a a e + 2 a + 2σ 2 2 ( a+ a + σ )( T t) a + 2σ e = a + 2σ ( T t) 2 2 ( a 2σ a) ( e ) a + 2σ 2ab σλ σ 2 (2.20) The deduction of the formulas are not presented. For further details see Cox, Ingersoll Ross (985). It is easy to recognize that, the zero-coupon bond prices depend only on the instantaneous short rate. The value of r t determines the whole term structure of spot yields at time t. 7 Altering (2.5) to correspond with the model of Vasicek it looks like: f da = r + λσ A dt+ σ dw, ( ) Applying the same derivation, (2.7) would become: t A t A t t = λ + t t dw A dt dw Following the same logic in the CIR framework: f da = r + λσ A dt+ σ A dw ( ) t A t A t t dw t = λ At dt+ dwt As a result the estimated market prices of risk in the different frameworks will be comparable according to the following: λ λ r, where r stands for the mean of r t t -s. Vasicek CIR t

13 2.7. Estimation by GMM The Generalized Method of Moment is a universal estimation technique. Its main advantage is that it does not require the distribution of the dependent variable to be normal. The asymptotic justification for the GMM procedure requires only that the distribution of the interest rate changes is stationary, ergodic, and that the relevant expectations exist [Chan, Karolyi, Longstaff, Sanders (992)]. The relaxation of the normality assumption is essential when dealing with data of short rate changes. As another benefit of GMM, one can mention that the estimators and their standard errors are consistent, even if the disturbances, T heteroskedastic. Define the parameter vector Θ as [ β, β, σ] ε t -s are conditionally Θ = 0, where 0 β, β and σ are the values from equation (2.2), and (2.8), for the Vasicek, and the CIR model, respectively. Denote the empirical residuals of (2.2), and (2.8) by u t. The moment restrictions for estimating both the Vasicek, and the CIR model are similar to the ones, used by Chan, Karolyi, Longstaff and Sanders (992). In case of the Vasicek model: [ t] [ ] E u E u r E u t t = 0 = 0 σ = t 2 2 ( t σ ) E u r t = 0 The CIR moment conditions are similar, only the volatility structure has to be adjusted: [ t] [ ] E u E u r E u t t = 0 = t ( σ r t) 2 2 ( ( σ ) ) = 0 E ut rt r t = 0 (2.2) (2.22) We have 4 moment conditions to estimate 3 parameters in each case. Therefore the systems might be overidentified. Appendix A introduces, how the GMM works once having more moment conditions than parameters, and how to test the overidentification. For further details see Greene (2003). 2

14 2.8. Generating future r t -s The aim is to determine, what would be the highest loss on a given portfolio in one year with 99.5% probability, as a consequence of yield curve movements. The procedure starts with generating one year ahead interest rates. If the dataset has τ observations a year, it means that knowing the value of the short rate today r t, we project the τ steps ahead instantaneous yield, rt + τ. Hereafter, the method in case of the CIR framework will be introduced, the Vasicek is a simplified version of the preceding. Once being familiar with the necessary parameters, namely a, b, and σ, the model itself, (2.7) can be used to generate τ steps (= year) ahead short rates under the real world measure. Obviously, first a discrete Wiener process has to be simulated, then the change between each discrete time step can be used for forecasting a period ahead. The accumulation of the movements will show the change of the yield on a year horizon. Assume, the vector of the generated Wiener process is as follows: [,,..., ] T dw dwt dwt + dwt + τ =. Afterwards, the evolution of r t year ahead ( t+ ) can be calculated by a recursive approach as it follows, where the discrete time horizon: ( ) ( ) dr = a b r dt+ σ r dw t t t t dr = a b r dt+ σ r dw t+ t+ t+ t+ ( ) dr = a b r dt+ σ r dw t+ τ t+ τ t+ τ t+ τ r τ year dt= symbolize one step on τ (2.23) Here, t is the last observable date, r = t dr + + t r and t rt + τ is the value in one year. Note, that we used the real world parameters for projection. Therefore, as a result the real world evolution process of the short rate is depicted. As already known, the spot yields can be calculated by the formula (2.9) naturally in line with (2.20). Apparently, beside the real world parameters a, b and σ, the market price of risk also shows up in (2.20), but λ is already known from (2.5) and assumed to stay unchanged on the forecast horizon. ( ) ( ) ( τ, ) t,, B t + T r+ τ P t+ τ T = A t+ τ T e (2.24) The procedure can be repeated several times. In our case it would be 0000, meaning that for the upcoming year 0000 different scenarios are generated. Each individual case can be used to evaluate a given portfolio and to determine the present value of the cash flows PV + τ ). The results are compared to the value when the last observed spot rates ( ( t, T ) 3

15 (Base Curve, BC) are used. The possible loss is the difference in the present value of the portfolio according to the BC and the previously obtained yield r t+ τ. PV + PV (2.25) ( t τ, T ) BC Applying (2.25) for the 0000 different PV( + τ, ), a loss/gain distribution can be depicted. The 99.5 th percentile of the distribution shows the amount of loss we are looking for. t T 3. Multi-factor approach 3.. The HJM framework Heath, Jarrow and Morton [Heath, Jarrow, Morton (992)] developed a framework for multi-factor interest rate modeling. The importance of the HJM theory lies in the fact, that virtually any interest-rate model can be derived within such a framework 8. According to their approach the instantaneous forward-rate f ( t, T ) evolves as the following diffusion process: ( ) α( ) σ( ) df t, T = t, T dt+ t, T dwt (3.) f ( t, T ) is the instantaneous forward rate at time t, meaning that the forward interest rate from time T to time T+δt, where δt tends to zero. ( t, T) α is an F dimensional row vector, and can be interpreted as a drift term, the function of both t and T. σ is also an F ( 2 F ) dimensional vector σ σ ( t, T), σ ( t, T) σ ( t, T) =, standing for the volatilities corresponding to the different underlying uncertainty expressed by W, which is an F dimensional Brownian motion W ( W W W ) Wiener process at time t, in a diagonal form. In that case, the product of ( ) =, 2 F. Let t dwt, dwt,2 dw t = 0 dw t, F σ t T dw will look like as follows:, t (, ) = (, ), (, ),..., (, ) dw denote the changes in the (, 2,2, ) σ t T dw σ t T dw σ t T dw σ t T dw t t t F t F (3.2) Heath, Jarrow and Morton showed, that to apply the above introduced framework in an arbitrage-free context, the function α cannot be arbitrarily chosen, but it must be a 8 Citation from [Brigo, Mercurio 200] 4

16 function of the vector volatility, σ ( t, T) relationship between α and σ in (3.) is as it follows.. In particular, under the risk neutral measure the T F T i (3.3) t i= t (, ) (, ) (, ) (, ) (, ) α t T σ t T σ t s ds σ t T σ t s ds = = Although the approach is able to fit exactly with any type of empirical yield curve, the application of such a complex model has shortcomings in practice. The first, and the most important is that the instantaneous forward rates that the model treats, are not observable in the real world. Furthermore, the estimation of such a system on empirical data requires further assumptions on the parameters, and fancy estimation techniques. From the view of this paper, the above drawn HJM framework has two questionable aspects. First, as mentioned, the instantaneous forward rates can only be calculated by using the government rates, which are originally observable on the market. Moreover, because of the lack of many tenor rates, the researcher should apply further assumptions on the shape of the forward curve between two observable points on the yield curve. Therefore, to keep the number of assumptions and transformations of yields as low as possible, we seek for a model, where zero coupon yields can be used as an input. In that case also the output will be the zero yields. This is important, when comparing the outputs of the different approaches investigated in this paper. Second, since ex ante we do not postulate stressed security values to be arbitrage free, to determine shocked values of interest rates on empirical basis, the usage of the criteria of no arbitrage world is not necessary. The model is calibrated on historical data, thus it might forecast stressed curves, which are not free of arbitrage. Nevertheless, we do not question the validity of history, even if arbitrage opportunities occurred. Therefore, we do not ignore models and projections with arbitrage possibilities. Taking into account the previous statement to achieve the goal of this paper, based on the HJM general framework, a simpler model can be built, estimated, and applied to forecast worst-case movements The volatility structure The model presented in this chapter, postulates the idea that there exists a common factor across the movements of the interest rates at the different tenor dates. Literally, if for example the 3 month rate increases due to an unexpected event, it has an impact on all the other maturities to some extent. It is likely, that closer the maturities are, the larger the correlation between the interest rate movements. Therefore, it is assumed that the effect is the highest on the close maturities (in this case the 6 month rate), and diminishes as we investigate further standing points ( year, 3 years, 5 years etc.) of the yield curve. As 5

17 another example, the movement of the n year rate has the highest impact on the n- and the n+ years point, and affects the rest of the curve to a lower extent. To ease the tractability of such a model, the above mentioned features of the yield curve will be captured by one number, κ. A plausible and a simple specification of the covariance structure, which takes into account the above mentioned criteria s, can be drawn as: i j Ω ij = κ 9 i,j=,2 n, the number of the different tenors, and κ <. Ω ij is one element of the covariance matrix Ω, in row i, and column j. In this case, the construction of the covariance matrix takes the following shape: 2 n κ κ κ ω ωn κ κ 2 Ω= = κ κ (3.4) ωij κ ωn ωnn n n 2 κ κ κ Moreover, the volatility structure will also include the feature of level dependent volatility, similarly as it is the case in the CIR model. As an expansion, the volatility of the zero rate of a given maturity, besides its own level, will be allowed to be dependent also on the level of the zero rates of different maturities. Instead of using simply the square root method (as in case of the CIR approach), each tenor rate ( Z, Z2... Zn ) will have an own so called variance scale up factor ( λ, λ2... λn ). λ measures the extent of the contribution of every different maturity to the volatility of the other maturities of the yield curve. For example, the volatility of tenor will be a function of λ 2 Z, λ Zλ2 Z2, λ Zλ3 Z3,..., λ Z λn Z n. As a summary the entire covariance structure is assumed to take the shape of the following formula: ( λ Z) ( λ Z) here ( λ Z) diag( λ Z, λ2 Z2,..., λn Zn) =, a diagonal matrix. Ω (3.5) We can outline a comparison between the volatility structure in (3.5) and the general HJM framework presented in (3.). The former, deals with zero rates maturing at n 9 [Kamizono, Kariya (996)] 6

18 different times. In fact, it models n different zero rates at the same time. If we consider (3.) to be the stochastic differential equation of n different rates, namely the T T T, ( t, T) instantaneous forward rates at time, 2,... n dimensional matrix, and form the following shape: σ ( t, T) σ in (3.) would become an n F ( t, T) ( t, T ) F( t, TF) ( t, T) ( t, T ) ( t, T ) σ σ2 2 σ σ σ σ σ F F = ( t, T) σ ( t, T ) n nf F (3.6) Here, the rows describe the volatility structure of the corresponding instantaneous forward rate. If we additionally assume that the number of the underlying factors is equal to the number of different forward rates modeled, so to say that ( t, T) σ is an n n dimensional matrix, one can make the comparison between the volatility structures drawn in (3.5) and (3.6): ( t, T) = ( Z) Ω ( Z) σ λ λ As an example, the volatility structure belonging to the first modeled maturity depicts as: 2 ( (, ), (, ),..., n(, n) ) σ t T σ t T σ t T = λ Z + κλ λ Z Z κ λλ Z Z κ λλ Z Z 2 n 3 3 n Once being multiplied by the driving uncertainty, the Wiener processes mentioned in (3.2), the final covariance structure looks like as follows: ( t T) ( λ, λ... 2 λ κ), ( ) ( ) ( ), t t n σ t T dw = λ Z Ω λ Z dw (3.7) 2 (, ) σ t T dw = λ Z dw + κλλ Z Z dw + t t, 2 2 t,2 + κ λλ Z Z dw κ λλ Z Z dw 2 n 3 3 t,3 n n t, n σ represents the first row of (3.6). Here, the unknown parameters are θ =,. κ will be estimated directly from the correlation matrix of residuals, n obtained from univariate AR equations, while ( λ, λ2... λn ) will be estimated by GMM in a system of equations Univariate AR modeling As mentioned, the applied model does not have to suit the no arbitrage condition as presented in (3.3). Therefore, the drift term, α ( t,t) in (3.) can take various shapes. One 7

19 plausible approach is to suppose, that the historical zero rate changes 0 at each tenor follow a univariate autoregressive process with different lags. The general p-order autoregressive process looks like as follows: p Z = c + ψ Z + ε (3.8) t, i i k, i t k, i t, i k= Subscript i stands for the different maturities, while t is the period in time. c is the constant, ψ i is the vector of the autoregressive coefficients. ψ k, i belongs to the k th lagged value at maturity i. ε i is the vector of residuals, theoretically coming from an iid. sample, if the fitted AR process matches exactly with the underlying process. To fit the above mentioned structure in the HJM framework introduced in (3.), the dimensions of ( t, T) α have to be adjusted, to shape a matrix form. Each row, i=,2, n corresponds to a different maturity. The columns represent the different factors a given maturity is dependent on. Since every tenor is only the function of the constant and its own lagged values up to the order of p, the dimensions of ( t, T) ( p ) n +. c c ψ ψ p ψ 2 2 Ψ= α( t, T) = ψ ik α will be equal to (3.9) c ψ n ψ n np The reader might recognize, that the drift structure Ψ could be even more generalized, allowing for mutual dependence of one tenor on the lagged values of the other. However, this feature is assumed to be captured already in the volatility structure, σ ( t, T) residuals in (3.8), ε -s, are the corresponding outcomes of the uncertainty; ( ) σ t T dw, t. The = ε (3.0) 3.4. Merging the volatility structure and the AR together Merging (3.7), (3.8), (3.9), (3.0) and (3.) all together, the final model can be constructed according to the next formula. 0 Since the level itself, i Z is not a stationary process (it has a unit root), it cannot be econometrically modeled. Therefore, similarly to the HJM framework in (3.), the change of the zero rate is used. 8

20 Z ψ Z t, t, i Z t,2 ψ 2 Z t 2, i t, n n Z t p, i ( λ ) ( λ ) = + Zt Ω Zt dwt Z ψ (3.) Note, that ψ, ψ n are the corresponding rows of matrix Ψ in (3.9). This notation had to be introduced, because each row has to be multiplied by the corresponding lagged values of maturity i. The estimation procedure of the unknown parameters will be carried out in two steps Estimating kappa First, the individual AR processes will be estimated according to (3.8). This step is needed to obtain the vectors of the residuals ε i -s, since the estimation of the common factor, κ will be designed in such a way, that the sum of the squared differences within Ω and the correlation matrix of εi -s is minimal. κ is known by solving the following minimization problem. n n min ε l= m= 2 ( C( ) Ω ) C ( ε) is the correlation matrix of the residuals εi -s, obtained from the individual univariate AR modeling in (3.8). Ω is simply the hypothetical correlation matrix, as it was configured in (3.4). lm lm 3.6. Estimating the system of equations in (3.) As mentioned already, (3.) is estimated by applying GMM. Only those autoregressive coefficients are included in the system, which are significant when estimating univariate autoregressive processes in section 3.3. A plausible way to estimate κ would be to simply include it in the GMM estimation of (3.) as a variable. To estimate the system of equations Eviews was applied. Once treating κ as variable in the system, the software sends an error message, meaning that the moment restrictions are too close to each other, and the system cannot be estimated. This means that the regression model has high collinearity, therefore effectively there are more parameters than independent moment equations. That is the reason why alternative method has to be invented. 9

21 A general approach is to use moment conditions on the expected value of the error term, the error term explanatory variables and some additional ones, used by Chan, Karolyi, Longstaff and Sanders (992) when testing short rate models in application. To have a better understanding, which moment restrictions were used, a simple notation will be applied first. Let ν denote the error term, and X the explanatory set of variables in a regression. ϑ is the assumed volatility structure of the residuals. To suit this notation with the one in (3.), the following comparisons can be made: Z ψ Z t, t, i Z t,2 ψ 2 Z t 2, i ν = Z t, n ψ n Z t p, i (,..., t t p) X = Z Z ( ) ( λ ) ϑ= λ Z Ω Z dw t t t Then the moment restrictions, used in (3.) can be outlined as it follows: [ ν] = 0 [ X] = 0 E n moment restrictions Eν h for the significant AR parameters These are the moment restrictions used in a least square estimation. 2 2 [ ν ϑ ] = 0 2 [( ϑ ) ] 2 X = 0 ( ν 2 ϑ 2 ) X 2 E n E ν h E = 0 h The first two were proposed by Chan, Karolyi, Longstaff and Sanders (992), and an additional which incorporates the second power of the explanatory variables is included. They all intend to fit the predetermined volatility structure with the residuals. All in all 2n+3h moment restrictions were mentioned to estimate 2n+h unknown parameters. The number of over identifying restrictions is 2h. The system similarly to what we could see when estimating the Vasicek, and the CIR, might be overidentified. For further details see Appendix A How to use the estimates for forecasts As in section 2.8, one year ahead forecasts of the yield curve will be generated by simulation. Instead of one, as in the chapter before, now we have n underlying 20

22 uncertainty factors driving the system. It means that n different Wiener processes have to year be simulated. Again considering τ time steps a year ( dt= ), the dimensions of the τ generated Wiener process will be τ n. dwt, dwt,2 dwt, n dwt, dw + = dwt+ τ, dwt+ τ,2 dw t+ τ, n The evolution of the zero rates, Zt, i -s one year ahead will be calculated by using a recursive approach. Applying (3.) every time one step ahead, the zero-coupon rate in one year can be obtained. Z = Z + Z Here t+ t t Z = Z + Z t+ τ t+ τ t+ τ (3.2) Zt is calculated by equation (3.), where the applied uncertainty at a given time is the corresponding row of the matrix (3.2). t is the last observable date and Z t+ τ is the value of the zero rate in one year. The procedure is repeated 0000 times, to have enough number of results to be able to determine a 99.5% with a reliable accuracy. By applying this method, zero coupon rates will be disposable only for the maturities, which were used as an input to the model. Therefore, for the evaluation of a given portfolio, which has cashflows at any date in time up until 30 years, the interpolation and the extrapolation of the zero coupon curve has to be done for each outcome. For interpolation the cubic spline method was used, while for extrapolation the exponentially decaying discount factor model was applied 2. The present value of a given portfolio can be calculated according to 0000 different forecasts of the yield curve. The results, as in the previous chapter, are compared to the value obtained by using the last observed zero coupon curve (Base Curve, BC), see formula (2.25). The possible loss is similarly the difference between the present value today and the present value according to the 0000 different forecasts. 2 For further detail see technical appendix in section Appendix B. 2

23 4. PCA approach 4.. The common factors Spot rates are determined by expectations of market participants about the future shortterm interest rates. These in turn are determined by their expectations about the future path of the economy: output, prices, and money supply. Assume that r 0 is the current short-term nominal interest rate, r t is the currently expected future short-term interest rate at time t, and r is the long-term expected future interest rate. Then, assuming that present interest rate expectation is a function of r 0 and r, it can be written as: ( ) r r r r e δ t t = + 0 (4.) In the given framework, a parallel shift occurs when both short term and long term expectations change at once by the same amount. A slope shift occurs, when the shortterm expectation changes but the long-term one stays, or vice versa. The third type movement, the curvature shift, has a variety of different reasons. A change in the mid yield can be a result of a change in the expectation of the market volatility, or a change in the term premium for the interest rate risk, or due to market segmentation caused by temporary supply/demand imbalances at specific maturities [Phoa (2000)]. Principal Component Analysis (PCA) is a statistical method, which is able to explore the most common movements of the yield curve, mentioned in section. It needs historical yield curve movements for several maturities as input. Basically, it is a dimension reduction process. Initially, one can identify as many different risk factors of the yield curve as the number of different maturity points that are observable. The key thought, when using PCA is that these different points of the yield curve do not vary separately. There exist some mutual driving factors (the before mentioned three common movements). These factors can describe most of the variation of the curve. It has been empirically tested, [Phoa (2000)] and [Soto (2004)], that usually the parallel shift on its own explains 90% of the yield curve variation. Including the second and the third components, the twist and the inflection factor, the explanatory power reaches 95% or more. Instead of having to deal with all the observable maturity points, one has only three different risk factors of the yield curve. A further benefit of such a technique is, that the obtained risk factors are orthogonal, which means that they are uncorrelated with each other (they have zero correlation between them). It makes the creation of worst-case 22

24 scenarios a lot easier. One can now derive shock events on each of the driving factors separately, and simply sum the results. No correlation issue has to be considered (for detailed explanation, see the following section Mathematical background As it was mentioned before, PCA needs historical data of the yield curve movements as an input. Since this method is exceptionally sensitive on the quality of the input, the downloaded data for different maturities has to be adjusted to mutually accord. For detailed description on data cleaning, see section 5. First we calculate the relative changes, needed as input for the covariance matrix as follows: Z ln Zt, n = ln Z t+, n t, n. (4.2) The frequency, τ of the data has to be chosen according to a tradeoff between two measures. On one hand relative changes had to be independent, on the other hand in order to obtain robust results, as many observations are needed as possible. Weekly, τ = 52 (each Friday) frequency seemed to be decent, since autocorrelation was found 3 up to eight lags only. The next step is to calculate the variance covariance matrix of the relative changes. ( Z Z ) ( Z Z ) cov ln, ln cov ln, ln n C= cov( ln Zn, ln Z) cov( ln Zn, ln Zn) Here,, 2 n are the different maturities. (4.3) It is very likely that the above written C matrix would indicate significant co-movements of the different yields. Basically to come up with a shocked yield curve all the comovements of the different maturities would have to be considered. In order to simplify the process of drawing the worst-case term structure, PCA is used to reduce the dimension of the above-mentioned matrix C. A great extent of co-variation is desired to be captured in three factors. 3 Detailed description of correction for autocorrelation can be found in section

25 Consider the relative change of the whole zero curve at a given date t, as one data point. It has n different maturity points, so basically it can be fully determined by a vector in n R. Each of the n coordinates will correspond to the original n axes, which are the different yields themselves. The goal is to reduce the dimension of this hyperspace. Eventually, by reducing the dimensions, the movements will not be able to be fully described, but only with a decent accuracy, which will be called the threshold level. First, one has to find an axis, e in the hyperspace along which most of the variation of the co-movements of the relative changes can be drawn. Theoretically, that axis will describe the level movements. The elements of vector e will correspond to the direction of the axis in the hyperspace, as a linear combination of the original axes. Since a normalized vector is needed, the maximization problem can be written as: λ = = (4.4) max e ' Ce s.t. e ' e e λ is the variance of the observations along axis e. With further maximizations similar to formula (4.4) e2, e3... e n can be determined. These will be the axis along which the second, the third most variation is estimated, theoretically the slope, the curvature movements respectively. An important feature of these axes is, that they are mutually orthogonal. The elements of e 2 and e 3 will still describe the position of the new axes based on the original ones. λ = max e ' Ce s.t. e ' e =, e e 2 3 e2 e λ = max e ' Ce s.t. e ' e =, e, e, e orthogonal λ = max e ' Ce s.t. e ' e =, e n en n n n n i e i j j (4.5) The following method describes how to obtain the desired vectors of the linear combinations e, e2, e 3, and the variances of the different types of movements λ, λ2, λ 3. Since C is a square matrix, one can calculate the eigenvalues and the eigenvectors in such a way that: C e = λ e (4.6) l Here, l=,2, n, e l is a particular eigenvector, and λ l is the corresponding eigenvalue. Note, that that: l λl -s and el -s are in accordance with the maximization equations. It follows ( λ ) 0 l C I e l = (4.7) 24