An efficient method for simulating interest rate curves

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1 BANCO DE MÉXICO An efficient method for simulating interest rate curves Javier Márquez Diez-Canedo Carlos E. Nogués Nivón Viviana Vélez Grajales February 2003

2 Summary The purpose of this paper is to present a methodology for simulating temporal interest rate structures which preserve curve shape, using a minimum number of parameters based on historical data. For this methodology we use Nelson and Siegel s parsimonious modeling, which is able to capture commonly observed multiple yield curve shapes using only four parameters. The process only requires a random sampling of occurrences of the values of these four parameters, and any other point in the curve is obtained directly using Nelson and Siegel s formula. 1 Historical series of parameters generated by these structures are obtained along with their combined probability distribution. Random series of parameters can thus be generated, which behave according to the historical distribution and interest rate curve structures for the corresponding maturities. I. Introduction The market risk of a portfolio of financial assets is estimated according to adjustments in the value of the portfolio originated by changes in risk factors. Regarding debt instruments, the most important risk factor is interest rates. Interest rates are also a risk factor when valuing other instruments like options, futures, and forwards. As for bonds, the valuation method commonly used consists of discounting future flows to the present value. In order to do so, both the flows and their payment dates must be known and the applicable discount factors as well. While flows and dates on which they are generated are specified in a contract, discount factors are the result of a market consensus in relation to the temporal structure of future interest rates applicable to the bond s valuation. Since this structure is uncertain, bond portfolio risk analysis often resorts to the Monte Carlo simulation in order to generate multiple temporal rate structures and obtain the risk profile of the portfolio and its VaR. 2 1 Nelson, C. R. and A. F. Siegel. Parsimonious Modeling and Yield Curves, Journal of Business 60 (October 1987), pp

3 This simulation is usually performed by previously setting the number of points (maturities) on the curve and undertaking a point-to-point simulation. Besides the potential difficulty involved and the fact that it can lead to inconsistent results, it also involves a complicated code and computational over-exertion. This paper examines the use of the Nelson and Siegel s yield curve generation model using a random sample of only four parameters, far fewer than the points on the curve that need to be simulated. 3 II. Spot and forward rates 4 In this section we review algebraic expressions that relate spot rates with forward rates. Since handling compound annual yields using algebraic expressions is more complicated, compound rates will be handled throughout this section. Studies of interest rates with different maturities are normally based on information from debt instruments or bonds. Theoretically speaking, in a complete and tax-free market, the price of a bond is the present value of the coupons paid plus the present value of the principal. Discount rates for different maturities used in the calculation correspond to a spot rate temporal structure; in other words, the yield structure at a given date t is represented by a graph of the spot rate for each maturity date. 5 2 Value at Risk. 3 Around thirty. 4 Svensson, Lars E. O., Estimating and Interpreting Forward Interest Rates: Sweden , Working Paper no. 4871, National Bureau of Economic Research, Spot rates at a given maturity correspond to the interest rate paid on a zero-coupon bond with that maturity. 2

4 Let r (t,t ) be the continuous compound interest rate of a zero-coupon bond bond negotiated at time t and maturing ont > t. L et m = T t be the time it takes to mature. The price at t of this zero-coupon bond with principal equal to 1 peso will denote the discount function d(t,t ). The discount function and the spot rate are therefore related as follows: r (t,t )(T t ) d(t,t ) = P(t,T ) = e Consider a bond with a nominal value of 100 pesos paying annual coupons of c pesos and maturing within m years. The present value at time t of the coupon payment that took place in year k, k=1,2,.., m, will be c d (t, t + k ), and the present value of the principal payable in year m will be 100 d (t, t + m). The price of the bond on the transaction date will therefore be: m P(t,t + m) = cd(t,t + k ) + 100d(t,t + m) k =1 Let f (t,t ',T ) be the forward rate continuously comprised of a forward contract negotiated at time t and whose underlying transaction begins at time t ' and ends at T where T > t ' > t. 3

5 r (t,t ) = u =t T t The forward rate thus relates to the spot rate as follows: f (t,t ',T ) = (T t )r (t,t ) (t ' t )r (t,t ' ) T t ' The instant forward rate is the forward rate of a contract maturing at infinity and is defined as follows: f (t,t ' ) = lim f (t,t ',T) T t ' The forward rate f (t,t ',T ) is the average of instant forward rates: T f (t,u)du f (t,t ',T ) = u =t ' T t ' Similarly, the spot rate r (t,t) ) at time T and maturity T is the average of instant forward rates: T f (t,u)du III. Nelson and Siegel s model Nelson and Siegel developed an adjustment model for interest rate temporal structures, flexible enough to represent the different shapes these curves generally adopt: Monotonous Humped S-shaped 4

6 Nelson and Siegel s parametric model assumes that the instantaneous forward rate is the solution of a second order differential equation with two equal roots. Thus, following the same notation as in the previous section, assuming that the negotiation of a forward contract is t = 0, and that the underlying transaction begins on day m, the function of the instantaneous forward rate is defined as follows: 5

7 where ( ) is the vector of parameters that determines the shape of the curve. As explained in the previous section, the spot rate r (0,m) is obtained by integrating the instantaneous forward rates from 0 to m and dividing them by m. This produces the spot rate function: [ ] To appreciate the flexibility of this model in adapting to the shapes that temporal rate structures commonly adopt, the function s coefficients are examined. Note how these coefficients denote the weight given to the short, medium, and long-term sections of the forward curve. First, when m tends towards infinity, the limit of r (m) is. The constant parameter therefore denotes the weight given to the long-term component. Similarly, denotes the weight given to the short-term component and denotes relative medium-term importance in the temporal rate structure. Forward curve components Long term 0.6 Short term (exp(-m)) 0.4 Medium term (m*exp(-m)) Maturity The chart above shows the different components of the forward curve. The longterm component is a constant that does not reach zero at the limit. The medium- 6

8 term component is associated with the only maturity that begins with zero (and thus it is not short term) and reaches zero (and thus it is not long term). The shortterm maturity is the only one that falls repeatedly to zero. At the same time, parameter " " determines the speed at which maturities that include it in the equation tend towards their limits. Therefore, with a small value of " ", curve approximations are better for short-term maturities than for long-term maturities. Similarly, with large values of " ", a better adjustment is achieved in the long term than in the short term. The chart below shows the feature of variable " ". The curve adjusted with =90 represents better the curve of the original data in the short term, unlike the curve adjusted with =150, which differs more from the real curve in the short term but adjusts better to long-term data. Note how if it were not for " ", the model is linear in other parameters Adjusted data and curves Original data Adjusted with tao = 90 Adjusted with tao = Maturity in days Figure 2 IV. Methodology for adjusting temporal rate structures As described in the corresponding paragraph, simple interest rates for different maturities observed at a given point in time must be converted to continuously compound rates. A yield structure corresponding to the Nelson and Siegel s model is applied to these original data using least squares. 7

9 Although one can simply try to estimate the parameters using a group of conventional non-linear least squares, the fact the model is linear in all parameters except for can be used with some degree of efficiency. The form used by the authors consists of iterating on the model s nonlinear parameter and successively solving linear regressions on the other parameters until convergence by maximizing R 2 is obtained. A one dimensional search procedure 6 is therefore used on the nonlinear section of the model, and efficiency of the estimation in the linear part is exploited. To facilitate this procedure, we chose the Golden Section method for the one dimensional search. 7 First, Nelson and Siegel s equation is noted as follows: [ ] where r (m) represents the continuously compounded rate at maturity m. The iterative process is as follows. Assuming there is data on spot rates, compounded at different maturities m1,m2,..., mn, and for a given value of parameter τ, parameters a, b and c are obtained as least squares solutions of the system of equations presented below in matrix form. Thus: ( ) [ ] [ ] ( [ ] ) 6 On a single variable. 7 Generally speaking, it is easier to undertake one-dimensional searches than to solve the problem all at once. This also results in the need to choose good starting points for the iterative process, which is easier using the method proposed. 8

10 ( ) [ ] [ ] ( [ ] ) Thus, the system of equations simply described by X = MC is: ( ) [ ] [ ] ( [ ] ) which corresponds to a multiple linear regression problem represented by wellknown normal equations ; in other words: C * = (M T M ) 1 M T X. Next, the vector of estimated rates Xˆ is calculated for the same maturities as for the original data using the values obtained; in other words: Xˆ = MC. The sum of the squares for error and the adjusted R 2 corresponding to the Nelson and Siegel model are subsequently calculated as follows: [( ) ( )] After setting parameters a, a, b and c, is optimized using the golden section method within an interval [u,v ].With the new value of parameter the process is repeated until convergence is established at R 2. The vector of parameters (τ, β_0, β_1, β_2) is therefore obtained, which represents the structure of 9

11 data-adjusted rates. The golden method section for finding a global maximum requires the function to be optimized in order for it to be unimodal and quasiconcave. 8 Otherwise, only convergence to a local optimum can be guaranteed. When this occurs, there are normally two local optimums at the limits of the interval, and convergence to one or the other will very likely depend on the starting point. Unfortunately, the function to optimize cannot be guaranteed, which in this case is R 2 adjusted to the Nelson and Siegel model, either unimodal or quasiconcave in relation to parameter. 9 For the four cases examined in this paper (Cetes, Udibonos, Libor, and T-bill curves) mixed concave/convex patterns were found, and a general way of choosing the variability interval to carry out the search was not found. For this reason, prior to optimizing to choose the function domain, target functions were charted at several equidistant points within a broad interval. Depending on the shape of the curves and the maturities of the original data, an initial interval was chosen where the optimum value of was located. Below are the charts of the type of functions produced by the research Adjusted R squared Tao Adjusted R squared Tao Figures 3 and 4 10

12 The good or poor performance of R 2 as a function of tao has nothing to do with the algorithm used to optimize. In the first two functions, R 2 versus are unimodal and quasiconcave and their global optimum is within the interval. Since they behave well, no other precaution is required. In the subsequent one, a function that is also quasi-concave can be observed, but since its rises monotonously in, it is not unimodal within the interval. The optimum would appear to be (which makes no practical sense). The interval in this case is thus truncated once R 2 has been accurately obtained, and the corresponding limit is chosen as the optimum value of. The interval should not be overly extended, since besides from it causing numerical stability problems, the model would adjust well in the long term but at the expense of a poor short-term adjustment. 0.5 Adjusted R squared Tao Figure 5 The chart below shows a case in which after the local optimum around = 50, the function decreases before rising indefinitely again, although it appears to never surpass the local maximum mentioned. In this case, the interval was narrowed in order for the search to converge to the aforementioned local optimum. 11

13 Adjusted R squared Tao Figure 6 Finally, curves that were not quasi-concave and with local optimums at the two interval limits were also found. In this case, the larger of the two is chosen (the one on the left). Adjusted R squared Tao Figure 7 Each case should be carefully analyzed in order to obtain the best parameter adjustment. For example, in the case of the Cetes curve, a golden section algorithm was run initially allowing to vary in interval (1,1000), but resulted in negatives; in other words, over the long-term, estimated yields had negative values while short-term yields were excessively high. After several tests, the problem was resolved by narrowing the interval to (10,364), as 364 is the longest maturity for 12

14 which historical data are available. On the basis of this observation, the decision was taken to run the model using a longer maturity, and data was obtained for each of the remaining instruments. V. Simulation of temporal structures In order to assess the technique and illustrate it, we present the results of adjustments to the temporal structures of Cetes, Udibonos, Libor, and T-bill yield curves. The four series of the corresponding parameters were built based on daily data from January 2001 to January A histogram of frequencies per parameter was obtained for the four series of parameters, and the variance-covariance matrix was calculated for the four parameters (,,, ) in that order, and then decomposed using the Cholesky A factorization such that = ΑΑ'. The histograms of the parameters were considered as empirical probability distributions. The simulation model consists simply of applying the following formula: In this equation, denotes the vector of the simulated parameters, is the vector of middle values of the parameters, A is the Cholesky factorization of the variance-covariance matrix of the parameters, and is a vector of random samples of the corresponding empirical distributions of such parameters

15 In order to use this simulation process, the empirical distribution of the parameters had to be standardized by deducting their mean at each interval and dividing it between the distribution s standard deviation in order to not duplicate the effects of multiplying the vector by the matrix using Cholesky s factorization. This process can be used to generate as many temporal structure curves as required. V.1. Examples of curve and parameter distribution adjustments Below are adjustments of the temporal structure on January 28, 2002 for Cetes data. The figures show the initial data based on which Nelson and Siegel s model was adjusted and the values the adjusted curve adopts for certain maturities. The original points are charted and the curve that is formed when substituting the optimum value of the parameters. Subsequently, the proportions of the curves generated by the simulators and of the original curves for which the model was adjusted are presented by grouping them according to the most commonly observed shapes, namely: normal, inverted and mixed. s of yield structures Figure Maturity in days 10 This is achieved by generating random numbers evenly distributed between zero and one (i.e. ) and associating them with the corresponding interval of the accumulated empirical distribution of each parameter using the Monte Carlo method. 13

16 The 1 structure is known as a normal structure in which the yield is an incremental function of the maturity; in other words, the greater the maturity, the higher the yield, as uncertainty increases over time. The 2 structure is known as a mixed structure; it increases up to a certain point during the maturity and then decreases in the long term. The last type of curve is known as an inverted structure because the yield decreases throughout the maturity indicating that shortterm rates are very high and investors expect them to decrease in the future to more normal levels. Subsequently, only the most relevant results obtained for other instruments whose temporal yield structures were adjusted are shown as well as the process used to make the adjustment and the outcome of the Monte Carlo simulation. V.2. Example: Cetes V.2.1. Curve adjustment In the case of the Cetes curve, a decision was made to work with maturities generally observed in the market usually corresponding to 28, 91, 182, and 364 days. The table on the left shows the Cetes yield data at the close of January 28, The optimum parameter vector for the curve at this date obtained using the technique described in paragraph IV is: (,,, ) = ( ) 11 Data obtained from the curve file of Valmer, the price supplier. The file shows both market yields and estimated supplier prices. Note that the estimates of the supplier in the case of long-term Cetes are very similar every day of the year. 14

17 Table 1 Cetes data for 28/01/2002 Maturity Simple yield Continuous yield Cetes adjustment Maturity Simple yield Continuous yield The box on the right and the graph below show the adjustment obtained for the Nelson and Siegel model both numerically and graphically. Simple yield Cetes Maturity in days Cetes data Cetes adj. Figure 9 V.2.2. Empirical distribution of parameters 15

18 From the series of 4 parameters, and the mean vector, and variance and co-variance matrix was obtained. In the case of Cetes, the following results were obtained: Matrix A resulting from Cholesky s factorization of is as follows: Histograms corresponding to each parameter were also carried out and normalized as described. The figures below show the histograms obtained for the Cetes yield curve parameters. V.2.3. Simulation process Below are graphs of both the proportions of the structures generated by the simulation and the original structures the model was adjusted to according to the yield structure type: Cetes Cetes Figures 10 and 11 16

19 Frequency Both graphs are very similar, 1 being the most frequently observed structure. There is also a graph showing only some of the structures the simulation generated. Cetes Cetes Figures 12 and 13 Shape of the original curves Shape of the curves generated by the simulation 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Curve type 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Curve type 17

20 V.3. Examples: Udibonos, Libor and T-bill V.3.1. Summary Below is a table summarizing Cholesky s Mean Vector and Variance-Co-variance matrixes, derived from the parameter estimates used for their respective structure simulations. The complete results of the adjustments, histograms, and simulation process can be found in the appendices. Table 2 Cetes Udibonos Mean Var-Covar

21 V.3.2. Obtaining data and estimating parameters Udibonos Data for Udibonos were obtained from the Valmer vector file. 12 This file contains the Udibonos yield observed in the market and the price supplier s estimated yield. Unlike the Valmer file from which Cetes data were obtained, in the case of Udibonos it is not known which data correspond to market transactions and which are Valmer estimates. For this reason, all maturities published were used. Maturities for daily data w were between 10 and 3700 days. 12 Valmer is a price provider. 19

22 The optimization algorithm was run allowing to vary at interval (10,3700). During the first six months of 2001, Udibonos yields for 15-day and longer maturities were published. In the second half of the year, the maturities published begin as of around 200 days. In other words, the model is adjusted using data as of 200 days and is used to estimate short-term yields. Libor Data for the Libor rate was obtained from Bloomberg. The published rates correspond to 28, 91,182, 273 and 365-day maturities. Running the algorithm and allowing vary within a very large interval resulted in negative values of to for estimated longterm yields while short-term yields took on very high values in relation to observed ones. After several tests, and based on the results found, the problem was solved by reducing the interval to (10,150). The optimum value of 100. was therefore generally below T-bill Data corresponding to T-bill yields were also obtained from Bloomberg. It is important to note that, in general terms, there is data for 91, 182, 365, 730, 1825, 3650 and days. There are very long-term yields and the optimum is high (the mean is 2857). This means that the curve adjusts better in the long term than in the short term. If adjustment is required, the short term may have to reduce the interval of parameter, or an optimization of parameters that ignores long-term data may be needed. In the beginning, the model was adjusted by letting vary in interval (1,1000), with a result in which the parameter always assumed an extreme value of 999; in other words, the interval was too short. By considering a greater interval of (500,6000), parameter generally found optimum values within the interval.

23 V.3.3. Results obtained Since the initial histograms of adjusted parameters resulted in very regular shapes, thought was given to adjusting a known distribution to the estimated parameters, by simplifying the Montecarlo Simulation process even further. However, this is not generally the case, as when you review the histograms for instruments like LIBOR and Udibonos, shown in figures 17 and 18 and further on, the same smooth performance of the histogram was not observed. For this reason, it was decided to simulate parameters directly from their empirical distribution. Udibonos Libor As can be observed, the most problematic parameters were, and ; in other words, those associated with the short-, mid- and long-term components of the forward curve. 21

24 VI. Conclusions Considering the aforementioned, as is usually the case in such empirical studies certain difficulties were encountered when estimating parameters which were solved by examining the data and by making thoughtful use of the non-linear least squares optimization algorithm. In order to validate the yield curve simulation methodology, in this paper it was only necessary to simulate 200 temporal structures to obtain distribution parameter convergence. For the purpose of calculating the VaR of a portfolio of financial debt instruments using the Monte Carlo simulation, as many as required should be simulated to guarantee that convergence is as accurate as possible. At present, with the typical portfolios of Mexican banks, this may imply simulating a number of curves greater than 10,000. This is why the Nelson and Siegel model turns out to be relevant, as it implies large savings since the curve is fully specified using four parameters. Perhaps the major flaw of the Nelson and Siegel model is that the adjustment may not be at all convincing in terms of certain maturities, especially on curves contemplating very long maturities. This is because, as seen in section three, depending on the value of parameter, the curve adjusts well in the short term or in the long term, but not both. An interesting hypothesis is to include a new term in the equation in order to solve the problem. Also, this term could provide the function with better concavity properties, thus facilitating optimization by paying a small price by increasing only one parameter. Finally, another pending issue is finding correlations between the parameters of curves associated with different instruments. This would complete the process, as although the dimension of the simulation process considerably increases, it would allow the simulation of congruent groups of curves in order to achieve a better representation of interest rate scenarios. However, this simply implies obtaining the co-variance matrix across parameters and the simulation model remains the same.

25 Appendix 1 Golden section method The Golden section search is an iterative method in which the interval for searching the optimum of a variable s function is successively narrowed. In the case of maximizing R 2 to adjust the Nelson and Siegel equation on, the search was performed until an interval less than or equal to 1 was found. Below we explain the optimization algorithm for a minimization by taking [10,m] as the starting interval, where m represents the longest maturity for which data for each of the temporal structures of the yields examined could be obtained. Let [u 1,v 1 ] be the initial interval on which the search for the optimum point will be performed. The subindexes indicate the number of iterations the search is found in. k denotes the iteration counter. Thus, in this initial iteration k=1. Let 1 = u 1 + (1 )(v 1 u 1 ) and 1 = a 1 + (v 1 u 1 ) where = Regressions are calculated for = 1 and for = 1, and the following steps are repeated: 1. If v k u k < 1 stop, the optimum solution is in interval [u k,v k ]. Otherwise, if 3. R 2 > R 2,go to step2 and if R 2 R 2, go to step k k k k 2. u k +1 = k and v k +1 = v k. Also, let k +1 = k y k +1 = u k +1 + (v k +1 u k +1 ). Calculate R 2 ( k +1 ) and go to step u k +1 = u k and v k +1 = k. Also, let k +1 = k and k +1 = u k +1 + (1 )(v k +1 u k +1 ). Calculate R 4. Substitute k for k + 1 and go to step 1. 2 ( k +1 ) and go to step 4. 23

26 Levels Udibonos Appendix 2 Adjustment of yield structures 28/01/2002 Vector of parameters (,,, ) = ( ) Udibonos data for 28/01/2002 Maturity Simple yield Continuous yield Udibonos adjustment Maturity Simple yield Continuous yield Simple yield Udibonos Udibonos data Udibonos adjustment Maturity in days 24

27 Libor Vector of parameters (,,, ) = ( ) Libor data for 28/01/2002 Maturity Simple yield Continuous yield Simple yield Continuous yield Simple yield Libor Libor data Libor adjustment Maturity in days 25

28 T-bill Vector of parameters (,,, ) = ( ) T-bill data for 28/01/2002 Maturity Simple yield Continuous yield Simple yield Continuous yield Simple yield T-bill Maturity in days T-bill data T-bill adjustment 26

29 Frequency Udibonos Appendix 3 s Udibonos Udibonos Udibonos Udibonos

30 Frequency Frequency Frequency Libor Libor Libor Libor Libor

31 T-bill T-bill T-bill Frequency Frequency T-bill Frequency Frequency T-bill 29

32 Frequency Frequency Frequency Appendix 4 Simulations Udibonos Shape of original curves Shape of curves generated by simulation 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% % Curve type Curve type Libor Shape of original curves Shape of curves generated by 80% 70% 70% 60% 60% 50% 50% 40% 40% 30% 30% 20% 20% 10% 10% 0% % Curve type Curve type 30

33 Frequency T-bill Shape of original curves Shape of curves generated by simulation 80% 70% 60% 50% 40% 30% 70% 60% 50% 40% 30% 20% 20% 10% 10% 0% Curve type 0% Curve type 31

34 Bibliography Bliss Robert R., Testing Term Structure Estimation Methods, Working Paper 96-12a, Federal Reserve Bank of Atlanta, November Márquez Diez-Canedo, Javier. Fundamentos de Teoría de Optimización, Limusa Mokhtar S Bazaraa and C. M. Shetty, Nonlinear Programming, Theory and Algorithms, John Wiley & Sons, USA,1979. J.P. Morgan and Reuters, RiskMetrics-Technical Document, New York, Fourth Edition, Nelson, C. R. and A. F. Siegel. Parsimonious Modeling and Yield Curves, Journal of Business 60 (October 1987), Svensson Lars E. O., Estimating and Interpreting Forward Interest Rates: Sweden , Working Paper No. 4871, National Bureau of Economic Research,

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