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Yield curve estimation in Costa Rica

This document discusses the estimation of the yield curve for Costa Rica through metaheuristic optimization methods, including the Nelson-Siegel and Svensson models. Various optimization techniques such as genetic algorithms, ant colonies, simulated annealing, and adaptive barrier methods are examined for their effectiveness in improving yield estimation. Results demonstrate the performance of these methods in achieving accurate yield curve estimations using historical data from the country's national stock exchange.

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Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos School of Mathematics – CIMPA University of Costa Rica March 9, 2017
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Contents Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Interest Rates Introduction ◮ Payment of a debtor to a creditor by use of capital ◮ Some factors that affect interest rates: ◮ Inflation risk ◮ Uncertainty ◮ Quality of information ◮ Random fluctuations ◮ Long term investments
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Introduction ◮ Valuation of fixed rent assets ◮ Indicator of economic agents expectations related to future interest rates and inflation ◮ Reliable indicator of economic activity of a country ◮ Bank of Canada (1999)1: introduction of parametric models → the optimization process can be improved ◮ Bank of International Settlements- BIS (2005): compared methodologies and models of 13 countries (→ parametric models Nelson-Siegel & Svensson) 1 Bolder & Streliski, “Yield curve at the Bank of Canada, T.R. 84.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Spot rate ◮ Spot interest rate: for t periods, it is the money that will be received in period t after borrowing money today ◮ Spot rate is the implicit interest rate got after investing in a zero coupon bond that expires in period t ◮ It is obtained at solving for it: Pr = F (1 + it)t where Pr is the bond price and F is its facial value ◮ Yield to maturity (investment on a bond that periodically pays coupons with interest r): Pr = t X k=1 c (1 + r)k + F (1 + r)t where c: value of coupons, t: period.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Forward rate ◮ Forward rate: an interest rate defined today for a future transaction ◮ It is an expectation on what will happen to the spot rate in the future ◮ For continuous interest rates δt and δs, for t and s days (s < t), the continuous forward rate is: ft,x = tδt − sδs t − s ◮ Instantaneous forward rate is the limit: f(t) = lim s→t ft,s
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Nelson-Siegel Model ◮ Description of the instantaneous forward rate: f(t) = β0 + β1e−λt + β2λteλt (1) where θ θ θ = (β0, β1, β2, λ) is a parameter vector ◮ From (1) it is obtained a continuous function for the spot rate: i(t) = β0 + β1 1 − e−λt λt + β2 1 − e−λt λt − e−λt (2) with ◮ β0 0 is the long term interest rate, ◮ β1 is the short term interest rate, ◮ λ 0, specifies position of concavity, ◮ β2 sign establishes form of the curve.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Svensson Model ◮ Extension of the Nelson-Siegel model ◮ Introduction of parameters β3 and λ2 ◮ Forward rate: f(t) = β0 + β1e−λ1t + β2λ1teλ1t + β3λ2te−λ2t (3) with parameter vector θ θ θ = (β0, β1, β2, λ1, λ2) ◮ Corresponding spot rate: i(t) = β0 + β1 1 − e−λ1t λ1t + β2 1 − e−λ1t λ1t − e−λ1t +β3 1 − e−λ2t λ2t − e−λ2t (4) ◮ sign of β3 determine a second form of concavity ◮ λ2 0 specifies position of second concavity
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Yield Curve Estimation Duration ◮ Duration measures the average time (in years) expected for receiving benefits from a bond: D = 1 Pr t X k=1 kc (1 + r)k + tF (1 + r)t ! ◮ It quantifies the change relation between yield and bond price at different terms ◮ It is usual to use a modification of duration: D∗ = D 1 + r ◮ This can be used as a weight for the objective function (square of the deviations of observed prices and estimated prices for n observations): n X k=1 Prk − c Prk PrkD∗ k !2 (5)
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Regression ◮ Prices are estimated through non linear regression, using a parametric model (Nelson-Siegel or Svensson) ◮ Parameters satisfy b θ θ θ = arg min θ θ θ n X k=1 Prk − c Prk PrkD∗ k !2 ◮ The stock market in Costa Rica is small, therefore many of the existing methods are not feasible to implement → use of historical data. We suppose that variability is proportional to the number of days since the bond was negotiated2 ◮ Weights: use the bid-ask spread Hk, b θ θ θ = arg min θ θ θ n X k=1 (Prk − c Prk)2 Hk(1 + NDk) where NDk is the number of days since transaction of bond k 2 We suppose price increments independent and stationary
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Regression ◮ In non-linear regression for estimating the yield curve, it is usually used a Gauss-Newton type method ◮ It is well known that it can converge to a local minimum (or may not converge at all) −→ use of optimization metaheuristics
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Genetic Algorithm ◮ Algorithm based on ideas of genetic evolution and biology. ◮ It starts with a population of 100 solutions chosen randomly, in each iteration a new population is obtained from the previous one by pairing, mating and mutation. ◮ Chromosomic representation is based on numerical vector of parameters (4 parameters for Nelson-Siegel model, 6 parameters for Svensson model). ◮ In this work roulette wheel rank weighting is used as selecting pairing. ◮ 50% as natural selection, 10% as mutation. ◮ For mating it is used the algorithm based on [2], a mix of extrapolation method with crossover method.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Ant Colonies ◮ Metaheuristic that takes its ideas from the way ants get food. ◮ In this study it is used the version for continuous domains presented in [7]. ◮ The pheromones are used by means of an array that stores a number of solutions and new solutions are built sequentially using the information of the array. ◮ From [7] it was taken the parameters of the method: 50 solutions store, 2 ants, 0.4 locality of the search and 1.1 speed of convergence.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Particle Swarms ◮ Based on the behavior of some groups of animals. ◮ The performance of an individual is influenced by its best historical performance and the best performance of the group up to the present iteration. ◮ We used 47 particles, -0.1832 of inertia, 0.5287 as cognitive parameter and 3.1913 as social parameter.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Simulated Annealing ◮ Based on the physical process named annealing, which takes a solid to a high temperature and then let it cool very slowly in order to get a more resistant and pure state of the solid. ◮ Also it uses the Metropolis criterion of acceptation whose purpose is to get out of local minimum zone. ◮ We used the version named very fast simulated reannealing [3] which allows to work with restrictions. ◮ The size of the Markov chain was established in 100, and the temperature is updated with the factor 0.95.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details An Adaptive Barrier Method ◮ The objective function depends on the vector θ θ θ which has to satisfy (7) or (6), so that the constrained optimization problem in changed into an unconstrained problem, and an adaptive barrier method is used [5]. ◮ In this case a logarithmic barrier is added to the objective function in order to handle the constraints (7) or (6). ◮ If the minimum lies on the boundary the barrier will not allow to reach it, to deal with this the logarithmic barrier has a component that changes in each iteration [4].
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Constraints A set of constraints similar to those used in [1] have been implemented with two goals, results economically feasible and speed in the optimization process. The following constraints are for the Nelson-Siegel model: 0% β0 25% −20% β1 20% 0% β2 25% (6) 1/300 λ 12 0 β0 + β1. Constraints for the Svensson model: 0% β0 25% −20% β1 0% 0% β2 25% 0% β3 25% (7) 1/300 λ1 12
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details The Data ◮ Historical data were provided by the Bolsa Nacional de Valores (BNV, National Stock Exchange). ◮ These are bonds and zero coupon bonds issued by the Central Bank and the Ministry of Finance Costa Rica, which are called tp0, tp, bem0 and bem. ◮ Data are for the period of February 23th, 2015 to March 12th, 2015, only emissions in colones3 were considered and there is no restriction on the amounts of transactions. 3 National currency of Costa Rica
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results ◮ Multistart strategy of size 2,000. ◮ The way of comparison is as follows: the best objective function value for the metaheuristics is the expected value from their multistart, in the case of the adaptive barrier the best objective function value is the minimum value that was achieved from its multistart. ◮ The following values are reported: ◮ the objective function value, ◮ the coefficient of variation in the multistart, ◮ the goodness of fit P k(rk − r̂k)/m, and ◮ the average time of running the R-code measured in seconds.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Nelson-Siegel model Table: Summary metaheuristics performance in estimating the Nelson-Siegel model. Algorithm Objective Coefficient Goodness Average function of variation of fit time value (m) Particle S. 441.5018 1% 0.003345% 00:22 Simulated A. 441.5034 1% 0.003353% 00:28 Adaptive barrier 441.5243 240% 0.003354% — Genetic A. 1,206.0571 18% 0.066502% 01:16 Ant C. 1,207.6136 36% 0.066541% 00:18
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Nelson-Siegel Model Figure: Yield curves estimated for Nelson-Siegel model and yield curve of the Costa Rican Central Bank as of 17-March-2015.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Svensson Model Table: Summary metaheuristics performance in estimating the Svensson model. Algorithm Objective Coefficient Goodness Average function of variation of fit time value (m) Particle S. 251.5805 1% 0.012147% 00:43 Simulated A. 251.6899 1% 0.012550% 00:46 Ant C. 254.6444 84% 0.012345% 01:32 Adaptive barrier 441.6267 317% 0.003164% — Genetic A. 1,138.3852 12% 0.052407% 00:13
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Svensson Model Figure: Yield curves estimated for Nelson-Siegel model and yield curve of the Costa Rican Central Bank as of 17-March-2015.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Concluding Remarks ◮ Two metaheuristics were better in both models: ◮ Particle swarm ◮ Simulated annealing ◮ These metaheuristcs besides of having the best results, their algorithms are easy to implement, the execution time is acceptable and the outcomes are very stable. ◮ Thus, Particle swarm and Simulated annealing are recommended for getting the parameters of the Nelson-Siegel and Svensson models.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Further Research ◮ For future research it is suggested to repeat this study with other sets of sample data so as to confirm the results obtained so far. ◮ For another data set, similar restrictions for (6) or (7) which are adjusted to the Costa Rican market characteristics, should be determined. ◮ Finally, a review of the configuration used in Genetic algorithm and Ant colony could also be made in order to obtain satisfactory parameters that could make compete these metaheuristics with the better ones.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details References Bolder, D.; Streliski, D. (1999) “Yield curve modeling at the Bank of Canada”, Technical Report No. 84, Bank of Canada: 69. Haupt, R.; Haupt, S. (2004) Practical Genetic Algorithms, 2nd Ed. Wiley, New York. Jang, J.; Sun, C.; Mizutani, E. (1997) Neuro-Fuzzy and Soft Computing. Prentice Hall, New Jersey. Lange, K. (1999) Numerical Analysis for Statisticians. Springer, New York. Luenberger, D.; Ye, Y. (2008) Linear and Nonlinear Programming, 3rd Ed. Springer, New York. Nelson, C.; Siegel, A. (1987) “Parsimonious modeling of yield curves”, Journal of Business 60(4): 473–489. Socha, K.; Dorigo, M. (2008) “Ant colony optimization for continuous domains”, European Journal of Operational Research 185: 1155–1173. Svensson, L. (1994) “Estimating and Interpreting Forward Interest Rates: Sweeden 1992–1994”, Institute for International Economic Studies 579.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Objective Function Bid-ask Spread Hk = Pm i=1 OVk,ifVk,i fVk,total − Pm i=1 OCk,ifCk,i fVk,total (8) where ◮ OVk,i is the i-th sell offer, ◮ OCk,i is the i-th buy offer, ◮ fVk,i is the facial amount of the i-th sell offer, ◮ fCk,i is the facial amount of the i-th buy offer, ◮ fVk,total is the total amount of facial sell offers, and ◮ fCk,total is the total amount od facial buy offers. Bonds with high volatility will not sum high vallues to the objective function.
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Genetic Algorithm Crossover ◮ Choose two parents at random f, m ◮ Choose a position k at random ◮ Define allele and sons:
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Ant Colony Details
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Particle Swarms Details
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Simulated Annealing Details
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Adaptive Barrier Details
Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details BFGS Broyden-Fletcher-Goldfarb-Shanno algorithm — Details

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Yield curve estimation in Costa Rica

  • 1.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Estimation of the Yield Curve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos School of Mathematics – CIMPA University of Costa Rica March 9, 2017
  • 2.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Contents Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details
  • 3.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Interest Rates Introduction ◮ Payment of a debtor to a creditor by use of capital ◮ Some factors that affect interest rates: ◮ Inflation risk ◮ Uncertainty ◮ Quality of information ◮ Random fluctuations ◮ Long term investments
  • 4.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Introduction ◮ Valuation of fixed rent assets ◮ Indicator of economic agents expectations related to future interest rates and inflation ◮ Reliable indicator of economic activity of a country ◮ Bank of Canada (1999)1: introduction of parametric models → the optimization process can be improved ◮ Bank of International Settlements- BIS (2005): compared methodologies and models of 13 countries (→ parametric models Nelson-Siegel & Svensson) 1 Bolder & Streliski, “Yield curve at the Bank of Canada, T.R. 84.
  • 5.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Spot rate ◮ Spot interest rate: for t periods, it is the money that will be received in period t after borrowing money today ◮ Spot rate is the implicit interest rate got after investing in a zero coupon bond that expires in period t ◮ It is obtained at solving for it: Pr = F (1 + it)t where Pr is the bond price and F is its facial value ◮ Yield to maturity (investment on a bond that periodically pays coupons with interest r): Pr = t X k=1 c (1 + r)k + F (1 + r)t where c: value of coupons, t: period.
  • 6.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Yield Curve Estimation Forward rate ◮ Forward rate: an interest rate defined today for a future transaction ◮ It is an expectation on what will happen to the spot rate in the future ◮ For continuous interest rates δt and δs, for t and s days (s < t), the continuous forward rate is: ft,x = tδt − sδs t − s ◮ Instantaneous forward rate is the limit: f(t) = lim s→t ft,s
  • 7.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós & Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks & Further Research References Details Nelson-Siegel Model ◮ Description of the instantaneous forward rate: f(t) = β0 + β1e−λt + β2λteλt (1) where θ θ θ = (β0, β1, β2, λ) is a parameter vector ◮ From (1) it is obtained a continuous function for the spot rate: i(t) = β0 + β1 1 − e−λt λt + β2 1 − e−λt λt − e−λt (2) with ◮ β0 0 is the long term interest rate, ◮ β1 is the short term interest rate, ◮ λ 0, specifies position of concavity, ◮ β2 sign establishes form of the curve.
  • 8.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Svensson Model ◮ Extension of the Nelson-Siegel model ◮ Introduction of parameters β3 and λ2 ◮ Forward rate: f(t) = β0 + β1e−λ1t + β2λ1teλ1t + β3λ2te−λ2t (3) with parameter vector θ θ θ = (β0, β1, β2, λ1, λ2) ◮ Corresponding spot rate: i(t) = β0 + β1 1 − e−λ1t λ1t + β2 1 − e−λ1t λ1t − e−λ1t +β3 1 − e−λ2t λ2t − e−λ2t (4) ◮ sign of β3 determine a second form of concavity ◮ λ2 0 specifies position of second concavity
  • 9.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Yield Curve Estimation Duration ◮ Duration measures the average time (in years) expected for receiving benefits from a bond: D = 1 Pr t X k=1 kc (1 + r)k + tF (1 + r)t ! ◮ It quantifies the change relation between yield and bond price at different terms ◮ It is usual to use a modification of duration: D∗ = D 1 + r ◮ This can be used as a weight for the objective function (square of the deviations of observed prices and estimated prices for n observations): n X k=1 Prk − c Prk PrkD∗ k !2 (5)
  • 10.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Regression ◮ Prices are estimated through non linear regression, using a parametric model (Nelson-Siegel or Svensson) ◮ Parameters satisfy b θ θ θ = arg min θ θ θ n X k=1 Prk − c Prk PrkD∗ k !2 ◮ The stock market in Costa Rica is small, therefore many of the existing methods are not feasible to implement → use of historical data. We suppose that variability is proportional to the number of days since the bond was negotiated2 ◮ Weights: use the bid-ask spread Hk, b θ θ θ = arg min θ θ θ n X k=1 (Prk − c Prk)2 Hk(1 + NDk) where NDk is the number of days since transaction of bond k 2 We suppose price increments independent and stationary
  • 11.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Regression ◮ In non-linear regression for estimating the yield curve, it is usually used a Gauss-Newton type method ◮ It is well known that it can converge to a local minimum (or may not converge at all) −→ use of optimization metaheuristics
  • 12.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Genetic Algorithm ◮ Algorithm based on ideas of genetic evolution and biology. ◮ It starts with a population of 100 solutions chosen randomly, in each iteration a new population is obtained from the previous one by pairing, mating and mutation. ◮ Chromosomic representation is based on numerical vector of parameters (4 parameters for Nelson-Siegel model, 6 parameters for Svensson model). ◮ In this work roulette wheel rank weighting is used as selecting pairing. ◮ 50% as natural selection, 10% as mutation. ◮ For mating it is used the algorithm based on [2], a mix of extrapolation method with crossover method.
  • 13.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Ant Colonies ◮ Metaheuristic that takes its ideas from the way ants get food. ◮ In this study it is used the version for continuous domains presented in [7]. ◮ The pheromones are used by means of an array that stores a number of solutions and new solutions are built sequentially using the information of the array. ◮ From [7] it was taken the parameters of the method: 50 solutions store, 2 ants, 0.4 locality of the search and 1.1 speed of convergence.
  • 14.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Particle Swarms ◮ Based on the behavior of some groups of animals. ◮ The performance of an individual is influenced by its best historical performance and the best performance of the group up to the present iteration. ◮ We used 47 particles, -0.1832 of inertia, 0.5287 as cognitive parameter and 3.1913 as social parameter.
  • 15.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Simulated Annealing ◮ Based on the physical process named annealing, which takes a solid to a high temperature and then let it cool very slowly in order to get a more resistant and pure state of the solid. ◮ Also it uses the Metropolis criterion of acceptation whose purpose is to get out of local minimum zone. ◮ We used the version named very fast simulated reannealing [3] which allows to work with restrictions. ◮ The size of the Markov chain was established in 100, and the temperature is updated with the factor 0.95.
  • 16.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details An Adaptive Barrier Method ◮ The objective function depends on the vector θ θ θ which has to satisfy (7) or (6), so that the constrained optimization problem in changed into an unconstrained problem, and an adaptive barrier method is used [5]. ◮ In this case a logarithmic barrier is added to the objective function in order to handle the constraints (7) or (6). ◮ If the minimum lies on the boundary the barrier will not allow to reach it, to deal with this the logarithmic barrier has a component that changes in each iteration [4].
  • 17.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Constraints A set of constraints similar to those used in [1] have been implemented with two goals, results economically feasible and speed in the optimization process. The following constraints are for the Nelson-Siegel model: 0% β0 25% −20% β1 20% 0% β2 25% (6) 1/300 λ 12 0 β0 + β1. Constraints for the Svensson model: 0% β0 25% −20% β1 0% 0% β2 25% 0% β3 25% (7) 1/300 λ1 12
  • 18.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details The Data ◮ Historical data were provided by the Bolsa Nacional de Valores (BNV, National Stock Exchange). ◮ These are bonds and zero coupon bonds issued by the Central Bank and the Ministry of Finance Costa Rica, which are called tp0, tp, bem0 and bem. ◮ Data are for the period of February 23th, 2015 to March 12th, 2015, only emissions in colones3 were considered and there is no restriction on the amounts of transactions. 3 National currency of Costa Rica
  • 19.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results ◮ Multistart strategy of size 2,000. ◮ The way of comparison is as follows: the best objective function value for the metaheuristics is the expected value from their multistart, in the case of the adaptive barrier the best objective function value is the minimum value that was achieved from its multistart. ◮ The following values are reported: ◮ the objective function value, ◮ the coefficient of variation in the multistart, ◮ the goodness of fit P k(rk − r̂k)/m, and ◮ the average time of running the R-code measured in seconds.
  • 20.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Nelson-Siegel model Table: Summary metaheuristics performance in estimating the Nelson-Siegel model. Algorithm Objective Coefficient Goodness Average function of variation of fit time value (m) Particle S. 441.5018 1% 0.003345% 00:22 Simulated A. 441.5034 1% 0.003353% 00:28 Adaptive barrier 441.5243 240% 0.003354% — Genetic A. 1,206.0571 18% 0.066502% 01:16 Ant C. 1,207.6136 36% 0.066541% 00:18
  • 21.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Nelson-Siegel Model Figure: Yield curves estimated for Nelson-Siegel model and yield curve of the Costa Rican Central Bank as of 17-March-2015.
  • 22.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Svensson Model Table: Summary metaheuristics performance in estimating the Svensson model. Algorithm Objective Coefficient Goodness Average function of variation of fit time value (m) Particle S. 251.5805 1% 0.012147% 00:43 Simulated A. 251.6899 1% 0.012550% 00:46 Ant C. 254.6444 84% 0.012345% 01:32 Adaptive barrier 441.6267 317% 0.003164% — Genetic A. 1,138.3852 12% 0.052407% 00:13
  • 23.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Results Svensson Model Figure: Yield curves estimated for Nelson-Siegel model and yield curve of the Costa Rican Central Bank as of 17-March-2015.
  • 24.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Concluding Remarks ◮ Two metaheuristics were better in both models: ◮ Particle swarm ◮ Simulated annealing ◮ These metaheuristcs besides of having the best results, their algorithms are easy to implement, the execution time is acceptable and the outcomes are very stable. ◮ Thus, Particle swarm and Simulated annealing are recommended for getting the parameters of the Nelson-Siegel and Svensson models.
  • 25.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Further Research ◮ For future research it is suggested to repeat this study with other sets of sample data so as to confirm the results obtained so far. ◮ For another data set, similar restrictions for (6) or (7) which are adjusted to the Costa Rican market characteristics, should be determined. ◮ Finally, a review of the configuration used in Genetic algorithm and Ant colony could also be made in order to obtain satisfactory parameters that could make compete these metaheuristics with the better ones.
  • 26.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details References Bolder, D.; Streliski, D. (1999) “Yield curve modeling at the Bank of Canada”, Technical Report No. 84, Bank of Canada: 69. Haupt, R.; Haupt, S. (2004) Practical Genetic Algorithms, 2nd Ed. Wiley, New York. Jang, J.; Sun, C.; Mizutani, E. (1997) Neuro-Fuzzy and Soft Computing. Prentice Hall, New Jersey. Lange, K. (1999) Numerical Analysis for Statisticians. Springer, New York. Luenberger, D.; Ye, Y. (2008) Linear and Nonlinear Programming, 3rd Ed. Springer, New York. Nelson, C.; Siegel, A. (1987) “Parsimonious modeling of yield curves”, Journal of Business 60(4): 473–489. Socha, K.; Dorigo, M. (2008) “Ant colony optimization for continuous domains”, European Journal of Operational Research 185: 1155–1173. Svensson, L. (1994) “Estimating and Interpreting Forward Interest Rates: Sweeden 1992–1994”, Institute for International Economic Studies 579.
  • 27.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Objective Function Bid-ask Spread Hk = Pm i=1 OVk,ifVk,i fVk,total − Pm i=1 OCk,ifCk,i fVk,total (8) where ◮ OVk,i is the i-th sell offer, ◮ OCk,i is the i-th buy offer, ◮ fVk,i is the facial amount of the i-th sell offer, ◮ fCk,i is the facial amount of the i-th buy offer, ◮ fVk,total is the total amount of facial sell offers, and ◮ fCk,total is the total amount od facial buy offers. Bonds with high volatility will not sum high vallues to the objective function.
  • 28.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Genetic Algorithm Crossover ◮ Choose two parents at random f, m ◮ Choose a position k at random ◮ Define allele and sons:
  • 29.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Ant Colony Details
  • 30.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Particle Swarms Details
  • 31.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Simulated Annealing Details
  • 32.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details Adaptive Barrier Details
  • 33.
    Estimation of the YieldCurve for Costa Rica using Metaheuristic Optimization Andrés Quirós Javier Trejos Introduction Yield curve estimation Nelson-Siegel Model Svensson Model Regression Optimization Methods Genetic Algorithm Ant Colonies Particle Swarms Simulated Annealing An Adaptive Barrier Method The Data Results Concluding Remarks Further Research References Details BFGS Broyden-Fletcher-Goldfarb-Shanno algorithm — Details