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Zero coupon yield curve construction for a low liquidity bond market: a new approach

Zero coupon yield curve construction for a low liquidity bond market: a new approach. Dr. Sergey Smirnov. Need of the y ield curve fitting for the credit risk modeling.

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Zero coupon yield curve construction for a low liquidity bond market: a new approach

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  1. Zero coupon yield curve construction for a low liquidity bond market: a new approach Dr. Sergey Smirnov EBC Meeting, Amsterdam, June 2004

  2. Need of the yield curve fittingfor the credit risk modeling • “Yield curve smoothing has long been the Rodney Dangerfield of risk management analytics.  In spite of the importance of yield curve smoothing technology, the discipline has not gotten the respect that it deserves.” (Donald R. van Deventer, January 2004) • The accuracy of yield curve smoothing techniques has taken on an increased importance in recent years because of the intense research focus among both practitioners and academics on credit risk modeling. • In particular, the reduced form modeling approach of Duffie and Singleton [1999] and Jarrow [2001] has the power to extract default probabilities and the “liquidity premium” (the excess of “credit spread” above and beyond expected loss) from bond prices and credit default swap prices. EBC Meeting, Amsterdam, June 2004

  3. Two ways of credit spreads estimation • The first method, which is generally considered to be the most precise ( but it is not necessary the case), is to use the closed form solution for zero coupon credit spreads in the respective credit model and to solve for the credit model parameters that minimize the sum of squared pricing error for the observable bonds or credit default swaps.  • The second method, which is used commonly in academic studies of credit risk, is to calculate credit spreads on a “credit model independent basis” in order to later study which credit models are the most accurate. This will be our way. Note: liquidity premium is included in the spreads and cannot be separated from the premium for the credit risks EBC Meeting, Amsterdam, June 2004

  4. “Best practice” credit spread construction: Step 1 • 1.      For each of the M payment dates on the chosen corporate bond, calculate the continuously compounded zero coupon bond price and and the smooth risk free zero coupon yield (in the USA market it is the U.S. Treasury smoothed yield curve).  Note: These yields will be to actual payment dates, not scheduled payment dates, because the day count convention associated with the bond will move scheduled payments forward or backward (depending on the convention) if they fall on weekends or holidays EBC Meeting, Amsterdam, June 2004

  5. “ Best practice” credit spread construction: Step 2 • 2.      Guess a continuously compounded credit spread of x that is assumed to be the same for each payment date (no credit risk term structure). Note: this assumption is not always meaningful from the economic point of view, but is inevitable element of the modeling technique EBC Meeting, Amsterdam, June 2004

  6. “ Best practice” credit spread construction: Step 3 • 3.      Calculate the present value of the chosen corporate bond using the M continuously compounded zero coupon bond yields y(t) + x, where y(t) is the zero coupon bond yield to the payment date t on the risk free curve.  EBC Meeting, Amsterdam, June 2004

  7. “ Best practice” credit spread construction: Step 4 • 4.      Compare the present value calculated in Step 3 with the value of the chosen corporate bond (price plus accrued interest) observed in the market. • Note: for the low liquidity market the bond price can be not directly observable EBC Meeting, Amsterdam, June 2004

  8. 5.      If the theoretical value and observed value are within a tolerance e, then stop and report x as the credit spread.  If the difference is outside the tolerance, improve the guess of x using standard methods and go back to Step 3.[4]6.      Spreads calculated in this manner should be confined to non-callable bonds or used with great care in the case of callable bonds.5.      If the theoretical value and observed value are within a tolerance e, then stop and report x as the credit spread.  If the difference is outside the tolerance, improve the guess of x using standard methods and go back to Step 3.[4]6.      Spreads calculated in this manner should be confined to non-callable bonds or used with great care in the case of callable bonds.“Best practice” credit spread construction: Step 5 • 5.      If the theoretical value and observed value are within a tolerance e, then stop and report x as the credit spread.  If the difference is outside the tolerance, improve the guess of x using standard methods and go back to Step 3. • Note: Spreads calculated in this manner should be confined to non-callable bonds or used with great care in the case of callable bonds EBC Meeting, Amsterdam, June 2004

  9. The importance of the yield curve smoothing technology • Yield curve smoothing technology is at the heart of this credit spread calculation. The reason is that the M payment dates on the corporate bond require zero coupon risk free yields on dates that are unlikely to be payment dates or maturity dates observable in the market ( for example U.S. Treasury).  • Yield curve smoothing is even more important (a) in countries where the number of risk free bonds observable is far fewer (like Japan ) or (b) when smoothing is being done directly on the risky bond issuer’s yield curve itself.  The chosen Company may have, say, only 3 bonds with observable prices, for example, compared to more than 200 in the U.S. Treasury market.  EBC Meeting, Amsterdam, June 2004

  10. Possible definition of risk free rate in Eurozone The inversion of the described above algorithm would imply the following requirements: • The zero coupon yield curve for each country, fitted to the (coupon) sovereign bonds, must have credit spread near constant (in maturity) • Risk free rate curve must be sufficiently smooth • Risk free rate curve must be less then zero coupon yield curve for any country • A parallel shift of risk free rate curve increasing the level of rates leads to an intersection with zero coupon yield curve for some country EBC Meeting, Amsterdam, June 2004

  11. Study of the Russian governmental bond market (ruble denominated) EBC Meeting, Amsterdam, June 2004

  12. GKO-OFZ Market in 2003 • 47 bonds (11 GKO, 36 OFZ) • Average/max/min bonds outstanding: 43/47/37 • Average/max/min bonds traded: 16/25/6 EBC Meeting, Amsterdam, June 2004

  13. GKO-OFZ Market in 2003 Average number of deals per day Trade intensity (% of traded days) EBC Meeting, Amsterdam, June 2004

  14. Data Filtering • Whole issues exclusion (having very low liquidity) • Short term maturity filtering • “Out of the market deals” filtering EBC Meeting, Amsterdam, June 2004

  15. “Out of the market deals” filtering Duration – YTM graphs February, 20. Bond 27016. July, 31. Bond46003. EBC Meeting, Amsterdam, June 2004

  16. The impact of the “out of the sample” bond on the yield curve behavior Yield curves on July, 31. Bond 46003 is not excluded. EBC Meeting, Amsterdam, June 2004

  17. Excluding “out of the sample” bonds increases accuracy and smoothness of the yield curve Yield curves on July, 31. Bond 46003 is excluded. EBC Meeting, Amsterdam, June 2004

  18. Methods used • Static methods - yield curve fitting • Parametric methods (Nelson-Siegel, Svensson) • Spline methods (Vasicek-Fong, Sinusoidal-Exponential splines) • Dynamic methods • 3-factor Vasicek model with Kalman filter estimation for parameters • General affine term structure model (to be implemented) • Bond price dynamics approach EBC Meeting, Amsterdam, June 2004

  19. Parametric methods of yield curve fitting • Svensson – 6 parameters Instantaneous forward rate is assumed to have the following form: • Nelson-Siegel – special case of Svensson, 4parameters: Assuming specific functional form for yield curve is arbitrary and has no economic ground EBC Meeting, Amsterdam, June 2004

  20. Vasicek-Fong method for yield curve fitting Discount function is approximated by exponential splines of the form: EBC Meeting, Amsterdam, June 2004

  21. 3-factor dynamic Vasicek model • Short rate is assumed an affine function of factors: • Factors satisfy SDE: • Parameters are estimated using non-linear Kalman filter EBC Meeting, Amsterdam, June 2004

  22. Particularities of the low-liquidity markets • Systematic liquidity premium for certain instruments • Large bid/ask spread • Inactive trading on some days • Highly volatile market data • Unreliable data (“non-market” trades) EBC Meeting, Amsterdam, June 2004

  23. Typical situations in low-liquidity markets Missing market data. This problem is an issue not only for low-liquidity markets. EBC Meeting, Amsterdam, June 2004

  24. Missing data problem– no “long maturities” (trading day 1) Term structure on September, 5 EBC Meeting, Amsterdam, June 2004

  25. Missing data problem– no “long maturities” (trading day 2) Term structure on September, 8 EBC Meeting, Amsterdam, June 2004

  26. Missing data problem– no “long maturities” (trading day 3) Term structure on September, 9 EBC Meeting, Amsterdam, June 2004

  27. Missing data problem– no “long maturities” (trading day 2 with forecasting) Term structure on September, 8 with predictions for non-traded bonds EBC Meeting, Amsterdam, June 2004

  28. Missing data problem– no “long maturities” (trading day 3 with forecasting) Term structure on September, 9 with predictions for non-traded bonds EBC Meeting, Amsterdam, June 2004

  29. Missing data problem– no “short maturities”) Term structure on November, 14 EBC Meeting, Amsterdam, June 2004

  30. Missing data problem (example 1 – no “short-end”) Term structure on November, 14 with predictions for non-traded bonds EBC Meeting, Amsterdam, June 2004

  31. Two-stage approach to constructing the yield curve in low-liquidity market Stage 1: making predictions for unobserved prices Stage2:fitting yield curve on basis of observed market prices and predictions for unobserved market prices EBC Meeting, Amsterdam, June 2004

  32. Missing data problem Evident solution – use past price information We need to mix current and past prices in a meaningful way We need a stochastic term-structure model EBC Meeting, Amsterdam, June 2004

  33. Stochastic term-structure models • Standard approach – modeling the dynamics of the short rate Following D.Duffie andK.Singleton defaultable zero-coupon prices could be represented in the form of risk-neutral expectation: • Bond price dynamics approach – modeling directly the market prices - default arrival intensity - loss given default - liquidity premium EBC Meeting, Amsterdam, June 2004

  34. Bond price dynamics approach The simplest stochastic dynamics: - independent standard Wiener processes - parameters How to estimate parameters ? EBC Meeting, Amsterdam, June 2004

  35. Bayesian Approach Prior density Posterior density Likelihood function - parameters (random variables) - observed data EBC Meeting, Amsterdam, June 2004

  36. Pros and cons of Bayesian approach Pros: • Formal mechanism to incorporate prior information • More precise estimations for small samples • All analysis (point and interval estimations, test of hypothesis) follows directly from posterior distribution • Proper Bayesian methods are insensitive to dimension of the parameter space Cons: • Subjective results • High computational requirements EBC Meeting, Amsterdam, June 2004

  37. Bayesian estimation – the case of complete data Sample from multivariate normal distribution with unknown mean vector and covariance matrix Conjugative priors Close-formsolutions. High computational speed. EBC Meeting, Amsterdam, June 2004

  38. Bayesian estimation – the case of incomplete data Variable dimension of observations (depending on how many bond prices are observed) No conjugative priors Numericsolutions. Much lower computational speed. EBC Meeting, Amsterdam, June 2004

  39. Markov Chain Monte-Carlo Methods - All data (observed+missing) - Observed data - Missing data - Complicated distribution - Simple distribution EBC Meeting, Amsterdam, June 2004

  40. Markov Chain Monte-Carlo Methods (improving computational efficiency) 1) Imputation Step Sampling missing data 2) Posterior Step Sampling parameters from posterior distribution Markov chain EBC Meeting, Amsterdam, June 2004

  41. Model Testing • 200 normally-distributed vectors • 2000 iterations • Missing data ratio (0%, 20%, 50%) EBC Meeting, Amsterdam, June 2004

  42. Joint Posterior Distributions (Missing Ratio 0 %) EBC Meeting, Amsterdam, June 2004

  43. Joint Posterior Distributions (Missing Ratio 20 %) EBC Meeting, Amsterdam, June 2004

  44. Joint Posterior Distributions (Missing Ratio 50 %) EBC Meeting, Amsterdam, June 2004

  45. Application to a subset of bonds of Russian GKO-OFZ market 5 bonds were chosen: - 45001 (5% of missing data) - 46001 (12%) - 28001 (10%) - 27015 (31%) - 46002 (30%) for the periodMarch,1 – December,31 EBC Meeting, Amsterdam, June 2004

  46. Estimations of parameters EBC Meeting, Amsterdam, June 2004

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