ebc meeting, amsterdam, june 2004 zero coupon yield curve construction for a low liquidity bond...
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EBC Meeting, Amsterdam, June 2004
Zero coupon yield curve construction for a low liquidity bond market: a new approach
Dr. Sergey Smirnov
EBC Meeting, Amsterdam, June 2004
Need of the yield curve fitting for 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
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
“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
“ 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
“ 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
“ 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
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
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
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
Study of the Russian governmental bond market (ruble denominated)
EBC Meeting, Amsterdam, June 2004
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
0 20 40 60 80 100 120
SU45001RMFS3
SU27008RMFS0
SU27010RMFS6
SU27011RMFS4
SU27006RMFS4
SU46002RMFS0
SU27012RMFS2
SU27018RMFS9
SU45002RMFS1
SU27017RMFS1
SU27023RMFS9
SU21167RMFS0
SU21161RMFS3
SU21168RMFS8
SU27016RMFS3
SU21164RMFS7
SU21169RMFS6
SU21166RMFS2
SU21170RMFS4
SU21171RMFS2
SU46009RMFS5
SU26197RMFS2
SU26198RMFS0
Average number of deals per day Trade intensity (% of traded days)
0 5 10 15 20 25
SU46001RMFS2
SU46002RMFS0
SU27008RMFS0
SU27006RMFS4
SU45002RMFS1
SU28001RMFS4
SU27007RMFS2
SU27013RMFS0
SU27012RMFS2
SU27023RMFS9
SU21165RMFS4
SU21161RMFS3
SU27021RMFS3
SU21162RMFS1
SU26003RMFS2
SU21163RMFS9
SU21164RMFS7
SU26001RMFS6
SU46003RMFS8
SU26002RMFS4
SU46009RMFS5
SU27020RMFS5
SU26197RMFS2
GKO-OFZ Market in 2003
EBC Meeting, Amsterdam, June 2004
Data Filtering
Whole issues exclusion (having very low liquidity)
Short term maturity filtering “Out of the market deals” filtering
EBC Meeting, Amsterdam, June 2004
“Out of the market deals” filtering
February, 20. Bond 27016. July, 31. Bond 46003.
Duration – YTM graphs
EBC Meeting, Amsterdam, June 2004
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
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
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
Parametric methods of yield curve fitting
Svensson – 6 parameters
Instantaneous forward rate is assumed to have the following form:
Assuming specific functional form for yield curve is arbitrary and has no
economic ground
Nelson-Siegel – special case of Svensson, 4 parameters:
EBC Meeting, Amsterdam, June 2004
Vasicek-Fong method for yield curve fitting
Discount function is approximated by exponential splines of the form:
EBC Meeting, Amsterdam, June 2004
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
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
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
Missing data problem – no “long maturities” (trading day 1)
Term structure on September, 5
EBC Meeting, Amsterdam, June 2004
Missing data problem – no “long maturities” (trading day 2)
Term structure on September, 8
EBC Meeting, Amsterdam, June 2004
Term structure on September, 9
Missing data problem – no “long maturities” (trading day 3)
EBC Meeting, Amsterdam, June 2004
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
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
Term structure on November, 14
Missing data problem– no “short maturities”)
EBC Meeting, Amsterdam, June 2004
Missing data problem (example 1 – no “short-end”)Term structure on November, 14 with predictions for non-traded bonds
EBC Meeting, Amsterdam, June 2004
Stage 1: making predictions for unobserved prices
Stage 2: fitting yield curve on basis of observed market prices and predictions for unobserved market prices
Two-stage approach to constructing the yield curve in low-liquidity market
EBC Meeting, Amsterdam, June 2004
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
Missing data problem
EBC Meeting, Amsterdam, June 2004
Stochastic term-structure models Standard approach – modeling the dynamics of the short rate
Following D.Duffie and K.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
- independent standard Wiener processes
Tktttt WWWW ),...,,( 21
kttt WWW ,...,, 21
kkk RR , - parameters
How to estimate parameters ?
Bond price dynamics approachThe simplest stochastic dynamics:
EBC Meeting, Amsterdam, June 2004
Posterior density
Likelihood function
Prior density
- parameters (random variables)
- observed data
Bayesian Approach
EBC Meeting, Amsterdam, June 2004
Pros and cons of Bayesian approachPros:
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
Bayesian estimation – the case of complete data
Conjugative priors
Close-form solutions. High computational speed.
Sample from multivariate normal distribution with unknown mean vector and covariance matrix
EBC Meeting, Amsterdam, June 2004
Bayesian estimation – the case of incomplete data
No conjugative priors
Numeric solutions. Much lower computational speed.
Variable dimension of observations (depending on how many bond prices are observed)
EBC Meeting, Amsterdam, June 2004
Markov Chain Monte-Carlo Methods
- All data (observed+missing)
- Observed data
- Missing data
- Complicated distribution
- Simple distribution
EBC Meeting, Amsterdam, June 2004
Markov Chain Monte-Carlo Methods (improving computational efficiency)
1) Imputation Step
2) Posterior Step
Sampling missing data
Sampling parameters from posterior distribution
Markov chain
EBC Meeting, Amsterdam, June 2004
Model Testing
200 normally-distributed vectors
2000 iterations
Missing data ratio (0%, 20%, 50%)
EBC Meeting, Amsterdam, June 2004
Joint Posterior Distributions (Missing Ratio 0 %)
EBC Meeting, Amsterdam, June 2004
Joint Posterior Distributions (Missing Ratio 20 %)
EBC Meeting, Amsterdam, June 2004
Joint Posterior Distributions (Missing Ratio 50 %)
EBC Meeting, Amsterdam, June 2004
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 period March,1 – December,31
EBC Meeting, Amsterdam, June 2004
Estimations of parameters
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