EBC Meeting, Amsterdam, June 2004 Zero coupon yield curve construction for a low liquidity bond...

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EBC Meeting, Amsterdam, June 200 4 Zero coupon yield curve construction for a low liquidity bond market: a new approach Dr. Sergey Smirnov

Transcript of EBC Meeting, Amsterdam, June 2004 Zero coupon yield curve construction for a low liquidity bond...

Page 1: 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

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

Dr. Sergey Smirnov

Page 2: 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.

Page 3: 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

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

Page 4: 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

“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

Page 5: 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

“ 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

Page 6: 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

“ 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. 

Page 7: 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

“ 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

Page 8: 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

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

Page 9: 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

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. 

Page 10: 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

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

Page 11: 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

Study of the Russian governmental bond market (ruble denominated)

Page 12: 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

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

Page 13: EBC Meeting, Amsterdam, June 2004 Zero coupon yield curve construction for a low liquidity bond market: a new approach Dr. Sergey Smirnov.

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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

Page 14: EBC Meeting, Amsterdam, June 2004 Zero coupon yield curve construction for a low liquidity bond market: a new approach Dr. Sergey Smirnov.

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Data Filtering

Whole issues exclusion (having very low liquidity)

Short term maturity filtering “Out of the market deals” filtering

Page 15: 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

“Out of the market deals” filtering

February, 20. Bond 27016. July, 31. Bond 46003.

Duration – YTM graphs

Page 16: 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

The impact of the “out of the sample” bond on the yield curve behavior

Yield curves on July, 31. Bond 46003 is not excluded.

Page 17: 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

Excluding “out of the sample” bonds increases accuracy and smoothness of the yield curve

Yield curves on July, 31. Bond 46003 is excluded.

Page 18: 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

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

Page 19: 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

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:

Page 20: 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

Vasicek-Fong method for yield curve fitting

Discount function is approximated by exponential splines of the form:

Page 21: 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

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

Page 22: EBC Meeting, Amsterdam, June 2004 Zero coupon yield curve construction for a low liquidity bond market: a new approach Dr. Sergey Smirnov.

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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)

Page 23: 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

Typical situations in low-liquidity markets

Missing market data. This problem is an issue not only for low-liquidity markets.

Page 24: 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

Missing data problem – no “long maturities” (trading day 1)

Term structure on September, 5

Page 25: 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

Missing data problem – no “long maturities” (trading day 2)

Term structure on September, 8

Page 26: 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

Term structure on September, 9

Missing data problem – no “long maturities” (trading day 3)

Page 27: 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

Missing data problem – no “long maturities” (trading day 2 with forecasting)

Term structure on September, 8 with predictions for non-traded bonds

Page 28: 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

Missing data problem – no “long maturities” (trading day 3 with forecasting)

Term structure on September, 9 with predictions for non-traded bonds

Page 29: 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

Term structure on November, 14

Missing data problem– no “short maturities”)

Page 30: 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

Missing data problem (example 1 – no “short-end”)Term structure on November, 14 with predictions for non-traded bonds

Page 31: 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

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

Page 32: 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

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

Page 33: 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

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

Page 34: 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

- independent standard Wiener processes

Tktttt WWWW ),...,,( 21

kttt WWW ,...,, 21

kkk RR , - parameters

How to estimate parameters ?

Bond price dynamics approachThe simplest stochastic dynamics:

Page 35: 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

Posterior density

Likelihood function

Prior density

- parameters (random variables)

- observed data

Bayesian Approach

Page 36: 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

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

Page 37: 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

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

Page 38: 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

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)

Page 39: 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

Markov Chain Monte-Carlo Methods

- All data (observed+missing)

- Observed data

- Missing data

- Complicated distribution

- Simple distribution

Page 40: 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

Markov Chain Monte-Carlo Methods (improving computational efficiency)

1) Imputation Step

2) Posterior Step

Sampling missing data

Sampling parameters from posterior distribution

Markov chain

Page 41: 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

Model Testing

200 normally-distributed vectors

2000 iterations

Missing data ratio (0%, 20%, 50%)

Page 42: 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

Joint Posterior Distributions (Missing Ratio 0 %)

Page 43: 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

Joint Posterior Distributions (Missing Ratio 20 %)

Page 44: 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

Joint Posterior Distributions (Missing Ratio 50 %)

Page 45: 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

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

Page 46: 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

Estimations of parameters