zero coupon yield curve

The zero coupon yield curve:How well does it work?

Susan Thomas

http://www.igidr.ac.in/˜susant

susant@mayin.org

IGIDR

Bombay

The zero coupon yield curve:How well does it work? – p. 1

Questions

How well does the ZCYC work:How bad is the difference between prices fromthe ZCYC and the market?What would happen if we used the ZCYC pricesfor speculation and hedging?Or, what is the correlation between returns on theZCYC prices and market prices?

What improvements can be made, to give greaterimportance to ‘the benchmark bonds’?

The zero coupon yield curve:How well does it work? – p. 2

Data is the plural of anecdote

We should be careful about drawing inferences fromcasual perusal of a few days of data.

All the work shown here uses daily data from Jan2002 to end August 2003: a 20 month period.

Bond market liquidity has improved over this period,so we do see some meaningful time trends in this.

The zero coupon yield curve:How well does it work? – p. 3

What this talk does not address

Concerns about market manipulation

Impediments to arbitrage - we are implicitlyassuming perfect arbitrage.

The zero coupon yield curve:How well does it work? – p. 4

Difference between market andmodel prices

The zero coupon yield curve:How well does it work? – p. 5

Problems with the current marketprice used

We use the NSE WDM value weighted averageprice.

We therefore accumulate trade prices during theentire trading day, morning and evening.

This is not the same as the “closing price” from theelectronic exchange.

Therefore, all our results will be weak: we cannotexpect a solution of the quality that we are used toseeing on the equity market.

The zero coupon yield curve:How well does it work? – p. 6

Performance evaluation based onprices

Simplest method: Pick model with the smallest sumof squared errors across all bonds.

The average pricing error (not just σ) for liquidbonds is of great interest - it reflects liquidity premia.

Here, we also calculate the average error across a setof liquid (“benchmark”) bonds.

A model focused on benchmark bonds will have asmall average error and a small σ of the error.

The zero coupon yield curve:How well does it work? – p. 7

Defining errors between market andmodel prices

We define error,e = 100∗(model price−market price)/market price.

We only use the market prices for T + 0 trades forthe evaluation of the ZCYC performance.

Please note: For this exercise, the calculations aremade using prices, not YTMs.

The zero coupon yield curve:How well does it work? – p. 8

Liquidity premia

NSE ZCYC − averaging across all bonds

ZCYC with perfect liquidity

ZCYC focused on benchmark bonds

Time to maturity, t

Inte

rest

rate

The zero coupon yield curve:How well does it work? – p. 9

Interpreting E(e), the average pric-ing error

E(e) tends to be the liquidity premium of a highlyliquid bond when compared to the ZCYC.

If the ZCYC hugs the benchmark bonds, E(e) willbe small for benchmark bonds but large for otherbonds.

A ZCYC that is an “average” off all trades will yieldE(e) = 0 for all bonds put together, butE(e) < 0 for liquid bonds.

The zero coupon yield curve:How well does it work? – p. 10

Interpreting σe

It is the standard deviation of the percentage error.

It is the sum of noise in the WDM VWA and noisein the ZCYC.

As long as there are problems in the design of thebond market, the error variance will not fully goaway.

The zero coupon yield curve:How well does it work? – p. 11

Defining “benchmark bonds”

The zero coupon yield curve:How well does it work? – p. 12

Distribution of turnover

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160

Cum

ulat

ive

Tra

ded

Vol

ume

Top Bonds by Traded Volume

Cumulative Traded Volume Vs Top Bonds for June 2002

CDF

The zero coupon yield curve:How well does it work? – p. 13

Selecting the benchmark bonds

We see that 10 bonds capture 60% of the tradedvolume in the month.

We select “benchmark bonds” on the 1st of everymonth as follows:

We calculate “traded volume” as the total traded volume

over the last 3 months.

We sort this and take the top ten bonds.

The benchmark bonds are held fixed for theremainder of the month and the model error iscalculated for these benchmark bonds every day.

E(e) is the average error for the month, σe is thestandard deviation of these errors for the month.

The zero coupon yield curve:How well does it work? – p. 14

Stability in “benchmark bonds”?

Rank Feb ’03 Mar ’03 Apr ’03 May ’03

1 8.07% 2017 7.40% 2012 8.07% 2017 8.07% 2017

2 7.40% 2012 8.07% 2017 7.40% 2012 7.40% 2012

3 9.39% 2011 7.46% 2017 9.81% 2013 9.81% 2013

4 7.46% 2017 9.81% 2013 7.46% 2017 7.46% 2017

5 11.50% 2011 9.39% 2011 9.85% 2015 9.85% 2015

The zero coupon yield curve:How well does it work? – p. 15

Illiquidity in “benchmark bonds”

“Benchmark bonds” have days of missing data: anon-zero probability of non-trading.

On average, these bonds have a 78% probability oftrading!In comparison, Nifty stocks almost never have a daywithout trading.

The zero coupon yield curve:How well does it work? – p. 16

Issues in improving models

The zero coupon yield curve:How well does it work? – p. 17

Factors that might improve modelperformance

Select bonds for the estimation by liquidity criteria –number of trades, traded volume in the day.

Greater weights for liquid bonds – weight the errorsby turnover ratio or traded volume weights.

The zero coupon yield curve:How well does it work? – p. 18

Alternatives

M1 : Existing NSE ZCYC.

M2 : NS ZCYC with filtered input data.

M3 : NS ZCYC with Turnover Ratio (TR) weightsin estimation.

M4 : NS ZCYC with Traded volume (TV) weightsin estimation.

M5 : NS ZCYC with filtered input data and TRweights.

M6 : NS ZCYC with filtered input data and TVweights.

The zero coupon yield curve:How well does it work? – p. 19

Performance of errors betweenmarket and model prices of liquid

bonds

The zero coupon yield curve:How well does it work? – p. 20

Performance from January 2002 toAugust 2003

For benchmark bonds:

Model No. E(e) σe

(in basis points)

NSE -17.95 53.84

NS with filtered inputs -7.35 58.22

NS with TR weights -0.62 56.69

NS with TV weights 2.69 58.28

NS with both filter and TR weights -1.54 54.25

NS with both filter and TV weights 2.09 51.38

The zero coupon yield curve:How well does it work? – p. 21

Performance

(basis points)

Benchmark bonds All bonds

Model E(e) σe σe

Full period (1/2002 – 8/2003)

NSE -17.94 53.84 34.74

NS with filters and TV weights 2.09 51.38 31.12

2003 only (1/2003 - 8/2003)

NSE -27.05 44.25 30.05

NS with filters and TV weights -1.24 40.04 29.13The zero coupon yield curve:How well does it work? – p. 22

Time variation in E(e)

-40

-20

0

20

40

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Mea

n of

Mon

thly

Err

or T

erm

s (

Bas

is P

oint

s)

Month

Benchmark by TV

NSE

The zero coupon yield curve:How well does it work? – p. 23

Time variation in E(e)

-40

-20

0

20

40

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Mea

n of

Mon

thly

Err

or T

erm

s (

Bas

is P

oint

s)

Month

Benchmark by TV

NSEFiltered TV Weighted NS

The zero coupon yield curve:How well does it work? – p. 24

Time variation in σe

0

20

40

60

80

100

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Sd

of M

onth

ly E

rror

Ter

ms

(B

asis

Poi

nts)

Month

Benchmark by TV

NSE

The zero coupon yield curve:How well does it work? – p. 25

Time variation in σe

0

20

40

60

80

100

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Sd

of M

onth

ly E

rror

Ter

ms

(B

asis

Poi

nts)

Month

Benchmark by TV

NSEFiltered TV Weighted NS

The zero coupon yield curve:How well does it work? – p. 26

Market YTM vs. model YTM

Bond market participants tend to focus on the yieldto maturity (YTM), rather than the price, as aindicator of value.

The YTM is a single rate at which all cashflows of abond is discounted.

Thus, the YTM notion assumes that the yield curveis flat.

However, given the weight that market participantsgive the YTM, it might be useful to ask how theYTM of a given bond calculated when it is pricedusing the ZCYC performs against the YTM of thebond calculated when using the market’s price.

The zero coupon yield curve:How well does it work? – p. 27

The definition of errors as YTM dif-ferences

Here, error is defined ase = 100 ∗ (model YTM − market YTM).

The average of the error is calculated for the 10benchmark bonds in every month (similar to thecalculations that we did for the price errors earlier).

We only use the market prices for T + 0 trades forthe evaluation of the ZCYC performance.

The σe is similarly calculated using these YTMerrors.

The zero coupon yield curve:How well does it work? – p. 28

Description of the YTM error graphs

In the following graphs, the models depicted are:1. The NSE NS model.2. The TV weighted NS model.3. The TV weighted NS model estimated with

filtered input data.4. The TV weighted NS model estimated with

filtered input data with constraints on the shortterm rate.

The zero coupon yield curve:How well does it work? – p. 29

Time variation in E(e)

-6

-4

-2

0

2

4

6

8

10

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Mea

n of

Mon

thly

Err

or T

erm

s (

Bas

is P

oint

s)

Month

E(errors): Benchmark Bonds selection by TV Criterion

NSEFiltered TV Weighted NS

Filtered TV Weighted MIB ConstrainedTV Weighted NS

The zero coupon yield curve:How well does it work? – p. 30

Time variation in σe

2

4

6

8

10

12

14

16

18

Jan02

Feb02

Mar02

Apr02

May02

Jun02

Jul02

Aug02

Sep02

Oct02

Nov02

Dec02

Jan03

Feb03

Mar03

Apr03

May03

Jun03

Jul03

Aug03

Sd

of M

onth

ly E

rror

Ter

ms

(B

asis

Poi

nts)

Month

Sigma(errors): Benchmark Bonds selection by TV Criterion

NSEFiltered TV Weighted NS

Filtered TV Weighted MIB ConstrainedTV Weighted NS

The zero coupon yield curve:How well does it work? – p. 31

Choice of model: summary

We have explored several avenues of research to dobetter on the basic NSE model, to go closer to themost liquid bonds.

We find that the TV weighted NS model estimatedwith filtered data is the best in having the lowest σe

for the most liquid bonds.

The liquidity premium of the NSE ZCYC modelprices also goes away once this is done.

For the remaining analysis, we focus only on thisbest model.

The zero coupon yield curve:How well does it work? – p. 32

Hedging single bond exposures: oneexample

The zero coupon yield curve:How well does it work? – p. 33

Thinking in returns

Suppose there is a market price of 100 and a modelprice of 110.

Suppose the next day, the market price goes up to110 and the model price goes up to 121.

The model is apparently doing very badly, but it is avery useful tool for hedging and speculation!

The zero coupon yield curve:How well does it work? – p. 34

An example of hedging

Suppose we have a cash settled futures market which uses the

NSE NS ZCYC model price for the 11.50% 2011A bond.

We focus on the worst one-week loss on the bond in the last six

months (last week of January, 2003):

Market Model

Price on 24th Jan, 2003 139.858 140.611

Price on 31st Jan, 2003 135.908 137.412

% change -2.5446 -2.3017

Unhedged: we’d have lost 2.55%.

Hedged: we’d have lost 0.24%.The zero coupon yield curve:How well does it work? – p. 35

Results using a better model

The above results used the present NSE ZCYC. We have some

progress on a better model.

Once again, we focus on the worst one-week loss on the bond:

Market Model

Price on 24th Jan, 2003 139.8580 139.9221

Price on 31st Jan, 2003 136.3441 136.7203

% change -2.5446 -2.3149

Unhedged: we’d have lost 2.55%.

Hedged: we’d have lost 0.23%.

The zero coupon yield curve:How well does it work? – p. 36

Correlation in returns: one bond,one day

The zero coupon yield curve:How well does it work? – p. 37

Motivation

For the futures market, the real key is correlationsbetween market price and model price.

You’d be willing to trade a futures contract as longas movements in the futures price are correlated withthe spot price.

This requires focusing on the correlation betweenreturns on the model price versus returns on themarket price.

We do this only for the top 3 bonds: GS 11.50%2011A, GS 8.07% 2017, GS 7.40%2012.

Perfect model : All points will fall on 45 degree line.The zero coupon yield curve:How well does it work? – p. 38

How good do the correlations have tobe?

India’s best futures market is the Nifty futures - vastretail market, anonymous order matching, typicallyRs.3,000 crore per day.

Daily returns on the Nifty spot and daily returns onthe Nifty futures have a correlation of 0.95.

The zero coupon yield curve:How well does it work? – p. 39

Scatter diagram of model vs. marketreturns for 11.50% 2011A

-5 0 5

Market returns

-5

0

5

NS

retu

rns

GS 11.50% 2011A

The zero coupon yield curve:How well does it work? – p. 40

Scatter diagram of model vs. marketreturns for 8.07% 2017

-5 0 5

Market returns

-5

0

5

NS

retu

rns

GS 8.07% 2017

The zero coupon yield curve:How well does it work? – p. 41

Scatter diagram of model vs. marketreturns for 7.40% 2012

-5 0 5

Market returns

-5

0

5

NS

retu

rns

GS 7.04% 2012

The zero coupon yield curve:How well does it work? – p. 42

Overall correlations

Bond ρmarket,model (%) ρ without

two outliers

GS 11.50% 2011A 85.73 87.74GS 8.07% 2017 89.00 90.31GS 7.40% 2012 86.66 87.55

The zero coupon yield curve:How well does it work? – p. 43

Time series of correlations in modelvs. market returns for 2011A

Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r 25

0 da

ys GS 11.50% 2011A

The zero coupon yield curve:How well does it work? – p. 44

Time series of correlations in modelvs. market returns for 2017

Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r 25

0 da

ys GS 8.09% 2017

The zero coupon yield curve:How well does it work? – p. 45

Time series of correlations in modelvs. market returns for 2012

Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r 25

0 da

ys GS 7.40% 2012

The zero coupon yield curve:How well does it work? – p. 46

Correlations in returns: one bond,weekly returns

The zero coupon yield curve:How well does it work? – p. 47

Motivation

For traders with horizons bigger than a day, weshould not focus on correlations of daily returns.

There are problems with data from the bond marketwhich implies that changes in the VWA prices aremore trustworthy over longer periods as comparedwith shorter periods.

We examine scatter plots and time series variation incorrelations for weekly rather than daily returnsnext.

The zero coupon yield curve:How well does it work? – p. 48

Scatter diagram of model vs. marketweekly returns for 2011A

-5 0 5

Weekly market returns

-5

0

5

Wee

kly

NS

retu

rns

GS 11.50% 2011A

The zero coupon yield curve:How well does it work? – p. 49

Scatter diagram of model vs. marketweekly returns for 2017

-5 0 5

Weekly market returns

-5

0

5

Wee

kly

NS

retu

rns

GS 8.07% 2017

The zero coupon yield curve:How well does it work? – p. 50

Scatter diagram of model vs. marketweekly returns for 2012

-5 0 5

Weekly market returns

-5

0

5

Wee

kly

NS

retu

rns

GS 7.40% 2012

The zero coupon yield curve:How well does it work? – p. 51

Overall correlations

Bond ρmarket,model

GS 11.50% 2011A 96.65GS 8.07% 2017 94.19GS 7.40% 2012 97.28

The zero coupon yield curve:How well does it work? – p. 52

Time series of correlations in modelvs. market weekly returns: 2011

Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r th

irty

wee

ks

GS 11.50% 2011A

The zero coupon yield curve:How well does it work? – p. 53

Time series of correlations in modelvs. market weekly returns: 2017

Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r th

irty

wee

ks

GS 8.07% 2017

The zero coupon yield curve:How well does it work? – p. 54

Time series of correlations in modelvs. market weekly returns: 2012

Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003 0.7

0.8

0.9

Rol

ling

corr

elat

ions

ove

r th

irty

wee

ks

GS 7.40% 2012

The zero coupon yield curve:How well does it work? – p. 55

Insights

The zero coupon yield curve:How well does it work? – p. 56

What have we learnt

The ZCYC does seem to work rather well.

The bond market has important design flaws. Weshould not expect solutions of the quality of theequity market.Yet, we have quite some strength - broadcorrelations of 90-95%.

Not hedging is much worse than hedging with a“weak contract”.

Not speculating is much worse than speculating witha “weak contract”.

Reminder: A perfect ZCYC does not ensure perfectfutures pricing. That will require a vibrant arbitragebusiness.

The zero coupon yield curve:How well does it work? – p. 57