This document discusses the performance of the zero coupon yield curve (ZCYC) in pricing government bonds in India. It finds that a ZCYC that weights bonds by traded volume and uses filtered input data performs best, with the lowest error levels for the most liquid bonds. The analysis shows that such a model can effectively hedge exposures to single bonds, protecting against losses compared to an unhedged position. Overall, the document evaluates different ZCYC specifications and bond selection criteria to determine how well the ZCYC works in pricing bonds and its usefulness for hedging and speculation activities.

1. 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

2. Questions
How well does the ZCYC work:
How bad is the difference between prices from
the ZCYC and the market?
What would happen if we used the ZCYC prices
for speculation and hedging?
Or, what is the correlation between returns on the
ZCYC prices and market prices?
What improvements can be made, to give greater
importance to ‘the benchmark bonds’?
The zero coupon yield curve:How well does it work? – p. 2

3. Data is the plural of anecdote
We should be careful about drawing inferences from
casual perusal of a few days of data.
All the work shown here uses daily data from Jan
2002 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

4. What this talk does not address
Concerns about market manipulation
Impediments to arbitrage - we are implicitly
assuming perfect arbitrage.
The zero coupon yield curve:How well does it work? – p. 4

5. Difference between market and
model prices
The zero coupon yield curve:How well does it work? – p. 5

6. Problems with the current market
price used
We use the NSE WDM value weighted average
price.
We therefore accumulate trade prices during the
entire trading day, morning and evening.
This is not the same as the “closing price” from the
electronic exchange.
Therefore, all our results will be weak: we cannot
expect a solution of the quality that we are used to
seeing on the equity market.
The zero coupon yield curve:How well does it work? – p. 6

7. Performance evaluation based on
prices
Simplest method: Pick model with the smallest sum
of squared errors across all bonds.
The average pricing error (not just σ) for liquid
bonds is of great interest - it reflects liquidity premia.
Here, we also calculate the average error across a set
of liquid (“benchmark”) bonds.
A model focused on benchmark bonds will have a
small average error and a small σ of the error.
The zero coupon yield curve:How well does it work? – p. 7

8. Defining errors between market and
model prices
We define error,
e = 100∗(model price−market price)/market price.
We only use the market prices for T + 0 trades for
the evaluation of the ZCYC performance.
Please note: For this exercise, the calculations are
made using prices, not YTMs.
The zero coupon yield curve:How well does it work? – p. 8

9. Liquidity premia
NSE ZCYC − averaging across all bonds
ZCYC focused on benchmark bonds
Interest rate
ZCYC with perfect liquidity
Time to maturity, t
The zero coupon yield curve:How well does it work? – p. 9

10. Interpreting E(e), the average pric-
ing error
E(e) tends to be the liquidity premium of a highly
liquid bond when compared to the ZCYC.
If the ZCYC hugs the benchmark bonds, E(e) will
be small for benchmark bonds but large for other
bonds.
A ZCYC that is an “average” off all trades will yield
E(e) = 0 for all bonds put together, but
E(e) < 0 for liquid bonds.
The zero coupon yield curve:How well does it work? – p. 10

11. Interpreting σe
It is the standard deviation of the percentage error.
It is the sum of noise in the WDM VWA and noise
in the ZCYC.
As long as there are problems in the design of the
bond market, the error variance will not fully go
away.
The zero coupon yield curve:How well does it work? – p. 11

12. Defining “benchmark bonds”
The zero coupon yield curve:How well does it work? – p. 12

13. Distribution of turnover
Cumulative Traded Volume Vs Top Bonds for June 2002
1
CDF
0.9
0.8
Cumulative Traded Volume
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 20 40 60 80 100 120 140 160
Top Bonds by Traded Volume The zero coupon yield curve:How well does it work? – p. 13

14. Selecting the benchmark bonds
We see that 10 bonds capture 60% of the traded
volume in the month.
We select “benchmark bonds” on the 1st of every
month 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 the
remainder of the month and the model error is
calculated for these benchmark bonds every day.
E(e) is the average error for the month, σe is the
standard deviation of these errors for the month.
The zero coupon yield curve:How well does it work? – p. 14

15. 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

16. Illiquidity in “benchmark bonds”
“Benchmark bonds” have days of missing data: a
non-zero probability of non-trading.
On average, these bonds have a 78% probability of
trading!
In comparison, Nifty stocks almost never have a day
without trading.
The zero coupon yield curve:How well does it work? – p. 16

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

18. Factors that might improve model
performance
Select bonds for the estimation by liquidity criteria –
number of trades, traded volume in the day.
Greater weights for liquid bonds – weight the errors
by turnover ratio or traded volume weights.
The zero coupon yield curve:How well does it work? – p. 18

19. Alternatives
M1 : Existing NSE ZCYC.
M2 : NS ZCYC with filtered input data.
M3 : NS ZCYC with Turnover Ratio (TR) weights
in estimation.
M4 : NS ZCYC with Traded volume (TV) weights
in estimation.
M5 : NS ZCYC with filtered input data and TR
weights.
M6 : NS ZCYC with filtered input data and TV
weights.
The zero coupon yield curve:How well does it work? – p. 19

20. Performance of errors between
market and model prices of liquid
bonds
The zero coupon yield curve:How well does it work? – p. 20

21. Performance from January 2002 to
August 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

22. 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 29.13
The zero coupon yield curve:How well does it work? – p. 22

23. Time variation in E(e)
Benchmark by TV
NSE
40
Mean of Monthly Error Terms
20
(Basis Points)
0
-20
-40
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Febzero coupon yield curve:How well doesAug – p. 23
The Mar Apr May Jun Jul it work?

24. Time variation in E(e)
Benchmark by TV
NSE
Filtered TV Weighted NS
40
Mean of Monthly Error Terms
20
(Basis Points)
0
-20
-40
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Febzero coupon yield curve:How well doesAug – p. 24
The Mar Apr May Jun Jul it work?

25. Time variation in σe
Benchmark by TV
100
NSE
80
Sd of Monthly Error Terms
60
(Basis Points)
40
20
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Junwell doesAug – p. 25
The zero coupon yield curve:How Jul it work?

26. Time variation in σe
Benchmark by TV
100
NSE
Filtered TV Weighted NS
80
Sd of Monthly Error Terms
60
(Basis Points)
40
20
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Junwell doesAug – p. 26
The zero coupon yield curve:How Jul it work?

27. Market YTM vs. model YTM
Bond market participants tend to focus on the yield
to maturity (YTM), rather than the price, as a
indicator of value.
The YTM is a single rate at which all cashflows of a
bond is discounted.
Thus, the YTM notion assumes that the yield curve
is flat.
However, given the weight that market participants
give the YTM, it might be useful to ask how the
YTM of a given bond calculated when it is priced
using the ZCYC performs against the YTM of the
bond calculated when using the market’s price.
The zero coupon yield curve:How well does it work? – p. 27

28. The definition of errors as YTM dif-
ferences
Here, error is defined as
e = 100 ∗ (model YTM − market YTM).
The average of the error is calculated for the 10
benchmark bonds in every month (similar to the
calculations that we did for the price errors earlier).
We only use the market prices for T + 0 trades for
the evaluation of the ZCYC performance.
The σe is similarly calculated using these YTM
errors.
The zero coupon yield curve:How well does it work? – p. 28

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

30. Time variation in E(e)
E(errors): Benchmark Bonds selection by TV Criterion
10
NSE
Filtered TV Weighted NS
Filtered TV Weighted MIB Constrained
8 TV Weighted NS
6
Mean of Monthly Error Terms
4
(Basis Points)
2
0
-2
-4
-6
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Febzero coupon yield curve:How well doesAug – p. 30
The Mar Apr May Jun Jul it work?

31. Time variation in σe
Sigma(errors): Benchmark Bonds selection by TV Criterion
18
NSE
Filtered TV Weighted NS
Filtered TV Weighted MIB Constrained
16 TV Weighted NS
14
Sd of Monthly Error Terms
12
(Basis Points)
10
8
6
4
2
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Febzero coupon yield curve:How well doesAug – p. 31
The Mar Apr May Jun Jul it work?

32. Choice of model: summary
We have explored several avenues of research to do
better on the basic NSE model, to go closer to the
most liquid bonds.
We find that the TV weighted NS model estimated
with filtered data is the best in having the lowest σe
for the most liquid bonds.
The liquidity premium of the NSE ZCYC model
prices also goes away once this is done.
For the remaining analysis, we focus only on this
best model.
The zero coupon yield curve:How well does it work? – p. 32

33. Hedging single bond exposures: one
example
The zero coupon yield curve:How well does it work? – p. 33

34. Thinking in returns
Suppose there is a market price of 100 and a model
price of 110.
Suppose the next day, the market price goes up to
110 and the model price goes up to 121.
The model is apparently doing very badly, but it is a
very useful tool for hedging and speculation!
The zero coupon yield curve:How well does it work? – p. 34

35. 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

36. 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

37. Correlation in returns: one bond,
one day
The zero coupon yield curve:How well does it work? – p. 37

38. Motivation
For the futures market, the real key is correlations
between market price and model price.
You’d be willing to trade a futures contract as long
as movements in the futures price are correlated with
the spot price.
This requires focusing on the correlation between
returns on the model price versus returns on the
market 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

39. How good do the correlations have to
be?
India’s best futures market is the Nifty futures - vast
retail market, anonymous order matching, typically
Rs.3,000 crore per day.
Daily returns on the Nifty spot and daily returns on
the Nifty futures have a correlation of 0.95.
The zero coupon yield curve:How well does it work? – p. 39

40. Scatter diagram of model vs. market
returns for 11.50% 2011A
GS 11.50% 2011A
5
NS returns
0
-5
-5 0 5
Market returns
The zero coupon yield curve:How well does it work? – p. 40

41. Scatter diagram of model vs. market
returns for 8.07% 2017
GS 8.07% 2017
5
NS returns
0
-5
-5 0 5
Market returns
The zero coupon yield curve:How well does it work? – p. 41

42. Scatter diagram of model vs. market
returns for 7.40% 2012
GS 7.04% 2012
5
NS returns
0
-5
-5 0 5
Market returns
The zero coupon yield curve:How well does it work? – p. 42

43. Overall correlations
Bond ρmarket,model (%) ρ without
two outliers
GS 11.50% 2011A 85.73 87.74
GS 8.07% 2017 89.00 90.31
GS 7.40% 2012 86.66 87.55
The zero coupon yield curve:How well does it work? – p. 43

44. Time series of correlations in model
vs. market returns for 2011A
GS 11.50% 2011
Rolling correlations over 250 days
0.9
0.8
0.7
Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003
The zero coupon yield curve:How well does it work? – p. 44

45. Time series of correlations in model
vs. market returns for 2017
GS 8.09% 2017
Rolling correlations over 250 days
0.9
0.8
0.7
Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003
The zero coupon yield curve:How well does it work? – p. 45

46. Time series of correlations in model
vs. market returns for 2012
GS 7.40% 2012
Rolling correlations over 250 days
0.9
0.8
0.7
Dec 2002 Jan 2003 Feb 2003 Mar 2003 Apr 2003 May 2003 Jun 2003
The zero coupon yield curve:How well does it work? – p. 46

47. Correlations in returns: one bond,
weekly returns
The zero coupon yield curve:How well does it work? – p. 47

48. Motivation
For traders with horizons bigger than a day, we
should not focus on correlations of daily returns.
There are problems with data from the bond market
which implies that changes in the VWA prices are
more trustworthy over longer periods as compared
with shorter periods.
We examine scatter plots and time series variation in
correlations for weekly rather than daily returns
next.
The zero coupon yield curve:How well does it work? – p. 48

49. Scatter diagram of model vs. market
weekly returns for 2011A
GS 11.50% 2011A
5
Weekly NS returns
0
-5
-5 0 5
Weekly market returns
The zero coupon yield curve:How well does it work? – p. 49

50. Scatter diagram of model vs. market
weekly returns for 2017
GS 8.07% 2017
5
Weekly NS returns
0
-5
-5 0 5
Weekly market returns
The zero coupon yield curve:How well does it work? – p. 50

51. Scatter diagram of model vs. market
weekly returns for 2012
GS 7.40% 2012
5
Weekly NS returns
0
-5
-5 0 5
Weekly market returns
The zero coupon yield curve:How well does it work? – p. 51

52. Overall correlations
Bond ρmarket,model
GS 11.50% 2011A 96.65
GS 8.07% 2017 94.19
GS 7.40% 2012 97.28
The zero coupon yield curve:How well does it work? – p. 52

53. Time series of correlations in model
vs. market weekly returns: 2011
Rolling correlations over thirty weeks
0.9
0.8
GS 11.50% 2011A
0.7
Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003
The zero coupon yield curve:How well does it work? – p. 53

54. Time series of correlations in model
vs. market weekly returns: 2017
Rolling correlations over thirty weeks
0.9
0.8
GS 8.07% 2017
0.7
Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003
The zero coupon yield curve:How well does it work? – p. 54

55. Time series of correlations in model
vs. market weekly returns: 2012
Rolling correlations over thirty weeks
0.9
0.8
GS 7.40% 2012
0.7
Sep 2002 Nov 2002 Jan 2003 Mar 2003 May 2003
The zero coupon yield curve:How well does it work? – p. 55

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

57. What have we learnt
The ZCYC does seem to work rather well.
The bond market has important design flaws. We
should not expect solutions of the quality of the
equity market.
Yet, we have quite some strength - broad
correlations of 90-95%.
Not hedging is much worse than hedging with a
“weak contract”.
Not speculating is much worse than speculating with
a “weak contract”.
Reminder: A perfect ZCYC does not ensure perfect
futures pricing. That will require a vibrant arbitrage
business. The zero coupon yield curve:How well does it work? – p. 57