Methodology for Discount Curve Construction – India

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1. Overview

Numerica constructs a zero-coupon Government of India yield curve each trading day from outright trades in dated government securities and treasury bills on CCIL’s NDS-OM platform. The methodology is the one we use for our Saudi curve, with parameters set for the Indian market: volume-weighted curve fitting, a short end observed from treasury bills with the RBI policy repo rate as a light guard rail, historical aggregation on the rare day with too few trades, and temporal smoothing on a thin day only; on a full day the day’s fit stands as it is. Each day’s curve is then checked against the zero-coupon curves published independently by CCIL and FBIL. The result on each day is a set of annually compounded spot rates by maturity, which is what discounting under Ind AS 19, IAS 19 and AS 15 requires.

Numerica’s constructed curve is published from 28 September 2026. For earlier dates the page shows CCIL’s published Zero Coupon Yield Curve, marked as such.

2. Data Collection

2.1 Data Source

Primary Source: CCIL NDS-OM market watch, central government securities segment
Collection Frequency: Daily at 5:30 PM IST, after the close of trading, with a second attempt at 6:30 PM if the first fails
Instruments: Dated Government of India securities and treasury bills
Policy rate: RBI policy repo rate, collected daily
Published curves: CCIL’s Zero Coupon Yield Curve and FBIL’s zero coupon curve, collected daily for the reference check in section 8

Market holidays follow the exchange calendar. A collection that finds the page stamped with a different date than the one requested is treated as a failure, never as that day’s data.

2.2 Trade Validation

Only executed trades enter the curve. The NDS-OM market watch reports outright trades with their last traded yield and the day’s traded amount for each security; a security with no trade on the day carries no data and is not used. Quotes and indicative levels play no part.

2.3 Bond Selection

Included:

• Fixed-rate dated Government of India securities
• Treasury bills, as zero-coupon instruments: a bill is one redemption flow, so its spot rate follows exactly from its traded price
• Valid maturity dates

Excluded:

• Bills maturing within 0.1 years (about 36 days): their yields reflect settlement and liquidity rather than term structure, and nothing is published below that tenor
• Bills traded without a price
• Floating-rate bonds and other special securities
• State development loans and other non-central-government paper
• Securities with missing or invalid data

3. Data Validation and the RBI Anchor

3.1 Outlier Detection

Yields are validated using both hard limits and a statistical filter:

The filter measures distance from the day’s curve, not from the day’s average yield: on a curve with bills near 5.5% and long bonds near 7.5% a test against the average would flag the bills for being short, not for being wrong. The residual is studentised, so a bill that alone sets the short-end slope cannot pass its own error to its neighbour. The test runs once, on days with at least eight yields.

3.2 RBI Short-Rate Anchor (Soft Anchor)

Treasury bills give the short end of the curve its level. To stop the fitted curve running away inside the first month on a day without bill trades, the RBI policy repo rate is added as a soft anchor:

Property	Value	Notes
Tenor	0.1 years (~36 days)	Short-end anchor point
Rate	RBI policy repo rate	No spread
Weight	0.02	A guard rail for the overnight tenor, not a level: the bills set the level
Floor	None	Treasury bills routinely trade through the repo rate, so the short rate is not floored at it

The anchor influences the fit with a small fixed weight and does not force the curve through it: the short end is set by the treasury bills that trade, and the anchor only stops the fitted curve running away inside the first month on a day without them.

4. Historical Aggregation for Thin Trading

The Indian market is deep: several dozen securities, bonds and bills, trade on a typical day, so the curve is normally fitted from the day’s trades alone. The aggregation below exists for the exceptional day and is the same rule used in Saudi Arabia.

IF traded_bonds_today < 3:
    days_back = 0
    WHILE total_bonds < 3 AND days_back < 7:
        days_back += 1
        Fetch bonds from (today - days_back)
        Add securities not already in the set
        IF total_bonds >= 3:
            BREAK

• Aggregates only when necessary (fewer than 3 securities today)
• Adds earlier days one at a time and stops as soon as 3 are reached; maximum lookback 7 calendar days
• Each security at most once, at its most recent trade
• The chart marks securities carried from earlier days and names the dates used

5. Curve Fitting: Nelson-Siegel Model

5.1 Bootstrapping Zero-Coupon Spot Rates

Traded securities are coupon-bearing, and a yield to maturity is not a discount rate. Before fitting, each security’s cash flows are laid out under the G-Sec convention (semi-annual coupons on a 30/360 basis, with accrued interest on the traded price) and zero-coupon spot rates are bootstrapped from the shortest maturity outwards: each security’s coupons are discounted at the spot rates already found and its own maturity’s spot rate is solved from its price. Treasury bills carry no coupon, so each bill’s node is its own annually compounded rate from its traded price, exact, and the short end is observed rather than inferred. Ind AS 19 and IAS 19 require discount rates based on spot rates, not yields to maturity, which makes this step essential.

The output is a set of annually compounded spot rates on an ACT/365.25 basis. That is the convention of every rate we publish. CCIL publishes its curve with continuous compounding; it is restated to annual compounding before it is compared with ours or drawn on the chart.

5.2 Nelson-Siegel

The bootstrapped spot rates are then fitted with the Nelson-Siegel model, used by central banks and financial institutions for yield curve construction. It provides smooth, economically sensible curves with four parameters:

Where:

• y(τ) = spot rate at maturity τ
• β₀ = long-term level (as τ → ∞)
• β₁ = short-term component (decays quickly)
• β₂ = medium-term component (hump)
• λ = decay parameter controlling where the hump sits

5.3 Estimation

For a given λ the model is linear in the three β parameters, so they are estimated by weighted least squares in closed form, with the objective measured in basis points; there is no iterative optimiser to under-converge. λ is held fixed at 3.5 years, following the practice recommended by Diebold and Li, so that the curve’s shape does not jump between days for reasons unrelated to the market. The value was calibrated on the first week of Indian trades (21 to 25 September 2026) by minimising the pooled weighted fit error with λ held and the β parameters free; a free λ moved between 2.6 and 4.3 from one day to the next, which is the jumping a fixed value prevents. The calibration is provisional and will be repeated on a month of history, as a new parameter version.

5.4 Volume-Weighted Fitting

Not all trades are equally informative. Large trades in liquid securities provide more reliable price signals than small trades in illiquid ones. Each observation is weighted by the square root of its traded value relative to the day’s largest, so that a benchmark trading fifty times another counts seven times as much rather than fifty. Weighting by value directly was tried and rejected: on a typical day the ten-year benchmark carried all the weight and the fit beyond ten years was flat, whatever the long bonds traded at.

Data Point Type	Weight	Notes
Securities with traded value	√(value traded / largest value traded that day)	Normalised to [0, 1]
RBI anchor	0.02	Soft anchor, section 3.2
Securities without traded value	0.5	Neutral default
Statistical outliers	weight × 0.5	Penalty applied after volume weighting (section 3.1)

The estimation minimises the weighted sum of squared errors:

6. Temporal Smoothing

6.1 The Need for Smoothing

Temporal smoothing blends today’s fitted parameters with the previous day’s, in proportion to how much the day’s data can be trusted. In the Indian market the day’s data can usually be trusted a great deal: on a day with 20 or more securities traded there is no blend at all, and the mechanism matters only on a thin or unusual day.

6.2 Alpha: The Confidence Score

The smoothing parameter α determines how much weight today’s data receives against the previous curve:

Alpha Calculation: α is a weighted score of five factors, each between 0 and 1:

Component	Weight	What It Measures
Bond Count	20%	Number of securities in the fit, scored on a smooth curve centred at 6
Volume	50%	Total value traded, on a log scale, against a threshold of INR 500 crore a day
Liquidity	15%	Bid-ask spreads where reported: narrow spreads score higher
Quality	10%	Goodness of fit (R²)
Coverage	5%	Maturity range covered and its spread across the curve

The full-confidence rule: on a day with 20 or more securities traded (bonds and bills, that day only, never carried from earlier days) α is set to 1 and the day’s fit is used as it is. The five scores are still recorded, and the chart says “unsmoothed” with the number of securities traded in place of α.

6.3 Interpreting Alpha

Alpha Range	Interpretation	Typical Scenario
α = 1.0	Full confidence	20 or more securities traded: the usual Indian day, unsmoothed
α > 0.8	High confidence	10 to 19 securities traded with high volume
0.5 < α < 0.8	Moderate confidence	5-10 securities, decent volume
0.2 < α < 0.5	Low confidence	3-5 securities, thin volume, lookback used
α = 0.0	No confidence	No trades within the lookback: the previous curve is carried forward and the chart says so

Each published curve shows its α, R² and fit error on the chart, so a reader can judge how much of the day’s curve is new information.

7. Long-End Extrapolation

Curves are published to 50 years. Indian trades reach 40 years, so most of the curve is observed. Beyond 30 years (or the longest observed maturity, if later) the curve’s slope decays by 10% per year, so the curve continues in its direction and flattens towards a stable level without overshooting. Rates beyond the last traded maturity are an extrapolation and should be read as such.

8. Reference Check Against Published Curves

Two zero-coupon curves are published daily for the Indian market by independent bodies: CCIL’s Zero Coupon Yield Curve and FBIL’s zero coupon curve. Both are collected, restated to annual compounding, and compared with each day’s constructed curve over the maturities both cover. Two numbers are recorded for each comparison: the root mean square difference and the largest absolute difference, in basis points.

Verdict	Condition	Effect on the website
Pass	Within 25 bps RMS and 50 bps maximum of at least one published curve	Published
Warn	Outside tolerance of both, while the two published curves disagree with each other by more than the tolerance	Published; the disagreement is between the publishers
Fail	Outside tolerance of both published curves, and they agree with each other	CCIL’s published curve is shown for that day instead, marked as a published reference curve, until the day has been reviewed and either accepted or rebuilt

A constructed curve far from both publishers is more likely to reflect a problem in the day’s data than a genuine divergence, and the check exists to keep such a day off the page. Because FBIL publishes with a lag, a verdict can settle some days after the curve is first built. The chart shows CCIL’s and FBIL’s curves as overlays on every day, so the comparison is visible to the reader as well. The published curves are checks and overlays only; they are not inputs to the construction.

9. Parameters in Force

Parameter	Value
Currency	INR
Trade source	CCIL NDS-OM market watch, 5:30 PM IST
Instruments	Fixed-coupon dated Government of India securities and treasury bills; bills maturing within 0.1 years excluded
Cash-flow convention	Semi-annual coupons, 30/360; bills as single redemption flows
Yield limits and outlier filter	0-20%; studentised residual from the day’s term structure beyond 2, weight halved
Minimum securities and lookback	3 securities; up to 7 days
Short-rate anchor	RBI policy repo rate at 0.1 years, weight 0.02, no floor
Smoothing on a full day	None: 20 or more securities traded means the day’s fit is used as it is
Fit weighting	Square root of relative value traded
Nelson-Siegel λ	3.5 years, fixed; calibrated on 21 to 25 September 2026, to be recalibrated on a month of history
Smoothing volume threshold	INR 500 crore a day (initial)
Long-end extrapolation	Beyond 30 years, slope decay 10% a year, to 50 years
Reference check	CCIL ZCYC and FBIL zero curve; 25 bps RMS and 50 bps maximum; a fail needs both publishers to agree
Published compounding	Annual, ACT/365.25
Published from	28 September 2026; earlier dates show CCIL’s published curve

10. Versioning and Change Control

Every published curve records the method version it was built with and the version of the market’s parameters, and both are visible on request. A change to the mathematics is a new method version; a change to a parameter is a new parameter version with an effective date; earlier curves are never rebuilt silently under a later version. Changes are recorded in a numbered design log with the reason for each. The λ recalibration and the volume threshold above are the two parameters expected to change once Indian history has accumulated.

11. Limitations and Appropriate Use

11.1 Market Structure Limitations

• Short End: the short end is set by the treasury bills that trade on the day; on a day without bill trades it rests on the RBI anchor and the shortest traded bonds
• Long End: trades beyond 40 years are rare; rates beyond the last traded maturity are extrapolated
• Published Curves: CCIL’s and FBIL’s curves are produced under their own methodologies and can differ from ours and from each other; the reference check is a guard, not a reconciliation
• History: Numerica’s constructed curve is published from 28 September 2026; earlier dates are CCIL’s published curve, produced by a different method

11.2 Model Limitations

• Nelson-Siegel assumes a smooth curve and may miss local anomalies
• It cannot capture market segmentation or arbitrage opportunities
• A fixed λ trades some day-to-day fit for stability of shape

11.3 Appropriate Use Cases

11.4 Professional Judgment Required

These curves should be one input among several in your decision-making process:

• For Auditors: verify the methodology is appropriate for the client’s circumstances, check α, the fit statistics and the reference-check verdict for data quality, and consider whether adjustments are needed
• For Finance Teams: cross-reference with other market indicators, understand the confidence level (α), and maintain internal documentation

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References

1. Nelson, C. R., & Siegel, A. F. (1987). “Parsimonious Modeling of Yield Curves”. Journal of Business, 60(4), 473-489.
2. Diebold, F. X., & Li, C. (2006). “Forecasting the term structure of government bond yields”. Journal of Econometrics, 130(2), 337-364.
3. Svensson, L. E. (1994). “Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994”. NBER Working Paper No. 4871.
4. The Clearing Corporation of India Ltd – NDS-OM and the Zero Coupon Yield Curve: https://www.ccilindia.com
5. Financial Benchmarks India Pvt Ltd – G-Sec zero coupon curve: https://www.fbil.org.in
6. Reserve Bank of India – policy repo rate: https://www.rbi.org.in
7. Ind AS 19 Employee Benefits; IAS 19 Employee Benefits; AS 15 Employee Benefits.

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