Methodology for Discount Curve Construction – Saudi Arabia

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1. Overview

Our yield curve construction methodology is designed specifically for emerging markets with thin trading volumes. The Saudi government bond market presents unique challenges: not all bonds trade every day, and when they do trade, volumes can be modest. Our approach addresses these challenges through intelligent historical aggregation and volume-weighted curve fitting.

2. Data Collection

2.1 Data Source

Primary Source: Saudi Exchange (Tadawul) Sukuk Market Watch page
Collection Frequency: Daily at 5:00 PM KSA (after market close at 3:00 PM KSA)
Instruments: Government bonds and sukuk only (corporate bonds excluded from curve construction)

2.2 Trade Validation

A bond is considered “traded” only if it passes a rigorous two-step validation process:

Step	Validation Criteria	Purpose
Step 1 Main Page Check	• Last traded price is not null • Bid yield is not null • Ask yield is not null	Candidate identification
Step 2 Detail Page Verification	• Last traded nominal > 0 • Last trade date = curve date	Confirm actual transaction occurred today

This two-step process prevents stale quotes or indicative prices from being treated as actual trades.

2.3 Bond Selection

Included:

• Saudi government bonds and sukuk
• Fixed-rate coupon instruments only
• Valid maturity dates

Excluded:

• Corporate bonds (collected for reference but not used)
• Floating-rate instruments
• Bonds with missing or invalid data

3. Data Validation and Quality Control

3.1 Outlier Detection

Yields are validated using both hard limits and statistical methods:

Outliers are typically caused by data entry errors, stale quotes, or illiquid bonds with wide bid-ask spreads.

3.2 SAMA Short-Rate Anchor

To improve curve fitting at the short end, we incorporate the SAMA reverse repo rate as an anchor point:

Anchor Point	Source	Weight
Overnight rate (0.003 years maturity)	SAMA official reverse repo rate	Fixed at 1.0

This anchor prevents unrealistic short-rate extrapolation when no short-dated bonds trade.

4. Historical Aggregation for Thin Trading

4.1 The Challenge

On any given day, only 5-15 government bonds typically trade out of 50+ outstanding issues. With so few observations, a parametric curve fit would be unreliable. Our solution: intelligently aggregate recent historical trades.

4.2 Aggregation Logic

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 bonds with unique ISINs IF total_bonds >= 3:
            BREAK

Key Features:

• Only aggregates when absolutely necessary (< 3 bonds today)
• Adds bonds incrementally, day by day
• Stops as soon as minimum threshold (3 bonds) is reached
• Maximum lookback: 7 days (configurable)
• Unique ISINs only – no duplicate bonds

5. Curve Fitting: Nelson-Siegel Model

5.1 Bootstrapping Zero-Coupon Spot Rates

Before fitting the Nelson-Siegel model, we first bootstrap zero-coupon spot rates from the observed bond yields-to-maturity (YTM):

Process:

• Input: Observed YTM from traded bonds (coupon-bearing instruments)
• Method: Piecewise-constant forward rate assumption
• Output: Zero-coupon spot rates at each bond’s maturity

Bootstrapping strips out the coupon effect to derive the pure discount function. IAS 19 requires discount rates based on spot rates, not yields-to-maturity, making this step essential for compliance.

5.2 Nelson-Siegel

The spot rates derived from bootstrapping are then used to fit a smooth curve. We use Nelson-Siegel model which is widely used in central banks and financial institutions for yield curve construction. It provides smooth, economically sensible curves with just four parameters:

Where:

• y(τ) = yield at maturity τ
• β₀ = long-term level (as τ → ∞)
• β₁ = short-term component (decays quickly)
• β₂ = medium-term component (hump)
• λ = decay parameter controlling curve shape

5.3 Volume-Weighted Fitting

Not all trades are equally informative. Large trades in liquid bonds provide more reliable price signals than small trades in illiquid bonds. We weight each observation by its trading volume:

Data Point Type	Weight Formula
Bonds with volume data	weight = volume / max_volume_that_day
SAMA anchor	weight = 1.0 (fixed)
Bonds without volume	weight = 0.5 (neutral default)
Statistical outliers	weight × 0.5 (penalty)

The optimization minimizes weighted mean squared error:

6. Temporal Smoothing

6.1 The Need for Smoothing

Day-to-day fluctuations in yield curves can be dramatic in thin markets, even when underlying fundamentals haven’t changed. A single large trade can shift the entire curve. Temporal smoothing addresses this by blending today’s fitted curve with yesterday’s curve.

6.2 Alpha: The Confidence Score

The smoothing parameter α (alpha) determines how much weight to give today’s data vs. yesterday’s curve:

Alpha Calculation: Alpha is calculated based on five factors, each scored 0-1:

Component	Weight	What It Measures
Bond Count	20%	Number of bonds traded (more = higher confidence)
Volume	50%	Total trading volume (higher = more reliable)
Liquidity	15%	Spread of maturities across curve
Quality	10%	Model fit quality (R²)
Coverage	5%	Proportion of bonds that actually traded today

6.3 Interpreting Alpha

Alpha Range	Interpretation	Typical Scenario
α > 0.8	High confidence	10+ bonds traded with high volume
0.5 < α < 0.8	Moderate confidence	5-10 bonds, decent volume
0.2 < α < 0.5	Low confidence	3-5 bonds, thin volume, used lookback
α = 0.0	No confidence	Zero trades, using previous curve

7. Limitations and Appropriate Use

7.1 Market Structure Limitations

The Saudi government bond market has structural characteristics that affect curve reliability:

• Thin Trading: Many bonds don’t trade daily, requiring historical aggregation
• Buy-and-Hold Bias: Saudi banks hold bonds to maturity, reducing secondary market activity
• Limited Maturity Range: Most bonds mature within 5-15 years; long-end extrapolation is less reliable
• Sukuk vs. Bonds: We treat sukuk and conventional bonds identically for curve purposes, though their structures differ

7.2 Model Limitations

The Nelson-Siegel model makes simplifying assumptions:

• Assumes smooth, continuous curve (may miss local anomalies)
• Cannot capture arbitrage opportunities or market segmentation
• Parametric form may not perfectly fit all curve shapes

7.3 Appropriate Use Cases

7.4 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 alpha values for data quality, and consider whether adjustments are needed
• For Finance Teams: Cross-reference with other market indicators, understand the confidence level (alpha), 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. Svensson, L. E. (1994). “Estimating and Interpreting Forward Interest Rates: Sweden 1992-1994”. NBER Working Paper No. 4871.
3. Saudi Exchange (Tadawul) – Sukuk Market Watch: https://www.saudiexchange.sa
4. International Accounting Standards Board (IASB). IAS 19 Employee Benefits.

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