Building Yield Curves for ALM: From Market Quotes to Discount Factors

Every IRRBB calculation begins with a yield curve. Cashflow projections, present values, duration, ΔEVE — all depend on the quality of the discount factors you extract from market data. A poorly built curve doesn't just introduce noise; it systematically biases every downstream result.

Why the Curve Matters

The yield curve is the foundation of the entire IRRBB measurement framework. It serves three essential functions:

• Discounting cashflows — Computing the present value of every future payment on every asset and liability. This is the basis of EVE and all sensitivity measures (duration, DV01, key rate duration).
• Projecting forward rates — Determining the expected floating rates at future reset dates. This drives the cashflow projection for floating-rate instruments and, by extension, ΔNII.
• Defining the base scenario — The curve is the "as of today" reference point. All six BCBS scenarios and any internal stress scenarios are defined as perturbations to this base.

An error of 5 basis points in a 10-year discount rate might seem small, but when applied to a $10 billion mortgage portfolio with ~7-year duration, it translates to a $35 million ΔEVE bias — potentially the difference between passing and failing the outlier test.

What Curve to Build

The first decision is which instruments to use as inputs. This choice has changed significantly since the LIBOR transition.

For IRRBB purposes, most banks now build a SOFR OIS curve (or the local equivalent) as their primary risk-free discounting curve. Credit spreads are layered on top for specific asset classes, but the base curve is OIS.

The input instruments typically include:

• Short end (0–1Y): Overnight rate, 1-week to 12-month OIS
• Medium term (1–5Y): OIS swaps at annual tenors
• Long end (5–50Y): OIS swaps at 5Y, 7Y, 10Y, 15Y, 20Y, 30Y, and sometimes 40Y/50Y

Bootstrapping: From Par to Zero

Market quotes are typically expressed as par rates — the coupon rate at which a swap has zero net present value at inception. But for discounting purposes, we need zero rates (also called spot rates) — the yield on a single cashflow at each maturity.

Bootstrapping is the iterative process of extracting zero rates from par rates. The key insight: a par instrument can be decomposed into a series of zero-coupon cashflows, and if we know the zero rates for all but the last cashflow, we can solve for the missing one.

The algorithm proceeds from short to long maturities. At each step, all previously bootstrapped zero rates are used to discount the intermediate cashflows, and the equation is solved for the one remaining unknown — the zero rate at the new maturity.

Between quoted maturities (e.g., between the 2Y and 3Y quotes), we need an interpolation method to fill in the gaps.

Interpolation Methods

Bootstrapping gives us zero rates at the exact tenors where we have market quotes. But we need discount factors at arbitrary maturities — every cashflow date for every instrument on the balance sheet. Interpolation fills the gaps.

Three methods dominate ALM practice:

Day-Count and Compounding Conventions

Two seemingly minor technical choices have material impact on curve construction:

Day-count conventions determine how time fractions are computed between dates. The most common:

• ACT/360 — Actual days divided by 360. Standard for money markets and SOFR.
• ACT/365 — Actual days divided by 365. Common in GBP markets and some ALM systems.
• 30/360 — Assumes 30 days per month. Used in some bond markets.

A mismatch between the day-count convention used to build the curve and the convention assumed by the cashflow engine introduces systematic bias. Five basis points of error at the 10Y tenor can arise from this alone.

Compounding conventions determine how rates are converted to discount factors:

• Continuous: DF = e^−r·t — Mathematically convenient; standard in academia and some derivatives pricing.
• Simple: DF = 1 / (1 + r·t) — Used for short-term instruments (under 1 year).
• Discrete (annual/semi-annual): DF = 1 / (1 + r/n)^n·t — Bond market convention.

The key principle: be consistent. The same compounding convention must be used when building the curve and when discounting cashflows. Mixing conventions is a surprisingly common source of ΔEVE errors in practice.

A Worked Example

The bootstrapped zero rates will differ from the par rates — especially at longer tenors where coupon reinvestment effects accumulate. For an inverted curve (short rates above long rates), zero rates will be slightly lower than par rates at the long end. For a normal curve, slightly higher.

Common Pitfalls

Yield curve construction is conceptually straightforward but operationally fragile. Common sources of error:

1. Stale quotes. If market quotes are not updated daily (or intraday for volatile markets), the curve reflects yesterday's — or last week's — rates. For IRRBB measurement at month-end, this may be acceptable. For daily risk monitoring, it's not.

2. Thin liquidity at the long end. Beyond 30 years, market quotes become sparse and bid-ask spreads widen. For banks with 40-year or 50-year assets (pension-related liabilities, long-duration securities), the extrapolation method for the ultra-long end can move ΔEVE materially.

3. Missing tenors. If you only have quotes at 1Y, 2Y, 5Y, 10Y, and 30Y, the interpolation between 10Y and 30Y spans 20 years — a large gap where the assumed shape can diverge significantly from reality. Adding intermediate quotes (15Y, 20Y) improves precision.

4. Negative forward rates. Some interpolation methods (particularly cubic spline) can produce negative instantaneous forward rates between vertices, even when all input rates are positive. This is mathematically undesirable and can cause pricing anomalies. Flat forward interpolation avoids this by construction.

5. Convention mismatches. Building the curve with ACT/365 but discounting cashflows with ACT/360, or mixing continuous and discrete compounding, are among the most common (and hardest to detect) errors in production IRRBB systems.

The Bigger Picture

The yield curve is not just an input — it is the lens through which every IRRBB number is viewed. Duration, DV01, ΔEVE, the fair value of a swap hedge — all are derivative of the discount factors extracted from this curve. An error in curve construction propagates into every downstream calculation.

For banks operating in multiple currencies, curve construction multiplies in complexity — each currency needs its own OIS curve, with cross-currency basis adjustments for foreign-currency positions. But the principles remain the same: start with clean market data, bootstrap consistently, interpolate thoughtfully, and validate relentlessly.