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Understanding ARRs / RFRs with Calculation Examples & System Implementation – Part 2 of 2

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I. Recap & Expanded Context

We already covered:

  • Why LIBOR was phased out (manipulation, illiquidity, regulatory pressure)
  • The shift to transaction-based overnight Risk-Free Rates (RFRs) as Alternative Reference Rates (ARRs)
  • The absence of bank credit risk and forward-looking term structure in plain RFRs
  • The development of TERM rates (e.g. Term SOFR, Term CORRA) to provide forward-looking tenors
  • The existence of other benchmark rates (EURIBOR, NIBOR, etc.), some reformed rather than replaced
  • The role of governance, oversight bodies, free vs paid access, fallback frameworks, legal and operational impacts

This section drills into calculation methodologies, formula examples, and system behavior related to overnight ARR-based interest calculations.

II. Interest Calculation Approaches for ARRs / RFRs

Many ARRs are backward-looking, and thus the interest is often calculated only near or after the end of the interest period. Several methods exist:

  1. Simple Interest (Simple ARR)
  2. Compounded in Arrears
  3. Daily Rate with Compounding
  4. Simple Average

Definitions of Terms Used in Formulas

  • d_b: number of business days in the interest period
  • d_c: number of calendar days in the interest period
  • r_i: published ARR rate for business day i
  • n_i: number of calendar days for which r_i applies
  • N: day count convention (360, 365, etc.)
  • P_m: compounding factor on day m (P_0 = 1)

III. Formula Examples

1. Simple Interest (Simple ARR)

Formula:

Accrued Interest = (1/N) * Σ(r_i * n_i) * Notional

Alternatively expressed as:

R_Simple = (1/N) * Σ(r_i * n_i)

This method uses raw ARR rates and does not account for compounding. It’s administratively simple but may not accurately reflect reinvestment effects.

2. Compounded in Arrears

Formula (Aggregate):

R_Compounded = ∑[(1 + (r_i * n_i) / N)] - 1 * (N / d_c)

Alternative Expression:

Accrued = [Product over i of (1 + r_i * n_i / N) - 1] * Notional

3. Daily Rate with Compounding

Formula:

P_m = P_{m-1} * (1 + r_m * n_m / N)
Accrued = (P_d_b - 1) * Notional

Similar to compounded in arrears, but built incrementally over each day.

4. Simple Average

Formula:

R_Avg = (1/d_b) * Σ(r_i)
Interest = (R_Avg * d_c / N) * Notional

IV. System Parameters for Calculation

1. Lookback Days

Offsets rate observation window so earlier published rates are used when newer rates are not yet available.

2. Lockout Days

Freezes the rate for the final days of a period to provide certainty ahead of payment.

3. Pricing Lag Days

Accounts for the delay in publishing overnight rates (e.g. T+1). E.g., SOFR is published at 8:00 AM the next business day.

4. Payment Lag Days

Delays payment obligation beyond interest end date. This allows time for invoice generation and reconciliation.

5. Rate Basis

Rate basis (e.g., Actual/360 or Actual/365 or Actual/Actual) must be consistent across interest calculation and accruals.

6. Accrual Rules

Special rules apply for leap years, boundary-crossing accruals, and prohibited combinations with payment lag.

V. Example with New Numbers

Scenario:

  • Notional: $40,000,000
  • Period: Aug 1 to Aug 16 (16 calendar days)
  • Business days: 12
  • N = 360
  • Sample RFRs (r_i in decimal): 0.0470 to 0.0500 incrementally

Example Calculation: Simple Interest

Avg r_i = 0.0485
R_Simple = (0.0485 * 16) / 360 = 0.00215556
Interest = 40,000,000 * 0.00215556 = $86,222.22

Example Calculation: Compounded in Arrears (Hypothetical Product M = 1.0019)

R_Compounded = (1.0019 - 1) * (360 / 16) = 0.04275
Interest = 40,000,000 * 0.04275 = $1,710,000

VI. Additional Insights: Term Rates & IBOR Hybrids

Term Rates

  • Term SOFR, Term CORRA: Forward-looking rates derived from futures curves
  • Formula:
Term Rate = (1/Δ) * [1/DF_end - 1/DF_start]

Used in cash markets but often excluded from derivatives.

Reformed Benchmarks

  • EURIBOR (hybrid methodology)
  • NIBOR, HIBOR, TIBOR: Still used in some regional markets with reforms

VII. Summary & Practical Tips

  1. Align compounding logic with spread and day count
  2. Carefully document fallback language
  3. Test edge cases (month-end, leap year, holiday)
  4. Balance simplicity vs accuracy
  5. Monitor rate publication delays and data integrity
  6. Use lookback and lockout configurations to ensure payment readiness
  7. Educate borrowers that interest may not be known until late in the period

Cheers,

Gage Gorman

www.gagegorman.com

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