Overnight Index Swap Calculation Example

Comprehensive Guide to Overnight Index Swap (OIS) Calculations

An Overnight Index Swap (OIS) is an interest rate swap where the periodic floating rate is equal to the geometric average of an overnight index (such as SOFR, SONIA, or €STR) over the payment period. OIS contracts have become the standard for hedging and speculating on short-term interest rates since the 2008 financial crisis, as they reflect the credit and liquidity risk-free rates more accurately than traditional interbank rates like LIBOR.

Key Components of OIS Calculations

1. Notional Amount: The hypothetical principal amount on which interest payments are calculated. This amount is not exchanged but serves as the basis for computing interest payments.
2. Fixed Rate: The predetermined rate paid by one counterparty, typically set at the swap’s inception based on market expectations of future overnight rates.
3. Floating Rate Index: The overnight benchmark rate (e.g., SOFR) that determines the floating leg payments. This rate is compounded over the payment period.
4. Compounding Frequency: How often the overnight rates are compounded (daily, weekly, etc.). More frequent compounding increases the effective rate.
5. Day Count Convention: OIS typically uses an “Actual/360” convention for the fixed leg and “Actual/360” or “Actual/Actual” for the floating leg, depending on the currency.

Step-by-Step OIS Calculation Process

The calculation involves two legs:

• Fixed Leg: Computed as:
  Fixed Payment = Notional × Fixed Rate × (Days / Day Count Basis)
• Floating Leg: Computed as:
  Floating Payment = Notional × [(1 + r₁) × (1 + r₂) × … × (1 + rₙ) – 1]
  where r₁, r₂, …, rₙ are the daily overnight rates over the period.

The net payment is the difference between the floating and fixed legs, paid by the party owing more. The effective rate is the annualized rate that equates the present value of both legs.

Compounding Methods in OIS

Compounding significantly impacts the floating leg’s value. The three primary methods are:

Method	Formula	When Used
Daily Compounding	(1 + r₁)(1 + r₂)…(1 + rₙ) – 1	Most common for USD SOFR swaps
Periodic Compounding	[∏(1 + rᵢ)]^(1/n) – 1	Used for some non-USD OIS
Simple Average	(r₁ + r₂ + … + rₙ)/n	Rare; used in legacy contracts

Real-World Example: SOFR OIS Calculation

Consider a 1-year SOFR OIS with:

• Notional: $10,000,000
• Fixed Rate: 2.50%
• SOFR rates over the year average 2.30% (daily compounded)

Fixed Leg:
$10,000,000 × 2.50% × (365/360) = $253,472.22

Floating Leg:
$10,000,000 × [(1 + 0.0230)^(365/365) – 1] = $230,000.00

Net Payment:
$230,000.00 – $253,472.22 = -$23,472.22 (fixed receiver pays floating payer)

OIS vs. LIBOR Swaps: Key Differences

Feature	OIS (SOFR)	LIBOR Swap
Credit Risk	Near risk-free (secured)	Includes bank credit risk
Tenor	Overnight (compounded)	Term rates (1M, 3M, etc.)
Volatility	Lower (less speculative)	Higher (forward-looking)
Collateralization	Typically fully collateralized	Often uncollateralized
Market Size (2023)	$150+ trillion	$50 trillion (declining)

Regulatory Framework for OIS

The shift from LIBOR to OIS reflects regulatory priorities post-2008. Key regulations include:

• Dodd-Frank Act (2010): Mandated central clearing for standardized swaps, reducing systemic risk. OIS became the preferred instrument due to its transparency.
• Basel III: Increased capital requirements for uncollateralized derivatives, making OIS (typically collateralized) more capital-efficient.
• SEC & CFTC Rules: Enhanced reporting requirements for swaps, favoring OIS due to its simpler valuation.

For further reading, consult the Federal Reserve’s OIS data and the BIS report on risk-free rates.

Advanced Topics: OIS Discounting

Since 2008, OIS rates have replaced LIBOR as the standard for discounting collateralized derivatives. This shift reflects:

1. Collateralization: OIS rates approximate the risk-free rate when collateral is posted daily.
2. Multi-Curve Framework: Post-crisis, markets use separate curves for discounting (OIS) and forwarding (LIBOR/SOFR).
3. Cross-Currency Basis: OIS rates are used to compute the cost of funding in different currencies.

The ARRC’s SOFR conventions (New York Fed) provide standardized methodologies for SOFR OIS calculations.

Common Pitfalls in OIS Calculations

• Day Count Mismatches: Using “Actual/365” instead of “Actual/360” can distort payments by ~0.8% annually.
• Holiday Adjustments: Failing to account for non-business days in the overnight rate series.
• Compounding Errors: Incorrectly applying simple averaging instead of geometric compounding.
• Curve Construction: Using linear interpolation between OIS tenors without convexity adjustments.

Practical Applications of OIS

OIS contracts are used for:

• Hedging: Corporates hedge floating-rate liabilities (e.g., commercial paper) by paying fixed in an OIS.
• Speculation: Traders bet on central bank policy shifts by taking positions in OIS futures.
• Regulatory Arbitrage: Banks optimize capital requirements by replacing LIBOR swaps with OIS.
• Cross-Currency Basis Swaps: OIS rates serve as the floating leg in FX hedges.

The transition from LIBOR to SOFR OIS has been documented by the CFTC’s Interest Rate Swap Report, which shows SOFR OIS volume surpassing LIBOR swaps since 2021.