Let's now assume that this 12-month, 3.85% fixed-rate, $50 million notional principal, quarterly settlement interest rate swap is collateralized. Cash that is posted to meet the collateral obligation earns the OIS rate. To get the OIS discount factors, we need to assume some things about that market. Suppose the 3-month fixed rate is 0.10% on an OIS for a notional principal of $50 million. At settlement, the payoff will be based on the difference between the fixed and floating legs on the swap. Assuming 90 days for the three months (i.e., for simplicity, the 30/360 day-count convention), the fixed leg is:
The floating leg depends on the sequence of realized daily reference rates.
EFFV EFF2, ... , EFF90 are the reported daily observations for the effective fed funds rate. (Note that this neglects the odd manner in which the Friday fed funds rates is used for Saturday and Sunday – rather than being compounded for three days, simple interest is used.) Net settlement on the OIS is the difference between the two legs. The 3-month OIS fixed rate determines the 0×3 discount factor.
Suppose that the fixed rate on a 6-month OIS is 0.62%. Given 180 days for the time period and $50 million in notional principal, the fixed and floating legs are:
The 6-month discount factor for the OIS curve is:
In general, the divisor is “Year/Days,” where Year is 360 or 365 days, and Days is the actual or assumed number of days in the time period.
OIS fixed rates for other maturities out to one year typically are quoted in the same manner, that is, on a simple interest basis common for money market instruments. Suppose that the fixed rates for 9 and 12 months are 1.10% and 1.64% and that these apply to 270 and 360 days given the simplifying assumption of the 30/360 day-count. The 0×9 and 0×12 OIS discount factors are:
OIS contracts maturing in more than one year usually are designed to have periodic settlements against the fixed rate, as is standard for LIBOR swaps. To stay with the simplistic design of these numerical examples, assume that the annual OIS fixed rates for quarterly settlement are: 1.98% for 15 months, 2.32% for 18 months, 2.63% for 21 months, and 2.90% for 24 months. For these, the discount factors are obtained using a bootstrapping technique equivalent to that shown in Chapter 5. The difference in presentation here is that the discount factors are used directly rather than the implied spot rates.
Consider again the valuation of the seasoned LIBOR swap from the previous section. It has a fixed rate of 3.85%, a notional principal of $50 million, and 12 months remaining until maturity. Using the LIBOR swap discount factors, the market value is shown above to be $856,523. Now, using the slightly higher OIS discount factors, the market value of the swap goes up to $859,019.
Clearly, the difference is not large but that is due to the low level of interest rates and the relatively short time frame in the example. The impact of using OIS rather than LIBOR discount factors is greater for longer-term swaps and when the difference between the contractual rate and the current market swap rate is larger. What matters is that this market value better captures the minimal credit risk on a collateralized interest rate swap. It's better bond math.