# TRADITIONAL LIBOR DISCOUNTING

To see the impact of OIS discounting, let's first do another example of swap valuation under traditional LIBOR discounting. Suppose that a 3.85% fixed-rate, $50 million notional principal, quarterly settlement interest rate swap on 3-month LIBOR has 12 months remaining. This swap might originally have had a tenor of five years and four years have gone by. Its current market value is based on a comparison to the 2.12% fixed rate on the 12-month swap. The annuity is the difference between the contractual and the current market fixed rates, times the notional principal and the day- count factor.

The present value of this annuity can be calculated using the sequence of bootstrapped implied spot rates shown in Table 8.1. The value of the swap is $856,523.

The bootstrapped discount factors give the same result.

This interest rate swap is an asset worth $856,523 to the fixed-rate receiver and a liability for the same amount to the fixed-rate payer because market rates are lower than when the swap was originated.

Another approach to swap valuation is to use the interpretation of the contract as a long/short combination of floating-rate and fixed-rate bonds. The implicit $50 million, floating-rate bond pays interest quarterly based on 3-month LIBOR. In principle, the unknown levels for future LIBOR can be hedged using FRAs or Eurodollar futures to lock in the sequence of forward rates. Therefore, the value of the floater is the present value of the projected cash flows. Using the LIBOR discount factors, this present value is $50 million.

The implicit, $50 million, 3.85% fixed-rate bond pays interest in the amount of $481,250 each quarter (= $50,000,000 * 0.0385/4). The value of this bond, also using the LIBOR discount factors, is $50,856,523. (All of the calculations are done on a spreadsheet using unrounded discount factors.)

The value of the swap is just the difference in the bond prices, $856,523.

The reason for illustrating the two methods to value an interest rate swap is that with traditional LIBOR discounting, the same result is obtained. That will not be true in a world of OIS discounting. That's an important theme in the remainder of this chapter.

The two key assumptions to this calculation of market value are: (1) The swap is not collateralized (or, if it is, the collateral is not considered in the valuation methodology), and (2) the fixed-rate payer is a “LIBOR-flat” borrower. The second assumption means that the owing counterparty, here the fixed-rate payer, has credit quality consistent with the banks that are used to establish the LIBOR index. In other words, this counterparty can borrow funds for 12 months at LIBOR flat (meaning a margin of zero above the reference rate) on a quarterly payment floating-rate basis or at 2.12% fixed. In sum, the LIBOR- based implied spot rates and discount factors are appropriate to get the present value of its future obligations. Usually, this corresponds to an investment-grade borrower having a quality rating of A+ to AA- on its debt.

Suppose instead that the fixed-rate payer is a financially distressed company that has had its debt liabilities downgraded to noninvestment grade. If the fixed- rate receiver requested early termination of the swap, the payer would offer to settle the obligation for something less than $856,523. That counterparty to the contract would argue that the present value of the (unambiguous) $216,250 annuity should be calculated with discount factors that reflect its higher-than- LIBOR-flat or higher-than-2.12%-fixed 12-month cost of borrowed funds. In sum, the fair value of the swap would be overstated at $856,523.

While using default-risk-adjusted discount factors is appropriate in principle for an early termination, it would be unwieldy for routine valuations carried out daily by swap dealers having a multitude of open contracts. The advantage to using the LIBOR swap curve is that there are good publically available data for a full range of maturities. Importantly, the bootstrapped numbers are “internal” to the valuation problem. In this traditional approach, one can start with either the LIBOR forward curve or fixed rates on at-market swaps and easily infer the implied spot rates and discount factors needed to value the swap book.