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THE LIBOR FORWARD CURVE FOR OIS DISCOUNTINGA useful application for the OIS discount factors is to calculate the implied LIBOR forward curve that is consistent with the observed rates on collateralized interest rate swaps. To see the difference between LIBOR and OIS discounting, assume that the fixed rates on the sequence of collateralized swaps are the same as before. That is, the fixed rates are 1.04%, 1.58%, 2.12%, 2.44%, 2.76%, 3.08%, and 3.40% for quarterly settlement swaps against 3month LIBOR for maturities ranging from 6 to 24 months. Given that 3month LIBOR is assumed to be 0.50%, the 3×6 implied LIBOR forward is calculated using the 0×3 and 0×6 OIS discount factors and the 6month swap fixed rate. This equation follows the principle of swap pricing – the fixed rate, here 1.04% for the 6month maturity, is the “average” of the forward curve in that the present values are the same after the rates are monetized by multiplying by the notional principal (100) and the daycount factor (0.25). Previously, we used the relevant segment of the forward curve to get the swap fixed rate. Now, we use the known swap fixed rate to solve for the incremental forward rate. The 6×9 implied forward rate further illustrates the property that OIS discounting lowers the implied LIBOR forward curve when the LIBOROIS spread is positive and the forward curve is upwardly sloped. In Table 8.1, the 6x9 implied forward rate for 3month LIBOR is shown to be 2.6694%. That rate is consistent with LIBOR discount factors. Here it is 2.6671% for OIS discounting. The difference in the implied forward rates becomes a bit larger moving out along the curve. These are the remaining implied rates for LIBOR using the OIS discount factors. Notice that in each equation, the incremental forward rate is the unknown variable. The known inputs are the forward rates up to that time period and the swap fixed rate. You probably can envision the spreadsheet that would do these calculations for you. In this example, the use of OIS rather than LIBOR discount factors does not make for a large difference in the implied LIBOR forward curve. It is only 1.29 basis points, 5.7427% compared to 5.7298%, for the 21 x 24 time period. Nevertheless, these are the correct forward (or projected) rates to use on collateralized swaps when determining the fixed rate on a nonvanilla design, for instance, a varying notional principal, forwardstarting swap, or when pricing options on swaps (i.e., “swaptions”). These implied forward rates are also important when valuing a swap using the combinationofbonds approach. Consider again the 3.85%, $50 million notional principal, 12month quarterly settlement swap shown earlier to have a market value of $859,019 when collateralized and marked against a 2.12% atmarket contract. That amount is the annuity of $216,250 for the difference in the fixed swap rates discounted using the OIS curve. The equation is repeated here. The implicit 3.85% fixedrate bond pays a quarterly coupon of $481,250. It can be valued using the OIS discount factors as if the bond has been upgraded to riskfree status. This is greater than the bond price of $50,856,523 found earlier, where it is calculated using the (lower) LIBOR discount factors. The key point is that it would be incorrect to assume that the implicit floatingrate bond continues to be priced at par value. That would value the swap wrongly at $1,104,920 (= $51,104,920 – $50,000,000). This “floater” should be treated as a riskfree security having a price greater than $50,000,000 because it is collateralized. To get that price, assume that the implied forward rates based on the OIS discount factors are the fixed rates on collateralized FRAs. That means the cash flows on the LIBOR floatingrate bond can be fixed via hedging. The value of the implicit floater is $50,245,902. The difference in the (unrounded) values for the two implicit bonds is $859,019, which is the same as found directly by discounting the $216,250 annuity with the OIS discount factors. This valuation exercise assumes the same series of swap fixed rates whether the contracts are uncollateralized or collateralized. But that is not likely to happen in reality. Other things being equal, the fixed rate on the collateralized swap should be higher than when it is uncollateralized. The reasoning is similar to the adjustment between interest rates on exchange traded futures and overthecounter forwards. The idea is that posting cash collateral is costly to the counterparty for which the swap is underwater, meaning it has a negative market value. Either the funds need to be borrowed or are diverted from other uses, thereby imposing a financial or, at least, an opportunity cost to the entity. Moreover, the rate earned on cash collateral nowadays is the OIS rate whereas the bank's cost of funds is LIBOR (or above). The reason for the higher fixed rate on a collateralized swap is that the impact of having to post costly collateral is not symmetric between the two counterparties – the fixedrate receiver suffers from interest rate volatility while the payer benefits. Suppose the contract is underwater to the fixedrate receiver because swap rates have risen since entrance. If rates rise further, more collateral is needed; if rates fall, less is required. In contrast, suppose the swap is underwater to the fixedrate payer because rates have fallen. If rates then rise somewhat, less collateral is needed; and if rates fall further, more is required. Systematically, the fixedrate receiver posts more costly collateral when rates go up and the fixedrate payer posts more when rates go down. This asymmetry, other things being equal, leads to a higher fixed rate on the collateralized swap. CONCLUSIONInterest rate swaps demonstrate how once again we can employ a toolkit of bond math techniques to go beyond basic fixedrate and zerocoupon securities. Interest rate swaps combine aspects of fixedrate and floating rate bonds. They can be interpreted as holding a long position in one and a short position in the other. That allows us to get reasonable estimates of the modified duration and basispointvalue for a swap. Most important, pricing swaps at initiation and valuing them thereafter are direct applications of spot and forward curve analysis. Getting the “source” forward curve from futures markets is a difficult technical matter, and one that requires an adjustment factor drawn from a mathematical model of the term structure of interest rates. Alternatively, we can work with observed fixed rates on standard swaps and interpolate rates for the intermediate maturities. Then we can bootstrap the implied spot and forward rates. Once we have a forward curve, our swap pricing and valuation analysis is merely a spreadsheet away. We can do this in the traditional world of LIBOR discounting or in the brave new world of OIS discounting. 
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