By Kevin Samborn, Sapient Global Markets

Historically, financial institutions used a single standard curve to value derivatives. Recently market participants have started to move away from a single curve for both discounting and forecasting. Instead, they are using multiple curves, each playing a specific role in valuation. Forecast curves continue to be based on Libor, but are built specifically for different tenors. Also, a significant number of participants construct discount curves based on overnight indexed swaps rates.

**After The Crisis**

Historically, all derivative valuation was performed assuming a single standard curve. This methodology was based on the belief all market participants had equal credit risk, the firm could fund itself with Libor and the embedded credit risk for rates of different maturities was negligible. This methodology made pricing easy, since calculation libraries, systems, and reports only required one curve. During the crisis, the assumption each institution had equal credit risk was clearly invalidated. This observation was plainly shown in several market trends. For example, the spread between Libor and 'risk free' OIS, such as those based on Fed Funds Effective Rate and Euro OverNight Index Average widened. Also, the basis spread between Libor rates of different maturities widened.

Market participants noticed a dramatic increase in the Libor-overnight indexed swaps spread during the credit crunch. In Aug. 2007, this spread was around 10 basis points. However, by Oct. 2008, following the collapse of **Lehman Brothers**, the spread had widened to 365 basis points. (*"The LIBOR-OIS Spread as a Summary Indicator", Rajdeep Sengupta and Yu Man Tam, Federal Reserve Bank of St. Louis, Economic Synopses, 2008, Number 25.*) The previously negligible single currency basis swap spread between Libor rates of different maturities also became significant. For example, the three-month versus six-month USD Libor five-year swap spread widened from a very stable sub-1 basis point spread for the first six months of 2007 to a high of almost 21 basis points in March 2009. (*Bloomberg, LP, USBC1 Currency HP, January 2, 2007 to May 26, 2010.*)

To adapt to the new conditions of unequal credit risk, market participants are adopting a two-curve framework for valuing derivatives. One curve is used for discounting and a curve that matches the maturity of the underlying floating rate is used for projection.

**The Discount Curve**

The discount curve is the foundation for all other calculations, including construction of the forecast curve. A basic concept in all valuation is net present value. NPV is the value of future cash flows according to their worth today. This concept is derived through the application of a discount curve. The curve is a mathematical function of discount factors for each point in time from today into the future. Each discount factor is the value of one unit of currency at a future point in time, relative to its value today. For example, if the one-year interest rate is 3%, the value of USD1.0 in one year is approximately USD0.97 today. This is because the 0.97 can be invested at the one-year rate, and 1.0 will be returned. The curve that contains all the discount factors is referred to as the discount curve. To adapt to the issue that Libor no longer reflects equal credit risk, many market participants are now using a discount curve built from OIS.

The underlying reasons for the move from a Libor-based curve to an OIS-based standard curve are twofold. First, the intense focus on collateral led the market to understand that the discounting methodology used to value derivatives must match the calculation of interest paid on collateral. Second, during the credit crisis, banks refused to lend to each other because of counterparty credit risk. This observation resulted in a perceived breakdown of the reliability of LIBOR as a benchmark, as it is a consensus composite.

When a derivative is in-the-money, the counterparty with positive mark-to-market collects collateral from the other counterparty. Interest is paid on posted collateral, including both bilateral bilateral **International Swaps and Derivatives Association** Credit Support Annexes and through centrally cleared **LCH.Clearnet** SwapClear margin accounts. The rate used is a standard overnight rate, such as Fed Funds, EONIA, or Sterling OverNight Index Average. These rates are considered as close to 'risk free' as possible since the rates exist only for a single day. This process protects the positive counterparty in case of default. Since the in-the-money counterparty is paying interest on posted cash collateral, the counterparty is essentially funding the position with the overnight rate.

Therefore, it is natural to present value-collateralized derivatives with a funding curve built from OIS. Opinions vary on how far to take the new methodology. Some participants believe that an OIS curve should be used to discount all trades. Others believe that it should only be used for collateralized trades with the bank's unsecured cost of funding used for uncollateralized trades. This cost of funding, however, is subjective and can lead to many different prices in the market based on relative value. Still, others continue to use the legacy way, the old standby Libor curve--using Libor deposits, futures, and swap rates. These banks claim that this method still captures the most efficient and liquid market. While the first and third options are attractive for their simplicity, the second option is probably the most accurate, but is much more subjective and complex.

**The Forecast Curve**

As well as being used for all discounting, the Libor-based curve was used to calculate forward rates to project unknown Libor resets for future cash flows on the floating leg of swaps.

Derivatives, such as interest rate swaps, have both fixed and floating cash flows. To establish the present value of a trade, it is necessary to obtain values for all future cash flows, including floating cash flows. Forward rates are an estimation of future interest rates, given current market conditions. With floating cash flows, actual values are unknown in the future. A forward curve must therefore be used to estimate the future floating rate.

As noted above, before the crisis, it was easy and straightforward to use one curve for both discounting and projecting forward rates. The simplification meant that one curve was logically used twice. However, the perception was that only one curve was required, and market participants typically used the same Libor-based discounting curve to calculate forward rates. Since we now should use the OIS curve for discounting, a forecast curve for projection is still required. This should still be a Libor curve, but we need to make sure it is constructed consistently with the OIS discount curve.

In addition, separate forecast curves must now be constructed for each Libor tenor used in floating rate derivative legs. For example, swaps indexed with three-month Libor must use a different forecast curve than those indexed with six-month Libor.

The reason is that during the crisis, the basis spread--the spread added to one side of a swap between two floating index legs--between Libor rates of different maturities also widened. For example, a USD three-month versus a six-month Libor swap will have a spread added to the three-month side. This spread is indicative of the additional credit risk carried for the longer-maturity six-month loan. In order for the three-month side to equal the six-month side at inception, a spread must be added. Before the crisis, when all credit risk was considered equal, these spreads were negligible and typically ignored.

There are two possible ways to bootstrap, or build, forecast curves for each Libor tenor. The preferred and most direct way is to choose instruments that use the underlying of the proper tenor for the entire curve. For example, to strip the three-month USD Libor curve, choose USD swaps that pay three-month Libor. To strip the six-month USD Libor curve, choose USD swaps that pay six-month Libor. However, the instrument quotes may not be available or there are not enough liquid instruments for the entire curve.

The alternative is to strip a 'base' Libor curve, using the de facto vanilla tenor, along with a spread built from basis swaps.

A key issue currently being debated in the market is whether the swap quotes themselves use OIS discounting or the traditional LIBOR-based discounting. Our observation is that vanilla swaps are quoted assuming OIS funding. Since interdealer vanilla swaps are centrally cleared at LCH.Clearnet SwapClear, this is consistent with the view that collateralized trades should be discounted with OIS.

**Systems & Procedure Requirements**

To implement fully the changes needed to address new post-crisis market conditions, it is important to ensure that valuation methods, systems, and reports can handle both a discount curve and a forecast curve.

Wide-scale adoption of OIS-based discount curves and multiple forecast curves is still in its early stages. Among dealers that choose OIS discounting, some report all pricing carried out systematically with full integration into systems. Others employ OIS-based curves on an ad-hoc basis on the trading desk and are carrying out IT projects in order to systematically accommodate their use.

Changing systems and procedures can be challenging. Although commercial systems have long technically supported separated discount and forecast curves, proprietary systems need to be adapted to accommodate multiple curves. Analytics software vendors are also upgrading their offerings to incorporate the multiple curve framework for valuing exotics using sophisticated option pricing models.

Perhaps more complex than systems updates, is the need to adapt middle- and back-office procedures to multiple curves when they were specifically designed to support a simplified single curve framework. First, even when systems can technically support multiple curves, the configuration needs to be changed for each curve, as well as the pricing policies and model setup. As such, it can be challenging to identify which trades are collateralized and which are not. Next, market data needs to be loaded for a greater number of instrument quotes, perhaps even involving new sources. The middle office needs to be prepared to analyze market data exceptions and profit and loss explanation when multiple curves are in place. Also, any reports that assumed a single yield curve must be updated.

In addition, most of the work so far has been on relatively vanilla interest rates products. Systems and procedures for exotics and such asset classes as equities, credit, foreign exchange, and commodities must all be adapted to support the new discount curve, and forecast curves as applicable.

**Testing the New Approach**

To test the effects of the new curves on valuation, participants typically calculate sensitivities to the various curves and then estimate the effect of the new curve. After revaluing the book, the new value is compared with the estimate and any differences above a specified tolerance are investigated.

The estimate is typically calculated with a Taylor approximation using sensitivities to both the funding and forecast curves. This requires an implementation where a single curve is used, but applied in two roles as a discount curve and a forecast curve.

The curve is perturbed by bucket and sensitivity and is calculated for both discounting and forecasting. The sensitivity buckets are then stored at either the trade or portfolio level.

To compile the estimate of the effects of switching discount curves, an OIS-Libor spread is then used to multiply the sensitivities along the curve. The total is an approximation of the effects of switching from Libor to an OIS-based funding curve. When the new OIS curve is used to revalue the book, the new valuations are compared with the estimate and any discrepancies are investigated.

A similar approach is used to evaluate the effects of the forecast curve.

**Conclusion**

The credit crunch has clearly had major ramifications on the conduct of the financial industry. One of the results is the way sophisticated institutions have challenged the means by which vanilla interest rate swaps are valued based upon the issue of unequal credit risk. As this new methodology is being implemented and understood, it is important that less-sophisticated institutions also understand the changes. It is not the first time that the market has witnessed a change in a standard methodology--a similar scenario also occurred at the height of the crisis in the structured credit market, when traders moved away from Gaussian copula for valuing super-senior tranches in their correlation book. However, what past history has shown is that changes in methodology can be very costly if ignored and, while the system and back office may be expensive to change, ignoring these new market developments could be even more costly.

*This week's Learning Curve was written by ***Kevin Samborn**, a v.p. of valuation and risk management initiatives at **Sapient Global Markets **in Boston.