The valuation of financial instruments is the foundation of all investing activity. Valuations are necessary to determine the fair market value of an investment, as well as for financial reporting and risk management purposes. By far, the most common method of valuation is the discounted cash flow (DCF) method, which involves the estimation of future cash flows and the discount rate applicable to those cash flows.

While cash flows are often easy to estimate, the discount rate can be tricky to ascertain. However, it is helpful to keep in mind that in order for the market value of an investment to be fair the discount rate must be the same as the rate of return on that investment. (To learn more about the DCF method, see: *Introduction to Discounted Cash Flow Valuation*.)

### Derivative Valuation: Before and After 2007

Financial derivatives are useful for altering portfolio risks and diversifying risk by gaining exposure to different markets. Derivatives are valued under the assumption that the return on all underlying assets is the risk-free rate, and thus that same risk-free rate is the appropriate discount rate for derivative cash flows. This is known as risk-neutral valuation and relies on the presumption that it is not possible to make risk-free profits in the markets due to investors having access to similar information. An implicit assumption of risk-neutral valuation is that there is no counterparty risk, only market risk. (For related reading, see: *How Risk Free Is The Risk-Free Rate Of Return?*)

Prior to the 2007-08 financial crisis, the yields on government bonds were assumed to be risk-free since it is near impossible for a government to default on its own debt denominated in its own currency. The reason for this is that the government treasury has the ability to print money in order to meet its debt obligations. In addition, it was thought that swap rates based on interbank lending rates (e.g. LIBOR, EURIBOR etc.) were essentially risk-free. Swap rates were considered more appropriate for risk-neutral valuation than bond yields because proceeds from derivative transactions are generally invested in the interbank market and not the bond market.

After the 2007-08 financial crisis, the failure of some banks proved that interbank lending rates were not risk-free. In fact, it also emerged that there was significant counterparty risk in derivative transactions that were not subject to collateral or margin calls. This was most evident when Lehman Brothers failed at a time when (according to Summe, K. (2011)) it was a counterparty to 930,000 derivative transactions, representing approximately 5% of global derivative transactions. (To learn more about the failure of Lehman Brothers, see: *Case Study: The Collapse of Lehman Brothers*.)

### Collateralized Versus Uncollateralized Derivatives

Derivatives that trade over-the-counter (OTC) make use of a standard ISDA agreement. Many ISDA agreements include a credit support annex (CSA), which is an agreement that outlines permissible credit mitigants for a transaction such as netting and collateralization. Collateralized transactions pose less counterparty risk because the collateral can be used to recoup any losses. When the collateral falls below a certain threshold, additional collateral can be called for.

Due to the difference in risk between collateralized and uncollateralized transactions, the discount rates for valuation must differ. There are various, well-understood discount rate choices for risky uncollateralized derivatives. Less risky collateralized derivatives, however, must be valued using a risk-free rate. The choice of risk-free rate presents a challenge since it is now known that swap rates are not risk-free.

### OIS Discounting and Risk-Free Rates for Collateralized Derivatives

Standard Credit Support Annex (CSA) agreements stipulate daily collateral calls, failing which the transaction with that counterparty would be closed out, thereby limiting further losses. Given daily collateral calls, the natural choice for the risk-free discount rate is some kind of overnight rate. This is referred to as "OIS discounting" or "CSA discounting." An overnight yield curve can be derived from overnight index swaps (OIS).

Prior to the financial crisis, there was little difference between the overnight yield curve and the yield curve derived from swap rates. During the crisis, the spreads between the two yield curves widened. Although overnight index swaps were introduced fairly recently, some developed markets now have highly liquid OIS trading activity (most notably in the U.S., the EU, the U.K., Japan and Switzerland), which enable reliable valuations.

*Historical Spread between 3-month USD LIBOR and the Fed Funds Effective overnight rate*

(Data source: www.treasury.gov; research.stlouisfed.org)

In countries with an insufficiently liquid OIS market or no OIS market at all, valuations have become a tremendous challenge. One possible workaround is to create synthetic cross-currency OIS utilizing a liquid OIS curve as base and applying a cross-currency basis spread to derive a local currency OIS curve. Another possible solution is to make use of historical spreads between overnight rates and swap rates and attempt to model an OIS curve.

### OIS Discounting Impact on Interest Rate Swaps and Cross-Currency Swaps

In the case of interest rate swaps and cross-currency swaps, the difference in valuation will depend on whether the swap is in- or out-the-money. Under OIS discounting – provided that swap cash flows are determined independently of the overnight curve – a swap that is in-the-money will be deeper in-the-money, while a swap that is out-the-money will be deeper out-the-money. It is also possible that swaps may have a non-zero day one P&L if market makers still make use of interbank lending rates (i.e. LIBOR for instance as noted above) discount curves for pricing purposes (this is the case in some less developed markets). If market makers make use of OIS discounting when pricing, the swap will price to par and there will be no P&L on trade date.

### OIS Discounting and Options

If options are collateralized and the option margin earns an overnight rate, OIS discounting can be applied. Call options (or interest rate caplets) will have lower values and put options (or interest rate floorlets) will have higher values. Again, this relationship is true only if the option cash flows are determined independently of the overnight curve. (To learn more about options, see: *Options Basics Tutorial*).

### The Bottom Line

In order to determine a market value for an investment, a valuation must usually be performed using the DCF method. For the value to be fair, the discount rate must be the same as the rate of return. Derivatives were considered to be free of counterparty risk, and being traded in the interbank market, were valued using interbank lending rates as discount rates.

The 2007-08 financial crisis showed that these rates were not good proxies for the risk-free rate and that derivatives have counterparty risk. Uncollateralized derivative trades retain counterparty risk and can be valued using well-understood methods. Collateralized derivative trades reduce counterparty risk and therefore must be valued using a risk-free rate.

Daily collateral calls suggest that OIS discounting is the logical choice for such types of valuation; however, this can prove to be a challenge in markets where the OIS market is insufficiently liquid.