## Using OIS Swap valuation – Overnight Indexed Swap rates versus LIBOR

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# Using OIS vs LIBOR.

## Overnight Indexed Swap Rates and Interest Rate Swap valuations

The conventional way for pricing interest rate swaps (IRS) is to discount the cash flows of the swap by using the discount factor, which is further calculated from the 3-month London Interbank Offer Rate (LIBOR). However, after the Global Financial Crisis of 2007-2009, there has been a change from using the LIBOR to the Overnight Indexed Swap (OIS) discount factor in order to price interest rate swaps.

*The following presentation is a summary of the paper “**Valuing Interest Rate Swaps Using OIS Discounting” by Donald J. Smith (July 2012- Boston University School of Management Research Paper Series)** which illustrates how swaps are valued using LIBOR and Overnight Indexed Swap rates.*

### What is an Overnight Indexed Swap?

An overnight indexed swap is a derivative contract based on the total return on a reference rate that is compounded daily over a specific time period. In the US, this reference rate is the effective federal funds rate, i.e. the weighted average of brokered trades between banks for overnight ownership of bank reserves. This rate is calculated and released daily by the Federal Reserve in its H.15 report.

## Reasons for Switching from LIBOR to OIS Discounting.

The major reason for switching from using LIBOR to the OIS as a term structure for pricing interest rate swaps is that OIS discounting provides a better method for measuring collateral and counterparty credit risk in an interest rate swap.

In recent years, the use of collateralization in the interest rate swap market has become popular in order to nullify counterparty credit risk. During the 1990?s, after the introduction of the Credit Support Annex (CSA) in the standard International Swap and Derivatives Association (ISDA) master agreement, it became a norm to post collateral in the form of market securities or cash. Nowadays, bilateral CSAs with zero thresholds are common in the market. These CSAs only require the counterparty which has a credit risk to post collateral. Thus, as a result of these developments, credit risk on swaps has become very low, due to which the discount factors used to price these swaps needs to be based on (near) risk-free interest rates.

The ideal discount factors which should be used to price interest rate swaps must be based on securities which have the same liquidity, tax status, and volatility as the interest rate swaps. Plus, these securities should have their credit risk approaching zero. Before 2007, traders and analysts considered the LIBOR to be a good proxy to produce the risk-free yield curve required for pricing interest rate swaps. In recent years, the LIBOR-OIS spread has persistently widened, particularly after August 2007. Dealers now prefer to use the OIS to value collateralized interest rate swaps because the OIS curve removes the bank credit and liquidity risk that is inherently priced in LIBOR. Thus, OIS rates can now be seen as (near) risk-free interest rates with credit risk approaching zero.

## Pricing Interest Rate Swaps Using LIBOR.

The paper assumes that there is a 2 year interest rate swap with USD 100 million notional principal, 5.26% fixed vs 3-month LIBOR quarterly settled. The comparable at-market fixed rate is 3.40%. Discounting the 8-period annuity of USD 465,000 using the at-market fixed rate gives a value of USD 3,581,649, as detailed below.

The above calculation assumes a flat swap curve since each payment has been discounted by the same interest rate. In practice, LIBOR discount factors for the entire term structure are used to integrate the shape of the swap curve.

The valuation method for calculating the discount rates uses cash market rates for the first 12 months, followed by at-market rates for the months after that. The following table illustrates the discount rates for the entire term structure:

The general formula for calculating the quarterly LIBOR discount factors is the following,

where Aj is the fraction of the year for the jth period, given that in the US, the actual/360 day count convention is used. For example, for the first 92 day time period in the first quarter, the LIBOR discount factor is as follows:

After the first 12-month period, discount factors are calculated by bootstrapping fixed rates on at-market swaps. The fifth discount factor is calculated in the following way:

The swap is treated as a 15-month, 2.44% fixed-rate, non-amortizing bond priced at a par value of 1.

A similar methodology is used to calculate the sixth discount factor, while now treating the swap as an 18-month,2.76% fixed rate, non-amortizing bond.

Thus, the general formula for bootstrapping LIBOR discount factors from at-market swap fixed rates(SFR’s) is:

Another important term is that of the implied forward rate, which is also known as the projected forward rate for a 3-month LIBOR between n-1 and n periods. It is calculated via the following formula:

The implied LIBOR forward curve is useful in pricing options on swaps and non-standard interest rate swaps. An example of a non-standard interest rate swap is of a swap whose notional principal varies over its lifespan.

The value of the interest rate swap is determined by calculating the value of the two implicit bonds which make up the interest rate swap. First, the price of the Floating Rate Note(FRN) is calculated. This is calculated via the following formula:

The price of the 5.26% fixed rate note is calculated, via the following formula:

The price of both of the implicit bonds using LIBOR discount factors has now been calculated. The difference in the price of these bonds is equal to the market value of the swap, which is as follows:

Thus, the value of this swap using LIBOR discount factors is USD 3,662,844.

The main assumption made for the above calculations was one of the following:

- The swap was uncollateralized and for the fixed-rate payer, the swap is a liability. Thus, the fixed-rate payer is a LIBOR flat credit risk.
- The swap was collateralized, and the LIBOR rates can be effectively used as a term structure for pricing the swap.

However, the above two assumptions are nullified owing to the following two recent scenarios:

- Collateralization is nowadays prevalent throughout the market.
- The LIBOR-OIS spread is significant and cannot be ignored.

Thus, due to these two conditions, it has become necessary to use OIS discounting to price interest rate swaps.

## Pricing Interest Rate Swaps Using OIS Discounting.

The same 2-year interest rate swap on a USD 100 million notional amount is assumed. Suppose the 3-month fixed rate is 0.10% on an overnight indexed swap of the aforementioned notional amount. Assuming 92 days for a quarter and an actual/360 day count, the payoff of the fixed leg of the swap is USD 25,556, calculated as follows:

The payoff of the floating leg of the swap depends on the following sequence of realized daily reference rates:

The following table lists the discount rates for the entire term structure:

OIS discount factors for the first four periods are calculated via the following formula:

Similar to LIBOR, OIS rates for tenors longer than 12 months involve periodic settlement payments. The general formula for bootstrapping the OIS discount factors beyond 12-months is the following:

For example, the OIS discount factor for the fifth quarter is calculated as the following:

The price of the swap is determined by finding out the market value of both, the fixed rate note and the floating rate note (FRN) of the swap.

The market value for the implicit 2-year, 5.26% fixed rate note using the OIS discounting factors is as follows:

In order to find the value of the implicit FRN note it is essential that a new LIBOR forward curve is generated. This is important because OIS discount rates are now being used to price the swap. Therefore a forward curve which prices LIBOR deposits and at-market LIBOR swaps with OIS discount factors is needed.

For the money market segment of the swap curve, i.e. from n=1 to n=4, the following formula is used to calculate the implied forward rates:

For n>4, the at-market LIBOR swap fixed rates are used:

The market value for the implied FRN is then calculated as follows:

Finally, the value of the 5.26%, 2 year collateralized swap is now equal to:

## OIS Discounting vs LIBOR – Conclusion.

The market value of the swap using OIS discount rates is higher at USD 3,681,573, compared with the market value of the swap priced at USD 3,662,844 using LIBOR discount rates.

This higher price is a reflection of the reduced credit risk on a collateralized interest rate swap. OIS discounting is a more accurate way of stating the price of a collateralized interest rate swap as a LIBOR term structure can no longer be considered a risk free yield curve proxy.

## OIS vs LIBOR Example Two.

While the first example is the original reproduction from Donald Smith’s paper, we now present a second illustrated example with a little more detail and color.

We work with two, 2 year interest rate swap with USD100,000/= and EUR 100,000/= notional principal, 5.5% fixed vs 3-month LIBOR quarterly settled. **The comparable at-market fixed rate will be the last quarter swap rate**.

# Libor Discounting.

The valuation method for calculating the discount rates uses cash market rates for the first 12 months, followed by at-market rates for the months after that. The general formula for calculating the quarterly LIBOR discount factors is the following,

Where Aj is the fraction of the year for the jth period, IFRn is the Implied Forward Rate, SFRn is Swap Fixed Rate, given that in the US, 90/360 day count convention is used. For example, for the first 90 day time period in the first quarter for USD, the LIBOR discount factor is as follows:

The discount rates for USD and EUR used in the example are therefore given by the following tables:

After the first 12-month period, discount factors are calculated by bootstrapping fixed rates on at-market swaps. The fifth onwards discount factor is calculated in the following way:

This simplifies into:

For instance for the 5th discount step the calculation would be:

The swap is treated as a 15-month, 1.162% fixed-rate, non-amortizing bond priced at a par value of 1.

The next step is to calculate the implied forward rate, which is also known as the projected forward rate for a 3-month LIBOR between n-1 and n periods. Using the IFR formula shared above, we calculate the implied forward rates for the 4×5 IFR and then extend the calculations for the rest of the terms, shared below.

The implied LIBOR forward curve is traditionally used in pricing options on swaps and non-standard interest rate swaps. The two implied forward curves are also plotted graphically below:

The value of the interest rate swap is determined by calculating the value of the two implicit bonds which make up the interest rate swap. First, the price of the Floating Rate Note (FRN) is calculated. This is calculated via the following formula:

Then the price of the fixed rate note is calculated, using the formula presented just above. The price of both of the implicit bonds using LIBOR discount factors has now been calculated. The difference in the price of these bonds is equal to the market value of the swap.

The main assumptions made for the above calculations were:

- The swap is uncollateralized and for the fixed-rate payer, the swap is a liability. Thus, the fixed-rate payer is a LIBOR flat credit risk.
- The swap is collateralized, and LIBOR rates can be effectively used as a term structure for pricing the swap.

However, the two assumptions are nullified given:

- Collateralization is now prevalent throughout the market.
- The LIBOR-OIS spread is significant and cannot be ignored.

## OIS Swap Valuation.

The same instruments are now valued using OIS rates versus LIBOR rates. Here are the new rates used for valuation.

The resultant implied forward rates (IFR) and forward curves:

The implied forward curve plots

The valuation model uses:

And the revised valuation results

As well as a side by side comparison of the two approaches.