Derivative Products

Swaps Pricing

Time Value of Money

Before we delve into swaps pricing, this section reviews the Time Value of Money, introduces compounding & continuous compounding. Then we’ll have all of the math necessary for this section and your further studies at MH.

The Variables of Pricing a Cash Flow

1. Dollar amount at the start of the calculation
2. Rate: fixed or floating
   • Fixed rate: will be in force for the entire length of the calculation
   • Floating rate: will change periodically on a pre-agreed basis
3. How often the rate will be paid (daily, weekly, quarterly, etc. )
4. Day count convention of the cash flow (30/360, ACT/365, etc)

Present Value & Future Value

1. FUTURE VALUE calculates the value of a cash flow in the future for an investment made today. 2. PRESENT VALUE calculates the value of a cash flow today, presumed to be paid in the future.

Compounded Values

Compounding assume reinvestment of a cash flow received during the year. If I own a bond paying 10% and interest is paid semi-annually, the annual rate at year end will be more than 10%. The cash flow received at 6 months is reinvested for the remainder of the year.

Compounded Present Value

To calculate the compounded present value, we use the following calculation where R = the annual rate, C = compounding periods per year & t = time to maturity.

Compounded Future Value

To calculate the compounded future value, we use the following calculation where R = the annual rate, C = compounding periods per year & t = time to maturity.

Continuously Compounded Values

We use continuous compounding to convert all rates to a single compounding period so we can compare the rates apples to apples.

Continuously Compounded Present Values

To calculate a continuously compounded cash flow, we use the Exponent e notated (and used in Excel) as EXP where r = rate per year & t = time to maturity. The only difference between continuously compounded present value and future value is the use of a negative rate -r*t, to calculate a present value, continuously compounded. The right hand column in the table below are Present Values continuously compounded using the same assumptions asa above: $100 invested for one year at 10%

Continuously Compounded Future Values

To calculate a continuously compounded cash flow, we use the Exponent e notated (and used in Excel) as EXP where r = rate per year & t = time to maturity. Since we’re calculating the continuously compounded FUTURE cash flow, we leave the RATE ( r) as a positive number.

Interest Rate Swaps Pricing

The “First Commandment” of Derivatives pricing is using the most liquid instrument whose terms and conditions most closely resemble what you’re trying to price. The logic is:

1. if the contract is liquid enough to hedge the swap it will be liquid enough to price the swap.
2. IF YOUR CONTRACT ISN’T LIQUID ENOUGH TO HEDGE THE SWAP OVER TIME, IT’S NOT LIQUID ENOUGH TO USE TO PRICE THE SWAP.

The Timeline Diagram With Dates

The Timeline diagram shows the cash flows of a swap. on the right-hand side you can see the floating rates we’ll use to price the following swap: $1 million; MATURITY: 3 years; PAY fixed; RECEIVE floating (both legs netted & paid annually) in order to price a new At-Market swap, the Net Present Value of all cash flows must equal zero. In order to find the fixed rate of the swap, we use the floating rates. After bringing the floating forward rates to present value, we iterate and solve for the FIXED rate which results in an NPV of zero.

Forward Rate Agreements (FRA’s)Are A Cornerstone Product for Swaps Pricing

Forward Rate Agreements (FRA’s) are agreements to pay (receive) a fixed rate of interest and receive (pay) a floating rate at some time in the future (maturity). FRA’s are really single period interest rate swaps.

• In a payer FRA the payer agrees to pay a fixed rate at some date in the future, in exchange for receiving whatever rate is in force for that term to maturity. In a payer FRA you make money as rates go higher (prices fall).

• In a Receiver FRA the receiver agrees to receive a fixed rate and pay the floating rate in force for that term at maturity. You make money on a receiver FRA as rates go DOWN (price goes up)

Pricing An Interest Rate Swap using Forward Rate Agreements

We have our LIBOR rates so we can calculate the Forward Rate Agreements:

The first FRA is the 12 x 24 (12 month libor = 24-12; forward start date: 12 months)

Then we need to price the 24 x 36 FRA (12 month libor = 36-24; forward start = 24 months forward)

Now we have all of our rates & forward rates to price the swap. Here’s the resulting timeline showing the Rate Agreement (rate reset today) and the 2 forward rates 12 x 24 & 24 x 36.

Next we’ll use a cash flow diagram and continue on with pricing each cash flow and then solving for the fixed rate on the swap. We use the FRA’s to get the correct forward rates for the swap, but for the swap to be priced fairly, we need all of the cash flow to come back to a NET PRESENT VALUE OF ZERO.

Pricing Interest Rate Swaps using A Cash Flow Diagram

The steps to calculate a new At-Market Interest Rate Swap

1. Collect the necessary LIBOR rates
2. LIBOR rates converted to forward rates (for floating leg)
3. Forum your cash flow diagram in pieces
   1. Columns for spot & Forward rates
   2. Columns Present value of floating leg
   3. Columns Present value of fixed leg
   4. Last column of Net present values
   5. Bottom of last column = NPV
      • Goal seek to solve for NPV -0-

The Cash Flow Diagram in one Piece

Here’s a snapshot of the whole swap. Now we’ll go through the columns piece by piece.

The Cash Flow Diagram Step by Step

The first several columns are for Forward Rate Calculations. The forward rates can be seen in the right hand column with their title to the left. i.e. , 7.755% = 1 yr. LIBOR, 1 year forward. The next four columns are used to calculate the present value of the floating leg of the swap.

The future Value is the notional &1,000,000 * the forward rates. The present value brings these cash flows back using the LIBOR rate for 1, 2 & 3 years. Not the sub-diagram Discount Factors. A Discount factor is the zero coupon rate or present value of one unit of currency. The discount rate used is the same as above 1, 2 & 3 year LIBOR deposit rates.

The next two columns calculate the Present Value of the fixed leg of the swap, followed by the last column showing the NPV (Floating PV – Fixed PV).

Notice, the YELLOW highlight around the cell atop the FIXED FV Column. Also note the column NET PV. To solve for a Par swap, use Goal Seek (Data > what if analysis) the second choice down should be Goal Seek, if you don’t see this choice you will need to install your analysis toolkit. Using the goal seek functions as shown above, will give you the fixed rate at which all of the cash flow solve for an NPV of zero.

The Cash Flow Spreadsheet for Pricing Interest Rate Swaps

Here is the spreadsheet in one piece. Note the forward rates are on the row above the cash flow. This tells us the floating rate is set in advance and paid in arrears.

Repricing Interest Rate Swaps – Two Curves Necessary

Both cleared and un-cleared derivatives are repriced the same way, using the OIS curve & the LIBOR Curve.

• The LIBOR curve is used to calculate the market-to-market, but
• The OIS curve is used to calculate collateral calls.

During the Great Credit Crisis and the reporting dealers forced the LIBOR reset rate lower, Since the LIBOR reset rate is used to reset all swaps & loans resetting that day, the rate impacts a large dollar amount. Market Participants decided, given the volatility of LIBOR a more stable rate should be used. The OIS Swap (LIBOR minus Effective Fed Fund) was chosen as the rate for COLLATERAL POSTING. Please note the difference: LIBOR curve is used to calculate Profit & Loss, but the OIS Spread to calculate the collateral one counterparty needs to post.

What is the OIS Spread?

The OIS Spread or OIS Swaps is the spread between LIBOR and the effective fed funds rate (EFF).

Why don’t we just use Fed Funds Then?

First, recall the Fed Funds are loans made between banking institutions CAN ONLY LEND THEIR FREE RESERVES IN THE FED FUNDS MARKET, and the market is short dated, it’s only liquid out 270 days.

Where do we find OIS Spreads to Reprice Long-Dated Swaps?

We use the OIs Swaps Market rates. Why? Because they’re so liquid. However there are other sources in the links section is you prefer to do your own work on the curve.

What are OIS Swaps?

The OIS swap is a floating for floating swap, LIBOR versus Effective Fed Funds. As with most floating – floating swaps, the maturity of the swap is long (i.e. , 5 yr swap) but the two floating rates are observed very frequently, with a net cash flow periodically.