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.
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.
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.
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.
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.
We use continuous compounding to convert all rates to a single compounding period so we can compare the rates apples to apples.
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%
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.
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:
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 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.
|Using the following 12 ML%|
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.
Here’s a snapshot of the whole swap. Now we’ll go through the columns piece by piece.
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.
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.
Both cleared and un-cleared derivatives are repriced the same way, using the OIS curve & the LIBOR Curve.
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.
The OIS Spread or OIS Swaps is the spread between LIBOR and the effective fed funds rate (EFF).
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.
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.
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.