Pricing and Valuation of Interest Rate Swaps

Derivatives 4 min readModule 5 of 7Save

What you'll learn

LOS 28.e
Describe how interest rate swaps are priced, and calculate and interpret their no arbitrage value.

The distinction between the price and the value of a swap is the same one we made for forward contracts. Remember that the price of a forward contract is the forward rate that yields a zero value for the contract at initiation. After contract initiation, as rates or prices change, the contract will likely have value to either the long or the short.

Consider an interest rate swap. One party agrees to pay floating (borrow at the floating rate) and receive fixed (lend at a fixed rate). At initiation of the swap, the fixed rate is selected so that the present value of the floating rate payments is equal to the present value of the fixed rate payments, which means the swap value is zero to both parties.

This fixed rate is called the swap rate or the swap fixed rate. Determining the swap rate is equivalent to “pricing” the swap.

As short term rates (and expected future short term rates) change over the term of the swap, the value of the swap to one of the parties will be positive. The swap position is an asset to that party.

If market interest rates increase during the life of the swap, the swap will take on a positive value from the perspective of the fixed rate payer. The value of the swap contract to the other party (the pay floating side) will thus be negative.

Computing the Swap Fixed Rate

The swap fixed rate is derived from the MRR curve corresponding to the swap tenor. Consider a two year, semiannual interest rate swap. The swap fixed rate underlying this swap will be determined based on the MRR rates corresponding to the four settlement dates of this swap.

Using those four MRRs, we calculate the discount factors (Zs) for each.

Z = \frac{1}{1 + M R R ( days /360 )}

The periodic swap fixed rate SFR(periodic) can then be calculated as:

S F R_{periodic} S F R_{annual} = \frac{1 - final discount factor}{sum of discount factors} = S F R_{periodic} \times number of settlement periods per year

Worked example: Calculating the fixed rate on a swap with quarterly payments

Annualized MRR spot rates today are:

R_{90} = 0.030, R_{180} = 0.035, R_{270} = 0.040, R_{360} = 0.045

You’re analyzing a 1 year swap with quarterly payments and a notional principal amount of $5,000,000. Calculate:

The fixed rate in percentage terms.

The quarterly fixed payments in $. The first net payment.

Solution

First calculate the discount factors. Don’t forget to convert from annualized rates to per period rates.

Z_{90} Z_{180} Z_{270} Z_{360} = \frac{1}{1 + 0.030 ( 90/360 )} = 0.99256 = \frac{1}{1 + 0.035 ( 180/360 )} = 0.98280 = \frac{1}{1 + 0.040 ( 270/360 )} = 0.97087 = \frac{1}{1 + 0.045 ( 360/360 )} = 0.95694

The quarterly fixed rate on the swap is:

\frac{1 - 0.95694}{0.99256 + 0.98280 + 0.97087 + 0.95694} = 0.011 = 1.1%

The quarterly fixed rate payments, assuming a notional principal of $5,000,000, are:

$5, 000, 000 \times 0.011 = $55, 000

The fixed rate on the swap in annual terms is:

1.1% \times \frac{360}{90} = 4.4%

We have priced a swap in which one side pays quarterly MRR and the other side pays 4.4% fixed annually (1.1% quarterly).

The first floating rate payment will be based on the 90 day rate known at inception (i.e., 3%). So the first net payment of 1.4% adjusted for quarterly time intervel will be paid by the fixed rate payer:

0.014 \times \frac{90}{360} \times 5, 000, 000 = $17, 500

Calculating the Market Value of an Interest Rate Swap

For the purpose of the exam, we are only responsible for valuation of an interest rate swap on settlement dates (after settlement has occurred), and not between settlement dates.

After the initiation of an interest rate swap, the swap will take on a positive or negative value as interest rates change. The party that is the fixed rate payer benefits if rates increase because they are paying the older (and lower) fixed rate and receiving the newer (and higher) floating rate. Similarly, the fixed rate receiver benefits if rates decrease because they are receiving the older (and higher) fixed rate while paying a newer (and lower) floating rate.

The value is calculated as present value of the difference in payments (under the new current rate, relative to the payments under the older, locked in rate).

V_{payer} = \sum Z \times (S F R_{new} - S F R_{old}) \times \frac{days}{360} \times notional principal

$\sum Z$ is the sum of discount factors associated with the remaining settlement periods. Days is the number of days in the settlement period.

Worked example: Valuing a swap on payment date

Let’s continue our previous example of a one year, quarterly settlement, $5 million notional swap with a swap fixed rate at initiation of 4.4%. Suppose that after 180 days, the MRR term structure is flat at 3.5% over the next year. Calculate the value of the swap to the fixed rate receiver.

Solution

Because the MRR term structure is flat at 3.5%, the new swap fixed rate must be 3.5%. At t = 180, there are two more settlement dates remaining: 90 and 180 days in the future.

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