Price and Value of Swaps

Remember that a swap contract involves a series of periodic settlements with a final settlement at maturity. Swap price (or par swap rate) is a periodic fixed rate that equates the present value (PV) of all future expected floating cash flows to the PV of fixed cash flows.

The swap rate is equivalent to the forward rate, \(F_0(T)\); it satisfies no-arbitrage conditions. On the other hand, the current market reference rate (MRR) is the “spot” price. Therefore, from the fixed-rate payer perspective, the periodic value is given by:

$$\text{Periodic settlement value}=(\text{MRR}-\text{S}_{\text{N}})\times\text{Notional amount}\times\text{Period}$$

The swap value on any settlement date is calculated as the current settlement value using the above formula plus the present value of all the remaining future swap settlements.

Like all other forward commitments, the value of a swap contract at initiation is zero.

Note that it’s our assumption that MRR is set at the beginning of each interest period and has the same periodicity and day count as the swap rate. In addition, the net of fixed and floating differences is exchanged at the end of each period.

Examples: Calculating the Swap Value and Effect of Varying MRRs

FinnLay LTD has entered a 4-year interest rate swap with a financial institution with a notional amount of USD 100 million. The contract states that FinnLay signed to receive a semiannual USD fixed rate of 2.5% and, in turn, pay a semiannual market reference rate (MRR).

The MRR is expected to equal the respective implied forward rates (IFRs).

Scenario 1

If at the beginning of the sixth month, the MRR is 0.85%, the first swap settlement value from Finnlay’s perspective is closest to:

Solution

$$\begin{align*}\text{Periodic settlement value}&=(\text{MRR}-\text{S}_{\text{N}})\times\text{Notional Amount}\times\text{Period}\\&=(2.5\%-0.85\%)\times\text{USD 100m}\times0.5\\&=\text{USD 0.825m}\end{align*}$$

Scenario 2

If implied forward rates remain constant as set at trade inception, how will this affect the MTM value from Finnlay’s perspective immediately after the first settlement?

Solution

The swap price (or fixed swap rate) of 2.5% is set at the initiation of the trade, which equates to the PV of fixed versus floating payments.

If there is no change in interest rate expectations, the PV of remaining floating payments rises above the PV of fixed payments.

As such, Finnlay, as a fixed receiver, realizes an MTM loss on the swap because:

$$\sum\text{PV}(\text{Floating payments paid})>\sum\text{PV}(\text{Fixed payments received})$$

Scenario 2

If implied forward rates decline just after initiation, how will this affect the MTM value from Finnlay’s perspective ?

Solution

A decrease in expected forward rates just after initiation will reduce the PV of floating payments while the fixed swap rate will remain constant.

Since FinnLay is the fixed-rate receiver, it will realize an MTM gain because:

$$\sum\text{PV}(\text{Floating payments paid})<\sum\text{PV}(\text{Fixed payments received})$$

Question

Invest Capital Inc has signed a three-year swap contract to receive a fixed interest rate of 2.5% on a semiannual basis and pay a 6-month USD MRR. The notional amount of the swap contract is USD 100,000.

Assume that the initial 6-month MRR sets at 0.56%, and MRR is expected to be upward sloping. The first settlement value in six months from Invest Capital is closest to:

A. $970.

B. $1,940.

C. $2,500.

Solution

The correct answer is A.

From the fixed-rate payer perspective, the periodic value is given by:

$$\begin{align*}\text{Periodic settlement value}&=(\text{S}_{\text{N}}-\text{MRR})\times\text{Notional Amount}\times\text{Period}\\&=(2.5\%-0.56\%)\times\text{USD 100,000}\times0.5\\&=$970\end{align*}$$

B is incorrect. It is calculated as \(=2.5\%-0.56\%\times\text{USD 100,000}\). It omits the period in the formula.

C is incorrect. It is the amount of the fixed interest amount after six months.