Swaps are typically derivative contracts in which two parties exchange (swap) cash flows or other financial instruments over multiple periods for a give-and-take benefit, usually to manage risk.
Both swap contract parties have future obligations. Thus, similar to forwards and futures, swaps are forward commitments as both parties are committed in the future. The net initial value of a swap to each party should be zero, and as one side of the swap contract gains, the other side loses by the same amount.
An interest rate swap allows the parties involved to exchange their interest rate obligations (usually a fixed rate for a floating rate) to manage interest rate risk or to lower their borrowing costs, among other reasons.
Interest rate swaps have two legs, a floating leg (FLT) and a fixed leg (FIX). The floating rate cash flows are expressed in the following equation:
On the other hand, the fixed-rate cash flows are given by:
\(r_=\) Observed floating rate appropriate for the time i
\(r_=\) Fixed swap rate
\(NAD_=\) Number of accrued days during the payment period
\(NTD_=\) Total number of days during the year applicable to cash flow \(i\)
Suppose that the accrual periods are constant. Then the receive-fixed, pay-floating net cash flow can be determined as:
On the other hand, the receive-floating, pay-fixed net cash flow can be expressed as:
Suppose that the fixed rate is 5%, and the floating rate is 4.25%. Given that the accrual period is 60 days based on a 360-day year, the payment of a receive-fixed, pay-floating swap is closest to:
The value of a swap to the receiver of a fixed rate and payer of a floating rate is given by:
\(C=\) Coupon payment for the fixed-rate bond
\(PV_>\)= Appropriate present value factor for the i th fixed cash flow.
The value of a floating rate bond is par, assumed to be I. The assumption is that we are on a reset date, and the interest payment matches the discount rate.
At the contract inception, the fixed rate is determined such that the present value of the floating rate payments equates to the present value of the fixed-rate payments. The fixed-rate is known as the swap rate. Determining the fixed (swap) rate is similar to pricing the swap.
In other words, the fixed swap rate is simply one minus the final present value term divided by the sum of present values.
Consider a one-year LIBOR based interest rate swap with quarterly resets. The annualized Libor spot rates are given below:
The swap rate is closest to:
Recall that the swap rate is equivalent to the fixed rate.
We first need to calculate the discount factor:
The quarterly swap rate is then calculated as:
We the calculate the annualize fixed rate as follows:
Notice that the swap rate (fixed rate) is very close to the last spot rate. You can use this tip to check whether your resulting swap rate is close to the last spot rate. Additionally, the swap rate should lie within the spot rates range as it is seen as the average of spot rates.
The value of a fixed-rate swap at some future point in time t is determined as the sum of the present value of the difference in fixed swap rates times the notional amount.
The swap value to the receive fixed party is:
Note that the above equation provides the value to the party receiving fixed.
A bank entered a $500,000, five-year receive-fixed LIBOR-based interest rate swap, which is reset annually one year ago. Suppose that the fixed rate in the swap contract entered one year ago was 1.5%. The estimated discount factors are given in the following table;
$$\begin<|c|c|>\hline\textbf & \textbf\\ \hline1 & 0.9723 \\ \hline2 & 0.9667\\ \hline3 & 0.9625 \\ \hline4 & 0.9569\\ \hline\end$$
a) The fixed rate of the swap is closed to:
b) The value for the party receiving the floating rate will be closest to:
The equivalent receive-floating swap value is simply the negative of the receive-fixed swap value.
The swap value to the receive fixed party is:
Therefore, the swap value to the receive floating party is -$7,389.
Since the fixed rate exceeds the floating rate, the party that receives fixed (and pays floating) would receive this amount from the party that pays fixed (and receives floating).
A currency swap is an agreement between two counterparties to exchange future interest payments in different currencies. The payments can be based on either a fixed interest rate or a floating interest rate. By swapping future interest obligations, the two parties can manage currency risk.
Currency swaps may also involve exchanging notional amounts at both the starting of the contract and the contract expiration. The counterparties can exchange payments denominated in one currency to equivalent payments denominated in another currency.
Pricing a currency swap involves solving the appropriate notional amount in one currency, given the notional amount in the other currency, and determining the two fixed interest rates such that the currency swap value is zero at the initiation.
Similar to interest rate swaps, currency swaps are priced by determining the fixed swap rate. The equilibrium fixed swap rate equation for a currency X is given as:
A France company needs to borrow 500 million dollars ($) for one year for one of its American Subsidiaries. The company decides to issue Euro-denominated bonds in an amount equivalent to $500 million. The company enters into a one-year currency swap that reset quarterly and agrees to exchange notional amounts at the contract inception and maturity. The following spot rates and present values are observed at time 0.
Given that the spot exchange rate of EUR/USD is 0.8163;
$$\small<|c|c|c|>\hline\textbf & \textbf & \textbf \\ \hline90 & 2.13\% & 0.09\% \\ \hline180 & 2.21\% & 0.13\% \\ \hline270 & 2.30\% & 0.17\% \\ \hline360 & 2.38\% & 0.21\% \\ \hline\end>$$
The annual fixed swap rates for EUR and USD are closest to:
The present values for each reset date are calculated as follows:
EUR present values:
Annual Fixed rate for EUR: