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* 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. Therefore, similar to forwards and futures, swaps are forward commitments since both parties are bound by a future obligation. 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). Interest rate swap allows the parties to manage interest rate risk or lower their borrowing costs, among other benefits.

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:

$$ S_i = \left(\frac{{NAD}_{FLT,i}}{{NTD}_{FLT,i}}\right)r_{FLT,i} $$

On the other hand, the fixed-rate cash flows are given by:

$$ FS=\left(\frac{{NAD}_{FIX,i}}{{NTD}_{FIX,i}}\right)r_{FIX} $$

Where:

- \(r_{FLT}\) = Observed floating rate appropriate for the time \(i\).
- \(r_{FIX}\) = Fixed swap rate.
- \(NAD_i\) = Number of accrued days during the payment period.
- \(NTD_i\) =Total number of days during the year applicable to cash flow \(i\).

In a case where the accrual periods are constant, the receive-fixed, pay-floating net cash flow can be determined as:

$$ FS – S_i = AP(r_{FIX} – r_{FLT,i}) $$

On the other hand, the receive-floating, pay-fixed net cash flow can be expressed as:

$$ S_i – FS = AP(r_{FLT,i} – r_{FIX}) $$

30/360 and ACT/ACT are the most popular day count methods. The 30/360 suggests that each month has a total of 30 days, making a 360-day year. The ACT/ACT treats accrual periods as having the actual number of days in the year. The floating interest rate is assumed to be advanced set and settled in arrears. Therefore, it is set at the beginning and paid when the period ends.

Assume 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:

$$ \begin{align*} FS – S_i & = AP(r_{FIX} – r_{FLT,i}) \\ &= \left(\frac{60}{360}\right)5\% – 4.25\%= 0.00125 \text{ per notional of 1} \end{align*} $$

The value of a swap to the receiver of a fixed rate and payer of a floating rate is given by:

$$ V = \text{Value of fixed bond} – \text{Value of floating bond} = FB – VB $$

Where:

\(\text{Value of fixed bond (FB)} = C \sum_{i=1}^{n}{{PV}_{0,t_i}\left(1\right)+{PV}_{0,t_n}\left(1\right)}\)

Where:

- \(C\) = Coupon payment for the fixed-rate bond.
- \({PV}_{0,t_i}\) = Appropriate present value factor for the i
^{th}fixed cash flow.

The value of a floating rate bond is par. 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 to deliberately equate the present value of the floating rate payments 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:

$$ r_{FIX}=\frac{1-{PV}_{0,t_n}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i}\left(1\right)}} $$

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:

$$ \begin{array}{c|c} \textbf{Year} & \textbf{Spot rates} \\ \hline \text{90-day LIBOR} & 1.90\% \\ \hline \text{180-day LIBOR} & 2.30\% \\ \hline \text{270-day LIBOR} & 2.60\% \\ \hline \text{360-day LIBOR} & 3.00\% \end{array} $$

The swap rate is *closest to*:

Recall that the swap rate is equivalent to the fixed rate:

$$ r_{FIX}=\frac{1-{PV}_{0,t_n}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i}\left(1\right)}} $$

We first need to calculate the discount factors:

$$ \begin{align*} D_{90} &=\frac{1}{1+\left(0.019\times\frac{90}{360}\right)}=0.9953 \\ D_{180} & =\frac{1}{1+\left(0.023\times\frac{180}{360}\right)}=0.9886 \\ D_{270} &=\frac{1}{1+\left(0.026\times\frac{270}{360}\right)}=0.9809 \\ D_{360} &=\frac{1}{1+\left(0.03\times\frac{360}{360}\right)}=0.9709 \end{align*} $$

The quarterly swap rate is then calculated as:

$$ r_{FIX}=\frac{1-0.9709}{(0.9953+0.9886+0.9809+0.9709)}=0.0074=0.74\% $$

We then calculate the annualized fixed rate as follows;

$$ \text{Annual fixed rate} =0.74\%\times\frac{360}{90}=2.96\% $$

* Note to candidates: *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:

$$ V=NA\left({FS}_0-{FS}_t\right)\sum_{i=1}^{n^{\prime}}{PV}_{t,t_i} $$

Note that the above equation provides the value to the party receiving fixed.

## Question

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:$$ \begin{array}{c|c} \textbf{Year} & \textbf{Discount factor} \\ \hline 1 & 0.9723 \\ \hline 2 & 0.9667 \\ \hline 3 & 0.9625 \\ \hline 4 & 0.9569 \end{array} $$

The value for the party receiving the floating rate is

closest to:

- −$7,389.
- $7,500.
- $7,389.
## Solution

The correct answer is A.We need first to calculate the fixed rate of the swap as follows.

$$ \begin{align*} r_{FIX} &=\frac{1-{PV}_{0,t_n}\left(1\right)}{\sum_{i=1}^{n}{{PV}_{0,t_i}\left(1\right)}} \\ r_{FIX} &=\frac{1-0.9569}{0.9723+0.9667+0.9625+0.9569}=1.117\% \end{align*} $$

$$ \begin{align*} V &=NA\left({\rm FS}_0-{FS}_t\right)\sum_{i=1}^{n^\prime}{PV}_{t,t_i} \\ &=$500,000\left(1.5\%-1.117\%\right)\times( 0.9723+0.9667+0.9625+0.9569) \\ & =$ 7,389 \end{align*} $$

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).

Reading 33: Pricing and Valuation of Forward Commitments

*LOS (e) Describe how interest rate swaps are priced, and calculate and interpret their no-arbitrage value.*