How to Value Interest Rate Swaps

A wide variety of swaps are utilized in the over-the-counter (OTC) market in order to hedge risks, including interest rate swaps, credit default swaps, asset swaps, and currency swaps. In general, swaps are derivative contracts through which two private parties—usually businesses and financial institutions—exchange the cash flows or liabilities from two different financial instruments.

A plain vanilla swap is the simplest type of swap in the market, often used to hedge floating interest rate exposure. Interest rate swaps are a type of plain vanilla swap. Interest rate swaps convert floating interest payments into fixed interest payments (and vice versa).

The two parties in an interest rate swap are often referred to as counterparties. The counterparty making payments on a floating rate typically utilizes benchmark interest rates, such as the Secured Overnight Financing Rate (SOFR). Payments from fixed interest rate counterparties are benchmarked to U.S. Treasury Bonds.

Two parties may decide to enter into an interest rate swap for a variety of different reasons, including the desire to change the nature of the assets or liabilities in order to protect against anticipated adverse interest rate movements. Like most derivative instruments, plain vanilla swaps have zero value at the initiation. This value changes over time, however, due to changes in factors affecting the value of the underlying rates. And like all derivatives, swaps are zero-sum instruments, so any positive value increase to one party is a loss to the other.

How Is the Fixed Rate Determined?

The value of the swap at the initiation date will be zero for both parties. For this statement to be true, the values of the cash flow streams that the swap parties are going to exchange should be equal. This concept is illustrated with a hypothetical example in which the value of the fixed leg and floating leg of the swap will be V_fix and V_fl respectively. Thus, at initiation:

$V_{fix} = V_{fl}$Vfix​=Vfl​

Notional amounts are not exchanged in interest rate swaps because these amounts are equal; it does not make sense to exchange them. If it is assumed that parties also decide to exchange the notional amount at the end of the period, the process will be similar to an exchange of a fixed rate bond to a floating rate bond with the same notional amount. Therefore, such swap contracts can be valued in terms of fixed-rate and floating-rate bonds.

For example, suppose that Apple Inc. decides to enter a one-year, fixed-rate receiver swap contract with quarterly installments on a notional amount of $2.5 billion. Goldman Sachs is the counterparty for this transaction that provides fixed cash flows that determine the fixed rate. Assume the LIBOR rates (in dollars) are as follows:

Let’s denote the annual fixed rate of the swap by c, the annual fixed amount by C, and the notional amount by N.

Thus, the investment bank should pay c/4*N or C/4 each quarter and will receive the LIBOR rate multiplied by N. c is a rate that equates the value of the fixed cash flow stream to the value of the floating cash flow stream. This is the same as saying that the value of a fixed-rate bond with the coupon rate of c must be equal to the value of the floating rate bond.

 $\begin{aligned} &\beta_fl = \frac{c/q}{(1 + \frac{libor_{3m}}{360} \times 90)} + \frac{c/q}{(1 + \frac{libor_{6m}}{360} \times 180)} + \frac{c/4}{(1 + \frac{libor_{9m}}{360} \times 270)} + \frac{c/4 + \beta_{fix}}{(1 + \frac{libor_{12m}}{360} \times 360)} \\ &\textbf{where:}\\ &\beta_{fix}=\text{the notional value of the fixed rate bond that is equal to the notional amount of the swap—\$2.5 billion}\\ \end{aligned}$​βf​l=(1+360libor3m​​×90)c/q​+(1+360libor6m​​×180)c/q​+(1+360libor9m​​×270)c/4​+(1+360libor12m​​×360)c/4+βfix​​where:βfix​=the notional value of the fixed rate bond that is equal to the notional amount of the swap—$2.5 billion​

Recall that at the issue date—and immediately after each coupon payment—the value of the floating rate bonds equals the nominal amount. That is why the right-hand side of the equation is equal to the notional amount of the swap.

We can rewrite the equation as:

 $\beta_{fl} = \frac{c}{4} \times \left( \frac{1}{(1 + \frac{libor_{3m}}{360} \times 90)} + \frac{1}{(1 + \frac{libor_{6m}}{360} \times 180)} + \frac{1}{(1 + \frac{libor_{9m}}{360} \times 270)} + \frac{1}{(1 + \frac{libor_{12m}}{360} \times 360)}\right) + \frac{\beta_{fix}}{(1 + \frac{libor_{12m}}{360} \times 360)}$βfl​=4c​×((1+360libor3m​​×90)1​+(1+360libor6m​​×180)1​+(1+360libor9m​​×270)1​+(1+360libor12m​​×360)1​)+(1+360libor12m​​×360)βfix​​

On the left-hand side of the equation discount factors (DF) for different maturities are given.

Recall that:

 $DF = \frac{1}{1 + r}$DF=1+r1​

So if we denote DF_i for i-th maturity, we will have the following equation:

 $\beta_{fl} = \frac{c}{q} \times \sum_{i = 1}^n DF_i + DF_n \times \beta_{fix}$βfl​=qc​×∑i=1n​DFi​+DFn​×βfix​

Which can be re-written as:

 $\begin{aligned} &\frac{c}{q} = \frac{\beta_{fl} - \beta_{fix} \times DF_n}{\sum_i^n DF_i } \\ &\textbf{where:}\\ &q=\text{the frequency of swap payments in a year}\\ \end{aligned}$​qc​=∑in​DFi​βfl​−βfix​×DFn​​where:q=the frequency of swap payments in a year​

We know that in interest rate swaps, parties exchange fixed and floating cash flows based on the same notional value. Thus, the final formula to find the fixed rate will be:

 $\begin{aligned} &c= q \times N \times \frac{1 - DF_n}{\sum_i^n DF_i } \\ &\text{or}\\ &c= q \times \frac{1 - DF_n}{\sum_i^n DF_i}\\ \end{aligned}$​c=q×N×∑in​DFi​1−DFn​​orc=q×∑in​DFi​1−DFn​​​

Now let’s go back to our observed LIBOR rates and use them to find the fixed rate for this hypothetical interest rate swap.

The following are the discount factors corresponding to the LIBOR rates given:

 $c = 4 \times \frac{(1 - 0.99425)}{(0.99942 + 0.99838 + 0.99663 + 0.99425)} = 0.576 \%$c=4×(0.99942+0.99838+0.99663+0.99425)(1−0.99425)​=0.576%

Thus, if Apple wishes to enter into a swap agreement on a notional amount of $2.5 billion in which it seeks to receive the fixed rate and pay the floating rate, the annualized swap rate will be equal to 0.576%. This means that the quarterly fixed swap payment that Apple is going to receive will be equal to $3.6 million (0.576%/4* $2.5 billion).

Now assume that Apple decided to enter the swap on May 1, 2019. The first payments would have been exchanged on August 1, 2019. Based on the swap pricing results, Apple will receive a $3.6 million fixed payment each quarter. Only Apple’s first floating payment is known in advance because it’s set on the swap initiation date, and it's based on the 3-month LIBOR rate on that day: 0.233%/4* $2.5 billion = $1.46 million.

The next floating amount payable at the end of the second quarter will be determined based on the 3-month LIBOR rate effective at the end of the first quarter. The following figure illustrates the structure of the payments.

Suppose that 60 days had elapsed after this decision. The date is July 1, 2019; there is only one month left until the next payment, and all other payments are now two months closer. What is the value of the swap for Apple on this date? A term structure is needed for one, four, seven, and 10 months. Suppose that the following term structure is given:

It is necessary to revalue the fixed leg and floating leg of the swap contract after the interest rates change, and then compare them in order to find the value for the position. We can do so by re-pricing respective fixed and floating rate bonds.

Thus, the value of fixed rate bond is:

 $v_{fix} = 3.6 \times (0.99972 + 0.99859 + 0.99680 + 0.99438) + 2500 \times 0.99438 = \$2500.32 \text{mill.}$vfix​=3.6×(0.99972+0.99859+0.99680+0.99438)+2500×0.99438=$2500.32mill.

And the value of floating rate bond is:

 $v_{fl} = (1.46 + 2500) \times 0.99972 = \$2500.76 \text{mill.}$vfl​=(1.46+2500)×0.99972=$2500.76mill.

 $v_{swap} = v_{fix} - v_{fl}$vswap​=vfix​−vfl​

From Apple’s perspective, the value of the swap on July 1, 2019 was $ -0.45 million (the results are rounded). This number is equal to the difference between the fixed rate bond and floating rate bond.

 $v_{swap} = v_{fix} - v_{fl} = -\$0.45 \text{mill.}$vswap​=vfix​−vfl​=−$0.45mill.

The swap value was negative for Apple (under these hypothetical circumstances). This makes sense because the decrease in the value of the fixed cash flow is higher than the decrease in the value of the floating cash flow.

The Bottom Line

Swaps have increased in popularity due to their high liquidity and ability to hedge risk. In particular, interest rate swaps are widely utilized in fixed income markets such as the bond market. While history suggests that swaps have contributed to economic downturns, interest rate swaps can prove to be valuable tools when financial institutions utilize them effectively.