# How to Value Interest Rate Swaps

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To hedge risks, the over-the-counter (OTC) market employs a broad range of swaps, including interest rate swaps, credit default swaps, asset swaps, and currency swaps. Swaps are derivative transactions in which two private parties—typically corporations and financial institutions—exchange cash flows or liabilities from two separate financial instruments.

A plain vanilla swap is the most basic sort of swap on the market, and it is often used to hedge floating interest rate risk. Plain vanilla swaps include interest rate swaps. Floating interest payments are converted into fixed interest payments using interest rate swaps (and vice versa).

### Key Takeaways

• Swaps are derivative transactions in which two parties—typically corporations and financial institutions—exchange the cash flows or liabilities of two distinct financial instruments.
• In finance, a broad range of swaps are used to hedge risks, including interest rate swaps, credit default swaps, asset swaps, and currency swaps.
• Floating interest payments are converted into fixed interest payments using interest rate swaps (and vice versa).
• In an interest rate swap, the two parties are referred to as counterparties; the counterparty making payments on a floating rate often uses a benchmark interest rate.
• Payments from fixed-rate counterparties are compared to US Treasury bonds.
• When used correctly, interest rate swaps may be important instruments for financial firms.

Counterparties are the two parties involved in an interest rate swap. The counterparty making floating-rate payments often uses benchmark interest rates, such as the London Interbank Offered Rate (LIBOR).Payments from fixed-rate counterparties are compared to US Treasury bonds.

Two parties may opt to engage into an interest rate swap for a number of reasons, including the desire to modify the character of assets or obligations to safeguard against expected unfavorable interest rate fluctuations. Plain vanilla swaps, like other derivative products, have no value at the outset. This value, however, fluctuates over time owing to changes in variables impacting the value of the underlying rates. Swaps, like other derivatives, are zero-sum instruments, which means that any positive value rise for one side is a loss for the other.

LIBOR is being phased out due to recent scandals and issues about its legitimacy as a benchmark rate. LIBOR will be phased out by June 30, 2023, according to the Federal Reserve and UK authorities, and will be replaced with the Secured Overnight Financing Rate (SOFR).LIBOR one-week and two-month USD LIBOR rates will no longer be reported as part of this phase-out after December 31, 2021.

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## How Is the Fixed Rate Determined?

The swap’s value at the start date will be zero for both sides. The values of the cash flow streams that the swap parties will exchange must be identical for this statement to be accurate. This is shown using a hypothetical scenario in which the fixed leg and floating leg of the swap have values of Vfix and Vfl, respectively. Thus, at the start:

﻿Vfix=VflV_{fix} = V_{fl} V​=V​﻿

Notional sums are not exchanged in interest rate swaps since they are equal; exchanging them makes no sense. If the parties agree to swap the notional amount at the end of the term, the procedure will be comparable to exchanging a fixed rate bond for a variable rate bond with the same notional amount. As a result, such swap contracts may be valued using fixed-rate and floating-rate bonds.

Assume AppleInc. chooses to sign into a one-year, fixed-rate receiver swap contract with quarterly payments totaling $2.5 billion. Goldman Sachs is the transaction’s counterparty, and it delivers fixed cash flows that set the fixed rate. Assume the following LIBOR rates (in dollars): Let’s call the swap’s annual fixed rate c, the yearly fixed amount C, and the notional amount N. Thus, the investment bank should pay c/4*N or C/4 each quarter and get the LIBOR rate multiplied by N.c is a rate that translates the fixed cash flow stream’s value to the floating cash flow stream’s value. This is equivalent to stating that the value of a fixed-rate bond with a coupon rate of c must match the value of a floating-rate bond. f l = c / q ( 1 + l I b o r 3 m 3 6 0 9 0 ) + c / q ( 1 + l I b o r 6 m 3 6 0 1 8 0 ) + c / 4 ( 1 + l I b o r 9 m 3 + c / 4 + f I x ( l I b o r 1 2 m 3 6 0 ) where: f I x = the nominal value of the fixed rate bond equal to the nominal amount of the swap—$2.5 billion beginaligned fracc/q &beta fl = fracc/q ((1 + fraclibor 3m360 multiplied by 90) plus fracc/q ((1 + fraclibor 6m360 multiplied by 180) + \frac{c/4} ((1 + fraclibor 9m360 multiplied by 270) plus fracc/4 + beta fix ((1 + fraclibor 12m360 times 360)) &textbf where: &beta fix=textthe nominal value of the fixed rate bond equal to the nominal value of the swap—2.5 billion endaligned ​​ Remember that the value of the variable rate bonds matches the nominal amount on the issuance date and immediately after each coupon payment. As a result, the right-hand side of the equation equals the swap’s nominal amount. Index Trading Strategy & Education We can rewrite the equation as: f l = c 4 ( 1 ( 1 + l I b o r 3 m 3 6 0 9 0 ) + 1 ( 1 + l I b o r 6 m 3 6 0 1 8 0 ) + 1 ( 1 + l I b o r 9 m 3 6 0 2 7 0 ) beta fl = fracc4 times remaining (frac1(1 + fraclibor 3m360 multiplied by 90) + frac1(1 + fraclibor 6m360 multiplied by 180) + \frac{1} ((1 + fraclibor 9m360 multiplied by 270) + frac1(1 + fraclibor 12m360 multiplied by 360)right) + fracbeta fix ((1 + fraclibor 12m360 times 360)) β​=​×(​+​+​+​)+ ​ On the left side of the equation, discount factors (DF) for various maturities are provided. Recall that: ﻿DF=11+rDF = \frac{1}{1 + r} DF=​﻿ So, if we designate DFiffor the i-th maturity, we have the following equation: f l = c q I = 1 n D F I + D F n f I x beta fl = fraccq times sum i = 1n DF i + DF n times beta fix = 1n DF i + DF n times beta fix β​=​×∑​DF​+DF​×β ​ Which can be re-written as: c q = f l f I x D F n I n D F I n D F I n D F I n D F I n D F I n D F I n D F I n where q = the frequency of swap payments in a year begins to be aligned &fraccq = fracbeta fl – beta fix times DF nsum in DF i &textbfwhere: &q=textthe frequency of swap payments in a year endaligned year ​​ We know that participants in interest rate swaps exchange fixed and floating cash flows for the same notional value. As a result, the final formula for calculating the fixed rate will be: c=q×N×1−D F n I n D F I or c = q 1 D F n I n D F I Fn∑ InD F I began to align &c= times q N times frac1 – DF nsum iin DF i &textor &c= q times frac1 – DF nsum iin DF i &c= q times frac1 – DF nsum iin DF i endaligned ​​ Returning to our observed LIBOR rates, we can now utilize these to calculate the fixed rate for this hypothetical interest rate swap. The discount factors that correlate to the provided LIBOR rates are as follows: c=4×(1−0.99425) (0.99942+0.99838+0.99663+0.99425)=0.576% c = 4 frac(1 – 0.99425)(0.99942 + 0.99838 + 0.99663 + 0.99425) = 0.576 % c=4×​=0.576% Thus, if Apple enters into a swap agreement for2.5 billion in which it intends to receive the fixed rate and pay the floating rate, the yearly swap rate will be 0.576%. This implies that Apple will get a quarterly fixed swap payment of $3.6 million (0.576%/4*$2.5 billion).

Assume Apple has opted to engage the swap on May 1, 2019. On August 1, 2019, the first payments would have been made. Apple will get a $3.6 million fixed payment each quarter based on the swap price outcomes. Only Apple’s first floating payment is known in advance since it is based on the 3-month LIBOR rate on the swap start date: 0.233%/4*$2.5 billion = $1.46 million. Institutional Traders vs. Retail Traders: What's the difference? The next floating amount due at the end of the second quarter will be calculated using the three-month LIBOR rate in place at the end of the first quarter. The payment arrangement is shown in the diagram below. Assume 60 days have passed after this judgment. The date is July 1, 2019, and there is only one month before the next payment, which is now two months away. What is the current value of the swap for Apple? For one, four, seven, and ten months, a term structure is required. Assume the following term structure is given: After interest rates change, the fixed and floating legs of the swap contract must be revalued and compared in order to determine the position’s worth. We may do this by re-pricing the corresponding fixed and floating rate bonds. Thus, the value of fixed rate bond is: v f I x = 3. 6 ( 0. 9 9 9 7 2 + 0. 9 9 8 5 9 + 0. 9 9 6 8 0 + 0. 9 9 4 3 8 ) + 2 5 0 0 0. And the value of floating rate bond is: v f l = ( 1. 4 6 + 2 5 0 ) 0. 9 9 9 7 2 =$ 2 5 0 0. 7 6 mill. v fl = (1.46 + 2500) multiplied by 0.99972 equals $2500.76 textmill. v =(1.46+2500)0.99972=$2500.76mill.

v s w a p = v f I x v f l v swap = v fix – v fl v swap = v fix – v fl v​=v​−v ​

According to Apple, the value of the exchange on July 1, 2019 was $-0.45 million (the results are rounded).This figure represents the difference between the fixed and variable rate bonds. v s w a p = v f I x v f l =$ 0 v swap = v fix – v fl = – $0.45 textmill. v​=v​−v​=−$0.45mill.

Apple’s swap value was negative (under these hypothetical circumstances).This makes sense since the value of the fixed cash flow decreases faster than the value of the floating cash flow decreases.

## The Bottom Line

Swaps have grown in popularity as a result of their high liquidity and capacity to hedge risk. Interest rate swaps, in particular, are commonly used in fixed income markets such as the bond market. While history implies that swaps have contributed to economic downturns, interest rate swaps may be helpful instruments when used properly by financial firms.

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