﻿ Interest rate swap - valuation, how does it work

# Interest rate swap - Definition, valuation

## Definition:

An interest rate swap is a financial derivative instrument in which two parties agree to exchange interest rate cash flows. It is used in order to hedge against or speculate on changes in interest rates.

## Example of use of interest rate swaps:

In order to fix the future interest expenses relative to a debt (hedging of the interest rate risk), a corporate can enter into a swap: the debt is finally at fixed rate. However, if the interest rates decline, the corporate will not benefit from low rates.

Therefore, the net debt of corporate should be sufficiently fixed to secure interest expenses. However, the corporate can benefit from falling rates with a residual floating net debt.

As already mentioned, interest rate swaps can be used for speculation ends: if a bank anticipates a drop of rates, it can enter into a swap to pay floating rates and to receive fixed rates. As a consequence, if the interest rates really drop, the bank will pay less interest expenses (meanwhile, the bank will continue to receive the same fixed cash flows).

## Characteristics:

In an interest rate swap traded by two parties, each counterparty agreed to pay either a fixed or floating rate to the other counterparty. The swap has two legs: one is related to the cash flows paid by the counterparty A to the counterparty B ; the other is related to the cash flows paid by the counterparty B to the counterparty A. Characteristics of an interest rate swap are the following:
Notional: this notional amount is only used for calculating the size of cash flows to be exchanged. The notional amount is not exchanged if the 2 legs have the same currency
Currency: typically, currencies are the same for both legs (for instance: euro, dollar, etc.). By trading another financial derivative instrument, the Cross Currency Swap, 2 counterparties agreed to exchange cash flows in 2 different currencies.
Trade date: this is the date at which the swap is traded.
Value date: this is the date at which the swap is really effective, that is to say the date from which cash flows are calculated.
End date: this is the maturity date of the swap.

For each leg of a swap, the following characteristics are determined:
Rate type: fixed rate or floating rate. For example, the counterparty A pays a fixed rate to B (fixed leg) and B pays a floating rate to A (floating leg).
Frequency: this is the frequency at which cash flows are paid or received (often 3 months, 6 months or 1 year). The frequency of each leg can be different.
Time basis: this is the basis on which the cash flows calculation is based. For example, it can be 30/360: in this case, we consider that a year is equivalent to 360 days and a month is equivalent to 30 days.

## Interest rate swap valuation:

The valuation of an interest rate swap is based not only on its characteristics (mentioned above), but also on market data (interest rates, foreign exchange rates, etc.). This is what we usually call "Mark-to-Market". At inception date, the rate of the fixed leg is generally determined in order to calculate a valuation equal to 0 at this date. If the valuation is not equal to 0, a cash payment will occur (the counterparty for which the valuation is positive will pay the other party).

To valuation an interest rate swap, several yield curves are used:

The zero-coupon yield curve, used to calculate the discount rates of future cash flows, paid or received, fixed or floating. Cash flows of each leg have to be discounted.

The forward rate curve, used to calculate the size of the floating cash flows paid (or received). If the rate of the floating leg is 6 month Libor, this curve will inform on the level of the 6 month Libor at each fixing date (we calculate therefore the size of the cash flows). This curve can be deducted from the zero-coupon yield curve, or collect directly on a data market provider (Bloomberg, Reuters, etc.).

Once cash flows calculated, we have to sum each discounted cash flow on each leg.

Finally, the swap valuation is the difference between the sum of the discounted received cash flows and the sum of the discounted paid cash flows.

## Example of the valuation of an interest rate swap which the following characteristics:

- Trade date: December 31, 2014
- End date: December 31, 2019
- Valuation date: June 30, 2015
- Notional: 100 000 000 EUR
- Payment frequency: 6 months for the fixed leg and for the floating leg
- Fixed rate paid: 2%
- Floating rate received: Euribor 6 mois - Basis: 30/360 (a month is equivalent to 30 days)

Please find below the market data on valuation date (June 30, 2015). Figures are the following:

- Discount rate: used to discount the future cash flows
- Distance from the valuation date: used to calculate the discount factor
- Discount factor: future cash flows has to be multiplied by this factor to be discounted. It is equal to: 1/(1+Discount rate)^(Distance from the valuation date)
- Rates of floating cash flows (calculated from the forward rate curve): used to calculated the size of the floating cash flows
Date Discount rate
(zero-coupon curve)
Distance from
June 30, 2015 (basis 30/360)
Discount
factor
Rate of floating cash flows
(forward rate curve)
31/12/20150,0510,50,999750,05
30/06/20160,073410,999270,0954
30/12/20160,09681,50,998550,1437
30/06/20170,122420,997560,1992
29/12/20170,16292,49720,995940,3261
29/06/20180,21462,99720,99360,4721
31/12/20180,27533,50,990420,6358
28/06/20190,34063,99440,986510,7998
31/12/20190,41394,50,981580,9906

Please find below the calculation detail of the discounting of the future fixed cash flows (fixed leg):

Fixed leg (interest rate equals to 2%)
Coupon
date
Interest rate
(annual basis)
Cash flowNotional
cash flow
Sum of cash flowsDiscount
factor
Discounted cash flow
31/12/20152%-1 000 000-1 000 0000,99975-999 745
30/06/20162%-1 000 000-1 000 0000,99927-999 267
30/12/20162%-1 000 000-1 000 0000,99855-998 550
30/06/20172%-1 000 000-1 000 0000,99756-997 556
29/12/20172%-994 444-994 4440,99594-990 411
29/06/20182%-1 000 000-1 000 0000,9936-993 595
31/12/20182%-1 005 556-1 005 5560,99042-995 926
28/06/20192%-988 889-988 8890,98651-975 549
31/12/20192%-1 011 111-100 000 000-101 011 1110,98158-99 150 952
Sum of cash flows-107 101 551

The cash flows are calculated by multiplying the notional of the swap (100 million EUR) by the interest rate (2%) and by the coupon duration (about 0,5 in our example).

Please find below the calculation detail of the discounting of the future floating cash flows (floating leg):

Floating leg (interest rates calculted from the forward rate curve)
Coupon
date
Interest rate =
forward rate
Cash flowNotional
cash flow
Sum of cash flowsDiscount
factor
Discounted cash flow
31/12/20150,05%25 00025 0000,9997524 994
30/06/20160,0954%47 70047 7000,9992747 665
30/12/20160,1437%71 85071 8500,9985571 746
30/06/20170,1992%99 60099 6000,9975699 357
29/12/20170,3261%162 144162 1440,99594161 486
29/06/20180,4721%236 050236 0500,9936234 538
31/12/20180,6358%319 666319 6660,99042316 605
28/06/20190,7998%395 457395 4570,98651390 122
31/12/20190,9906%500 803100 000 000100 500 80398 650 042
Sum of cash flows99 996 555

The valuation of the swap is the sum of the discounted (and signed) future cash flows of each leg. As of June 30, 2015, the interest rate swap valuation is negative: -7,1 million EUR.

Discounted cash flows
Fixed leg (paid cash flows)- 107 101 551
Floating leg (received cash flows)99 996 555
Swap valuation- 7 104 996
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