swaps and interest rate derivatives

7/28/2019 Swaps and Interest Rate Derivatives

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Chris Dzera


2/20

A swap is an agreement between twocompanies to exchange cash flows in the future,defining the date the cash flows will be paid and

the way they will be calculated This usually involves the future value of an

interest rate, foreign exchange rate, equityprice, commodity price, or another marketvariable

Swaps are heavily traded - according to theInternational Swaps and Derivatives Associationthere was $426.7 trillion in interest rate andcurrency swaps outstanding in 2009


3/20

There are many kinds of swap contracts,including interest rate swaps, and fixed-for-

fixed currency swaps which are two of themost common types

Other kinds of swaps include commodityswaps, equity swaps, total return swaps,swaptions, variance swaps, and Amortisingswaps


4/20

In this case a company would purchase or sell aswap contract if, for example, they want to buyor sell an underlying asset in 1 year and again in

2 years This contract could be paid up for upfront by

paying the present value of guaranteed prices,say $100 for the first year and $110 for thesecond, at the risk free interest rate

The swap could also be paid for after the twoyear period if the buyer does not want to pay theseller upfront due to potential credit risk of theseller


5/20

To price a prepaid swap, we take into accountrisk-free interest rates

Assume the risk free interest rate withmaturity 1 year is 4.0%, 2 years is 4.5%, 3years is 5.0%, and 4 year is 5.5%

The price of a prepaid swap contract for theprices outlined before would be 100/1.04 +110/(1.0452) or $196.88


6/20

The postpaid swap contract also takes into account risklessinterest rates, and the $196.88 figure that we calculated earlier

Typically, we determine the level annual payment that isequivalent to the prepaid amount, so we take X to be the levelannual payment and solve: X/1.04 + X/(1.0452) = 196.88

In this case we get X = $104.88, which becomes the swap price forboth year 1 and year 2

In this case the buyer essentially lends the seller $4.88 the firstyear of the contract, and the buyer underpays the seller by $5.12

the second year, but the accumulated value of the $4.88 becomes$5.12 when taking into account the effective interest rate earnedbetween the first and second years

This rate is calculated as follows: (1.0452)/1.04 - 1 = 5.0024%


7/20

When a this contract is created it has no value despite implicitborrowing and lending

However, the value of a contract can change if forward priceschange

Say forward prices rise to $105 in year 1 and $115 in year 2 We get a prepaid price of $206.27, and a level annual swap price of

$109.88 calculating the same way we did earlier After the first year the buyers net cash flow could be 109.88-

104.88 = $5.00, and the net cash flow is the same at time 2 The new contract added to the old contract gets no value since the

new contract has net value of 0, but the market value of theoriginal contract is the present value of 2 payments of $5.00,which is: 5/1.04 + 5/(1.0452) = $9.39


8/20

An interest rate derivative is a derivativewhose payoffs are dependent on future

interest rates The interest rate derivatives market is the

largest derivatives market in the world


9/20

There are many kinds of interest rate derivatives, basicinterest rate derivatives include: interest rate swap,interest rate cap/floor, interest rate swaption, bondoption, forward rate agreement, interest rate future,

money market instruments, and cross currency swaps Less basic derivatives include: range accrual

swaps/notes/bonds, in arrears swap, constant maturity ortreasury swap derivatives, interest rate swap

And exotic interest rate derivatives include: power reverse

dual currency note, target redemption note, CMSsteepener, snowball, inverse floater, strips of collateralizeddebt obligations, ratchet caps and floors, Burmudanswaptions, and cross currency swaptions


10/20

There are many methods to price interest rate derivatives, more simple onesinclude a model by Black that can work for bond options, caps, and swap options

More complex models include equilibrium models (assumptions about economicvariables and derive a process for the short rate r, and explore what the processfor r means about bond and option prices) by Rendleman and Bartter; Vasicek;and Cox, Ingersoll, and Ross

There are also two-factor models by Brennan and Schwartz (short rate reverts tolong rate) and Longstaff and Schwartz (volatility) that follow stochastic processes

Ho and Lee proposed a no arbitrage model using a binomial tree of bond priceswith the parameters being the short-rate standard deviation and the marketprice of risk of the short rate, Hull and White proposed a one factor model similarto the Vasicek model and the Ho-Lee model, Black and Karasinski created amodel that allows only positive interest rates which was an advantage over the

Ho-Lee and Hull-White models, though it did not have as many analyticapplications, and Hull and White have a two factor model similar to the onecreated by Brennan and Schwartz, but arbitrage free

These are only a few of the many pricing methods for interest rate derivatives


11/20

The most common type of swap is a vanillainterest rate swap, where a company agrees topay cash flows equal to interest at a

predetermined fixed rate on principal for a fewyears, while receiving interest at a floating rateon the same principal for the same time period

Typically the floating rate in interest rate swapagreements is the LIBOR rate LondonInterbank Offered Rate, the rate of interest atwhich a bank is prepared to deposit money withother banks in the Eurocurrency market


12/20

Two companies agree to a 4 year swap initiatedlast Friday, October 22, 2010 Apple agrees topay Microsoft .5% annually on a principal of

$400 million and in return Microsoft pays Applethe 6 month LIBOR rate on the same principal

Apple is the fixed-rate payer, and Microsoft isthe floating-rate payer

In this case the payments are to be exchangedevery 6 months and the .5% interest rate iscompounded semi-annually


13/20

The first exchange of payments would take place April22, 2011

Apple would pay Microsoft $1 million, the interest on$400 million principal for 6 months at .5%, andMicrosoft would pay Apple interest on the principal atthe 6-month LIBOR rate from 6 months before thepayment, or on October 22, 2010 this rate is .45%

Then Microsoft pays Apple (.5)(.0045)($400), or$900,000

There is no uncertainty about this first exchange ofpayments, because they are determined by the LIBORrate at the time the contract is entered into


14/20

The second exchange of payments would take placeOctober 22, 2011

Apple would pay Microsoft $1 million again, and assumingthe LIBOR rates are .52% on April 22, 2011 Microsoft

would pay Apple $1.04 million the same way we calculatedthe $900,000

This swap would have a total of eight exchanges ofpayment, with the fixed payments always being $1 millionand the floating payments calculated by using the 6-

month LIBOR rate from 6 months before the paymentdate The way the swap is structured not all money is

exchanged, money only goes one way each payment date


15/20

To change a liability for Apple they could transform afloating rate loan into a fixed rate loan and for Microsoft todo the opposite

Each company would have three cash flows, two under the

terms of the swap and another to outside lenders To change an asset Apple and Microsoft could want to

transform a fixed rate asset into a floating rate asset orvice versa

Again each company would have three cash flows, one

coming in from the asset and two from the swap terms In each of these cases it is most likely that the borrowed

amount is the same amount as the principal agreed to inthe terms of the swap that never actually changes hands


16/20

An interest rate swap is always worth zero, orclose to it, when it is first initiated

After it has been in existence for some timeits value could be positive or negative

We have two approaches to value the swap The first views the swap as a difference of

two bonds The second regards the swap as a portfolio of

forward rate agreements


17/20

From the point of view of the floating ratepayer, a swap can be regarded as a long

position in a fixed rate bond and a shortposition on a floating rate bond, and thereverse from the perspective of the fixed ratepayer:

Vswap = Bfix Bfloat (floating rate payer)

Vswap = Bfloat Bfix (fixed rate payer)


18/20

A financial institution agreed to pay 6-month LIBOR and receive .5% peryear (compounded semi-annually) on a principal of $400 million and theswap has 1.25 years left

LIBOR rates for 3, 6, 9, and 15 month maturities are .5%, .52%, .54%, and.58%

The cash flows are $1 million, $1 million, and $401 million for the fixedrate payer at each upcoming payment date, and the discount factors forthese cash flows are e-0.005*.25, e-0.0054*.75, e-0.0058*1.25

Here we have principal of $400 million, interest due of .5*.0052*400 =$1.04 million, and a time of .25, so the floating rate bond can be valuedlike it produces a cash flow of $401.04 million in 3 months using the firstdiscount factor it has present value of $400.539 million

The discount factor applied to all of the fixed rate cash flows results in avalue of $400.098 million, so the value of the swap difference is -441,000dollars for the financial institution paying the 6-month LIBOR

In this case the swap would be worth 441,000 to the fixed rate payer


19/20

A swap can also be characterized as a portfolio of forward rateagreements lets consider the interest rate swap example we did earlierbetween Apple and Microsoft

We had a 4 year deal entered into October 22, 2010, with semiannualpayments with the first exchange known at the time that the swap was

negotiated The other seven exchanges can be regarded as forward rate agreements,

the exchange on October 22, 2011 can be regarded as a FRA whereinterest at .5% is exchanged for interest at the 6 LIBOR rate observedApril 22, 2011, and so on

A FRA can be valued by assuming forward interest rates are realized, sowe do this by using the LIBOR/swap zero curve to calculate forward ratesfor each of the LIBOR rates that will determine swap cash flows, thencalculate swap cash flows on the assumption that LIBOR rates will equalforward rates, and discount these swap cash flows using the LIBOR/swapzero curve to obtain swap value


20/20

Lets consider the same example from valuing the swap in terms of bond prices Each of the next 3 payments has a fixed cash flow of $1 million, the first payment

has a floating cash flow of $1.04 million, with the second cash flow we need tocalculate a forward rate corresponding to the period between 3 and 9 months: ((.0054)(.75)-(.005)(.25))/.5 = .0056

The forward rate would change under typical circumstances but the formula: 2(e(.0056/2) - 1) gives us essentially the same value (.0056078473)

The cash outflow for the floating rate payer is therefore $1.12 million the seconddate, and calculating the same way we get $1.28 million the third

Thus , using the same discount factors, the present value for the exchange in 3months is $39,950.03 , the present value for the exchange in 9 months is$121,078.10, and the present value for the exchange in 15 months is $280,012.72 all in favor of the fixed rate payer, so the present value net cash flow is the sum

of these values, $441,010.85 in favor of the fixed rate payer At the outset of the interest rate swap the fixed rate is chosen so that the swap is

worth zero initially, meaning the sum of the values of the FRAs underlying theswap is zero although each individual FRA would not be zero

swaps and interest rate derivatives

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