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Swaps and Interest Rate Derivatives

Chris Dzera. Swaps and Interest Rate Derivatives. Swap. A swap is an agreement between two companies to exchange cash flows in the future, defining the date the cash flows will be paid and the way they will be calculated

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Swaps and Interest Rate Derivatives

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  1. Chris Dzera Swaps and Interest Rate Derivatives

  2. Swap • A swap is an agreement between two companies 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, equity price, commodity price, or another market variable • Swaps are heavily traded - according to the International Swaps and Derivatives Association there was $426.7 trillion in interest rate and currency swaps outstanding in 2009

  3. Types of swap contracts • There are many kinds of swap contracts, including interest rate swaps, and fixed-for-fixed currency swaps which are two of the most common types • Other kinds of swaps include commodity swaps, equity swaps, total return swaps, swaptions, variance swaps, and Amortising swaps

  4. Commodity swaps • In this case a company would purchase or sell a swap contract if, for example, they want to buy or 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 the second, at the risk free interest rate • The swap could also be paid for after the two year period if the buyer does not want to pay the seller upfront due to potential credit risk of the seller

  5. Pricing of a prepaid commodity swap contract • To price a prepaid swap, we take into account risk-free interest rates • Assume the risk free interest rate with maturity 1 year is 4.0%, 2 years is 4.5%, 3 years is 5.0%, and 4 year is 5.5% • The price of a prepaid swap contract for the prices outlined before would be 100/1.04 + 110/(1.0452) – or $196.88

  6. Pricing of a postpaid swap contract • The postpaid swap contract also takes into account riskless interest rates, and the $196.88 figure that we calculated earlier • Typically, we determine the level annual payment that is equivalent to the prepaid amount, so we take X to be the level annual payment and solve: • X/1.04 + X/(1.0452) = 196.88 • In this case we get X = $104.88, which becomes the swap price for both year 1 and year 2 • In this case the buyer essentially “lends” the seller $4.88 the first year 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 earned between the first and second years • This rate is calculated as follows: • (1.0452)/1.04 - 1 = 5.0024%

  7. Market value of this commodity swap contract • When a this contract is created it has no value despite implicit borrowing and lending • However, the value of a contract can change if forward prices change • 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 buyer’s 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 the original contract is the present value of 2 payments of $5.00, which is: • 5/1.04 + 5/(1.0452) = $9.39

  8. Interest rate derivatives • An interest rate derivative is a derivative whose payoffs are dependent on future interest rates • The interest rate derivatives market is the largest derivatives market in the world

  9. Types of interest rate derivatives • There are many kinds of interest rate derivatives, basic interest rate derivatives include: interest rate swap, interest rate cap/floor, interest rate swaption, bond option, 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 or treasury swap derivatives, interest rate swap • And exotic interest rate derivatives include: power reverse dual currency note, target redemption note, CMS steepener, snowball, inverse floater, strips of collateralized debt obligations, ratchet caps and floors, Burmudanswaptions, and cross currency swaptions

  10. Interest rate derivative pricing methods • There are many methods to price interest rate derivatives, more simple ones include a model by Black that can work for bond options, caps, and swap options • More complex models include equilibrium models (assumptions about economic variables and derive a process for the short rate r, and explore what the process for 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 to long rate) and Longstaff and Schwartz (volatility) that follow stochastic processes • Ho and Lee proposed a no arbitrage model using a binomial tree of bond prices with the parameters being the short-rate standard deviation and the market price of risk of the short rate, Hull and White proposed a one factor model similar to the Vasicek model and the Ho-Lee model, Black and Karasinski created a model 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 analytic applications, and Hull and White have a two factor model similar to the one created by Brennan and Schwartz, but arbitrage free • These are only a few of the many pricing methods for interest rate derivatives

  11. Interest rate swaps • The most common type of swap is a “vanilla” interest rate swap, where a company agrees to pay cash flows equal to interest at a predetermined fixed rate on principal for a few years, while receiving interest at a floating rate on the same principal for the same time period • Typically the floating rate in interest rate swap agreements is the LIBOR rate – London Interbank Offered Rate, the rate of interest at which a bank is prepared to deposit money with other banks in the Eurocurrency market

  12. Example • Two companies agree to a 4 year swap initiated last Friday, October 22, 2010 – Apple agrees to pay Microsoft .5% annually on a principal of $400 million and in return Microsoft pays Apple the 6 month LIBOR rate on the same principal • Apple is the fixed-rate payer, and Microsoft is the floating-rate payer • In this case the payments are to be exchanged every 6 months and the .5% interest rate is compounded semi-annually

  13. Initial payments • The first exchange of payments would take place April 22, 2011 • Apple would pay Microsoft $1 million, the interest on $400 million principal for 6 months at .5%, and Microsoft would pay Apple interest on the principal at the 6-month LIBOR rate from 6 months before the payment, 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 of payments, because they are determined by the LIBOR rate at the time the contract is entered into

  14. Second payments and beyond • The second exchange of payments would take place October 22, 2011 • Apple would pay Microsoft $1 million again, and assuming the LIBOR rates are .52% on April 22, 2011 Microsoft would pay Apple $1.04 million the same way we calculated the $900,000 • This swap would have a total of eight exchanges of payment, with the fixed payments always being $1 million and the floating payments calculated by using the 6-month LIBOR rate from 6 months before the payment date • The way the swap is structured not all money is exchanged, money only goes one way each payment date

  15. Why do this? • To change a liability – for Apple they could transform a floating rate loan into a fixed rate loan and for Microsoft to do 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 or vice 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 in the terms of the swap that never actually changes hands

  16. Valuation of interest rate swaps • An interest rate swap is always worth zero, or close to it, when it is first initiated • After it has been in existence for some time its 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. Valuation of an interest rate swap in terms of bond prices • From the point of view of the floating rate payer, a swap can be regarded as a long position in a fixed rate bond and a short position on a floating rate bond, and the reverse from the perspective of the fixed rate payer: • Vswap= Bfix – Bfloat(floating rate payer) • Vswap= Bfloat – Bfix(fixed rate payer)

  18. Example • A financial institution agreed to pay 6-month LIBOR and receive .5% per year (compounded semi-annually) on a principal of $400 million and the swap 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 fixed rate payer at each upcoming payment date, and the discount factors for these 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 valued like it produces a cash flow of $401.04 million in 3 months – using the first discount factor it has present value of $400.539 million • The discount factor applied to all of the fixed rate cash flows results in a value of $400.098 million, so the value of the swap difference is -441,000 dollars 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. Valuation of an interest rate swap in terms of forward rate agreements • A swap can also be characterized as a portfolio of forward rate agreements – lets consider the interest rate swap example we did earlier between Apple and Microsoft • We had a 4 year deal entered into October 22, 2010, with semiannual payments – 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 where interest at .5% is exchanged for interest at the 6 LIBOR rate observed April 22, 2011, and so on • A FRA can be valued by assuming forward interest rates are realized, so we do this by using the LIBOR/swap zero curve to calculate forward rates for each of the LIBOR rates that will determine swap cash flows, then calculate swap cash flows on the assumption that LIBOR rates will equal forward rates, and discount these swap cash flows using the LIBOR/swap zero curve to obtain swap value

  20. Example • 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 to calculate 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 second date, 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 3 months 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 FRA’s underlying the swap is zero – although each individual FRA would not be zero

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