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# Swaps

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1. Swaps

2. Introduction • An agreement between two parties to exchange cash flows in the future. • The agreement specifies the dates that the cash flows are to be paid and the way that they are to be calculated. • A forward contract is an example of a simple swap. With a forward contract, the result is an exchange of cash flows at a single given date in the future. • In the case of a swap the cash flows occur at several dates in the future. In other words, you can think of a swap as a portfolio of forward contracts.

3. Mechanics of Swaps • The most commonly used swap agreement is an exchange of cash flows based upon a fixed and floating rate. • Often referred to a “plain vanilla” swap, the agreement consists of one party paying a fixed interest rate on a notional principal amount in exchange for the other party paying a floating rate on the same notional principal amount for a set period of time. • In this case the currency of the agreement is the same for both parties.

4. Notional Principal • The term notional principal implies that the principal itself is not exchanged. If it was exchanged at the end of the swap, the exact same cash flows would result.

5. An Example • Company B agrees to pay A 5% per annum on a notional principal of \$100 million • Company A Agrees to pay B the 6 month LIBOR rate prevailing 6 months prior to each payment date, on \$100 million. (generally the floating rate is set at the beginning of the period for which it is to be paid)

6. The Fixed Side • We assume that the exchange of cash flows should occur each six months (using a fixed rate of 5% compounded semi annually). • Company B will pay: (\$100M)(.025) = \$2.5 Million to Firm A each 6 months.

7. Summary of Cash Flows for Firm B Cash Flow Cash Flow Net Date LIBOR Received Paid Cash Flow 3-1-98 4.2% 9-1-98 4.8% 2.10 2.5 -0.4 3-1-99 5.3% 2.40 2.5 -0.1 9-1-99 5.5% 2.65 2.5 0.15 3-1-00 5.6% 2.75 2.5 0.25 9-1-00 5.9% 2.80 2.5 0.30 3-1-01 6.4% 2.95 2.5 0.45

8. Swap Diagram LIBOR Company A Company B 5%

9. Company A Borrows (pays) 5.2% Pays LIBOR Receives 5% Net LIBOR+.2% Company B Borrows (pays) LIBOR+.8% Receives LIBOR Pays 5% Net 5.8% Offsetting Spot Position Assume that A has a commitment to borrow at a fixed rate of 5.2% and that B has a commitment to borrow at a rate of LIBOR + .8%

10. Swap Diagram Company A Company B The swap in effect transforms a fixed rate liability or asset to a floating rate liability or asset (and vice versa) for the firms respectively. LIBOR LIBOR+.8% 5.2% 5% 5.8% LIBOR +.2%

11. Role of Intermediary • Usually a financial intermediary works to establish the swap by bring the two parties together. • The intermediary then earns .03 to .04% per annum in exchange for arranging the swap. • The financial institution is actually entering into two offsetting swap transactions, one with each company.

12. Swap Diagram Co A FI Co B A pays LIBOR+.215% B pays 5.815% The FI makes .03% LIBOR LIBOR 5.2% LIBOR+.8% 4.985% 5.015%

13. Day Count Conventions • The above example ignored the day count conventions on the short term rates. • For example the first floating payment was listed as 2.10. However since it is a money market rate the six month LIBOR should be quoted on an actual /360 basis. • Assuming 184 days between payments the actual payment should be 100(0.042)(184/360) = 2.1467

14. Day Count Conventions II • The fixed side must also be adjusted and as a result the payment may not actually be equal on each payment date. • The fixed rate is often based off of a longer maturity instrument and may therefore uses a different day count convention than the LIBOR. If the fixed rate is based off of a treasury note for example, the note is based on a different day convention.

15. Role of the Intermediary • It is unlikely that a financial intermediary will be contacted by parties on both side of a swap at the same time. • The intermediary must enter into the swap without the counter party. The intermediary then hedges the interest rate risk using interest rate instruments while waiting for a counter party to emerge. • This practice is referred to as warehousing swaps.

16. Why enter into a swap? • The Comparative Advantage Argument Fixed Floating A 10% 6 mo LIBOR+.3 B 11.2% 6 mo LIBOR + 1.0% Difference between fixed rates = 1.2% Difference between floating rates = 0.7% B Has an advantage in the floating rate.

17. Swap Diagram Co A FI Co B A pays LIBOR+.065% instead of LIBOR+.3% B pays 10.965% instead of 11.2% The FI makes .03% LIBOR LIBOR 10% LIBOR+1% 9.935% 9.965%

18. Spread Differentials • Why do spread differentials exist? • Differences in business lines, credit history, asset and liabilities, etc…

19. Valuation of Interest Rate Swaps • After the swap is entered into it can be valued as either: • A long position in one bond combined with a short position in another bond or • A portfolio of forward rate agreements.

20. Relationship of Swaps to Bonds • In the examples above the same relationship could have been written as • Company B lent company A \$100 million at the six month LIBOR rate • Company A lent company B \$100 million at a fixed 5% per annum

21. Bond Valuation • Given the same floating rates as before the cash flow would be the same as in the swap example. • The value of the swap would then be the difference between the value of the fixed rate bond and the floating rate bond.

22. Fixed portion • The value of either bond can be found by discounting the cash flows from the bond (as always). The fixed rate value is straight forward it is given as: • where Q is the notional principal and k is the fixed interest payment

23. Floating rate valuation • The floating rate is based on the fact that it is a series of short term six months loans. • Immediately after a payment date Bfl is equal to the notional principal Q. Allowing the time until the next payment to equal t1 • where k* is the known next payment

24. Swap Value • If the financial institution is paying fixed and receiving floating the value of the swap is Vswap = Bfl-Bfix • The other party will have a value of Vswap = Bfix-Bfl

25. Example • Pay 6 mo LIBOR & receive 8% 3 mo 10% 9 mo 10.5% 15 mo 11% Bfix = 4e .-1(.25)+4e -.105(.75)+104e -.11(1.25)=98.24M Bfloat = 100e -.1(.25)+ 5.1e -.1(.25)=-102.5M -4.27 M

26. A better valuation • Relationship of Swap value to Forward Rte Agreements • Since the swap could be valued as a forward rate agreement (FRA) it is also possible to value the swap under the assumption that the forward rates are realized.

27. To do this you would need to: • Calculate the forward rates for each of the LIBOR rates that will determine swap cash flows • Calculate swap cash flows using the forward rates for the floating portion on the assumption that the LIBOR rates will equal the forward rates • Set the swap value equal to the present value of these cash flows.

28. Swap Rate • This works after you know the fixed rate. • When entering into the swap the value of the swap should be 0. • This implies that the PV of each of the two series of cash flows is equal. Each party is then willing to exchange the cash flows since they have the same value. • The rate that makes the PV equal when used for the fixed payments is the swap rate.

29. Example • Assume that you are considering a swap where the party with the floating rate will pay the three month LIBOR on the \$50 Million in principal. • The parties will swap quarterly payments each quarter for the next year. • Both the fixed and floating rates are to be paid on an actual/360 day basis.

30. First floating payment • Assume that the current 3 month LIBOR rate is 3.80% and that there are 93 days in the first period. • The first floating payment would then be

31. Second floating payment • Assume that the three month futures price on the Eurodollar futures is 96.05 implying a forward rate of 100-96.05 = 3.95 • Given that there are 91 days in the period. • The second floating payment would then be

32. Example Floating side

33. PV of Floating cash flows • The PV of the floating cash flows is then calculated using the same forward rates. • The first cash flow will have a PV of:

34. PV of Floating cash flows • The PV of the floating cash flows is then calculated using the same forward rates. • The second cash flow will have a PV of:

35. Example Floating side

36. PV of floating • The total PV of the floating cash flows is then the sum of the four PV’s: \$2,040,622.0013

37. Swap rate • The fixed rate is then the rate that using the same procedure will cause the PV of the fixed cash flows to have a PV equal to the same amount. • The fixed cash flows are discounted by the same rates as the floating rates. • Note: the fixed cash flows are not the same each time due to the changes in the number of days in each period. • The resulting rate is 4.1294686

38. Example: Swap Cash Flows

39. Swap Spread • The swap spread would then be the difference between the swap rate and the on the run treasury of the same maturity.

40. Swap valuation revisited • The value of the swap will change over time. • After the first payments are made, the futures prices and corresponding interest rates have likely changed. • The actual second payment will be based upon the 3 month LIBOR at the end of the first period. • Therefore the value of the swap is recalculated.

41. Currency Swaps • The primary purpose of a currency swap is to transform a loan denominated in one currency into a loan denominated in another currency. • In a currency swap, a principal must be specified in each currency and the principal amounts are exchanged at the beginning and end of the life of the swap. • The principal amounts are approximately equal given the exchange rate at the beginning of the swap.

42. A simple example Assume that company A pays a fixed rate of 11% in sterling and receives a fixed interest rate of 8% in dollars. Let interest payments be made once a year and the principal amounts be \$15 million and L10 Million Company A Dollar Cash Sterling Cash Flow (millions) Flow (millions) 2/1/1999 -15.00 +10.00 2/1/2000 +1.20 -1.10 2/1/2001 +1.20 -1.10 2/1/2002 +1.20 -1.10 2/1/2003 +1.20 -1.10 2/1/2004 +16.20 -11.10

43. Intuition • Suppose A could issue bonds in the US for 8% interest, the swap allows it to use the 15 million to actually borrow 10million sterling at 11% (A can invest L 10M @ 11% but is afraid that \$ will strength it wants US denominated investment)

44. Comparative Advantage Again • The argument for this is very similar to the comparative advantage argument presented earlier for interest rate swaps. • It is likely that the domestic firm has an advantage in borrowing in its home country.

45. Example using comparative advantage Dollars AUD (Australian \$) Company A 5% 12.6% Company B 7% 13.0% 2% difference in \$US .4% difference in AUD

46. The strategy • Company A borrows dollars at 5% per annum • Company B borrows AUD at 13% per annum They enter into a swap Result Since the spread between the two companies is different for each firm there is the ability of each firm to benefit from the swap. We would expect the gain to both parties to be 2 - 0.4 = 1.6% (the differences in the spreads).

47. Swap Diagram Co A FI Co B A pays 11.9% AUD instead of 12.6% AUD B pays 6.3% \$US instead of 7% \$US The FI makes .2% AUD 11.9% AUD 13% 5% AUD 13% 5% 6.3%

48. Valuation of Currency Swaps • Using Bond Techniques • Assuming there is no default risk the currency swap can be decomposed into a position in two bonds, just like an interest rate swap. • In the example above the company is long a sterling bond and short a dollar bond. The value of the swap would then be the value of the two bonds adjusted for the spot exchange rate.

49. Swap valuation • Let S = the spot exchange rate at the beginning of the swap, BF is the present value of the foreign denominated bond and BD is the present value of the domestic bond. Then the value is given as Vswap = SBF – BD The correct discount rate would then depend upon the term structure of interest rates in each country

50. Other swaps • Swaps can be constructed from a large number of underlying assets. • Instead of the above examples swaps for floating rates on both sides of the transaction. • The principal can vary through out the life of the swap. • They can also include options such as the ability to extend the swap or put (cancel the swap). • The cash flows could even extend from another asset such as exchanging the dividends and capital gains realized on an equity index for a fixed or floating rate.