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Topic Six Valuing swap contracts

Topic Six Valuing swap contracts. How do you value a swap contract?. Topic Six Sub-Topics. Characteristics of swaps The comparative advantage theory of swaps Theoretical LIBOR/swap rates and curves Valuation of interest rate swaps Valuation of currency swaps.

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Topic Six Valuing swap contracts

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  1. Topic SixValuing swap contracts How do you value a swap contract?

  2. Topic SixSub-Topics • Characteristics of swaps • The comparative advantage theory of swaps • Theoretical LIBOR/swap rates and curves • Valuation of interest rate swaps • Valuation of currency swaps

  3. Topic SixStudent learning outcomes After completing this unit you should be able to do the following: • Describe the characteristics of swap contracts • Define and give examples of various types of swaps: currency, interest rate, commodity, equity • Discuss the comparative advantage theory of swaps • Explain the relevance of swaps and Eurodollar futures in determining theoretical zero rate curves • Construct a theoretical LIBOR/swap zero rate curve using the bootstrapping method • Calculate the value of an interest rate swap using bond prices • Calculate the value of an interest rate swap using FRA prices • Calculate the value of a currency swap using bond prices • Calculate the value of a currency swap using forward prices • Explain how credit risk arises in a swap

  4. Topic SixReferences • Hull, J.C. (2011). Fundamentals of Futures and Option Markets. New Jersey: Pearson Education. Chapter 7.

  5. Characteristics of swaps Describe characteristics of swaps • An agreement between two parties to exchange a series of future cashflows • Typically one party makes floating payments, such as an interest rate, a currency rate, an equity or a commodity price and the other makes payments that are either fixed or floating and whose value is determined by another factor • Swaps typically have a value of zero at initiation as, with the exception of currency swaps no cashflow is exchanged at initiation of the swap

  6. Characteristics of swaps Describe characteristics of swaps • On the settlement dates on which the cashflows are to be exchanged, typically one party makes a payment of the net amount owing to the other • It is rare for swaps to call for actual physical delivery of the underlying asset

  7. Characteristics of swaps Define different types of swaps • Interest rate swaps: one party pays a fixed rate and the other pays a floating rate with both sets of payments in the same currency • Commodity swaps: one party pays a fixed rate and the other pays a rate equal to the change in price of a commodity • Equity swaps: one party pays a fixed rate and the other pays a rate equal to the return on an equity index • Currency swaps: each party makes interest payments to the other in different currencies

  8. Our Toy Company What currency are their loans in? What currency are their payment in? What currency are their receipts in?

  9. Characteristics of swapsDescribe characteristics of swaps Example 6.1 • Microsoft and Intel have initiated a 3-year swap on March 5, 2007. • Microsoft has a floating rate loan. Intel have a fixed rate loan. • Microsoft agrees to pay Intel an interest rate of 5% pa, paid and compounding semi-annually. The notional principal is $100 million. In return Intel agrees to pay Microsoft the 6-month LIBOR rate on the same notional principal. • Assume the 6-month LIBOR rates as quoted on Mar-5, 2007 and then after at six monthly intervals turn out to be 4.2%, 4.8%, 5.3%, 5.5%, 5.6%, 5.9% and 6.4%. Calculate the payments made/received by Microsoft.

  10. Characteristics of swapsDescribe characteristics of swaps Example 6.1 • Assume the 6-month LIBOR rates as quoted on Mar-5, 2007 and then after at six monthly intervals turn out to be 4.2%, 4.8%, 5.3%, 5.5%, 5.6%, 5.9% and 6.4%. Calculate the payments made/received by Microsoft. • Sept 5 cashflows: • Microsoft pays Intel at the agreed fixed rate 5% pa • Intel would pay Microsoft at the LIBOR 6-mth rate as at Mar 5.

  11. ---------Millions of Dollars--------- LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, 2007 4.2% Sept. 5, 2007 4.8% +2.10 –2.50 –0.40 Mar.5, 2008 5.3% +2.40 –2.50 –0.10 Sept. 5, 2008 5.5% +2.65 –2.50 +0.15 Mar.5, 2009 5.6% +2.75 –2.50 +0.25 Sept. 5, 2009 5.9% +2.80 –2.50 +0.30 Mar.5, 2010 6.4% +2.95 –2.50 +0.45 Characteristics of swapsDescribe characteristics of swaps Example 6.1

  12. Characteristics of swaps Describe characteristics of swaps 5% Intel MS LIBOR

  13. Characteristics of swapsDescribe characteristics of swaps Example 6.2 • IBM and British Petroleum entered into a 5-year currency swap on February 1, 2007. IBM has agreed to pay a fixed rate of interest of 5% pa in sterling and will receive a fixed rate of interest of 6% pa in US dollars from British Petroleum. Interest payments are made annually and the principal amounts are USD18 million and STG10 million. • Calculate the payments made/received by IBM.

  14. Characteristics of swapsDescribe characteristics of swaps Example 6.2 • Calculate the payments made/received by IBM. • The Feb 1 2007 cashflows • IBM pays BP USD18m and receives STG10m • The Feb 1 2005 cashflows • IBM pays BP • BP pays IBM

  15. Characteristics of swaps Describe characteristics of swapsExample 6.2 Dollars Pounds $ £ Year ------millions------ 2007 –18.00 +10.00 +1.08 2008 –.50 +1.08 –.50 2009 2010 +1.08 –.50 +1.08 –.50 2011 2012 +19.08 -10.50

  16. Characteristics of swaps Describe characteristics of swaps Dollars 6% IBM BP Sterling 5%

  17. Characteristics of swaps Describe characteristics of swaps • Swap counterparties usually deal through a financial intermediary • “Plain vanilla” fixed-for-floating swaps on interest rates are usually structured so that the financial institution earns about 3 to 4 basis points on a pair of offsetting transactions • Where previously the parties dealt directly and there was only one swap agreement, involving a financial intermediary means there are two swap agreements, one between each counterparty and the intermediary

  18. Characteristics of swaps Describe characteristics of swaps 4.985% 5.015% Intel F.I. MS LIBOR LIBOR

  19. Characteristics of swaps Describe characteristics of swaps USD 6% USD 6.015% IBM F.I. BP STG 5% STG 5.015%

  20. Characteristics of swaps Describe characteristics of swaps • Many financial institutions act as market makers for swaps, entering into swaps without having an offsetting swap with another counterparty • Market Makers use bonds, FRAs, interest rate futures and other derivative instruments to hedge their exposure to swaps they enter into as principal • Market makers post bid – offer quotes for swaps with the bid-offer spread typically 3 – 4 basis points • Swap rate is the mid rate • As the swap is worth zero at initiation Pfix = Pflt

  21. Characteristics of swaps Describe characteristics of swaps • A LIBOR-based floating rate cashflow on a swap payment date is calculated where L is the principal, R is the relevant LIBOR rate and n is the number of days since the last payment • The fixed rate is usually quoted as actual/365 or 30/360, in which case to equate the fixed and floating rates either the fixed is multiplied by 360/365 or the floating by 365/360

  22. The comparative advantage theory of swaps Discuss comparative advantage theory • The comparative advantage theory of swaps suggests that the attractiveness of swaps is due to the ability of a swap to reduce the cost of borrowing to each party to the swap, where each party has a comparative advantage in a different debt market • Each party could borrow in the market in which they have a comparative advantage and so minimise their cost of funds by borrowing in their lowest cost market • The parties could then enter into a swap so as to pay a fixed or floating rate as desired

  23. The comparative advantage theory of swaps Discuss comparative advantage theoryExample 6.3 • Two companies, AAACorp and BBBCorp, both wish to borrow $10 million for five years. AAACorp can borrow fixed at 10.0% and floating at 6-mth LIBOR plus 0.3%, whereas BBBCorp can borrow fixed at 11.2% and floating at 6-mth LIBOR plus 1.00%. AAACorp has a AAA credit rating and wants to borrow floating, while BBBCorp has a BBB credit rating and wants to borrow fixed. • How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement?

  24. The comparative advantage theory of swaps Discuss comparative advantage theoryExample 6.3 FIXED FLOATING AAACorp 10.00 L+0.3 BBBCorp 11.2 L+1.0 _____ _____ 1.2 0.7 BBBCorp has relative advantage in floating but wants fixed, AAACorp has relative advantage in fixed and wants floating

  25. The comparative advantage theory of swaps Discuss comparative advantage theoryExample 6.3 • How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement? • AAACorp agrees to pay BBBCorp at 6-mth LIBOR and receives from BBBCorp at fixed rate of 9.95% on $10m • AAACorp interest rate cashflows: • It pays 10% pa to outside lenders • It receives 9.95% pa from BBBCorp • It pays LIBOR to BBBCorp Overall • 10% - 9.95% = .05% worse off • L+.3 –L =.3 better off • Total .3 -.05 = .25 better off

  26. The comparative advantage theory of swaps Discuss comparative advantage theoryExample 6.3 • How might each company achieve their objectives and minimise their cost of borrowing via a swap agreement? • BBBCorp interest rate cashflows: • It pays LIBOR + 1% to outside lenders • It receives LIBOR from AAACorp • It pays 9.95% to AAACorp Overall • 11.2 - 9.95% = 1.25% better off • L- L+1.0 =1.0 worse off • Total 1.25 -1.00 = .25 better off

  27. The comparative advantage theory of swaps Discuss comparative advantage theory 9.95% 10% AAA BBB LIBOR+1% LIBOR

  28. The comparative advantage theory of swaps Discuss comparative advantage theory 9.93% 9.97% 10% AAA F.I. BBB LIBOR+1% LIBOR LIBOR

  29. The comparative advantage theory of swaps Discuss comparative advantage theory • The spreads between the rates offered to AAACorp and BBBCorp are a reflection of the extent to which BBBCorp is more likely to default than AAACorp • The fact that the spread between the companies at the 5-year rate is larger than at the 6-month rate is indicative of the perception that the probability of default rises faster for a lower rated company than it does for a higher rated company

  30. The comparative advantage theory of swaps Discuss comparative advantage theory • It should be noted that the fixed rate is fixed for 5-years, whereas the floating rate is reset every six months. Hence, every six months the lender to BBBCorp will reset the floating rate to match the current 6-month LIBOR rate and may review the margin charged over LIBOR to BBBCorp • BBBCorp appears in Example 6.3 to pay 10.95%, however it will do so only so long as it can continue to borrow at a 1% margin over LIBOR.

  31. The comparative advantage theory of swaps Discuss comparative advantage theory • Should the margin charged to BBBCorp over LIBOR increase its comparative advantage will reduce and BBBCorp could end up paying a higher rate of interest under the swap as the margin is not “swapped”, only the LIBOR rate • AAACorp has fixed a rate of LIBOR + 0.05%, which is a good deal, though it has done so by accepting a counterparty risk with the financial intermediary, which it would not have borne if it had simply issued a bond

  32. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • LIBOR is the rate of interest at which AA-rated banks borrow for period between 1 and 12 months from other banks • Forward LIBOR rate is derived from Eurodollar futures rates for a three-month period in the future • Swap rate is the average of the bid rate a market maker pays in exchange for receiving LIBOR and the offer rate the market maker is prepared to accept for paying LIBOR

  33. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • LIBOR rates are used as proxies for risk-free rates when valuing derivatives, though they are not entirely risk-free • Swap rates similarly have a low credit risk • A financial institution can earn the 5-year swap rate on a notional principal by lending and re-lending the principal for 10 consecutive 6-month periods out to five years, then entering into a swap to exchange the LIBOR income for the 5-year swap rate • The credit risk on a 5-year swap corresponds to that of a series of 6-month LIBOR loans to AA-rated banks

  34. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • Swap rates define a set of par yield bonds • The value of a newly issued floating-rate bond that pays 6-month LIBOR is always equal to its principal or par value, when the LIBOR/swap zero curve is used for discounting, as LIBOR constitutes both the rate of interest on the bond and the discount rate of the bond’s cashflow, hence Pflt = Ppar • For a newly issued swap where the fixed rate equals the swap rate, Pfix = Pflt, therefore if Pflt = Ppar, Pfix = Ppar

  35. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • A LIBOR/swap zero curve can be constructed using LIBOR rates for the first 12-months, Eurodollar futures for the next 24-months and swap rates for maturities of greater than 36-months.

  36. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate Example 6.4 • The 6-month, 12-month and 18-month LIBOR/swap zero rates have been determined as 4%, 4.5% and 4.8% pa with continuous compounding and the 2-year swap rate for a swap with semiannual payments, is 5%. • Calculate the 24-month LIBOR/swap zero rate.

  37. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rateExample 6.4 • Calculate the 24-month LIBOR/swap zero rate.

  38. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • Eurodollar futures contracts maturing in March, June, September and December are sometimes used to help construct the LIBOR/swap zero curve, particularly for the maturities between 12-months and 36-months. • The Eurodollar futures rates are converted into forward rates for future 3-month periods using a convexity adjustment

  39. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rateExample 6.5 • The annual volatility of the short-term interest rate is typically observed to be 1.2% and the price of an 8-year Eurodollar futures contract is quoted as 94. • Calculate the forward rate for the 3-month period 8 years hence.

  40. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rateExample 6.5 • Calculate the forward rate for the 3-month period 8 years hence.

  41. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rate • If F1 is the forward rate calculated from the ith Eurodollar futures contract and Ri is the zero rate for a maturity Ti we have so that

  42. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rateExample 6.6 • The 400-day LIBOR zero rate has been calculated as 4.8% pa with continuous compounding and, from a Eurodollar futures quote, it has been calculated that the forward rate for a 91-day period beginning in 400 days is 5.30% pa with continuous compounding. • Calculate the 491-day LIBOR/swap zero rate.

  43. Theoretical LIBOR/swap rates and curvesConstruct a LIBOR/swap zero rateExample 6.6 • Calculate the 491-day LIBOR/swap zero rate.

  44. Valuation of an Interest Rate SwapCalculate value of Interest Rate Swap • The value of an interest rate swap at initiation is zero • After some time its value may rise or fall depending on underlying prices and rates • Interest rate swaps can be valued either: • as the difference between the value of a fixed-rate bond and the value of a floating-rate bond, or • as a portfolio of forward rate agreements (FRAs)

  45. Valuation of an Interest Rate SwapCalculate value of Interest Rate Swap • If we assume that a notional principal is exchanged at maturity, its net cashflows are unchanged as is its value • Hence a swap may be valued based on the value of a replicating portfolio of a long position and an offsetting short position in a fix rate bond and a floating rate bond • The fixed rate bond is valued in the usual way • The floating rate bond is valued based on the principal and the next floating payment, k*

  46. Valuation of an Interest Rate SwapCalculate value of Interest Rate Swap • The value of a interest rate swap where the holder has a long position in the fixed rate bond and a short position in the floating rate bond is given • The value of a interest rate swap where the holder has a short position in the fixed rate bond and a long position in the floating rate bond is given

  47. Valuation of an Interest Rate SwapCalculate value of Interest Rate SwapExample 6.7 • A financial institution has agreed to pay 6-month LIBOR and receive 8% pa compounding semi-annually on a notional principal of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates with continuous compounding are 10%, 10.5% and 11%, respectively for maturities 3-months, 9-months and 15-months. The 6-month LIBOR rate at the last payment date was 10.2% pa compounding semi-annually. • Calculate the value of the swap to the financial institution using bond pricing formulas.

  48. Valuation of an Interest Rate SwapCalculate value of Interest Rate SwapExample 6.7 • Calculate the value of the swap to the financial institution using bond pricing formulas.

  49. Valuation of an Interest Rate SwapCalculate value of Interest Rate SwapExample 6.7 • Calculate the value of the swap to the financial institution using bond pricing formulas.

  50. Valuation of an Interest Rate SwapCalculate value of Interest Rate Swap • A swap can be characterised as a portfolio of FRAs • An FRA can be valued using the forward interest as an estimate of the future spot interest rate • Use the LIBOR/swap zero curve to calculate forward rates for each of the LIBOR rates that will determine the swap cashflows • Calculate the swap cashflows on the assumption that the LIBOR spot rates will equal the forward rates • Discount these swap cashflows (using the LIBOR/swap zero curve) to obtain the swap value

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