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MSA 736 Fixed Income & Derivatives II

MSA 736 Fixed Income & Derivatives II. Chapters 5 Swaps Markets and Contracts. Important Concepts in Chapter 12. The concept of a swap Different types of swaps, based on underlying currency, interest rate, or equity Pricing and valuation of swaps Strategies using swaps. Swap Basics.

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MSA 736 Fixed Income & Derivatives II

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  1. MSA 736Fixed Income & Derivatives II Chapters 5 Swaps Markets and Contracts

  2. Important Concepts in Chapter 12 • The concept of a swap • Different types of swaps, based on underlying currency, interest rate, or equity • Pricing and valuation of swaps • Strategies using swaps

  3. Swap Basics • Four types of swaps • Currency • Interest rate • Equity • Commodity (not covered in this book) • Characteristics of swaps • No cash up front • Fixed rate is set such that market value of swap = 0 • Notional principal • Settlement date, settlement period • Credit risk

  4. Swap Contract Terminology • Notional principal: Amount used to calculate periodic payments • Swap rate: The fixed rate on a interest rate swap • Floating rate: Usually US LIBOR • Tenor: Time period covered by swap • Settlement dates: Payment due dates

  5. Plain Vanilla Interest Rate Swap • Fixed interest rate payments are exchanged for floating-rate payments • Notional amount is not exchanged at the beginning or end of the swap (both loans are in same currency and amount) • Interest payments are netted - On settlement dates, both interest payments are calculated and only the difference is paid by the party owing the greater amount

  6. Plain Vanilla Interest Rate Swap – Key Point • Floating rate payments are typically made in arrears, payment is made at end of period based on beginning-of-period LIBOR! (The floating rate at time 0 dictates the floating payment at time 1!) • Thus: First net payment is known at swap initiation!

  7. Plain Vanilla Interest Rate Swap The settlement payment made by the fixed-rate payer at time t (if negative, the fixed-rate payer will receive this amount): FR = fixed rate or swap rate T = # days in settlement period NP = notional principal

  8. Vanilla Interest Rate Swap: Example A 2-year interest rate swap with semiannual-settlement, floating payments based upon LIBOR, notional principal of $10 million with a swap rate (the fixed rate) of 6%. Thus, semiannual fixed payments are: (0.06 / 2) × $10 million = $300,000. Assume the following LIBOR term structure: LIBOR t0 =5%, t1 =5.8%, t2 =6.2%, t3 =6.6% 1st payment: Fixed-rate payer pays $50,000 net (0.06 – 0.05 )(180/360)(10 million) = $50,000 Again, first net payment is known at swap initiation (t=0)!

  9. Interest Rate Swap Example Continued LIBOR 5% at t0, 5.8% at t1, 6.2% at t2, 6.6% at t3 ************************************************** 2nd payment: Fixed rate payer pays $10,000 net (0.06 – 0.058)(180/360)(10 million) = $10,000 3rd payment: Floating rate payer pays $10,000 net (0.06 – 0.062)(180/360)(10 million) = –$10,000 4th payment: Floating rate payer pays $30,000 net (0.06 – 0.066)(180/360)(10 million) = –$30,000

  10. Pricing and Valuation of Interest Rate Swaps • How is the swap rate (the fixed rate) determined? • What is the market value of an interest rate swap at a date between the swap initiation date and the next settlement date? • We answer these by replication: An pay-fixed side of a swap can be replicated by issuing a fixed rate bond and using the proceeds to buy a LIBOR-based floating rate bond; where the face value of the two bonds equals the notional principal; similarly, pay-floating side can be replicated by issuing a floating rate bond and using the proceeds to buy a fixed rate bond

  11. The Swap Fixed Rate • Idea: Swap fixed rate must be set so swap value at initiation is zero (must be 0 if no money changes hands at swap inception) • Method: Value the swap as a combination of fixed-rate bond and floating rate bond • Key insight: At each settlement date, the market value of the floating rate bond will always reset to par (interest rates “adjust” to market rates)

  12. The Swap Fixed Rate – by Replication Consider the replication of a 4-year annual pay swap: • The fixed rate bond will pay coupon every year for 3 years and the final coupon and return of principal at year 4 • The floating rate bond will pay a coupon every year for 3 years (unknown in years 2 and 3) and the final coupon and return of principal at year 4

  13. The Swap Fixed Rate by Replication • Recall that the swap rate is set such that no money changes hands at initiation • For this to hold, the market value of the fixed rate bond must equal the present value of the floating-rate bond • For a floating rate bond at initiation or settlement date, the bond’s rate is reset so that the bond trades at par

  14. The Swap Fixed Rate • Assume $1,000 par bonds • At initiation, the floating rate is valued at $1,000 (since the rate is reset on day 1) • Since no money changes hands at swap inception, this also means that the fixed rate must be worth $1,000. • Summary, set the floating rate bond value to the fixed rate bond value and solve for the fixed coupon rate – this is the swap rate!

  15. The Swap Fixed Rate by Replication • The coupon rate for a 4-period fixed-rate bond must be: This “C” is the swap fixed rate!

  16. The Swap Fixed Rate by Replication • Let Zi= 1/(1+Ri); Zi is PV factor for the ith period • We can write: 1000 = Z1C + Z2C+ Z3C+ Z4C++ Z4(1000) • Simplify: 1000 – Z4 =C(Z1 + Z2+ Z3+ Z4) • Solve for C:

  17. Price of n-period $1 zero coupon bond = n-period discount factor The Swap Fixed Rate • Per $1 par, we get the formula for the swap rate:

  18. Swap Fixed Rate: Example • Calculate the swap rate and the fixed payment on a 1-year, quarterly settlement swap with a notional principal of $10MM 1/(1+0.045(90/360)) 1/(1+0.06)

  19. The Swap Fixed Rate: Example • Applying the formula for C, we get the quarterly swap rate: Note: This is a quarterly rate! Need to annualize!

  20. The Swap Fixed Rate: Example • Quarterly fixed-rate payment is: $10M × 0.0146 = $146,000 • Annualizing the quarterly rate gives us the annual swap rate of1.46% × 4 =5.84% Notional principal

  21. Valuing a Swap at date after inception(Fixed-Rate Payer Side) • Swap value: Difference in PV of fixed and floating payments • To calculate the value, we must find: • PV of “replicating” fixed rate bond • PV of “replicating” floating rate bond: • Fixed-rate payer swap value = PVfloat – PVfixed Received by fixed payer Paid by the fixed payer

  22. Valuing a Swap: Example • Consider the same swap priced at5.84%: • Assume 240 days have passed • 90-day LIBOR at last settlement (day 180) was 3.5% • Note that the two remaining settlement dates are now 30 and 120 days away

  23. Valuing a Swap: Example • Given that the swap rate is 5.84%, the quarterly payments from the fixed side are: • The fixed-rate payer will pay $0.0146 per $ of NP at the two remaining settlement dates (30 and 120 days from now) Per $ of notional principal

  24. Valuing a Swap: Example • The replicating fixed rate bond has a: • Coupon payment of $0.0146 in 30 days • Coupon and principal payment totaling $1.0146 ($1.000 + $0.0146) in 120 days • The value of the replicating bond is the present value of these two payments • Discounted using the 30- and 120-day discount factors

  25. Valuing a Swap: Example PV of fixed rate bond!

  26. Valuing a Swap: Example • Next, find PV of the replicating floating rate bond: • Now the trick: We know the value of the FRN at all settlement dates will be $1, so the value of the floating rate bond is the PV of $1.00875 in 30 days • Next coupon + $1 par value = $1.00875 From last settlement date (given)

  27. Valuing a Swap: Example It doesn’t matter what the floating rate payment is on day 360 because the bond will reprice to par on day 270 PV of floating bond!

  28. Valuing a Swap: Example • Finally! We can calculate the value to the pay-fixed side(per $ of notional principal):

  29. Valuing a Swap: Example • Assuming a $10M notional principal, the value to the fixed rate payer is: • Make sense? Yes, rates decreased and the pay-fixed side lost in value

  30. Equity Swaps • Characteristics • One party pays the return on an equity, the other pays fixed, floating, or the return on another equity • Rate of return is paid, so payment can be negative • Payment is not determined until end of period

  31. Equity Swaps (continued) • The Structure of a Typical Equity Swap • Setttling party paying stock and receiving fixed • Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be

  32. Equity Swaps (continued) • The Structure of a Typical Equity Swap (continued) • The fixed payment will be • $25,000,000(.0345)(90/360) = $215,625 • The first equity payment is: • So the first net payment is IVM pays $285,657.

  33. Equity Swaps (continued) • The Structure of a Typical Equity Swap (continued) • If IVM had received floating, the payoff formula would be: • If the swap were structured so that IVM pays the return on one stock index and receives the return on another, the payoff formula would be:

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