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Interest Rate Swaps. 國際財管 Dr. Chi-Sheng Hsu 東海大學國貿系. LIBOR deposits pay interest by multiplying the rate by the number of days over 360 and adding that interest to the amount deposited. A deposit of $1 at 9% for 6 months would pay back $1*(1+ 0.09*(180/360)) = $1.045

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## Interest Rate Swaps

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**Interest Rate Swaps**國際財管 Dr. Chi-Sheng Hsu 東海大學國貿系**LIBOR deposits pay interest by multiplying the rate by the**number of days over 360 and adding that interest to the amount deposited. • A deposit of $1 at 9% for 6 months would pay back • $1*(1+ 0.09*(180/360)) = $1.045 • To have$1 in 6 months at the 6-month LIBOR of 9% p.a., we need to deposit • $1/[1+ 0.09*(180/360)] = $1/1.045 = 0.9569. Video - Personal Finance Minute: All About Libor - WSJ.com LIBOR**An FRA’s settlement and maturity are defined by its name:**• A 3x9 FRA is signed today, will expire in 3 months, and will allow the investor to lock in 6-month LIBOR. • A 12x18 FRA is signed today, will expire in 12 months, and will allow the investor to lock in 6-month LIBOR FRA’s Settlement & Maturity t1 t2 0 Loan period origination date (fixed rate is set) settlement date, or delivery date (floating rate is set) end of forward period**3-month LIBOR**3-month LIBOR (in Month 3) (in Month 3) Company Y FRA Dealer Company Z 4.81% (bid rate) 4.85% (ask rate) Receive 3-month LIBOR Pay 3-month LIBOR A Pair of 3x6 FRA Transactions Borrower Lenders • The market makerpays a fixed rate of 4.81% for receipt of 3-month LIBOR and receives a fixed rate of 4.85% for payment of LIBOR.**Manage the impact of short-term interest rate fluctuations:**• Floating-rate debt+expect interest rates to rise • Buy an FRA (pay fixed-rate) to lock in short-term financing cost. • 由銀行bid(買,付低固定)與ask (賣,收高固定)的角度來看 • Fixed-rate debt + expect interest rates to drop • Sell an FRA (receive fixed-arte) to convert the fixed-rate obligation into a floating-rate one. How Markets Use FRAs**Sale of Forward Rate Agreement (FRA)**• In the near future, you will be making a deposit for a specific (short) period. In order to hedge the interest-rate risk, you sell an FRA. • If the Euribortwo business days prior to the settlement date < FRA rate • ING pays the discounted difference between the interest at the Euribor and the FRA rate, calculated on a pro rata temporis basis on the notional amount at the commencement of the hedging period. • If the Euribor two business days prior to the settlement date > FRA rate • You pay the discounted difference between interest at the Euribor and FRA rate, calculated on a pro rata temporis basis on the notional amount at the commencement of the hedging period. • http://www.ingfm.com/PDF/Productlibrary/SaleFRA.pdf • Euro Interbank Offered Rate Sale of Forward Rate Agreement (1) (Example from ING)**Terms and conditions**(http://www.ingfm.com/PDF/Productlibrary/SaleFRA.pdf) • Duration: 2 years maximum; Charges: zero cost. • Minimum amount: a minimum notional amount of EUR 250,000 • Standardized FRAs are quoted on the market. • Example: FRA 1x4: • the figure on the left indicates the waiting period, in this example: one month. • the difference between the figure on the right and that on the left indicates the hedging period, in this example: 4 – 1 = 3 months. • the figure on the right indicates the total period between the commencement and expiry of the agreement, in this example: 4 months. Sale of Forward Rate Agreement (2) (Example from ING)**A firm plans to borrow $20 million in 30 days. The loan is a**90-day loan with the rate of 3-month LIBOR plus 1%in 30 days. • Since the firm worried about rising rates, it buys an FRA with its bank and the bank agrees on a rate of 10%. The FRA will pay off according to 3-month LIBORin 30 days.(1x4) • The holder of the long FRA will have: (Notionalprincipal)(LIBOR-10%)(90/360)/[1 + LIBOR*(90/360)] An Example of the FRA**The formula for pricing an FRA is just the formula for a**LIBOR forward rate, given the interest payment conventions in the FRA market. • Recall: (1+r1)(1+1f2)=(1+r2)2 1f2 = (1+r2)2/(1+r1) –1 • (1+r2)2(1+2f3)=(1+r3)3 2f3 = (1+r3)3/(1+r2)2 –1 • Consider a 3x9 FRA. The term structure is given as: 3-month Libor = 5.6%, and 9-month Libor= 6%. What is the FRA rate? • [1+0.056*(90/360)]*[1+90f270*(180/360)] = 1+0.06*(270/360) • FRA = 90f270 = 0.0611 (p.a.) • Assume that we go long the 3x9 FRA, and it is 25 days later. The term structure shows that 65-day Libor = 5.9%, and 245-day Libor= 6.5%. What is the value of the FRA? Determine the FRA Rate (fixed rate)**Long an FRA Pay fixed (6.11%) Rate, Receive floating.**• -25 0 65 245 • |----------|---------------|------------------------| • Pay fixed -(1+0.0611*180/360) = -1.0306 • -1.0306/(1+0.065*245/360) = -0.9869 • Rec. Libor (1+Libor*180/360) • (1+Libor*180/360)/(1+Libor*180/360) =1 • 1/(1+0.059*65/360) = 0.9895 • Value = -0.9869+0.9895 = 0.0026*Notional Principal Value an FRA Contract**An agreement between two parties in which each party makes a**series ofinterest payments to the other at specified dates at different rates. • At least one rate must be floating. Usually the parties are an end user and a dealer, which is a commercial bank or an investment bank. • Plain vanilla swap: one party pays a fixed rate and the other pays a floating rate. • Example: A 3-year swap initiated on Mar. 1, 2008, in which Company B agrees to pay to Company A a fixed rate of 5% compounded semiannually, and to receive the 6-monthLIBORrate from Company A on a notional principal of $100million. Interest Rate Swaps**---------Millions of Dollars---------**LIBOR FLOATING FIXED Net Date Rate Cash Flow Cash Flow Cash Flow Mar.1, 2008 4.2% Sept. 1, 2008 4.8% +2.10 –2.50 –0.40 Mar.1, 2009 5.3% +2.40 –2.50 –0.10 Sept. 1, 2009 5.5% +2.65 –2.50 +0.15 Mar.1, 2010 5.6% +2.75 –2.50 +0.25 Sept. 1, 2010 5.9% +2.80 –2.50 +0.30 Mar.1, 2011 6.4% +2.95 –2.50 +0.45 Cash Flows to Company B**Recall that Unilever has a AAA credit rating, and would**prefer to borrow at floating rates. Xerox has a BBB credit rating, and would prefer to borrow at fixed rates. • Unilver (UK) can borrow fixed-rate funds 1.0% cheaper than Xerox (US), and can borrow floating-rate funds 0.5%cheaper. • This means that Unilever’s relative comparative advantage is to borrow fixed-rate funds, and therefore Xerox should borrow floating-rate funds. Relative Comparative Advantage**XYZ is borrowing $50 millionat 3-month LIBOR with rate reset**at beginning of each quarter. It would prefer a fixed rate. Payment is made at end of quarter. • XYZ enters into a pay-fixed (7.5%), receive-floating swap with dealer ABSwaps. • 一般稱付固定利率端為買方，收固定利率端為賣方。 • Notional principal is never exchanged (等值交換). Hence, a swap should have zero initial value. • Each Payment: Notional*(LIBOR-Fixed)*actual number of days/360 Another Example of the IRS**For a2-year swap, there are 8 quarterly floating-rate**payments. • While the first quarterly payment is known, the next sevenpayments are not. • Those floating rate payments may be estimated by examining the shape of the yield curve, or more practically, by referencing the rates associated with Eurodollar futures prices which reflect the shape of the curve. • Eurodollars as Risk Management Tools, CME Group, March 31, 2011. • Page 691 at Fabozzi (2007), Bond Markets, Analysis, and Strategies, 6th ed.. Interest Rate Swap2-year interest rate swap with quarterly payments**因係等值交換，IRS契約建立時價值為零。**• An IRS is transacted such that the presentvalue of the estimatedfloating rate payments equalsthe present value of the fixed rate payments, nomonetary consideration is passed on the basis ofthis initial transaction. • 那麼，設立IRS時，固定利率(fixed rate, swap rate) 應訂定為多少? 才能等值交換。 • 別忘了未來各期之浮動利率係採3MLIBOR，故可用Eurodollar futures 去估計(因它的標的即是3MLIBOR)。 Zero Initial Value**Given the spot rates equals 8% and equals 10%, what should**a 5% coupon, two-year bond cost? • PV = 50/(1+) + 1050/(1+)2 = 50/1.08 + 1050/1.102= $914.06 • We now want to calculate a single rate for the bond. We do this by solving for y in the following equation: • $914.06 = 50/(1+y) + 1050/(1+y)2 y= 9.95% • We call y the yield to maturity on the bond. • Spot interest rate and forward interest rate • (1+r1)(1+1f2)=(1+r2)2 1.08*(1+1f2)=(1.10)2 1f2 = 12.04% • PV = 50/(1+) + 1050/(1+)2 = 50/1.08 + 1050/(1.08*1.1204) = $914.06 讓我們複習YTM觀念**df1 = 1/(1+r1) (已知)**• df2 = 1/[(1+r1)(1+f2)] = df1/(1+f2) • df3 = 1/[(1+r1)(1+f2)(1+f3)] = df2/(1+f3) • df4 = 1/[(1+r1)(1+f2)(1+f3)(1+f4)] = df3/(1+f4) • dft = dft-1/(1+ft) Discount Factor**As of 3/9/2011, 名目本金：$1,000,000**下表並未利用折現因子公式，可點擊下表觀看Excel內容。 PV of Floating Rate Payments**As of 3/9/2011, 名目本金：$1,000,000**PV of Floating Rate Payments**As of 3/9/2011, 名目本金：$1,000,000**PV of Fixed Rate Payments**swap rate* 20409.21 = 18027.12**• swap rate = 0.8833% Find the Swap rate

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