资源描述
单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Lecture # 5:,Swaps,1,A s an agreement between two or more parties to exchange sets of cash flows over a period in the future. The parties that agree to the s known as counter-parties. The cash flows that the counter-parties make are generally tired to the value of debt instruments or to the value of foreign currencies. Therefore, the two basic kinds of swaps are interest rate swaps and currency swaps.,2,The Swaps Market,-,Swaps are custom tailored to the needs of the counter-parties.,-,The swaps market has virtually no government regulation.,-,Default risk,-,Value of Outstanding Swaps ($ Billion of Principal),3,Year,Total,Interest Rate Swap,Total,Currency Swap,1987,88,89,90,91,92,93,94,95,682.9,1,010.2,1,539.3,2,311.5,3,065.1,3,850.8,6,177.8,8,815.6,10,617.4,182.8,316.8,434.8,577.5,807.2,860.4,899.6,914.8,993.6,4,-,Plain Vanilla Swaps,Interest rate swaps,Currency Swaps,-,Motivations for swaps,5,Commercial needs: As an example of prime candidate for an interest rate swaps, consider a typical savings and loan association. Savings and loan associations accept deposits and lend those funds for long-term mortgages. Because depositors can withdraw their funds on shot notice deposit rates must adjust to changing interest rate conditions. Most mortgagors wish to borrow at a fixed rate for a long time in US. Is there any interest risk? Can swaps contract help?,6,Comparative advantage: In many situations, one firm may have better access to the capital market than another firm. For example, a U.S. firm may be able to borrow easily in the U.S., but it might not have such favorable access to the capital market in Germany. Similarly, a German firm may have good borrowing opportunities domestically but poor opportunities in the States.,Firm,USD rate,GEM rate,German firm,10%,7%,US firm,9%,8%,7,Interest Rate Swaps,-,Two Parties exchange periodic interest payments over a period. Typically, one partys payments are based on a fixed rate whereas its counterpartys payments are based on a floating rate. Interest payments are computed using a notional principal.,8,-,Example: Both A and B need to borrow $100 million for 3 years. The financing rates facing them are summarized as follows:,9,-,It is comparatively cheaper for A to use the floating rate debt. For B, fixed rate borrowing will be cheaper. Why?,1.,If A desires the floating rate debt and B prefers the fixed rate debt, there is no need for them to engage in a swap.,2.,If A desires the fixed rate debt and B prefers the floating rate debt, A should still borrow floating rate and B borrow fixed rate. They can then enter a s better both parties.,10,6.3%,Company,Company,| |,LIBOR+ A,B 6.3%,0.85% LIBOR,a.,Company A: Borrows floating rate and enters the above swap.,b.,Company B: Borrows fixed rate and enters the above swap,11,-,The results,Company A: On a semiannual basis, receives (LIBOR-6.3%)*50m from the swap, and pays the floating rate debt service (LIBOR+0.85%)*50m. The net payment is 7.15%*50m, which is less than 7.5%*50m.,Company B: On a semiannual basis, receives (6.3%-LIBOR)*50m from the swap, and pays the fixed rate debt service 6.3%*50m. The net payment is LIBOR*50m, which is less than (LIBOR+0.25%)*50m.,12,-,Note: S refers to fixed rate swap.,-,Swaps through an intermediary,6.4% 6.25%,Company,Swap,Company,| |,LIBOR+ A,Dealer,B 6.3%,0.85% LIBOR LIBOR,13,-,The results,Company A: On a semiannual basis, receives (LIBOR-6.4%)*50m from the swap, and pays the floating rate debt service (LIBOR+0.85%)*50m. The net payment is 7.25%*50m, which is less than 7.5%*50m.,Company B: On a semiannual basis, receives (6.25%-LIBOR)*50m from the swap, and pays the fixed rate debt service 6.3%*50m. The net payment is (LIBOR+0.05%)*50m, which is less than (LIBOR+0.25%)*50m.,S: Makes (6.4%-6.25%)*$50m=$75,000,14,-,Pricing Schedules,The fixed rate in the s quoted as a certain number of basis points above the T-note yield.,Table: Indication pricing for interest rate swaps at 1:30pm, New York Time on May 11, 1995,15,Maturity,(years),Bank Pays Fixed Rate,Bank receives Fixed Rate,Current TN,Rate (%),2,3,4,5,7,10,2-yrTN+17bps,3-yrTN+19bps,4-yrTN+21bps,5-yrTN+23bps,7-yrTN+27bps,10-yrTN+31bps,2-yrTN+20bps,3-yrTN+22bps,4-yrTN+24bps,5-yrTN+26bps,7-yrTN+30bps,10-yrTN+34bps,6.23,6.35,6.42,6.49,6.58,6.72,16,-,Netting: interest payments are made by one counter-party to the other after netting out the fixed and floating interest payments. Assume: Notional amount = Q; fixed rate payment = k; Floating rate used in time t=R,t-1,(LIBOR at time t-1). NET payment at time t: Fixed rate at time t: Fixed-rate payer receives (R,t-1,Q-k) and floating-rate payer receives (k-R,t-1,Q). The following is a possible scenario of cash flows for the fixed-rate payer under a $100 million, 5-year s 5.6% with semiannual cash flow exchanges.,17,#,Time (years),LIBOR,Floating Payment,Fixed Payment,Net,0,1,2,3,4,5,6,7,8,9,10,0.0,0.5,1.0,1.5,2.0,2.5,3.0,3.5,4.0,4.5,5.0,5.2,5.7,6.1,5.8,5.5,5.6,5.3,5.7,5.9,5.8,5.5,2.60,2.85,3.05,2.90,2.75,2.80,2.65,2.85,2.95,2.90,-2.80,-2.80,-2.80,-2.80,-2.80,-2.80,-2.80,-2.80,-2.80,-2.80,-0.20,+0.05,+0.25,+0.10,-0.05,+0.00,-0.15,+0.05,+0.15,+0.10,18,-,What is the implication of netting about credit (default risk)?,-,Pricing interest rate swaps:,Set the fixed rate of s that the s a zero value at the time of initiation. This is called par swap.,Suppose that payment dates are t,1,t,2,t,n,. The value of a s time t,V,t, from the perspective of the floating-rate payer:,V,t,=B,1t,-B,2t,19,B,1t,: value of fixed-rate bond underlying the s,t,i,t,t,i,+1, B,1t,=,n,j,=i+1,ke,-r(t,tj,)(,tj,-t),+,Qe,-r(t,tn,)(,tn,-t),.,B,2t,: value of floating-rate bond underlying the swap. At the floating rate resetting day, i.e., t=t,1,t,2,t,n, immediately after the payment is made, B,2t,=Q. Why? In between, i.e.,t,i,t,t,i,+1, B,2t,=(Q+k,*,)exp-r(t,t,i,+1,)(,t,i,+1,-t), where k,*,is the floating rate payment at time,t,i,+1,already known at time t. Determining the s at time 0: V,0,(k) =,n,i,=1,kexp-r(0,t,j,),t,j,+,Qexp,-r(0,t,n,),t,n, - Q=0,Q=,n,i,=1,kexp-r(0,t,j,),t,j,+,Qexp,-r(0,t,n,),t,n,. That is, set an appropriate coupon rate so that the bond is priced at par.,20,Example: Counter-party A in a three-year s 6-month LIBOR and receives a fixed rate on a notional principal of $100 million. The s 1.25 years to maturity. (The s was determined one year and nine-month ago.) At the time of initiation, 3-year 8% bond was priced at par. The LIBOR at the last payment date was 10.2% (semiannual compounding). Discount rates for 3-month, 9-month and 15-month maturities are 10%, 10.5%, and 11%, respectively. The fixed rate =8% per annum. B,1,=4e,-0.25*0.1,+4e,-0.75*0.105,+104e,1.25*0.11,=98.24, B,2,=(100+5.1)e,-0.25*0.1,=102.51.V=98.24-102.51=-4.27(million) to A and 4.27million B.,21,Portfolio of forwards: A swap (semiannual interest exchanges) can be viewed as a sequence of forwards with maturities: t,1,t,2,t,n,with a common forward price. Define P,t,(,) as the time-t value of zero-coupon bond maturing at time,for $1 face value.,For,t,i,t,t,i,+1,22,1.,At t,i+1,: k-k,*, evaluated at t, (k-k,*,)P,t,(t,i+1,),2.,At t,i+2,: k-0.5R(t,i+1,)Q, evaluated at t, PV,t,t(i+2),k-0.5R(t,i+1,)Q=k-0.5R,(t,i+1,t,i+2,)Qexp-r(t,t,i+2,)(t,i+2,-t), where R,(t,i+1,t,i+2,) is the forward rate (semiannual compounding) at time t over t,i+1,t,i+2,. Why?,3.,Similarly for t,i+3,t,i+4,4.,The total value of the s time t:,(k-k,*,)exp(-r(t,t,i+1,)(t,i+1,-t)+,n,j=i+1,k-0.5R,(t,j,t,j+1,)Qexp-r(t,t,j+1,)(t,j+1,-t),23,-,Example: Continue the previous example,R,(3m,9m)=2exp0.5*(0.75*0.105-0.25*0.1)/(0.75-0.25)-1=11.04%,R,(9m,15m)=2exp0.5*(1.25*0.11-0.75*0.105)/(1.25-0.75)-1=12.10%,V=(4-5.1)e,-0.1*0.25,+(4-0.5*0.1104*100)e,-0.105*0.75,+ (4-0.5*0.121*100)e,-0.11*1.25,=-4.27,24,Variation of interest rate swaps,-,Index amortized swaps: the notional principal is reduced over the life of the swap.,-,Constant yield swaps: both parts are floating. For example, one part may be linked to the yield on the 30-year T-bond and the other may be linked on the 10-year T-note.,25,-,Rate-capped swaps: floating rate is capped.,-,Putable and Callable swaps: one or both counter-parties have the right to cancel the s certain times without additional costs.,-,Forward swaps: the s is set but the s not commence until a later date.,26,Currency swaps,-,Two parties exchange periodic interest payments and principals in two currencies.,-,Example: Both A and B need to borrow USD50 million (or DEM equivalent of 84 million based on 1.68DEM/USD) for three-year. The financing rates facing them are summarized as follows:,27,It is comparatively cheaper for A to use the DEM debt. For B, USD borrowing will be cheaper. Why?,1.,If A desires the DEM debt and B prefers the USD debt, there is no need for them to engage in a swap.,2.,If A desires the USD debt and B prefers the DEM debt, A should still borrow DEM and B borrow USD. They can enter a currency s better both parties.,28,Interest payment flows,6.9%USD,Company,Company,|,|,4.2%DEM A,B 6.9%USD,3.9%DEM,29,b.,Initial principal flow,84m DEM,Company,Company,| |,84DEM A,B 50m USD,50m USD,30,b.,Terminal principal flow,84m DEM,Company,Company,| |,84DEM A,B 50m USD,50m USD,b.,Company A: Borrows DEM debt and enters the above swap.,c.,Company B: Borrows USD debt and enters the above swap.,d.,The results:,31,1.,Company A: Beginning: Exchange DEM84 million for USD50 million, a fair transaction at the current exchange rate (DEM168/USD1).,In-between: On a semiannual basis, receives DEM4.2m*3.9% and pays USD25m*6.9% due to the swap, and pays DEM42m*4.2% due to its DEM debt. The net payment is USD25m*6.9%+DEM42m*0.3%, comparing to USD25m*7.5%.,End: Exchange USD50m for DEM84m, not a fair exchange at the prevailing exchange rate.,32,2.,Company B: Beginning: Exchange USD50 million for DEM84 million, a fair transaction at the current exchange rate (DEM168/USD1).,In-between: On a semiannual basis, receives USD25m*6.9% and pays DEM4.2m*3.9% due to the swap, and pays USD25m*6.9% due to its USD debt. The net payment is DEM42m*3.9%, which is less than DEM42m*4.0%,End: Exchange DEM84m for USD50m, not a fair exchange at the prevailing exchange rate.,33,-,S an intermediary,7.4%$ 6.9%$ Company,Swap,Company,| |,4.2%DM A,Dealer,B 9%USD,4.2%DM 3.9%DM,34,-The results,1.,Company A: Beginning: Exchange DEM84 million for USD50 million, a fair transaction at the current exchange rate (DEM168/USD1).,In-between: On a semiannual basis, receives DEM4.2m*4.2% and pays USD25m*7.4% due to the swap, and pays DEM42m*4.2% due to its DEM debt. The net payment is USD25m*7.4%, which is less than USD25m*7.5%.,End: Exchange USD50m for DEM84m, not a fair exchange at the prevailing exchange rate.,35,2.,Company B: Beginning: Exchange USD50 million for DEM84 million, a fair transaction at the current exchange rate (DEM168/USD1).,In-between: On a semiannual basis, receives USD25m*6.9% and pays DEM4.2m*3.9% due to the swap, and pays USD25m*6.9% due to its USD debt. The net payment is DEM42m*3.9%, which is less than DEM42m*4.0%,End: Exchange DEM84m for USD50m, not a fair exchange at the prevailing exchange rate,3.,S: On a semiannual basis, earns USD(7.4%-6.9%)*25m and loss DEM(4.2%-3.9%)*$2m.,36,-,Pricing currency swaps,Set the two fixed rates of a s that the s a zero value at the time of initiation.,Suppose that payment dates are t,1,t,2,t,n,. The value of a s time t,V,t, based on the domestic currency:,V,t,=,S,t,B,Ft,-,B,Dt,S,t:,exchange rate (domestic price of one unit foreign currency) at time t.,37,B,Dt,: value of domestic fixed-rate bond underlying the s,t,i,t,t,i,+1,B,Dt,=,n,j,=i+1,k,D,e,-rd(t,tj,)(,tj,-t),+,Q,D,e,-rd(t,tn,)(,tn,-t), where,k,D,is the payment in the domestic currency, Q,D,is the principal amount in the domestic currency.,B,Ft,: value of foreign fixed-rate bond underlying the swap (measured in the foreign currency) when,t,i,t,t,i,+1,B,Ft,=,n,j,=i+1,k,F,e,-,rf,(t,tj,)(,tj,-t),+,Q,F,e,-,rf,(t,tn,)(,tn,-t), where,k,F,is the payment in the foreign currency, Q,F,is the principal amount in the foreign currency.,38,Determining the fixed rate at time 0,Set,k,D,and,k,F,such that,Q,D,=,n,j,=1,k,D,e,-rd(0,tj,),tj,+,Q,D,e,-rd(0,tn,),tn,Q,F,=,n,j,=i+1,k,F,e,-,rf,(0,tj,),tj,+,Q,F,e,-,rf,(0,tn,),tn,This implies V,0,= S,0,B,F0,-B,D0,=S,0,Q,F,-Q,D,=0,That is, set two appropriate coupon rates so that both bonds are priced at par.,39,Example: Counter-party A in a three-year s a fixed rate on a principal of USD100m and receives a fixed rate on a principal of DEM168m. The payments are made on a semiannual basis. The principals were set according to the exchange rate at the time of initiation. The current exchange rate is 1.52DEM/USD. The s 1.25 years to maturity. (The s was determined one year and nine- month ago.) At the time of initiation, 3-year 7.2% USD bond was priced at par, and 3-year 4.2% DEM bond was also priced at par. The current term structure for USD and DEM are both flat at 8% and 4% respectively.,40,B,D,=3.6e,-0.25*0.08,+3.6e,-0.75*0.08,+103.6e,-1.25*0.08,=100.66m,B,F,=1.68*2.1e,-0.25*0.04,+2.1e,-0.75*0.04,+102.1e,-1.25*0.04, =170.08m,To A: V=1708/1.52-100.66=USD11.23m and to B: V=-USD11.23m,Portfolio of forwards: A currency s be viewed as a sequence of forwards with maturities: t,1,t,2,t,n,with a common forward price. For,t,i,t,t,i,+1,41,1.,At t,i+1,:S,t(i+1),k,F,-k,D, evaluated at t, it has a value equal to F,t,(t,i+1,)k,F,-k,D,exp-r,Dt,(t,i+1,)(t,i+1,-t),2.,At t,i+2,:S,t(i+2),k,F,-k,D, evaluated at t, it has a value equal to F,t,(t,i+2,)k,F,-k,D,exp-r,Dt,(t,i+2,)(t,i+2,-t),3.,Similarly for t,i+3,t,i+4,t,i+n-1,4.,At t,i+n,: S,t(i+n),(k,F,+Q,F,) -(k,D,+Q,D,), evaluated at t, it has a value equal to F,t,(t,i+n,)(k,F,+Q,F,)-(k,D,+Q,D,)exp-r,Dt,(t,i+n,)(t,i+n,-t),5.,The total value of the s time t is the sum of all the terms.,42,-,Example,: Continue the previous example. F(0.25)=1/1.52,exp(0.08-0.04),0.25=0.6645; F(0.75)=1/1.52,exp(0.08-0.04),0.75=0.6679; F(1.25)=1/1.52,exp(0.08-0.04),1.25=0.6916,V = (0.6645*2.1*1.68-3.6)*e,-0.08*0.25,+(0.6779*2.1*1.68-3.6)e,0.08*0.75,+,(0.6916*102.1*1.68-103.6)e,-0.08*1.25,=- 1.2308-1.1380+13.5986=USD11.2298m,43,Equity swaps,-,Two parties exchange periodic payments over a fixed duration. Typically, one partys payments are based on a stock index return whereas its counter-partys payments are based on a benchmark-floating rate. Payments are computed using a notional principal.,44,-,Example: Notional principal $100m. Counter-party A receives 3-month LIBOR and pays S&P500 index return plus a s of -0.1%.,S&P500 return-0.1%,Company,Company,A,B,LIBOR,45,46,-,The value of equity swap,The value of this equity s zero on Jan 2, the time of initiation. The same is true for April 2, July 2, Oct 2 and Jan 2 immediately after the payment is made. Why?,The value of this equity s, say March 1, will not be zero, however. Assume that the futures price of S&P500 index futures contract maturing in April contract finished at 460.1 on that day. The discount rate on March 1 for the maturity of April 2 is 9.1%.,47,What is the value of s the LIBOR payer? The LIBOR payment on April 2 is known to be 225,000. Its present value is 225,000*exp(-0.091*32/365)=223,212. The receipt on April 2 subject to the S&P500 index performance is (I,A2,-I,J2,)/I,J2,-0.1%*100m. Its present value is (460.1-469.75-0.001*469.75)*100m/469.75*exp(-0.091*32/365)=-2,137,166. The total value =-223,212-2,137,166=-2,360,378.,48,Commodity swaps,-,In a typical commodity swap, one counter-party makes periodic payments to the second counter-party at a fixed price per unit for a given notional quantity of some commodity. The second counter-party pays the first counter-party a floating price for a given notional quantity of some commodity. The commodities are usually the same. The floating price is usually calculated as an average price.,49,Credit Default Swaps,Will be discussed in the section of credit risk.,50,Procter & Gamble Bankers Trust Leveraged Swap,51,1,The story,On November 2, 1993, P&G and BT entered a five year, semiannual settlement, $200 million notional principal interest rate s known as the “5/30” swap. BT pays a fixed rate of 5.30% and P&G pays a floating rate depends on thirty-day commercial paper (CP) daily average rate less then 75 basis points, plus some spread. The key factors in the agreement are the spread and the 75 basis points a plain vanilla s have been 5.3% versus the CP daily average rate flat. The s scheduled to lock in on May 4, 1994. Because the spread on the lock-in-date was 2,750 basis points, P&G experienced significant losses and filed a lawsuit. An out-of-court settlement was reached in May 1996. BT agreed to absorb $157 million.,52,2,The P&G-BT leveraged swap,Term: 5 year,Frequency: Semiannual payments,Fixed rate payer: Bankers Trust at 5.3%,Floating payer: P&G at 30-day commercial paper daily average rates less 75 Basis points plus a spread.,53,3,The spread,The spread is zero for the first 6-month settlement period, and then would be fixed for the remaining nine semiannual periods, depending on Treasury yields and prices on the first settlement date, May 4, 1994, according to the formula.,Spread = max0, 98.5(5-year CMT%/5.78% - (30-year TYS Price)/100 ,54,5-year CMT% is the yield on the 5-year constant-maturity Treasury note. The 30-year Treasury (TSY) bond price is the midpoint of the bid and offer prices on the 6.25% T-bond maturing in August 2023, not including accrued interest.,The spread on November 2, 1993 was zero because,98.5*5.02%/5.78% - 102.57811/100 = -0.1703,55,The spread on May 4, 1994 was,Max 0, 98.5*6.71%/5.78% - 86.84375/100 = 0.2750,Thus in return for receiving a fixed rate of 5.3%, the P&G would have been obligated to pay the 30-day CP daily average rate plus 26.75% (27.50%-0.75%) for the next four and one half years on the $200 million s the formula had not been amended prior to May 1994.,56,4,The amendment,The s amended in January 1994 to move the determination date of the spread from May 4, 1994 to May 19, 1994 in exchange for 13 basis points improvement in the floating rate side of the swap, i.e., 75 basis points has been changed to 88 basis points.,Interestingly, there is a Federal Open Market Committee meeting scheduled on May 17, two days before the new spread determination date.,57,P&G decided in March 1994 to lock in the spread, instead of waiting for May 19 determination. This was done in three stages with $50 million on March 10, $50 on March 14, and the remaining $100 million on March 29 All in all, the spread was locked in at 15%. The loss can be estimated at about $106.541 million in present value terms.,58,5,Was corporate treasury group at P&G able to ascertain the risk it was beating upon entering the transaction?,A loss of over $100 million on a
展开阅读全文