Lecture_5Swaps(衍生金融工具人民银行研究院,何佳)

Lecture#5:SwapsA swap is 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 swap are 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.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)YearTotal Interest Rate SwapTotalCurrency Swap19878889909192939495682.91,010.21,539.32,311.53,065.13,850.86,177.88,815.610,617.4182.8316.8434.8577.5807.2860.4899.6914.8993.6-Plain Vanilla Swaps1.Interest rate swaps2.Currency Swaps-Motivations for swaps1.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?2.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.FirmUSD rateGEM rateGerman firm10%7%US firm 9%8%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.-Example:Both A and B need to borrow$100 million for 3 years.The financing rates facing them are summarized as follows:FixedFloatingA7.5%6-month LIBOR+0.85%B 6.3%6-month LIBOR+0.25%-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 swap to better both parties.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-The resultsa.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.b.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.-Note:Swap rate 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-The resultsc.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.d.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.Swap dealer:Makes(6.4%-6.25%)*$50m=$75,000-Pricing SchedulesThe fixed rate in the swap is 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 11,1995Maturity(years)Bank Pays Fixed Rate Bank receives Fixed Rate Current TNRate(%)2345710 2-yrTN+17bps3-yrTN+19bps4-yrTN+21bps5-yrTN+23bps7-yrTN+27bps10-yrTN+31bps 2-yrTN+20bps3-yrTN+22bps4-yrTN+24bps5-yrTN+26bps7-yrTN+30bps10-yrTN+34bps 6.236.356.426.496.586.72-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=Rt-1(LIBOR at time t-1).NET payment at time t:Fixed rate at time t:Fixed-rate payer receives(Rt-1Q-k)and floating-rate payer receives (k-Rt-1Q).The following is a possible scenario of cash flows for the fixed-rate payer under a$100 million,5-year swap at 5.6%with semiannual cash flow exchanges.#Time(years)LIBOR Floating Payment Fixed Payment Net 0123456789100.00.51.01.52.02.53.03.54.04.55.05.25.76.15.85.55.65.35.75.95.85.5 2.602.853.052.902.752.802.652.852.952.90-2.80-2.80-2.80-2.80-2.80-2.80-2.80-2.80-2.80-2.80-0.20+0.05+0.25+0.10-0.05+0.00-0.15+0.05+0.15+0.10-What is the implication of netting about credit(default risk)?-Pricing interest rate swaps:a.Set the fixed rate of swap so that the swap has a zero value at the time of initiation.This is called par swap.b.Suppose that payment dates are t1,t2,tn.The value of a swap at time t,Vt,from the perspective of the floating-rate payer:Vt=B1t-B2tc.B1t:value of fixed-rate bond underlying the swap when titti+1,B1t=nj=i+1ke-r(t,tj)(tj-t)+Qe-r(t,tn)(tn-t).d.B2t:value of floating-rate bond underlying the swap.At the floating rate resetting day,i.e.,t=t1,t2,tn,immediately after the payment is made,B2t=Q.Why?In between,i.e.,titti+1,B2t=(Q+k*)exp-r(t,ti+1)(ti+1-t),where k*is the floating rate payment at time ti+1 already known at time t.Determining the swap rate at time 0:V0(k)=ni=1kexp-r(0,tj)tj+Qexp-r(0,tn)tn-Q=0 Q=ni=1kexp-r(0,tj)tj+Qexp-r(0,tn)tn.That is,set an appropriate coupon rate so that the bond is priced at par.e.Example:Counter-party A in a three-year swap pays 6-month LIBOR and receives a fixed rate on a notional principal of$100 million.The swap has 1.25 years to maturity.(The swap rate 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.B1=4e-0.25*0.1+4e-0.75*0.105+104e 1.25*0.11=98.24,B2=(100+5.1)e-0.25*0.1=102.51.V=98.24-102.51=-4.27(million)to A and 4.27million B.f.Portfolio of forwards:A swap(semiannual interest exchanges)can be viewed as a sequence of forwards with maturities:t1,t2,tn with a common forward price.Define Pt()as the time-t value of zero-coupon bond maturing at time for$1 face value.For titti+1,1.At ti+1:k-k*,evaluated at t,(k-k*)Pt(ti+1)2.At ti+2:k-0.5R(ti+1)Q,evaluated at t,PVt,t(i+2)k-0.5R(ti+1)Q=k-0.5R(ti+1,ti+2)Qexp-r(t,ti+2)(ti+2-t),where R(ti+1,ti+2)is the forward rate(semiannual compounding)at time t over ti+1,ti+2.Why?3.Similarly for ti+3,ti+4,4.The total value of the swap at time t:(k-k*)exp(-r(t,ti+1)(ti+1-t)+nj=i+1k-0.5R(tj,tj+1)Qexp-r(t,tj+1)(tj+1-t)-Example:Continue the previous exampleR(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=-4.27Variation 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.-Rate-capped swaps:floating rate is capped.-Putable and Callable swaps:one or both counter-parties have the right to cancel the swap at certain times without additional costs.-Forward swaps:the swap rate is set but the swap does not commence until a later date.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: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 swap to better both parties.USDDEMAB7.5%6.9%4.2%4.0%a.Interest payment flows 6.9%USD Company Company|4.2%DEM A B 6.9%USD 3.9%DEM b.Initial principal flow 84m DEM Company Company|84DEM A B 50m USD 50m USDb.Terminal principal flow 84m DEM Company Company|84DEM A B 50m USD 50m USDb.Company A:Borrows DEM debt and enters the above swap.c.Company B:Borrows USD debt and enters the above swap.d.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*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.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.-Swap through an intermediary 7.4%$6.9%$Company Swap Company|4.2%DM A Dealer B 9%USD 4.2%DM 3.9%DM-The results1.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.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.Swap dealer:On a semiannual basis,earns USD(7.4%-6.9%)*25m and loss DEM(4.2%-3.9%)*$2m.-Pricing currency swapsa.Set the two fixed rates of a swap so that the swap has a zero value at the time of initiation.b.Suppose that payment dates are t1,t2,tn.The value of a swap at time t,Vt,based on the domestic currency:Vt=StBFt-BDtc.St:exchange rate(domestic price of one unit foreign currency)at time t.d.BDt:value of domestic fixed-rate bond underlying the swap when titti+1,BDt=nj=i+1kDe-rd(t,tj)(tj-t)+QDe-rd(t,tn)(tn-t),where kD is the payment in the domestic currency,QD is the principal amount in the domestic currency.e.BFt:value of foreign fixed-rate bond underlying the swap(measured in the foreign currency)when titti+1,BFt=nj=i+1kFe-rf(t,tj)(tj-t)+QFe-rf(t,tn)(tn-t),where kF is the payment in the foreign currency,QF is the principal amount in the foreign currency.f.Determining the fixed rate at time 0Set kD and kF such that QD=nj=1kDe-rd(0,tj)tj+QDe-rd(0,tn)tnQF=nj=i+1kFe-rf(0,tj)tj+QFe-rf(0,tn)tnThis implies V0=S0BF0-BD0=S0QF-QD=0That is,set two appropriate coupon rates so that both bonds are priced at par.g.Example:Counter-party A in a three-year swap pays 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 swap has 1.25 years to maturity.(The swap rate 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.h.BD=3.6e-0.25*0.08+3.6e-0.75*0.08+103.6e-1.25*0.08=100.66mi.BF=1.68*2.1e-0.25*0.04+2.1e-0.75*0.04+102.1e-1.25*0.04=170.08mj.To A:V=1708/1.52-100.66=USD11.23m and to B:V=-USD11.23mk.Portfolio of forwards:A currency swap can be viewed as a sequence of forwards with maturities:t1,t2,tn with a common forward price.For titti+1,1.At ti+1:St(i+1)kF-kD,evaluated at t,it has a value equal to Ft(ti+1)kF-kDexp-rDt(ti+1)(ti+1-t)2.At ti+2:St(i+2)kF-kD,evaluated at t,it has a value equal to Ft(ti+2)kF-kDexp-rDt(ti+2)(ti+2-t)3.Similarly for ti+3,ti+4,ti+n-14.At ti+n:St(i+n)(kF+QF)-(kD+QD),evaluated at t,it has a value equal to Ft(ti+n)(kF+QF)-(kD+QD)exp-rDt(ti+n)(ti+n-t)5.The total value of the swap at time t is the sum of all the terms.-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=USD11.2298mEquity 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.-Example:Notional principal$100m.Counter-party A receives 3-month LIBOR and pays S&P500 index return plus a swap spread of-0.1%.S&P500 return-0.1%Company Company A B LIBORDate Days LIBOR%SP500 SP500 return LIBOR Payment S&P Payment Net Payment Jan 2 9.00469.75 Apr 2909.15479.152.00%225,000190,106-34,894Jul 2919.35507.425.90231,292580,003348,711Oct 2928.65491.70-3.10238,944-319,803-558,747Jan 292 499.101.50221,056140,498-80,558-The value of equity swapa.The value of this equity swap was 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?b.The value of this equity swap on,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%.1.What is the value of swap to 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(IA2-IJ2)/IJ2-0.1%*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.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.Credit Default SwapsWill be discussed in the section of credit risk.Procter&GambleBankersTrustLeveragedSwap1The storyOn November 2,1993,P&G and BT entered a five year,semiannual settlement,$200 million notional principal interest rate swap contract 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 swap would have been 5.3%versus the CP daily average rate flat.The swap was 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.2The P&G-BT leveraged swapTerm:5 yearFrequency:Semiannual paymentsFixed 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.3The spreadThe 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 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=-0.1703The spread on May 4,1994 wasMax 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 swap if the formula had not been amended prior to May 1994.4The amendmentThe swap was 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.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.5Was 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 swap with a notional principal of$200 million is extraordinary,a real testament to the power of leverage.The spread formula can be simplified to Spread=max(0,0.170415*5-year-CMT%-0.01*30-year-TSY price)Spread=0.170415*CMT%-0.01*TSY Price.Using 6.15%as the six-month forward yield in Nov.1993 on the T-bond on August 1,2023,we haveTSY Price=-13.6442*YLD%Spread=0.170415*CMT%+0.136442*YLD%.A parallel shift in term structure(yield curve)will cause the spread to move by a factor of 0.306857.One basis point increase raises the spread by 30.6857 basis points,a very big leverage.A move of only 58 basis points from November 2,1993 curve would be able to cover the 75 basis point cushion.Each additional basis point move drives the market value of the leverage swap down by about$2.389 million.In addition,P&G is facing both the CP rate and the spread.Both components are positively related to the interest rte level.P&G get hit twice if the interest rate moves together.This swap is,therefore,poorly structured.6Is the swap a fair deal?In its“5/30”leveraged swaps with BT,P&G stood to gain if five-year and thirty year Treasury yields remained steady between Nov.1993 and May 1994.P&Gs upside gains were limited to the 75 basis points per period(later amended to 88 basis points).Its downside risk was unlimited.This payoff pattern is consistent with an interpretation that the leveraged swap consists of a plain vanilla receive fixed interest rate swap and written options on Treasury securities.P&Gs strategy was to use the premium on the written option to reduce it cost of borrowed funds.Using information on put options on T-bond futures one can see the deal was not fair to P&G.For the amount of risk borne on transaction.P&G received no more than one-third the value of the written options via the swap than if the puts had been written on the exchange directly.7Aggressive corporate financeP&Gs。