# Manual for SOA Exam FM/CAS Exam 2.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 2009 Edition, available at 1/78

2 Swaps Definition 1 A swap is a contract between two counterparts to exchange two similar financial quantities which behave differently. The two things exchanged are called the legs of the swap. A common type of swap involves a commodity. Another common type of swap is an interest rate swap of a fixed interest rate in return for receiving an adjustable rate. Usually, one leg involves quantities that are known in advance, known as the fixed leg. The other involves quantities that are (uncertain) not known in advance, known as the floating leg. Usually, a swap entails the exchange of payments over time. 2/78

3 LIBOR. The (London Interbank office rate) LIBOR is the most widely used reference rate for short term interest rates world wide. The LIBOR is published daily the (British Bankers Association) BBA. It is based on rates that large international banks in London offer each other for inter bank deposits. Rates are quoted for 1 month, 3 month, 6 month and 12 month deposits. 3/78

4 The following table shows the LIBOR interest rates for a loan in dollars during a week on June, 2007: Date 1 month 3 month 6 month 9 month 12 month % % % % % % % % % % % % % % % % % % % % % % % % % The previous rates are from 4/78

5 A LIBOR loan is an adjustable loan on which the interest rate is tied to a specified Libor. The interest rate is the most recent value of the LIBOR plus a margin, subject to any adjustment cap. LIBOR is used in determining the price of interest rate futures, swaps and Eurodollars. The most important financial derivatives related to LIBOR are Eurodollar futures. Traded at the Chicago Mercantile Exchange (CME), Eurodollars are US dollars deposited at banks outside the United States, primarily in Europe. The interest rate paid on Eurodollars is largely determined by LIBOR. Eurodollar futures provide a way of betting on or hedging against future interest rate changes. 5/78

6 Structure of interest rates. Let P(0, t) be the price of a \$1 face value zero coupon bond 1 maturing on date t. Notice that P(0,t) is the interest factor from time zero to time t, i.e. \$1 invested at time 0 accumulates to 1 P(0,t) at time t. P(0, t) is the discount factor from time zero to time t. The implied interest factor from time t j 1 to time t j is P(0,t j 1 ) P(0,t j ). The implied forward rate from time t j 1 to time t j is r 0 (t j 1, t j ) = P(0,t j 1) P(0,t j ) 1. Let s n be the n year spot rate. Then, (1 + s n ) n = 1 P(0,n). 6/78

7 A quantity which appear later is the coupon rate R for a n year bond with annual coupons and face value, redemption value and price all equal to one. The price of this bond is 1 = n RP(0, j) + P(0, n). j=1 Hence, R = 1 P(0, n) n j=1 P(0, j). 7/78

8 Example 1 The following table lists prices of zero coupon \$1 face value bonds with their respective maturities: Number of years to maturity Price 1 \$ \$ \$ \$ (i) Calculate the 1 year, 2 year, 3 year, and 4 year spot rates of interest. (ii) Calculate the 1 year, 2 year, and 3 year forward rates of interest. (iii) Calculate the coupon rate R for a j year bond with annual coupons whose face value, redemption value and price are all one. 8/78

9 Solution: (i) Note that the price of j th bond is P(0, j) = (1 + s j ) j 1. Hence, s j = 1. In particular, P(0,j) 1/j 1 s 1 = P(0, 1) 1 = = % 1 s 2 = P(0, 2) 1/2 1 = 1 1 = % /2 1 s 3 = P(0, 3) 1/3 1 = 1 1 = % /3 1 s 4 = P(0, 4) 1/4 1 = 1 1 = % /4 9/78

10 Solution: (ii) To get the j 1 year forward rate f j, we do f j = (1+s j ) j 1 = P(0,j 1) (1+s j 1 ) j 1 P(0,j) 1. We get that: f 2 = = %, f 3 = = %, f 4 = = %. 10/78

11 Solution: (iii) We have that R j = P 1 P(0,j) j Hence, we get that: k=1 P(0,k). R 1 = = %, R 2 = = %, R 3 = = %, R 4 = = %. 11/78

12 Example 2 Suppose the current LIBOR discount factors P(0, t j ) are given by the table below. LIBOR discout rates P(0, t j ) time in months Calculate the annual nominal interest rate compounded quarterly for a loan for the following maturity dates: 3, 6, 9 and 12 months. 12/78

13 Solution: Let s (4) j be the annual nominal interest rate compounded quarterly for a loan maturing in 3j months, j = 1, 2, 3, 4. Then, (1 + s (2) j /4) j 1 = P(0,j/4). So, ( ) s (4) 1 1 = = %, ( ( ) s (4) 1 1/2 2 = 4 1) = %, ( ( ) s (4) 1 1/3 3 = 4 1) = %, s (4) 4 = ) 1/4 1) ( ( = %. 13/78

14 We denote by r t0 (t 1, t 2 ) to the nonannualized interest rate for the period from t 1 to t 2 using the interest rates at t 0, i.e. 1 + r t0 (t 1, t 2 ) is the interest factor for the period from t 1 to t 2 using the interest rates at t 0. We denote by P t0 (t 0, t 1 ) to the price at time t 0 of a zero coupon bond with face value \$1 and redemption time t 1. So, 1 + r t0 (t 1, t 2 ) = P t 0 (t 0, t 1 ) P t0 (t 0, t 2 ). Notice that we abbreviate P 0 (0, t) = P(0, t). 14/78

15 Forward rate agreement. Suppose that a borrower plans to take a loan of \$L at time t 1, where t 1 > 0. He will repaid the loan at time t 2, where t 2 > t 1. The amount of the loan payment depends on the term structure of interest rates at time t 1. Let r t1 (t 1, t 2 ) be interest rate from t 1 to t 2 with respect to the structure of interest rates at time t 1, i.e. a zero coupon bond with face value F and redemption time t 2 costs F 1+r t1 (t 1,t 2 ) at time t 1. To pay the loan, the borrower needs to pay \$L(1 + r t1 (t 1, t 2 )) at time t 2. 15/78

16 Since r t1 (t 1, t 2 ) is unknown at time zero, the borrower does not know how much it will have to pay for the loan. In order to hedge against increasing interest rates, the borrower can enter into a (FRA) forward rate agreement. A FRA is a financial contract to exchange interest payments for a notional principal on settlement date for a specified period from start date to maturity date. Usually one of the interest payments is relative to a benchmark such as the LIBOR. This is a floating interest rate, which was described in Subsection 2. The other interest payment is with respect to a fixed rate of interest. An FRA contract is settled in cash. The settlement can be made either at the beginning or at the end of the considered period, i.e. either at the borrowing time or at the time of repayment of the loan. 16/78

17 The two payments involved in an FRA are called legs. Both payments are made at time t 2. Usually FRA s are floating against fixed. One leg consists of an interest payment with respect to a floating rate. The interest payment of the floating rate leg is Lr t1 (t 1, t 2 ). The side making the floating rate leg payment is called either the floating rate leg party, or the floating rate side, or the floating rate payer. The interest payment of the fixed rate leg is Lr FRA, where r FRA is an interest rate specified in the contract. The side making the fixed rate leg payment is called either the fixed rate leg party, or the fixed rate side, or the fixed rate payer. 17/78

18 If the FRA is settled at the time of the repayment of the loan, we say that the FRA is settled in arrears. Suppose that the FRA is settled at time t 2 (in arrears). The FRA is settled in two different ways: 1. If L(r FRA r t1 (t 1, t 2 )) > 0, the fixed rate side makes a payment of L(r FRA r t1 (t 1, t 2 )) to the floating rate side. 2. If L(r FRA r t1 (t 1, t 2 )) < 0, the floating rate side makes a payment of L(r t1 (t 1, t 2 ) r FRA ) to the fixed rate side. 18/78

19 Usually an FRA is mentioned as an exchange of (interest payments) legs. By interchanging their legs, it is meant that: 1. The floating rate leg party makes a payment of Lr t1 (t 1, t 2 ) to its counterpart. 2. The fixed rate leg party makes a payment of Lr FRA to its counterpart. The combination of these two payments is: the fixed rate leg party makes a payment of L(r FRA r t1 (t 1, t 2 )) to the floating rate leg party. This means that if L(r FRA r t1 (t 1, t 2 )) is a negative number, the floating rate side makes a payment of L(r t1 (t 1, t 2 ) r FRA ) to the fixed rate side. 19/78

20 Example 3 Company A pays \$75,000 in interest payments at the end of one year. Company B pays the then current LIBOR plus 50 basis points on a \$1,000,000 loan at the end of the year. Suppose that the two companies enter into an interest payment swap. Suppose that in one year the current LIBOR rate is 6.45%. Find which company is making a payment at the end of year and its amount. 20/78

21 Example 3 Company A pays \$75,000 in interest payments at the end of one year. Company B pays the then current LIBOR plus 50 basis points on a \$1,000,000 loan at the end of the year. Suppose that the two companies enter into an interest payment swap. Suppose that in one year the current LIBOR rate is 6.45%. Find which company is making a payment at the end of year and its amount. Solution: Company B s interest payment is ( )( ) = To settle the forward interest agreement, company A must make a payment of = 5500 to Company B. 21/78

22 The fixed rate side payment is L(r FRA r t1 (t 1, t 2 )). Assuming that the fixed rate side borrows L at time t 1 his total interest payment at time t 2 is Lr t1 (t 1, t 2 ) + L(r FRA r t1 (t 1, t 2 )) = Lr FRA. A borrower can enter into an FRA as a fixed rate side to hedge against increasing interest rates. If the FRA is settled at time t 1 (at borrowing time), to settle the FRA the floating rate side makes a payment of L(rt 1 (t 1,t 2 ) r FRA ) 1+r t1 (t 1,t 2 ) to the fixed rate side. This number could be negative. If r FRA > r t1 (t 1, t 2 ), the fixed rate side makes a (positive) payment of L(r FRA r t1 (t 1,t 2 )) 1+r t1 (t 1,t 2 ) to the floating rate side. 22/78

23 Since the interest factor from time t 1 to time t 2 is 1 + r t1 (t 1, t 2 ), the previous payoffs are equivalent to the ones for an FRA paid in arrears. In this case, the fixed rate side can apply the FRA payment to the principal he borrows. He takes a loan of L + L(r FRA r t1 (t 1, t 2 )) 1 + r t1 (t 1, t 2 ) = L(1 + r FRA) 1 + r t1 (t 1, t 2 ) at time t 1. The principal of the loan at time t 2 is (1 + r t1 (t 1, t 2 )) L(1 + r FRA) 1 + r t1 (t 1, t 2 ) = L(1 + r FRA). Again, it is like the fixed rate side is able to borrow at the rate r FRA. 23/78

24 Usually the floating rate side is a market maker. The FRA agreement transfers interest rate risk from the fixed rate side to the floating rate side. In order to hedge this interest rate risk, the market maker could create a synthetic reverse FRA. 24/78

25 Suppose that the FRA is settled in arrears. The scalper buys a zero coupon bond maturing at t 1 with face value L and short sells a zero coupon bond with face value LP(0,t 1) P(0,t 2 ) maturing at t 2. The scalper cashflow at time zero is LP(0, t 1 ) LP(0, t 1) P(0, t 2 ) P(0, t 2) = 0. At time t 1, the scalper gets L from the first bond, which invests at the current interest rate. He gets L(1 + r t1 (t 1, t 2 )) at time t 2 from this investment. His total cashflow at time t 2 is L(1 + r t1 (t 1, t 2 )) + L(r FRA r t1 (t 1, t 2 )) LP(0, t 1) P(0, t 2 ) =L(1 + r t1 (t 1, t 2 )) + L(r FRA r t1 (t 1, t 2 )) L(1 + r 0 (t 1, t 2 )) =L(r FRA r 0 (t 1, t 2 )). Hence, the no arbitrage rate of an FRA is r 0 (t 1, t 2 ), which is the current nonannualized interest rate from t 1 to t 2. 25/78

26 Example 4 Suppose that the current spot rates are given in the following table (as annual nominal rates convertible semiannually) spot rate 6% 7.5% maturity (in months) 6 12 Timothy and David enter into separate forward rate agreements as fixed rate sides for the period of time between 6 months and 12 months. Both FRA s are for a notional amount \$ Timothy s FRA is settled in 12 months. David s FRA is settled in 6 months. In six months, the annual nominal interest rate compounded semiannually for a six month loan is 7%. (i) Find the no arbitrage six month rate for an FRA for the period of time between 6 months and 12 months. (ii) Calculate Timothy s payoff from his FRA. (iii) Calculate David s payoff from his FRA. 26/78

27 Solution: (i) The no arbitrage six month rate for a FRA for the period of time between 6 months and 12 months is r FRA = ( ) = %. 2 (ii) Timothy s payoff is (iii) David s payoff is (10000)( ) = ( ) (10000) ) = /78

28 Interest rate swaps. An interest rate swap is a contract in which one party exchanges a stream of interest payments for another party s stream. Interest rate swaps are normally fixed against floating. Interest rate swaps are valued using a notional amount. This nominal amount can change with time. We only consider constant nominal amounts. The fixed stream of payments are computed with respect to a rate determined by the contract. The floating stream of payments are determined using a benchmark, such as the LIBOR. 28/78

29 Suppose that a firm is interested in borrowing a large amount of money for a long time. One way to borrow is to issue bonds. Unless its credit rating is good enough, the firm may have trouble finding buyers. Lenders are unwilling to absorb long term loans from a firm with a so and so credit rating. So, the firm may have to borrow short term. Even if a company does not need to borrow short term, usually short term interest rates are lower than long term interest rates. As longer the maturity as more likely the default. Anyhow, suppose that a firm is interested in borrowing short term, but needs the cash long term. The firm takes a short term loan. At maturity, the firm pays this loan and takes another short term loan. This process will be repeated as many times as needed. Current short term rates are known. But, the short term interest rates which the firm may need to take in the future are uncertain. The firm has an interest rate risk. If short term rates increase, the company may get busted. To hedge this risk, the firm may enter into an interest rate swap, which we describe next. 29/78

30 Suppose that a borrower takes a loan of L paying a floating interest rate according a benchmark such as the LIBOR. Suppose that the interest is paid at times t 1 < t 2 < < t n. The principal owed after each payment is L. This means that at time t j, the borrower pays Lr tj 1 (t j 1, t j ) in interest, where 1 + r tj 1 (t j 1, t j ) is the interest factor from time t j 1 to time t j, calculated using the LIBOR at time t j 1. This rate is the rate which the borrower would pay, if he borrows at time t j 1, pays this loan at time t j and takes a new loan for L at time t j. The borrower is paying a stream of floating interest rate payments. The borrower can hedge by taking several FRA s. If the borrower would like to have a single contract, he enters into a interest rate swap. 30/78

31 The borrower would like to enter into an interest rate swap so that the current interest payments plus the payments to the swap will add to a fixed payment. This situation is similar to that of having several FRA s. The borrower can enter an interest rate swap with notional amount L. The borrower would like to have a fixed rate leg in its contract: Payment LR LR LR Time t 1 t 2 t n where R is the swap interest rate in the contract. The borrower would like that its counterpart has a floating rate leg: Payment Lr 0 (0, t 1 ) Lr t1 (t 1, t 2 ) Lr tn 1 (t n 1, t n ) Time t 1 t 2 t n 31/78

32 An interest rate swap consists of an interchange of interest payments. The total outcome of this interchange is that at every time t j the floating leg side makes a payment of L(r tj 1 (t j 1, t j ) R) to the fixed leg side. Again L(r tj 1 (t j 1, t j ) R) could be negative. If L(r tj 1 (t j 1, t j ) R) < 0, the fixed leg side makes a payment at each time t j of L(R r tj 1 (t j 1, t j )) to the floating leg side. 32/78

33 By exchanging legs, the borrower makes a payment at each time t j when L(R r tj 1 (t j 1, t j )) to his counterpart. The borrower is also making interest payments of Lr tj 1 (t j 1, t j ). The total borrower s interest payments add to LR. By entering a swap, a borrower is hedging against increasing interest rates. The total borrower s cashflow is that of a company issuing bonds. Often the borrower has poor credit rating and it is unable to issue bonds. In some sense, some borrowers enter into an interest rate swap so that its counterpart issues a bond to them. Sometimes the borrower uses a interest rate swap to avoid to issue a fixed rate long term loan. Takers of this loan could require a higher interest to borrow. 33/78

34 Usually, the borrower s counterpart is a market maker, which must hedge its interest rate risk. The (market maker) fixed rate payer gets a payment of L(r tj 1 (t j 1, t j ) R) at each time t j. The fixed rate payer profit by entering the swap is n P(0, t j )L(r tj 1 (t j 1, t j ) R). j=1 34/78

35 By using bonds, a market maker can create a synthetic cashflow of payments equal to the swap payments. The cost of these bonds is its present value according with the current term structure of interest rates. Hence, if there is no arbitrage, a market maker can arrange so that the cost of payments he receives is n LP(0, t j )(r 0 (t j 1, t j ) R), j=1 i.e. instead of using the uncertain rates r tj 1 (t j 1, t j ), the scalper can use the current forward rates. Therefore, the no arbitrage swap rate is n j=1 R = P(0, t j)r 0 (t j 1, t j ) n j=1 P(0, t. j) The swap rate R when there is no arbitrage is called the par swap rate. Notice that the par swap rate R is a weighted average of implied forward rates r 0 (t j 1, t j ). The weights depend on the present value of a payment made at time t j. 35/78

36 Using that P(0,t j 1) P(0,t j ) = 1 + r 0 (t j 1, t j ), we get that ( ) n j=1 P(0, t P(0,tj 1 ) j) P(0,t j ) 1 R = n j=1 P(0, t j) n j=1 = P(0, t j 1) n i=1 P(0, t j) n j=1 P(0, t = 1 P(0, t n) n j) j=1 P(0, t j). Notice that R is the coupon rate for a bond with price, face value and redemption all equal, using the current term structure of interest rates. It is like that the floating rate party enters the swap to use the market maker credit rating to issue a bond. 36/78

37 Example 5 Suppose the LIBOR discount factors P(0, t j ) are given in the table below. Consider a 3 year swap with semiannual payments whose floating payments are found using the LIBOR rate compiled a semester before the payment is made. The notional amount of the swap is LIBOR discount rates P(0, t j ) time (months) (i) Calculate the par swap rate. (ii) Calculate net payment made by the fixed rate side in 18 months if the six month LIBOR interest rate compiled in 12 months is 2.3%. 37/78

38 Solution: (i) The par swap rate is R = 1 P(0,tn) P n j=1 P(0,t j ) = = = %. (ii) The payment made by the fixed rate side is L(R r tj 1 (t j 1, t j )) = 10000( ) = /78

39 Notice that the floating interest payment use the interest rates compiled one period before the payment. These interest rates are called realized interest rates. Since interest rates change daily, we may be interested in the market value of a swap contract. One of the parties in the swap contract may sell/buy his position in the contract. The market value of a swap contract for the fixed rate payer is the present value of the no arbitrage estimation of the payments which he will receive. The market value of a swap contract for the fixed rate payer immediately after the k the payment is n P(t k, t j )L(r tk (t j 1, t j ) R). j=k+1 If this value is positive, the fixed rate payer has exposure to interest rates. Current interest rates are higher than when the swap was issued. The market value of a swap is the no arbitrage price to enter this contract. One of the counterparts in the contract may be interested in selling/buying his position on the contract. 39/78

40 Example 6 Suppose current LIBOR discount factors P(0, t j ) are given by the table below. An interest rate swap has 6 payments left. The swap rate is 3.5% per period. The notional principal is two million dollars. The floating payments of this swap are the realized LIBOR interest rates. LIBOR discount rates P(0, t j ) time (months) Calculate market value of this swap for the fixed rate payer. 40/78

41 Solution: The market value of the swap per dollar is n P(0, t j )(r tk (t j 1, t j ) R) j=k+1 ( ) ( ) = ( ) ( ) ( ) ( ) = ( ) ( ) ( ) ( ) ( ) ( ) = The market value of the swap is ( )( ) = /78

42 A deferred swap is a swap which begins in k periods. The swap par rate is computed as n j=k R = P(0, t ( ) n j)r 0 (t j 1, t j ) j=k P(0, t P(0,tj 1 ) j) P(0,t j ) 1 n j=k P(0, t = n j) j=k P(0, t j) = n j=k P(0, t j 1) n j=k P(0, t j) n j=k P(0, t j) = P(0, t k 1) P(0, t n ) n j=k P(0, t. j) 42/78

43 Example 7 Suppose the current annual nominal interest rates compounded quarterly from the LIBOR are given by the table below. Two counterparts enter into a fixed against floating swap using the LIBOR rate compiled a quarter before the payment is made. The notional principal is \$ The times of the swap are in 12, 15 and 18 months. i (4) 4.5% 4.55% 4.55% 4.6% 4.6% 4.65% maturation time in months (i) Calculate the par swap rate. (ii) Calculate the payment made by the fixed rate party in 18 months if in 15 months the spot annual nominal interest rate compounded quarterly is 4.65%. 43/78

44 Solution: (i) The par swap rate is R = P(0, t k 1) P(0, t n ) n j=k P(0, t j) ( /4) 3 ( /4) 6 = ( /4) 4 + ( /4) 5 + ( /4) 6 = = %. (ii) The payment made by the fixed rate party is L(R r tj 1 (t j 1, t j )) = (50000)( /4) = /78

45 The par swap rate R = 1 P(0,tn) P n j=1 P(0,t j ) is a weighted average of implied forward rates. If the current interest rate does depend on the maturing time, then P(0, t) = (1 + i) t, for some constant i > 0. In this case, R = 1 P(0, t n) 1 (1 + i) tn n j=1 P(0, t = n j) j=1 (1 +. i) t j If the periods in the swap have the same length, then t j = jh, 1 j n, for some h > 0, and = R = 1 (1 + i) nh n j=1 (1 + i) nj = 1 = (1+i) h 1 (1+i) h 1 (1 + i) nh (1+i) h (1+i) (n+1)h 1 (1+i) h 1 (1 + i) h (1 + i) h = (1 + i) h 1. (1 + i) h 1 is the effective rate for a period of length h. Notice that the assumption P(0, t) = (1 + i) t, for some constat i > 0, almost never happens. 45/78

46 Example 8 Suppose the current annual nominal interest rates compounded quarterly from the LIBOR is 5.4% independently of the maturity of the loan. Two counterparts enter into a fixed against floating swap using realized LIBOR rates. The times of the swap are in 3, 6 and 12 months. Calculate the par swap rate. 46/78

47 Example 8 Suppose the current annual nominal interest rates compounded quarterly from the LIBOR is 5.4% independently of the maturity of the loan. Two counterparts enter into a fixed against floating swap using realized LIBOR rates. The times of the swap are in 3, 6 and 12 months. Calculate the par swap rate. Solution: The par swap rate is R = 1 P(0, t n) n j=1 P(0, t j) 1 ( /4) 4 = ( /4) 1 + ( /4) 2 + ( /4) 4 = = %. 47/78

48 Notice that in the previous problem the swap periods do not have the same length. Even if annual interest rate is constant over time, the interest payments vary over time. If the annual interest rate remains constant over time, the floating rate payments are L ((1 + i) t 1 1) L ((1 + i) t 2 t 1 1) L ((1 + i) tn t n 1 1) t 1 t 2 t n Suppose that we take a loan of L at time zero. We make payments of LR at t j, for 1 j n. The par swap rate R is the constant periodic rate such that the final outstanding in this loan is L. Notice that the present value of the payments is LR n j=1 P(0, t j). If the final principal is L = L LR P n j=1 P(0,t j ) P(0,t n), then, R = P 1 P(0,tn) n j=1 P(0,t j ). 48/78

49 Commodity swaps. A commodity swap is a swap where one of the legs is a commodity and the other one is cash. Hence, there are two counterparts in a swap: a party with a commodity leg and another party with a cash leg. Since the spot price of a commodity changes over time, the commodity leg is floating. Certain amount of a commodity is delivered at certain times. In some sense is like to combine several forward contracts. But, swaps are valuated considering the total deliveries. Hence, changes in interest rates change the value of a swap. Usually, the swap payment for this commodity is constant. The commodity leg party is called the short swap side. The cash leg party is called the long swap side. 49/78

50 Suppose that a party would like to sell a commodity at times 0 < t 1 < t 2 < < t n. Suppose that the nominal amounts of the commodity are Q 1, Q 2,, Q n, respectively. The commodity leg is Amount of delivered commodity Q 1 Q 2 Q n Time t 1 t 2 t n Usually the cash leg is either or Payments C Time 0 t 1 t 2 t n Payments C 1 C 2 C n Time t 1 t 2 t n 50/78

51 If two parties enter into a commodity swap, one will be designed the party with the commodity leg and the other the party with the cash leg. The party with the commodity leg will deliver commodity to its counterpart according with the table above. The party with the cash leg will pay cash payments to its counterpart. From a practical point of view, the party with the commodity leg is a seller. The party with the cash leg is a buyer. A commodity swap contract needs to specify the type and quality of the commodity, how to settle the contract, etc. A swap can be settled either by physical settlement or by cash settlement. If a swap is settled physically, the commodity leg side delivers the stipulated notional amount to the cash leg side, and the cash leg side pays to the commodity leg side. If a swap is cash settled, one of the parties will make a payment to the other party. 51/78

52 Suppose that the current forward price of this commodity with delivery in T years is F 0,T. Let P(0, T ) be the price of a zero coupon with face value \$1 and expiration time T. Then, the present value of the commodity delivered is n P(0, t j )Q j F 0,tj. j=1 In a prepaid swap the buyer makes a unique payment at time zero. If there exists no arbitrage, the price of a prepaid swap is n j=1 P(0, t j)q j F 0,tj. 52/78

53 Usually the cash leg consists of series of payments made at the times when the commodity is delivered. Usually each swap payment per unit of commodity is a fixed amount. Hence, the cashflow of payments is Payment Q 1 R Q 2 R Q n R Time t 1 t 2 t 2 where R is the swap price per unit of commodity. The present value of the cashflow of payments is n j=1 P(0, t j)q j R. The no arbitrage price of a swap per unit of commodity is R = n j=1 P(0, t j)q j F 0,tj n j=1 P(0, t j)q j. R is a weighted average of forward prices. 53/78

54 Suppose that at time of delivery, the buyer pays a level payment of R at each of the delivery times. Then, the present value of the cashflow of payments is n j=1 P(0, t j)r. The no arbitrage level payment is n j=1 R = P(0, t j)q j F 0,tj n j=1 P(0, t. j) A commodity swap allows to lock the price of a sale. It can be used by a producer of a commodity to hedge by fixing the price that he will get in the future for this commodity. It also can be used by a manufacturer to hedge by fixing the price that he will pay in the future for a commodity. A commodity swap can be used instead of several futures/forwards. Since the price of a swap involves all the deliveries, a commodity swap involves loaning/lending somehow. 54/78

55 Example 9 Suppose that an airline company must buy 10,000 barrels of oil every six months, for 2 years, starting 6 months from now. The company enters into a long oil swap contract to buy this oil. The cash leg swap consists of four level payments made at the delivery times. The following table shows the annual nominal rate convertible semiannually of zero coupon bonds maturing in 6, 12, 18, 24 months and the forward price of oil with delivery at those times. F 0,T \$50 \$55 \$55 \$60 annual nominal rate 4.5% 5% 5% 5.5% expiration in months /78

56 (i) Find the no arbitrage price of a prepaid swap with the commodity leg which the airline needs. 56/78

57 (i) Find the no arbitrage price of a prepaid swap with the commodity leg which the airline needs. Solution: (i) The present value of the cost of oil is n Q j F 0,tj P(0, t j ) j= =(10000) (10000)( ) (10000)( ) (10000)( ) = = /78

58 (ii) Suppose that the airline company pays the swap by a level payment of R at each of the delivery times. Calculate R. 58/78

59 (ii) Suppose that the airline company pays the swap by a level payment of R at each of the delivery times. Calculate R. Solution: (ii) We have that n j=1 P(0, t j) = ( ) 2 ( ) 3 ( ) 4 = = and R = n j=1 Q jf 0,tj P(0, t j ) n j=1 P(0, t j) = = /78

60 (iii) Suppose that the airline company pays the swap by unique price per barrel. Calculate the price per barrel. 60/78

61 (iii) Suppose that the airline company pays the swap by unique price per barrel. Calculate the price per barrel. Solution: (iii) The price of the swap per barrel is = /78

62 Example 10 Suppose that an airline company must buy 10,000 barrels of oil in two months, 12,000 barrels of oil in four months and 15,000 barrels of oil in six months. The company enters into a long oil swap. The payment of the swap will be made at the delivery times. The following table shows the annual nominal rate convertible monthly of zero coupon bonds maturing in 6, 12, 18, 24 months and the forward price of oil with delivery at those times. Barrels of oil \$10000 \$12000 \$15000 F 0,T \$55 \$56 \$58 annual nominal rate 4.5% 4.55% 4.65% expiration in months /78

63 (i) Suppose that the cash leg swap consists of a level payment of R at each of the delivery times. Calculate R. 63/78

64 Solution: (i) We have that n Q j F 0,tj P(0, t j ) j= =(10000) ( ) (12000)( ) (15000)( = = , ) 6 n j=1 P(0, t j) = + + ( ) 2 ( ) 4 ( ) 6 = = and R = n j=1 Q jf 0,tj P(0, t j ) n j=1 P(0, t j) = = /78

65 (ii) Suppose that the cash leg swap consists of a unique payment per barrel made at each of the delivery times. Calculate the no arbitrage swap price per barrel. 65/78

66 (ii) Suppose that the cash leg swap consists of a unique payment per barrel made at each of the delivery times. Calculate the no arbitrage swap price per barrel. Solution: (ii) We have that n j=1 P(0, t j)q j = ( ) 2 ( ) 4 ( ) 6 = = and the swap price per barrel is R = n j=1 P(0, t j)q j F 0,tj n j=1 P(0, t = j)q j = /78

67 If a swap is cashed settled, then the commodity is valued at the current spot price. If the current value of the commodity is bigger than the value of the cash payment, then the (cash leg side) long swap pays this difference to the (commodity leg side) short swap. Reciprocally, if the current value of the commodity is smaller than the value of the cash payment, the (commodity leg side) short swap side pays this difference to the (cash leg side) long swap. 67/78

68 Suppose that the swap involves the sale of a commodity at times t 1 < t 2 < < t n. with notional amounts of Q 1, Q 2,, Q n, respectively. Suppose the swap payment is a fixed amount per unit of commodity. We saw that this amount is n j=1 R = P(0, t j)q j F 0,tj n j=1 P(0, t. j)q j When a swap is cash settled, the value of the commodity is found using the current spot price S tj. At time t j the long swap pays Q j R Q j S tj to its counterpart. Notice that Q j R Q j S tj could be a negative real number. If Q j R Q j S tj < 0, the short swap pays Q j S tj Q j R to its counterpart. 68/78

69 Changes in the forward contracts and interest rates alter the value of the swap. The market value of a swap is found using the present values of its legs using the current structure of interest rates. The market value of a long swap immediately after the settlement at time t k is n P(t k, t j )Q j (F tk,t j R). j=k+1 This is the price which an investor would pay to enter the swap as a long swap side. Immediately after the swap is undertaken the market value of the contract is n Q j ( P(0, t j ) F 0,tj P(0, t j )F 0,tj ), j=1 where P(0, t j ) and F 0,tj are the new market values. 69/78

70 Example 11 Suppose that an airline company must buy 1,000 barrels of oil every six months, for 3 years, starting 6 months from now. Instead of buying six separate long forward contracts, the company enters into a long swap contract. According with this swap the company will pay a level payment per barrel at delivery. The following table shows the annual nominal rate convertible semiannually of zero coupon bonds maturing in 6, 12, 18, 24 months and the forward price of oil at those times. F 0,T \$55 \$57 \$57 \$60 \$62 \$64 annual nominal rate 5.5% 5.6% 5.65% 5.7% 5.7% 5.75% expiration in months /78

71 (i) Find the price per barrel of oil using the swap. 71/78

72 (i) Find the price per barrel of oil using the swap. Solution: (i) We have that n j=1 P(0, t j )Q j F 0,tj =(1000) (1000)( (1000) ) 4 + (1000) ( ( = ) 2 + (1000)( ) ) 5 + (1000) ( = , ) 6 72/78

73 (i) Find the price per barrel of oil using the swap. Solution: (i) n P(0, t j )Q j j=1 = ( ) ) 4 + ) ( ( ( ( = ) = ) 6 We have that n j=1 R = P(0, t j)q j F 0,tj n j=1 P(0, t = j)q j = /78

74 (ii) Suppose the swap is settled in cash. Assume that the spot rate for oil in 18 months is \$57. Calculate the payment which the airline receives. 74/78

75 (ii) Suppose the swap is settled in cash. Assume that the spot rate for oil in 18 months is \$57. Calculate the payment which the airline receives. Solution: (ii) The airline gets a payment of Q j S tj Q j R = (1000)(57) (1000)( ) = /78

76 (iii) Suppose that immediately after the swap is signed up, the future prices of oil are given by the following table F 0,T \$55 \$58 \$59 \$61 \$62 \$63 expiration in months Calculate the value of the swap for the cash leg party. 76/78

77 Solution: (iii) The market value of the swap for the long party is n P(0, t j )(Q j Ft0,t j Q j R) j=1 = ( ) + ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) ( ) 2 = = /78

78 Suppose that immediately after the forward is signed, every future price increases by K. Then, the market value of the swap is n P(0, t j )(Q j (F 0,tj + K) Q j R) = j=1 n P(0, t j )Q j K. The swap counterpart is a scalper which hedges the commodity risk resulting from the swap. The scalper has also interest rate exposure. The scalper needs to hedge changes in the price of the commodity and in interest rates. j=1 78/78

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivative markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall

### Interest Rate and Currency Swaps

Interest Rate and Currency Swaps Eiteman et al., Chapter 14 Winter 2004 Bond Basics Consider the following: Zero-Coupon Zero-Coupon One-Year Implied Maturity Bond Yield Bond Price Forward Rate t r 0 (0,t)

### Hedging with Futures and Options: Supplementary Material. Global Financial Management

Hedging with Futures and Options: Supplementary Material Global Financial Management Fuqua School of Business Duke University 1 Hedging Stock Market Risk: S&P500 Futures Contract A futures contract on

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 5. Bonds. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 2009 Edition,

### Advanced forms of currency swaps

Advanced forms of currency swaps Basis swaps Basis swaps involve swapping one floating index rate for another. Banks may need to use basis swaps to arrange a currency swap for the customers. Example A

### Eurodollar Futures, and Forwards

5 Eurodollar Futures, and Forwards In this chapter we will learn about Eurodollar Deposits Eurodollar Futures Contracts, Hedging strategies using ED Futures, Forward Rate Agreements, Pricing FRAs. Hedging

### VALUATION OF PLAIN VANILLA INTEREST RATES SWAPS

Graduate School of Business Administration University of Virginia VALUATION OF PLAIN VANILLA INTEREST RATES SWAPS Interest-rate swaps have grown tremendously over the last 10 years. With this development,

### CFA Level -2 Derivatives - I

CFA Level -2 Derivatives - I EduPristine www.edupristine.com Agenda Forwards Markets and Contracts Future Markets and Contracts Option Markets and Contracts 1 Forwards Markets and Contracts 2 Pricing and

### Introduction to swaps

Introduction to swaps Steven C. Mann M.J. Neeley School of Business Texas Christian University incorporating ideas from Teaching interest rate and currency swaps" by Keith C. Brown (Texas-Austin) and Donald

### We first solve for the present value of the cost per two barrels: (1.065) 2 = 41.033 (1.07) 3 = 55.341. x = 20.9519

Chapter 8 Swaps Question 8.1. We first solve for the present value of the cost per two barrels: \$22 1.06 + \$23 (1.065) 2 = 41.033. We then obtain the swap price per barrel by solving: which was to be shown.

### FIN 472 Fixed-Income Securities Forward Rates

FIN 472 Fixed-Income Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU Interest-Rate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward

### 2 Stock Price. Figure S1.1 Profit from long position in Problem 1.13

Problem 1.11. A cattle farmer expects to have 12, pounds of live cattle to sell in three months. The livecattle futures contract on the Chicago Mercantile Exchange is for the delivery of 4, pounds of cattle.

### 550.444 Introduction to Financial Derivatives

550.444 Introduction to Financial Derivatives Week of October 7, 2013 Interest Rate Futures Where we are Last week: Forward & Futures Prices/Value (Chapter 5, OFOD) This week: Interest Rate Futures (Chapter

### A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall

### FIXED-INCOME SECURITIES. Chapter 10. Swaps

FIXED-INCOME SECURITIES Chapter 10 Swaps Outline Terminology Convention Quotation Uses of Swaps Pricing of Swaps Non Plain Vanilla Swaps Terminology Definition Agreement between two parties They exchange

### INTEREST RATE SWAP (IRS)

INTEREST RATE SWAP (IRS) 1. Interest Rate Swap (IRS)... 4 1.1 Terminology... 4 1.2 Application... 11 1.3 EONIA Swap... 19 1.4 Pricing and Mark to Market Revaluation of IRS... 22 2. Cross Currency Swap...

### FIXED-INCOME SECURITIES. Chapter 11. Forwards and Futures

FIXED-INCOME SECURITIES Chapter 11 Forwards and Futures Outline Futures and Forwards Types of Contracts Trading Mechanics Trading Strategies Futures Pricing Uses of Futures Futures and Forwards Forward

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 6. Variable interest rates and portfolio insurance. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam

### Forwards and Futures

Prof. Alex Shapiro Lecture Notes 16 Forwards and Futures I. Readings and Suggested Practice Problems II. Forward Contracts III. Futures Contracts IV. Forward-Spot Parity V. Stock Index Forward-Spot Parity

### 1. Buy Low, Sell High. 2. Be diversified. First note that diversification works

Hedging interest rate risk using financial futures Through DGAP and FGAP, a bank can assess its interest rate risk. Now we discuss how to reduce this exposure by using financial futures. First I present

DERIVATIVES Presented by Sade Odunaiya Partner, Risk Management Alliance Consulting DERIVATIVES Introduction Forward Rate Agreements FRA Swaps Futures Options Summary INTRODUCTION Financial Market Participants

### SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS This page indicates changes made to Study Note FM-09-05. April 28, 2014: Question and solutions 61 were added. January 14, 2014:

### Chapter 10 Forwards and Futures

Chapter 10 Forwards and Futures Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives.

### Understanding Cross Currency Swaps. A Guide for Microfinance Practitioners

Understanding Cross Currency Swaps A Guide for Microfinance Practitioners Cross Currency Swaps Use: A Currency Swap is the best way to fully hedge a loan transaction as the terms can be structured to exactly

### Learning Curve Forward Rate Agreements Anuk Teasdale

Learning Curve Forward Rate Agreements Anuk Teasdale YieldCurve.com 2004 Page 1 In this article we review the forward rate agreement. Money market derivatives are priced on the basis of the forward rate,

### Fina4500 Spring 2015 Extra Practice Problems Instructions

Extra Practice Problems Instructions: The problems are similar to the ones on your previous problem sets. All interest rates and rates of inflation given in the problems are annualized (i.e., stated as

### Finance 350: Problem Set 6 Alternative Solutions

Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas

### Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

### 2015 Exam 2 Syllabus Financial Mathematics Exam

2015 Exam 2 Syllabus Financial Mathematics Exam The syllabus for this exam is defined in the form of learning objectives that set forth, usually in broad terms, what the candidate should be able to do

### INTEREST RATE SWAPS September 1999

INTEREST RATE SWAPS September 1999 INTEREST RATE SWAPS Definition: Transfer of interest rate streams without transferring underlying debt. 2 FIXED FOR FLOATING SWAP Some Definitions Notational Principal:

### Fixed-Income Securities. Assignment

FIN 472 Professor Robert B.H. Hauswald Fixed-Income Securities Kogod School of Business, AU Assignment Please be reminded that you are expected to use contemporary computer software to solve the following

### Floating rate Payments 6m Libor. Fixed rate payments 1 300000 337500-37500 2 300000 337500-37500 3 300000 337500-37500 4 300000 325000-25000

Introduction: Interest rate swaps are used to hedge interest rate risks as well as to take on interest rate risks. If a treasurer is of the view that interest rates will be falling in the future, he may

### Fixed Income Portfolio Management. Interest rate sensitivity, duration, and convexity

Fixed Income ortfolio Management Interest rate sensitivity, duration, and convexity assive bond portfolio management Active bond portfolio management Interest rate swaps 1 Interest rate sensitivity, duration,

### Chapter 1 - Introduction

Chapter 1 - Introduction Derivative securities Futures contracts Forward contracts Futures and forward markets Comparison of futures and forward contracts Options contracts Options markets Comparison of

### Forward Contracts and Forward Rates

Forward Contracts and Forward Rates Outline and Readings Outline Forward Contracts Forward Prices Forward Rates Information in Forward Rates Reading Veronesi, Chapters 5 and 7 Tuckman, Chapters 2 and 16

### Callable Bonds - Structure

1.1 Callable bonds A callable bond is a fixed rate bond where the issuer has the right but not the obligation to repay the face value of the security at a pre-agreed value prior to the final original maturity

### Assumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk

Derivatives Why? Allow easier methods to short sell a stock without a broker lending it. Facilitates hedging easily Allows the ability to take long/short position on less available commodities (Rice, Cotton,

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support

### FIN 472 Fixed-Income Securities Forward Rates

FIN 472 Fixed-Income Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU Interest-Rate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward

### Eris Interest Rate Swap Futures: Flex Contract Specifications

Eris Interest Rate Swap Futures: Flex Contract Specifications Trading Hours Contract Structure Contract Size Trading Conventions Swap Futures Leg Conventions Effective Date Cash Flow Alignment Date ( CFAD

### Hot Topics in Financial Markets Lecture 1: The Libor Scandal

Hot Topics in Financial Markets Lecture 1: The Libor Scandal Spot and Forward Interest Rates Libor Libor-Dependent Financial Instruments The Scandal 2 Spot Interest Rates Bond Market The yield on a bond

### CALCULATOR TUTORIAL. Because most students that use Understanding Healthcare Financial Management will be conducting time

CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses

### Treasury Bond Futures

Treasury Bond Futures Concepts and Buzzwords Basic Futures Contract Futures vs. Forward Delivery Options Reading Veronesi, Chapters 6 and 11 Tuckman, Chapter 14 Underlying asset, marking-to-market, convergence

### Fundamentals of Finance

Euribor rates, forward rates and swap rates University of Oulu - Department of Finance Fall 2015 What next Euribor rates, forward rates and swap rates In the following we consider Euribor spot rate, Euribor

### ACI THE FINANCIAL MARKETS ASSOCIATION

ACI THE FINANCIAL MARKETS ASSOCIATION EXAMINATION FORMULAE 2009 VERSION page number INTEREST RATE..2 MONEY MARKET..... 3 FORWARD-FORWARDS & FORWARD RATE AGREEMENTS..4 FIXED INCOME.....5 FOREIGN EXCHANGE

Equity-index-linked swaps Equivalent to portfolios of forward contracts calling for the exchange of cash flows based on two different investment rates: a variable debt rate (e.g. 3-month LIBOR) and the

### Introduction to Derivative Instruments Part 1 Link n Learn

Introduction to Derivative Instruments Part 1 Link n Learn June 2014 Webinar Participants Elaine Canty Manager Financial Advisory Deloitte & Touche Ireland ecanty@deloitte.ie +353 1 417 2991 Christopher

### Learning Curve Interest Rate Futures Contracts Moorad Choudhry

Learning Curve Interest Rate Futures Contracts Moorad Choudhry YieldCurve.com 2004 Page 1 The market in short-term interest rate derivatives is a large and liquid one, and the instruments involved are

### Practice set #4 and solutions

FIN-465 Derivatives (3 credits) Professor Michel Robe Practice set #4 and solutions To help students with the material, seven practice sets with solutions will be handed out. They will not be graded: the

### NOTES ON THE BANK OF ENGLAND UK YIELD CURVES

NOTES ON THE BANK OF ENGLAND UK YIELD CURVES The Macro-Financial Analysis Division of the Bank of England estimates yield curves for the United Kingdom on a daily basis. They are of three kinds. One set

### Lecture 12. Options Strategies

Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same

### Financial Mathematics Exam

2014 Exam 2 Syllabus Financial Mathematics Exam The purpose of the syllabus for this examination is to develop knowledge of the fundamental concepts of financial mathematics and how those concepts are

### Lecture 09: Multi-period Model Fixed Income, Futures, Swaps

Lecture 09: Multi-period Model Fixed Income, Futures, Swaps Prof. Markus K. Brunnermeier Slide 09-1 Overview 1. Bond basics 2. Duration 3. Term structure of the real interest rate 4. Forwards and futures

### Fin 3312 Sample Exam 1 Questions

Fin 3312 Sample Exam 1 Questions Here are some representative type questions. This review is intended to give you an idea of the types of questions that may appear on the exam, and how the questions might

### CHAPTER 9 SUGGESTED ANSWERS TO CHAPTER 9 QUESTIONS

INSTRUCTOR S MANUAL MULTINATIONAL FINANCIAL MANAGEMENT, 9 TH ED. CHAPTER 9 SUGGESTED ANSWERS TO CHAPTER 9 QUESTIONS 1. What is an interest rate swap? What is the difference between a basis swap and a coupon

### Solutions to Practice Questions (Bonds)

Fuqua Business School Duke University FIN 350 Global Financial Management Solutions to Practice Questions (Bonds). These practice questions are a suplement to the problem sets, and are intended for those

### Creating Forward-Starting Swaps with DSFs

INTEREST RATES Creating -Starting Swaps with s JULY 23, 2013 John W. Labuszewski Managing Director Research & Product Development 312-466-7469 jlab@cmegroup.com CME Group introduced its Deliverable Swap

### Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

### n(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3

ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric

### ANALYSIS OF FIXED INCOME SECURITIES

ANALYSIS OF FIXED INCOME SECURITIES Valuation of Fixed Income Securities Page 1 VALUATION Valuation is the process of determining the fair value of a financial asset. The fair value of an asset is its

### Pricing Forwards and Swaps

Chapter 7 Pricing Forwards and Swaps 7. Forwards Throughout this chapter, we will repeatedly use the following property of no-arbitrage: P 0 (αx T +βy T ) = αp 0 (x T )+βp 0 (y T ). Here, P 0 (w T ) is

### Determination of Forward and Futures Prices

Determination of Forward and Futures Prices Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012 Short selling A popular trading (arbitrage) strategy is the shortselling or

### SOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Financial Economics

SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Financial Economics June 2014 changes Questions 1-30 are from the prior version of this document. They have been edited to conform

### Learning Curve An introduction to the use of the Bloomberg system in swaps analysis Received: 1st July, 2002

Learning Curve An introduction to the use of the Bloomberg system in swaps analysis Received: 1st July, 2002 Aaron Nematnejad works in the fixed income analytics team at Bloomberg L.P. in London. He graduated

### MONEY MARKET FUTURES. FINANCE TRAINER International Money Market Futures / Page 1 of 22

MONEY MARKET FUTURES 1. Conventions and Contract Specifications... 3 2. Main Markets of Money Market Futures... 7 3. Exchange and Clearing House... 8 4. The Margin System... 9 5. Comparison: Money Market

### 19. Interest Rate Swaps

19. Interest Rate Swaps Reading: Stigum 19 on Swaps. See also Hull who builds from the idea (mentioned in Stigum) that swaps are like a portfolio of forward contracts. Daily Financial Times includes bid-ask

### YIELD CURVE GENERATION

1 YIELD CURVE GENERATION Dr Philip Symes Agenda 2 I. INTRODUCTION II. YIELD CURVES III. TYPES OF YIELD CURVES IV. USES OF YIELD CURVES V. YIELD TO MATURITY VI. BOND PRICING & VALUATION Introduction 3 A

### Obligatory transactions on a specified date at a predetermined price

Obligatory transactions on a specified date at a predetermined price DERIVATIVE MARKET Bond Derivatives Bond Futures www.jse.co.za Johannesburg Stock Exchange A bond future is a contractual obligation

### Resident Money Market and Investment Funds Return (MMIF) Worked examples - derivatives, securities borrowing/lending and overdrafts

Resident Money Market and Investment Funds Return (MMIF) Worked examples - derivatives, securities borrowing/lending and overdrafts Version 2 March 2014 Email: sbys@centralbank.ie Website: http://www.centralbank.ie/

### Introduction, Forwards and Futures

Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35

### International Bond and Money Markets. Quiz Questions. True-False Questions

Chapter 9 International Bond and Money Markets Quiz Questions True-False Questions 1. The abolition of the Interest Equalization Tax, Regulation M, the cold war, and the US and UK foreign exchange controls

### SYLLABUS The ACI Dealing Certificate (Prometric Code: 3I0-008)

SYLLABUS The ACI Dealing Certificate (Prometric Code: 3I0-008) Examination delivered in ENGLISH and GERMAN The ACI Dealing Certificate is a foundation programme that allows candidates to acquire a working

### Chapter 3: Commodity Forwards and Futures

Chapter 3: Commodity Forwards and Futures In the previous chapter we study financial forward and futures contracts and we concluded that are all alike. Each commodity forward, however, has some unique

1.0 ALTERNATIVE SOURCES OF FINANCE Module 1: Corporate Finance and the Role of Venture Capital Financing Alternative Sources of Finance TABLE OF CONTENTS 1.1 Short-Term Debt (Short-Term Loans, Line of

### MONEY MARKET SUBCOMMITEE(MMS) FLOATING RATE NOTE PRICING SPECIFICATION

MONEY MARKET SUBCOMMITEE(MMS) FLOATING RATE NOTE PRICING SPECIFICATION This document outlines the use of the margin discounting methodology to price vanilla money market floating rate notes as endorsed

### Chapter 15 OPTIONS ON MONEY MARKET FUTURES

Page 218 The information in this chapter was last updated in 1993. Since the money market evolves very rapidly, recent developments may have superseded some of the content of this chapter. Chapter 15 OPTIONS

### ECO 4368 Instructor: Saltuk Ozerturk. Bonds and Their Valuation

ECO 4368 Instructor: Saltuk Ozerturk Bonds and Their Valuation A bond is a long term contract under which a borrower (the issuer) agrees to make payments of interest and principal on speci c dates, to

### Exotic options [April 4]

Exotic options [April 4] We ve looked at European and American Calls and Puts European options with general payoff functions From a theoretical point of view. This could be carried further, to more Exotic

### International Swaps and Derivatives Association, Inc. Disclosure Annex for Interest Rate Transactions

International Swaps and Derivatives Association, Inc. Disclosure Annex for Interest Rate Transactions This Annex supplements and should be read in conjunction with the General Disclosure Statement. NOTHING

### Analytical Research Series

EUROPEAN FIXED INCOME RESEARCH Analytical Research Series INTRODUCTION TO ASSET SWAPS Dominic O Kane January 2000 Lehman Brothers International (Europe) Pub Code 403 Summary An asset swap is a synthetic

### Bond Price Arithmetic

1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus \$250 1329.78 = \$332,445. The closing futures price for the March contract was 1364.00,

### Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics

Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock

### Percent, Sales Tax, & Discounts

Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that of implies multiplication. Suppose that the local sales tax rate is 7.5% and you purchase

Review for Exam 1 Instructions: Please read carefully The exam will have 21 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation

### Managing Interest Rate Exposure

Managing Interest Rate Exposure Global Markets Contents Products to manage Interest Rate Exposure...1 Interest Rate Swap Product Overview...2 Interest Rate Cap Product Overview...8 Interest Rate Collar

### Measurement Concepts for Banking, Trading, and Investing

Banking,, Banking,, for the MFM Orientation September 1, 2010 Banking,, If you are going to work with bankers, traders, or investment managers, it is important for you to understand the language and concepts

### David Bob Case Scenario

David Bob Case Scenario David Bob, CFA, is a derivatives analyst at Capital Inc. Capital Inc. deals mainly in arbitrage positions along with leveraged positions. David is following the options prices and

### Introduction to Eris Exchange Interest Rate Swap Futures

Introduction to Eris Exchange Interest Rate Swap Futures Overview Eris Exchange interest rate swap futures ( Eris contracts ) have been designed to replicate the net cash flows associated with plain-vanilla,

### Interest Rate Swap. Product Disclosure Statement

Interest Rate Swap Product Disclosure Statement A Product Disclosure Statement is an informative document. The purpose of a Product Disclosure Statement is to provide you with enough information to allow

### Chapter 16: Financial Risk Management

Chapter 16: Financial Risk Management Introduction Overview of Financial Risk Management in Treasury Interest Rate Risk Foreign Exchange (FX) Risk Commodity Price Risk Managing Financial Risk The Benefits

### Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates

Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Cash Flow Financial transactions and investment opportunities are described by cash flows they generate. Cash flow: payment

### Chapter 2 An Introduction to Forwards and Options

Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram

### The Basics of Interest Theory

Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

### SwapEx Contract Specifications. USD LIBOR Interest Rate Swaps: Fixedto-Floating

SwapEx Contract Specifications USD LIBOR Interest Rate Swaps: Fixedto-Floating There are two types of USD LIBOR Interest Rate Swaps; Fixed-to-Floating contracts available for trading on SwapEx: Fixed Rate

### Glossary of Common Derivatives Terms

DRAFT: 10/03/07 Glossary of Common Derivatives Terms American Depository Receipts (ADRs). ADRs are receipts issued by a U.S. bank or trust company evidencing its ownership of underlying foreign securities.