Financial Mathematics 2009

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1 MATH3090 MATH7039 Financial Mathematics 2009 Lecturer: Graeme Chandler Room , Consultation: Wed & Fri /MATH3090 Part 1.1

2 Part 1. The Money Market. Weeks 1-3. L Mon :00, (50-S201), Tues :00. (42-212) Tut Tues : (inc Week 1) Quiz Mon 23rd March 6.00pm, Notes and tutorials handed out in class, and on the web. Part 2. QTC Simulation Game. Weeks 4-8 L. Tues 4.00 (42-212) Tues Computer Labs. Teams of 4-6. Weekly performance & reports. Final presentations th April. See a previous game book on the web. 2/MATH3090 Part 1.1

3 Part 3. Shares and options. Weeks 9-13 L Mon :00, (50-S201), Tues :00. (42-212) Tut Tues : (inc Week 1) Assignment due 2nd June. Notes and tutorials handed out in class and on the web. 3/MATH3090 Part 1.1

4 Money Market Governments, banks, and big companies raise money for short and long term needs by issuing (i.e. selling) bonds and commercial paper both in Australia and overseas. Investors (i.e buyers) include superannuation funds, other companies, banks, retail investors, or investment funds. Queensland Treasury Corporation handles Qld state, local, and semi-government requirements. Currently it manages (i.e. has borrowed) $52B. (12/08 search for annual report ). First we learn how to price the most common financial instruments. 4/MATH3090 Part 1.1

5 Present Value I have $P now and invest at r% pa compounded quarterly. Each quarter I get interest i = r/4 % and my money increases by a factor of (1 + i) per quarter Time Q0 Q1 Q2 Q3 Q4 Interest P i P(1 + i)i P(1 + i) 2 i P(1 + i) 3 i Total P P(1 + i) P(1 + i) 2 P(1 + i) 3 P(1 + i) 4 After k quarters I have P(1 + i) k. Having $M to be paid k quarters in the future is equivalent to having $M/(1 + i) k now. 5/MATH3090 Part 1.1

6 I.e the present value (PV) of $M to be paid k quarters in the future is M/(1 + i) k (using interest rate r% pa compounded quarterly i.e. i = r/4%). Example: Suppose we compound quarterly at an interest rate of r = 8% pa. What is the future value of $100,000 in 1 year s time. What is the present value of a payment of $500,000 to be paid in 1 year s time. Here i =.08/4 =.02, so $100,000 grows to (1 +.02) 4 = in 1 year. The present value of $500,000 to be paid in 1 years time is $500000/(1+.02) 4 = $461, /MATH3090 Part 1.1

7 Commercial Paper (CP) Is a contract where the issuer receives the proceeds $P (aka price) now, and promises to repay the face value (FV) at maturity. More precisely Price = Proceeds = P = FV 1 + y ( d /365 ) where y = yield (% per annum), d = days to maturity. Example: A bank issues CP with FV=$10M with 92 days to maturity at a yield of 5.9%? What is the price, i.e. how much money do we get now? How much do we have to repay in 92 days time? 7/MATH3090 Part 1.1

8 Ans: Here y = 5.9% =.059, so P = (Ans: 9,853,466.80). Thus we get $9,853, now and repay $10M in 92 days time. CP usually has 3 or 6 months to maturity. The yield y is the interest rate we pay Note P is the present value of the repayment of FV at maturity using interest rate y. Equivalently we receive P and repay P+interest = P+P y d 365 = FV. 8/MATH3090 Part 1.1

9 Bonds A bond is a contract in which the issuer pays regular coupons, of c% pa, as well as returning the FV at maturity. Usually equal coupons of 1 2 c% are paid every 6 months with maturity of 3 to 20 years. In general c τ % is paid at times τ, 2τ,..., nτ; where nτ is the time to maturity. (Eg. τ = 1, 1 2, 1 4 ) Example: Munchkin Bank issuess a 5 year bond with FV =$100,000 and coupons of 8% pa paid every 6 months. (I.e MB pays $4,000 every 6 months and $100,000+$4,000 in 5 years to the buyer or investor).. The buyer and seller must agree on the yield. (Intuitively yield is the interest rate the buyer expects for an investment of this maturity with MB). This determines the price the buyer or investor will pay now 9/MATH3090 Part 1.1

10 for the promised repayments; or equivalently the proceeds Munchkin receives. Example: Suppose MB agrees with an investor to a yield of y = 6.0% p.a., what is the price of the bond, i.e. what price should an investor pay now The price or proceeds of the bond is the present value of the future repayments under this contract, calculated using the agreed yield. To calculate the price, the interest rate for period τ is i = yτ : The coupon FV c τ paid at time kτ has present value c τ FV/ (1 + yτ) k. The repayment of FV at time nτ has present value FV/ (1 + yτ) n. 10/MATH3090 Part 1.1

11 The total price of the bond is P = PV C + PV FV = = c τ FV (1 + yτ) = c τ FV (1 + yτ) Check =? c FV y n 1 n k=1 c τ FV (1 + yτ) k + FV (1 + yτ) n 1 k=0 (1 + yτ) k + FV (1 + yτ) n ( ) 1 1 (1+yτ) n FV (1 + yτ) n (1+yτ) ( ) 1 1 (1 + yτ) n + FV (1 + yτ) n ) = FV FV y c y ( 1 1 (1 + yτ) n [ Sum n terms GP r = 1/ (1 + yτ) ] 11/MATH3090 Part 1.1

12 Ans: P = Here τ =.5, c = 8% and yτ =.06/2 =.03; so ( Ans: $ 34, $ 74, =$ 108, ) As c > y this is an attractive contract so P > FV. Traders haggle over the yield y, as there are many different bonds with different coupons from different issuers. As yield goes up the price goes down. The coupon is usually different from the yield. par bond P = FV c = y premium bond P > FV c > y discount bond P < FV c < y 12/MATH3090 Part 1.1

13 If a 5 year bond is traded after one year, i.e. after exactly 2 coupon payments, then it may be priced as a 4 year bond. But the formula must be adjusted if the bond is traded between coupon payments. Suppose the yield is y and there are d τ days between the previous and the next coupon payment. Let P = P 0 days gone be the price calculated with the above formula when the full d τ days remain to the next coupon. When d days have gone and so d τ d remain to the coupon payment, the price is P d days gone = (1 + τy) d dτ P0 days gone The reason is the present value increases by a factor of (1 + τy) over the full period of d τ days. Compounded daily, this is (1 + τy) 1/d τ per day for each of the d τ days or (1 + τy) d/d τ for d days. 13/MATH3090 Part 1.1

14 Example (ctd): What is the price of the MB bond if it is traded after 40 days have gone out of the full coupon period of d τ = 183 days. The yield is unchanged at y = 6%. Ans: P 40 days gone = (Ans:$109,233.68). 14/MATH3090 Part 1.1

15 In Oz, bonds are traditionally issued by Federal, state (eg QTC), and semi government authorities. Now more recently by companies & foreign govts? Investors include fund managers, insurance companies, and banks. The total turnover in the Oz Bond market (Gov & Non Gov) is $1.3 Tr (06/08 AFMA Annual Report). Oz GDP A$1.07 Tr (2008 Wikipedia.). Oz share market capitalization A$1.2 Tr (July 08) Turnover on ASXA$1.6 Tr) 15/MATH3090 Part 1.1

16 Credit Foncier Loans In Credit Foncier Loans, the borrower receives the principal and pays back the loan in equal repayments. The PV of the repayments equals the principal. Example: What are the repayments on a credit foncier loan for $200,000 at 12.5%pa for 4 years with equal repayments every 6 months.? Ans: The PV of the repayments should equal the principal of $200,000. If each repayment is $p every τ =.5 years, the PV of the k th repayment after kτ years is p/( ) k. So we want Principal = PV all repayments. 16/MATH3090 Part 1.1

17 I.e. 200, 000 = = = = 8 k=1 p ( ) k p ( ( ) 7 ) 1 p 1 ( ) ( ) ( ) p ( ) 8 [ Sum GP 8 terms ratio = 1 ( ) ] $p = (Ans: = 32,526.59) 17/MATH3090 Part 1.1

18 Back of Envelope Check: Interest ( 1 2 Principal) time interest = 1 2 Principal = Principal.25. Thus repayment (Principal + Interest) /8 Principal , /MATH3090 Part 1.1

19 Bid Offer Spread Banks profit by quoting different yields for buying and selling. To remember which is which the golden rule is You always get screwed dealing with a bank. On the trial game sheet, the yield for 3 yr bonds is quoted as 5.08 or 5.12%. If I am issuing or selling a bond, I am borrowing. I must accept the higher yield of 5.12% (i.e. receive lower proceeds). If I am buying (back) bonds I am investing money and must accept the lower yield. An exception to the rule is the floating sided of swap payments. As this is a nominal calculation the mid point is used. 19/MATH3090 Part 1.1

20 Exercises 1. Use the rates from the trial game sheet. The date is 15/03/2001. denotes the answer may be found in the deal slip examples on the web. (a) You are issuing CP with face value of $250,000 that will mature in 3 mths time. What yield is applicable? What are the proceeds. From the game sheet, yields are 5.65/5.69%. (Ans: $ ) (b) You are issuing a $500,000 5 year bond with semi-annual coupons of 5.10% pa. From the rate sheet what yield do you use. From the bond pricing formula, what proceeds do you receive. The game sheet quotes these yields as 5.28/32 %, i.e. 5.28/5.32 %. (Ans: $ ). 20/MATH3090 Part 1.1

21 (c) The coupon in the previous bond is due on 15 Sept. What is the price to buy back the bond on 15th June if the yields are 5.44/5.48%. (Ans: $ ) 2. You have a Credit Foncier loan of $200,000 for 3 years at 6.05%pa with quarterly repayments. What are the quarterly repayments. (Ans: ) 21/MATH3090 Part 1.1

22 Quiz 1. You are issuing CP with maturity 91 days FV=$10M at a yield of 8%. You receive/pay (which) now about 1.) 9,804,448.26, 2.) 10M, 3.) 10,199, You are buying a 5 year bond with FV $1M, and with 8% coupons paid every 6 months. (a.) Which yield should you accept out of the quotes %, %, %, % Your colleague has negotiated a yield of 6%. (b.) About how much does your company pay/receive (which?) now? 1.085M, 1.00M,.915M. (c.) What is your next payment/receipt (which)? 40,000, 30,000, 200,000, 1,000, /MATH3090 Part 1.1

23 (d.) 90 days after buying the bond, yields have not changed. What is the price of the bond? 1.00M, 1.044M, 1.06M, 1.085M, 1.101M, 1.117M 23/MATH3090 Part 1.1

24 xx xx xx xx xx xx xx xx xx xx xx Bond Holder Original Bond. Euro Bond x x xxx x xx xx Pay fixed Me Receive fixed Pay Floating The Swap MUNCHKIN BANK xx xx xx xx xx xx xx xx xx xx xx Pay fixed US$ Me Fixed US$ Floating A$ The Cross Currency Swap MUNCHKIN BANK Swaps A swap is a contract to exchange interest commitments. 24/MATH3090 Part 1.1

25 Examples: (1.) You issued 10 year bonds at a yield of 8% but now think interest rates will fall. You could issue CP & buy back the bonds. Instead, arrange a swap with a bank. You will pay the bank the CP interest rates and the Bank will pay you the agreed fixed term coupons. I.e. receive fixed and pay floating. (2.) Small companies can only borrow short term, and would be forced to repeatedly refinance at fluctuating short term rates. Instead they enter a swap to recieve floating and pay fixed, effectively borrowing at fixed long term rates. The principal is not exchanged and so there is little default risk; so small companies can get swaps more easily. The market for swaps is now very active. According to the 2008 AFMA annual report the turnover for Oz swaps is A$ 6.1 T 25/MATH3090 Part 1.1

26 To price a swap to receive fixed and pay floating at an agreed swap yield of y%pa. The swap is equivalent to two contracts Fixed Side Buy a bond with FV F & coup c%pa, at yield y% pa At the beginning Pay the price of the Bond Recieve F. Every 6 mths you receive a coupon of F c/2 Recieve FV of bond, F. Floating side Issue CP with proceeds F every 3 mths. From then on Every 3 mths repay FV of CP and raise F from a new CP i.e. pay F (1 + y begin (d/365)) F = F y begin (d/365). At the end Repay F(1 + y begin (d/365)) 26/MATH3090 Part 1.1

27 Here d =days in the quarter and y begin =yield for 3 mth CP at the beginning of quarter. This will change unpredictably from quartern to quarter. Example: On July 15th you enter a 5 year swap in which you receive fixed and pay floating. The principal is $100, yield negotiated is 7.5%. The fixed side has coupon of 8%pa paid every 6 mths. 3 mth CP rates are now are 6%. In 3 mths on Oct 15th they are 5% and in a further 3 mths on Jan 15th they are Work out how much you receive or pay 1.) at the start on July 15th. 2.) in 3 months time on Oct 15th 3.) in 6 months time on Jan 15th. 27/MATH3090 Part 1.1

28 Ans: 1.) At the start of the swap we essentially buy the bond at a price of $102, and raise $100, by issuing CP (with FV of ( /365) = 101,512.33). I.e. we pay $ on 15th July. 2.) On Oct 15th we pay the FV of the CP (101,512.33) and recieve by issuing CP at a yield of 5% (with a FV of ( /365) = 101,260.27). i.e. we pay 1, ) On Jan 15th we pay the FV of our second round of CP, receive the coupon from the fixed side, and raise 100,000 by issuing CP at a yield of 4.75%. Thus we recieve 101, = /MATH3090 Part 1.1

29 Note the individual transactions are not actually carried out, we just pay or receive from our counterparty the small resultant cash flow. In the game you can issue a Bond, or issue CP and use a swap to pay fixed and receive floating. You can make profits by looking carefully at which is better! 29/MATH3090 Part 1.1

30 Exercises Use the rates from the trial game sheet. The date is 15/03/2001. denotes the answer may be found in the deal slip examples on the web. You have entering into a 3 year swap with nominal face value $50,000,000 in which you receive fixed and pay floating. From the game book the fixed side has coupons of 5.10% paid semi-annually starting 6 mths after the start of the swap. From the trial rate sheet, the 3yr swap rates are 5.08/ month CP rates are 5.65/5.69%. 1.) Fill in the deal slip correctly. ( ) 2.) What are your cash flows i. at the start of the contract, (Ans Pay $27, ). 30/MATH3090 Part 1.1

31 ii. in 3 months time. (Ans pay $714,575.34) iii. in 6 months time. (Ans Receive $544,041.10) (CP rates in 3 mths time are 5.78/5.82, and in 6 mths time 5.98/6.02). 31/MATH3090 Part 1.1

32 Quiz 1. You are entering a 3yr swap in which you receive fixed and pay floating. (a.) Do you want a higher yield or a lower yield? (b.) Which of the following should you accept? %, %, %, %. (c.) Having entered into the swap, you want CP yields increase/decrease over the next 3 years? 32/MATH3090 Part 1.1

33 Euro Bonds Euro bonds are bonds issued in foreign countries. All repayments to be made in the foreign currency. If the foreign currency appreciates against the AUD, the repayments will turn out to be very expensive. QTC borrows abroard to ensure liquidity, rather than chasing low interest rates. QTC, and you in the game, are not allowed to be exposed to FX fluctuations. If you issue a Euro bond, your foreign currency repayments must be hedged by a cross currency swap (CCS). The CCS pays all you FX obligations, and in return you make interest payments in AUD at floating CP rates. If you buy back the bond you must reverse the swap, i.e. pay fixed and receive floating. 33/MATH3090 Part 1.1

34 Suppose you issue a Euro bond with principal USD F US (or F Oz when converted to AUD at the mid spot rate.) you receive the bond proceeds (using high side of yield) converted to AUD at the mid spot rate, say P Oz. You must also arrange the CCS, which is made up of two transactions. you buy the fixed side of the swap with face value F US USD, at the low side of the CC swap yields and with proceeds converted to AUD at the mid spot rate, say P SW you raise F Oz by issuing CP in Oz. Your cash flow at the beginning is P Oz P SW + F Oz. Afterwards you only have to deal with the floating side of the swap, which operates like the floating side of a domestic swap. 34/MATH3090 Part 1.1

35 A Euro Bond Example Use the rates from the old game sheet. You are issuing a US$10,000,000 US bond, maturing in 10 years. The coupon is 6% pa paid semi-annually (info from the game book). Your institution s risk management policy is that you must enter into a cross currency swap (CCS) that covers all the $US repayments for the life of the bond. 1. What is the correct bond yield? (Hint: From the rate sheet your US 10yr bond yields are 6.03/07 %.). What are the proceeds in USD? What will these proceeds in AUD? (Hint: the spot rate is.5498/.5502 ). From the game book, the swap available to you is a cross currency swap in which the fixed side matches the maturity and coupon of the 35/MATH3090 Part 1.1

36 US bond. Receive Principal in USD Coupon 6.00% pa semi-annually in $USD Maturity 10 years Pay Principal in AUD Pay floating 3 mth Australian CPrates on principal Although you have issued a 10 year bond, effectively you will pay 3 mth CP rates. (What whould you do to pay long term rates?) 2. What are the principals on the fixed and floating side of the CCS? From the trial rate sheet the spot rate is.5498/.5502 ( ) 3. On the fixed side of the swap, what yield and exchange rate is 36/MATH3090 Part 1.1

37 used? (From the trial rate sheet, the spot rate is.5498/.5502 and the 10 year swap rate is 6.03/6.07). Hence what are the proceeds of the fixed side in AUD. ( ) 4. From the rate sheet what yield will be used on the floating side of the CCS? 5. What is your net cash flow now. ( ) What is your net cash flow in 3 mths time. ($259,845.58) You will have to keep paying interest at fluctuating 3 mth CP rates for 10 years i.e. you are borrowing short term. 37/MATH3090 Part 1.1

Math 441 Mathematics of Finance Fall Midterm October 24, 2006

Math 441 Mathematics of Finance Fall 2006 Name: Midterm October 24, 2006 Instructions: Show all your work for full credit, and box your answers when appropriate. There are 5 questions: the first 4 are