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 6, OFOD) Fall Break: October 14 th Class will meet on Tuesday the 15 th, instead Mid Term: (Oct 16, Wednesday) In Two Weeks: Swaps (Chapter 7, OFOD) HW will be returned at Section on Thurs-Friday 1.1 1.2 Assignment For This Week (October 7 th ) Read: Hull Chapter 6. Interest Rate Futures Problems (Due October 7 th ) Chapter 5: 2, 4, 6, 7, 12, 16, 17, 20; 24 Chapter 5 (7e): 2, 4, 6, 7, 12, 16, 17, 20; 24 Problems (Due October 15 th ) Chapter 6: 4, 6, 9, 11, 14, 21; 26, 27 Chapter 6 (7e): 4, 6, 9, 11, 14, 21; 23, 24 Assignment Next Week (October 15 th ) Problems (Due October 15 h ) Chapter 6: 4, 6, 9, 11, 14, 21; 26, 27 Chapter 6 (7e): 4, 6, 9, 11, 14, 21; 23, 24 Review (Tuesday, Oct 15 th ) and Mid-term (Oct 16 th ) 1.3 1.4 1

Assignment In Two Weeks (October 21 st ) Read: Hull Chapter 7. Swaps Problems (Due October 28) Chapter 7: 1, 3, 5,6, 9, 12, 18; 22, 23 Chapter 7 (7e): 1, 3, 5, 6, 9, 12, 18; 20, 21 Exams Final Exam: Dec 17 th ; 9:00am Noon (Mergen 111) Plan for This Week Review an item from previous The Forward Rate Agreement (FRA) Interest Rate Futures Pricing Interest Rate Instruments & Day Counts Eurodollar (ED) Futures Generating Forward and Spot Rates from ED Bond & Note Futures 1.5 1.6 Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a specific rate, R K, will apply to a principal, L, during a specified future time period Borrowing & Lending is usually at LIBOR R K : Rate of FRA R F : Forward LIBOR between T 1 and T 2 R M : LIBOR observed in the market at T 1 for period between T 1 & T 2 Normally the applicable rate at T 1 is R M ; so as a consequence of the FRA, there is a differential rate (of R K -R M ) for having the FRA as opposed to executing in the market this is the source by which the value of the FRA is measured at T 1 If R K >R M a lender will receive (at T 2 ) a cash flow of L( RK RM )( T 2 T1 ) in excess of the market rate; a borrower will pay this amount in excess of R M FRAs may be settled at T 1 rather than T 2 ; the settlement payoff at T 1 is L( RK RM )( T2 T1 ) 1 RM ( T2 T1 ) 1.7 Forward Rate Agreement An FRA can be thought of as an agreement where interest at a predetermined rate, R K, is exchanged for interest at the market rate, R M (though in fact, it is just a forward on the rate) An FRA can be valued at any time prior to T 1 by assuming that the forward interest rate, R F, is certain to be realized at T 1 Indeed, the forward interest rate can always be locked-in The present value of the FRA to a lender where a fixed rate, R K, will be received on principal L between times T 1 and T 2 : R2T2 L( RK RF )( T2 T1 ) e Today, a new FRA with R K = R F can be entered into without cost to either party 1.8 2

A B C Locking in the Forward Rate L t 0 T 1 T 2 L t 0 T 1 T 2 Lexp[-R 1 T 1 ] t 0 T 1 T 2 Lexp[-R 1 T 1 ] exp[r 2 T 2 ] = Lexp[R F (T 2 -T 1 )] Lexp[-R 1 T 1 ] exp[r 2 T 2 ] Borrowing in C and Depositing in B provides the lender with the funds L for the Forward loan in A at the rate R F in force at t 0 The value for this FRA at t 0 is zero when R K is established as R F 1.9 Day Count Conventions in the U.S. Number of Days between Dates Interest Earned in Reference Period Number of Days in Reference Period = Interest Earned between Two Dates Treasury Bonds: Corporate Bonds: 30/360 Money Market Instruments: Actual/Actual (in period) Actual/360 1.10 Day Count Conventions in the U.S. Day Count Conventions in the U.S. US Treasury Bond Interest (Actual/Actual) Coupon Rate is 8%; payment dates are 3/1 & 9/1 Interest Earned between 3/1 and 7/3 Reference Period is 3/1 to 9/1 = 184 days Actual Days = 124 Interest Earned = 4 x 124/184 = $2.6957 (on $100 face) US Corporate, Municipal and MBS (30/360) Coupon Rate is 8%; payment dates are 3/1 & 9/1 Interest Earned between 3/1 and 7/3 Reference Period is 3/1 to 9/1 = 180 days Actual Days = 122 ( = 4 x 30 + 2 ) US Money Market Instruments (Actual/360) Interest earned in 90 days is exactly ¼ of quoted rate; interest earned in a whole year of 365 days is 365/360 times the quoted rate Prices of money market instruments are sometimes quoted using a discount rate (the interest earned as a percentage of the final face value as opposed to a percentage of the initial price) Treasury Bills (assume 13-week/91-day T-Bill) Price Quoted as 8 means annualized rate of interest is 8% of face Interest of $2.0222 = ( $100 x.08 x 91/360) (on $100 final face) Corresponds to a True Rate of interest of 2.0222/(1-.020222) = 2.064; Indeed: Interest Earned = 4 x 122/180 = $2.7111 (on $100 face) 1.11 100 = x + 2.0222 => x = 97.9778; 97.9778 (1 + z) = 100 => z =.020639 1.12 3

Day Count Conventions in the U.S. US Money Market Instruments (Actual/360) Treasury Bills (quoted as rate, price is a discount) Price Quoted as 8 means annualized rate of interest is 8% of face Interest of $2.0222 = ( $100 x.08 x 91/360) (on $100 final face) Corresponds to a True Rate of interest of 2.0222/(1-.020222) = 2.064 100 = x + 2.0222 => x = 97.9778; 97.9778 (1 + z) = 100 => z =.020639 Relation between the cash price, P, and the quoted price, Y, is P = 100 Y x (n/360) If price quote on a 13-week T-Bill is 10, then P = 100 10x(91/360) = 100 2.527778 = 97.4722 The annualized cc return is: 91 365 100 2.527778 R 1 97.4722 e 97.4722 91 ln1 0.025933 R 365 365 R ln 1 0.025933 10.2692 91 1.13 Eurodollar Futures On 3-month (90-day) Eurodollar deposit rate This is equivalent to 3-month LIBOR Can lock-in rate on $1 million for a future 3-mo period Long contracts to invest (deposit & receive interest) Short contracts to borrow (pay interest) Chicago Mercantile Exchange (CME) Mar, Jun, Sep & Dec delivery for 10 years forward Delivery on third Wednesday of the delivery month A change of one bp or 0.01 in a Eurodollar futures quote corresponds to contract price change of $25 1.14 Eurodollar Futures Eurodollar Futures 1.15 1.16 4

Eurodollar Futures Eurodollar Futures 1.17 1.18 Eurodollar Futures Eurodollar Futures A Eurodollar futures contract is settled in cash When it expires (on third Wednesday of delivery month) the final settlement price (of the futures) is 100 minus the actual three month deposit rate If the price is 97.42, the 3-mo rate is 2.58% (annual) A long who contracts to deposit $1 million at 2.58% earns interest of: (.0258/4) x 1,000,000 = 6,450 dollars If the price were 97.41, then the 3-mo rate is 2.59% For $1 million deposited at 2.59% the depositor receives interest of $6,475 The depositor (long) receives $25 more on investment Losses $25 on the ED futures 1.19 Price decline of.01 represents a loss of $25 for the ED short on the borrowing (and a $25 gain for the long) Corresponds with rate increase of 1bp in rate which the short will suffer in the interest paid but which is gained on the contract since it declined.01 OTOH, the same decline of.01 represents a gain of $25 for the ED long on the loan (the same $25 loss for short) The rate increase of 1bp in rate would result in the borrower having to pay $25 more in interest to the lender over the term of his loan an amount offset by the $25 loss on the futures contract long position If the price goes up by.01, the long gains $25 Rate goes down on the investment at final settlement 1.20 5

Example (See Table, below) Suppose you buy (take a long position in) a contract on November 1 (to hedge plans to make a deposit on December 21 and presumably earn interest of 100 97.12 = 2.88% The contract expires on December 21 The daily settlement prices are as shown Date Quote Nov 1 97.12 Nov 2 97.23 Nov 3 96.98. Dec 21 97.42 1.21 Example continued If on Nov. 1 you know that you will have $1 million to invest for three months on Dec 21, the contract (through a long position) locks in a rate of: 100-97.12 = 2.88% When the contract settles on Nov. 2 at 97.23 (which would correspond to a lock-in of 2.77%): The margin account is credited with $275 on Nov. 2 Lets continue this analysis through the previous table to the final settlement 1.22 Example continued Formula for Contract Value In the example, for the Dec 21 st final settlement, you earn 100 97.42 = 2.58% on $1 million for three months when you make your deposit on Dec 21 From your deposit of $1 million at 2.58%, the market 3-mo LIBOR, you will earn $6,450 From your margin account, where you have accumulated P/L associated with daily contract pricemoves, you have made a net gain to the margin account of $750 by Dec 21 Alternatively, you could consider that on Nov. 1 you locked-in 2.88%, earning $7,200 (= 6,450 + 750, the return from the Dec 21 market rate + the ED lock ) 1.23 The CME defines the ($) value of one contract as Value = 10,000[100-0.25(100-Q)] Where Q is the quoted price of the ED contract (Q is like 98) Value = $1M interest earned over 3-months Note the contract value derived as a discount vs. the quoted price also being related to the actual 3-month Eurodollar interest rate ( 100 - Q ) / 100 Interest earned from the Eurodollar deposit (in 3-mo) = 1,000,000 x.25 x (100-Q)/100 1.24 6

Forward vs. Futures Eurodollar futures and the FRA are quite similar Both can lock-in a rate for a future 90-day period between T 1 & T 2 ED contract settles daily, with final settlement at T 1 FRA is not settled daily; final settlement (reflecting the realized rate between T 1 & T 2 ) is made at T 2 (though it can be made at T 1 using the PV of the differential that would be realized at T 2 ) Can ED be used to extended the LIBOR Zero Curve? Components of differences between ED and FRA (through a kind of separation theorem ) Daily Settlement Component: Assume both ED and FRA have their payoff at T 1, but maintain their daily settlement differences T 1 vs. T 2 Payoff Component: FRA difference when settlement is at T 1 vs. T 2. 1.25 Forward vs. Futures Daily Settlement Component Decreases value of forward rate for the FRA relative to ED Suppose you have the FRA on a deposit where there is a payoff on the differential R M -R F at T 1, where R F is a predetermined rate for the period T 1 to T 2 and R M is the realized market rate You have the option to switch to daily settlement When rates are high, the value of the payoff goes up, daily cash inflows When rates are low, the value goes down, daily cash outflows Attractive as more money goes into the margin account when rates are high The market sets a higher R F in the futures (compensated for this option) The long borrows at this rate, the short deposits; rate = 100 - price On the other hand, switching from daily settlement to an FRA that settles only at T 1 will correspond to an FRA with a lower R F ED will predict a higher rate for the forward rate, R F, than is real 1.26 Forward vs. Futures T 1 vs. T 2 Payoff Component Decreases value of forward rate for the FRA relative to ED Suppose the payoff on the deposit differential R M -R F is at T 2 rather than T 1 (the payoff at T 2 is the standard for vanilla FRA), where R F is a predetermined rate for the period T 1 to T 2 and R M is the realized market rate If R M is high, payoff is positive; as payoff is at T 2 rather than T 1, the cost is high since there is no opportunity for investment at the higher rate You would rather payoff at T 1 rather than T 2 ; therefore R F is lower (all other things being equal) Therefore, with ED s payoff at T 1 rather than T 2, ED will predict a higher rate for the forward rate, R F, than is real This is a smaller effect than daily settlement; as T 1 can be much longer into the future than the period from T 1 to T 2 1.27 Forward vs. Futures To resolve the difference between forward rates, F, and the futures rate (determining the forward rate from ED) so the Extended LIBOR Zero Curve, R, can be constructed Adjust the Futures Rate with a so-called convexity adjustment 1 2 Forward Rate, F = Futures Rate T1T 2 2 Where both rates are with continuous compounding and, T 1 : time to maturity of the futures contract T 2 : time to maturity of the rate underlying the futures contract : standard deviation of the change in the short-term interest rate over 1-year (use 1.2% or 0.012) Fi ( Ti 1 Ti ) RiTi Since Fi ( Ti1 Ti ) Ri1Ti1 RiTi, then Ri 1 T i1 1.28 7

Forward vs. Futures Convexity Adjustment when =0.012 (Table in Example 6.4, page 141) Maturity of Futures Convexity Adjustment (bps) 2 3.2 4 12.2 6 27.0 8 47.5 10 73.8 1.29 1.30 Extending the LIBOR Zero Curve SWAP ED Yield Curve Analysis October 01, 10 LIBOR deposit rates define the LIBOR zero curve out to one year Eurodollar futures can be used to determine forward rates and the forward rates can then be used to bootstrap the zero curve: From: Fi ( Ti 1 Ti ) Ri 1Ti 1 RiTi We have: Fi ( Ti 1 Ti ) RiTi Ri 1 Ti 1 Where the F i is the convexity-adjusted forward rate from the futures contact 1.32 1.33 8

SWAP ED Yield Curve Analysis October 01, 10 ED Synthetic Forward Rates October 01, 10 1.34 1.35 Treasury Bond Price Quotes in the U.S Cash price = Quoted price + Accrued Interest Full price = Flat price + Accrued Interest Dirty price = Clean price + Accrued Interest Treasury Bond Futures Traded on CME (CBOT) The Deliverable is any US Treasury bond with more than 15-years to maturity on the first delivery day of the delivery month Eclipsed by Note contract 6.5 to 10-years, the deliverable One contract is for $100,000 face value; quoted 100-16 = 100.50 One point change calls for a $1,000 change in the value of the contract 1.37 1.38 9

Treasury Bond Futures US T-Bond (USZ2) Basis Analysis October 05, 12 Cash price received by the short position = Bond Future Settlement Price CF + Accrued interest Example Settlement price of bond future = 90.00 Conversion factor for bond delivered = 1.3800 Accrued interest on bond = 3.00 Price received by short for bond is (1.3800 90.00)+3.00 = 127.20 per $100 of principal The conversion factor (CF) for a bond is approximately equal to the value of the bond assuming that the yield curve is flat at 6% with semiannual compounding 1.39 1.40 US T-Bond (USZ2) Basis Analysis October 05, 12 US T-Bond (USZ2) Basis Analysis October 05, 12 1.41 1.42 10

US T-Bond (USZ2) Basis Analysis October 05, 12 US T-Bond (USZ2) Basis Analysis October 05, 12 1.43 1.44 US T-Bond (USZ2) Conversion Factors Treasury Bond Futures Cheapest to Deliver into the Bond Futures Many bonds can be delivered; for each there is a conversion factor (the quoted price that the bond would have per dollar of principal on the first day of the delivery month on the assumption that the yield for all maturities is 6% - with lots of details) Consider a 10% coupon bond with 20-years and 2 months to maturity By convention, bond is assumed to have exactly 20-yrs to maturity for CF calculation (one of those details) Value of the bond is So the CF is 1.4623 40 i1 5 100 i (1.03) (1.03) 40 146.23 1.45 1.46 11

Treasury Bond Futures US T-Bond (USZ0) Basis Analysis October 05, 10 Cheapest to Deliver into the Bond Futures Because the short receives: (Settlement Price x CF ) + Accrued And the cost of the bond is: Quoted bond price + Accrued The cheapest-to-deliver bond is the one for which Quoted Price (Settlement Price x CF) is the lowest 1.47 1.48 US T-Bond (USZ0) Basis Analysis October 05, 10 US T-Bond (USZ2) Basis Analysis October 05, 12 1.49 1.50 12

CBOT T-Bonds & T-Notes Duration Matching Factors that affect the futures price: Delivery can be made any time during the delivery month Any of a range of eligible bonds can be delivered The wild card play Futures stop at 2pm central Cash until 4pm (central) Short has until 8pm to issue delivery notice If cash prices go down dramatically after 2pm Short can issue delivery notice and buy cheap This involves hedging against interest rate risk by matching the durations of assets and liabilities It provides protection against small parallel shifts in the zero curve Duration Based (Price Sensitivity) Hedge Ratio PDP F D P DP y P bonds to deliver into the 2pm futures price 1.51 1.53 F C : D F : P : D P : Remember: C F Contract price for interest rate futures Duration of asset underlying futures at futures expiration Value of portfolio being hedged Duration of portfolio at hedge maturity Example It is August: A fund manager has $10 million invested in a portfolio of government bonds with a duration of 6.80 years and wants to hedge against interest rate moves between August and December The manager decides to use December T-bond futures. The futures price is 93-02 or 93.0625 and the duration of the cheapest to deliver bond is 9.2 years The number of contracts that should be shorted is Limitations of Duration-Based Hedging Assumes that only parallel shifts in the yield curve takes place Assumes that yield curve changes are small 10,000,000 6.80 79 93,062.50 9.20 1.54 1.55 13