1

1.1

550.444Introduction to

Financial Derivatives

Week of October 8, 2012Interest Rate Futures

1.2

Where we are

Last week: Forward & Futures Prices/Value (Chapter 5, OFOD)

This week: Interest Rate Futures (Chapter 6, OFOD)

Fall Break: October 15th

Class will meet on Tuesday the 16th , insteadMid Term: (Oct 17, Wednesday) In Two Weeks: Swaps (Chapter 7, OFOD) HW will be returned at Section on Thurs-Friday

1.3

Assignment

For This Week (October 8th) Read: Hull Chapter 6. Interest Rate Futures Problems (Due October 8th)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 16h)Chapter 6: 4, 6, 9, 11, 14, 21; 26, 27 Chapter 6 (7e): 4, 6, 9, 11, 14, 21; 23, 24

1.4

Assignment

Next Week (October 16th) Review (Tuesday, Oct 16th) and Mid-term (Oct 17th)

In Two Weeks (October 22nd) Read: Hull Chapter 7. Swaps Problems (Due October 29)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: Thursday, Dec 20th, 9:00 – Noon; Gilman 132

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1.5

Plan for This Week

Review some items 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

1.6

Forward Rate Agreement A forward rate agreement (FRA) is an agreement that a specific rate,

RK, will apply to a principal, L, during a specified future time period Borrowing & Lending is usually at LIBOR RK: Rate of FRA RF: Forward LIBOR between T1 and T2

RM: LIBOR observed in the market at T1 for period between T1 & T2

Normally the applicable rate at T1 is RM; so as a consequence of the FRA, there is a differential rate (of RK- RM) 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 T1

If RK>RM a lender will receive (at T2) a cash flow of in excess of the market rate; a borrower will pay this amount in excess of RM

FRAs may be settled at T1 rather than T2; the settlement payoff at T1 is

)(1))((

12

12

TTRTTRRL

M

MK

))(( 12 TTRRL MK

1.7

Forward Rate Agreement An FRA can be thought of as an agreement where interest at

a predetermined rate, RK, is exchanged for interest at the market rate, RM (though in fact, it is just a forward on the rate)

An FRA can be valued at any time prior to T1 by assuming that the forward interest rate, RF, is certain to be realized at T1 Indeed, the forward interest rate can always be locked-in

The present value of the FRA to a lender where a fixed rate, RK, will be received on principal L between times T1 and T2 :

Today, a new FRA with RK= RF can be entered into without cost to either party

22))(( 12TR

FK eTTRRL

1.8

Locking in the Forward Rate

t0 T1 T2

t0 T1 T2

t0 T1 T2

L

L

Lexp[-R1T1]

Lexp[-R1T1] exp[R2T2]

Lexp[-R1T1] exp[R2T2] = Lexp[RF(T2-T1)]A

B

C

• Borrowing in C and Depositing in B provides the lender with the funds L for theForward loan in A at the rate RF in force at t0

• The value for this FRA at t0 is zero when RK is established as RF

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1.9

Day Count Conventions in the U.S.

Treasury Bonds: Actual/Actual (in period)

Corporate Bonds: 30/360

Money Market Instruments: Actual/360

Number of Days between Dates Interest Earned in Reference Period Number of Days in Reference Period= Interest Earned between Two Dates

1.10

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 ) Interest Earned = 4 x 122/180 = $2.7111 (on $100 face)

1.11

Day Count Conventions in the U.S.

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: 100 = x + 2.0222 => x = 97.9778; 97.9778 (1 + z) = 100 => z =.020639 1.12

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:

91365100 2.5277781

97.4722 97.472291ln 1 0.025933

365365 ln 1 0.025933 10.269291

Re

R

R

4

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 (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

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Eurodollar Futures

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Eurodollar Futures

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Eurodollar Futures

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1.17

Eurodollar Futures

1.20

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 lend $1 million at 2.58% will

earn 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 loaned at 2.59% the lender receives

interest of $6,475 The lender (long) receives $25 more on the investment Losses $25 on the ED futures

1.21

Eurodollar Futures

Price decline of .01 results in a loss of $25 for the ED short on the borrowing (or 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 results in a gain of $25 for the ED long on the loan (the same $25 loss of the 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 $25Rate goes down on the investment at final settlement 1.22

Example

Date QuoteNov 1 97.12Nov 2 97.23Nov 3 96.98……. ……

Dec 21 97.42

• 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

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1.23

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) 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 table to the

final settlement …

1.24

Example continued

In the example, for the Dec 21st 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 price-moves, 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”)

1.25

Formula for Contract Value

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)/1001.26

Forward vs. Futures Eurodollar futures and the FRA are quite similar Both can lock-in a rate for a future 90-day period between T1 & T2

ED contract settles daily, with final settlement at T1

FRA is not settled daily; final settlement (reflecting the realized rate between T1 & T2 ) is made at T2 (though it can be made at T1using the PV of the differential that would be realized at T2)

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 T1, but maintain their daily settlement differences T1 vs. T2 Payoff Component: FRA difference when settlement is

at T1 vs. T2 .

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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 RM- RF at T1, where RF is a predetermined rate for the period T1 to T2 and RM 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 RF 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 T1 will correspond to an FRA with a lower RF

ED will predict a higher rate for the forward rate, RF, than is real1.28

Forward vs. Futures T1 vs. T2 Payoff Component Decreases value of forward rate for the FRA relative to ED Suppose the payoff on the deposit differential RM- RF is at T2 rather

than T1 (the payoff at T2 is the standard for “vanilla” FRA), where RF is a predetermined rate for the period T1 to T2 and RM is the realized market rate If RM is high, payoff is positive; as payoff is at T2 rather than T1, the cost

is high since there is no opportunity for investment at the higher rate You would rather payoff at T1 rather than T2; therefore RF is lower (all

other things being equal) Therefore, with ED’s payoff at T1 rather than T2, ED will predict a

higher rate for the forward rate, RF, than is real This is a smaller effect than daily settlement; as T1 can be much

longer into the future than the period from T1 to T2

1.29

Forward vs. Futures To resolve the difference between forward rates, F, and the futures

rate (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

Forward Rate, F = Futures Rate

Where both rates are with continuous compounding and, T1: time to maturity of the futures contract T2: 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)

Since , then

212

21 TT

iiiiiii TRTRTTF 111 )(1

11

)(

i

iiiiii T

TRTTFR

1.30

Forward vs. Futures

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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

1.33

Extending the LIBOR Zero Curve

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: We have:

Where the Fi is the convexity-adjusted forward rate from the futures contact

iiiiiii TRTRTTF 111 )(

1

11

)(

i

iiiiii T

TRTTFR

1.34

SWAP – ED Yield Curve Analysis October 01, 10

1.35

SWAP – ED Yield Curve Analysis October 01, 10

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1.36

ED Synthetic Forward Rates October 01, 10

1.37

SWAP – ED Yield Curve Analysis October 05, 12

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SWAP – ED Yield Curve Analysis October 05, 12

1.39

ED Synthetic Forward Rates October 05, 12

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1.40

ED Synthetic Forward Rates October 05, 12

1.41

ED Synthetic Forward Rates October 05, 12

1.43

Treasury Bond Price Quotesin the U.S

Cash price = Quoted price + Accrued Interest

Full price = Flat price + Accrued Interest

Dirty price = Clean price +Accrued Interest

1.44

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

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1.45

Treasury Bond Futures

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

1.46

US T-Bond (USZ2) Conversion Factors

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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

401

5 100 146.23(1.03) (1.03)i

i

1.48

Treasury Bond Futures

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

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US T-Bond (USZ0) Basis Analysis October 05, 10

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US T-Bond (USZ0) Basis Analysis October 05, 10

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US T-Bond (USZ2) Basis Analysis October 05, 12

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US T-Bond (USZ2) Basis Analysis October 05, 12

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CBOTT-Bonds & T-Notes

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 centralCash 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”

bonds to deliver into the 2pm futures price 1.55

Duration Matching

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

FC: Contract price for interest rate futuresDF: Duration of asset underlying futures at futures expirationP : Value of portfolio being hedgedDP: Duration of portfolio at hedge maturity

Remember:

FC

P

DFPD

PP D y

P

1.56

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

7920.980.6

50.062,93000,000,10

1.57

Limitations of Duration-Based Hedging

Assumes that only parallel shifts in the yield curve takes place

Assumes that yield curve changes are small