chp05b interest rate futures

8/13/2019 Chp05b Interest Rate Futures

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K.Cuthbertson, D. Nitzsche 1

Version 1/9/2001

FINANCIAL ENGINEERING:

DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

LECTURE

Interest Rate Futures


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Topics

Cash Market for T-Bills

T-Bill Futures Contract

3m Sterling Futures Contract

Hedging

Arbitrage: Pricing a T-Bill Futures Contract


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Cash Market for T-Bills


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Spot Market: T- Bills,Market Price and Yield

Face or par value is FV= $100

n = days to maturity / days in year

Simpleyield ( y ~ proportion , p.a.):

P = 100 / [ 1 + y (n) ]

Compound yield ( y ~ proportion , p.a.):

P = 100 / ( 1 + y )n

Continuously compounded yield, y

P = 100 exp(- y . n )


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Price from discount Rate: T- Bills

The dollar(or sterling) discountis :

D = FV

The price is: P = FV - D

Also:

P = FV

Price and discount rate are inversely related

d m

a100

1

100

d m

a


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T-Bill Futures Contract


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What is a T-Bill Futures Contract ?

At expiry, (T), which may be in say 9m time

the (long) futures delivers a T-Bill which matures at

T+90, with face value M=$100.

This allows you to lock in at t=0, the forward rate, f12



Figure 5.2 : Interest rate (T-Bill) futures contract

t0 t* T=t1 t2

r1

r2

t12,

f12

Futures protection period = t12

Exposure period, t0to t1

T= t1= Maturity of futures contract



Buy one Sept, T-Bill

futures at F0= 98(no cash is exchanged

T = t10 T+90days = t2

Exposure period(2m) Protection period=t12

Receive a90-day T-Bill andpay F0

T-Bill

maturesat M =100

(M/F0)365/90 = ( 1 + annual interest earned over t12 )

(simple)annual interest earned is approx (2/98) x 4 = 8.16 %

T-Bill Futures Contracts



What is the known yield locked in at t=0, which applies between T (t1)and T+90 = t2)

F0 = M / ( 1 + annual interest earned over t12 )90/365

F0 = M / ( 1 + f12 )t12

Since f12(compound) is observable at t=0, then this is howwe price the futures contract (riskless arbitrage is hiddenin above)

Also F0 = M / ( 1 + f12 t12 ) - f12 is simple interest/yield

= M exp(- f12 t12) - f12is contin compound

Note: For all interest rate contracts, if f falls then F rises

Futures Price and futures Yield



3m Sterling Futures Contract



STERLING 3-MONTH CONTRACT (LIFFE)

Contract Size

DeliveryPrice Quotation

Tick Size (Value)

Settlement

Initial Margin

z = 500,000

Mar/June/Sept/DecF = (100 - futures rate)F = 0.01 (= 1-tick) (12.5)

Cash

500

Can lock in interest rate on 3-mth deposits

Tick value = 500,000 (0.01 / 100 ) (1/4) = 12.5

If F changes by 0.01 ie.1-tick (eg from 95.00 to95.01) then value of one contract changes by12.5

F = 100 - f where f in the futures /forwardrate(applicable from T to T+3m )



Calculation of Tick Value

500,000 ( 0.01 / 100 ) (1/4) = 12.5

z ( ( F1- F0) / 100 ) (1/4)

The 1/4 appears because a change of 1% pa

in f is equal to a change of 1/4 of 1% over 3-mnths

(the life of the deposit underlying this futurescontract)



Hedging



Simple Hedge :Short Sterling, Nave Hedge Ratio

Will receive 1m in 2m time and then wishes to place fundson deposit for 3m . Fears a fall in interest rates

15th April(today)

r0= 10% f

0= 10.5% F

0= 89.5

15th June(Hedge ends)

r1= 8% f1= 8.5% F1= 91.5

Nf= TVS0/FVF0= 1m/ (0.5m x 0.895) = 2.33 (=2)

Lose 2% in cash market and gain 2% on futures



Simple Hedge :Short Sterling

Loss of interest in cash market

= 0.02 x (1/4) x 1m = 5000

Profit on futures contract

= 2 x 200 ticks x 12.5 = $5000

Perfect hedge

Note:

Strictly the cash market loss is based on r0= 10% could nothave been achieved.

(Futures contract used matures in say, December)



Risks in a Hedge: Short Sterling

Example: 1st Jan and will receive 1.2m on 1st Sept

On 1st Sept wish to put proceeds into Commercial Bill for 6-months

Underlying in futures is a 3-month deposit

Futures matures at end of March, June, Sept, Dec

Potential Problems

Cash amount is not exact multiple of contract size

Margin calls may be required

Nearby contracts matures before Sept and would have to berolled over , otherwise use Sept contract

Underlying = Commercial bill, is not the same as the underlying

in the futures (ie. Eurosterling deposit) - Cross Hedge



Fig 5.3 : Hedge using US T-Bill Futures

Purchase T-Billfuture with Sept.delivery date

$1m cashreceipts

Maturity date Sept.T-Bill futures contract

3 monthexposure period

Desired investment/protectionperiod = 6-months

May Aug. Sept. Feb.Dec.

Maturity of Underlyingin Futures contract



Duration based Hedge Ratio

F = futures pricez = sizeof one futures contractFVF0= face value of onefutures contract = z F0Nf= number futures contracts heldys = spot yield, yF= futures yield (usually = f12 in textbook)

Using the min var hedge ratio but replacing(S, F)and 2(F) terms with duration and yields weget:

Fyos yy

)(0

0

yF

s

f D

D

FVF

TVS

N



Cross Hedge: US T-bill Futures

May (Today)Funds of $1m accrue in August to be invested for 6mmonths in bank or commercial billsUse Sept 3m T-bill Futures contract

Assume parallel shift in the yield curve

Qf= 89.2 (per $100 nominal) hence:

F0= 100 - (1/4)(100 - Qf) = 97.30

FVF0= $1m (F0/100) = $973,000

Nf= (TVS0 / FVF0) (Ds/Df)= ($1m / 973,000) ( 0.5/ 0.25) = 2.05 (=2)




Table 5.4 : Cross Hedge Using US T-Bill Futures

3 month US T-Bill Futures : Sept Maturity

Spot Market(May)

(T-Bill yields)

CME Index

Quote Qf

Futures Price, F

(per $100)

Face Value of $1m

Contract, FVF

May y0(6m) = 11% Qf,0= 89.2 97.30 $973,000

August y1(6m) = 9.6% Qf,1= 90.3 97.58 $975,750

Change -1.4% 1.10 (110 ticks) 0.28 $2,750

(per contract)Durations are : Ds= 0.5, Df= 0.25

Amount to be hedged = $1m. No. of contracts held = 2


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Gain on the futures position= $1m(0.97750.973)2 = $5,500

or (using tick value of $25 and Q = 0.01 is 1 tick)

= 110 ticks x $25 x 2 contracts = $5,500

The gain on the futures position of $5,500 when invested

over 6-months at y1= 9.6% is $5,764 hence (usingsimple interest):


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Hedged ReturnEffective (simple) interest

= y1+ = 0.096 + 0.0115 = 0.1075 (10.75%)

The 10.75% hedged return is substantially above theunhedged rate (y2) of 9.6% and

is reasonably close to the implied (simple) yield on the

September futures contract of 11.1% (= (100/97.31) 4).


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Arbitrage: Pricing a T-Bill Futures


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Figure 5.5 : Pricing the futures contract

Receive $100face value of2-year T-bill

Buy 2-yearT-bill for $S withface value $100

A.0 1

2

r2 r2

Receive $100 facevalue of T-billunderlying the F.C.

Buy 1-year T-billfor F/(1 + r1)

B.0 1 2

r1 f12

Maturity of T-billreceive $F

Portfolio A : 2-year T-billPortfolio B : 1-year T-bill plus interest rate futures contract

Go long a T-billfutures (at zero cost today)

Pay $Ffor F.C.on 1-year T-bill


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Figure 5.5 : Pricing the futures contract

The 1-year T-Bill with maturity value F must cost (at t=0) :

[5.30] Price 1-year T-Bill = F / (1+ r1)

The two portfolios payoff is the same at t=2 and hence

must cost the same today:[5.32] F / (1+ r1) = S

Price at t=0 of a 2-year T-Bill is :

[5.33] S = 100 / (1+r2)2= 100 / (1+r1) (1 + f12)

Substituting equation [5.33] into equation [5.32] :

[5.34] F = 100 / (1+ f12

)


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END OF SLIDES

chp05b interest rate futures

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