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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
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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
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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
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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
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3m Sterling Futures Contract
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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 )
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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)
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Hedging
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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
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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)
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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
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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
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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
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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)
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Cross Hedge: US T-bill Futures
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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Cross Hedge: US T-bill Futures
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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Cross Hedge: US T-bill Futures
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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