1. Lec 6 Interest Rate Futures
Lecture 6: Interest Rate Futures (Hull, Ch. 6)
▸ U.S. Treasury Bills and Futures Contracts on U.S. T-Bills
▸ EuroDollar Time-Deposits and EuroDollar Futures
▸ U.S. Treasury Bond Futures Contracts
▸ Hedging Strategies
U.S. T-Bills (Price Quotes in the Spot Market)
iask is the current interest rate on T-bills
T = time to maturity
Example, iask = 10%/year and T = 90 days, then
➟ $Interest over 90 days = $25,000
➟ Pask = 1M - $25,000 = $975,000 for $1M Face Value.
Lec 6 Interest Rate Futures
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2. Lec 6 Interest Rate Futures
U.S. T-Bill Futures Contract
Contract: 90-day U.S. Treasury bills
Exchange: IMM at CME
Delivery: Mar, Jun, Sep, Dec.
Contract Size: $1M (FV)
Example: Go long one Dec T-Bill futures at IMM index = 90.00
In English, I have agreed to:
• Buy $1M (FV) of U.S. T-Bills with 90 days to maturity
• actual trade will take place 3rd week in December
• IMM index = 90.00
➟ Forward Rate = (100-IMM index)/100 = 0.10
➟ Futures Price = $1M { 1 – (0.10/360) x 90} = $975,000
Thus, the Index is like a price:
Price of a zero coupon bond with Face Value = $100.
if IMM index ↑ Long gains, if IMM index ↓ Long loses.
Just like stocks.
Lec 6 Interest Rate Futures
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3. Lec 6 Interest Rate Futures
Daily Settlement (i.e., Daily Mark to Market)
Day
IMM Settle
Index
Forward interest rate
Implied Futures
Price
Margin for
long Short
Day 0, 9AM
90.00
10%
975,000
$2,000
Day 0, Close
90.01
9.99%
975,025
$2,025
$1,975
Day 1, Close
90.05
9.95%
975,125
$2,125
$1,875
Day 2, Close
89.95
10.05%
974,975
. . .
Expiration
92.50
7.75%
981,250
Lec 6 Interest Rate Futures
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4. Lec 6 Interest Rate Futures
What Happens at Expiration?
Suppose the last Settle IMM = 92.50
➟ Forward Rate = = 7.5% = Spot Rate
➟ Spot Price = Futures Price = $1M { 1 – (0.075/360) x 90} = $981,250
Cumulative Δ = $981, $975,000 = + $6,250
➟ Long gained $6,250, ➟ Short lost $6,250
A. Long Buys T-Bills thru futures.
▸ Long receives $1M (FV) of T-Bills from short.
▸ Long pays $981,250 to short.
▸ But, Long gained $6,250 in the futures market. Therefore
Actual cost = 981, ,250 = $975,000.
(Just as originally agreed)
Lec 6 Interest Rate Futures
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5. Lec 6 Interest Rate Futures
B. Close futures position, and buy in the spot mkt
▸ Go Short 1-Dec-futures at (to close long position).
▸ Long pays $981,250 in the spot market.
▸ But, Long gained $6,250 in the futures market. Therefore
Actual cost = 981, ,250 = $975,000.
(Just as originally agreed)
Lec 6 Interest Rate Futures
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6. Lec 6 Interest Rate Futures
Euro$ Time Deposits
▸ U.S. Banks (e.g., BoA) will accept only US dollars for deposits
▸ Banks in London will accept US$ deposits: Euro$ Deposits
▸ LIBOR is the interest rate on these funds
Euro$ Futures Contract
Contract: 90-day EuroDollar Time Deposit
Exchange: IMM at CME
Delivery: Mar, Jun, Sep, Dec. Settlement in Cash
Contract Size: $1M (FV)
Notes
1. The underlying asset is a 90day Eurodollar deposit with FV = $1M
2. Daily “Mark to Market” is exactly the same as for US T-Bill futures
3. At expiration, Euro$ Futures are settled in cash.
4. Futures contracts range in maturity from 1-month to 10 years.
Lec 6 Interest Rate Futures
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7. Lec 6 Interest Rate Futures
Hedging
Example 6.8: Use Euro$ Futures to Hedge Interest Rate Risk
▸ UTC needs to borrow $15M for 3-months (May, June and July).
Interest paid monthly.
▸ Monthly interest rate (set at the beginning of each month) = LIBOR + 1%
▸ On April 30, LIBOR = 0.08/yr.
➟ interest due on May 30 = $15M*( )/12 = $112,500
LIBOR ➀ 0.08
Borrow $15M ➀ $112, June Interest = ? July = ?
|–––––––––––––––––|–––––––––––––|–––––––––––––––|
Apr May Jun July 30
Problem: As of time 0, June and July LIBOR rates are unknown and random.
Lec 6 Interest Rate Futures
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8. Lec 6 Interest Rate Futures
For example, Assume LIBOR will evolve as follows:
➁ ➂ ➀ $112, ➁ $122, ➂$130,000
|–––––––––––––––|–––––––––––––––|––––––––––––––––|
Apr May 30 Jun 30 July 30
Under this scenario, interest payments will go UP
➁ June 30
At t=1 (Beginning of June) monthly LIBOR = 0.088/yr
Interest Pmt due June 30 = $15M*( )/12 = $122,500
➂ July 30
At t=2 (Beginning of July) monthly LIBOR = 0.094/yr
Interest Pmt due July 30 = $15M*( )/12 = $130,000
What to do? Set up a hedge strategy. How?
Lec 6 Interest Rate Futures
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9. Lec 6 Interest Rate Futures
Hedging Strategy for June:
▸ At t=0 (in April)
Short 5 June IMM_L Index = 91.88
➟ Forward Rate= ( )/100 = 8.12%
➟ Notional Value = $1M[ 1 - {0.0812/360} *90 ] = $979,700
▸ May 30 buy back these 5 contracts. Suppose IMM_L Index = 91.12
➟ Forward Rate=( )/100 =
➟ Notional Value = $1M[ 1 - {0.0888/360} *90 ] = $977,800
Gain/Loss on 5 Futures = - 5(977, $979,700) = + $9,500
Therefore,
Unhedged Interest Pmt for June (paid June 30)
= $15M*( )/12= $122,500
Hedged Interest Pmt = $122, $9,500 = $113,000
➟ Gains on short futures offset higher Interest Pmt.
Lec 6 Interest Rate Futures
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10. Lec 6 Interest Rate Futures
Hedging Strategy for July:
▸ At t=0 (in April)
Short 5 September IMM_L Index = 91.44
➟ Forward Rate= ( )/100 = 8.56%
➟ Notional Value = $1M[ 1 - {0.0856/360} *90 ] = $978,600
▸ June 30 buy back these 5 contracts. Suppose IMM_L Index = 90.16
➟ Forward Rate=( )/100 =
➟ Notional Value = $1M[ 1 - {0.0894/360} *90 ] = $975,400
Gain/Loss on 5 Futures = - 5(975, $978,600) = + $16,000
Therefore,
Unhedged Interest Pmt for July (paid July 30)
= $15M*( )/12= $130,000
Hedged Interest Pmt = $130, $16,000 = $114,000
➟ Gains on short futures offset higher Interest Pmt.
Lec 6 Interest Rate Futures
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11. Lec 6 Interest Rate Futures
In sum:
Q: in this example why short 5 contracts?
Time
1-month
Spot LIBOR
Interest
Payment
CF from
Futures
Net Int
0, Apr 30
0.08/12
➀ 1, May 30
0.088/12
$112,500
➁ 2, June 30
0.094/12
$122,500
+9,500
$113,000
➂ 3, July 30
$130,000
+16,000
$114,000
Lec 6 Interest Rate Futures
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12. Lec 6 Interest Rate Futures
Hedging Interest Rate Risk
Def’n:
Duration measures the sensitivity of the bond Price (B0 )
to changes in interest rates (Δy).
This relationship applies to Bond futures also. ⇨ ΔF = - (DF x F0 ) x Δy
To immunize a bond portfolio against changes in interest rates,
Set ΔB + NF (ΔF) = 0
Lec 6 Interest Rate Futures
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13. Lec 6 Interest Rate Futures
In our example, Interest rate changes monthly.
➟ Duration of the $15M bond (loan) =1/12 = years.
Duration of the Futures Contract: 90/360 = 0.25
For June Hedge
F0 = Notional Value (as of April 30) = $979,700
➟ ΔF = - (DF x F0 ) x Δy = - (0.25 x $979,700) x Δy
➟ ΔB = - (DB x B0 ) x Δy = - ( x $15M) x Δy
Set NF such that
[ Δ Bond Portfolio Value ] + NF (Δfutures) = 0
- ( x $15M) x Δy - NF (0.25 x $979,700) x Δy = 0
NF = Short 5 contracts
For July Hedge
F0 = Notional Value (as of April 30) = $978,600
- ( x $15M) x Δy - NF (0.25 x $978,600) x Δy = 0
NF = Short 5 contracts
Lec 6 Interest Rate Futures
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14. Lec 6 Interest Rate Futures
US Treasury Bonds : Spot (Cash) Market
▸ Prices in the spot market are quoted as
➟ /32 = $ (per $100 FV)
▸ Cash Price = Quoted Price + AI (seller wants cash for AI)
Example:
Spot Price is
CI = $5.50 (per $100) on Jan 10 and July 10 (181 days apart)
if you buy this bond March 5, then
Invoice Price = (5.50/181)*54 = $ per $100
If you buy this bond June 30, and Spot Price is still
Invoice Price = (5.50/181)*171 = $122,696 per $100,000 FV
Lec 6 Interest Rate Futures
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15. Lec 6 Interest Rate Futures
US Treasury Bond Futures
Contract: U.S. T-Bond with 15 years to maturity, and 6% CI rate
Exchange: CBOT
Contract Size: $100,000 Face Value
Delivery: Mar, Jun, Sep, Dec. Any day during delivery month
Example: Go long one Dec T-Bond futures at ,
In English, I have agreed to:
• Pay 97+18/32 = $97,562.50
for a U.S. T bond with $100,000 (FV), 15 yrs to maturity and 6% CI
• Short will decide the actual delivery day in Dec and the actual
bond to sell.
Lec 6 Interest Rate Futures
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16. Lec 6 Interest Rate Futures
Daily Mark to Market
Day
IMM Settle
Index
Implied Futures
Price
Margin for
long Short
Day 0, 9AM
97-18
97,562.50
$4,000
Day 0, Close
97-19
97,593.75
$4,031.25
$3,968.75
Day 1, Close
98-19
98,593.75
$5,031.25
$2,968.75
Day 2, Close
. . .
Expiration
Lec 6 Interest Rate Futures
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17. Lec 6 Interest Rate Futures
What Happens at Expiration?
Close futures position before Dec 1. Buy Bonds in the spot market. OR
(You may Skip the rest below)
B. Buy T-Bond thru the futures. Then, Long must pay the following price:
Invoice Price = Futures Price*Conversion Factor + AI
Explanations:
Suppose the last settle Futures price is
Then, the Futures Price = 94,468.75
Lec 6 Interest Rate Futures
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18. Lec 6 Interest Rate Futures
What exactly is a Conversion Factor?
Short must deliver
bonds with $100,000(FV), 6% CI AND 15 yrs Time to maturity
Most likely, Short cannot find one! What to do?
Solution: CBOT allows delivery from a wide variety of US T-Bonds.
Any coupon rate is okay. Only requirement: 15 or more YTM.
For example, consider these two bonds (CI paid semiannually)
Bond A: 6% CI (semiannual, FV = $100,000 and 15 YTM.
(a long Futures position is the obligation to buy this bond)
Bond B: 10% CI rate, FV = $100,000 and 20 years to maturity
Lec 6 Interest Rate Futures
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19. Lec 6 Interest Rate Futures
Assume a discount rate of 6%/yr. Then,
PriceA = $3,000 [PVIFA, 0.06/2, 15yrs*2 ] +
+ $100,000[PVIF, 0.06/2, 15yrs*2 ] = $100,000
PriceB = $5,000 [ PVIFA, 0.06/2, 20yrs*2 ]
+ $100,000 [ PVIF, 0.06/2, 20yrs*2 ] =$146,230
Suppose Short wants to deliver Bond B.
Should Long pay $100,000 or $146,230?
Clearly, Long should pay $146,230
➟ $146,230 = (same as) * $100,000
Lec 6 Interest Rate Futures
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20. Lec 6 Interest Rate Futures
Definition of Conversion Factor
Co Fac = {Value of deliverable bond priced to yield 6%} /
{Value of underlying bond (Bond A) priced to yield 6%} = = 146,230/ 100,000 =
NOTE:
The Conversion Factor is just a mechanical formula. It converts a bond with any coupon and any Term to Maturity into Bond A, Assuming a YTM of 6%.
Lec 6 Interest Rate Futures
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21. Lec 6 Interest Rate Futures
One more complication: Cheapest to Deliver Bond
At time of delivery, short has the option to deliver any long term bond.
Example: Suppose 3 bonds are available
Quoted (spot)
Bond Bond Price Conversion Factor
X $
Y $
Z $
At delivery, to buy any bond in the spot market Short must pay:
Invoice Price = Quoted Bond Price + AI
To settle the futures, Short delivers this bond and receives:
Invoice Price = Futures Price*Conv. Factor + AI
Suppose the last Settle Futures Price =
Then, the net CF for short is:
Net CFX = - $99, ($94,468.75*1.0382) = - $1,423
Net CFY = -$143,498 + ($94,468.75*1.5190) = $0
Net CFZ = -$119,750 + ($94,468.75*1.2615) = - $578
Short will buy and deliver $100,000 FV of bond Y
Lec 6 Interest Rate Futures
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22. Lec 6 Interest Rate Futures
Thank You
(a Favara)
Lec 6 Interest Rate Futures
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