Interest Rates Futures

1
Interest Rates Futures

• Fin 288
• Futures Options and Swaps

2
Interest Rate Future Contracts

• Traded on the CBOT
• 30 Year Treasury Bond 30 Yr Mini
• 10, 5, 2 year Treasury note futures
• 30 Day Fed Funds
• 5 10 year Swap
• German Debt
• Traded on CME
• Eurodollar Futures

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A quick look at contract Specifications

• Treasury Bonds and Notes-
• Range of delivery dates
• Fed Funds Futures
• Price
• Swaps
• Delivery
• Muni
• Underlying Asset

4
Treasury Securities

• Since a majority of the interest rate instruments
  we will use are related to treasury securities,
  we need to discuss some basics relating to the
  pricing of Treasury securities.

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Some Pricing Issues

• Day Count Conventions
• Used to determine the interest earned between two
  points in time
• Useful in calculating accrued interest
• Specified as X/Y
• X the number of days between the two dates
• Y The total number of days in the reference
  period

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Day Count Conventions

Day Count Convention Market Used
US Treasury Bonds
Corporate and Municipal Bonds
US T-Bills money Market Instruments
7
Price Quotes for Treasury Bills

• Let Yd annualized yield, D Dollar Discount F
  Face Value, t number of days until maturity
• Price F -D

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Price Quotes on T- Bills

• Note Return was based on face value invested,
  not the actual amount invested.
• 360 day convention makes it difficult to compare
  to notes and bonds.
• CD equivalent yield makes the measure comparable
  to other money market instruments

9
Accrued Interest

• When purchasing a bond between coupon payments
  the purchaser must compensate the owner for for
  interest earned, but not received, since the last
  coupon payment

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Price Quotations

• QuotationsThe quoted and cash price are not the
  same due to interest that accrues on the bond.
  In general

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Example

• Assume that today is March 5, 2002 and that the
  bond matures on July 10, 2004
• Assume we have an 11 coupon bond with a face
  value of 100. The quoted price is 90-05 (or 90
  5/32 or 90.15625)
• Bonds with a total face value of 100,000 would
  sell for 90,156.25.

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Example continued

• Coupons on treasuries are semiannual. Assume that
  the next coupon date would be July 10, 2000 or 54
  days from March 5.
• The number of days between interest payments is
  181 so using the actual/actual method we have
  accrued interest of
• (54/181)(5.50) 1.64
• The cash price is then
• 91.79625 90.15625 1.64

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Conversion Factors

• Since there are a range of bonds that can be
  delivered, the quoted futures price is adjusted
  by a conversion factor.

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Price based upon 6 YTM

• The conversion factor is based off an assumption
  of a flat yield curve of 6 (that interest rates
  for all maturities equals 6).
• By comparing the value of the bond to the face
  value, the CBOT produces a table of conversion
  factors.

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Conversion Factor Continued

• The maturity of the bond is rounded down to the
  nearest three months.
• If the bond lasts for a period divisible by 6
  months the first coupon payment is assumed to be
  paid in six months. (A bond with 10 years and 2
  months would be assumed to have 10 years left to
  maturity)

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Conversion Factor continued

• If the bond does not round to an exact six months
  the first coupon is assumed to be paid in three
  months and accrued interest is subtracted.
• A bond with 14 years and 4 months to maturity
  would be treated as if it had 14 years and three
  months left to maturity

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Example 1

• 14 coupon bond with 20 years and two months to
  maturity
• Assuming a 100 face value the value of the bond
  would equal the price valued at 6
• The conversion factor is then
• 1.92459/100 1.92459

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Example 2

• What if the bond had 18 years and four months
  left to maturity? The bond would be considered
  to have 18 years and three months left to
  maturity with the first payment due in three
  months.Finding the value of the bond three
  months from today

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Example 2 continued

• Assume the rate for three months is
• (1r)2 1.03 r .014889
• Using this rate it is easy to find the PV of the
  bond
• 187.329/1.014889 184.581
• There is one half of a coupon in accrued interest
  
• so we need to subtract 7/23.50
• 184.581 - 3.50 181.081
• resulting in a conversion factor of
• 181.081/100 1.81081

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Price Quote on T-Bills

• Quotes on T- Bills utilize the actual /360 day
  count convention.
• The quoted price of the treasury bill is an
  annualized rate of return expressed as a
  percentage of the face value.

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T- Bills continued

• The quote price is given by (360/n)(100-Y)
• where Y is the cash price of the bill
• with n days until maturity
• 90 day T- Bill Y 98
• (360/90)(100-98) 8.00

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Rate of Return

• The quote is not the same as the rate of return
  earned by the treasury bill.
• The rate of interest needs to be converted to a
  quarterly compounding annual rate.
• 2/98(365/90) .0828

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Quoted Price

• The price quote on a Treasury bill is then given
  by 100 - Corresponding Treasury bill price quote
  
• (quoted price 8 so futures quote 92)
• Given Z the quoted futures priceY the
  corresponding price paid for delivery of 100 of
  90 day treasury bills then Z 100-4(100-Y) or
  Y 100-0.25(100-Z) Z 100-4(100-98) 92

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Cheapest to Deliver Bond

• There are a large number of bonds that could be
  delivered on the CBOT for a given futures
  contract.
• The party holding a short position gets to decide
  which bond to deliver and therefore has incentive
  to deliver the cheapest.

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Cheapest to Deliver

• Upon delivery the short position receives
• The cost of purchasing a bond is
• Quoted bond price accrued interestBy
  minimizing the difference between the cost and
  the amount received, the party effectively
  delivers the cheapest bond

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Cheapest to deliver

• The bond for which
• is minimized is the one that is cheapest to
  deliver.

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Example Cheapest to Deliver

• Consider 3 bonds all of which could be delivered
• Quoted Conversion
• Bond Price Factor
• 1 99.5 1.0382 99.5-(93.25(1.0382))
  2.69
• 2 143.5 1.5188 143.5-(93.25(1.5188))
  1.87
• 3 119.75 1.2615 119.75-(93.25(1.2615)
  )2.12

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Impact of yield changes on CTD

• As yield increases bonds with a low coupons and
  longer maturities become relatively cheaper to
  deliver. As rates increase all bond prices
  decrease, but the price decrease for the longer
  maturity bonds is greater
• As yields decrease high coupon, short maturity
  bonds become relatively cheaper to deliver.

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Wild Card Play

• Trading at the CBOT closes at 2p.m. however
  treasury bonds continue to trade until 400pm and
  a party with a short position has until 8pm to
  file a notice of intention to deliver.
• Since the price is calculated on the closing
  price in the CBOT the party with a short position
  sometimes has the opportunity to profit from
  price movements after the closing of the CBOT.
• If the Bond Prices decrease after 2 pm it
  improves the short position.

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Eurodollar Futures

• Eurodollar dollar deposited in a foreign bank
  outside of the US. Eurodollar interest rate is
  the interest earned on Eurodollars deposited by
  one bank with another bank.
• London Interbank Offer Rate (LIBOR) Rate at
  which banks loan to each other in the London
  Interbank Market.

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Simple Hedge Example

• Assume you know that you will owe at rate equal
  to the LIBOR 100 basis points in three months
  on a notional amount of 100 Million. The
  interest expenses will be set at the LIBOR rate
  in three months.
• Current three month LIBOR is 7, Eurodollar
  futures contract is selling at 92.90.

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Simple Hedge Example

• 100 - 92.90 7.10
• The futures contract is paying 7.10
• Assume the interest rate may either
• increase to 8 or decrease to 6

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A Short Hedge

• Agree to sell 10 Eurodollar future contracts
  (each with an underlying value of 1 Million).
• We want to look at two results the spot market
  and the futures market. Assume you close out the
  futures position and that the futures price will
  converge to the spot at the end of the three
  months.

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Rates increase to 8

• Spot position
• Need to pay 8 1 9 on 10 Million 10
  Million(.09/4) 225,000
• Futures Position
• Fut Price 92 interest rates increased by .9
• Close out futures position
• profit (10 million)(.009/4) 22,500

35
Rates Increase to 8

• Net interest paid
• 225,000 - 22,500 202,500
• 10 million(.0810/4) 202,500

36
Rates decrease to 6

• Spot position
• Need to pay 6 1 7 on 10 Million 10
  Million(.07/4) 175,000
• Futures Position
• Fut Price 94 interest rates decreased by 1.1
• Close out futures position
• loss (10 million)(.011/4) 27,500

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Rates Decrease to 8

• Net interest paid
• 175,000 27,500 202,500
• 10 million(.0810/4) 202,500

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Results of Hedge

• Either way the final interest rate expense was
  equal to 8.10 or 100 basis points above the
  initial futures rate of 7.10
• Should the position be hedged?
• It locks in the interest rate, but if rates had
  declined you were better off without the hedge.

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Simple Example 2

• On January 2 the treasurer of Ajax Enterprises
  knows that the firm will need to borrow in June
  to cover seasonal variation in sales. She
  anticipates borrowing 1million.
• The contractual rate on the loan will be the
  LIBOR rate plus 1
• The current 3 month LIBOR rate is 3.75 and the
  Eurodollar futures contract is 4.25

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Simple Example 2 Continued

• To hedge the position assume the treasurer sells
  one June futures contract.
• Assume interest rates increase to 5.5 on June
  13.
• Assume that the expiration of the contract is
  June 13, the same day that the loan will be taken
  out. The futures price will be
• 100-5.50 94.50

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Rates increase to 5.5

• Spot position
• Need to pay 5.51 6.5 on 1 Million 1
  Million(.065/4) 16,250
• Futures Position
• Fut Price 94.50 interest rates
• increased by 1.25
• Close out futures position
• profit (1million)(.0125/4) 3,125

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Rates Increase to 5.5

• Net interest paid
• 16,250 - 3,125 13,125
• 1 million(.0525/4) 13,125
• which is the interest rate implied by the
  Eurodollar futures contract
• 4.25 1 5.25

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Assumptions

• The hedge worked because of three assumptions
• The underlying exposure is to the three month
  LIBOR which is the same as the loan
• The end of the exposure matches the delivery date
  exactly
• The margin account did not change since the rate
  changed on the last day of trading.

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Basis Risk revisited

• The basis is a hedging situation is defined as
  the Spot price of the asset to be hedged minus
  the futures price of the contract used. When the
  asset that is being hedged is the same as the
  asset underlying the futures contract the basis
  should be zero at the expiration of the contract.
  
• Basis Spot - Futures

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Basis Risk

• On what types of contracts would you expect the
  basis to be negative? Positive? Why?(-) Low
  interest rates assets such as currencies or gold
  or silver (investment type assets with little or
  zero convenience yield. F S(1r)T()
  Commodities and investments with high interest
  rates (high convenience yield)
• F S(1ru)T Implies it is more likely that
• F lt S(1ru)T

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Mismatch of Maturities 1

• Assume that the maturity of the contract does not
  match the timing of the underlying commitment.
• Assume that the loan is anticipated to be needed
  on June 1 instead of June 13.

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Simple Example Redone

• On January 2 the treasurer of Ajax Enterprises
  knows that the firm will need to borrow in June
  to cover seasonal variation in sales. She
  anticipates borrowing 1million.
• The contractual rate on the loan will be the
  LIBOR rate plus 1
• The current 3 month LIBOR rate is 3.75 and the
  Eurodollar futures contract is 4.25

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Simple Example 2 Continued

• To hedge the position assume the treasurer sells
  one June futures contract.
• Assume interest rates increase to 5.5 on June 1.
• Assume that the futures price has decreased to
  94.75 (before it had decreased to 94.50) implying
  a 5.25 rate (a 25 bp basis)

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Rates increase to 5.5

• Spot position
• Need to pay 5.51 6.5 on 1 Million 1
  Million(.065/4) 16,250
• Futures Position
• Fut Price 94.75 interest rates
• increased by 1.00
• Close out futures position
• profit (1million)(.0100/4) 2,500

50
Rates Increase to 5.5

• Net interest paid
• 16,250 - 2,500 13,750
• 1 million(.055/4) 13,750
• which is more than the interest rate implied by
  the Eurodollar futures contract
• 4.25 1 5.25

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Minimizing Basis Risk

• Given that the actual timing of the loan may also
  be uncertain the standard practice is to use a
  futures contract slightly longer than the
  anticipated spot position.
• The futures price is often more volatile during
  the delivery month also increasing the
  uncertainty of the hedge
• Also the short hedger could be forced to accept
  delivery instead of closing out.

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Mismatch in Maturities 2

• Assume that instead of our original problem the
  treasurer is faced with a stream of expected
  borrowing.
• Anticipated borrowing at 3 month LIBOR
• Date Amount
• Mach 1 15 Million
• June 1 45 Million
• September 1 20 million
• December 1 10 Million

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Strip Hedge

• To hedge this risk, it to hedge each position
  individually.
• On January 1 the firm should
• enter into 15 short March contracts
• enter into 45 short June contracts
• enter into 20 short Sept contracts
• enter into 10 short December contracts

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Strip Hedge continued

• On each borrowing date the respective hedge
  should be closed out.
• The effectiveness of the hedge will depend upon
  the basis at the time each contract is closed out.

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Rolling Hedge

• Another possibility is to Roll the Hedge
• January 2 enter into 90 short March contracts
• March 1 enter into 90 long March contracts
• enter into 75 short June contracts
• June 1 enter into 75 long June contracts
• enter into 30 short Sept contracts
• Sept 1 enter into 30 long Sept contracts
• enter into 10 short Dec contracts
• Dec 1 enter into 10 long Dec contracts

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Rolling the Hedge

• Again the effectiveness of the hedge will depend
  upon the basis at each point in time that the
  contracts are rolled over.
• This opens the from to risk from the resulting
  rollover basis.

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Example

• Now assume that the treasury has decided to
  borrow it the commercial paper market instead of
  from a financial institution.
• There is not a commercial paper futures contract
  so it must be decided what contract to use to
  hedge the possible interest rate change in the
  commercial paper market.
• Assume that the treasure wants to borrow 36
  million in June with a one month commercial paper
  issue.

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Number of contracts part 1

• You must choose what underlying contract best
  matches the 30 day commercial paper return.
• 90 Day T-Bill. 90 day LIBOR Eurodollar, 10 year
  treasury bond.
• Assume 90 day LIBOR Eurodollar has the highest
  correlation so it is chosen.
• Assume now that the treasurer for Ajax has ran
  the regression and that the beta is .75

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Number of contracts part 2

• We also need to consider the asset underlying the
  three month LIBOR futures contract and one month
  commercial paper rate have different maturities.
• A 1 basis point movement in 1,000,000 of
  borrowing is 1,000,000(.0001)(30/360) 8.33
• A one basis point change in 1,000,000 of the
  future contract is equal to
• 1,000,000(.0001)(90/360) 25

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Number of contracts part 2

• The change in the three month contract is three
  times the size of the change in the one month
  this would imply a hedge ratio of 1/3 IF the
  assets underlying both positions was the same.
• Both sources of basis risk need to be considered.

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Number of Contracts

• The treasurer will need to enter into
• 36(.75)(.33) 9 million
• Of short futures contracts

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The Cross Hedge

• On January 2
• 3 month LIBOR 3.75
• June Eurodollar Future price is 95.75 implying
  4.24 rate
• Spread between spot LIBOR rate and 1 month
  commercial paper rate is 60 basis points
• This implies a 4.35 commercial paper rate.

63
Expectations

• Previously Ajax hoped to lock in a 4.25 3 month
  LIBOR rate or an increase of 50 basis points form
  the current 3.75
• Keeping the 50 basis point increase constant and
  using our hedge ratio of .75 the goal becomes
  locking in a .75 (50) 37.5 basis point increase
  in the commercial paper rate.
• This implies a one month rate of 4.35 37.5BP
  4.725

64
Results Futures

• Assume that on June 1 the 3 month LIBOR rate
  increases to 5.5 (as it did in our previous
  example), also assume that the futures contract
  price falls to 94.75.
• Closing out the Futures contract resulted in a
  profit of 2,500 per 1million. Since we have 9
  1 million contracts our profit is
• 9(2,500)22,500

65
Results Spot

• LIBOR increased by 1.75 or 175 basis points,
  assuming our hedge ratio is correct this implies
  a .75(175) 131.25 basis point increase in the
  one month commercial paper rate.
• So the new expected one month commercial paper
  rate is 4.351.3125 5.6625
• However assume that the relationship was not
  perfect ant the actual one month rate is 5.75

66
Results

• Given the 5.75 commercial paper rate the cost of
  borrowing has increased by
• 36,000,00(.0575-.0435)(30/360) 42,000
• Subtracting our profit of 22,500 in futures
  market the net increase in borrowing cost is
• 42,000 - 22,500 19,200
• This is equivalent to an increase of
• 36,000,000(X)(30.360) 19,500 X 65 BP

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Results

• Using the 65 BP increase Ajax ended up paying 5
  for its borrowing.
• The treasurer was attempting to lock in 4.725 or
  27.5BP less than what she ended up paying.
• The 27.5 BP difference is the result of basis
  risk.

68
Basis Risk

• Source 1
• June 1 spot LIBOR was 5.5 the LIBOR rate implied
  by the futures contract was 5.25 a 25 BP
  difference
• Given the hedge ratio of .75 this should be a
  25(.75) 18.75 BP difference for commercial
  paper
• Source 2
• Expected 1 month commercial paper rate is
  5.6625, actual is 5.75 a 8.75 BP difference

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Basis Risk

• The result of the two sources of risk
• 18.75 8.75 27.5 basis points

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Duration The Big Picture

• Calculation Given the PV relationships, we need
  to weight the Cash Flows based on the time until
  they are received. In other words we are looking
  for a weighted maturity of the cash flows where
  the weight is a combination of timing and
  magnitude of the cash flows

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Calculating Duration

• One way to measure the sensitivity of the price
  to a change in discount rate would be finding the
  price elasticity of the bond (the change in
  price for a change in the discount rate)

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Duration MathematicsMacaulay Duration

• Macaulay Duration is the price elasticity of the
  bond (the change in price for a percentage
  change in yield).
• Formally this would be

73
Duration Mathematics

• Taking the first derivative of the bond value
  equation with respect to the yield will produce
  the approximate price change for a small change
  in yield.

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Duration Mathematics
The approximate price change for a small change
in r
75
Duration MathematicsMacaulay Duration
substitute
76
Macaulay Duration of a bond
77
Duration Example

• 10 30 year coupon bond, current rates 12, semi
  annual payments

78
Example continued

• Since the bond makes semi annual coupon payments,
  the duration of 17.3895 periods must be divided
  by 2 to find the number of years.
• 17.3895 / 2 8.69475 years
• Another interpretation of duration is shown here
  Duration indicates the average time taken by the
  bond, on a discounted basis, to pay back the
  original investment.

79
Using Duration to estimate price changes
Rearrange
Change in Price
Estimate the price change for a 1 basis point
increase in the yearly yield
Multiply by original price for the price change
-0.000820257(838.8357)-.688061
80
Using Duration Continued

• Using our 10 semiannual coupon bond, with 30
  years to maturity and YTM 12
• Original Price of the bond 838.3857
• If YTM 12.01 the price is 837.6986
• This implies a price change of -0.6871
• Our duration estimate was -0.6881 a difference of
  .0010

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Note

• Previously yield increased from 12 a year to
  12.01.
• We used the Duration represented in years,
  8.69475
• We could have also used duration represented in
  semiannual periods, 17.3895. The change in yield
  needs to be adjusted to .0001/2 .00005 however,
  the original yield (1r) stays at 1.06.

The estimated price change is then the same as
before -0.000820257(838.8357)-.688061
82
Modified Duration
The Change in price was given above as
Substitute DMOD
83
Modified vs Macaulay Duration
84
Duration - Continuous Time

• Using continuous compounding the bond value
  formula becomes
• And the Duration equation becomes

85
Change in Bond Price Continuous Time

• The estimated percentage change in the price of
  the bond is then given by letting value (V)
  price (P)
• By rearranging the actual price change is then

86
Duration Hedging

• You can also estimate the hedge ratio using
  duration.
• We know that the change in price can be estimated
  using duration. Assume that we have a bond
  portfolio with duration equal to DP
• DP-PDPDy
• Likewise the change in the asset underlying a
  futures contract should be estimated by
• DF-FDFDy

87
Duration Hedging

• You can combine the two to produce a position
  with a duration of zero.
• The optimal number of contracts is
• Must assume a bond to be delivered

88
Tailing the Hedge

• Adjustments to the margin account will also
  impact the hedge and need to be made.
• The idea is to make the PV of the hedge equal the
  underlying exposure to adjust for any interest
  and reinvestment in the margin account.
• For N contracts this becomes Ne-rT contracts
  where r is the risk free rate and T is the time
  to maturity.

89
Duration Hedging

• You can also estimate the hedge ratio using
  duration.
• We know that the change in price can be estimated
  using duration. Assume that we have a bond
  portfolio with duration equal to DP
• DP-PDPDy
• Likewise the change in the asset underlying a
  futures contract should be estimated by
• DF-FDFDy