Derivatives - Matt Will Web Page

Transcript Derivatives - Matt Will Web Page

Lecture 7

  Topics Pricing Delivery   Complications for both Multiple assets can be delivered on the same contract…unlike commodities The deliverable assets all have different prices

Product “Eligible” Maturity Face Amount Min. Tick Values Copyright: CME Group 2011

    Cheapest to Deliver Delivery = Treasury futures allow the short position to select which bond to deliver (or sell) to the long futures position. The short will deliver the bond which is the least costly for the short position to purchase.

This occurs since only 4 contracts are used to hedge all interest rate instruments. Thus, a real underlying asset does not exist.

Certain bonds are “eligible” for delivery

Copyright: Bloomberg Financial Services 2015

   Conversion Factor Bond prices vary for many reasons ◦ Higher coupons have higher prices ◦ Lower coupons have lower prices ◦ Longer maturities have higher prices ◦ Shorter maturities have lower prices If you deliver a more expensive bond, the amount you receive at delivery goes up If you deliver cheap bond, the amount you receive at delivery goes down

    Quoted price = Price of the bond as quoted in the paper Accrued interest = amount of coupon earned on a bond since the last coupon payment Bond Cash Price = (Quoted price of bond X notational amount) + accrued interest Invoice Amount = Amount of money that is exchanged when a futures contract bond is delivered

Accrued Interest  Days since last coupon Total days in period  Coupon paid during period

 Example What is the cash price of a bond that pays a 4% semiannual coupon and matures in 12 years and three months, if the YTM is 6.5%?

Price FV = 1000 Pmt = 20 int = 3.25

n = 24.50 Solve for PV = $781.20 Quoted Price = 78.12

 Example (continued) What is the cash price of a bond that pays a 4% semiannual coupon and matures in 12 years and three months, if the YTM is 6.5%?

Accrued Interest Bond Cash Price  90 180  10  20

Price



7 81.20



10



$ 791 .

20

 Conversion Factor Since the bond we deliver is not specified in the futures contract, the price of the bond must be standardized.   The conversion factor converts the futures price into a settlement or invoice price.  The conversion factor is the present value of $1 at YTM=6%, assuming coupons are paid semiannual. Repo Rate Difference between the conversion factor yield of 6% and the coupon on the bond.

 Used to convert futures prices to bond prices

 Quoted Price of bond @ YTM 100 = 6%  What is the cash price of a bond that pays a 4% semiannual coupon and matures in 12 years and three months, if the YTM is 6.5%?



81.837

100



0 .

81837

Using exact dates on a HP12c provides 82.824

 Also called the Adjusted Futures price  Cash Price = Futures Price x Conversion Factor Futures Price = Cash Price / Conversion Factor

Invoice Amount = Futures Price x Conversion factor x Contract Size + accrued Interest Total amount of money exchanged at delivery

   The price of a treasury futures contract. The price is merely the future value of the spot price of the treasury, less PV of the coupons. This assumes a flat yield curve. 

= present value of coupons

0  (

0 

)

e rT

 Example Compute the conversion factor of a bond with exactly 9 years to maturity a 5% coupon, paid semiannually, and a YTM of 4.8%.

 Quoted Price of bond @ YTM 100 = 6%  93 .

12 100  .

9312

 Example (continued) Compute the quoted price of the bond with exactly 9 years to maturity a 5% coupon, paid semiannually, and a YTM of 4.8%.

Price FV = 1000 Pmt = 25 int = 2.4

n = 18 Solve for PV = $1014.48

Quoted Price = 101.45

 Example (continued) Compute the price of the 9 month futures contract. Remember the next coupon payment will be made in 6 months.

 2 .

 (.

048  .

50 )  2 .

0  ( 101 .

45  2 .

44 )

048  .

75  102 .

How To Calculate Delivery Cost (steps) 1 - Look up the price (FP) 2 - Compute “Conversion Factor” (CF) 3 - CF x FP x (contract size) + (accrued interest) = Delivery cost

The CTD can be found three ways 1.

Quoted Bond Price – (Futures Price x CF) Also called the “Gross basis” Select the lowest Invoice Amount (lowest) Also called the “Delivery Cost” Highest Repo Rate The interest rate earned by short selling a security and buying it back later.

Theoretical Futures Price (FP)?

 Price of bond  ?

Accrued interest and others items

3 Ways to Derive CTD 1 – Highest Repo Rate ( The interest rate earned by short selling a security and buying it back later. ) 2 - Calculate Futures Delivery Spot Price 3 - Cost of Delivery (“Gross Basis”) 

QP CF



 [



]

Example Two bonds are eligible for delivery on the June 2012 T Bond Futures K 1 - 9.875Nov38 2 - 7.25May39

deliveries on 15th of maturity month On June 12, you announce to deliver a bond

Q: If YTM = 5%, which will you deliver and what is its price?

A: CF 9.875Nov38 1.51

7.25May39

1.17

Bond Price 171.05

133.09

Deliver 9.875 Nov38 FC Spot Price 113.28

113.75

Q: If YTM = 9%, which will you deliver & what is its price?

A: CF 9.875Nov38 1.51

7.25May39

1.17

Bond Price 108.76

82.36

Deliver 7 1/4 May39 FC Spot Price 72.03

70.39

Q: If YTM = 7% and the listed futures price is 110.50, which bond is CTD?

A: 9 7/8Nov38 CTD = 134.39 - (110.5 x 1.51) = -32.47

7 1/4May39 CTD = 103.00 - (110.5 x 1.17) = -26.29

Implied Repo Rate Cost of Carry

1 - The Duration Model 2 - Naive Hedging Model 3 - Conversion Factor Model 4 - Basis Point Model 5 - Regression Model 6 - Yield Forecast Model

 Duration Model HR = Cash Price Futures Price  Duration Cash Duration CTD  1 

Cash 1 + Er CTD 1 

Cash 1 + Er CTD  Usually tossed out due to poor forecsating

Duration Model    Your cash position is $1,000,000 10% coupon, 26year bonds, with YTM=12.64% and duration of 8.24 years. The 6%, 20year, TBill has a duration of 10.14 years, YTM=8.5% The FC on this bond is priced at 96.87

HR = 79.98x8.24 = 659.04 = .671

96.87x10.14 982.26

(1,000,000 / 100,000) x .671 = 6.71 or 7 contracts

   Duration Example In 3 months, you will receive $3.3 mil in cash and must invest it for 6 months. The current 6 month rate is 11.20%. You like that rate, and wish to lock it in. 6 month tbills have a .50 duration, while 3 month bills have a .25 duration. If the 3 month futures price is 97.36, what number of Ks are required to lock in the rate?

HR = 100 x .5 = 2.05 x (3.3 / .1) = 67.8 contarcts 97.36 x .25

Naive Model HR = 1.0 (all previous examples were naive hedges) Conversion Factor Model HR = conversion factor CF = Price of deliverable bond @ 6% YTM 100

Conversion Factor Model Example  You own a $1mil portfolio you wish to hedge. Your are considering a 3 month futures K. The bond that could be delivered against the contract is a 9.5%(semiannual) bond with a 30year maturity. The bond is callable in 15 years. How many Ks should you use to hedge the position?

CF = 134.30/100 = 1.34 x (1mil/.1) = 13 contracts

   Example - Conversion Factor Model You have a $1mil portfolio, containing 21.5 year 10 3/8 bonds. Price = 100.5363 (YTM = 10 5/16) CTD 20year, 8% bond has YTM = 10.43

Create the hedge.

  Assume that in 6 months YTM on your portfolio rises to 12 % and YTM on CTD rises to 12.217% Create a table showing your position/profit/loss

Example - Conversion Factor Model CF = PV of 5.1875 @ 3% for 43 periods / 100 = 1.52

1.52 x (1mil/100,000) = 15 Today 6 mths Cash Own $1mil @ 100.5363

($1,005,363) Sell @ 87.63

+ $876,301 (129,062) Futures Short 15 K @ 79.718 (given) + $1,195,770 buy 15 K @ 71.07 (given) ($1,066,050) +129,720

Basis Point Model



BVC CASH BVC CTD CF CTD B

# of Ks BVC cash = $ change in value per basis point of B = Relative yield volatility of cash to CTD = (V cash cash position / V ctd ) BVC ctd = $ change in value per basis point of CTD CF ctd =conversion factor of CTD

    Example YTM = 9% on semi-annual bonds Your cash portfolio consists $1mil of 26 year 9 7/8 bonds, that have a yield volatility of .60

Futures CTD is a 7.25% 26.5 year note with a yield volatility of .50

Use the basis point model to create a hedge and show the position table for a 3month time period and a change in YTM to 10%.

Basis Point Model Bond Value CTD Value 9

108 .

737 82.442

108 .

630 82.356

BVC

107 86 Use Calculator bond functions for calculations

example - continued Cash value @ 9% = 108.737

BVC cash = $107 (PV @ 9% - PV @ 9.01) BVC ctd = $86 B = .6 / .5 = 1.20

CF = .1.16 (PV of CTD @ 6% / 100) HR* = ( 107 ) x1.20 = 1.73

( 86 / 1.16) 1 mil / 100,000 x 1.73 = 17 contracts

example - continued (10%) Cash Today $1mil @ 108.737

-$1,087,370 3 months (YTM = 10%) $1 mil @ 98.82

+$ 988,212 Net Position $99,158 loss net gain of $2,672 Futures 17K @ 82.44 (given) +1,401,480 17K @ 76.45 (given) - $1,299,650 $101,830 gain

example - continued Assume YTM = 8% Cash Today $1mil @ 108.737

-$1,087,370 3 months (YTM = 8%) $1 mil @ 120.30

+$ 1,203,034 Net Position $115,664 gain net loss of $5,376 Futures 17K @ 82.44 (given) + 1,401,480 17K @ 89.56 (given) - $1,522,520 $121,040 loss

Regression Model HR = Covariance of Cash & Futures Variance of futures best model if HR = .90, then we know that a $1 change in futures prices correlates to a $0.90 change in cash value.

requires constant monitoring because HR changes with duration

Yield Forecast Model Given various yield forecasts, the HR changes Term Structure can forecast yields HR = CVdiff / FCV diff Example Cash Value = 97.94 & Futures = 72.50

Forecasted YTM YTM CV YTM FC 12.65

11.25

12.85

13.55

13.75

11.40

12.05

12.20

CV 101.72

100.14

94.99

93.62

FC 75.06

74.14

70.37

69.54

CVdiff 3.77

2.20

-2.95

-4.33

FCdiff 2.56

1.64

-2.13

-2.96

HR 1.48

1.34

1.36

1.47