Topics Pricing Delivery Complications for both Multiple assets
can be delivered on the same contractunlike commodities The
deliverable assets all have different prices
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Copyright: CME Group 2011 Product Eligible Maturity Face Amount
Min. Tick Values
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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
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Copyright: Bloomberg Financial Services 2015
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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
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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
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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
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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 InterestBond Cash Price
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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.
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Used to convert futures prices to bond prices 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%? Using exact dates on a
HP12c provides 82.824
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Also called the Adjusted Futures price Cash Price = Futures
Price x Conversion Factor Futures Price = Cash Price / Conversion
Factor
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Invoice Amount = Futures Price x Conversion factor x Contract
Size + accrued Interest Total amount of money exchanged at
delivery
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Futures Price Calculation 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. I = present value of coupons
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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%.
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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
= $1014.48Quoted Price = 101.45
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Example (continued) Compute the price of the 9 month futures
contract. Remember the next coupon payment will be made in 6
months.
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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
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The CTD can be found three ways 1. Quoted Bond Price (Futures
Price x CF) Also called the Gross basis Select the lowest 2.
Invoice Amount (lowest) Also called the Delivery Cost 3. Highest
Repo Rate The interest rate earned by short selling a security and
buying it back later.
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Theoretical Futures Price (FP)? 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) Accrued interest and
others items
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Example Two bonds are eligible for delivery on the June 2012 T
Bond Futures K 1 - 9.875Nov38 deliveries on 15th of maturity month
2 - 7.25May39 On June 12, you announce to deliver a bond
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Q: If YTM = 5%, which will you deliver and what is its price?
A: CFBond PriceFC Spot Price 9.875Nov381.51171.05113.28
7.25May391.17133.09113.75 Deliver 9.875 Nov38
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Q: If YTM = 9%, which will you deliver & what is its price?
A: CFBond PriceFC Spot Price 9.875Nov381.51108.7672.03
7.25May391.1782.3670.39 Deliver 7 1/4 May39
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Q: If YTM = 7% and the listed futures price is 110.50, which
bond is CTD? A: 9 7/8Nov38CTD = 134.39 - (110.5 x 1.51) = -32.47 7
1/4May39CTD = 103.00 - (110.5 x 1.17) = -26.29 Implied Repo Rate
Cost of Carry
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1 - The Duration Model 2 - Naive Hedging Model 3 - Conversion
Factor Model 4 - Basis Point Model 5 - Regression Model 6 - Yield
Forecast Model
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Duration Model
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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
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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
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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
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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
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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
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Example - Conversion Factor Model CF = PV of 5.1875 @ 3% for 43
periods / 100 = 1.52 1.52 x (1mil/100,000) = 15 CashFutures
TodayOwn $1mil Short 15 K @ 100.5363@ 79.718 (given) ($1,005,363)+
$1,195,770 6 mthsSell @ 87.63buy 15 K @ 71.07 (given) +
$876,301($1,066,050) (129,062)+129,720
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Basis Point Model BVC cash = $ change in value per basis point
of cash position B = Relative yield volatility of cash to CTD = (V
cash / V ctd ) BVC ctd = $ change in value per basis point of CTD
CF ctd =conversion factor of CTD
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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%.
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Basis Point Model Use Calculator bond functions for
calculations
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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
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example - continued (10%) CashFutures Today $1mil @ 108.73717K
@ 82.44 (given) -$1,087,370+1,401,480 3 months (YTM = 10%) $1 mil @
98.8217K @ 76.45 (given) +$ 988,212- $1,299,650 Net Position$99,158
loss$101,830 gain net gain of $2,672
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example - continued Assume YTM = 8% CashFutures Today $1mil @
108.73717K @ 82.44 (given) -$1,087,370+ 1,401,480 3 months (YTM =
8%) $1 mil @ 120.3017K @ 89.56 (given) +$ 1,203,034- $1,522,520 Net
Position$115,664 gain$121,040 loss net loss of $5,376
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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
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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
CVYTM FCCVFCCVdiffFCdiffHR 12.6511.25101.7275.063.772.561.48
12.8511.40100.1474.142.201.641.34
13.5512.0594.9970.37-2.95-2.131.36
13.7512.2093.6269.54-4.33-2.961.47
