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

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

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

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 = Solve for PV = $ 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 InterestBond Cash Price

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  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

 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

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

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%.

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 = $ Quoted Price =

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

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 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.

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

Example Two bonds are eligible for delivery on the June 2012 T Bond Futures K Nov38 deliveries on 15th of maturity month May39 On June 12, you announce to deliver a bond

Q: If YTM = 5%, which will you deliver and what is its price? A: CFBond PriceFC Spot Price 9.875Nov May Deliver Nov38

Q: If YTM = 9%, which will you deliver & what is its price? A: CFBond PriceFC Spot Price 9.875Nov May Deliver 7 1/4 May39

Q: If YTM = 7% and the listed futures price is , which bond is CTD? A: 9 7/8Nov38CTD = (110.5 x 1.51) = /4May39CTD = (110.5 x 1.17) = 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

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 years, YTM=8.5%  The FC on this bond is priced at HR = 79.98x8.24 = = x (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 x.25

Naive Model HR = 1.0 (all previous examples were naive hedges) Conversion Factor Model HR = conversion factor CF = Price of deliverable 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 = /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 = (YTM = 10 5/16)  CTD 20year, 8% bond has YTM =  Create the hedge.  Assume that in 6 months YTM on your portfolio rises to 12 % and YTM on CTD rises to %  Create a table showing your position/profit/loss

Example - Conversion Factor Model CF = PV of 3% for 43 periods / 100 = x (1mil/100,000) = 15 CashFutures TodayOwn $1mil Short (given) ($1,005,363)+ $1,195, buy (given) + $876,301($1,066,050) (129,062)+129,720

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

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 Use Calculator bond functions for calculations

example - continued Cash 9% = BVC cash = $107 9% ) BVC ctd = $86 B =.6 /.5 = 1.20 CF =.1.16 (PV of 6% / 100) HR* = ( 107 ) x1.20 = 1.73 ( 86 / 1.16) 1 mil / 100,000 x 1.73 = 17 contracts

example - continued (10%) CashFutures Today (given) -$1,087,370+1,401,480 3 months (YTM = 10%) $ (given) +$ 988,212- $1,299,650 Net Position$99,158 loss$101,830 gain net gain of $2,672

example - continued Assume YTM = 8% CashFutures Today (given) -$1,087,370+ 1,401,480 3 months (YTM = 8%) $ (given) +$ 1,203,034- $1,522,520 Net Position$115,664 gain$121,040 loss net loss of $5,376

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 = & Futures = Forecasted YTM YTM CVYTM FCCVFCCVdiffFCdiffHR