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• For T-bonds, a tick is 1/32nd. The resulting quote of 112-15 equals 112 and 15/32.
• But, for 5 and 10-year T-notes, a tick is ½ of a 32nd, or $15.625 per tick. The resulting quote, say, of 98.095 = 98 and 9.50/32.
• For 2-year T-Notes, however, tick sizes are ¼ of a 32nd. So,– A quote of 92.072 = 92 and 7.25/32.– A quote of 109.017 = 109 and 1.75/32. – But, the contract size is $200,000 of deliverable T-Notes, so a tick
= $15.625.
• For CBOT futures prices for T-bonds and T-notes, see: http://www.cbot.com/cbot/www/page/0,1398,12+31,00.html .
Using T-bond and T-note Futures to Hedge Interest Rate Risk
• Buy T-bond or T-note futures to hedge against falling interest rates. Sell them to hedge against rising interest rates. (Remember that when interest rates fall, bond prices rise, and when interest rates rise, bond prices fall.)
• Use T-bond futures to hedge against changes in long-term (15+ years) rates. Use 10-year T-note futures to hedge against changes in 8-10 year rates.
• Estimate the loss in value if the spot YTM adversely changes by one basis point, denoted VS.
• Estimate the YTM of the CTD Treasury security if the spot YTM changes by a basis point; assume the CTD’s YTM will change by ‘b’ basis points. Compute the change in the CTD’s price if its YTM changes by b basis points. Denote this as SCTD.
• Estimate the change in the futures price if the CTD’s price changes by SCTD, denoted F per $100 face value. It can be shown that:
• The profit, VF, is then $1000 F.• Compute the number of futures contracts to trade, N, so that
• U.S. Treasury bonds and notes are coupon bonds. Their values are computed using:
• C is the semiannual coupon payment.• F is the face value.• Y is the unannualized, or periodic, 6-month yield.• N is the number of 6-month periods to maturity.• This assumes that the first coupon payment is 6 months hence.
• To calculate the value of a bond , one must discount each cash flow at the appropriate zero rate.
• For example, let r(0,t) be the annual spot rate for a zero coupon bond maturing t years from today. Then,– Let r(0,0.5) = 2%, r(0,1) = 2.5%, r(0,1.5) = 3%, and r(0,2) = 3.3%.
– The semiannual coupon amount is $25, and face value is $1000.
• U.S. T-bond and T-note prices are in percent and 32nds of face value.
• For example (see fig. 9.1), on 12/01/00, the bid price of the 6 3/8% of Sep 01 T-note was 100 and 4/32% of face value.
• If the face value of the note is $1000, then the bid price is $1001.25.
• The asked price of this note is $1001.875.
• N.B. These prices are based on transactions of $1 million or more. In other words, a trader could buy $1 million face value of these notes for about $1,001,875 from a government securities dealer.
• These prices are quoted flat; i.e., without any accrued interest.
On December 1, 2000, the 6 3/8% of September 2001 was quoted to yield 6.12%.
• You can verify by using the YIELD function in Excel:=YIELD("12/01/00","9/30/01",0.06375,100.1875,100,2,1)
• To Calculate the cash price (quoted price + AI), you would pay:– The ask price is 100:06, or $1001.875 for a $1000 par bond.– Coupons on this note are paid every March 31 and September 30.– The last coupon was paid on September 30, 2000, which is 62 days
before December 1, 2000. – There are 182 days between the last coupon payment date and the
next one on March 31, 2001. – Interest accrues on an actual/actual basis. Thus the buyer pays the
seller accrued interest equal to:
AI = (62/182)*(63.75/2) = $10.859
The cash price (quoted price plus accrued interest) is: QP + AI = $1001.875 + $10.859= $1012.73.
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Hedging With T-Bond Futures: Changing the Duration of a Portfolio
• Hedging decisions are essentially decisions to alter a portfolio’s duration.
• By buying or selling futures, managers can lengthen or shorten the duration of an individual security or portfolio without disrupting the underlying securities (an “overlay”).
• That is, adding (buying) T-bond or T-note futures to a portfolio increases its interest rate sensitivity, while selling futures decreases the interest rate sensitivity of the portfolio.
• A portfolio manager will want to decrease (increase) the duration of the portfolio if the manager expects interest rates to increase (decrease).
• A completely hedged portfolio lowers the duration to the duration of a short-term riskless Treasury bill.
• The bond portfolio manager can change the duration of the existing portfolio to the duration of a target portfolio. This “immunizes” the portfolio against a change in interest rates.
• That is, if the portfolio manager knows:– how the current and target portfolios respond to interest rate
changes. – how T-bond (or T-note) futures contracts respond to interest
rate changes.
• Fortunately, if interest rates change by a small amount, say one basis point, the value of the portfolio will change predictably.
• Using the bond pricing formula, the duration formula, and some algebra, the change in the value of a bond or a portfolio of bonds if interest rates change by one basis point can be written:
• When dy = 0.0001 (1 basis point), then dB is called the basis point value (BPV).
• The portfolio manager chooses a target duration so that it will have a particular BPV; i.e., a targeted change in value if interest rates change by one basis point.
• Assuming that the CTD bond and the bond portfolio will both experience a one basis point change in yield, the goal is to choose to buy or sell NF futures contracts so that
BPV (target) = NF BPV (futures) + BPV (existing)
• Thus, the BPV of the existing portfolio, the target portfolio, and the futures contract must be computed.
• Facts:– On December 1, 2000, a fixed-income portfolio manager
expects a steep decline in bond yields over the next six weeks.
– Because of these strong beliefs and “aggressive” management, the manager decides to more than double the duration of his fixed-income portfolio.
– The manager wants to avoid disrupting his carefully constructed bond portfolio to profit from the belief that interest rates will decline only over the next six weeks.
– Therefore, the manager decides to buy T-bond futures to increase the duration.
2. Calculate the BPV of the T-Bond futures contract.
– It has a face value of $100,000.– Using well-known techniques, one can determine that the
CTD T-bond on December 1st for March futures is the 8.875% of August 2017.
– On December 1st, the conversion factor of this T-bond is 1.2957, duration is 9.83, and YTM is 5.80%. So,if interest rates change by one basis point, then the value of the CTD T-Bond will change by
• Delivery dates exist every 3 months.• Delivery months are in March, June, September, and
December. • On December 1, 2000, the Dec T-bond settle price is 102-02.
This equals 102 and 2/32% of face value, or $102,062.50. • The December 2000 futures price was down 17 ticks, or 17/32. • This means that on December 14, 2000, the December contract
settled at 102-19. • A price change of one tick (1/32) will result in a daily
• At its website, the CBOT lists deliverable T-bonds and T-notes, by delivery date.
• For example, as of November 29, 2000, there were 34 T-bonds deliverable into nearby T-bond futures contracts.
• By allowing several possible bonds to be delivered, the CBOT creates a large supply of the deliverable asset.
• This makes it practically impossible for a group of individuals who are long many T-bond futures contracts to “corner the market” by owning so many T-bonds in the cash market that the
• Recall that the invoice price equals the futures settlement price times a conversion factor plus accrued interest.
• The conversion factor is the price of the delivered bond ($1 par value) to yield 6 percent.
• The purpose of applying a conversion factor is to try to equalize deliverability of all of the bonds.
• If there were no adjustments made, the short would merely choose to deliver the cheapest (lowest priced) bond available.
• In theory, if the term structure of interest rates is flat at a yield of 6% then, by applying conversion factor adjustments, all bonds would be equally deliverable.
• In practice, however, there is a “cheapest to deliver” or CTD T-bond. This is the T-bond used to price futures contracts on T-bonds.