tbond

Long-term Interest Rates Futures

Phenomenal growth

Why?

Increase in bonds issued by governments
(Hedge interest rate risk associated with cash positions)

CASH T-BONDS
NOTES - 10 YRS OR LESS
BONDS - > 10 YRS TO 30 YRS
MARKETS - PRIMARY - U.S. TREASURY AUCTION
SECONDARY - AUTHORIZED DEALERS (BANKS & SECURITY FIRMS)
OVER THE COUNTER TELEPHONE MARKET

THREE BOND RULES(MALKIEL)

1. BOND PRICES MOVE INVERSELY TO INTEREST RATES

2.FOR LIKE BONDS (RISK, CALL CHARACTERISTICS, & MATURITY) EXCEPT COUPON, SMALL COUPON BOND HAS MORE PRICE VARIABILITY THAN LARGE COUPON BOND

3.FOR LIKE BONDS(COUPON, RISK, CALL CHARACTERISTICS) EXCEPT MATURITY, LONG-TERM BOND HAS MORE PRICE VARIABILITY THAN SHORT-TERM BOND

BOND PRICING FORMULA
n
P₀ = sum C_t/(1 + r)^t

WHERE P₀ = CURRENT PRICE
                C=CASH FLOW @t
                r=Yield to maturity
                t=TIME PERIOD
                n=number of periods until maturity

EXAMPLE

P = sum C_t/(1 + r)^t WHAT IS PRICE OF A THREE YEAR 6% ANNUAL NOTE($1000) WHEN
THE YIELD TO MATURITY IS 8%?
P = [60/1.08] + [60/(1.08)²] + [60/(1.08)³] + [1000/(1.08)³]
P = 55.55 + 51.44 + 47.63 +793.83 = 948.45

LIKEWISE, CAN USE FORMULA TO SOLVE FOR YIELD TO MATURITY.

EXAMPLE
PRICE OF 4 YEAR $10,000 NOTE WITH 4% ANNUAL COUPON IS $9,462. WHAT IS YIELD TO MATURITY?
$9,462= 400/(1+r)¹ + 400/(1+r)² + 400/(1+r)³ +10,400/(1+r)⁴
r = 5.53615%

Quoted Bond Prices

Treasury bonds, December 13, 1989

rate         maturity        bid             ask             bid change yield
12.00     may05k        134-17       134-23       +06           8.03
10.75     aug05k         123-25       123-31       + 04          8.03
9.37       feb06k          112-15       112-21       + 04         7.97
11.75     feb05-10k    131-29       132-03       + 05          8.05
7.25       may16k        92-12         92-16         + 05          7.93

asked price - price at which bonds purchased
bid price - price at which bond sold
bid-asked spread - compensation

prices quoted as percentage of par value in 32'nds

example - may 2016 bond w/ coupon of 7.25 has ask price of 92-16. means 92% of par + 16/32 of % point of par value

purchase price = $92,500
(.92 X 100000) + (.01 x 100000 X 16/32)
bid change = + 05
                 = 5/32
                 = 5 X $31.25
                 = $156.25

Bond Transaction Price
Accrued interest

ACCRUED INTEREST

[(B - tc)/ B] x coupon(k)

where B = total number of days in coupon period
tc= number of days until next coupon payment
P = Quoted price + accrued int.

example - 7.25% May2016 bond pays $3.625 per $100 face value on Nov 15 and May 15 of each year. On Dec 13, Quoted price is 92 16/32. Settle date is Dec 14 and bond has 152 days until next coupon payment.

P = 92,500 + [3625 X 29/181]
= 92,500 + 580.80
= 93,080

Duration

summary measure of price sensitivity to interest rate changes

High duration bond is more sensitive to interest rate changes than lower duration bond

Bonds
coupon         maturity         duration
low                 long                high
high                 short               low

Maculay's Duration
(assumes flat term structure)

weighted average of the maturities of the coupons and principal repayment cash flows, where weights are the fractions of the price that the cash flows in each period represent.

Maculay's Duration

m
D = 1/P x sum [(t*X_t)/(1+r)^t]
t=1

where D=duration of bond
            X_t=payment on bond in period t
            r=annual yield to maturity divided by number of payments in year
            m=number of payment periods
            P=transaction price of bond

Measures duration in periods, not years. To convert to duration in years, multiply by 1/f where f is the number of coupon payments/year.

Example
Two year note with 8% coupon(k), an annual yield to maturity of 10%, and a market price of 98.

D= 1/98[(1x4)/(1+.05)¹+ (2x4)/(1+.05)² + (3x4)/(1+.05)³ + (4x4)/(1+.05)⁴ + (4x100)/(1+.05)⁴]
    =1/98[(4/1.05)+(8/1.1025)+(12/1.1576)+(16/1.2155)+(400/1.2155)]
    =1/98[3.8095+7.2562+10.366+13.1633+329.0827]
    =3.7109
For annual duration multiple by 1/2. 3.7109/2 = 1.855

Modified Duration

provides percentage measure of price volatility

D_m= D/ (1+r/f)
    = 1.855/(1+(.10/2))
    = 1.855/1.05
    = 1.7671

modified duration measures the percentage change in a bond's price relative to a given percentage change in yield to maturity

-D_m = % of change in P/% of change in (1+r)

we can use this identity to forecast.

% of change in P = -D_m x (% of change in (1+r))

for example, if yield to maturity increases by .5%
then:

% of change in P = -1.7671 x [(.005/1.10) x 100%]
= -1.7671 x .4545
= -0.8031%

Thus, price of two-year note will decrease by
98 x .008031 = .79
= 25/32
price will be 97 7/32

U.S. T-Bond Futures Contract

One of more complex contracts
delivery provisions
wide variety of deliverable bonds

delivery of $100,000 face value of U.S. Treasury bonds maturing at least 15 years from delivery date with notional coupon of 6%. This contract used to have a notional coupon of 8%.

March, June, Sept, Dec
delivery dates

Delivery Process

Short initiates delivery process by choosing bond to deliver and when to deliver it during delivery month.

Delivery Process is 3 day sequence.

(1)position day - short declares intention to deliver(8:00pm).

(2)notice day - clearinghouse matches short with oldest outstanding long and notifies both parties. Short states what bond will be delivered and invoices long. Uses settlement price on position day.

(3)delivery day - exchange takes place

Deliverable bonds

To adjust for different market values of bonds, conversion factors are used to adjust invoice prices.

Delivery Options

Quality - right to choose which eligible bond to deliver. Short can maximize return by delivering cheapest-to-deliver bond.

timing - which day of month

wild card - 1)position day, close at 2:00 notify by 8:00
2)short can wait until 5:00 on notice day to choose bond.

Futures Price Quotations

Invoice Prices

(Decimal futures settlement price x Conversion factor x 100000) + accrued interest on deliverable bond

Example - On Dec. 13, 1989, short Dec 89 delivers the 14% Nov 2006-11 bond against short position. Dec89 t-bond futures settled at 99 16/32 on Dec 13(.995).
Conversion factor is 1.5481.
Accrued interest on Nov2006 bond is (100000x.07x(29/181)) = $1121.55

Invoice price is:
[.995 x 100000 x 1.5481] + 1121.55
$155,157.50

This price will generate 8% yield to maturity. Remember today the conversion factor will result a price that generates a 6% yield to maturity.

Pricing T-Bond Futures

T-bond futures prices should equal adjusted cash prices of cheapest-to-deliver bond plus net cost of carry on that bond.

At delivery, theoretical futures prices is :

FP_t,T= CP^*_t/CF^*

where Fp_t,T = theoretical futures price(decimal) at time t for a contract deliverable at time T.
CP^*_t= quoted cash price (decimal) of cheapest to deliver bond at time t
CF^* = conversion factor on cheapest to deliver bond

Any time prior to expiration, theoretical futures price is:

Fp_t,T= CP^*_t/CF^*+C^*_t,T

where C^*_t,T = net carrying cost from t to T of cheapest to deliver bond.

Cheapest to Deliver bond

Duration

If yields are > 6%, deliver bond with highest duration

If yields are < 6%, deliver bond with lowest duration (assumes flat yield curve)

Cheapest to Deliver bond

Implied Repo Rate(IRR)

Highest IRR = cheapest to deliver bond

IRR represents the riskless rate of return that can be earned on a cash and carry arbitrage without borrowing to finance it

Hedging with T-Bond Futures

1. Determine hedge ratio
        HR - change in CP/ change in FP
        Use duration or regression
    Duration - formula is
        (D_spot)     value_spot(1+Y_fut)
HR= ____   * ________ * _______   * (CF^*)
        D_fut         value_fut(1+Y_spot)

Regression
Use beta from regression of change in spot price as function of change in futures price.

2. Enter position

Using Duration to determine Hedge Ratios

HR=(CF^*)- D_cpxCp_tx(1+R_cp^*)x change in R_cp
_{__________________________________}
-D_cp^*xCp_t^*x(1+R_cp)x change in R_cp^*

        where CF^* = conversion factor for cheapest to deliver bond.
        D_cp= duration for cash T-Bond
        Cp_t= cash price of cash T-Bond
        R_cp^*=annual yield to maturity in cheapest to-deliver bond
        Change in R_cp= change in annual yield to maturity of cash bond

December 13, 1989, an investor buys $10 million of 12%Aug2013 bond.To protect against movements in interest rates, he goes short March 90 T-bond futures. Hedge ratio=1.33

12%Aug2013 bond has price of 137.6875, yield of 8.05%, and duration of 8.92. The cheapest to deliver bond is 10.375%Nov 2012 bond with price of 121.8125, yield of 8.04, duration of 9.288, and conversion factor of 1.2216.

Assuming the changes in yields are equal then

HR=1.2216 x(-8.926x137.6875x(1+.0804)
-9.288x121.8125x(1+.0805)
HR = 1.33

December 13, 1989, an investor buys $10 million of 12%Aug2013 bond.To protect against movements in interest rates, he goes short March 90 T-bond futures. Hedge ratio=1.33

date             cash                         futures
12/13           long 10m                 Short133
                    12%Aug2013         MarTbond
                    @137.68                99 17/32
2/13             short 10m                 long133
                    12%Aug2013           96 22/32
                    134.12
____________________________________
                    -3.56                         +2 27/32

Net     100x3.56x1000                 133x31.25x91
            -356,000                           378,218.75
            +22,218.75