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i. If yields > 6%, deliver low coupon, long maturity bonds; (worth less) If yields < 6%, deliver high coupon, short maturity bonds.
ii. If yield curve upward sloping, deliver long bonds; If yield curve downward sloping, deliver short bonds.
iii. Some bonds sell for more than their theoretical value; e.g., low-coupon bonds; bonds where coupons can be stripped. These have abnormal demand & will not likely be cheapest to deliver.
c. Wild card play (a timing option). On expiration day at CBOT,
T.Bond futures trading ends at 2 pm; T.Bonds themselves trade until 4 pm.Short futures position can announce intent to deliver 8pm.If bond prices after 2 pm, can announce intent to deliver& then buy cheapest bond for delivery. Otherwise, wait till next day.
d. NOTE: Delivery options are not free; Value is reflected in F (↓F).
2. Eurodollar Futures; (IMM & LIFFE). We’ll focus on this.
a. Eurodollar -- U.S. $ deposited outside the U.S.
b. Eurodollar rate -- LIBOR;
i. Rate earned on ED deposited by one bank with another bank.
ii. Determined by trading of deposits between banks on the Eurocurrency market.
iii. 1-month LIBOR is rate offered by one bank to another on 1-month deposits at a given time. If variable rate loan charges 1-month LIBOR, then rate is reset to 1-month LIBOR at 1-month intervals, with interest paid in arrears.
iv. Other analogous rates are 3-month & 6-month LIBOR.
c. LIBOR is generally higher than corresponding T.Bill rate.LIBOR is a commercial lending rate (not gov't borrowing).
d. Eurodollar futures contract similar to T. Bill futures contract.
e. Some important differences:
i. T. Bill futures contract delivers a 90-day T. Bill. Eurodollar futures contract is settled in cash.
ii. Final marking-to-market sets contract price equal to:
VF = 10,000( 100 - .25R ),
where R = quoted 90-day ED rate (LIBOR) at that time.
iii. Note that this quoted ED rate is the actual 90-day rate on ED deposits with quarterly compounding -- not a discount rate. Hence, ED contract is a futures on an interest rate, while T.Bill contract is a futures on a T. Bill.
4. Example: Long Hedge with ED futures for a Bank.
Jan. 6: Bank expects to receive $1 MM payment on May 11 (4 months). Anticipates investing funds in 3-month ED deposits.
Cash Market risk exposure:
Bank would like to invest @ today’s ED rate, but won’t have funds for 4 months. If ED rate , bank will realize opportunity loss (will have to invest the $1 MM at lower ED rates).
Long Hedge: Buy ED futures today (promise to deposit later @ R).
Then if cash rates , futures rates (R) will & futures prices (Q) will . so long futures position will to offset opportunity losses in cash mkt.
The best ED futures to buy is June contract; expires soonest after May 11.
Jan. 6 May 11 June 14 |__________________________________________|_____________|
$1 MM receivable due May 11. Cash: Plan to invest $1MM on May 11Invest the $1 MM in ED deposits. Futures: Buy 1 ED futures contract. Sell the futures contract.
A measure of how long, on average, bondholder must wait to receive cash.-- A Zero-coupon bond maturing in n years has D = n; A Coupon-bearing bond maturing in n years has D < n; some cash received < n.
A. Definition.
1. Assume: Today is t0. Bond pays coupons, ci, at times ti (1 i n).
Then Bond price, B, and yield, y, are related: B = Σ ci e -y (ti) .
n n ┌─ ─┐ Duration: D = [ Σ ti ci e -y (ti) ] / B = Σ ti │ ci e -y (ti)
/ B │
n=1 n=1
└─ ─┘
2. Notes:a. The term in brackets is NPV( payment at time ti ) divided by bond price, B, which is NPV( all payments ).
b. Thus, D is weighted avg of times when payments are made with the weight for time ti a measure of how important the payment is ( i.e., the proportion of B provided by the payment at time ti ).
1. Consider the following bond: Face Value = $100; C / F = 10% coupon (semi-annual payments of $5); n = 3 years maturity; y = 12% p.a.; (Note: coupon rate < mkt rate)
2. Notes: a. Column 3 shows NPV(payments) using y = 12%.b. Sum of Column 3 gives Bond price.c. Weights in Column 4 = (numbers in Column 3) B.d. Sum of Column 5 gives D.
A. Can express the optimal hedge in terms of Duration, for hedging a money market instrument or a bond portfolio. [Recall, optimal hedge for stock portfolio is in terms of β.]
Define: VF = value of one interest rate futures contract;
DF = duration of asset underlying futures at expiration of futures;
P = forward value of portfolio being hedged at expiration of hedge (assumed same as today’s value of portfolio being hedged);
DP = duration of asset being hedged at expiration of hedge.
Assume: Δy is same for all maturities (only parallel shifts in yield curve).
Then from ΔB = -BDΔy, we have: ΔP = -P DP Δy .
To a reasonable approx, we also have: ΔVF = -VF DF Δy .
For parallel shift in yield curve (Δy), these formulas give anticipated changes in
value of bond portfolio being hedged, & value of one interest rate futures contract.
Thus ratio, ΔP / ΔVF, reflects optimal number of contracts to hedge against Δy :
N* = P DP / VF DF .Called the duration-based hedge ratio or the price sensitivity hedge ratio.
** This hedge ratio makes the duration of combined hedged position zero.
[Recall optimal hedge for stock portfolio reduces β to 0.]
1. This hedge assumes only parallel shifts in yield curve.
2. Given nonparallel shifts in yield curve, hedge performance can be
disappointing, especially if there is a big difference between DS & DF.
Problem 6.15.
Suppose that a bond portfolio with a duration of 12 years is hedged using a futures contract in which the underlying asset has a duration of four years.
What is likely to be the impact on the hedge of the fact that the 12-year rate is less volatile than the four-year rate?
Duration-based hedging assume parallel shifts in the yield curve.
(i.e., long rates & short rates move same amount.)
Consider a hedge with futures on the 4-year rate. Since the 4-year rate tends to move by more than 12-year rate, likely to be over-hedged.