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無代寫-CHAPTER 17:
時間:2021-07-12
PRACTICE PROBLEMS
CHAPTER 17: TREASURY FUTURES CONTRACTS

Exercise 17.1: Consider a T-bond futures contract. The face value is $100,000, the
initial margin is $3,000 and the maintenance margin is $2,500. The daily futures
prices are 101.5, 101.84, 101.66, 102.03, 101.91. What’s the total gain or loss for
the long position? The short position? Which position requires a margin call?
What’s the final margin account balance for each position?

Exercise 17.2: Consider a ten-year T-note futures contract. The face value is
$100,000, the initial margin is $1,500 and the maintenance margin is $1,000. The
daily futures prices are 99.34, 99.88, 100.25, 100.31, 100.66. What’s the total
gain or loss for the long position? The short position? Which position requires a
margin call? What’s the final margin account balance for each position?

Exercise 17.3: The quoted discount rates of two U.S. T-bills and their maturities are
shown below.
(a) Find the forward rate (using annual compounding) that can be locked in for the
purposes of borrowing between July 23, 2014 and December 17, 2014.
(b) Find the rate that can locked in for the purposes of lending.
(c) Find the spread between the borrowing rate and the lending rate.
We’ll still assume that Price = 1 ? dn/360.
Discount Rates:
Settlement Maturity Ti Bid Ask (Offer)
5/18/2014 7/23/2014 66 days 5.04% 5.02%
12/17/2014 213 days 4.92% 4.90%
(d) Now, the yield curve in this example is downward sloping, and so the forward rate
should be less than the T2 = 213-day rate (intuitively, the forward rate has to be
low in order to “pull down” the long-term rate). However, the forward rates seem
to be higher than the T2-discount rates. What’s going on here?

Exercise 17.4: The quoted discount rates of two U.S. T-bills and their maturities are
shown below.
(a) Find the forward rate (using annual compounding) that can be locked in for the
purposes of borrowing between August 17, 2014 and November 15, 2014.
(b) Find the corresponding lending rates.

Discount Rates:
Settlement Maturity Ti Bid Offer
6/18/2014 8/17/2014 60 days 6.55 6.53
11/15/2014 150 days 6.70 6.68

(c) Repeat this question for the following data, i.e., find the borrowing and lending
rates that can be locked in between July 18, 2014 and October 16, 2014:
2

Discount Rates:
Settlement Maturity Ti Bid Offer
6/18/2014 7/18/2014 30 days 7.20 7.18
10/16/2014 120 days 7.18 7.16

Exercise 17.5: Suppose our T-bond futures price is 99.337 and we are short traders.
(a) For delivery, we have three T-bonds to choose from,
Coupon Quoted price Conversion factor
12.000% 139.333 1.3781
11.250% 138.625 1.3599
7.250% 95.000 0.9159
Which is the cheapest to deliver bond?
(b) With the same futures price, we have these three T-bonds to choose from:
Coupon Quoted price Conversion factor
11.875% 136.938 1.3516
7.875% 101.813 0.9890
10.000% 120.688 1.1891
Which is the cheapest to deliver bond?

Exercise 17.6: The cash prices of two T-bills and the cash forward price of a 90-day
T-bill that matures 90-days from now are (ignoring the bid-ask spread)
b(0,
365
90 ) = 0.98574, b(0,
365
180 ) = 0.96942, and G0(
365
90 ,
365
180 ) = 0.98370.
Is there an arbitrage opportunity? If so, how do you take advantage of it, assuming
zero net cash-flow at the maturity of the futures? How much money do you make per
forward contract—assuming a face value of $1 million?
From the forward price find the market forward rate, and compare this to the
calculated forward rate implied from the bond prices. Use effective annual
compounding.

Exercise 17.7: Suppose that in 4 months, your company will have to borrow $20
million for 180 days. You’ve chosen the 180-day Eurodollar futures contract to hedge
this cash-flow, and it has a cash price of $985,000. (What is the asset-to-be-delivered
for a Eurodollar futures contract? See the example in the lecture notes.) (i) How
should you hedge the cash flow? (ii) How would you hedge the same cash flow if
you needed to invest the $20 million rather than borrowing it?


3
Solutions:
Exercise 17.1: The futures price tends to go up, so let’s look at the details for the
short position:
Daily Daily Cumulative Margin
Futures Gain per Gain per Gain per Account Margin
Date Price $100 Face
Value
Contract Contract Balance Call
0 101.50 3000
1 101.84 -0.34 -340 -340 2660 0
2 101.66 0.18 180 -160 2840 0
3 102.03 -0.37 -370 -530 2470 530
4 101.91 0.12 120 -410 3120 0
The total loss for the short position is 410;
the total gain for the long position is 410.
The short position requires a margin call. Because the Cumulative Gain per Contract
is always negative here, it would be always positive for the long position, so there
is no chance that the long position would require a margin call.
The final margin account balance for the short position is 3,120. (Note that our final
balance is higher than our initial balance, yet we lost money. Note also that if the
balance is higher than the initial margin, we are allowed to remove money from
the margin account.)
The final margin account balance for the long position is (since there were no margin
calls) = initial margin + total gain = 3,000 + 410 = 3,410.

Exercise 17.2: The futures price tends to go up, so let’s look at the details for the
short position:
Daily Daily Cumulative Margin
Futures Gain per Gain per Gain per Account Margin
Date Price $100 Face
Value
Contract Contract Balance Call
0 99.34 1500
1 99.88 -0.54 -540 -540 960 540
2 100.25 -0.37 -370 -910 1130 0
3 100.31 -0.06 -60 -970 1070 0
4 100.66 -0.35 -350 -1320 720 780
The total loss for the short position is 1,320;
the total gain for the long position is 1,320.
The short position requires two margin calls; the long position doesn’t require a
margin call. (Since the daily losses are all negative here, they’d all be positive for
the long position, so there’s no chance that the long position would require a
margin call.)
The final margin account balance for the short position is 1,500. (After a margin call,
you must bring the account balance back up to the initial margin of 1,500, not just
the maintenance margin of 1,000.)
The final margin account balance for the long position is (since there were no margin
calls) = initial margin + total gain = 1,500 + 1,320 = 2,820.

Exercise 17.3: Note that prices are given by: Price = 1 ? dn/360, where the discount
rates, d, are given in the table. Here’s the table of prices:
4

Prices:
Settlement Maturity Ti Bid Ask (Offer)
5/18/2014 7/23/2014 66 days 0.99076 0.99080
12/17/2014 213 days 0.97089 0.97101

(a) For the borrowing rate, we want cash flows:
t = 0 T1 T2
0 +1 ? 12)],(1[ 210
TTb TTf ??
To get these cash flows, we basically need to
buy a T1-period T-bill (at the ask price); we’ll receive $1 at time T1;
sell x T2-period T-bills (at the bid price) and pay (1 + interest) at T2.
The forward rate for borrowing must be given by
[1 + bf0 (T1, T2) ]
T T2 1? =
),0(
),0(
2
1
Tb
Tb
B
A
=
1
2
)1(
)1(
1
2
TA
TB
R
R
?
? ,
where the superscript A represents an ask price (or rate) and the superscript B
represents the bid price (or rate).

Since T2 ? T1 = 147/365, bA(0,T1) = 0.99080, and bB(0,T2) = 0.97089, we have
bf0 (T1, T2) = (0.99080/0.97089)365/147 ? 1 = 5.17%.

(b) For the lending rate, we want cash flows:
t = 0 T1 T2
0 ?1 +[ ( , )]1 0 1 2 2 1? ?f T T T T?
To get these cash flows, we basically need to
sell a T1-period T-bill (at the bid price); we’ll owe $1 at time T1;
buy x T2-period T-bills (at the ask price) and receive (1 + interest) at T2.
The forward rate for lending must be given by
[1 + f0
? (T1, T2) ]T T2 1? =
b T
b T
B
A
( , )
( , )
0
0
1
2
=
( )
( )
1
1
2
1
2
1
?
?
R
R
A T
B T ,
where the superscript A represents an ask price (or rate) and the superscript B
represents the bid price (or rate).

Since T2 ? T1 = 147/365, bB(0,T1) = 0.99076, and bA(0,T2) = 0.97101, we have
f0
? (T1, T2) = (0.99076/0.97101)365/147 ? 1 = 5.13%.
(c) The spread is 5.17% ? 5.13% = 0.04%, or 4 basis points.
(d) The discount rate for the 213-day bond is roughly 4.9%, which is less than either
forward rate, but the discount rate is not an effective annual interest rate.
The bid price of the bond is, 1 ? (0.0492)?(213)/360 = 0.97089, so the effective
annual interest rate is given by solving 0.97089 = (1 + R)?213/365, that is
Rbid = (0.97089)?365/213 ? 1 = 5.19% > 5.17% = the forward bid rate; i.e., if dbid =
4.92% then Rbid = 5.19%.
Similarly, the ask price is 0.9710, and the ask rate is Rask = 5.17% > 5.13% = the
forward ask rate.

Exercise 17.4: (a) Here’s the table of prices:
5
Prices:
Settlement Maturity Ti Bid Ask (Offer)
6/18/2014 8/17/2014 60 days 0.98908 0.98912
11/15/2014 150 days 0.97208 0.97217

Now, [1 + f b0 (T1, T2) ]
T T2 1? =
b T
b T
A
B
( , )
( , )
0
0
1
2
=
( )
( )
1
1
2
1
2
1
?
?
R
R
B T
A T ,
so, f b0 (T1, T2) = (0.98912/0.97208)365/90 ? 1 = 7.30%.
(b) [1 + f0
? (T1, T2) ]T T2 1? =
b T
b T
B
A
( , )
( , )
0
0
1
2
=
( )
( )
1
1
2
1
2
1
?
?
R
R
A T
B T ,
so, f0
? (T1, T2) = (0.98908/0.97217)365/90 ? 1 = 7.24%.

(c) Now, the table of prices is:
Prices:
Settlement Maturity Ti Bid Ask (Offer)
6/18/2014 7/18/2014 30 days 0.99400 0.99402
10/16/2014 120 days 0.97607 0.97613
So,
f b0 (T1, T2) = (0.99402/0.97607)365/90 ? 1 = 7.67%,
and
f0
? (T1, T2) = (0.99400/0.97613)365/90 ? 1 = 7.63%.

Exercise 17.5: We just need to calculate the cost of deliver (basis) for each bond:
basis = quoted price ? (quoted futues price)?(conversion factor).

Suppose our T-bond futures price is 99.337 and we are short traders. (a) For delivery,
we have three T-bonds to choose from,
Coupon Quoted price Conversion factor Basis
12.000% 139.333 1.3781 2.44
11.250% 138.625 1.3599 3.54
7.250% 95.000 0.9159 4.02
The cheapest to deliver bond is the 12% bond.
(b) With the same futures price, we have these three T-bonds to choose from:

Coupon Quoted price Conversion factor Basis
11.875% 136.938 1.3516 2.67
7.875% 101.813 0.9890 3.57
10.000% 120.688 1.1891 2.57
The cheapest to deliver bond is the 10% bond.

6
Exercise 17.6: We have G0(
365
90 ,
365
180 )?b(0,
365
90 ) = 0.969672 > 0.96942 = b(0,
365
180 )
(assuming face value = $1; use $1 million face value below, so our profit will be
$1M(0.969672 ? 0.96942)).
So, we
short 1 forward contract; $1M?G0(
365
90 ,
365
180 ) = 983,700
buy 1 180-day T-bill for $1Mb(0,
365
180 ) = 969,420; at date T1 sell it for G0(
365
90 ,
365
180 );
sell y 90-day T-bills where y = G0(
365
90 ,
365
180 ) = 0.98370 (= “market” forward price);
$1Mb(0,
365
90 ) = 985,740.
Total profit at date t = 0 is
y×985,740 ? 969,420 = 969,672 – 969,420 = $252 (for face value = $1 million).
For the market forward rate, we know that the market forward price is
0.98370 = 365
90
21 )),(1(
?
? TTfm and so fm(T1,T2) = 6.89%;
For the calculated forward rate, the calculated forward price is

),0(
),0(
1
2
Tb
Tb =
0.98574
0.96942 = 0.983444 = 365
90
21 )),(1(
?
? TTfc ; so fc(T1,T2) = 7.01%.

Exercise 17.7: In 4 months, the $20 million will have a duration of 180 days. The
asset underlying the futures has a duration of 90 days (i.e., the asset-to-be-delivered
is, as usual, a 90-day T-bill). So, N = P
C F
P MD
F MD
? = 20 (180days)
985,000(90days)
M ? = 40.6 ? 41.
(i) If you are borrowing the $20 million, then that is like shorting a zero coupon bond,
so you want to short the futures contract. (Alternatively, if r ?, then the cost of
borrowing goes up, so you lose money. To hedge, you want a futures for which you
gain money when r ?. If r ?, the zero coupon bond price drops, and so does the
futures price. If you short the futures, you will make money when its price drops.)
So, short N = 41 futures contracts.
(ii) If you are investing (lending) the $20 million, then that is like buying a zero
coupon bond, so you want a long futures contract. (Alternatively, if r ?, then the
price of zero coupon bonds drops, so it is cheaper to invest, and you gain money.
You hedge with a contract that loses money as r ?. With a long futures, if r ?, then
bond prices drop, as does the futures price, so you lose money.) So, go long 41
futures contracts.


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