Solution to Arbitrage with Bonds1
It is February 15, 2000. Three bonds, as listed in Table 1 below, are for sale. Each
bond has a face value of $100. Every 6 months, starting 6 mouths from the
current date and ending at the expiration date, each bond pays
0.5 * (coupon rate) * (Face value).
At the expiration date the face value is paid. For example, the second bond pays

$2.75 on 8/15/00

$102.75 on 2/15/01
Bond
1
2
3
Table 1: Bond Data
Current Price
$10l.625
$10l.5625
$103.80
Expiration Date
8/15/2000
2/15/2001
2/15/2001
Coupon Rate
6.875
5.5
7.75
Given the current price structure, the question is whether there is a way to make
an infinite amount of money. To answer this, we look for an arbitrage. An
arbitrage exists if there is a combination of bond sales and purchases today that
yields

a positive cash flow today

nonnegative cash flows at all future dates
If such a strategy exists, then it is possible to make an infinite amount of money.
For example, if buying 10 units of bond 1 today and selling 5 units of bond 2
today yielded, say, $1 today and nothing at all future dates, then we could make
$k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today.
We would also be able to cover all payments at future dates from money
received on those dates. Clearly, we expect that bond prices at any point in time
will be set so that no arbitrage opportunities exist.
Based on 4-102 (p. 181) in Practical Management Science (2 nd ed., Winston and Albright, 2001
Duxbury Press). Solution by David Juran, 2001.
1
(a) Show that an arbitrage opportunity exists for the bonds in Table 1. (Hint: Set
up an LP that maximizes today's cash flow subject to constraints that cash
flow at each future date is nonnegative. You should get a "no convergence"
message from Solver.)
Managerial Formulation
Decision Variables
How much to buy or sell of each bond. (Selling a bond is conceptually the same
as buying a negative amount.)
Objective
Maximize cash flow at the end of the first period (today).
Constraints
Non-negative cash flow at the end of all future periods.
Mathematical Formulation
Decision Variables
Define Xi = quantity of bond i purchased today.
Define Cij = Cash flow per face value unit for bond i in period j, as shown in
Table 2 below.
j = periods
0 months from now 6 months from now 12 months from now
Bond 1
-$101.63
$103.44
$0.00
i = bonds Bond 2
-$101.56
$2.75
$102.75
Bond 3
-$103.80
$3.88
$103.88
Table 2: Cash Flows
3
3
 X C
i
i 1 j 1
3
X C
j 1
i
i 1
i
= total cash flows from all bonds over all three periods.
ij
= total cash flows from bond i over all three periods.
ij
= total cash flows from all bonds in period j.
3
X C
ij
B60.2350
2
Prof. Juran
Objective
Maximize Z =
3
X C
i 1
i
i1
Constraints
3
X C
i 1
i
ij
 0 for j = 2, 3.
Here’s the spreadsheet model. Two odd elements here will make more sense
later:
The stuff in row 22 will be used to create a special constraint.
Buying and selling are conceptually just the inverses of each other, so we really
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
A
Data on bonds
Bond
1
2
3
B
C
D
Current price
$
101.625
$
101.563
$
103.800
Expiration date
6
12
12
Coupon rate
0.06875
0.05500
0.07750
Face value of each bond
100
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Buy
1.00
1.00
1.00
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
$
$
$
$
$
F
G
H
If we maximize the cash flow in cell B22, without an
upper bound constraint on it (as suggested in cells
B23 and B24), the Solver does not converge. To
make it converge, we add this upper bound
constraint. This lets us make $1 (or any other value
you want to put in cell B24).
Sell
0.00
0.00
0.00
=IF($C5=D$16,1+$D5/2,IF($C5>D$16,$D5/2,0))*($B13-$C13)*$B$7
=B3*(C11-B11)
0
(101.63)
(101.56)
(103.80)
(306.99)
<=
1.00
E
$
$
$
$
$
6
103.44
2.75
3.88
110.06
>=
-
12
$
$
$
$
$
102.75
103.88
206.63
>=
-
=SUM(D17:D19)
The Solver parameters are quite simple; we want to maximize current period
cash flow, while never having negative cash flow in the future.
B60.2350
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Prof. Juran
When we try to solve this model, we get the following error message:
This is actually good news! It indicates an “unbounded” problem; one in which
there are no constraints that limit the value of the objective function. In the
context of this problem, it means that there is no limit on the amount of cash flow
in the first period. In other words, there is an arbitrage opportunity.
Unfortunately, because Solver couldn’t solve the problem, we don’t know which
bonds to buy and sell. We can get around this by playing a little trick; we
introduce a new constraint limiting the objective function artificially.
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Prof. Juran
Here is the optimized spreadsheet:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
A
Data on bonds
Bond
1
2
3
B
C
D
Current price
$
101.625
$
101.563
$
103.800
Expiration date
6
12
12
Coupon rate
0.06875
0.05500
0.07750
Face value of each bond
100
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Buy
0.21
20.30
0.00
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
Sell
0.00
0.00
20.08
0
$
$
$
$
$
(21.60)
(2,061.87)
2,084.48
1.00
<=
1.00
6
$
$
$
$
$
21.99
55.83
(77.82)
>=
-
12
$
$ 2,085.98
$ (2,085.98)
$
>=
$
-
Conclusion: Buying bonds 1 and 2 today, while selling bond 3, offers an arbitrage
opportunity.
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Prof. Juran
(b) Usually bonds are bought at an ask price and sold at a bid price. Consider the
same three bonds listed in Table 1 and suppose the ask and bid prices are as
listed in Table 2. Show that these bond prices admit no arbitrage
opportunities.
Bond
1
2
3
Ask Price
$101.6563
$101.5938
$103.7813
Bid Price
$101.5938
$101 5313
$103.7188
Table 2: Bid and Ask Prices
Using the same basic model, augmented with both ask and bid prices, we see
that the optimal solution is to buy no bonds at all:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
A
Data on bonds
Bond
1
2
3
B
C
Ask price
$ 101.6563
$ 101.5938
$ 103.7813
Bid price
$ 101.5938
$ 101.5313
$ 103.7188
Face value of each bond
100
Decisions: number of bonds to buy or sell now
Bond
1
2
3
Buy
0
0
0
Cash flows
Months from now
Bond 1
Bond 2
Bond 3
Total cash flow
E
Expiration date Coupon rate
6
0.06875
12
0.055
12
0.0775
Sell
0
0
0
0
$
$
$
$
D
6
-
$
$
$
$
12
-
$
$
$
$
-
$
>=
$
>=
-
This result indicates that no arbitrage opportunity exists. The only way to have
non-negative cash flows in the first period and zero cash flows in all future
periods is not to invest at all.
B60.2350
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Prof. Juran