Solution to Arbitrage with Bonds[1]

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 / Current Price / Expiration Date / Coupon Rate
1 / $10l.625 / 8/15/2000 / 6.875
2 / $10l.5625 / 2/15/2001 / 5.5
3 / $103.80 / 2/15/2001 / 7.75

Table 1: Bond Data

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.

(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

= total cash flows from all bonds over all three periods.

= total cash flows from bond i over all three periods.

= total cash flows from all bonds in period j.

Objective

Maximize Z =

Constraints

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

The Solver parameters are quite simple; we want to maximize current period cash flow, while never having negative cash flow in the future.

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.

Here is the optimized spreadsheet:

Conclusion: Buying bonds 1 and 2 today, while selling bond 3, offers an arbitrage opportunity.

(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 / Ask Price / Bid Price
1 / $101.6563 / $101.5938
2 / $101.5938 / $101 5313
3 / $103.7813 / $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:

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.

B60.23501Prof. Juran

[1]Based on 4-102 (p. 181) in Practical Management Science (2nd ed., Winston and Albright, 2001 Duxbury Press). Solution by David Juran, 2001.