Three bonds, as listed in the file P04_127.xlsx, are currently for sale. Each bond has a face value of $100. Every six months, starting six months 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 six months from now

■ $102.75 a year from now Given the current price structure, the question is whether there is a way to make an infinite amount of money. To answer this, you need to 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, you could make $k by purchasing 10k units of bond 1 today and selling 5k units of bond 2 today. You could also cover all payments at future dates from money received on those dates.

a. Show that an arbitrage opportunity exists for the bonds in the file P04_127.xlsx. (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.)

b. Usually bonds are bought at an ask price and sold at a bid price. Consider the same three bonds as before and suppose the ask and bid prices are as listed in the same file. Show that these bond prices admit no arbitrage opportunities.