## Related questions with answers

Broker Steve Johnson is currently trying to maximize his profit in the bond market. Four bonds are available for purchase and sale, with the bid and ask pricc of each bond as shown in Table 40. Steve can buy up to 1,000 units of each bond at the ask price or sell up to 1,000 units of each bond at the bid price. During each of the next three years, the person who sells a bond will pay the owner of the bond the cash payments shown in Table 41. Steve’s goal is to maximize his revenue from selling bonds less his payment for buying bonds, subject to the constraint that after each year’s payments are received, his current cash position (due only to cash payments from bonds and not purchases or sale of bonds) is nonnegative. Assume that cash payments are discounted, with a payment of $1 one year from now being equivalent to a payment of 90¢ now. Formulate an LP to maximize net profit from buying and selling bonds, subject to the arbitrage constraints previously described. Why do you think we limit the number of units of each bond that can be bought or sold? Table 40:

$\begin{matrix} \text{Bond} & \text{Bid Price} & \text{Ask Price}\\ \text{1} & \text{980} & \text{990}\\ \text{2} & \text{970} & \text{985}\\ \text{3} & \text{960} & \text{972}\\ \text{4} & \text{940} & \text{954}\\ \end{matrix}$

Table 41:

$\begin{matrix} \text{Year} & \text{Bond 1} & \text{Bond 2} & \text{Bond 3} & \text{Bond 4}\\ \text{1} & \text{100} & \text{80} & \text{70} & \text{60}\\ \text{2} & \text{110} & \text{90} & \text{80} & \text{50}\\ \text{3} & \text{1,100} & \text{1,120} & \text{1,090} & \text{1,110}\\ \end{matrix}$

Solution

VerifiedBroker Steve is trying to maximize product in bond market. Steve can buy up to $1000$ units of each bond at the ask price or sell them at the bid price. During each of the next three years, person selling bonds will pay the bond owner cash payments shown in the table in the assignment. After each year's payments are recieved Steve's current cash position is nonnegative. Let $x_i$ and $y_i$ be the units of bond $i$ bought at ask price and sold at bid price respectively. Since we are to maximize the net profit we observe the following function:

$z=980y_1+970y_2+960y_3+940y_4-990x_1-985x_2-972x_3-954x_4.$

However, $z$ is subject to the constraint that we mentioned that after each year's payments are recieved, current cash position is nonnegative. Cash payments are discounted, with a payment of $\$1$ a year from now equivalent to paying $0.9\$$ now. Therefore let$z_i$be a cash on hand after payments of year$i.$Therefore it is implied:$ $z_1=100(x_1-y_1)+80(x_2-y_2)+70(x_3-y_3)+60(x_4-y_4).$ $In the year two we have similar situation:$ $z_2=\frac{10}{9}z_1+110(x_1-y_1)+90(x_2-y_2)+80(x_3-y_3)+50(x_4-y_4).$ $In third situation we have:$ $z_3=\frac{10}{9}z_2+1100(x_1-y_1)+1120(x_2-y_2)+1090(x_3-y_3)+1110(x_4-y_4).$ $Since at most$1000$units of each bond could be sold or bought, it must be$ $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4\leq1000.$$

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