**LP
Modeling**

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**Cash
Matching Problem 2: **

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In most multiperiod planning problems the management of liquid or cash-like assets is an important consideration. In this model we add a “twist” from the standard cash matching problem in that we require that all “surplus cash” in any time period must also be iinvested. The key feature in this model is that for every period, there is a constraint which effectively says “(sources of cash) – (uses of cash)=0” The following simple but realistic example illustrates the major features of such models.

Suppose that as a result of a careful planning exercise you have concluded that to meet certain commitments you will need the following amounts of cash over the next 14 years:

__Year: 0
1 2 3 4 5
6 7 8 9 10
11 12 13 14 __

Cash(in $1,000s): $10 11 12 14 15 17 19 20 22 24 26 29 31 33 36

A common example where such a projection is made is in a
personal injury lawsuit. Both parties may reach an agreement that the injured
party should receive a stream of payments such as above or *its equivalent*.

For administrative simplicity both parties prefer an immediate single lump sum payment which is “equivalent” to the above stream of 15 payments. One possible solution is to determine the present value of the stream using a low interest rate such as that obtained in a very low risk investment such as a government guaranteed savings accounts. For example, if an interest rate of 4% is used, the present value of the stream of payments is $230,437. The party which must pay the lump sum, however, would like to argue for a much higher interest rate. To bee successful, such an argument must include evidence that such higher interest rate investments are available and are no riskier than savings accounts. The investments usually offered are government securities. Generally a broad spectrum of such investments are available on a given day. For simplicity assume that there are just two such investments available with the following features:

SECURITY |
CURRENT |
YEARLY RETURN |
YEARS TO MATURITY |
PRINCIPAL REPAYMENT AT MATURITY |

1 |
$980 |
$60 |
5 |
$1000 |

2 |
$965 |
$65 |
12 |
$1000 |

The paying party will offer a lump sum now with a recommendation of how much should be invested in securities 1, 2 and in a saving account such that the yearly cash requirements are met with the minimum lump sum payment.

Formulate the problem of determining the minimum initial lump sum payment as a linear program. In your model there will be a constraint for each year which forces the cash flows to net to zero. Assume that idle cash is invested at 4% in a savings account and all amounts are measured in $1000. Solve this LP.

How does this solution differ from the solution of the “standard” cash matching model were it is not required that idle cash be invested in each period? To answer this question you will have to formulate and solve the “standard” cash matching model for this data.