A portfolio manager for a bank has $10 million to invest. The securities availab

Question

A portfolio manager for a bank has $10 million to invest. The securities available for purchase, as well as their respective quality ratings, maturities, and yields are shown in the following table: The bank places the following policy limitations on the portfolio manager's actions: 1. Government and agency bonds must total at least $4 million. 2. The average quality of a portfolio cannot exceed 1.4 on the bank's quality scale. (A low number on this scale means a high-quality bond.) 3. The average years to maturity of the portfolio must not exceed 5 years. (a) Write a linear program to maximize after-tax earnings. The optimal solution of the LP should indicate to the portfolio manager how any dollars to invest in each bond. Do not solve your LP yet. (b) Solve your LP from Part a using AMPL. Specifically, write model and data files for the portfolio management linear program obtained in Part a and solve. What bonds should the portfolio manager purchase? (c) Suppose that it became possible to borrow up to $1 million at 2.75% after taxes. Modify your LP from Part a to include this option. What should the portfolio manager do?

Explanation / Answer

Expecting that the goal of the portfolio chief is to expand after-assess income and that the duty rate is 50 percent, what securities would it be a good idea for him to buy? On the off chance that it ended up noticeably conceivable to get up to $1 million at 5.5 percent

before charges, in what capacity should his determination be changed?

Leaving the subject of acquired subsidizes aside for the occasion, the choice factors for this issue are basically the dollar measure of every security to be obtained:

x A = Amount to be put resources into bond An; in a large number of dollars.

xB = Amount to be put resources into bond B; in a large number of dollars.

xC = Amount to be put resources into bond C; in a large number of dollars.

xD = Amount to be put resources into bond D; in a large number of dollars.

xE = Amount to be put resources into bond E; in a large number of dollars.

We should now decide the type of the goal work. Accepting that all securities are obtained at standard (confront esteem) and held to development and that the salary on city bonds is impose excluded, the after-charge profit are given by:

z = 0.043xA + 0.027xB + 0.025xC + 0.022xD + 0.045xE.

Presently let us consider each of the confinements of the issue. The portfolio director has just an aggregate of ten million dollars to contribute, and in this manner:

xA + xB + xC + xD + xE 10.

Further, of this sum in any event $4 million must be put resources into government and office bonds. Thus,

xB + xC + xD 4.

The normal nature of the portfolio, which is given by the proportion of the aggregate quality to the aggregate estimation of the portfolio, must not surpass 1.4:

2xA + 2xB + xC + xD + 5xE

        ------------------------------------------- 1.4.

xA + xB + xC + xD + xE

Note that the imbalance is not exactly or-break even with to, since a low number on the bank's quality scale implies an amazing bond. By clearing the denominator and re-masterminding terms, we find that this disparity is unmistakably equal to the straight requirement:

0.6xA + 0.6xB 0.4xC 0.4xD + 3.6xE 0.

The imperative on the normal development of the portfolio is a comparable proportion. The normal development must not surpass five years:

9xA + 15xB + 4xC + 3xD + 2xE

-------------------------------------------- 5,

xA + xB + xC + xD + xE

which is proportional to the straight limitation:

4xA + 10xB xC 2xD 3xE 0.

Note that the two proportion imperatives are, truth be told, nonlinear limitations, which would require refined computational techniques if incorporated into this frame. In any case, essentially increasing the two sides of every proportion requirement by its denominator (which must be nonnegative since it is the aggregate of nonnegative factors) changes this nonlinear limitation into a basic direct imperative. We can condense our definition in scene shape, as takes after:

xA xB xC xD xE Relation Limits

Money 1 10

Governments 1 4

Quality 0.6 0.4 3.6 0

Development 4 10 1 2 3 0

Objective 0.043 0.027 0.025 0.022 0.045 = z (max)

(Ideal arrangement) 3.36 0 6.48 0.16 0.294

The estimations of the choice factors and the ideal estimation of the target work are again given in the last line of the scene.

Presently consider the extra plausibility of having the capacity to get up to $1 million at 5.5 percent some time recently charges. Basically, we can expand our money supply over ten million by acquiring at an after-charge rate of 2.75 percent. We can characterize another choice variable as takes after:

y = sum obtained in a huge number of dollars.

There is an upper bound on the measure of assets that can be obtained, and henceforth

y 1.

The money requirement is then altered to mirror that the aggregate sum acquired must be not exactly or equivalent to the money that can be made accessible including getting:

xA + xB + xC + xD + xE 10 + y.

Presently, since the acquired cash costs 2.75 percent after assessments, the new after-charge profit are:

z = 0.043xA + 0.027xB + 0.025xC + 0.022xD + 0.045xE 0.0275y.

We abridge the plan when getting is permitted and give the arrangement in scene frame as takes after:

xA xB xC xD xE y Relation Limits

Money 1 1 10

Obtaining 1 1

Governments 1 4

Quality 0.6 0.4 3.6 0

Development 4 10 1 2 3 0

Objective 0.043 0.027 0.025 0.022 0.045 0.0275 = z (max)

(Ideal arrangement) 3.70 0 7.13 0.18 1 0.296

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