MANAGMENT SCIENCE HW

MS 3053 — Homework #2 Due Date: January 31, 2018

#1

An investment advisor wants to develop a model that can be used to allocate investment funds among four alternatives: stocks, bonds, mutual funds, and a money market account. For the next quarter, the advisor used historical data to develop estimates of the projected annual rate of return and the associated risk for each option, which are given in the table below. Risk is measured using an index between 0 and 1, with higher values denoting more volatility, which implies more uncertainty. For example, cash held in the money market account has a low annual return of 1.1%, but the risk is extremely low; on the other hand, while stocks yield a high rate of return, they are highly reflective of market volatility, and thus the associated risk index is high.

Investment Proj. RoR Risk

Stocks 10.7% 0.82 Bonds 3.2% 0.22 Mutual Funds 4.5% 0.33 Cash (Money Market) 1.1% 0.02

The advisor has specifies guidelines that must be applied to all clients, regardless of their affinity for or aversion to risk.

• No more than 75% of the total investment may be in stocks. • The amount invested in mutual funds must be at least as much as invested in bonds. • The amount of cash held in the money market account must be at least 10% but no more

than 30% of the total investment.

Suppose the maximum risk value for a particular client is 0.4. Develop a linear program that can be used to determine the optimal allocation of investment funds among stocks, bonds, mutual funds, and money market accounts. Solve the LP using the Solver feature and determine the annual rate of return for this portfolio.

MS 3053 (Panda) — Homework #2 (Spring 2018) 1

#2

As part of the settlement for a class action lawsuit, XTech must provide sufficient cash to make the following annual payments.

Year 1 2 3 4 5 6

Payments ($) 9,000 12,500 11,500 13,500 15,000 22,000

The annual payments are to made at the beginning of each year. The court will approve an amount to be paid to a trustee now that, along with earnings on its investment, will cover the annual payments. Investment of the funds will be limited to two classes of government securities, whose investment information is provided in the table below (with par values of $1,000 per unit), and any remaining funds will be placed into a savings account, which earns a 2.75% annual interest rate.

Security Market Price Annual Coupon Rate Maturity

Class A $1,045 6.850% Beginning of Year 4 Class B $1,015 5.325% Beginning of Year 5

Formulate the linear programming model to determine the minimum total amount of cash F that XTech must give to the trustee now to fund the settlement plan. Let A and B denote the number of units of Class A and Class B securities, respectively, to purchase now, and let Si denote the amount that is put into savings at the beginning of year i, i = 1, 2, 3, 4, 5. You do not need to solve the LP.

MS 3053 (Panda) — Homework #2 (Spring 2018) 2