An Introduction to Management Science 13th Edition by David Anderson,Dennis Sweeney ,Thomas Williams ,Jeffrey Camm, Kipp Martin

Exercise 1

Hauck financial services has a number of passive, buy and hold clients. For these clients. Hauck offers in investment account whereby clients agree to put their memory into a portfolio of mutual funds that is rebalanced once a year. When the rebalancing occurs, Hauck determine the mix of mutual funds in each investor's portfolio by solving an extension of the Markowitz portfolio model that incorporates transaction costs. Investors are charged a small transaction cost of the annual rebalancing of their portfolio. For simplicity, assume the following:
• At the beginning of the time period (in this case one year), the portfolio is rebalanced by using and selling Hauck mutual funds.
• The transaction costs associated with buying and selling mutual fund are paid at the beginning of the period when the portfolio is rebalanced, which in effect reduces the amount of money available to reinvest.
• No further transaction are made until the end of the time period, at which the point the new value of the portfolio is observed.
• The transaction cost is a linear function of the dollar amount of mutual funds bought or sold.
Jean Delgado is one of Hauk's buy and-hold- clients. We briefly describe the model as is used by Hauck for rebalancing her portfolio. The mix of mutual funds that are being considered for her portfolio are a foreign stock ( FS ) an intermediate-term bond fund ( IB ), a large-cap growth ( LG ), a small-cap growth fund( SG ) and a small-cap value funds( SV ). In the traditional Markowitz model, the variables are usually interpreted as the proportion of the portfolio invested in the asset represented by the variable. For example, FS is the proportion of the portfolio invested in the foreign stock fund. Then the FS = 25,000 implies $25,000 is invested in the foreign stock fund. Based on these assumptions, the initial portfolio value must equal the amount of money spent on transaction costs plus the amount invested in all the assets after rebalancing. That is,
The extension of the Markowitz model that Hauck uses for rebalancing portfolio requires a balance constraint for each fund. This balance constraint is
Using this balance constraint requires three additional variables for each fund: one for the amount invested prior to rebalancing, one for the amount sold, and one for the amount purchased. For instance, the balance constraint for the foreign stock fund is
Jean Delgado has $ 100,000 in her account prior to the annual rebalancing, and she has specified a minimum acceptable return of 10%. Hauck plans to use the following model to rebalance Ms. Delgado's portfolio. The complete model with transaction costs is
Notice that the transaction fee is set at I% in the model (the last constraint) and that the transaction cost for buying and selling funds shares of the mutual funds is a linear function of the amount bought and sold. With this model, the transactions costs are deducted from the client's account at the time of rebalancing and thus reduces the amount money invested. The LINGO solution for Ms. Delgado's rebalancing problem is shown in Figure 8 16.
Managerial Report
Assume you are a newly employed quantitative analyst hired by Hauck financial services. One of your first tasks is to review the portfolio rebalancing model in order to helps resolve a dispute with Jean Delgado has had Hauck passively managed portfolios for the last year and has complained that she is not getting the rate of return of 10% that the specified. After a review of her annual statement for the last years she is actually getting less than 10% on average.
1. According to the model solution in Figure 8.16, IB_BUY = $41,268.51. How much transaction cost associated with rebalancing Ms. Delgado's portfolio?
2. Based on the model solution given in Figure 8.16, what is the total transaction costs associated with rebalancing Ms Delgado's portfolio?
3. After paying transaction cost, how much did Ms Delgado have invested in mutual funds after her portfolio was rebalanced?
4. According to the model solution in Figure 8.16, IB = $51,268.51 how much can Ms. Delgado expected to have in the intermediate - term bound fund at the end of the year?
5. According to the model solution in Figure 8.16, the expected return of the portfolio is $10,000. What is the expected dollar amount in Ms. Delgado's at the end of the year? Can she expect to earn 10% on the $100,000 she had at the beginning of the year?
6. It is now time prepare a report to management to explain why Ms. Delgado did earn each year on her investment. Make a recommendation in terms of a revised portfolio that Jean Delgado can have expected portfolio balance of $110,000 at the end next year. Prepare a report that includes a modified optimization model that will given an expect of return 10% on the amount of money available at the beginning of the year before paying the transaction costs Explain why current model does not do this.
7. Solve the formulation in part (6) for Jean Delgado. How does the portfolio composition differ that shown in Figure 8.16?
FIGURE 8.16 SOLUTION TO HAUK MINIMUM VARIANCE PORTFOLIO WITH TRANSACTION COSTS