Essentials of Business Analytics 1st Edition by Jeffrey Camm,James Cochran,Michael Fry,Jeffrey Ohlmann ,David Anderson

Exercise 18

Portfolio Optimization with Transaction Costs
Hauck Financial Services has a number of passive, buy-and-hold clients. For these clients, Hauck offers an investment account whereby clients agree to put their money into a portfolio of mutual funds that is rebalanced once a year. When the rebalancing occurs, Hauck determines 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 for 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 buying and selling Hauck mutual funds.
• The transaction costs associated with buying and selling mutual funds 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 transactions are made until the end of the time period, at which 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 Hauck s buy-and-hold clients. We briefly describe the model asit is used by Hauck for rebalancing her portfolio. The mix of mutual funds that are beingconsidered for her portfolio are a foreign stock fund (FS), an intermediate-term bond fund(IB), a large-cap growth fund (LG), a large-cap value fund (LV), a small-cap growth fund(SG), and a small-cap value fund (SV). In the traditional Markowitz model, the variablesare usually interpreted as the proportion of the portfolio invested in the asset representedby the variable. For example, FS is the proportion of the portfolio invested in the foreignstock fund. However, it is equally correct to interpret FS as the dollar amount invested in theforeign stock fund. Then FS 5 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 moneyspent 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 portfolios requiresa balance constraint for each mutual fund. This balance constraint is

Using this balance constraint requires three additional variables for each fund: one forthe amount invested prior to rebalancing, one for the amount sold, and one for the amountpurchased. 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 shehas specified a minimum acceptable return of 10 percent. Hauck plans to use the followingmodel to rebalance Ms. Delgado s portfolio. The complete model with transactioncosts is


Notice that the transaction fee is set at 1 percent in the model (the last constraint) and that thetransaction cost for buying and selling shares of the mutual funds is a linear function of theamount bought and sold. With this model, the transactions costs are deducted from the client saccount at the time of rebalancing and thus reduce the amount of money invested. Thesolution for Ms. Delgado s rebalancing problem is shown as part of the Managerial Report.
Managerial Report. Assume you are a newly employed financial analytics specialist hired by Hauck FinancialServices. One of your first tasks is to review the portfolio rebalancing model in order toresolve a dispute with Jean Delgado. Ms. Delgado has had one of the Hauck passively managedportfolios for the last five years and has complained that she is not getting the rate ofreturn of 10 percent that she specified. After a review of her annual statements for the lastfive years, she feels that she is actually getting less than 10 percent on average.
1. According to the following Model Solution, IB_BUY 5 $41,268.51. How much intransaction costs did Ms. Delgado pay for purchasing additional shares of the intermediate-term bond fund

2. Based on the Model Solution, what is the total transaction cost associated with rebalancingMs. Delgado s portfolio
3. After paying transactions costs, how much did Ms. Delgado have invested in mutualfunds after her portfolio was rebalanced
4. According to the Model Solution, IB 5 $51,268.51. How much can Ms. Delgadoexpect to have in the intermediate-term bond fund at the end of the year
5. According to the Model Solution, the expected return of the portfolio is $10,000. Whatis the expected dollar amount in Ms. Delgado s portfolio at the end of the year Canshe expect to earn 10 percent on the $100,000 she had at the beginning of the year
6. It is now time to prepare a report to management to explain why Ms. Delgado did notearn 10 percent each year on her investment. Make a recommendation in terms of arevised portfolio model that can be used so that Jean Delgado can have an expectedportfolio balance of $110,000 at the end of next year. Prepare a report that includesa modified optimization model that will give an expected return of 10 percent on theamount of money available at the beginning of the year before paying the transactioncosts. Explain why the current model does not do this.
7. Solve the formulation in part 6 for Jean Delgado. How does the portfolio compositiondiffer from that of the Model Solution