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Hauck Financial Services has a number of passive, buy-and-hold clients.

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Case Problem 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 as it is used by Hauck for rebalancing her portfolio. The mix of mutual funds that are being considered 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 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. However, it is equally correct to interpret FS as the dollar amount invested in the foreign stock fund. Then FS = 25,000 implies that $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 portfolios requires a balance constraint for each mutual 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 1% in the model (the last constraint) and that the transaction cost for buying and selling shares of the mutual funds is a linear function of the amount bought and sold. With this model, the transaction costs are deducted from the client’s account at the time of rebalancing and thus reduce the amount of money invested. The solution for Ms. Delgado’s rebalancing problem is shown as part of the Managerial Report.

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