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 investors 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 Haucks 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. Delgados 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 clients account at the time of rebalancing and thus reduce the amount of money invested. The solution for Ms. Delgados rebalancing problem is shown as part of the Managerial Report.

Managerial Report

Assume that you are a financial analytics specialist newly hired by Hauck Financial Services. One of your first tasks is to review the portfolio rebalancing model in order to resolve a dispute with Jean Delgado. Ms. Delgado has had one of the Hauck passively managed portfolios for the past five years and has complained that she is not getting the rate of return of 10% that she specified. After reviewing her annual statements for the past five years, she feels that she is actually getting less than 10% on average.

According to the following Model Solution, . How much in transaction costs did Ms. Delgado pay for purchasing additional shares of the intermediate-term bond fund?

Based on the Model Solution, what is the total transaction cost associated with rebalancing Ms. Delgados portfolio?

After paying transactions costs, how much did Ms. Delgado have invested in mutual funds after her portfolio was rebalanced?

According to the Model Solution, IB = $51,268.51. How much can Ms. Delgado expect to have in the intermediate-term bond fund at the end of the year?

According to the Model Solution, the expected return of the portfolio is $10,000. What is the expected dollar amount in Ms. Delgados portfolio at the end of the year? Can she expect to earn 10% on the $100,000 she had at the beginning of the year?

MODEL SOLUTION

Optimal	Objective Value
	27219457.356
Variable	Value		Variable	Value
	18953.280		IB_START	10000.000
	10000.000		IB_BUY	41268.510
	11569.210		IB_SELL	0.000
	5663.961		LG_START	10000.000
	9693.921		LG_BUY	0.000
	4119.631		LG_SELL	5060.688
FS	15026.860		LV_START	40000.000
IB	51268.510		LV_BUY	0.000
LG	4939.312		LV_SELL	40000.000
LV	0.000		SG_START	10000.000
SG	0.000		SG_BUY	0.000
SV	27675.000		SG_SELL	10000.000
TRANS_COST	1090.311		SV_START	20000.000
FS_START	10000.000		SV_BUY	7675.004
FS_BUY	5026.863		SV_SELL	0.000
FS_SELL	0.000		TRANS_FEE	0.010

It is now time to prepare a report to management to explain why Ms. Delgado did not earn 10% each year on her investment. Make a recommendation in terms of a revised portfolio model that can be used so that Jean Delgado can have an expected portfolio balance of $110,000 at the end of next year. Prepare a report that includes a modified optimization model that will give an expected return of 10% on the amount of money available at the beginning of the year before paying the transaction costs. Explain why the current model does not do this.

Solve the formulation in part (6) for Jean Delgado. How does the portfolio composition differ from that of the Model Solution?