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Argosy University/ Phoniex University/
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Phoniex University
Oct2001  Nov2016
WITH TRANSACTION COSTS5
Hauck Financial Services has a number of passive, buyandhold clients. For these clients, Hauck offers an investment account whereby clients agree to put their money into a portfo lio of mutual funds that is rebalanced once a year. When the rebalancing occurs, Hauck de termines 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 rebal anced 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, re duces 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 buyandhold 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 intermediateterm bond fund (IB), a largecap growth fund (LG), a largecap value fund (LV), a smallcap growth fund (SG), and a smallcap 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 $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 rebalanc ing. That is,
Initial portfolio value = Amount invested in all assets after rebalancing +
Transaction costs
The extension of the Markowitz model that Hauck uses for rebalancing portfolios requires a balance constraint for each mutual fund. This balance constraint is
Amount invested in fund i = Initial holding of fund i +
Amount of fund i purchased  Amount of fund i sold
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 pur chased. For instance, the balance constraint for the foreign stock fund is
FS = FS_START + FS_BUY  FS_SELL
Jean Delgado has $100,000 in her account prior to the annual rebalancing, and she has spec ified 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
5
Min ¹⁄₅ a (Rs  R)2
s = 1
s.t.
0.1006FS + 0.1764IB + 0.3241LG + 0.3236LV + 0.3344SG + 0.2456SV = R1
0.1312FS + 3.25IB + 0.1871LG + 0.2061LV + 0.1940SG + 0.2532SV = R2
0.1347FS + 0.0751IB + 0.3328LG + 0.1293LV + 0.385SG  0.0670SV = R3
0.4542FS  0.0133IB + 0.4146LG + 0.0706LV + 0.5868SG + 0.0543SV = R4
0.2193FS + 0.0736IB  0.2326LG  0.0537LV  0.0902SG + 0.1731SV = R5
5
¹⁄₅ a Rs = R
s = 1
R Ú 10,000 FS + IB + LG + LV + SG + SV + TRANS–COST = 100,000
FS–START + FS–BUY  FS–SELL = FS
IB–START + IB–BUY  IB–SELL = IB LG–START + LG–BUY  LG–SELL = LG LV–START + LV–BUY  LV–SELL = LV SG–START + SG–BUY  SG–SELL = SG SV–START + SV–BUY  SV–SELL = SV
TRANS–FEE * (FS–BUY + FS–SELL + IB–BUY + IB–SELL +
LG–BUY + LG–SELL + LV–BUY + LV–SELL + SG–BUY + SG–SELL +
SV–BUY + SV–SELL) = TRANS–COST FS–START = 10,000
IB–START = 10,000
LG–START = 10,000
LV–START = 40,000
SG–START = 10,000
SV–START = 20,000
TRANS–FEE = 0.01 FS, IB, LG, LV, SG, SV Ú 0
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 transactions costs are deducted from the client’s account at the time of rebalancing and thus reduce the amount of money invested. The LINGO solution for Ms. Delgado’s rebalancing problem is shown in Figure 8.16.
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 help resolve a dispute with Jean Delgado. Ms. Delgado has had one of the Hauck passively managed portfolios for the last five years and has complained that she is not getting the rate of return of 10% that she specified. After a review of her annual statements for the last five years, she feels that 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 did Ms. Delgado pay for purchasing additional shares of the intermediateterm bond fund?
2. Based on the model solution given in Figure 8.16, what is the total transaction cost associated with rebalancing Ms. Delgado’s portfolio?
3. After paying transactions costs, how much did Ms. Delgado have invested in mu tual 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 expect to have in the intermediateterm bond 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 portfolio at the end of the year? Can she expect to earn 10% on the $100,000 she had at the begin ning of the year?
6. 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 transac tion costs. Explain why the current model does not do this.
7. Solve the formulation in part (6) for Jean Delgado. How does the portfolio compo sition differ from that shown in Figure 8.16?
FIGURE 8.16 SOLUTION TO HAUCK MINIMUM VARIANCE PORTFOLIO WITH TRANSACTION COSTS

R3 
5663.961 
0.000000 

R4 
9693.921 
0.000000 

R5 
4119.631 
0.000000 

FS 
15026.86 
0.000000 

IB 
51268.51 
0.000000 

LG 
4939.312 
0.000000 

LV 
0.000000 
418.5587 

SG 
0.000000 
149.1254 

SV 
27675.00 
0.000000 

TRANS_COST 
1090.311 
0.000000 

FS_START 
10000.00 
0.000000 

FS_BUY 
5026.863 
0.000000 

FS_SELL 
0.000000 
1.516067 

IB_START 
10000.00 
0.000000 

IB_BUY 
41268.51 
0.000000 

IB_SELL 
0.000000 
1.516067 

LG_START 
10000.00 
0.000000 

LG_BUY 
0.000000 
1.516067 

LG_SELL 
5060.688 
0.000000 

LV_START 
40000.00 
0.000000 

LV_BUY 
0.000000 
1.516067 

LV_SELL 
40000.00 
0.000000 

SG_START 
10000.00 
0.000000 

SG_BUY 
0.000000 
1.516067 

SG_SELL 
10000.00 
0.000000 

SV_START 
20000.00 
0.000000 

SV_BUY 
7675.004 
0.000000 

SV_SELL 
0.000000 
1.516067 

TRANS_FEE 
0.010000 
0.000000 
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