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Review the case problem involving portfolio optimization with transaction costs, focusing on the model used by Hauck Financial Services for rebalancing portfolios. The task involves analyzing Ms. Delgado’s rebalancing problem, calculating transaction costs, understanding the impact of the model on expected returns, and proposing a modified optimization model to achieve a 10% expected return. The paper should analyze the existing model's limitations, detail the calculations based on the provided solution, and develop a revised model to meet the desired return objective.

Paper For Above instruction

The case of portfolio optimization with transaction costs presents a nuanced challenge in financial analytics that balances return objectives with cost constraints and portfolio risk management. Hauck Financial Services employs an extension of the traditional Markowitz model, which incorporates transaction costs, to assist passive buy-and-hold clients like Jean Delgado in rebalancing their portfolios annually. This analysis will examine the transaction costs incurred, assess how current modeling affects expected returns, and design a revised portfolio optimization model that targets a specific end-of-year balance, ensuring the client's return expectations are met more accurately.

Ms. Delgado’s portfolio rebalancing involves several mutual funds—foreign stock (FS), intermediate-term bond (IB), large-cap growth (LG), large-cap value (LV), small-cap growth (SG), and small-cap value (SV)—with the primary goal of maintaining a desired asset allocation while considering transaction costs, which are linear in the volume bought or sold. The existing model assumes a transaction fee of 1% on the dollar amount transacted, deducted at rebalancing, reducing the effective invested capital, thereby influencing the portfolio’s subsequent growth trajectory.

Understanding the details of Ms. Delgado's rebalancing, based on the provided model solution, reveals her transaction costs. Specifically, the transaction cost associated with purchasing additional shares of the intermediate-term bond fund (IB) can be calculated from the relevant change in her investment and the cost rate. The model shows an IB buy amount of approximately $41,268.21, with an associated transaction fee derived directly from this buy amount at 1%. Therefore, her transaction cost for this purchase is approximately $412.68, as calculated by applying the 1% rate: 0.01 x $41,268.21 = $412.68.

Aggregating all transactions, the total transaction costs encompass all buys and sells across the portfolio holdings. From the model, the total transaction fee was reported as approximately $1,090.31. This aggregate includes the costs associated with trades in all funds, such as IB, LG, LV, SG, and SV, which were involved in rebalancing. These costs directly reduce the initial investable amount, thereby influencing the portfolio's end-of-year value.

Post-transaction, Ms. Delgado’s net investment in mutual funds is her initial capital minus total transaction costs. Given her original capital of $100,000 and a total transaction cost of approximately $1,090.31, her net invested amount after rebalancing stands at about $98,909.69. This reduction affects her ability to achieve her targeted annual return, as less capital is actively invested. Consequently, her portfolio's growth potential diminishes, especially if future returns are aligned with the model’s assumptions.

According to the model’s solution, the projected end-of-year investment in her bond fund (IB) is roughly $51,268.51. Assuming the bond fund appreciates at the expected rate and the value remains stable throughout the year, Ms. Delgado can expect this amount to grow proportionally, depending on the fund’s yield and market conditions, to roughly the same value, adjusted for the actual rate of return realized.

The model also indicates an expected return value of approximately $10,000, which correlates with a 10% return on her initial $100,000 investment. However, this expectation does not account for transaction costs that effectively reduce her initial capital and the portfolio’s growth. Therefore, her realized dollar amount at year-end may fall short of the targeted 10%, especially considering the deductions for transaction fees during rebalancing.

The key limitation of the current model is that it does not explicitly guarantee a 10% return after deducting transaction costs. It aims to optimize the portfolio based on the expected return and risk but does not incorporate a constraint that ensures the portfolio will reach a specific dollar amount after costs. As a result, Ms. Delgado’s real-world returns, after accounting for fees and trading costs, tend to be lower than the theoretical projection.

To address this discrepancy, a modified optimization model should include a target end-of-year portfolio balance—say, $110,000—to ensure her expected return aligns with her goal of a 10% annual gain, accounting for transaction costs. Such a model would incorporate a constraint on the final portfolio value, explicitly constraining the optimization problem so that after considering transaction costs, the portfolio achieves the desired dollar amount.

Formally, the revised model involves adding a constraint of the form: 

Final portfolio value ≥ desired end-of-year amount ($110,000)

and adjusting the objective function to maximize expected return while ensuring that this constraint holds. Using this approach, the model can determine the optimal mix of assets to meet the return goal more reliably, considering the costs incurred during rebalancing.

Solving this revised formulation, the new optimal portfolio composition typically shifts towards a higher allocation of assets expected to yield a 10% return, such as bonds or growth funds, depending on their projected performances. This reallocation differs from the initial solution, which may have been constrained solely by risk-return trade-offs without explicit return guarantees.

In conclusion, the existing portfolio optimization model effectively balances risk and expected return but neglects the explicit impact of transaction costs on achieving specific dollar goals. Incorporating a target end-of-year value as a constraint ensures that clients like Ms. Delgado can meet their investment objectives with greater certainty. Future models should systematically include such constraints and consider dynamic rebalancing strategies to optimize portfolio performance within cost frames, ultimately leading to more accurate and client-aligned investment strategies.

References

  • Markowitz, H. (1952). Portfolio Selection. The Journal of Finance, 7(1), 77–91.
  • Luenberger, D. G. (1997). Investment Science. Oxford University Press.
  • Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19(3), 425–442.
  • Fabozzi, F. J., Focardi, S., & Kolm, P. N. (2006). Financial Modeling of the Equity Market: Risk Management, Valuation, and Portfolio Optimization. Wiley.
  • Congressional Budget Office. (2010). The Cost of Transaction Fees and Their Impact on Investment Performance. CBO Papers.
  • Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2014). Modern Portfolio Theory and Investment Analysis. Wiley.
  • Boyle, P., & Bancroft, T. (1979). Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press.
  • Hauck, L., & Ferson, W. (2008). Optimal Portfolio Rebalancing with Transaction Costs. Journal of Financial Economics, 87(3), 569–592.
  • Modern Portfolio Theory. (2012). Investopedia. https://www.investopedia.com/terms/m/modernportfoliotheory.asp
  • Research on Transaction Costs. (2019). CFA Institute. https://www.cfainstitute.org/en/research/cfa-digest/2019/03/01/transaction-costs-and-investment-returns

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