Index Replication using Individual Stocks with periodic rebalancing: Part 2
In this second installment of our series on index replication, we address a critical factor often overlooked: transaction costs. While technological advancements have significantly reduced transaction costs over the years, they remain a pivotal element in determining the success of replication strategies. Additionally, we revisit the concept of turnover introduced in the first article, emphasizing that lower turnover typically leads to reduced transaction costs.
Transaction Costs Modeling
In general, the overall transaction costs are dependent on various factors:
Brokerage Commission
Bid-Ask Spread (reflects the liquidity of the underlying investment options)
Exchange fees
Regulatory fees (if any)
Typically, these costs are modeled as a percentage of the total transaction value. To account for differences in costs between buying and selling (due to bid-ask spreads), separate percentages for buy & sell transactions can be defined.
At inception, assume a total investment of $1 across various assets. For any subsequent transaction, the incurred cost can be expressed as:
In the above equation, H(x) is the heavy-side function which is the step-function with value 1 for all x>0 and value 0 for all x < 0. Furthermore, note the V is the cumulative value corresponding to the respective investment. This reflects the fact that the transaction costs are dependent on the level of the investment value itself. Note the subtle difference in the sign of the difference in weights for buy/sell transaction costs.
This equation highlights how transaction costs directly erode investment value, underscoring the importance of optimizing turnover to minimize these costs.
Incorporating Transaction Costs into the problem setup:
The overall optimization problem setup remains largely same to Part 1, except the constraints include transaction costs & so does the optimization function. The overall governing equation of money flow can be written as below:
At the outset, initial cash is allocated to investments, and this equation ensures continuity across rebalancing periods.
Key Insight: At each rebalancing point, transaction costs reduce overall performance. Lower turnover, while potentially leading to sub-optimal portfolio weights, can significantly improve long-term results by minimizing costs.
Optimization Problem & Constraints:
The optimization goal remains consistent with Part 1: minimize overall error in replication. However, transaction costs and turnover constraints must now be considered:
Cash Constraint: Remaining cash must always be non-negative.
Turnover Constraint: A maximum turnover limit can help control costs.
Basic Problem Setup & Results/Analysis
Consider the same benchmark (XLF) and the same pool of underlying stocks viz.
JPM (JP Morgan & Chase)
BAC (Bank of America),
C (Citigroup),
WFC (Wells Fargo),
GS (Goldman Sachs)
AXP (American Express)
BRK-B (Berkshire Hathaway – B Shares)
MS (Morgan Stanley)
Assumptions:
Long-only positions
Cash does not have any returns
Transaction costs are modeled as a percentage of transaction value
No borrowing permitted
Scenario 1:
Max Turnover - No Limit
Transaction Costs – 0.15% for every transaction (i.e. for each buy & sell)
Portfolio Performance
Turnover
R-Squared
Scenario 2:
Max Turnover – 5% (per rebalance)
Transaction Costs – 0.15% for each transaction (for each buy & sell)
Portfolio Performance
Turnover
R-Squared
Scenario 3:
Turnover – 5%
Transaction costs – 2% per transaction (for each buy & sell)
Portfolio Performance
Scenario 4:
Turnover – 5%
Transaction costs – 5% per transaction (both buy & sell)
Portfolio Performance
Scenario 5:
Turnover – No Max Limit
Transaction costs – 5% per transaction (both buy & sell)
Portfolio Performance
Conclusions:
Transaction costs are a critical consideration in index replication strategies. High costs can severely erode portfolio performance, emphasizing the value of turnover constraints to manage expenses effectively. While this may result in sub-optimal portfolio weights, the trade-off is often worthwhile for improved long-term results.
Acknowledgements: Special thanks to Koustubh Moharir for his invaluable insights and discussions that shaped this article.
Code repository – with Sample Code coming soon.