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Implementation
Subtracts Value

The Market
Impact Model™ -
Second in a Series

Should You Worry about
Transaction Costs?

The BARRA Brainteaser
for Spting 1998

Solution to the Winter
1998 Brainteaser

The BARRA Convertible
Bond System (CBS)

Seven Quantitative Insights into Active Management–Part 6

Implementation Subtracts Value

Every promising investment strategy on paper loses value on implementation. Constraints imposed during portfolio construction, sometimes by the client, lower value added. Transaction costs lower value added. Insight, especially into the relationship between transaction costs and value added, can help minimize this loss.

Investors use constraints in portfolio construction for two reasons. Often they face external regulations–for example, lists of stocks they cannot own. Sometimes they use constraints to limit the impact of any flaws in their inputs: return and risk forecasts. These constraints lower realized value added relative to that of paper portfolios not subject to constraints. Much of this loss is either beyond the manager’s control or illusory (that is, constraints to counteract flawed inputs should increase value added).

Transaction costs lower value added, and the manager does have some control over these. In a utility function balancing expected returns, risk, and transaction costs, the transaction costs are particularly vexing because we incur them with certainty, in contrast to the expected returns we can only hope to achieve. There are two ways to control or reduce transaction costs: trade smarter (i.e., more cheaply trade by trade) or trade less. We will not deal here with approaches to reducing transaction costs trade by trade, though BARRA has pioneered methods here. Trading less doesn’t sound appealing on the surface, because less trading means acting on less of our superior information. But as this insight will show, we can trade substantially less without giving up much value added.

Value added versus turnover

We want to understand the tradeoff between value added and turnover. Let’s take a minute to develop this relationship. If you want to avoid the technical details, skip down to EQUATION 16 and the insight.

First let’s return to INSIGHT 2: Information ratios determine value added. There we saw that the information ratio acted as a budget constraint: achieving higher alpha required taking on more risk:

                                            (1)

Defining value added as:

                                           (2)

                                           (3)

leads to FIGURE 1, showing value added as a function of risk. FIGURE 1 is just a graph of EQUATION 3.

FIGURE 1
Value added by active management as a function of risk

That previous insight also derived and discussed formulas for the optimal level of risk,, and the optimal valued added, . We simply reprint those results here:

                                           (4)

                                           (5)

But we can rewrite EQUATION 4 as:

                                           (6)

and then substitute this into EQUATION 5, to find:

                                           (7)

Substituting these back into EQUATION 3, we can rewrite value added as a function of risk as:

                                           (8)

Relationship to turnover

EQUATION 8 connects value added to risk. Can we connect risk to turnover? Imagine that the optimal solution involves a set of purchases and sales, , with total turnover, , and optimal alpha, risk, and value added. Now the simplest strategy to cut turnover by a fraction x:

                                           (9)

is simply to reduce each trade by the same proportion.

Then:

                                       (10)

                                           (11)

                                           (12)

In this strategy:

                                           (13)

and so:

                    (14)

In a more sophisticated strategy, reducing turnover from by conducting the most valuable trades first, we would expect:1

                                           (15)

and hence:

                       (16)

FIGURE 2
The Value Added/Turnover Frontier

Insight

FIGURE 2 graphs this value added/turnover frontier. According to this result, we can achieve at least three-quarters of the value added with only half the turnover. And the key qualifier is "at least." An extremely simple strategy, just reducing each trade by exactly the same fraction, can achieve three-quarters of the value added with half the turnover (the lighter line in FIGURE 2). You can do even better by distinguishing transaction costs between stocks and scheduling the most valuable trades first (the darker line in FIGURE 2).

This insight has two practical implications. First, do not necessarily dismiss promising strategies with high turnover; you may be able to capture much of the value added with significantly less turnover. Second, transaction cost research is extremely valuable precisely for this reason: to distinguish trades and realize the most value at the least cost.

Define and , and show that if y < x and x < 1, then _.