Cost decomposition for a VWAP execution algorithm: Buy-side perspective

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Buy-side firms use a plethora of trading algorithms to implement their investment strategies. One of the oldest and most popular, is the Volume-Weighted Average Price (VWAP) algorithm. It is therefore important to be able to properly evaluate the associated cost. In this article, we linearly decompose the cost in a scheduling and selection component, which allows for more granular and faster cost evaluation on a small number of orders.

VWAP execution algorithm

Almgren and Chriss (2001) study the optimal execution problem for an order with an arrival price benchmark. Kato (2015) introduces a trading volume process in the Almgren-Chriss model and shows that the VWAP strategy is optimal for a risk-neutral trader. A VWAP strategy executes proportional to the market’s trading volume, i.e., a constant participation rate. Let us explain the cost of a VWAP algorithm by defining the following two sets of trades:

Set definitions for market and algorithm trades.

We define the VWAP in Eq. 1 as the volume-weighted average price for the trades in the set of market trades.

Eq. 1: Definition of VWAP during execution interval.

We define the algorithm’s execution price in Eq. 2 as the volume-weighted average price for the trades in the set of algorithm trades.

Eq. 2: Definition of execution price for VWAP algorithm.

Cost decomposition

Let us explain the cost decomposition for a buy order.¹ We discretize the execution interval in N volume-time buckets based on the realized market volume. We subsequently measure the cost relative to VWAP in each bucket, as shown in Figure 1.

Figure 1: Discretization of execution interval in volume-time and associated cost relative to VWAP. Source: Author.

We show in Eq. 3 that the algorithm has a cost relative to the interval VWAP and a deviation from the market’s relative volume for each volume-time bucket

Eq. 3: Definition of expanded execution price for VWAP algorithm.

which we can expand in Eq. 4:

Eq. 4: Definition of expanded execution price for VWAP algorithm (continued).

Hence, we can write the expected cost² in Eq. 5

Eq. 5: Definition of expected cost for VWAP algorithm.

We show in Eq. 6 that the expected cost consists of three components. We attribute the first to the algorithm’s scheduling logic and the latter two to the algorithm’s selection logic.

Eq. 6: Definition of expected cost for VWAP algorithm (continued).

Explaining the cost decomposition

For exemplary purposes, we introduce the price path in volume-time in Figure 2. During this interval, we execute a buy order using a VWAP algorithm.

Figure 2: Example price path in volume-time. Source: Author.

Scheduling cost

In the green volume-time bucket, the interval VWAP is strictly above the VWAP, which implies:

Eq. 7: Effect of interval VWAP being strictly above VWAP in cost decomposition.

If the algorithm trades more than the relative market volume in the green volume-time bucket, i.e.:

the scheduling cost are positive as shown in Eq. 8:

Eq. 8: Example calculation for positive scheduling cost in VWAP decomposition.

The green volume-time bucket is a period for which in hindsight prices are unfavorable. In practice, the scheduling cost can be non-zero due to a variety of reasons, e.g., a more lenient trading schedule or opportunistic order placement logic.

Selection cost

We show in Eq. 9 the first component of the selection cost, which measures the price deviation from VWAP in each volume-time bucket. In practice, this component is fairly stable throughout time and proportional to the spread of the security.

Eq. 9: Definition of first component of selection cost.

The second component of the selection cost is shown in Eq. 10. In practice, the expectation of the component is positive because of market impact of the additional traded volume. However, the magnitude of the component is relatively small because it is an interaction effect.

Eq. 10: Definition of second component of selection cost.

In Figure 4, we show an example in which we evaluate the selection cost intraday. In this example, we observe that algorithm B incurs a relatively high selection cost at the end of the execution.

Figure 4: Example intraday evaluation of selection cost for two competing VWAP algorithms. Source: Author.

Algorithm evaluation using cost decomposition

In Figure 5, we show an example cost decomposition for two competing VWAP algorithms A and B. We can infer that algorithm B is better mainly due to the significantly lower scheduling cost.

Figure 5: Example cost decomposition for two competing VWAP execution algorithms. Source: Author.

In Figure 6, we show an example in which we historically evaluate the scheduling and selection cost for a VWAP algorithm. You see that besides a cost attribution, the decomposition allows you to identify stable cost drivers.

Figure 6: Example of historical cost decomposition for VWAP algorithm. Source: Author.

Footnotes

¹ To measure cost for sell orders, multiply cost by -1.

² In practice, execution cost are measured in basis points or as a fraction of the spread.

References

[1]. Almgren, R., & Chriss, N. (2001). Optimal execution of portfolio transactions. Journal of Risk, 3, 5–40.

[2]. Kato, T., 2015. VWAP execution as an optimal strategy. JSIAM Letters, 7, 33–36.

If you’re keen on reading more, see a selection of my articles below:

Multi-armed bandits applied to order allocation among execution algorithms

Introduction to Probabilistic Classification: A Machine Learning Perspective

Beyond traditional asset return modelling: Embracing thick tails.