While executing all shares in the closing auction can guarantee the closing price, a large order can lead to significant market impact. As shown in Figure 1, a larger the participation rate in the closing auction will likely to move the price further away from its equilibrium. One solution is to start trading earlier in the continuous trading session. However, trading earlier will lead to tracking error: when an order is executed earlier from the matching time, the price volatility vs. the closing price will become higher (Figure 2). This leads to higher uncertainty when trader wants to benchmark the closing price.

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**Bloomberg Tradebook**
December 13, 2016
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**Bloomberg Tradebook**December 13, 2016

*Figure 1: Transaction cost of trading in the closing auction as a function of closing auction participation rate. Transaction cost is calculated by comparing to the last traded price in the continuous session. Positive cost represents underperformance. Sample period: 2015 H2.*

*Figure 2. Standard deviation of the difference between intraday price and end of day closing price. Standard deviations are calculated across sample days for a fixed time (e.g. 8:00) of the day. Sample period: January 2016.*

Thus an optimal strategy should be a balance between *market impact* and *tracking error*. The balancing depends on different variables:

The key function of the **Smart Close algorithm** is to calculate the ideal balance between market impact and tracking error, then translate it into an actionable trading schedule. In the first step, we define the trading cost as the slippage between the average executed price of the order and the closing price: positive means underperformance and vice versa. While the slippage is an unknown variable when entering the order, we consider (i) the expected value and (ii) variance of the slippage.

In the second step, we define the *cost* of a strategy that benchmarks the closing price as:

*Cost = (Expected Value of Slippage) + (Risk Weight) * (Variance of Slippage)*

The risk weight is a positive parameter which represents the urgency of a trader: If the trader wants to deviate less from benchmarking the closing price, he would increase the risk weight in the calculation. Tradebook’s Smart Close strategy condenses the risk weight into three urgency levels: low, medium and high (Figure 3). Higher urgency would imply a larger risk weight. Given a risk weight parameter, Smart Close allocation is calculated by minimizing this cost function.

*Figure 3. Urgency level drop-down selection of Smart Close ticket on GTMQ.*

To understand how this minimization problem leads to a reasonable trading schedule, we consider the cases of different urgency levels. When urgency is low, the cost is dominated by the expected value of slippage. In other words, fluctuation/variance of the slippage is less of a concern and we are allowed greater volatility risk (tracking error) to minimize market impact. Thus, the optimal strategy will start early. In the extreme case when risk weight equals zero, it becomes a VWAP strategy which has minimal market impact. On the other hand, when urgency is high, the cost is dominated by the variance of slippage. In this case, we are allowed to make as much market impact as we can to minimize the uncertainty of slippage. Thus, the optimal strategy becomes allocating all shares into the closing auction.

This observation has several implications. First, the Smart Close trading schedule is more back-loaded than a VWAP (Figure 4). Higher the urgency level would imply a more back-loaded allocation (Figure 5). Second, VWAP is not an ideal strategy, even if it starts at a later time. One way to understand it is that VWAP has a flat participation rate while the optimal schedule should have an increasing participation rate that compensates the price volatility (Figure 4).

To highlight the difference between a Smart Close and VWAP strategy, we consider the so-called *efficient frontier*, a two-dimensional plot that represents the expected value and variance of the slippage. In Figure 6, we compare Smart Close with different urgency to VWAP strategies that start at different times. Notice that the VWAP strategy is always inferior to Smart Close, in a sense that given an expected value of slippage, Smart Close can always achieve a lower variance of slippage than VWAP.

*Figure 4. A sample of Smart Close allocation vs. VWAP allocation for 400,000 shares of ANZ AU at medium urgency level. VWAP strategy is assumed to start time at the same time as the Smart Close strategy.*

*Figure 5. A sample of Smart Close allocation for 400,000 shares of ANZ AU at medium and high urgency level.*

*Figure 6. An example of efficient frontier of Smart Close vs VWAP strategy for benchmarking the closing price.*