Transaction cost analysis and execution benchmarks

Definition Transaction cost analysis (TCA) measures the cost of executing an order against a stated benchmark. A measurement is defined only once it fixes the order, meaning its side, its intended quantity $Q$ and its decision time, together with the reference price, the fills with quantities $q_i$ and prices $p_i$, the explicit costs such as commissions and fees, the comparison time $T$ with its price $p_T$, the treatment of shares that were never traded and the cash convention. Explicit costs are paid directly. Implicit costs arise from execution prices that differ from the reference price and from trades that were not completed. A cost can be reported in currency units, per share, or in basis points of a notional, with $1 \text{ bp} = 0.0001$. Two benchmarks applied to the same order answer different questions and need not agree.

Definition For a decision to buy $Q > 0$ shares at the decision price $p_0$, the paper portfolio buys all $Q$ shares at $p_0$ at the decision time, with no fees and no market impact, and holds them to the comparison time $T$. Its gain at the comparison price $p_T$ is

The actual portfolio holds the shares that were really bought, the fills $q_i$ at prices $p_i$ with filled quantity $Q_f = \sum_i q_i \le Q$, marked at the same price $p_T$, and it pays the explicit fees. Cash that was not spent earns no interest under the convention of this lesson, so

Both gains use the same intended quantity, comparison time, comparison price and cash convention. The paper portfolio is a reference construction. It is not a claim that $Q$ shares could actually have been bought at $p_0$.

Definition Implementation shortfall is the paper gain minus the actual gain, $\operatorname{IS} = G_{\text{paper}} - G_{\text{actual}}$. For a buy order, write $Q = Q_f + (Q - Q_f)$ in the paper gain and split $p_T - p_0 = (p_T - p_i) + (p_i - p_0)$ for each fill. The terms $q_i (p_T - p_i)$ cancel against the actual gain, which leaves

The first term is the filled slippage, the cost of the shares bought measured from the decision price. The second is the opportunity cost of the unfilled shares, and the third the explicit fees. For a sell order the paper portfolio sells all $Q$ shares at $p_0$, unsold shares stay in the actual portfolio at $p_T$, and the same accounting reverses the price differences,

A positive shortfall means that the actual portfolio did worse than the paper one. When some shares are unfilled, the opportunity cost is positive when the price moved against the order and negative when it moved in the order’s favor, for example when the price fell after a buy decision and not buying helped. The shortfall is an identity between two gains under the stated conventions. It does not show that the full quantity could have traded at the decision price, and it does not say which part was caused by the execution.

Definition The arrival price $p_a$ is the reference price, for example the midpoint, at the moment the order reaches the execution process, which can be later than the decision. Adding and subtracting $p_a$ in every price difference of the shortfall of a buy order gives

The first term is the delay cost, and it applies to the whole intended quantity, because the pieces $p_a - p_0$ from the filled and the unfilled shares add up to $Q(p_a - p_0)$. The second term is the execution cost of the fills measured from arrival, and the third is the opportunity cost measured from arrival. The decomposition redistributes the same total without dropping any shares. Using only the filled quantity in the delay term while measuring the unfilled opportunity from arrival would leave out the delay on the unfilled shares. Which desk is responsible for each term is a separate question that the algebra does not answer.

Definition The volume-weighted average price of trades $j$ with volumes $v_j > 0$ and prices $p_j$ over a specified interval is

The market VWAP $\bar p_{\text{mkt}}$ uses the market’s trades in the benchmark interval, under a stated rule for which trades are included, and the execution VWAP $\bar p_{\text{exec}}$ uses the order’s own fills, with weights $q_i$ and prices $p_i$. For a buy, the VWAP slippage of the filled shares is

and a negative value means that the fills beat the benchmark. The comparison uses only the filled shares, so it leaves out the unfilled shares and usually the delay before arrival, and it depends on the chosen interval and inclusion rule. A favorable VWAP comparison can therefore come together with a large implementation shortfall.

Definition Benchmark bias is a systematic difference between what a benchmark comparison reports and the execution quality it is meant to measure, caused by the way the benchmark or the sample of orders is built. One source is endogeneity, when the order’s own trades move the benchmark. Another is selection, when only completed orders are evaluated and abandoned decisions, whose opportunity costs can be large, are left out. A third is confounding, when brokers or algorithms receive orders of different size, urgency, timing or market conditions, so that differences in average cost mix the difficulty of the orders with the quality of their execution. The shortfall decomposition is an exact accounting identity, but it does not identify these causal differences. Evaluating a broker or an algorithm causally needs a separate research design, for example random assignment of comparable orders.