Execution Cost of Block Sale vs. Sliced Orders

Market Microstructure · Medium · Free problem

You need to sell 100,000 shares of a stock that trades at $\$50$ with an average daily volume (ADV) of 500,000 shares. The bid-ask spread is $\$0.05$, and the permanent price impact is $0.1\%$ of the stock price per

\%$ of ADV traded.

Compare two execution strategies:

  1. All-at-once: sell all 100,000 shares in a single trade.
  2. Sliced: split into 10 equal trades of 10,000 shares executed sequentially over the day.

For each strategy, estimate the total execution cost (spread cost plus permanent impact cost). What is the cost savings from slicing?

Hints

  1. The spread cost is the same regardless of how you split the order -- focus on what changes between the two strategies.
  2. Permanent price impact scales linearly with trade size as a fraction of ADV. For the sliced strategy, each subsequent trade faces the accumulated impact of all prior trades.
  3. Compute the cumulative impact cost as $\sum_{k=1}^{10} k \times \delta$ where $\delta$ is the per-slice impact. Use the triangular number formula $\sum_{k=1}^{n} k = n(n+1)/2$.

Worked Solution

How to Think About It: This is a bread-and-butter market microstructure question. Every time you trade, you pay two kinds of costs: the spread (a temporary cost you pay for immediacy) and permanent price impact (you move the market against yourself). The spread cost is the same whether you trade in one block or ten slices -- you cross the spread on every share regardless. The whole game is about permanent impact: it scales with trade size, so splitting your order into smaller pieces reduces the per-slice impact. But here is the catch -- permanent impact accumulates. Each subsequent slice executes at a worse price because all prior slices have already pushed the market. The question is: does slicing still save you money despite this accumulation?

Quick Estimate: The block trade is 20% of ADV, so permanent impact is $0.1\% \times 20 = 2\%$ of the stock price, which is $\

.00$ per share on a $\$50$ stock. That is $\
00{,}000$ in impact alone. Each slice is only 2% of ADV, so each slice's incremental impact is $0.1\% \times 2 = 0.2\%$, or $\$0.10$ per share. Ten slices accumulate to a total impact of only $0.2\% \times 10 = 2\%$ on the last trade, but earlier trades execute at lower accumulated levels. The average impact across slices is roughly half the final level, so total impact cost is around $\$0.10 \times 55{,}000 / 100{,}000 \times 100{,}000$ -- ballpark $\$55{,}000$. So slicing saves roughly $\$45{,}000$. Let's verify.

Approach: Compute spread and permanent impact costs separately for each strategy, then compare.

Formal Solution:

Assume the stock price is $S = \$50$.

Spread cost (same for both strategies):

You cross the half-spread on every share you sell:

$C_{\text{spread}} = 100{,}000 \times \frac{\$0.05}{2} = \

{,}500$

This cost is identical whether you trade in one block or ten slices.

Strategy 1 -- All-at-once:

Trade size as a fraction of ADV:

$f = \frac{100{,}000}{500{,}000} = 20\%$

Permanent impact:

$\Delta S = 0.1\% \times 20 \times S = 2\% \times \$50 = \

.00 \text{ per share}$

Total permanent impact cost:

$C_{\text{impact}}^{\text{block}} = 100{,}000 \times \

.00 = \
00{,}000$

Total execution cost:

$C_{\text{block}} = \

{,}500 + \
00{,}000 = \
02{,}500$

Strategy 2 -- Sliced into 10 trades of 10,000 shares:

Each slice is

0{,}000 / 500{,}000 = 2\%$ of ADV.

Incremental permanent impact per slice:

$\delta = 0.1\% \times 2 \times S = 0.2\% \times \$50 = \$0.10 \text{ per share}$

Permanent impact is permanent -- it does not revert. So the $k$-th trade executes at a cumulative adverse price shift of $k \times \$0.10$. The cost of the $k$-th slice is:

$C_k = 10{,}000 \times k \times \$0.10$

Total permanent impact cost:

$C_{\text{impact}}^{\text{sliced}} = \sum_{k=1}^{10} 10{,}000 \times k \times \$0.10 = 10{,}000 \times 0.10 \times \sum_{k=1}^{10} k = 1{,}000 \times \frac{10 \times 11}{2} = \$55{,}000$

Total execution cost:

$C_{\text{sliced}} = \

{,}500 + \$55{,}000 = \$57{,}500$

Cost savings from slicing:

$\Delta C = \

02{,}500 - \$57{,}500 = \$45{,}000$

Answer: Slicing the order into 10 equal trades saves approximately $\$45{,}000$ in execution costs. The block trade costs $\

02{,}500$ (dominated by $\
00{,}000$ in permanent impact), while the sliced strategy costs $\$57{,}500$. The spread cost of $\
{,}500$ is identical in both cases -- all the savings come from reducing permanent price impact. The sliced strategy's impact cost is $\$55{,}000$ versus $\
00{,}000$ for the block, a 45% reduction, because the cumulative impact grows as a triangular sum (
+ 2 + \cdots + 10$) rather than hitting the full 20%-of-ADV impact on every share.

Intuition

This problem illustrates the fundamental trade-off in optimal execution: trading faster gives you certainty (you are done, no more market risk) but costs more in impact. Trading slower reduces impact but exposes you to adverse price moves while you wait. The key insight is that permanent impact costs grow quadratically with trade size -- doubling your trade size doubles the impact per share AND doubles the number of shares, so total impact cost quadruples. Splitting into $n$ equal slices turns one big square ($n^2$) into a triangular sum (

+ 2 + \cdots + n = n(n+1)/2$), which is roughly half. This is exactly the intuition behind the Almgren-Chriss optimal execution framework.

In practice, the missing piece here is timing risk. While you slice your order over the day, the stock price might move against you for reasons unrelated to your trading. A risk-averse trader faces a real trade-off: accept higher impact costs to finish quickly, or accept more variance to reduce impact. The Almgren-Chriss model makes this precise by introducing a risk-aversion parameter that interpolates between "trade it all now" (zero variance, maximum impact) and "spread it out evenly" (minimum impact, maximum variance). Most real execution algorithms live somewhere in between.

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