2.3.3 Implementation Shortfall

Basic Prerequisites: 02 - TWAP-VWAP-POV and basic probability.

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1. Intuition & Practical Objective

VWAP/TWAP are day benchmarks: they ask “how close to the day’s average did you get?” Implementation shortfall (IS) asks a harder and more honest question: how much worse is your realized result than the paper decision you made at the moment you decided? Perold (1988) invented it as “paper versus reality.”

The idea is blunt. The portfolio manager’s decision is a paper transaction: buy $q$ shares at the decision-time price $m_0$. The reality: you filled only a fraction $\kappa$ at average price $\bar{p}$, and by the time you finished, the price had moved to $m_t$. IS is the dollar gap between those two worlds, decomposed into two costs you can manage separately:

• Execution cost - you filled shares, but at a price worse than the decision price.
• Opportunity cost - you didn’t fill shares, and the price ran away while you waited.

The practical objective of this page: define IS precisely, reproduce the corpus-verified Perold example, then drive home the critical distinction between benchmarks - IS (arrival price) is what a portfolio manager should judge execution by; VWAP is what a broker reports.

The one-sentence essence. “Implementation shortfall compares the executed portfolio to the decision, not to the day - it is execution cost on what you traded plus opportunity cost on what you failed to trade.”

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2. Mathematical Ground Truth & Derivations

Definition (Perold 1988; Foucault eq 2.29; Hasbrouck eq 14.1). Let $q$ = desired shares, $\kappa$ = fraction actually filled, $\bar{p}$ = average execution price of the filled part, $m_0$ = decision (arrival) price, $m_t$ = terminal price. Then

$$\boxed{\text{IS}=\kappa q(\bar{p}-m_0)+(1-\kappa )q(m_t-m_0)}=\underset{\kappa q(\bar{p}-m_0)}{\underbrace{\text{execution cost}}}+\underset{(1-\kappa )q(m_t-m_0)}{\underbrace{\text{opportunity cost}}}.$$

For a buy, $\bar{p}>m_0$ and $m_t>m_0$ are losses (positive IS). Hasbrouck writes the same idea in portfolio-vector form,

$$IS=(n_1-n_0{)}^{\prime }(p-{\pi }_0)+(v-n_1{)}^{\prime }({\pi }_1-{\pi }_0),$$

execution cost on actual shares traded plus opportunity cost on the shortfall $(v-n_1)$ - and notes the deep structural fact: execution cost is zero-sum across users sharing the benchmark ${\pi }_0$ (one trader’s slippage is another trader’s edge), while opportunity cost is not a transfer; it is pure tracking error.

The delay decomposition. Perold/Foucault further split the execution cost into the part attributable to delay (drift from decision to first fill) and the part attributable to the fills themselves:

$$\kappa q(\bar{p}-m_0)=\underset{\text{market impact on fills}}{\underbrace{\kappa q(\bar{p}-m_{\tau })}}+\underset{\text{delay cost}}{\underbrace{\kappa q(m_{\tau }-m_0)}}.$$

Why IS beats VWAP as a decision benchmark. VWAP is realized-volume-weighted over the whole day, so a patient broker that sets the day’s volume “always beats” it (page 02). IS is anchored to the one price the PM controlled - the decision price $m_0$ - so it measures what execution actually cost relative to the decision, which is what matters. This is the arrival-price benchmark.

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3. Computational Implementation - reproduce Perold and fresh cases

Reproduce the corpus-verified Perold/Foucault example ($24,000$, 2.4% of paper value) and then compute a fresh partial-fill IS with a full decomposition. Stdlib only; reproduced exactly.

Read the numbers. The untouched-Foucault case reproduces $24,000$ exactly. The fresh case: execution cost on the 40,000 filled shares is $16,000=0.8\cdot 50,000\cdot (50.40-50)$; opportunity cost on the 10,000 unfilled shares is $12,000=0.2\cdot 50,000\cdot (51.20-50)$. Observe the knife-edge: if the trader had been perfectly patient and filled everything at the terminal price, IS would be $60,000$ (all opportunity, no estimate saved), while filling everything at the realized average price $\bar{p}=50.40$ gives $20,000$ (execution cost only, no opportunity cost) - the schedule is the map between these, which is exactly the Almgren–Chriss trajectory of the sibling folder.

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4. Failure Modes & First-Principles Breakdowns

1. Benchmark choice changes your “score.” IS relative to decision price is what the PM cares about; VWAP slippage is what the broker reports. They routinely disagree, and - because VWAP is gameable - one shows “alpha” that is really a benchmarking artifact (Harris 2003; Hasbrouck Ch 14).
2. Opportunity cost is real but invisible in glossy reports. A broker that fills only half your order shows a great execution cost on the filled half while the unfilled half’s drift dominates. Perold’s whole point is IS must charge the miss.
3. IS is a random variable before you trade. $\bar{p}$ and $m_t$ are unknown ex-ante; only their distribution is. Optimizing IS therefore means optimizing $\mathbb{E}[\text{IS}]$ and its variance (the AC objective) - never a single static guess.
4. Execution cost is a transfer, not a loss. Because it nets to zero across users sharing ${\pi }_0$, a desk can look “low cost” by being on the right side of a transfer. Aggregate IS is a selection statistic, not a quality statistic (Hasbrouck Ch 14).

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5. References

• Perold, André F. - “The implementation shortfall: Paper versus reality,” Journal of Portfolio Management 14(3), 4-9 (1988). The origin of the IS benchmark.
• Foucault, Pagano, Roëll - Market Liquidity (2013)
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Almgren & Chriss - “Optimal execution of portfolio transactions,” Journal of Risk 3(2), 5-40 (2000). The arrival-price / IS objective as a mean-variance optimization.

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6. Connected Graph Bridges

• Continue: 04 · Scheduling & Volume Profiles · 05 · Failure Modes
• Model link: Optimal Execution · The Execution Problem
• Transaction costs context: Constraints & Transaction Costs