Contents

QCE-FND-F02-001, draft 0.3.1

When a positive estimate is still not permission

An estimate can be positive and still leave a decision unresolved. You will calculate net expected value for one fictional candidate, test how the result moves when assumptions change, and explain why a positive number does not authorize new exposure.

Module: Foundations F02, Outcomes and costs
Study time: 25 to 35 minutes, a planning estimate not yet measured with learners
Data: Synthetic. Fictional units, no market data
Status: Draft. Technical review and learner testing pending. Edition details

In this lesson

You will calculate net expected value and explain why a positive result is not sufficient permission.

Assumption

0.56 is an estimate in a fictional example. It is not a known market probability.

Units

All amounts are fictional educational units, on the same scale for gains, losses and costs.

What you will be able to do

Calculate net expected value for a two-outcome case, including total round-trip cost.
Calculate the break-even probability for a given gain, loss and cost.
Recalculate a candidate after one assumption changes, and say whether the decision changes.
Name the evidence that is missing before a positive estimate could support permission.
Explain why an existing position needs a separate risk-reduction process.

Before you start

Three ideas carry the whole lesson.

Probabilities as decimals. 56% is written 0.56. The probability of the other outcome is 1 − 0.56 = 0.44.

Average versus outcome. Expected value is a probability-weighted average under a model, not a prediction of the next outcome. You can picture it as the average over many imagined repetitions of the same situation. A fair bet that wins 1 or loses 1 with equal probability has an expected value of 0, yet every individual bet wins or loses 1.

Round-trip cost. Costs include everything paid to enter and exit: fees, spread and expected slippage. They are counted once for the complete round trip, not once per side.

The variables, and two separate states

p probability estimate: The estimated probability that the outcome is favourable. Some method produced it; it is not a known fact.
G gross gain: Units gained if the outcome is favourable, before costs.
L gross loss: Units lost if the outcome is unfavourable, before costs, written as a positive magnitude.
c round-trip cost: Total expected cost of entering and exiting, counted once.
EV net expected value: EV = pG − (1 − p)L − c, in estimated units.
Break-even probability: The value of p at which EV is exactly zero: (L + c) ÷ (G + L).

Two states stay separate throughout. The arithmetic check passes only when EV is strictly greater than zero; exactly zero does not pass. The decision status says what may happen next. Its default here is Evidence incomplete, and it changes only when a named rule applies. Each rule has a reason code:

COST_EXCEEDS_ESTIMATED_EDGE: Total cost is equal to or larger than the estimated gross edge, so estimated value is not above zero.
INPUT_STALE: Inputs are older than the policy allows. New risk is refused whatever the number says.
EVIDENCE_INCOMPLETE: The arithmetic may be positive, but the evidence needed for permission is missing.
POLICY_CHECKS_PASSED: Every named check in a defined policy has passed. None of the four cases in this lesson qualifies.

The worked calculation

The baseline candidate has p = 0.56, G = 12, L = 10 and c = 1.80.

1. Probability of the unfavourable outcome1 − 0.56 = 0.44
2. Probability-weighted gain0.56 × 12 = 6.72
3. Probability-weighted loss0.44 × 10 = 4.40
4. Estimated gross edge6.72 − 4.40 = 2.32
5. Subtract total round-trip cost2.32 − 1.80 = +0.52

The arithmetic check passes: +0.52 estimated units is above zero. Notice how much of the edge the cost consumes: 1.80 of 2.32, about 78%. A modest change in either number can remove what is left.

Why a positive estimate is not permission

The calculation is correct. It is still not a reason to act, for three reasons.

The probability is an estimate. Nothing in this case says how 0.56 was produced, from what data, or how much it could move. Lowering it by only 0.03, to 0.53, already makes the value negative.

Expected value is not an outcome. This example has exactly two mutually exclusive outcomes and the same fixed round-trip cost in either outcome. Real trades can have many outcomes and variable costs. Under these assumptions, each trade ends either at 12 − 1.80 = +10.20 units or at −10 − 1.80 = −11.80 units. A positive expectation does not bound a particular loss, prevent a run of losses or guarantee a positive result over any sample.

Permission is a separate decision. It means a defined policy has checked the evidence, the risk limits and the freshness of the inputs, and has issued a bounded authorization for one action. The arithmetic is one input to that policy.

Before a positive estimate could support permission you would need, at minimum: a documented method and data for the estimate, with its uncertainty; a cost model tied to the intended size and market conditions; inputs shown to be current at decision time; evidence that the result survives reasonable changes in assumptions; and limits on the size of the exposure.

Four cases

The same candidate under four sets of assumptions. Gain is 12 and loss is 10 in every case.

Synthetic cases. EV and break-even are computed exactly and rounded to two decimals.
Case	p	Cost c	Inputs	EV, units	Break-even	Reason code and decision
Baseline	0.56	1.80	Fresh	+0.52	53.64%	`EVIDENCE_INCOMPLETE`Arithmetic positive; evidence incomplete
Higher costs	0.56	2.60	Fresh	−0.28	57.27%	`COST_EXCEEDS_ESTIMATED_EDGE`Reject under the stated cost scenario
Lower probability assumption	0.49	1.80	Fresh	−1.02	53.64%	`COST_EXCEEDS_ESTIMATED_EDGE`Reject under this sensitivity scenario
Stale inputs	0.56	1.80	Stale	+0.52	53.64%	`INPUT_STALE`Refuse new risk; the number does not override freshness

Baseline. +0.52. The arithmetic passes; the decision is Evidence incomplete (EVIDENCE_INCOMPLETE).

Higher costs. Cost rises to 2.60: 6.72 − 4.40 − 2.60 = −0.28. The estimate did not change; the cost now exceeds the 2.32 gross edge. Reject under the stated cost scenario (COST_EXCEEDS_ESTIMATED_EDGE).

Lower probability assumption. With p = 0.49: 5.88 − 5.10 − 1.80 = −1.02. Reject under this sensitivity scenario (COST_EXCEEDS_ESTIMATED_EDGE). The 0.49 is chosen to test how fragile the result is. It is not a computed confidence bound, and not a better estimate.

Stale inputs. The numbers match the baseline, but the inputs are older than the policy allows. The arithmetic still gives +0.52; the decision is to refuse new risk (INPUT_STALE). A value calculated from stale inputs cannot override the freshness rule.

Break-even probability

Set EV to zero and solve for p:

pG − (1 − p)L − c = 0

p(G + L) = L + c

p* = (L + c) ÷ (G + L)

With c = 1.80: (10 + 1.80) ÷ (12 + 10) = 11.80 ÷ 22, about 53.64%. With c = 2.60: 12.60 ÷ 22, about 57.27%. Raising the cost by 0.80 units lifts the break-even probability by about 3.6 percentage points, and the baseline estimate of 0.56 sits only 2.36 points above it.

This is algebra on fictional inputs, not a measured trading threshold. If G + L is zero, the break-even probability is not defined. If it comes out above 100%, no probability estimate can break even under those inputs, and the calculator says so instead of capping the number.

Try the calculator

Change one input at a time and watch the arithmetic check and the decision status separately. Try the cost stress case: raise the cost to 2.60 units. What changes, and what evidence is still missing?

F02 calculator

All data synthetic. Fictional units. No market data.

Case

BaselineHigher costsLower probability assumptionStale inputs

Values match the selected case.

Use a point or a comma as the decimal separator, for example 1.80 or 1,80. Do not type thousands separators.

Probability estimate p

Between 0 and 1. An estimate, not a known probability.

Gross gain G

Units if the outcome is favourable, before costs.

Gross loss L

Units if unfavourable, before costs. Enter as a positive number.

Total round-trip cost c

Entry and exit together, counted once.

Input freshness

Fresh: within the stated age limitStale: older than the stated age limit

Expected value

EV = pG − (1 − p)L − c

EV = 0.56 × 12 − 0.44 × 10 − 1.80

= 6.72 − 4.40 − 1.80

+0.52estimated units, rounded to two decimals

Break-even probability, (L + c) ÷ (G + L)

(10 + 1.80) ÷ (12 + 10) = 53.64%

Estimate minus break-even: +2.36 percentage points. A small margin means a small change in the estimate reverses the sign.

Arithmetic check: PassedPasses only when estimated value is strictly above zero.
Input freshness: PassedInputs are within the stated age limit.
Evidence for permission: IncompleteNo documented method or uncertainty for the probability estimate in this lesson.
Decision status: Evidence incompleteEVIDENCE_INCOMPLETEPositive arithmetic is not authorization. Nothing here shows how the estimate was produced or how much it could change.

If a position already exists

Refusing new risk is not the same as closing an existing position. Actions that can increase exposure (NEW_RISK) are denied by default. Actions that reduce or close known exposure (RISK_REDUCTION) follow their own, separately defined controls.

When inputs are stale, the freshness rule stops fresh exposure, but it does not justify blindly sending an unverified market exit. First reconcile: confirm the actual position from an authoritative record. If the position is unknown or the records disagree, escalate to an operator under the written failure plan. An entry rule must never block a defined risk-reduction process.

Exercise

Calculate each case and choose its reason code, then answer the questions on evidence, existing positions and guaranteed profit. Feedback and worked answers appear after you check.

Worksheet and answer key

The two-page worksheet repeats the exercise for paper and includes an empty five-part decision dossier. Use the answer key after your attempt.

WorksheetQCE-FND-F02-001-WS, draft 0.3.1. PDF, 2 pages, 43 KB.Answer keyQCE-FND-F02-001-AK, draft 0.3.1. PDF, 2 pages, 45 KB.

Sources

Grinstead and Snell, Introduction to Probability, 2nd edition, American Mathematical Society. Chapter 6, Expected Value and Variance. Freely redistributable under the GNU Free Documentation License. Book page
Bailey, Borwein, López de Prado and Zhu, The Probability of Backtest Overfitting, Journal of Computational Finance 20(4), 2017: why an estimate selected from many trials can be fragile. Open-access copy (eScholarship)
QCE-WP-001 v0.3, Quant Capital Engineering concept whitepaper and product specification, 3 October 2026. Internal document; source of the four cases and decision rules.

Edition and status

Lesson: QCE-FND-F02-001, draft 0.3.1, 4 October 2026
Worksheet and key: QCE-FND-F02-001-WS and QCE-FND-F02-001-AK, draft 0.3.1
State: Drafted. Not released.
Arithmetic: Checked by automated tests: all four cases and both break-even values agree across this lesson, the calculator, the worksheet and the answer key.
Technical review: Not yet performed. No reviewer has been named.
Learner testing: Not yet performed.
Drafting: Prepared for Quant Capital Engineering with language-model assistance. Human verification of the equations and wording is pending and is required before release.
Data: Synthetic, created for this lesson.
Corrections: Draft 0.3.1, 4 October 2026: the primer describes expected value as a model average rather than a prediction of the next outcome; the two-outcome, fixed-cost assumption is stated before individual outcomes are discussed; the worksheet and answer key were reissued as draft 0.3.1 with unchanged numbers. Draft 0.3, 3 October 2026: first draft.

See where F02 sits in the Foundations syllabus