# Trading Arithmetic and Financial Numeracy Calculate exposure and trading results with consistent units. Use this workbook alongside the course. Write your answers before opening the solutions. Practical work is self-reviewed; scored knowledge checks are in the Academy. ## 1. Numbers and units ### Percent versus percentage points Percent means per hundred: 2% is 0.02, not 2. A change from 3% to 4% is one percentage point and a relative increase of one third. State which calculation you mean before comparing rates or performance. ### Basis points and price ticks One basis point is 0.01 percentage point. A price tick is the minimum quoted increment under a contract's specification and has its own monetary value. Basis points of a rate and ticks of a trading price are different units. ### Decimals and rounding Keep enough precision during intermediate calculations and round the final tradable quantity according to the allowed increment. Rounding position size up can exceed a cash budget. Display rounding should not alter the underlying accounting calculation. ### Troy ounces and contract multipliers Gold prices are often quoted per troy ounce, while an order may represent several ounces per contract or lot. The multiplier converts price movement into cash movement. Read the actual instrument specification; identical-looking symbols can represent different multipliers. ### Worked example A hypothetical contract represents 10 ounces and moves by $0.10 per ounce. The cash movement per contract is 10 × $0.10 = $1. Five contracts move $5 before costs. This is a unit calculation, not a claim about a broker's symbol. ### Independent exercise A hypothetical multiplier is 25 units and the price rises by 0.20 currency units per unit. Calculate the gross change for three contracts and label the units. My inputs and assumptions: My calculation or decision: Evidence that would change my conclusion: ## 2. Trading arithmetic ### Long and short P&L For a long position, gross P&L is exit minus entry, multiplied by quantity and instrument multiplier. For a short position, reverse the price difference. Record the direction explicitly rather than relying on a sign convention that may change between files. ### Gross versus net P&L Net P&L subtracts all applicable costs from gross P&L. Commissions, financing and conversion charges may be reported separately. Avoid subtracting a spread again when it is already reflected in actual entry and exit fills. ### Weighted average entry For entries at different prices, a quantity-weighted average is total price-times-quantity divided by total quantity. A simple average is correct only when quantities are equal. Partial exits require a consistent matching method and reconciliation with the provider statement. ### Currency conversion Convert a P&L amount into account currency using the appropriate rate and convention. Multiplying when the rate is quoted in the inverse direction produces a unit error. Preserve the rate, timestamp and any conversion fee so the result can be reconstructed. ### Worked example Buy one unit at 100 and three at 104: average entry is (100 + 312)/4 = 103. Exit all four at 105 with multiplier one: gross P&L is 8. If total separately charged costs are 3, net P&L is 5. ### Independent exercise For a short of two units entered at 108 and exited at 105 with multiplier ten, compute gross P&L and then subtract 7 of total costs. My inputs and assumptions: My calculation or decision: Evidence that would change my conclusion: ## 3. Risk arithmetic ### Cash risk and stop distance A planned cash budget divided by loss per unit gives an upper bound on quantity under the stated assumptions. Include multiplier, stop distance and a cost allowance. A stop is not a guaranteed price, so also calculate a worse-execution scenario. ### Reward-to-risk versus realized R Planned reward-to-risk compares a target distance with a stop distance after relevant costs. Realized R divides the actual trade result by the initial recorded risk budget. Moving the denominator after a loss would make comparisons inconsistent and hide the original decision. ### Drawdown and recovery asymmetry Drawdown measures decline from a previous equity peak. Recovery is measured from the smaller remaining balance. A 20% loss leaves 80% of the starting amount, requiring a 25% gain on the remaining amount to recover, before other cash flows. ### Aggregate planned losses Add planned losses across simultaneous positions to understand aggregate exposure. Related positions can lose together. The sum is a scenario under assumptions, not a guaranteed worst-case bound when gaps, correlations or funding conditions change. ### Worked example With a cash budget of 100, stop distance 2 and multiplier 10, price risk is 20 per contract. Add 3 estimated cost per contract: floor(100/23) gives four contracts, with planned cost-inclusive risk of 92. ### Independent exercise Repeat the example with a worse exit that creates a price distance of 3. Keep four contracts and costs of 3 per contract. Does the realized scenario still fit 100? My inputs and assumptions: My calculation or decision: Evidence that would change my conclusion: ## 4. Workbook checks ### Dimensional analysis Dimensional analysis checks whether the units on both sides of a calculation match. Price per ounce times ounces per contract times contracts should produce currency. A formula that still contains ounces where cash is expected has a missing or inverted conversion. ### Independent recalculation An independent recalculation uses a second method, not a copy of the same formula. Compare a spreadsheet result with a hand calculation and a broker statement where available. Agreement between duplicated formulas cannot reveal a shared mistake. ### Boundary and zero-value cases Boundary cases include zero stop distance, negative quantity, missing multiplier, insufficient funds and a minimum size larger than the calculated maximum. Treat undefined or invalid inputs as errors rather than silently replacing them with plausible-looking defaults. ### Explain a rejected position size A rejected position size is a valid calculation outcome. If the minimum order risks more than the permitted budget, the arithmetic does not authorize increasing the budget. Change the hypothetical instrument, revise the plan on its merits, or stay out. ### Worked example A worksheet divides 50 by a missing stop distance that has been converted to zero. The resulting infinite quantity is not a large opportunity. It indicates invalid input and must block the order-size output until corrected. ### Independent exercise Design three checks for a sizing worksheet: one unit check, one invalid-input check and one minimum-size check. Explain what the user should see when each fails. My inputs and assumptions: My calculation or decision: Evidence that would change my conclusion: ## Course project Produce a checked workbook of ten sizing and P&L cases, including losing trades. ### Self-review rubric - Concepts and reasoning: 25% - Calculations, data and evidence: 30% - Process and risk controls: 25% - Limitations and communication: 20% Record one correction and one next practice task. This rubric is not automatically graded. ## Worked solutions ### Exercise 1 Each contract changes by 25 × 0.20 = 5 currency units. Three contracts change by 15. The multiplication cancels the per-unit denominator; fees and direction still need to be included in the final trade result. ### Exercise 2 A short gains from the decline: (108 − 105) × 2 × 10 = 60 gross. Net P&L is 53 after subtracting costs. Reversing the subtraction would incorrectly turn the winning short into a loss. ### Exercise 3 The scenario loss is four × (3 × 10 + 3) = 132. It exceeds 100. The original position-size calculation respected a planned stop assumption, but that assumption was not a guaranteed execution outcome. ### Exercise 4 Check that price distance times multiplier produces cash; reject missing or nonpositive distance; compare the minimum tradable size with the computed maximum. Each failure should show a specific message and no executable quantity rather than a fabricated estimate. ## Further reading - https://www.cmegroup.com/education/courses/introduction-to-futures