Quick-reference shortcuts for risk sizing, roots, options math, and market intuition.
risk sizing
Risk sizing: max loss → share count
Decide how many dollars you can lose, then let the stop distance set the position size.
Daily risk capacity: how many full-size trades fit?
Divide your daily loss limit by per-trade risk to find the maximum number of full-size losing trades.
bps percent
BPS ↔ percent ↔ dollars: quick conversions
1 basis point = 0.01%. 100 bps = 1%. Use these anchors to convert between bps, percent, and dollars in your head.
Percent change shortcuts
Break any percent into easy building blocks: 10%, 5%, 1%, and 25%.
r multiples
R multiples: sizing outcomes in units of risk
1R is your planned stop distance in dollars. Express P&L as multiples of that unit to compare trades on any ticker.
Break-even win rate from reward:risk
The minimum win rate at which a setup earns zero, before fees. Lower reward:risk requires a higher hit rate.
spread cost
Spread in bps: how expensive is the market?
Express the bid-ask spread as bps of midpoint to compare execution costs across different-priced assets.
Round-trip execution cost: spread + fees
Total cost of entering and exiting = spread drag + explicit fees. Know your breakeven move before placing a trade.
everyday math
Tip calculation: the 10% building-block method
Any tip can be built from 10% (move decimal left) plus halves and doubles.
Discount and sale price: two paths to the same answer
Find the savings first (price × discount%), or compute the pay-fraction directly (price × remaining%).
Unit price: comparing products regardless of package size
Convert every product to the same unit (cents per oz, per 100g, per sheet) before comparing.
Fuel cost and cost per mile
Compute cost per mile first, then scale to any trip distance without needing to think in gallons.
cooking math
Recipe scaling: multiply every ingredient by one factor
Find the single scale factor (new servings ÷ old servings), then apply it uniformly to every ingredient.
Temperature conversions: F ↔ C anchors and shortcuts
Memorize 5 anchor pairs and interpolate. The exact formulas are on a 28°C-per-50°F grid.
Kitchen measurement conversions: the hierarchy
US cooking uses a strict hierarchy: 3 tsp = 1 tbsp, 16 tbsp = 1 cup, 2 cups = 1 pint, 2 pints = 1 quart, 4 quarts = 1 gallon.
fermi estimation
Fermi estimation: anchor, decompose, chain
Every Fermi estimate is a product of components. Break the problem into knowable pieces and multiply them.
Order-of-magnitude arithmetic: powers of 10
Working in powers of 10 lets you multiply huge numbers mentally and check if an answer is in the right ballpark.
physics math
Kinetic energy: ½mv²
Speed matters more than mass — doubling speed quadruples kinetic energy.
Ohm's law: V = IR
Voltage equals current times resistance — the fundamental circuit relationship.
science facts
Speed of light and other universal constants
Light covers 300,000 km in one second — use this to estimate cosmic distances.
geography anchors
World population anchors
Know the top-10 countries by population to anchor any global estimation.
Distance estimation anchors
Key distances to memorize for geographic and travel estimation.
health fitness
BMI: Body Mass Index
BMI = weight(kg) ÷ height(m)². Normal range: 18.5–24.9.
sports stats
Baseball stats: BA, OBP, SLG, ERA
Key baseball statistics and their benchmarks for quick estimation.
tech estimation
Network bandwidth and download time
Convert between bits and bytes, then calculate download time from file size and bandwidth.
math roots
Mental square root estimation
Three methods: bracket + linear interpolation, factor out perfect squares, and one Newton refinement step.
Mental cube root estimation
Bracket between consecutive perfect cubes, then apply the linear interpolation formula using 3a².
real estate
Cap rate, NOI, and property value
Cap rate is the un-levered yield on a property. Invert it to find the value multiple.
CRE debt sizing: DSCR, LTV, and debt yield
Lenders use three overlapping tests to size a commercial real estate loan. All three must clear.
RE return metrics: CoC, EM, YoC, and IRR shortcuts
Four metrics that together describe the full return picture: cash yield, total return, development profit, and time-weighted return.
REIT metrics: FFO, AFFO, NAV, and P/FFO
REITs use adjusted earnings metrics because GAAP depreciation overstates expenses for appreciating assets.
probability
Expected value: the core formula
EV = Σ p_i × x_i. Probability-weight every outcome and sum. This is the single most tested concept in prop firm interviews.
Conditional probability and Bayes' theorem
P(A|B) = P(A∩B) / P(B). Update your prior when you receive new information.
Geometric distribution: expected trials to success
If each trial succeeds with probability p, the expected number of trials is 1/p.
Combinatorics shortcuts: C(n,k) and permutations
C(n,k) = n! / (k!(n−k)!) counts unordered selections. Permutations count ordered. Know the symmetry and small values cold.
Poisson distribution: rare event counts
X ~ Poisson(λ) models the count of rare events in a fixed window when λ = expected count. E[X] = Var[X] = λ.
Normal approximation and the 68-95-99.7 rule
For large n, many distributions approximate normal. Know 1σ / 2σ / 3σ coverage by heart — it appears in every statistics interview.
mental arithmetic
Multiply any 2-digit number by 11
Add the two digits and insert the sum in the middle. Works instantly for any 2-digit number.
Square any number ending in 5
Multiply the leading digits by (themselves + 1), then append 25. Instant.
Near-round multiplication via difference of squares
Multiply two numbers near a round number using (a+b)(a−b) = a² − b². Eliminates hard multiplication.
Powers of 2: the reference table every quant needs
Memorize 2¹ through 2¹⁶. These appear in combinatorics, binary, computing capacity, and probability calculations.
Fraction to decimal conversions: the essential reference
Know halves, thirds, quarters, fifths, sixths, sevenths, and eighths as decimals. These appear in every probability calculation.
Percentage decomposition: compute any % in your head
Break any percentage into easy building blocks: 50%, 25%, 10%, 5%, 1%, and 0.5%. Assemble the answer from parts.
Sum of integers 1 to n: triangular numbers
The sum 1+2+3+…+n equals n(n+1)/2. Compute it instantly: multiply n by its neighbor, halve the result.
Sum of squares 1² to n²
1²+2²+…+n² = n(n+1)(2n+1)/6. Three consecutive-ish factors divided by 6 — memorize the structure, not the values.
Sum of cubes 1³ to n³
1³+2³+…+n³ = [n(n+1)/2]². The sum of cubes is the square of the sum of integers — a beautiful identity worth memorizing.
Divisibility rules: instant checks for 2 through 13
Test whether a number divides evenly without doing long division. Rules for 7, 11, and 13 share an elegant shortcut via 1001 = 7 × 11 × 13.
Last digit of a power: cyclic patterns
Only the last digit of the base determines the last digit of the power. Four base-digit patterns repeat with period 4; find exponent mod 4 to locate the position.
Squaring any 2-digit number: three fast methods
Round to the nearest multiple of 10, apply (a ± b)² = a² ± 2ab + b². Three cases cover all 2-digit numbers in under 5 seconds.
Casting out nines: instant arithmetic error detection
Replace every number with its digital root (keep summing digits until one digit), then verify the operation gives a matching digital root. Catches ~90% of arithmetic mistakes in seconds.
Work-rate problems and the harmonic mean trap
When two workers (or pipes, or rates) combine, add their rates — not their times. Average speed for equal distances is the harmonic mean, not the arithmetic mean. Most people get this wrong.
Pythagorean triples: recognize them instantly
Memorize the 5 primitive triples and their multiples. Recognizing a right triangle instantly saves 15–30 seconds per geometry problem — no Pythagorean theorem computation needed.
Pascal's triangle: instant C(n,k) without computing
Memorize rows 0–8. Read C(n,k) directly from row n, position k. For n > 8, use the multiplicative formula: C(n,k) = n×(n−1)×…/(k×(k−1)×…×1).
trading finance
Geometric series, perpetuities, and growing perpetuities
An infinite geometric series with ratio |r|<1 sums to a/(1−r). In finance this gives perpetuity value PV = C/r and growing perpetuity PV = C/(r−g). These appear in every valuation question.
Options Greeks: delta, gamma, vega, theta intuition
The Greeks measure how an option's price changes with respect to different inputs. Know them as intuitions, not formulas.
Kelly Criterion: optimal bet sizing
Kelly maximizes long-run wealth growth by betting a fixed fraction of wealth. Overbetting kills compounding; underbetting leaves returns on the table.
Rule of 72: compound interest mental math
Divide 72 by the annual rate to find how many years to double your money. Works for rates 1–20%.
Sharpe ratio: risk-adjusted return
Sharpe = excess return / volatility. Higher is better. Most serious strategies target Sharpe 1–2. Anything above 3 backtested is probably overfitted.
Yield and rate intuition: bonds and fixed income
Bond prices and yields move inversely. Duration tells you how much. 1 bp rise in yield ≈ −Duration% in price.
Market cap anchors: know the scale of major companies
Memorize a market cap ladder from $1B to $3T. These anchors let you immediately evaluate whether any market cap estimate is plausible.
GDP and macro anchors: global and US economic reference
Know the GDP ladder from small countries to global totals. These anchor Fermi estimates about markets, revenues, and policy.
US population anchors: cities, states, demographics
Know key US population numbers. These anchor estimates for market size, voter counts, consumer spending, and workforce calculations.
Logarithm shortcuts: log₁₀ and ln intuition
Know log₁₀ of common numbers. Logs appear in scoring, information theory, and order-of-magnitude calculations — including Fermiq's own scoring formula.
Scientific notation and order-of-magnitude arithmetic
Express large numbers as a × 10ⁿ. Multiply by adding exponents; divide by subtracting. Essential for Fermi estimation.