Finance & statistics drills

doubling time in one division

Divide 72 by the growth rate and you have the doubling time. It runs both ways, and stacking doublings gives you compound growth without a calculator.

discounting a single cash flow

One cash flow, one discount factor. Build the factor from the percentage blocks you already drill: 10% a year for two years is roughly a 17% haircut, not 20%.

CF / (r − g)

A stream forever is one division. Growth simply shrinks the denominator — and because it shrinks it, small changes in g swing the value enormously.

a fixed stream for n years

An annuity is a perpetuity minus a perpetuity that starts later. The factor is bounded by 1/r, which is the fastest way to check your arithmetic.

three years and an outlay

Learn three discount factors by heart for the rate in play and an NPV becomes three multiplications and a subtraction. At 10% they are 0.91, 0.83 and 0.75.

years to get the money back

One division tells you the payback, and flipping it gives the crude annual return. Interviewers use it as a screen before any discounting happens.

rf + β × ERP

Two multiplications and an addition. Beta scales the market premium, the risk-free rate sets the floor — this is the number that feeds every DCF you will build.

blended cost of capital

A weighted mean where only the debt side gets the tax shield. Do the after-tax debt cost first, then blend — it keeps the weighting arithmetic clean.

Hamada in one division

Peer betas carry peer debt. Divide out the leverage factor to get asset beta, then multiply it back at your own D/E before it enters CAPM.

the tail of a DCF

Grow the final forecast year once, then divide by (WACC − g). Convert the result into an implied multiple immediately — that is how you tell whether it is credible.

EV/EBITDA to equity value

Multiples are one multiplication; the trap is the bridge. Enterprise value belongs to everybody, equity value only to shareholders — net debt is the difference.

reverse the Gordon formula

Any multiple is a statement about growth. Flip it into a spread over the discount rate and you can say out loud what the market is assuming.

EBIT(1−t) + D&A − capex − ΔNWC

Four moves from operating profit to cash. Tax it, add back the non-cash charge, then pay for the assets and the working capital the growth consumes.

margin × turnover × leverage

Split any return on equity into three drivers you can attack separately: how much you keep, how hard the assets work, and how much of it is borrowed.

DSO + DIO − DPO

How many days the business funds itself before the customer pays. Every day removed releases roughly one day of revenue in cash.

price move for a yield move

Duration turns a yield move into a price move: percentage change equals minus duration times the change in yield. Basis points are just hundredths of a percent.

2 × s₂ − s₁, then exact

The forward is whatever makes rolling short equal to locking long. Approximate it linearly first, then run the exact compounding if you need the decimals.

breakeven, intrinsic, parity

A call pays what the share exceeds the strike, minus what you paid. Breakeven is strike plus premium; put-call parity ties the whole set together.

equity in, equity out, MoM

Three levers make the return: EBITDA growth, multiple change and debt paydown. Size the entry cheque, build the exit equity, divide — then convert the multiple into an IRR.

compare the earnings yields

In a share-for-share deal, forget the model: invert both P/Es. If the target's earnings yield beats your own cost of paper, the deal adds to EPS.

fixed / contribution

Contribution per unit is what each sale leaves after its own costs. Divide the fixed base by it and you know how many units the business must sell before it earns anything.

60/40 at 9% and 3%

A portfolio return is a weighted mean: each weight times its return, added up. Do it as percentage blocks — 60% of 9 is 5.4 — never as a plain average of the returns.

E[x²] − (E[x])²

Never chase deviations one by one. Average the squares, subtract the square of the average — that is the variance, and its root is σ.

(r − rf) / σ

Return per unit of risk. Subtract the risk-free rate, divide by σ. The coefficient of variation is the same shape without the risk-free leg — σ / mean.

(x − μ) / σ

Every normal question becomes easy once the value is expressed in sigmas. Subtract the mean, divide by σ, and read the answer off the 68/95/99.7 ladder.

sigma ladder and tails

Three numbers cover most normal questions asked in an interview. Learn the ladder, then split what is left in half to get a single tail.

ρ = cov / σxσy

Covariance has unreadable units. Divide by both standard deviations and it collapses into a number between −1 and 1 you can actually interpret.

σp with a correlation term

Risks do not add — variances do, plus a cross term that carries the correlation. That cross term is the whole story of diversification.

cov/var, or ρ·σᵢ/σm

Beta is the slope of the stock on the market. Given covariance, divide by market variance; given correlation, scale the sigma ratio by ρ. Same number.

b = ρ·σy/σx, R² = ρ²

A simple regression is correlation in disguise. The slope is ρ scaled by the ratio of the spreads, and R² is just ρ squared.

SE = σ / √n

The spread of a sample mean is the spread of the data divided by the square root of the sample size. Precision is expensive: four times the data for twice the precision.

mean ± 2 SE

Ninety-five percent means roughly two standard errors either side. Compute SE, double it, and you have the interval — 1.96 only matters when you are writing it down.

t = estimate / SE

Regression output is read with one division. Coefficient over standard error gives t; anything past 2 in absolute value is significant at the usual 5% bar.

independence in three moves

Independent events multiply for 'and'. For 'or', add and subtract the overlap. For 'neither', take the complement — often the fastest route of the three.

base rates beat intuition

Never manipulate Bayes' formula in your head. Imagine 100 (or 1,000) cases, fill the four boxes, and read the answer as a share of the row you are told you are in.

Σ p × payoff

Decision trees, options and scenario cases all reduce to the same sum: each outcome weighted by its probability. Check the probabilities add to one before you start.

σ_daily × √252

Returns scale with time; volatility scales with the square root of time. Memorise √252 ≈ 16 and √12 ≈ 3.5 and the conversion is one multiplication.

np and √(npq)

Counting successes out of n tries: the mean is np and the spread is the root of npq. For rare events the Poisson shortcut applies — mean and variance are both np.

ln(1 + r) and when it matters

Log returns add across time, which is why volatility work uses them. They sit below simple returns, and the gap widens fast once moves exceed 10%.