The whole theory, in one place

Start with a lemonade stand. You buy lemons and sugar for 40, you sell drinks for 100. Revenue is 100, cost is 40, profit is 60. That is the whole of accounting in one line — everything that follows is detail about when to count things and where to record them.

Two groups put money into a business. Lenders (the bank, bondholders) hand over money and want it back with interest, whatever happens. Owners (shareholders) put money in and get whatever is left after everyone else is paid. That is the whole reason the balance sheet is shaped the way it is: assets on one side, and on the other side the two groups with a claim on them.

Profit and cash come apart as soon as you let a customer pay later, or you pay a supplier later, or you buy a machine that lasts five years. Nothing dishonest is happening — the accounts are simply matching effort to the period that earned it, while the bank account only knows about money that actually moved.

Profit = Revenue − Costs
Assets = Liabilities + Equity
Cash at end = Cash at start + Cash in − Cash out

Income statement: covers a period ('for the year ended 31 December'). Balance sheet: covers a moment ('as at 31 December'). Cash flow statement: covers a period again, and reconciles the two. Reading the date line at the top of a statement is the first professional habit to build.

The two links are simple. First: profit that is not paid out as dividends is added to retained earnings inside equity, so a profitable year makes the balance sheet grow. Second: the bottom line of the cash flow statement is the change in the cash line on the balance sheet. If those two do not hold, someone has made a mistake.

Closing equity = Opening equity + Net income − Dividends
Closing cash = Opening cash + Operating + Investing + Financing

'Current' means within twelve months. Current assets — cash, receivables, inventory — are expected to turn into cash within a year. Current liabilities — payables, overdrafts, the next year of loan repayments — must be settled within a year. Everything else is non-current: factories, long-term loans, and so on.

The difference between current assets and current liabilities is working capital. It tells you whether the company can pay next year's bills out of next year's short-term assets. Negative working capital is not automatically bad — supermarkets run on it, because customers pay instantly and suppliers wait 60 days — but it is always worth explaining.

Assets = Liabilities + Equity
Equity = Assets − Liabilities
Working capital = Current assets − Current liabilities

A margin is just a line of the income statement divided by revenue. Gross margin says how much of each krona survives the direct cost of the product. Operating margin says how much survives running the business. Net margin says how much reaches the owners. Three numbers, three different diagnoses.

An expense is not the same as a payment. Depreciation is an expense with no payment attached this year — the money left when the machine was bought. Paying down a loan is a payment with no expense attached — only the interest is an expense. Keeping those apart is most of what separates a confident reader from a confused one.

Revenue − COGS = Gross profit
Gross profit − Operating expenses = Operating profit (EBIT)
EBIT − Interest − Tax = Net income
Margin = Profit line / Revenue

Depreciation is added back because it was subtracted on the income statement without any money leaving. The cash left years earlier when the asset was bought — and that purchase appears in investing, not operating. Adding it back is bookkeeping housekeeping, not a favour to the company.

Working capital changes hit cash directly. If receivables rise, you sold more than you collected: cash falls. If inventory rises, you bought more than you sold: cash falls. If payables rise, you are paying suppliers later: cash rises. Growth almost always eats cash for this reason — which is why fast-growing profitable companies still need financing.

Operating cash flow = Net income + Depreciation − Increase in working capital
Free cash flow = Operating cash flow − Capital expenditure

In an interview you will not have a calculator, so build every ratio out of 10% and 1% blocks. 10% of 480 is 48; 1% is 4.8. Any percentage is a small sum of those. To divide, round to a friendly number first and adjust: 38/500 is close to 40/500 = 8%, minus a bit, so about 7.6%.

Return on equity is profit divided by what owners put in; return on assets is profit divided by everything the company uses. The gap between them is leverage — borrowing raises ROE when things go well and deepens the hole when they do not. The current ratio (current assets over current liabilities) asks whether next year's bills are covered, and debt-to-equity asks how much of the company is borrowed.

Net margin = Net income / Revenue
Growth = (New − Old) / Old
ROE = Net income / Equity      ROA = Net income / Assets
Current ratio = Current assets / Current liabilities
Debt / Equity = Total debt / Equity

Simple interest pays on the original amount only: 100 at 10% for three years gives 130. Compound interest pays on the interest too: 110, then 121, then 133. Over one year they agree; over twenty years they are not remotely the same. Nearly all real finance compounds.

Discounting is the mirror image. If money grows at 10% a year, then 110 a year from now is worth 100 today — divide instead of multiply. The number you divide by is the discount rate, and it stands for what you could have earned elsewhere at similar risk. A higher rate means the future is worth less today, which is why rising interest rates push asset prices down.

Future value = PV × (1 + r)^n
Present value = FV / (1 + r)^n
Rule of 72: years to double ≈ 72 / interest rate in %

Lenders get paid first and get a fixed amount, so they take less risk and demand less return. Shareholders get paid last and get whatever remains, so they take more risk and demand more return. That single ranking explains nearly every financing decision a company makes.

'Cost of capital' sounds technical but is simply the return the people funding you expect. If lenders want 5% and shareholders want 12%, and the company is funded half and half, the blended expectation is around 8.5%. Any project earning less than that destroys value even if it makes an accounting profit — because the money could have been used for something that cleared the bar.

Cost of capital ≈ (share of debt × cost of debt) + (share of equity × cost of equity)
After tax, interest costs less: cost of debt × (1 − tax rate)

Everything in finance starts with reading the three financial statements. The balance sheet is a snapshot of what a company owns (assets), what it owes (liabilities) and the residual belonging to shareholders (equity) at a point in time — assets = liabilities + equity, always. The income statement measures performance over a period: revenue minus expenses yields net income. The cash flow statement reconciles net income to actual cash generated, split into operating, investing and financing. Net income is an opinion; cash flow is a fact.

You must be able to trace a transaction through all three statements. Buy equipment for 1M with cash: the balance sheet shows equipment up 1M and cash down 1M (net zero change in assets); the income statement is unaffected at purchase because depreciation hits later; the cash flow statement shows −1M in investing. This three-statement linkage is the foundation of every financial model.

Key ratios to internalise: ROE (net income / equity) measures return to shareholders, ROA (net income / total assets) measures how efficiently assets generate profit, and ROIC (NOPAT / invested capital) strips out capital structure and measures operating efficiency — the ratio that connects most directly to value creation. DuPont decomposition breaks ROE into margin × turnover × leverage, which tells you where profitability actually comes from.

Assets = Liabilities + Equity
ROE = Net income / Equity
ROIC = NOPAT / Invested capital
DuPont: ROE = (NI/Sales) × (Sales/Assets) × (Assets/Equity)

A krona today is worth more than a krona tomorrow because it can be invested. Present value discounts a future cash flow back to today; future value compounds a current amount forward. Net present value sums the present values of all cash flows from an investment, including the initial outlay. If NPV > 0 the investment creates value. This is the decision rule that governs all of corporate finance.

PV = CF / (1 + r)^t
NPV = −Investment + Σ [CF_t / (1 + r)^t]
Annuity PV = CF × [(1 − (1+r)^−n) / r]
Perpetuity PV = CF / r
Growing perpetuity PV = CF / (r − g)
Rule of 72: doubling time ≈ 72 / r%

Internal rate of return is the discount rate that makes NPV zero — it answers "what return does this earn?". Use it for ranking, not accept/reject: it misleads with non-conventional cash flows and mutually exclusive projects. Payback period measures how long until you recover the outlay; simple and intuitive but ignores time value and everything after payback. Discounted payback fixes the time-value problem and still ignores post-payback flows.

Corporate finance answers three questions: what to invest in (capital budgeting), how to fund it (capital structure) and how much to return to shareholders (payout policy). Capital budgeting applies NPV to real decisions — estimate incremental cash flows, discount at the project's cost of capital, accept if NPV > 0. Incremental means ignoring sunk costs, including opportunity costs, and accounting for cannibalisation of existing products.

WACC = (E/V) × Re + (D/V) × Rd × (1 − Tc)
CAPM: Re = Rf + β × (Rm − Rf)
Levered beta: βL = βU × [1 + (1 − Tc) × D/E]

Capital structure: Modigliani–Miller (1958) says structure does not affect firm value in a perfect market. In reality it does, because of taxes (interest is deductible, creating a tax shield), bankruptcy costs and agency conflicts. Trade-off theory balances the tax benefit of debt against expected distress costs. Pecking order theory says firms prefer internal funds, then debt, then equity, because issuing equity signals management thinks the stock is overvalued.

Payout policy: dividends versus buybacks. Miller–Modigliani says payout is irrelevant in perfect markets. In practice dividends signal stability while buybacks are flexible and tax-efficient. The signalling hypothesis explains why cutting a dividend hammers the stock — it signals management no longer expects to sustain earnings.

Fixed income first, because it is where discounting becomes mechanical. A bond's price is the present value of coupons plus principal. Price and yield move inversely. Duration measures price sensitivity to rate changes; DV01 converts it into a currency amount per basis point. Convexity captures the curvature that duration misses on large moves.

Bond price = Σ [C / (1+y)^t] + F / (1+y)^n
ΔPrice % ≈ −Modified duration × Δy
DV01 ≈ Modified duration × Price × 0.0001
Forward rate: (1+s2)² = (1+s1) × (1+f1,2)

The yield curve encodes expectations of rates, inflation and term premia; inversion has historically preceded recessions. On the equity side, prices are the present value of expected dividends or free cash flow; the Gordon growth model is the compact version. Market microstructure — bid-ask spreads, depth, order types — determines what you actually pay versus what the screen shows.

Markowitz showed that risk is a portfolio property, not an asset property. Combining assets with correlations below 1 reduces variance without giving up expected return — the only free lunch in finance. The efficient frontier is the set of portfolios with maximum return for each level of risk; adding a risk-free asset produces the capital market line and the tangency portfolio.

σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2
Sharpe = (Rp − Rf) / σp
CAPM: E(Ri) = Rf + βi × (Rm − Rf)
βi = Cov(Ri, Rm) / Var(Rm)

CAPM says only systematic risk is compensated because idiosyncratic risk is diversifiable. Empirically it explains less than it should, which is why factor models dominate practice: Fama–French three-factor adds size (SMB) and value (HML), Carhart adds momentum, and the five-factor version adds profitability and investment. These are what practitioners use for attribution and risk decomposition.

Derivatives derive value from an underlying. Four building blocks: forwards (bilateral contracts to trade at a future date), futures (standardised, exchange-traded, daily settled), options (the right but not the obligation) and swaps (exchanging cash-flow streams, most often fixed for floating).

Option pricing rests on no-arbitrage: two portfolios with identical payoffs in every state must have the same price. The binomial model builds a tree of up/down moves and prices by replication, working backwards to today. Black–Scholes is the continuous-time limit, assuming geometric Brownian motion and constant volatility.

C = S×N(d1) − K×e^(−rT)×N(d2)
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d2 = d1 − σ√T
Put–call parity: C − P = S − K×e^(−rT)

The Greeks measure sensitivities: delta to spot, gamma to delta, theta to time decay, vega to volatility, rho to rates. Implied volatility is the volatility that makes model price equal market price — forward-looking, and the number traders actually quote.

Valuation is where finance becomes applied, and you need all three approaches. DCF: project free cash flows for 5–10 years, calculate a terminal value (usually a growing perpetuity), discount at WACC to get enterprise value, subtract net debt for equity value, divide by shares for value per share. A DCF is only as good as its assumptions, so sensitivity on WACC and terminal growth is mandatory.

EV = Σ [FCF_t / (1+WACC)^t] + TV / (1+WACC)^n
TV = FCF_(n+1) / (WACC − g)
Equity value = EV − Net debt
Price per share = Equity value / Shares outstanding
FCF = EBIT×(1−Tc) + D&A − Capex − ΔNWC

Relative valuation compares a company's ratio to peers: P/E, EV/EBITDA, EV/Revenue for high-growth or unprofitable firms, P/B for banks and asset-heavy businesses. EV-based multiples are capital-structure neutral; P/E is not. Precedent transactions look at what acquirers actually paid, including a control premium of typically 20–40%.

Start with the price. The offer is made for equity, but you acquire the whole capital structure, so the purchase enterprise value is the offer equity value plus assumed net debt. On top of the target's unaffected share price sits a control premium — typically 20-40% in public deals — which is the buyer's payment for the right to direct the assets. Everything the buyer hopes to earn back sits in synergies and in better management of the same assets.

Synergies come in two families and they are not equally credible. Cost synergies (overlapping head office, procurement scale, plant consolidation, IT rationalisation) are specific, controllable and land within 12-24 months, so a diligence team can underwrite them. Revenue synergies (cross-selling, wider distribution, pricing power) require customers to behave, arrive later and are routinely overestimated. Always phase them: a synergy worth 100 a year, ramping 30/70/100 over three years with one-off integration costs of 1.5x the annual run-rate, is worth far less than 100/WACC.

Purchase EV = Offer equity value + Net debt assumed
Control premium = Offer price / Unaffected price − 1
Pro-forma NI = NI_acq + NI_tgt + Synergies×(1−t) − After-tax new interest − Incremental D&A
Pro-forma EPS = Pro-forma NI / (Shares_acq + New shares issued)
Accretive if Pro-forma EPS > Standalone EPS_acq
All-stock rule of thumb: accretive when P/E_acquirer > P/E_target (pre-synergy)
Cash deal breakeven: after-tax cost of debt < Target earnings yield (E/P)
PV of synergies = Σ [Phased synergy_t ×(1−t) / (1+WACC)^t] − Integration cost

Consideration structure carries the risk-sharing message. Cash is certain, uses balance-sheet capacity and signals that the acquirer believes its own shares are undervalued. Stock shares both upside and downside with target holders and signals the opposite, which is why acquirer shares usually fall on announcement of an all-stock deal. Fixed exchange ratios pass market risk to the seller; fixed value passes it to the buyer. Earn-outs bridge disagreement about the target's forecast, and contingent value rights do the same for a specific event such as a trial result.

Merger arbitrage is the market's own probability estimate of a deal closing. If a target trades at 92 against a 100 cash offer expected to close in six months, the spread compensates for deal risk: regulatory blocks, financing failure, a shareholder vote, or a material adverse change. Implied probability ≈ (Current − Standalone) / (Offer − Standalone). Quoting that number turns a qualitative 'will it close?' into a number the interviewer can argue with.

Peer selection decides the answer, so it is where you spend the effort. Screen on business model and end market first, then size, growth and margin, then geography and capital intensity. Six to ten tight comparables beat twenty loose ones. Write down explicitly why each name is in the set, because the first interviewer question is always 'why is that a comparable?'.

EV = Market cap + Debt + Minorities + Preferred − Cash
EV multiples pair with pre-interest metrics: Revenue, EBITDA, EBIT, unlevered FCF
Equity multiples pair with post-interest metrics: Net income (P/E), Book equity (P/B)
Implied EV = Metric × Peer multiple; Equity value = EV − Net debt
PEG = P/E / Growth rate (%)
Multiple from fundamentals: EV/EBIT ≈ (1 − t)(1 − g/ROIC) / (WACC − g)

Three mechanics separate a real comps table from a spreadsheet of ratios. Forward beats trailing, because value depends on the future and trailing multiples are distorted by whatever just happened. Calendarize so every company is on the same year-end — a company with a June year-end must be interpolated onto December before you take a median. And adjust the metrics: strip one-off restructuring, litigation and gains on disposal, capitalise or normalise leases consistently, and treat capitalised R&D the same way across the set.

Precedent transactions are the third leg. They use actual prices paid for whole companies, so they embed a control premium and, usually, buyer-specific synergies — which is why precedent multiples sit above trading comps. They are backward-looking and cycle-dependent: a deal struck at the top of a credit cycle tells you little about today. Use the most recent, most similar three to five deals, note the announcement date, and adjust for the market environment before quoting them.

Build in a fixed order. Sources & uses first: uses are the purchase enterprise value, refinanced debt, and fees; sources are the debt tranches plus the sponsor equity that plugs the gap. Then the operating case: revenue, margin, capex and working capital. Then the debt schedule with each tranche's rate, amortisation and cash sweep. Then the exit and the returns. Any model that computes returns before it has a debt schedule is guessing.

Entry EV = Entry EBITDA × Entry multiple
Uses = Entry EV + Refinanced debt + Fees;  Sources = Debt tranches + Sponsor equity
Sponsor equity = Uses − Debt raised
Cash available for debt service = EBITDA − Cash interest − Cash taxes − Capex − ΔWorking capital
Cash sweep = (Cash available − Mandatory amortisation) × Sweep %
Exit EV = Exit EBITDA × Exit multiple;  Exit equity = Exit EV − Net debt at exit
MoM = Exit equity / Sponsor equity;  IRR = MoM^(1/n) − 1
Leverage = Net debt / EBITDA;  Interest coverage = EBITDA / Cash interest

Sensitivity is where the interview happens. IRR is most sensitive to the exit multiple, then entry leverage, then margin improvement, then growth. A one-turn move in the exit multiple on 255 of exit EBITDA is 255 of equity value — in the example above roughly 3-4 points of IRR. Build the two-way table (entry multiple against exit multiple, and leverage against margin) and know which cell is your base case. Sponsors typically underwrite to a 20-25% IRR and a 2.0-2.5x MoM over five years.

Value at Risk answers: what is the maximum loss over a horizon at a confidence level? A 1-day 95% VaR of 2M means a 5% chance of losing more than 2M in one day. Three methods: historical simulation, parametric variance-covariance, and Monte Carlo. VaR says nothing about the size of losses beyond the threshold and assumes stable correlations that break in crises.

Parametric VaR = z_α × σ × Portfolio value
σ_annual = σ_daily × √252
Expected shortfall = E[loss | loss > VaR]

Expected shortfall (CVaR) fixes the tail problem by averaging losses beyond VaR — which is why Basel III moved to ES for market-risk capital. Stress testing complements statistical measures by asking what happens under a specific extreme scenario: a rate spike, a liquidity freeze, a counterparty default. Good risk management uses all three.

Financial analysis is diagnosis, not description. The sequence that professionals actually follow is: wealth creation (is the company growing profitably?), then investment (how much capital does that growth consume?), then financing (who funded it and on what terms?), then returns (does the return exceed the cost of the capital?). Anyone can quote a ratio; the skill is ordering ratios into an argument about whether the business model works.

Start with the margin ladder. Gross margin tells you about pricing power and input costs. EBITDA margin adds the fixed cost base and tells you about operating leverage. EBIT margin adds the capital intensity hidden in depreciation. Net margin adds financing and tax. When margins move, isolate which rung moved — a gross-margin fall is a commercial problem, an EBIT-margin fall with flat gross margin is an overhead or capacity problem.

Then look at capital. Capital employed is fixed assets plus working capital; ROCE is EBIT after tax divided by capital employed. Growth that raises revenue but raises capital employed faster destroys value even as profits rise — this is the single most common trap in growth cases. Working capital is the quiet killer: a company can be profitable and still fail because receivables and inventory absorb cash faster than earnings produce it. Read the cash conversion cycle as days: inventory days plus receivable days minus payable days.

Capital employed = Fixed assets + Working capital
ROCE = EBIT × (1 − t) / Capital employed
Working capital = Inventory + Receivables − Payables
Cash conversion cycle = DIO + DSO − DPO
Interest cover = EBIT / Interest expense
Net debt / EBITDA  (leverage);  FFO / Net debt  (repayment capacity)

Solvency analysis asks the reverse question: can the company survive its own balance sheet? Three lenses. Liquidity — will cash arrive before obligations fall due (current ratio, quick ratio)? Leverage — how much debt relative to earnings power (net debt / EBITDA, gearing)? Service — can earnings cover the interest (interest cover, FFO / net debt)? Lenders look at the last two; equity analysts often forget them until the covenant breaks.

A bond is a contract: fixed coupons for a period, then the face value back. Its price is simply the present value of that promised stream at the yield the market currently demands. Because the cash flows are fixed and only the discount rate moves, bond pricing is the cleanest possible application of time value of money — and it makes the inverse relationship unavoidable: when required yields rise, prices fall.

P = Σ C/(1+y)^t + F/(1+y)^n
Current yield = Annual coupon / Price
Coupon rate > y → premium bond;  coupon rate < y → discount bond
Approx. price change = −Modified duration × Δy + ½ × Convexity × (Δy)²

Yield to maturity is the single discount rate that makes the present value of the promised cash flows equal the market price. It is an internal rate of return, so it silently assumes coupons are reinvested at that same rate — which is why realised return rarely equals YTM. Duration then converts yield changes into price sensitivity: Macaulay duration is the weighted average time to cash flow, and modified duration is the percentage price change per one percentage point of yield. Convexity is the curvature correction that matters once moves get large.

The yield curve plots yield against maturity. An upward slope is normal and reflects term premium plus growth expectations; a flat curve signals uncertainty; an inverted curve — short rates above long rates — has preceded most recessions because it means the market expects rate cuts. Spreads over the government curve price credit risk: default probability, loss given default and liquidity. Credit ratings summarise that judgement, and the investment-grade to high-yield boundary drives large forced flows because many mandates cannot hold below it.

A share is a claim on residual cash flows with no maturity date, so its value is the present value of everything shareholders will ever receive. The dividend discount model states this directly, and the Gordon growth form collapses it into one line: value equals next year's dividend divided by the required return minus the growth rate. Its power is not precision but sensitivity analysis — it shows exactly how much of a share price is a bet on growth.

P₀ = D₁ / (r − g)
g = ROE × retention ratio (sustainable growth)
P/E = payout ratio / (r − g)
Justified P/B = (ROE − g) / (r − g)
Earnings yield = 1 / P/E;  Total return ≈ dividend yield + growth

Multiples are shorthand for the same discounted logic. A P/E of 20 embeds assumptions about growth, risk and payout; if you can rewrite a multiple as its implied growth rate you can argue about the assumption instead of the number. Use EV-based multiples (EV/EBITDA, EV/EBIT, EV/Sales) when capital structures differ, and equity multiples (P/E, P/B) when comparing within a sector with similar leverage. Always match the numerator to the claim: enterprise value pairs with pre-interest earnings, price pairs with post-interest earnings.

Two-stage models handle real companies: an explicit high-growth phase, then a terminal Gordon phase once growth converges toward the economy's rate. Terminal growth above long-run GDP growth implies a company eventually becomes the entire economy, which is where most amateur models break. Discipline the terminal value — if it exceeds roughly 75% of your total value, your answer is a growth assumption wearing a spreadsheet.

Modigliani–Miller is the reference point, not the answer. In a world with no taxes, no bankruptcy costs and no information asymmetry, capital structure is irrelevant: the value of a firm comes from its assets, and slicing the claims differently only reallocates risk. The theorem is useful precisely because it tells you where to look — every real-world capital structure decision is an argument about which MM assumption fails.

MM I (no tax):  V_L = V_U
MM II:  r_E = r_A + (r_A − r_D) × D/E
With tax:  V_L = V_U + t × D  (value of the tax shield)
β_E = β_A × (1 + (1 − t) × D/E)
WACC = E/V × r_E + D/V × r_D × (1 − t)

Add corporate tax and debt gains value because interest is deductible: the tax shield is worth roughly the tax rate times the debt. Add financial distress and the gain reverses at high leverage — customers leave, suppliers tighten terms, managers make short-horizon decisions, and refinancing risk becomes existential. Trade-off theory sets the optimum where the marginal tax shield equals the marginal expected distress cost. Pecking order theory adds the practical observation that managers prefer internal funds, then debt, then equity, because issuing equity signals overvaluation.

Payout policy carries the same information logic. Dividends are sticky, so raising them signals confidence in sustainable cash flow, and cutting them is read as distress. Buybacks are flexible and signal that management thinks the shares are cheap — they also raise EPS mechanically by shrinking the share count, which is why EPS accretion alone is never evidence of value creation. A company returning cash is telling you it has no reinvestment opportunities above its cost of capital; whether that is discipline or decline is the strategic question.

An exchange rate is a price with two directions, and most FX errors are direction errors. Fix the convention first: in EUR/USD 1.10, one euro buys 1.10 dollars — the base currency is quoted first. A rise in the number means the base currency strengthened. Get this wrong in an interview and every subsequent number is inverted.

Covered interest parity:  F/S = (1 + i_quote) / (1 + i_base)
Uncovered parity:  E[ΔS] ≈ i_quote − i_base
Relative PPP:  ΔS ≈ inflation_quote − inflation_base
Cross rate:  A/C = A/B × B/C
Real exchange rate = Nominal × (Price level foreign / Price level domestic)

Interest rate parity is the arbitrage backbone: a currency with higher interest rates trades at a forward discount, otherwise a riskless profit would exist. Purchasing power parity is the long-run anchor — currencies drift toward equalising the price of tradable goods — but it holds over years, not quarters. In the short run, capital flows, rate expectations and risk appetite dominate, which is why carry trades work until they suddenly do not.

Corporates face three exposures. Transaction exposure is contracted cash flows in foreign currency, hedged with forwards, futures or options. Translation exposure is the accounting effect of consolidating foreign subsidiaries, managed by matching the currency of assets and liabilities. Economic exposure is the competitive effect of currency moves on volumes and prices — the largest and least hedgeable. For valuation, either discount foreign cash flows at a foreign-currency rate and convert at spot, or convert cash flows using forward rates and discount at the domestic rate; never mix the two.

Berk and DeMarzo organise corporate finance around three decisions: what assets to buy (investment), how to pay for them (financing), and how much cash to return to shareholders (payout). Every other topic — governance, reporting, capital structure, options, M&A, valuation — is a tool for making those three decisions correctly. The unifying rule is NPV: take projects whose incremental cash flows, discounted at the opportunity cost of capital, create value.

NPV = Σ CF_t / (1 + r)^t  −  Initial investment
IRR: the r that makes NPV = 0; accept if IRR > hurdle rate
Payback: years to recover initial outlay; ignores time value and tail cash flows
Profitability index = PV of future cash flows / Initial investment

The NPV rule is robust because it measures value creation directly. IRR is popular for communicating a percentage return, but it fails when cash flows change sign more than once or when projects are mutually exclusive — a higher IRR can hide a lower NPV. Payback is a liquidity screen, not a value metric. Profitability index helps rank projects under a capital constraint, but the constraint itself is usually softer than it looks.

Corporate governance exists because managers, shareholders and debtholders do not share the same interests. Agency costs arise when managers empire-build, shirk, or invest in pet projects with negative NPV. Boards, concentrated ownership, debt covenants, executive pay and the market for corporate control are the mechanisms that align incentives. Graham's presidential address stresses that the clean models are a starting point; the real world adds taxes, behavioural biases, financial frictions and institutions that vary across countries and time.

Free cash flow problem: managers retain cash and over-invest when growth opportunities are poor
Debt as discipline: mandatory interest payments reduce discretionary cash
WACC = (E/V) × Re + (D/V) × Rd × (1 − Tc)
Project cost of capital ≠ firm WACC when project risk differs from average firm risk

Corporate reporting is the language of those decisions. The income statement, balance sheet and cash flow statement are linked by the fundamental identity and by the change in retained earnings. A consultant who cannot trace a transaction through all three statements cannot build a reliable model. Reporting quality matters because accrual accounting gives managers discretion: revenue recognition, depreciation choices, inventory methods and off-balance-sheet vehicles can all shift reported earnings without shifting cash.

Capital budgeting puts the NPV rule into practice. Estimate incremental revenues, costs, taxes, working capital and capex; choose a discount rate that reflects project risk; compute NPV, IRR and payback; then stress-test with sensitivity and scenario analysis. The cost of capital is usually the most fought-over number: CAPM gives the equity cost, the debt cost is observable from yields or credit spreads, and the weights should be target-market-value weights, not book weights.

Long-term financial planning links the investment plan to the sources of funds. If a firm grows faster than its internal cash generation and target leverage allow, it needs external equity or debt. The sustainable growth rate — ROE times retention — is the speed limit of growth without changing capital structure. Financial planning is not a forecast; it is a consistency check between strategy, operations and the balance sheet.

Sustainable growth rate = ROE × retention ratio
External financing needed = ΔAssets − ΔSpontaneous liabilities − Retained earnings
Retention ratio = 1 − payout ratio
Plowback drives future earnings; payout returns cash shareholders can reinvest themselves

Capital structure is where the textbook meets reality. Modigliani–Miller says value is independent of financing in a perfect world; the real debate is about which market imperfection matters most. The trade-off theory weighs the tax shield of debt against expected distress costs. Pecking order theory says firms prefer internal funds, then debt, then equity, because issuing equity signals overvaluation. Graham's evidence shows that firms do not always optimise taxes, that leverage is more influenced by market timing and institutional factors than by a single target, and that the 'right' debt ratio varies by industry, profitability and volatility.

Payout policy is the mirror of investment policy. If a firm has no positive-NPV projects, returning cash is the right answer. Dividends are sticky and signal stability; buybacks are flexible and signal undervaluation. In perfect markets payout is irrelevant; in practice it conveys information, affects leverage, and has tax consequences. The key interview insight is that a buyback raises EPS mechanically but only creates value if the shares were undervalued.

Options, real options and risk management extend NPV by recognising that managers can adapt. A real option is the right, but not the obligation, to make a future investment decision — to expand, abandon, delay or switch. Option pricing is useful because standard NPV ignores flexibility; it can also be abused by calling every uncertainty an option. Risk management with forwards, futures, swaps and options does not eliminate risk but converts it into a known cost, which stabilises cash flows and protects investment capacity.

Mergers and acquisitions are capital-budgeting decisions applied to whole companies. The acquirer pays a control premium in exchange for synergies, governance changes or strategic optionality. Accretion/dilution analysis compares pro-forma EPS, but the value test is whether the present value of synergies exceeds the premium paid. Valuation methods — DCF, comparable companies, precedent transactions — answer different questions and should be triangulated, not chosen to justify a predetermined answer.

Premium paid = Offer price / Target pre-bid price − 1
Synergy value = PV(revenue synergies) + PV(cost synergies) − integration costs
Value created for acquirer = Synergy value − Premium paid
Accretion = pro-forma EPS rises; dilution = pro-forma EPS falls

The single most useful check takes thirty seconds: compare cumulative net income to cumulative operating cash flow over three to five years. In a healthy business the two track each other, with growth absorbing some cash into working capital. When net income runs persistently ahead of operating cash flow, profit is being recognised before cash arrives, and receivables or inventory should show where it is sitting.

Cash conversion = Operating cash flow / Net income  (want ≈ 1 or better over a cycle)
Accrual ratio = (Net income − Operating CF − Investing CF) / Average net operating assets
Days sales outstanding = Receivables / Revenue × 365
Days inventory = Inventory / COGS × 365;  Days payable = Payables / COGS × 365
Cash conversion cycle = DSO + DIO − DPO
Free cash flow = Operating CF − Capex  (the number that cannot be accrued away)

The frictions produce four working theories. Signalling: raising a dividend is credible because it is costly to reverse, so it conveys management's confidence in sustainable cash flow. Clientele: income investors, pension funds and growth investors self-select into different payout profiles, so consistency matters more than level. Agency: paying cash out removes the temptation to fund weak projects, the same discipline leverage provides. Taxes: buybacks defer the tax event to the shareholder's choosing, which is why they have overtaken dividends in most markets.

Payout ratio = Dividends / Net income;  Retention b = 1 − payout
Sustainable growth g = ROE × b
Dividend yield = DPS / Price;  Total shareholder yield = (Dividends + Net buybacks) / Market cap
Buyback: new EPS = Net income / (Shares − Cash used / Price)
Buyback accretive when Earnings yield (E/P) > After-tax return on the cash used
Sustainable dividend ≈ Free cash flow to equity, not net income

Start with the fixed/variable split, because every operational decision hinges on it. Contribution margin — price minus variable cost — is what each extra unit contributes to covering fixed cost. Once fixed costs are covered, contribution falls straight to profit, which is why operating leverage magnifies both directions: a business with high fixed cost turns a 10% volume swing into a much larger profit swing.

Contribution margin = Price − Variable cost per unit
Break-even volume = Fixed costs / Contribution margin
Break-even revenue = Fixed costs / Contribution margin %
Degree of operating leverage = Contribution / Operating profit = %Δprofit / %Δvolume
Price variance = (Actual price − Budget price) × Actual volume
Volume variance = (Actual volume − Budget volume) × Budget contribution
Mix variance = Σ (Actual mix % − Budget mix %) × Actual total volume × Budget contribution per unit
ABC rate = Activity cost pool / Total cost-driver units

Activity-based costing replaces one plant-wide overhead rate with a rate per activity: setups, orders processed, deliveries, engineering changes, invoices. Cost is traced to the activity, then to the product or customer that consumes it. The result is usually a whale curve — cumulative profit rises to a peak well above 100% of reported profit and then falls back, because a tail of small, complex, high-service customers costs more to serve than they pay. That chart is the starting point for pricing, service-level and rationalisation decisions.

Start here. Everything earlier in this track assumed the firm behaves like one rational decision-maker: it takes positive-NPV projects and returns the rest of the cash. Real firms are run by people who do not own them. Agency theory is the study of that wedge — what happens when the person deciding and the person paying are not the same person — and it is the single most useful lens for explaining why a company that looks good on a spreadsheet destroys value in practice.

Agency cost  =  Monitoring cost  +  Bonding cost  +  Residual loss
Monitoring: audits, boards, reporting systems, analysts, covenants (paid by the principal)
Bonding: the agent voluntarily constrains himself — equity co-investment, debt, dividends, guarantees
Residual loss: the value still lost after both, because perfect alignment is unaffordable
Firm value  =  First-best value  −  Agency cost of equity  −  Agency cost of debt

Start here. If agency theory says what can go wrong, governance is the list of things that stop it. Split the list in two: internal mechanisms the firm designs itself (board, committees, pay, ownership, internal control) and external mechanisms the environment imposes (takeover threat, labour market, product-market competition, lenders, auditors, analysts, regulators). No single mechanism is decisive; they substitute for each other, which is why the same board structure looks effective in one country and useless in another.

Votes held  =  10 × A shares  +  1 × B shares
Wedge  =  vote share  −  capital share
Example: 2M A shares and 8M B shares → votes = 20M + 8M = 28M
A holder owning all A shares: capital share = 2/10 = 20%, vote share = 20/28 = 71%, wedge = 51 points
Control premium in a transaction reflects the value of the vote, not the cash flow

Start here. Managers know more about their firm than investors do, and investors know that. Every financing decision is therefore read as a message, not just as a transaction. This phase turns that observation into three usable results: the lemons discount on equity issuance, the cost-of-capital benefit of credible disclosure, and the rating-driven behaviour that dominates real capital-structure choices.

Pecking order:  retained earnings  →  secured debt  →  unsecured debt  →  convertibles  →  equity
Reason: information sensitivity of the claim. Debt payoff is flat, so mispricing matters less than for equity
Announcement effects (typical): equity issue ≈ −2% to −3%,  debt issue ≈ 0%,  buyback ≈ +2% to +3%
Cost of an issue  =  underwriting fee  +  underpricing  +  announcement value loss
Signalling condition: an action signals quality only if it is more costly for a weak firm to imitate

Start here. Agency theory has two parties; real decisions have a dozen. This phase gives you a way to rank claimants instead of listing them, and then connects that to Williamson's question of where the firm's boundary should sit. The link is contracting: when a relationship cannot be governed by a contract, it has to be governed by ownership or by the board — and stakeholders whose contracts are incomplete are exactly the ones who become powerful.

Buy cost  =  price × volume  +  contracting cost  +  expected hold-up loss
Make cost  =  cash operating cost  +  capital charge (WACC × invested capital)  +  bureaucracy / lost focus
Expected hold-up loss  =  probability of renegotiation × value of specific investment at risk
Integrate when Make cost < Buy cost, and revisit whenever specificity or volume changes

Three approaches, and a serious valuation uses at least two. The income approach discounts cash flows, with a cost of equity built up rather than read off a beta. The market approach applies multiples from listed peers or private transactions, adjusted for size and liquidity. The asset approach values net assets at fair value and sets the floor — the number below which an owner would rather liquidate.

Build-up cost of equity = Risk-free + ERP + Size premium + Industry premium + Company-specific risk
DLOM (discount for lack of marketability): typically 10-30% on a minority interest
Control premium ≈ 1/(1 − minority discount) − 1;  minority discount = 1 − 1/(1 + control premium)
Value of controlling, marketable interest → less DLOM → controlling, non-marketable
Normalised EBITDA = Reported ± owner compensation, related-party rent, non-recurring items
Equity waterfall: EV → secured debt → unsecured debt → preferred → equity

Order of operations matters. Start from a marketable, minority basis if you used trading comps, add a control premium if you are valuing a controlling stake, then apply the marketability discount last, because it reflects the time and cost of exit for whatever interest you hold. Applying a control premium and no marketability discount, or both to the same base, is the most common error in private valuation reports.

Now the distressed case. When ROIC sits below WACC for long enough, the enterprise is worth less than its debt and the capital structure has to be rewritten. Out-of-court workouts — amend-and-extend, covenant waivers, debt-for-equity swaps — are faster and cheaper but need near-unanimous creditor consent. Formal processes (Chapter 11 in the US, företagsrekonstruktion in Sweden) impose a stay on creditors, allow DIP or super-priority financing that jumps the queue, and can bind dissenting classes through a cram-down.

Layout first. Three zones: inputs (blue font, one cell per assumption, units labelled), calculations (black, left to right in time), outputs (the two or three numbers the client actually reads). Never repeat an input in two places. Never mix a constant into a formula — 'Revenue × 1.03' hides a growth assumption nobody can find. One formula, copied cleanly across the row, is both faster to build and far easier to check.

INDEX(range, MATCH(key, keys, 0))            — robust two-way lookup
XLOOKUP(key, keys, values, "n/a")            — modern replacement, exact-match by default
VLOOKUP breaks when columns move; INDEX/MATCH and XLOOKUP do not
SUMIFS / COUNTIFS / AVERAGEIFS               — aggregate by criteria instead of nested IFs
IFERROR(value, "check")                      — trap errors without hiding them
OFFSET / INDIRECT                            — powerful, volatile, and unauditable: avoid
Data → What-If → Data Table                  — one- and two-way sensitivity on any output
Named ranges: WACC, TaxRate, ExitMultiple     — formulas read like the model, not like coordinates

Circular references are the classic modelling trap: interest depends on average debt, debt depends on cash flow, cash flow depends on interest. Three ways out. Use opening-balance interest, which is simple, defensible and almost always accurate enough. Or enable iterative calculation, which resolves it but makes the model fragile and hard to audit. Or break the loop with a small manual convergence — a copy-paste of the interest figure. Say which one you chose and why; interviewers ask.

Porter's Five Forces is the workhorse for industry analysis: threat of new entrants, supplier power, buyer power, threat of substitutes, and rivalry. The output is not a score — it is a diagnosis of which forces constrain profitability and what a company can do about them. In technology-intensive industries, network effects and switching costs reshape the standard analysis.

PESTEL maps the macro environment (political, economic, social, technological, environmental, legal). It is context-setting, not conclusion-drawing: combine it with Five Forces for the full external picture. The industry life cycle — introduction, growth, maturity, decline — changes the playbook at each stage: growth strategy chases share, maturity strategy chases efficiency and positioning.

The Resource-Based View argues that sustained advantage comes from resources and capabilities that are valuable, rare, inimitable and non-substitutable (VRIN, later VRIO where O = organised to capture value). The implication: strategy should be built on what the firm uniquely does well, not on which market looks attractive.

Value chain analysis (Porter, 1985) decomposes a firm into primary activities — inbound logistics, operations, outbound logistics, marketing and sales, service — and support activities. The question is where in the chain the firm creates more value than rivals, and where activities should be outsourced. Dynamic capabilities extend RBV: the ability to sense, seize and reconfigure matters more than any static resource when the environment moves.

Porter's generic strategies — cost leadership, differentiation and focus — force a choice: being stuck in the middle means competing on price without the cost base to survive it. The Ansoff matrix maps growth along existing versus new products and existing versus new markets: market penetration, product development, market development, diversification, in ascending order of risk.

The BCG matrix classifies business units by market growth and relative share — stars, cash cows, question marks, dogs — and treats the portfolio as a cash-allocation problem. Blue Ocean Strategy argues the most profitable move is to create uncontested market space by changing what the industry competes on, rather than out-executing rivals on the same dimensions.

A strategy without execution is a presentation. The Balanced Scorecard (Kaplan and Norton) translates strategy into four measurement perspectives — financial, customer, internal process, learning and growth — so that leading indicators are tracked alongside lagging financial results. Strategy maps make the causal chain explicit: capability investments drive process quality, which drives customer outcomes, which drive financial results.

Organisational design follows strategy: functional structures favour efficiency, divisional structures favour responsiveness, matrix structures trade clarity for coordination. Game theory formalises competitive interaction. A Nash equilibrium is a set of strategies where no player gains by deviating unilaterally; the prisoner's dilemma explains why price wars break out even when everyone would prefer higher prices. Repeated games allow cooperation through reputation and tit-for-tat, and credible commitments change the game itself.

Capital allocation is strategy. Every investment decision is a strategic bet, and a DCF is a quantified strategic thesis. Companies that push capital toward projects with ROIC > WACC — and away from projects that do not clear the bar, however strategically appealing — consistently outperform.

M&A and value creation: most acquisitions destroy value for the acquirer's shareholders. A 20–40% control premium means synergies must exceed the premium just to break even. Strategic rationale must translate into quantifiable, achievable synergies, and the discipline is to walk away when the numbers do not work.

Financial strategy shapes competitive position. A fortress balance sheet lets a company invest through downturns while rivals retrench — a deliberate choice to accept lower returns in good times for optionality in bad times. High leverage amplifies returns and constrains strategic flexibility.

EVA = NOPAT − (WACC × Invested capital)
    = (ROIC − WACC) × Invested capital
EVA > 0 ⇒ value creation
EVA < 0 ⇒ value destruction, even if profitable

The interview and exam skill is not naming the lenses. It is choosing the one that fits the question and saying why the others are weaker here. An entry question in a concentrated, regulated market is an IO question. A question about why a leader lost share to a smaller rival after a technology shift is a dynamic-capabilities question. A make-or-buy question is transaction cost economics. A 'why did the board approve this obviously value-destroying deal' question is agency theory.

Head office plays four possible roles: portfolio manager (buy, hold, sell, allocate capital), restructurer (fix underperformers then exit), skills transferrer (move a capability across units) and activity sharer (run shared operations or functions). Only the last two require ownership rather than a fund; the first two must be justified against what an investor could do alone.

Value of group = Σ standalone unit values + synergies − corporate costs
Parenting advantage exists if that sum > best alternative ownership
Conglomerate discount = (Σ SOTP values − market cap) / Σ SOTP values
Break-up creates value when discount > separation and dis-synergy costs

Two economic effects push toward integration for different reasons. Hold-up: a supplier who has invested in equipment usable only for you can be squeezed, so it under-invests unless protected by ownership or a long contract. Double marginalisation: when an upstream monopolist and a downstream monopolist each add a margin, the final price is above the level that maximises joint profit, so integration raises volume, lowers price and raises combined profit.

Buy cost = supplier price × volume + contracting and monitoring cost
Make cost = variable cost × volume + fixed cost + capital charge
Capital charge = invested capital × WACC
Integrate if Make cost + flexibility loss < Buy cost + hold-up cost
Break-even volume = fixed cost + capital charge / (price − variable cost)

Framework criticism in one line each: Five Forces assumes stable industry boundaries and ignores complementors; RBV struggles to say how a resource is built; BCG assumes independent units and that share drives cost; disruption theory is often applied retrospectively; the Balanced Scorecard can multiply metrics without prioritising them. Saying one of these, and why it matters here, is what separates a top answer from a competent one.

Start with the distinction that everything else rests on. The population is every unit you care about: all customers, all trading days, all invoices. The sample is the part you actually measured. Statistics exists because you almost never have the population, and it works only when the sample was drawn in a way that does not systematically favour some units over others. A biased sample of a million is worse than a random sample of a hundred, because size shrinks noise but never touches bias.

Know your data type before you compute anything. Categorical data (industry, region, yes/no) can be counted and shared as proportions, but has no meaningful average. Ordinal data (satisfaction 1–5, credit rating) is ordered but the gaps are not equal, so a median is safer than a mean. Numeric data (revenue, price, days) supports the full toolkit. Half the bad charts in a case interview are a mean computed on something that has no middle.

For the middle: the mean is the balance point and uses every observation, which is also why one outlier moves it. The median is the middle observation and ignores how extreme the extremes are. The mode is the most common value, useful for categories. If mean and median differ a lot, the distribution is skewed — and in business data it usually is, because revenue per customer, deal size and firm size all have a long right tail.

For the spread: the range is fragile, the interquartile range is robust, and the standard deviation is the one everything downstream uses. Variance is the average squared distance from the mean; standard deviation is its square root, which puts it back into the units of the data. The coefficient of variation (sd ÷ mean) makes spread comparable across things measured on different scales — the correct way to say whether a 2% move in a bond is bigger news than a 2% move in a small-cap stock.

Mean  x̄ = Σx / n
Variance  s² = Σ(x − x̄)² / (n − 1)
Standard deviation  s = √s²
Coefficient of variation = s / x̄
z-score  z = (x − x̄) / s
Weighted mean = Σ(wᵢ xᵢ) / Σwᵢ
Empirical rule (normal-ish data): ≈68% within ±1s, ≈95% within ±2s, ≈99.7% within ±3s

Probability is the language of uncertainty, and finance is the business of pricing uncertainty. A random variable maps outcomes to numbers: discrete variables have probability mass functions, continuous ones have densities, and the CDF gives P(X ≤ x) for both. Bayes' theorem updates probabilities with new evidence — it is how credit models revise default probabilities, how fraud detection works, and how diagnostic tests are interpreted.

P(A|B) = P(B|A) × P(A) / P(B)
E(X) = Σ p_i × x_i
Var(X) = E[(X − μ)²] = E(X²) − μ²
P(A∪B) = P(A) + P(B) − P(A∩B)

Key distributions: normal (returns over short intervals), log-normal (prices, because returns are roughly normal and prices cannot go negative), binomial (successes in n trials, and the basis of binomial option pricing), Poisson (rare-event counts such as defaults or claims) and Student's t (heavier tails, for small samples and fat-tailed returns). The Central Limit Theorem says sample means converge to normal regardless of the underlying distribution; the Law of Large Numbers says they converge to the population mean. CLT tells you the shape, LLN the target.

Three return definitions, and interviewers do notice the difference. The simple return is (P₁ − P₀ + income)/P₀. The arithmetic mean return averages those period returns and answers "what do I expect next period?". The geometric mean compounds them and answers "what did I actually earn over the whole period?". The geometric mean is always the lower of the two, and the gap widens with volatility — which is precisely why a volatile fund can post a positive average return and still leave you poorer.

Log returns exist to make time arithmetic easy. Because ln(P₂/P₀) = ln(P₂/P₁) + ln(P₁/P₀), log returns add up across periods while simple returns must be multiplied. For small moves the two are nearly identical (a 2% simple return is a 1.98% log return); for large moves they diverge badly, and only the log version keeps a price from going negative in a simulation.

Volatility is quoted annually, so you have to scale. Under independence, variances add across periods, which means volatility grows with the square root of time: multiply a daily standard deviation by √252, a monthly one by √12. This single move — the square-root-of-time rule — turns up in option pricing, value-at-risk and every risk report you will ever read.

Then the honest caveat. Real return distributions have fatter tails and more negative skew than the normal distribution, and their volatility clusters: calm weeks follow calm weeks, and crashes arrive in clusters. So a 95% value-at-risk computed from a normal assumption is a floor, not a bound, and it tells you nothing about how bad the bad day is. Expected shortfall — the average loss given that you breached VaR — answers that, and is the reason regulators moved toward it.

Simple return  R = (P₁ − P₀ + D) / P₀
Log return  r = ln(P₁ / P₀) ≈ R for small R
Geometric mean = [(1+R₁)(1+R₂)…(1+Rₙ)]^(1/n) − 1
Annualised vol  σ_ann = σ_period × √(periods per year)
   daily × √252, weekly × √52, monthly × √12
Sharpe ratio = (R_p − R_f) / σ_p
VaR (normal, 95%) = μ − 1.645σ     (99%: μ − 2.326σ)
Expected shortfall = average loss beyond the VaR threshold

Covariance measures whether two variables deviate from their means in the same direction. Its sign is informative and its size is not, because it carries the units of both variables. Divide by the two standard deviations and you get correlation, bounded between −1 and +1, which is comparable across any pair. Correlation of 1 means an identical straight line, 0 means no linear relationship, −1 means a perfect mirror.

The two-asset variance formula is worth being able to write from memory, because it makes the diversification benefit visible: the third term carries the correlation, so lower correlation directly lowers portfolio risk. With equal weights and equal volatilities, correlation 1 leaves risk unchanged, correlation 0 cuts it by about 29%, and correlation −1 can eliminate it entirely. Nothing about expected return changed — only the variance did.

Diversification has a floor. Adding assets removes the idiosyncratic part of risk but not the shared part, so portfolio variance approaches the average covariance between holdings and stops there. That residue is systematic risk, and because it cannot be diversified away it is the only risk the market pays you to bear — which is the entire logic of CAPM, where beta measures exposure to it and alpha is what is left over.

Beta is a regression slope, so everything you know about regression applies: it is estimated with error, it depends on the window and frequency you chose, and R² tells you how much of the asset's variance the market explains. A beta of 1.4 with an R² of 0.15 is a weak claim about a mostly idiosyncratic stock. And correlations are not stable — they rise in crises, exactly when diversification was supposed to help.

Cov(x,y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1)
ρ = Cov(x,y) / (σₓ σᵥ)          −1 ≤ ρ ≤ 1
Two-asset variance:
  σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂
Beta  β = Cov(i, m) / Var(m) = ρ_im × (σᵢ / σ_m)
Portfolio beta = Σ wᵢ βᵢ
CAPM  E(Rᵢ) = R_f + βᵢ (E(R_m) − R_f)
Large-n limit: σ²_p → average covariance

Inference draws conclusions about a population from a sample. Two frameworks: frequentist (classical testing and confidence intervals) and Bayesian (updating priors with data). Coursework and interviews are mostly frequentist; cutting-edge applications lean Bayesian. Hypothesis testing: state H0 (usually no effect) and H1, compute a test statistic, compare to a critical value or compute a p-value. The p-value is not the probability H0 is true — it is the probability of data at least as extreme, assuming H0.

SE = σ / √n
95% CI = x̄ ± 1.96 × SE
t = (x̄ − μ0) / SE
Power = 1 − P(Type II error)

Type I error (false positive) is rejecting a true H0, controlled by α. Type II error (false negative) is failing to reject a false H0. Low-power studies produce unreliable results, so always consider sample size and effect size. Key tests: t-test for means, chi-square for categorical independence or goodness of fit, F-test for variances and in ANOVA, and ANOVA itself for comparing more than two group means.

Linear regression is the most important statistical tool for applied work. OLS minimises the sum of squared residuals, and under the Gauss–Markov assumptions — linearity, exogeneity, homoscedasticity, no autocorrelation, no perfect multicollinearity — it is BLUE: best linear unbiased estimator.

y = β0 + β1x1 + β2x2 + … + ε
R² = 1 − (SS_residual / SS_total)
Adjusted R² = 1 − [(1−R²)(n−1) / (n−k−1)]
t for βj = β̂j / SE(β̂j)

R² is the share of variance explained and always rises with more variables, so use adjusted R² for model comparison. But R² is not the point: the coefficients — signs, magnitudes, significance — answer the question. Diagnose heteroscedasticity (robust standard errors), multicollinearity (check VIF), autocorrelation (Durbin–Watson) and endogeneity (omitted variables, reverse causality, measurement error; fix with instruments or natural experiments). Logistic regression models binary outcomes through log-odds: ln(p/(1−p)) = β0 + β1x, so exponentiate coefficients to read odds ratios.

β̂₁ = Cov(x, y) / Var(x)          β̂₀ = ȳ − β̂₁x̄
ŷ = Xβ̂,  β̂ = (X'X)⁻¹X'y
SST = SSE + SSR      R² = SSE/SST = 1 − SSR/SST
SE(β̂ⱼ) = σ̂ / √(Var(xⱼ)·n·(1 − R²ⱼ))     VIF ⱼ = 1/(1 − R²ⱼ)
t = β̂ⱼ / SE(β̂ⱼ)      F = [(R²_full − R²_restr)/q] / [(1 − R²_full)/(n − k − 1)]
log–log slope = elasticity;  log–level slope ≈ % change in y

Time series data violate the independence assumption, so ordinary tools mislead. Stationarity — constant mean, variance and autocovariance — is the precondition for most models; test with augmented Dickey–Fuller and difference the series if it fails. ACF and PACF plots identify the order of AR and MA terms, which combine into ARIMA(p,d,q).

AR(1): y_t = c + φy_(t−1) + ε_t
ARIMA(p,d,q): differenced d times, p AR and q MA terms
GARCH(1,1): σ²_t = ω + αε²_(t−1) + βσ²_(t−1)
RMSE = √(Σ(ŷ − y)² / n)

Financial returns show volatility clustering: calm periods follow calm periods and turbulent ones cluster too. ARCH and GARCH model conditional variance directly, which is why they underpin risk and option-pricing work. Cointegration handles the case where individually non-stationary series share a long-run relationship — the statistical basis of pairs trading. Always evaluate forecasts out of sample and respect temporal order.

ARIMA(p,d,q): Δ^d y_t = c + Σφ_i Δ^d y_(t−i) + Σθ_j ε_(t−j) + ε_t
ADL(p,q):     y_t = c + Σφ_i y_(t−i) + Σβ_j x_(t−j) + ε_t
VAR(p):       Y_t = c + A_1 Y_(t−1) + … + A_p Y_(t−p) + ε_t
ECM:          Δy_t = α(y_(t−1) − βx_(t−1)) + γΔx_t + ε_t   (α < 0)
GARCH(1,1):   σ²_t = ω + αε²_(t−1) + βσ²_(t−1),  α + β < 1
Long-run variance = ω / (1 − α − β)
Forecast interval = ŷ ± z × σ̂_h

Model selection: identify candidate orders from the ACF and PACF (an AR(p) has PACF cutting off at p, an MA(q) has ACF cutting off at q), then compare by AIC or BIC — BIC penalises complexity harder and is the safer choice for forecasting. Diagnose the residuals: a Ljung–Box test on the residuals should fail to reject white noise, and an ARCH-LM test tells you whether you still need a volatility model. None of that substitutes for a genuine out-of-sample test: hold back the final 20% of the sample, or roll the estimation window forward one period at a time, and compare RMSE and MAE against a naive random-walk benchmark. If you cannot beat the random walk, say so — for many financial price series that is the honest finding.

ML is not a replacement for statistics; it is an extension that trades interpretability for predictive power when the prediction problem demands it. Supervised learning predicts labels: decision trees split recursively on the most informative feature and overfit aggressively, random forests average many bootstrapped trees to fix that, and gradient boosting (XGBoost, LightGBM) builds trees sequentially, each correcting the last. These are the best tabular models in practice — credit scoring, fraud detection, churn.

Regularisation penalises complexity: ridge (L2) shrinks coefficients toward zero, lasso (L1) can zero them out and so selects variables, elastic net combines both, and λ is tuned by cross-validation. Unsupervised learning has no labels: k-means partitions by minimising within-cluster variance, and PCA finds directions of maximum variance — applied to a return correlation matrix, the first component is usually "the market". For time series use walk-forward validation; never train on future data.

Bayesian statistics treats parameters as random variables with priors. Combine prior and likelihood to get the posterior — a full distribution of plausible values, not a point estimate. With small samples informative priors help; with large samples the prior washes out and Bayesian and frequentist answers converge. Bayesian methods underpin probabilistic programming, A/B testing and many risk models.

Deep learning stacks layers: feedforward networks for tabular data (often beaten by gradient boosting), RNNs and LSTMs for sequences, transformers for attention-based modelling. In finance it shows up in trading signals, NLP on earnings calls and filings, and non-linear modelling where accuracy beats interpretability. NLP extracts information from text — sentiment on calls and news, topic modelling, named-entity recognition on filings. High-dimensional analysis handles p ≥ n, where standard regression breaks: regularise, reduce dimensions, or select variables.

Python is the primary language for data work in finance: pandas for manipulation, numpy for numerics, scipy for distributions and tests, statsmodels for econometrics and GARCH, scikit-learn for machine learning, matplotlib and seaborn for charts. R remains dominant in academic statistics and some quant shops — tidyverse, forecast, tidymodels, rugarch.

SQL is mandatory: SELECT, all JOIN types, GROUP BY and HAVING, subqueries, CTEs, and especially window functions (ROW_NUMBER, RANK, LAG, LEAD, running aggregates) for rolling averages, rankings and year-on-year comparisons. Excel is still the lingua franca of corporate finance: INDEX-MATCH/XLOOKUP, SUMPRODUCT, IF/IFS, pivot tables, data tables for sensitivity, Goal Seek and Solver, named ranges — and keyboard speed, which interviewers do notice.

Visualisation is not decoration, it is analysis: a good chart answers a question. Match the chart type to the relationship — comparison is a bar, trend is a line, distribution is a histogram, correlation is a scatter, composition is a stacked bar (or a pie only for five categories or fewer). Remove chartjunk, use colour to highlight the point rather than everything, label directly instead of using legends, and respect the data-ink ratio.

In finance specifically: candlesticks for price action, heatmaps for correlation matrices, waterfalls for revenue and EBITDA bridges, tornado charts for sensitivity, and sparklines for inline trends in dashboards. In a case interview the same discipline applies verbally — state what the chart shows, then the so-what, before anyone asks.