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Quantitative Methods 14 items

R-squared (coefficient of determination)

R^{2} = \frac{R S S}{S S T} = 1 - \frac{S S E}{S S T}

RSS = regression sum of squares, SSE = error sum of squares

SST = total sum of squares. Fraction of variation explained.

Adjusted R-squared

R ˉ^{2} = 1 - \frac{n - 1}{n - k - 1} (1 - R^{2})

n = observations, k = number of independent variables

Penalizes for adding irrelevant predictors

F-statistic (overall regression)

F = \frac{R S S / k}{S S E / ( n - k - 1 )} = \frac{M S R}{M S E}

k = predictors, n = observations

Tests

H_{0} : a l l s l o p ecoe f f i c i e n t s = 0

Standard error of regression (SEE)

S E E = \frac{S S E}{n - k - 1} = M S E

SSE = sum of squared errors, n = observations, k = independent variables

Measures typical prediction error of the model

AR(1) model one-period forecast

x_{t} = b_{0} + b_{1} x_{t - 1} + ε_{t}

, x_t = value at time t, b_0 = intercept, b_1 = lag coefficient (|b_1|<1 for stationarity), ε_t = error

Multiple regression model equation

Y = b_{0} + b_{1} X_{1} + b_{2} X_{2} + \dots + b_{k} X_{k} + e

, Y = dependent variable, b0 = intercept, bj = partial slope on Xj, k = number of independent variables, e = error term

t-statistic for an individual regression coefficient

t = \frac{b _{j}}{s _{b_{j}}}

, bj = estimated slope coefficient, s_bj = standard error of the coefficient; df = n - k - 1, n = observations, k = independent variables

Partial F-test for a group of regression variables

F = \frac{( S S E _{R} - S S E _{U} ) / q}{S S E _{U} / ( n - k _{U} - 1 )}

, SSE_R = restricted SSE, SSE_U = unrestricted SSE, q = variables tested, n = observations, k_U = unrestricted predictors

Bias-variance decomposition of total prediction error

Total error = bias^{2} + variance + irreducible noise

, bias = error from oversimplification, variance = error from sensitivity to training data, irreducible noise = inherent randomness

Classification precision and recall

precision = \frac{T P}{T P + F P}, recall = \frac{T P}{T P + F N}

, TP = true positives, FP = false positives, FN = false negatives

Breusch-Pagan test statistic

B P = n \times R^{2}

, n = number of observations,

R^{2}

= R-squared from regressing squared residuals on the original X variables; compared to chi-squared with k df

Variance inflation factor

V I F_{j} = \frac{1}{1 - R _{j}^{2}}

,

R_{j}^{2}

= R-squared from regressing

X_{j}

on all other independent variables; VIF > 10 = severe multicollinearity

Odds ratio from a logistic coefficient

O R = e^{b_{j}}

, OR = odds ratio (multiplier on odds per unit increase in

X_{j}

),

b_{j}

= logistic coefficient on

X_{j}

; OR>1 increases odds, OR<1 decreases

Logistic regression probability from log-odds

p = \frac{1}{1 + e ^{- (b_{0} + b_{1} X_{1} + b_{2} X_{2} + \dots)}}

, p = probability of outcome, b = coefficients, X = predictors, log-odds =

b_{0} + b_{1} X_{1} + \dots

Economics 7 items

Covered Interest Rate Parity (CIP)

\frac{F}{S} = \frac{1 + r _{d}}{1 + r _{f}}

F = forward rate (d/f), S = spot rate (d/f)

r_d = domestic rate, r_f = foreign rate

No-arbitrage; holds in practice

International Fisher effect

r_{n o m, d} - r_{n o m, f} \approx E (π_{d}) - E (π_{f})

Nominal interest rate differentials reflect expected inflation differentials

Combines Fisher effect with PPP

Relative PPP

E (%Δ S_{d / f}) \approx π_{d} - π_{f}

Expected % change in spot rate ≈ inflation differential

Holds better over long horizons

Unhedged foreign asset return (in domestic currency)

(1 + R_{DC}) = (1 + R_{FC}) (1 + R_{F X})

; approx

R_{DC} \approx R_{FC} + R_{F X}

.

R_{F X}

= foreign-currency appreciation vs. domestic.

Labor productivity growth

g_{L P} = g_{Y} - g_{L}

, g_LP = growth in output per worker, g_Y = GDP growth, g_L = labor force growth

Uncovered interest rate parity expected spot change

E [%Δ S_{p r i ce / ba se}] \approx r_{p r i ce} - r_{ba se}

, E[%ΔS] = expected change in spot (price per base), r_price = price-currency rate, r_base = base-currency rate

Taylor rule policy rate

i = r_{n} + π + 0.5 (π - π^{*}) + 0.5 (y - y^{*})

, i = policy rate, r_n = neutral real rate, π = inflation, π* = target inflation, y - y* = output gap

Financial Statement Analysis 13 items

FCFF from EBIT

F C F F = E B I T (1 - t) + D & A - Δ W C - C a pE x

t = tax rate, D&A = depreciation & amortization

Δ W C

= change in working capital

Net pension expense components (ASC 715)

NPPC = Service Cost + Interest Cost - Expected Return + Amort PSC \pm Amort Gain/Loss

. IFRS (IAS 19): net interest on net pension liability replaces expected return.

US GAAP net periodic pension cost

P E = S C + I C - E R \pm A

, SC = service cost, IC = interest cost on beginning PBO, ER = expected return on plan assets, A = corridor amortization of actuarial gains/losses

Total periodic pension cost

T P P C = - Δ F S + C

,

Δ F S

= change in funded status (plan assets − PBO), C = employer contributions; equals SC + IC − actual return ± actuarial gains/losses

Bank efficiency ratio

Efficiency = \frac{Non-interest expense}{NII + Non-interest income}

, NII = net interest income; provisions deliberately excluded; lower is better

Full goodwill under the acquisition method

G W = (C + N C I_{F V}) - F V_{N A}

, GW = goodwill, C = consideration transferred, NCI_FV = non-controlling interest at fair value, FV_NA = fair value of identifiable net assets

Equity method investment carrying value (CAID)

C V = B e g C V + s \times N I - A m or t - s \times D i v + s \times O C I

, CV = carrying value, s = ownership share, NI = investee net income, Amort = amortization of excess purchase price, Div = dividends

Pre-CTA translated equity build-up (current rate method)

E_{p r e} = E_{b e g} + N I_{a v g} - D i v_{t x n}

,

E_{b e g}

= beginning equity,

N I_{a v g}

= net income at average rate,

D i v_{t x n}

= dividends at transaction rate

Cumulative translation adjustment plug (current rate method)

C T A = E_{r e q} - E_{p r e}

,

E_{r e q}

= translated net assets (assets − liabilities at current rate),

E_{p r e}

= pre-CTA translated equity

Balance sheet accruals ratio

Accruals ratio = \frac{Δ N O A}{Average N O A}

, NOA = net operating assets = (total assets − cash & investments) − (total liabilities − financial debt); ΔNOA = change in NOA over period

Cash flow accruals ratio

Accruals ratio = \frac{N I - C F O - C F I}{Average N O A}

, NI = net income, CFO = cash flow from operations, CFI = cash flow from investing, NOA = net operating assets

LIFO-to-FIFO equity adjustment

E q u i t y_{F I F O} = E q u i t y_{L I F O} + L R \times (1 - t)

, LR = LIFO reserve, t = tax rate; inventory rises by full LR and COGS falls by the change in LR

Five-component DuPont decomposition of ROE

R O E = \frac{N I}{E B T} \times \frac{E B T}{E B I T} \times \frac{E B I T}{R e v} \times \frac{R e v}{A sse t s} \times \frac{A sse t s}{E q u i t y}

, NI = net income, EBT = pretax income, EBIT = operating income, Rev = revenue, Assets = avg total assets, Equity = avg equity

Corporate Issuers 10 items

FCFE from FCFF

F C F E = F C F F - I n t (1 - t) + Δ Debt

Int = interest expense, t = tax rate,

Δ Debt

= net new borrowing.

FCFE from net income

F C F E = N I + N C C - Δ W C - CapEx + Δ Debt

NI = net income, NCC = non-cash charges (D&A, deferred tax),

Δ Debt

= net new borrowing.

FCFE constant-growth valuation

V_{0} = \frac{F C F E _{1}}{r _{e} - g}

. Requires

r_{e} > g

; g = sustainable FCFE growth. Gordon-growth analog using FCFE instead of dividends.

MM Propositions I & II (with taxes)

Prop I:

V_{L} = V_{U} + t D

— debt tax shield adds value.

Prop II:

r_{e} = r_{0} + (r_{0} - r_{d}) (1 - t) \frac{D}{E}

; WACC declines with leverage. Theoretical optimum: 100% debt.

Pure-play unlevered (asset) beta via Hamada equation

β_{a sse t} = \frac{β _{e q u i t y}}{1 + ( 1 - t ) ( D / E )}

, β = beta, t = comparable's tax rate, D/E = comparable's debt-to-equity ratio

Weighted average cost of capital (WACC)

W A C C = w_{d} r_{d} (1 - t) + w_{p} r_{p} + w_{e} r_{e}

, w = target weight, r = component cost, t = tax rate; subscripts d=debt, p=preferred, e=equity

Lintner target payout adjustment model expected dividend

D_{1} = D_{0} + a \times (TPR \times EPS_{1} - D_{0})

, D_1 = expected dividend, D_0 = current dividend, a = adjustment factor, TPR = target payout ratio, EPS_1 = projected EPS

Post-repurchase earnings per share

EPS_{n e w} = \frac{E}{N - C / P}

, E = total earnings, N = original shares, C = cash spent on repurchase, P = repurchase price per share

LBO multiple of invested capital

M O I C = \frac{E x i t E q u i t y}{I ni t ia l E q u i t y}

, Exit Equity = Exit EV − remaining debt, Initial Equity = sponsor's equity contribution at entry

Acquirer's gain in an acquisition

G ain = P V (S y n er g i es) - P r e mi u m

, PV(Synergies) = present value of net after-tax synergies, Premium = offer value above target's pre-announcement market value

Equity Valuation 16 items

Residual income model

V_{0} = B_{0} + \sum_{t = 1}^{\infty} \frac{( R O E _{t} - r _{e} ) B _{t - 1}}{( 1 + r _{e} ) ^{t}}

B_0 = book value, ROE = return on equity, r_e = cost of equity.

Sustainable growth rate

g = R O E \times b

b = retention ratio = 1 − payout ratio

ROE = net income / equity

Growth rate achievable without changing capital structure or issuing equity

Two-stage DDM

V_{0} = \sum_{t = 1}^{n} \frac{D _{0} ( 1 + g _{S} ) ^{t}}{( 1 + r ) ^{t}} + \frac{V _{n}}{( 1 + r ) ^{n}}

V_{n} = \frac{D _{n + 1}}{r - g _{L}}

g_S = high growth (stage 1), g_L = long-run growth (stage 2)

Build-up method for cost of equity (private company)

r_{e} = r_{f} + ERP + Size premium + Specific-company premium \pm Industry premium

. Used when CAPM fails for illiquid/private firms.

Sum-of-the-parts enterprise value

V_{S O T P} = \sum_{i} V_{i} - (m \times O H)

,

V_{i}

= standalone value of segment i, m = overhead capitalization multiple, OH = annual corporate overhead

Expected return as dividend yield plus capital gains yield

E (r) = \frac{D _{1}}{P _{0}} + \frac{V _{0} - P _{0}}{P _{0}}

,

D_{1}

= next dividend,

P_{0}

= current price,

V_{0}

= intrinsic value

H-model value

V_{0} = \frac{D _{0} ( 1 + g _{L} )}{r - g _{L}} + \frac{D _{0} \cdot H ( g _{S} - g _{L} )}{r - g _{L}}

,

D_{0}

= current dividend,

g_{S}

= initial high growth,

g_{L}

= long-run growth, H = half the high-growth period, r = required return

Residual income continuing value with persistence factor

C V = \frac{ω \times R I _{T}}{1 + r - ω}

, ω = persistence factor, RI_T = final-year residual income, r = cost of equity

Residual income (spread form)

R I = B \times (R O E - r)

, B = beginning book value, ROE = return on equity, r = cost of equity

Capitalized cash flow method value

V = \frac{C F}{r - g}

, V = firm value, CF = normalized cash flow, r = required return, g = stable growth rate

FCFF from net income

F C F F = N I + D e p + I n t (1 - t) - C a pE x - Δ W C

, NI = net income, Dep = depreciation, Int = interest expense, t = tax rate, CapEx = capital expenditures, ΔWC = change in working capital

Single-stage FCFF enterprise value

E V = \frac{F C F F _{1}}{W A C C - g}

, EV = enterprise value, FCFF₁ = next-period free cash flow to firm, WACC = weighted average cost of capital, g = stable growth rate

Gordon Growth Model value

V_{0} = \frac{D _{1}}{r - g}

,

V_{0}

= intrinsic value,

D_{1} = D_{0} (1 + g)

= next-period dividend, r = required return on equity, g = constant growth (g < r)

Discount for lack of control from a control premium

D L O C = 1 - \frac{1}{1 + C P}

, DLOC = discount for lack of control, CP = control premium

Justified price-to-book ratio

\frac{P}{B} = \frac{R O E - g}{r - g}

, ROE = return on equity, g = growth rate, r = required return on equity

Justified leading price-to-earnings ratio

\frac{P}{E _{1}} = \frac{1 - b}{r - g}

, b = retention ratio (1-b = payout), r = required return on equity, g = growth rate

Fixed Income 11 items

Option-Adjusted Spread (OAS)

O A S = Z -spread - Option value (bps)

Callable: OAS < Z-spread. Putable: OAS > Z-spread.

Effective duration

D_{eff} = \frac{P _{- Δ y} - P _{+ Δ y}}{2 P _{0} Δ y}

. Used for bonds with embedded options (callable, putable, MBS).

Portfolio effective duration (market-value weighted)

D_{eff, port} = \sum_{i} w_{i} D_{eff, i}

;

w_{i}

= market-value weight. Assumes parallel yield-curve shift; use KRDs for non-parallel.

Credit spread decomposition (term structure)

Observed Spread = Expected Loss + Credit Risk Premium + Liquidity Premium

. Expected loss = PD × LGD.

Hazard rate from credit spread (reduced-form model)

λ = \frac{spread}{1 - R}

, λ = hazard rate (default intensity), spread = credit spread, R = recovery rate, (1 − R) = loss given default

Value of a callable bond

V_{c a l l ab l e} = V_{o pt i o n - f r ee} - V_{c a l l}

, V_option-free = value of identical option-free bond, V_call = value of the embedded call option held by the issuer

Value of a putable bond

V_{p u t ab l e} = V_{o pt i o n - f r ee} + V_{p u t}

, V_option-free = value of identical option-free bond, V_put = value of the embedded put option held by the bondholder

Rolldown return from riding the yield curve

R_{r o l l} = Δ y \times D

,

Δ y

= yield change from maturity shortening, D = duration (assumes a stable curve over the holding period)

CDS mark-to-market gain to protection buyer

MTM = (s_{c u r} - s_{co n t r a c t}) \times D \times N

, s_cur = current spread, s_contract = contracted (entry) spread NOT fixed coupon, D = effective spread duration, N = notional

Backward induction node value in a binomial interest rate tree

V_{n o d e} = \frac{0.5 \times ( V _{u p} + V _{d o w n} ) + C}{1 + r _{n o d e}}

, V = bond value, C = coupon, r = node rate, 0.5 = risk-neutral probabilities

Lognormal up and down rate relationship in a binomial tree

r_{u p} = r_{d o w n} \times e^{2 σ}

, r_up = higher node rate, r_down = lower node rate, σ = annual std dev of ln of short rate

Derivatives 13 items

Binomial option pricing (one period)

c = \frac{π _{u} c _{u} + π _{d} c _{d}}{1 + r}

;

π_{u} = \frac{( 1 + r ) - d}{u - d}

,

π_{d} = 1 - π_{u}

(risk-neutral probs). u, d = up/down factors.

Put-call parity

c + \frac{X}{( 1 + r ) ^{T}} = p + S_{0}

. Continuous:

c + X e^{- r T} = p + S_{0}

. Same strike/expiry; European options.

Option gamma

Γ = \frac{\partial ^{2} V}{\partial S ^{2}} = \frac{\partial Δ}{\partial S}

. Highest ATM near expiry; long options are positive gamma, short are negative.

Option vega

ν = \frac{\partial V}{\partial σ}

. Price change per 1-pp vol move. Long call or put = positive vega. Highest ATM with longer expiry.

Delta-gamma-vega option price approximation

Δ C \approx δ d S + \frac{1}{2} γ (d S)^{2} + ν d σ

, δ = delta, γ = gamma, ν = vega, dS = stock move, dσ = vol change

Black-Scholes-Merton call price

C = S N (d_{1}) - K e^{- r T} N (d_{2})

, C = call value, S = spot, K = strike, r = risk-free rate, T = time, N = standard normal CDF,

d_{1}, d_{2}

= moneyness terms

Forward rate agreement settlement payment

P = \frac{N A \times ( r _{f} - r _{F R A} ) \times D C F}{1 + r _{f} \times D C F}

, NA = notional, r_f = floating reference rate, r_FRA = contract rate, DCF = day-count fraction of reference period

Fixed swap rate from discount factors

s = \frac{1 - Z ( n )}{\sum _{t = 1}^{n} Z ( t )}

, s = periodic fixed swap rate, Z(t) = discount factor for period t, n = number of settlement periods

Option theta (time decay)

Θ = \frac{\partial V}{\partial t}

, the change in an option value per unit of calendar time. Usually negative for long options (value erodes toward expiry); time decay is largest for at-the-money options near expiration.

Payer swaption payoff

Payer = N \cdot max (R_{mkt} - R_{K}, 0) \cdot A

, N = notional, R_mkt = market swap rate, R_K = strike rate, A = annuity factor (PV of $1 over swap tenor)

Caplet payoff

Caplet = N \cdot max (R_{ref} - R_{cap}, 0) \cdot \frac{days}{360}

, N = notional, R_ref = reference rate, R_cap = cap strike rate, days/360 = day-count fraction

Risk-neutral probability in a binomial tree

p = \frac{1 + r - d}{u - d}

, p = risk-neutral (up) probability, r = periodic risk-free rate, u = up factor, d = down factor

Binomial hedge ratio (option delta)

Δ = \frac{c _{u} - c _{d}}{S _{u} - S _{d}}

, Δ = shares per option, c_u/c_d = option payoff up/down, S_u/S_d = stock price up/down

Alternative Investments 14 items

TVPI (Total Value to Paid-In)

T V P I = \frac{Distributions + Residual NAV}{Paid-in}

= DPI + RVPI (realized + unrealized multiples).

Direct capitalization (real estate)

V = \frac{N O I _{1}}{Cap Rate}

;

N O I_{1}

= stabilized first-year NOI (after opex, before debt service/tax). Implicit: cap rate = r − g.

Equity REIT NAV per share

NAV/share = \frac{Property value + Other assets - Total liabilities}{Shares outstanding}

Property value typically from cap-rate or DCF on stabilized NOI. Price/NAV reveals premium or discount.

Pre-money and post-money valuation

Post-money = Pre-money + New investment

Investor share = \frac{Investment}{Post-money}

Price per share = pre-money / pre-money shares. PE / VC funding rounds.

Total return on a commodity futures position

R_{t o t a l} = R_{s p o t} + R_{r o l l} + R_{co l l a t er a l}

, R_spot = spot price return, R_roll = roll yield, R_collateral = interest earned on posted collateral

Commodity forward price under cost-of-carry

F = S \times e^{(r + c - y) \times T}

, F = forward price, S = spot price, r = risk-free rate, c = storage cost rate, y = convenience yield, T = time in years

Hard-hurdle incentive fee

F ee = p \times (R - h) \times A U M

, p = incentive rate, R = gross return, h = hurdle rate, AUM = assets under management (only when R > h)

Unsmoothed variance correction for serially correlated returns

σ_{u n s m oo t h e d}^{2} = \frac{σ _{r e p or t e d}^{2}}{1 - ρ ^{2}}

, σ²_reported = reported return variance, ρ = first-order autocorrelation of returns

Funds from operations for a REIT

F F O = N I + D e p_{R E} - G_{s a l e}

, NI = GAAP net income, Dep_RE = real estate depreciation/amortization, G_sale = gains on sale of properties

Adjusted funds from operations for a REIT

A F F O = F F O - C a p e x_{main t} - S L R

, FFO = funds from operations, Capex_maint = recurring maintenance capital expenditures, SLR = straight-line rent adjustment

Net operating income for a property

N O I = P G I - V - O pE x

, PGI = potential gross income at full occupancy, V = vacancy and collection losses, OpEx = operating expenses (excludes debt service, capex, income tax)

Cash-on-cash return on a leveraged property

C o C = \frac{N O I - D S}{E}

, NOI = net operating income, DS = annual debt service, E = equity invested

Alpha of an equity long/short fund

α = R_{g r oss} - (E_{n e t} \times R_{m})

,

R_{g r oss}

= fund gross return,

E_{n e t}

= net market exposure,

R_{m}

= market return

Net and gross exposure of a long/short fund

E_{n e t} = L - S

,

E_{g r oss} = L + S

,

L

= long exposure (% of capital),

S

= short exposure (% of capital); net = direction, gross = leverage

Portfolio Management 16 items

Treynor ratio

T_{p} = \frac{R _{p} - R _{f}}{β _{p}}

R_p = portfolio return, R_f = risk-free rate,

β_{p}

= portfolio beta

Excess return per unit of systematic risk

Jensen's alpha

α_{p} = R_{p} - [R_{f} + β_{p} (R_{m} - R_{f})]

Actual return minus CAPM expected return

Positive

α

= outperformance after adjusting for systematic risk

Fundamental Law of Active Management

I R = I C \times B R

IC = information coefficient (skill), BR = breadth (independent bets), IR = information ratio.

M-squared (Modigliani-Modigliani)

M^{2} = (R_{p} - R_{f}) \times \frac{σ _{m}}{σ _{p}} + R_{f}

Levered/delevered portfolio return at market's risk level

Expressed in % — directly comparable across portfolios

Sharpe-information ratio relationship

S R_{P} = S R_{B}^{2} + I R^{2}

, SR_P = portfolio Sharpe ratio, SR_B = benchmark Sharpe ratio, IR = information ratio (Pythagorean, never additive)

Optimal level of active risk

σ_{A}^{*} = \frac{I R}{S R _{B}} \times σ_{B}

, σ_A* = optimal active risk, IR = information ratio, SR_B = benchmark Sharpe ratio, σ_B = benchmark standard deviation

Break-even inflation rate

B E I = y_{n o m} - y_{r e a l}

, BEI = market expected average inflation,

y_{n o m}

= nominal government bond yield,

y_{r e a l}

= inflation-linked bond yield of same maturity

Forward-looking equity risk premium (DDM)

E R P = (D Y + g) - r_{f}

, DY = current dividend yield, g = expected nominal earnings growth,

r_{f}

= risk-free (government bond) rate

Authorized participant creation arbitrage profit

π = (P_{mk t} - N A V - c) \times Q

, P_mkt = ETF market price per share, NAV = net asset value per share, c = transaction cost per share, Q = creation unit size (shares)

Total cost of ETF ownership

T C = (E R \times H) + S_{r t} + P D

, ER = annual expense ratio, H = holding period in years, S_rt = round-trip bid-ask spread, PD = premium/discount impact

Arbitrage Pricing Theory expected return

E (R_{i}) = R_{f} + \sum_{k = 1}^{K} β_{i, k} λ_{k}

, R_f = risk-free rate, β = factor sensitivity, λ = factor risk premium, K = number of factors

Active specific risk from active risk decomposition

σ_{s p ec i f i c} = σ_{t o t a l}^{2} - σ_{f a c t or}^{2}

, σ_total = total active risk (tracking error), σ_factor = active factor risk, σ_specific = active specific risk

Square root of time VaR scaling

V a R_{T} = V a R_{d ai l y} \times T

, VaR_T = VaR over T days, VaR_daily = 1-day VaR, T = number of trading days (variance scales linearly, so σ scales with √T)

Parametric (variance-covariance) Value at Risk

V a R = V \times z \times σ

, V = portfolio value, z = z-score (95% = 1.65, 99% = 2.33), σ = periodic return standard deviation

Annual transaction cost drag from turnover

D r a g = T \times c

, T = annual portfolio turnover, c = cost per trade (including market impact, in bps or %)

Bonferroni correction adjusted significance threshold

α_{a d j} = \frac{α}{m}

,

α_{a d j}

= per-test threshold,

α

= desired overall significance level, m = number of tests run

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