Formula Book

NPV

NPV = −I₀ + CF1/(1+r) + CF2/(1+r)² + … + CFT/(1+r)^T

Value created after paying the price, at the required return.

I₀ = price today, CFt = cash flow in year t, r = required return

Use when: You know the required return and want to decide whether to invest.

Trap: Forgetting −I₀. Decision: NPV > 0 → accept.

IRR

IRR = r such that NPV(r) = 0

The return the investment actually earns.

Use when: Compare with required return: IRR > required → accept.

Trap: Compare IRR with the required return, not 0%. For MCQs, test the alternatives.

Terminal value (sale price)

TV_T = NOI(T+1) / exit cap

Sale proceeds at end of year T.

NOI(T+1) = next year's NOI

Use when: Property sold at the end of the holding period.

Trap: Use NOI in year T+1, not year T. Add TV to year-T cash flow.

Growing perpetuity

PV = CF1 / (r − g)

Value of a cash flow growing at g forever.

CF1 = cash flow one year from now

Use when: Income forever with constant growth.

Trap: Use CF1, not CF0. Requires r > g.

Solve for sale price

PV(sale) = Price + target NPV − PV(NOIs); Sale = PV(sale) × (1+r)^T

Backs out the required terminal value.

Use when: “What sale price gives NPV = X?”

Annuity payment

PMT = L × i / [1 − (1+i)^(−n)]

Constant payment on a fully amortizing loan.

L = loan, i = periodic rate, n = periods

Use when: Mortgage payment questions.

Trap: Monthly loans: i = annual/12, n = years × 12.

Interest-only payment

Payment = L × i

Only interest is paid; balance unchanged.

Use when: Interest-only / balloon loans.

LTV & DCR

LTV = Loan / Value DCR = NOI / Debt service

Leverage and debt coverage.

Use when: Lending constraints.

After-tax real borrowing rate

after-tax nominal = i(1 − t); real = (1 + i(1−t)) / (1 + π) − 1

True cost of borrowing after tax deduction and inflation.

t = tax rate, π = inflation

Use when: Mortgage cost questions with tax deduction.

Total return

Total return = (P1 − P0)/P0 + Income/P0

Capital gain + income yield.

P0 = beginning value

Use when: Price change + NOI/dividend in one period.

Trap: Divide by the BEGINNING value.

Exact real return (Fisher)

Real = (1 + nominal)/(1 + inflation) − 1 (approx: nominal − inflation)

Return in purchasing power.

Use when: Nominal vs real.

Trap: Exact vs approximate — check which the question asks.

Geometric mean

G = [(1+r1)(1+r2)…(1+rn)]^(1/n) − 1

Compound average return per period.

Use when: Several period returns, average growth.

Trap: Geometric ≤ arithmetic. A −50% loss needs +100% to recover.

CAGR from index

g = (End / Start)^(1/n) − 1

Compound growth per period.

n = number of periods

Use when: Index or price over many years/quarters.

Missing return given target

(1+G)^n = (1+r1)…(1+r(n−1))(1+x) → solve x

Final-period return needed to reach a target compound return.

Use when: “What return is needed in the last year…?”

Expected return (3 scenarios)

E[R] = p1·R1 + p2·R2 + p3·R3

Probability-weighted average.

p's sum to 1

Use when: Scenario analysis — expected IRR / NPV.

Trap: Don't take a simple average.

Variance & SD

Var = p1(R1−E)² + p2(R2−E)² + p3(R3−E)² σ = √Var

Risk of the scenarios.

Use when: Standard deviation of IRR/NPV.

Trap: Don't divide by n — probabilities do the weighting.

Coefficient of variation

CV = σ / E[R]

Risk per unit of return (lower is better).

Use when: Comparing projects with different means.

Bond price

P = C×[1 − (1+y)^(−n)]/y + FV/(1+y)^n

PV of coupons + PV of face value.

C = coupon, y = yield, n = years

Use when: Bond valuation.

Trap: Yield ↑ → price ↓. Coupon = yield → price = par.

Direct capitalization

V = NOI / cap rate; cap ≈ r − g

Property value from one year's NOI.

Use when: Cap-rate valuation; growth expectations.

Trap: Higher expected growth → lower cap rate.

Cap-rate spread

Spread = Cap rate − Government bond yield

Property risk premium proxy.

Use when: Bonds vs property questions.

Prediction & residual

Ŷ = b0 + b1X1 + … + bkXk; e = Y − Ŷ

Fitted value and error.

Use when: Plug in X values.

Trap: Residual = actual − predicted (not the reverse).

R² and adjusted R²

R² = 1 − SSR/SST; adj R² = 1 − (1 − R²)(n − 1)/(n − k − 1)

Share of variation in Y explained.

k = number of regressors

Use when: Model-fit questions.

Trap: High R² ≠ causal; adj R² penalizes extra regressors.

t-statistic

t = (b − β_H0) / SE(b)

How many SEs b is from the H0 value.

Use when: Testing a coefficient.

Trap: Usually β_H0 = 0. Critical 1.645 / 1.96 / 2.576 for 10/5/1%.

p-value rule

p < α → reject H0

Decision rule.

Use when: Any significance test.

Trap: Fail to reject ≠ H0 proven true.

Confidence interval

b ± t_crit × SE (90%: 1.645, 95%: 1.96, 99%: 2.576)

Range for the true β.

Use when: CI questions; if 0 is outside the 95% CI → significant at 5%.

Trap: SE = b / t if only t is shown.

Omitted variable bias — two conditions

Bias if Z affects Y AND Z is correlated with an included X

Both must hold.

Use when: “Will omitting Z bias b1?”

Trap: Only one condition → no bias.

Log interpretations

level-level: ΔY = b·ΔX log-level: %ΔY ≈ 100b·ΔX level-log: ΔY ≈ (b/100)·%ΔX log-log: b = elasticity

How to read coefficients.

Use when: Any ln() in the model.

Dummy in log-Y model

approx = 100b%; exact = 100(e^b − 1)%

Percentage effect of the dummy.

Use when: Dummy with ln(Y).

Interaction effect

Y = b0 + b1X + b2D + b3(X·D) → ∂Y/∂X = b1 + b3·D

Effect of X depends on D.

Use when: Interaction terms.

Quadratic

ME = b1 + 2·b2·X; turning point = −b1/(2b2)

Non-constant marginal effect.

Use when: X and X² both in the model.

Standardization

z = (x − μ) / σ; P(X < x) = Φ(z); P(X > x) = 1 − Φ(z)

Convert to standard normal.

Use when: Normal probabilities and percentiles (x = μ + zσ).

Trap: N(μ, σ²): second parameter is variance — use σ = √σ².

RMSE

RMSE = √[(e1² + e2² + … + en²) / n]

Typical prediction error.

Use when: Hold-out / test sample comparisons.

Trap: Lower hold-out RMSE → better out-of-sample. Higher adj R² → better in-sample.

Information criteria

Lower AIC / SC (BIC) → preferred model

Fit penalized by complexity.

Use when: Model selection output.

Probit

z = Xβ; P = Φ(z); ME_j = β_j·φ(z)

Probability and marginal effect.

Φ = CDF, φ = PDF

Use when: Binary Y with probit output.

Trap: Discrete change: compute P_new − P_old.

Linear probability model

P(Y=1|X) = Xβ; ΔP = β·ΔX (percentage points)

Coefficient = change in probability.

Use when: OLS with binary Y.

Trap: Percentage points, not percent.

Logit

P = 1/(1 + e^(−z)); ME_j = β_j·P(1 − P)

Probability and marginal effect.

Use when: Binary Y with logit output.

Fixed vs random effects

FE: within variation, drops time-invariant X. Hausman small p → FE

Panel estimator choice.

Use when: Panel data questions.

Trap: FE cannot estimate a time-invariant variable's coefficient.